Bd the
UNIVERSITY OF MICHIGAN
MAIOLI RISPENİMSELAM A MAINAN
4
We
BALZ
SCIENTIA
ARTES
VERITAS
LIBRARY
TIDA
CIRCUMERIO
GIFT OF
REGENT LA HUBBARD
EXECUTIVE COUNCIL
The
AFFILIATED CLUBS
FRANK A. VANDERLIP
FORMER PRESIDENT OF THE NATIONAL CITY BANK OF N. Y.
THE ECONOMIC CLUB OF NEW YORK (N. Y.)
STATISTICIAN
ROGER W. BABSON
JOHN HAYS HAMMOND
THE ECONOMIC CLUB OF BOSTON (Mass.)
MINING ENGINEER
National
Eronomir League
THE ECONOMIC CLUB OF WORCESTER (MASS.)
A. LAWRENCE LOWELL
PRESIDENT HARVARD UNIVERSITY
THE ECONOMIC CLUB OF PORTLAND (ME.)
NICHOLAS MURRAY BUTLER
PRESIDENT OF COLUMBIA UNIVERSITY
THE ECONOMIC CLUB OF PROVIDENCE (R. 1.)
GEORGE B. CORTELYOU
FORMER SECRETARY OF THE TREASURY
THE ECONOMIC CLUB OF SPRINGFIELD (MASS.)
6 Beacon St., Boston, Mass.
FRANK O. LOWDEN
FORMER GOVERNOR OF ILLINOIS
THE ECONOMIC CLUB OF INDIANAPOLIS (IND.)
LINDLEY M. GARRISON FORNER SECRETARY OF WAR
THE ECONOMIC CLUB OF SAN FRANCISCO (CAL.)
EDWARD A. FILENE
MERCHANT
OBJECT
THE ECONOMIC CLUB OF PHILADELPHIA (PA)
GEORGE W. WICKERSHAM
FORMER ATTORNEY GENERAL
THE EDUCATION AND EXPRESSION
THE ECONOMIC CLUB OF WASHINGTON (D. C.)
SECRETARY AND TREASURER
J. W. BEATSON
6 BEACON STREET, BOSTON
OF PUBLIC OPINION
THE ECONOMIC CLUB OF BUFFALO (N. Y.)
AND OTHERS
October 18th, 1923.
Dear Sir:
As a member of our National Council,
will you kindly answer and return the enclosed
questionnaire relating to subjects for future
consideration by The National Economic League?
The returns are to be classified under general
headings and voted on by our National Council,
after which more definite questions on the
subjects receiving the highest vote are to be
submitted.
Sincerely yours,
J. W. BEATSON
Secretary
To the members of the National Council
of The National Economic League.
.
॥ - /
|s2
3 4 .
Is Sunday
With May 27
365d, = 52 uks Iday
ਦੇ ਉਪ
San ,
/, 24 , 77, 72,ਤੇ
Jan. I
14 . 7 ) ,
Sandy
03
.
e
.
ਨੂੰ
ਸਨ ਪਰ
buck Noll Ho
,
. ਹੈ ਹੈ
Moth
Sunday
ਦੇ ॥
ਦੇ ਉਹੈ ਜੋ
کی {
0
"
''..": '.'' , ' ' ' '
stur
Hubbard
Imag. Voy.
ران
1
مل
L
21
W: ing doing
4 Go
c
"G
ind
Blow to Br
இதயம்
if
Morridoonialls f #4
zach H Mottast og lotion
ay
from
G
ff
M
Lect
thonol
ve
Bedfos
7
Proakor
or
3
Sant
3
2447
630
y
nay month
any
opravde
simpt
Fath in doop 373
Taj off
2
nellvé
387589
799341
45
13 4711
3 87389 29
m
m
m
19.727
1:11:
PHIM
1
1
12:}u
23:21
#
}s:51
H
meridian 37
Zo
Is gëzor
406349
1:97474
32394
4th,
Rob: Catbot
a
diff Laht 54
732394
farfars
I do
& Puff Lochendencia flesh
2
2:24234740
Platt
ylure, promijmerican
-344 Ls су г.
iff of hat in meredio
1490
/
I
3
1468
1
7)
T
13672
>27 86 1824
14.0 3902
>
13092
7815
.
:
.
!
a
1960
Sturmy Ætatis
captaine samuel
Tue 36
Effigies of
true
Anno
Domini
The
1669
BE
way
HES
WWW
ON
WWW
Home
www.
mode
legal
ws
w
WE'RE
WE
WOW
..
ws
.
ORI
.
38 W
WE
Am
wy
om
We are
CE
記
​.
1
What here you ſee is but a grauen face,
Only, the pịđure of that brittle Café,
Whole, Soules the magazine of all theſe Arts
,
which here most freely, he to all imp'arts
Nimasi
LL ANDPRACTICALL ARTS
and fortune
Art
pon
akan dengan
TURMYS
SUB
1
IN
THE
MARINERS MAGAZINE
or
VIT
and Containing
HARE
Chaos
W
Sturmys Mathematicall
and Practicallrts.
The Deſcription makeing andule of
the most usefull Inftrument for at fifts
and Navigators
The Arts of Navigation at Large
a New way of Surveying of Land
Gaging, Gunery, Aftronomy and
Dyalling, performed Geome-
Vtrically, Inſtrumentally
, and
by Calculation
By
Cap:Samuel Sturmy
M
1669:
1
Houres
1
W
Inclin,,ation of Mer.
20
30
of
영
​og
4
Gnomon
OL
20
1: The Greater Pole
70.
Bofill 8
Points
4
1
innan
Chords
the lefler pole
Leag:
10
OT
of
G 넹
​90
80
60
50
ont
30
20
10
Sines
10
20
30
40
Half Tangents
700 30
LILLE
빙
​Substite
ot
8
60
Secants
of
70
5
20
30
42
del
Tangent
10
oE
40
50
бо
7°
75
Stile
06
80
70
60
30
1
1
Edward Fage at the Suger Loaf in Hoſier Lane London Fecit, who makes all ſorts of Mathematicall Inſtruments, .
140
130
120
110
100
90
80
70
60
50
40
30
20
10
Werged Tangent 80
Longitud or Part of the Æquator.
Mercaters Meridian Line.
70
60
50
40
20
30
10
..
sites
1
3 Z
5
6
8 9 10
20
30
to
501
6
A 70 80 90 Sines Chordr
Tangents
89
88
86
87
70]
82
85
84
60
Tangents
83
30/60
2070
4050
84 1080
45
50
Numbers
1
3
6
41
5.
8
110
9
zol
20
300 aNumbers
410
50
2000
100
60
80 go
70
Inches
10
220
30
410
510
60
710
80
40
90
30
210
10 Reduction
262
The SCALE of SCALES
Thefteturall and Artficiall Scales are deſcribed at large with their Fundamentall Diagrams in the Second Booke of this Treatiſe, and their use exemplified in the Reſolution of all
Mathematicall concluſions, with many other Instruments, the makeing and necesſary uſe whereof is Demonſtrated.
ng
1
6.r: #4rt, THE вое 4.
re
SA )
Mariners Magazine;
1
alin
le
STURMY's Mathematical and Practical ARTS.
CONTAINING,
The Deſcription and uſe of the SCALE of SCALES;
it being a Mathematical Ruler,that reſolves moſt Mathematical Concluſions:
And likewiſe the Making and Uſe of the Croftaff, Quadrant, and the Quadrat,
Nocturnals and other mof Uſeful Inſtruments for all Artiſts and Navigators,
The ART of NAVIGATION, Reſolved Geometrically,
Inſtrumentally, and by Calculation, and by that lace Excellent Invention of
Logarithms, in the Three Principal kinds of Sailing ; with New Tables of the
Longitude and Latitude of the moſt Eminent Places round the World, froin
the Meridian of the Lizard: And New Exact Tables of the Sun's Declina-
tion, newly Calculated ; and of the Longitude and Latitude, Declination and
Right Aſcenſion of ſome Eminent Fixed Stars.
TO GETHER WITH.
A Diſcourſe of the Practick Part of NAVIGATION, in Working a
Ship in all Wcachers and Condicions at Sca.
A New Way of Surveying of Land, by che Mariners Azimuth or Amplitude
Compaſs; very eaſie and delightful to all ſorts of Navigators, Mariners, or others.
The ART of GAUGING all Sorts of VESSELS; and the Meaſuring
of Timber, Glaſs, Board, Stone, Walls, Cielings, and Tylings.
The ART of GUNNERY, Geometrically, Inſtrumentally, and by a
way of Calculation, by the Logarithm Tables, by Addition and Subſtraction,
in the place of other Mens way of Arithmetick, of Multiplication and Diviſion: Alſo,
Artificial Fire-Works for Sea and Land Service, as alſo for Recreation and Delight, with
Figures.
ASTRONOMY, Geometrical, Inſtrumental, and by Calculation.
The ART of DIALLING by a Gnomonical Scale, and likewiſe by
Calculation ; Making all forts of Dials both without Doors and within, upon any Wall,
Çieling, or Floor, be they never ſo Irregular, whereſoever the Director Reflect Beams of
the Sun may come.
WHEREUNTO IS ANNEXED,
An Abridgment of the Penaltics and Forfeitures, by Acts of Parliaments
appointed, relating to the Cuſtoms and Navigation.
ALSO,
A COMPENDIUM of FORTIFICATION, both Geometrically
and Inſtrumentally.
By Capt. SAMUEL STURMY.
LONDON, Printed by E. Cotes for G. Harlock, W.Fiſker, E. Thomas, and D. Page :
and are to be ſold at chcir
Shops over againſt St. Magnus Church neer
London-bridge, 20
che Postern-gate uicer Tower-bill, at the Adam and Eve in Little Britain, and at
the Anchor and Mariner in Eaſt-Smithfield. MDCLXIX.
moſt eaſie
4
-44
Homes
P
THT
1
f
ܢ
h
ܪ
n
ܐ
ܠܐ
ܪ
'-
f ;
1
ܙ
ܙ
1
h
1
1
1
ܪ
1
{ {
ܢ
+°
1
ܙ
ܠܐ. ܡ
Histz Scouce
soilinan
8.7.41
43637
To the moſt Auguſt and Moft Serenę
M A JE S TY
OF
CHARLES II.
King of Great-Britain, France, and Ireland,
Defender of the Faith, c.
meilt.
Mof Gracious Sovereign,
Our Majeſties Royal Grandfather
King Fames, the Peace-maker,was
by the Wiſeſt of His Age juftly
reputed worthy to be entituled
Great Britain's Solomon, for His
Wiſdom, Learning, and Proſperous Govern-
.
Your Pious and Judicious Father King
Charles the Firſt His eſteem of the more Inge-
nuous and Politer Arts, Sculpture, Picture, and
whatever elſe was Ancient or Excellent, drew
hither into His own Galleries and Cabinets,
and other Noble Palaces of this happx Jand,
the beſt Monuments of Old Greece, and of
Modern Rome : And His own Incomparable
Writings will demonſtrate to ſucceeding Ages,
how richly he was furniſhed with the bet kinds
of Knowledge, and with the truly Celeſtial
and Divine Arts, which appeared in the whole
Courſe
A 2
tor
+
The Epiftle Dedicatory.
Courſe of His Life, but moſt of all in His laſt
Sufferings.
And now Your Majeſty hath, not only an
Hereditary Claim to Your Anceſtors Virtues
and Bleſſings, being by God's miraculous Pro-
vidence and Protection reſtored, and redeemed
from Your Fathers Calamities; but herein You
have infinitely excelled them all, and that in
the main, in that You are the firſt Founder of
a Royal Inſtitution for the Advancement and
Propagation of all Noble Arts and Beneficial
Inventions : So that all the Kings in the Chri-
ſtian World do emulate Your Great Example,
and do value themſelves in nothing more, than
that ſome of their wiſelt Subjects do follow and
adore the Foor-ſteps of Your Royal Society,
And by this that Everlaſting Renown is alrea-
dy ſpread over the face of the whole Earth
3
it hath ſurrounded the Globe by Sea and Land;
and we who are Borderers on the Shores, and
all they who inhabite on the deep Seas, are wit-
neſſes of the perpetual Ecchoes of your Fame,
with this Averment, That the better Spirits in
all the Univerſities of Chriſtendom are allured by
Your Majeſties Luſtre to embrace and rejoyce
in this Brighter Light, which can never hence-
forth be extinguiſhed. And this conſtrains as
well all Your Loyal and I rue-hearted Sub-
jects, as all Intelligent and Friendly Foreiners,
to offer their Applauſes, Veneration, and Ad-
miration
But it is far above the reach of my Abilities
1
i
to
The Epiſtle Dedicatory.
to make juſt report of theſe Great Deſigns, and
of all Your Majeſties more than Heroical Ac-
compliſhments. And yet I have a more Obli-
ging and Perſonal Concernment: For as the
Amplitude and Proſperouſneſs of Your Ma-
jeſties Dominions are chiefly maintained by
Your Majeſties Countenance upon the Mathe-
matical Arts, and eſpecially of Navigation ;
ſo Your Majeſty doth apparently excel all the
Kings that ever were, in this Noble kind of
Knowledge of the Naval Arts and Sciences. And
for there, and many other good reaſons, I hold
it my bounden duty to lay theſe Eſſays of
my
beft Endeavours at Your Royal Feet; and with
the l'roteſtation of a tryed and truſty Sea-man,
to avouch, That not only my beſt Skill in Arts,
. But my Heart and Life alſo, are entirely devo-
ted for Your Majeſties Service: And all my fel-
low Sea-men, and all true-hearted Engliſhmen,
do joyn in this one Voice, God ſave King Charles
the Second
*
Moſt Gracious Sovereign,
Your Majeſties moft Dutiful
and moſt Obedient Subject.
SAMUEL STURMY.
9
{
To the Honourable SOCIETY of
MERCHANT-ADVENTURERS
Of the CITY of
1
1
B R I S T O L.
• Τ Ο Τ Η Ε
i
MASTER, WAR DE NS, and ASSISTANTS
OF THE SAID
S O CI Ε Τ Υ. .
To my Honoured Friends
Sir Robert Cannand Sir Robert Yeomans,Knights
and Baronets, Sir Humfry Hooke, Sir Henry
Creſwick, Sir Fohn Knight, and Sir Thomas
Langton,
Knights, fohn WV illoughby and fohn
Knight, Eſquires.
Honourable Gentlemen,
Ut of the great Refpet I bear to you all of this
Society, in regard from my Touch I have lived
among you, and am a Burgeſs of your City, and
have been commanded, and a Commander ,
bave failed to ſeveral Parts of the World out of
this Port, I cannot commit thefe Productions of
my Pen to the wide Ocean of flu&tuating Opinions, without the aſs-
rance of the Prote&ion of ſome Honourable Patronage, as your
ſelves; without the which I might haply appear mis ſhapen, and mon-
strous to the eye of the World, and uneaſily eſcape ſubmerſion; ſince
that as the Year conſisting of more foul than fair Days, ſo the World
in truth affords more bitter Blaſts and virulent Cenſures of Detra-
Etion, than candid, firene , unprejudiced Judgment. Therefore I
muſt repair to you, as to Seth's Pillar, which in deſpight of whole
Torrents of Oppoſers, and Cataracts of Zoilus bis furious off-
Spring,
and
A A
-
1
I
.
1
1
Spring, will (media inter prælia) ſecure theſe my poor Labours,
and perpetuate them as indelible as the Stars; or yodo deferved Ho-
nours, mbo of your Funktion are the Pillars of a Country, Jupa
pòrting all manner of Trade and Commerce in the World, and, like
the induſtrious Bee, bringing, Honey to repery ones Hive, in adven-
turing jour Eſtates to ſeveral Parts of the World, for tbe Good of
many thouſands, wbich live and depend upon your Profperity. I could,
with many others, deſire, That this City and Society
. had in all re-
ſpects the ſame Laws, and Customs in, all Maritime Affairs, as the
Flonourable City of London: then would much Loſs, "Charge, and
Damage be prevented, that many times befals your Ships and Goodsis
a competent Company of Sea-men, for half. Pay, would attend on
board at the Ships going out, and coming home, until diſcharged;
and that Pay would relieve-themſelves, their wives, and Children,
and fit them for the Voyage, and be great ſatisfaction to them for
their attendance on your Service : for
by tļse Rules of Charity, The
Labourer is worthy of his Hire ; and none deſerve it more than
Sea-men. Then would your Sea-mens Courage be fortified, Honeſty
incouraged, and deſerving Mein rewarded
, and the Reformers of an ill
Cuſtomi be had in everlaſting
memory, for the Good they did in their
Generation. i humbly beg pardon for this:Digreßion; and humbly
deſire that you would take it into conſideration. For my part lever
did, and ſhall to the utmoſt of my poor abilities endeauöur to honour
this City, and will adventure my Life to ſerve this Society in any
Parts of the World.
Accept therefore this off-ſpring of ſomie ſpare Hours, improved
more with an intent for the Publick Good, than for any Private
Benefit.
I ſhall conclude this my Humble Addreſs with a Temporal and
Spiritual wiſh, viz. That the Encreaſe of your Treaſures may anſwer
your Hazards and Deſires, and that your Virtues and Graces may
exceed your Treaſures in this Life, and in that to come, may your
Glories as far tranſceud both, as Heaven doth Earth: And for
this you have not only the earneſt Wiſhes, but tbe cordial Prayers of,
0
Your Honours moſt humble Servitor
1
Briſol, Noven.be
1 6 6 7
to be Commanded,
SAMUEL STURMY,
1
.
TO THE
t
Courteous Reader.
I
M
Shall make his
Athematical Studies have for theſe many Years been
much neglected, if not contemned; yet have chere
been ſo many rare Inventions found, even by Men of
our own Nation, that nothing now ſcems almoſt pof-
fible to be added more. As in other Studies, ſo we
may ſay in theſe, Nil di£tum quod non dictum prius :
We at the leaſt muſt needs acknowledge, That in this
we have preſented thee with nothing new, nothing
that is our own. Ex integra greca, integram Comædiam hodie ſum ačturus, Qui edeficer
faith Terence, that moſt excellent Comedian, in his Heautontimorumenon. publick place,
Tranſlation was his Apology; Tranſcription,Collection,and compoſition, ours. trop haue on
This only we have endeavoured, That the firft Principles and Foundations trop balſe.
of thoſe Studics (which were not to be known until now, but by being ubo builds i'ch
'
acquainted with niany Books) might in a due method, and a perſpicuous way where all
manner, be as it were at oncc preſented to thy view.
paſs by,
The Matter, being Mathematical and Practical Arts of my own pra- houſe too low or
Eticc, I can the better avouch the caſe and rruth of thcoi to all ingenuous high).
Pra&ticioners, and unto ſuch as have as yet Icarned nothing but i Arith-
metick.
To that purpoſe, we have at firſt laid down ſuch Propoſitions, as all
young Seamen are or ſhould be perfect in, concerning the Compaſs, and
the Moon's Motion, Inſtrumentally and Arithmetically; and by it; in the
ſame manner, how to know the Rúlcs af the Ebbing and Flowing of the
Sea, with the Rules of Time of Flood and High-water in any Port in the
World; with a Diſcourſe of the Practick Pari of Navigation, in working
of a Ship in all Caſes and Conditions of Weacher ar Sea, to the beſt of
my Experience.
And the ABG of Geometry, its Definitions and Geometrical Pro-
blems, out of Euclid and others, as muſt be known to ſuch as would
know the Nature and Menſuration of Triangles. Next,We have proceeded
to the Deſcriptions of all the moſt uſeful inſtruments for Artiſts and Na-
vigators; as the scale of Scales, which is a Mathematical Ruler, that re-
ſolves all Mathematical Rules whatſoever : And wc our ſelves have fitted
Tables and Diagrams in that inanner, as we preſune has not been done
in that plainnels, and ſo caſic to be underltood, by any Man before.
Therç is the Diagrams and Tables together, both Natural and Artificial
and the Scale, and its Making and uſe follows. Secondly, The Making
and Uſe of the Traverſe-Scale of Artificial Points and Quarters; The Ma-
king of the Quadrant and Index, and their ready uſe in Aſtronomy and Na-
vigation; and the Protractor ; The Projection and uſe of the Nocturnal,
and new Tables of the North Stars Declination. And on the back-ſide are
32 of the moſt uſeful Stars in the Heaven for Navigators, and its Ule;
with
3
а
}
1
1
To the Reader.
with Tables of the Longitude and Latitude, Right Aſcenſion and Declinati-
on; The Deſcription and uſe of the Foreſtaff, Davis his Quadrant ; as
allo a new Quadrant and Quadrat, that I uſe my ſelf at Land and Sea; A
Conſtant Kalendar, joyned-with the Tables of the Suns Declination, for
32 years to come. And, Thirdly, The Nature and Quality of Triangles.
And, Fourthly. Of Sailing by the Plain Sca-Chart, and the Uacertainty
thereof; and of Navigation by Mercator, or Mr. Wright's Projection,
and by the necreſt way of Sailing by the Globe, or Arch of a Great Circle ;
with the making of the True Sea-Chart, Geometrically, Arithmetically,
and Inſtrumentally, as the true way of keeping a Sca-Journal at Sca, very
caſic at once by plain, Mercator, or Great Circle Sailing, with new Ta-
bles of Longitude and Latitude round the World, from the Meridiat, of
tho Lizard, terminating at 180 deg. Eaft and weſt of tbat Meridian.
The Fifth Book, The Art of Surveying of Land by the Sea Azimistkor
Amplitude Compaſs, very eaſie and uſeful for Sea-men : The Art of Gam-
ging of all ſorts of veſſels
, and Meaſuring of Timber, Stone, and Glaſs,
and Ships, Geometrically, Inſtrumentally, arid Arithmetically; and a
nioſt excellent Gunners Scale; with the caſieſt way of Gunnery char hath
been writ by any : For what Nathaniel Nye hath done by Arithmetick, by
the square and Cube, and their Roots, which is the hardeſt ſort of Arith-
metick, by Maltiplication and Diviſion, I have done by the Logarithme
Tables, by Addition and Subtraction, and likewiſe Geometrically and In-
ſtrumentally. The Scale (héws at once, in a moment, the ready Dimenifi-
ons of twenty ſorts of moſt uſeful Ordnance, from a Bafe to a Cannon-
Rayxb;' their length, and weight of the Gun, Powder, and Shot, and Ta-
blés of weight of Shor of Lead, Iron, and Stone'; with a Table of Righie
Ranges and Point Blanks; with a Plain Scale and Dialling Scale, Quadrant,
and Quadrat, for taking of Heights and Diſtances; with a Line of Inches
and Numbers, for the ready working all other Proportions of Solids, or
otherwiſe ; being a moſt uſeful Inſtrument for all Land and Sea Gunners :
But moſt eſpecially I do adviſe all Sca Gunners to carry one of thoſe molt
uſefal Inſtruments in bis Pocket, and by our Directions learn the Uſe per-
fc&ly of them. I am aſhamed to hear how fenflelly many Sea Gunners
will talk of rhe Art, and know little or nothing therein, but only how to
ſpunge, lade, and fire a Gun at Random,. without any Rules of finding
the Diſpart, thickneſs of the Mettle in all places, and proportion any
Charge of Powder thereto, and other Rules which ſhould be known.
Hercin how many of them are defective ? And to ſupply that defect, I
have taken this pains in the Aft, to the end to help all ſuch as are ingenu.
ous, and willing to learn : As allo, all manner of Artificial Fire-works
and Rockets, with their Figures and Fiery Arrows, Granadoes, and Pots.
The Sixth Book is the Art of Aſtronomy, containing the Definition of the
Circle of the spheres, with the manner how to reſolve all the moſt ne-
ceſſary Propoſitions thereto belonging, by a Line of Chords and sines, and
Chords and Targents, and half Tangents, Geometrically and by Calcula-
tion, by the Logarithme Tables of Artificial Sines and Tangents: And all
uſeful Aſtronomical Propoſitions appertaining to the Firſt Motion ;
and
Tables for finding always the Suns truc place; being all of extraordinary
uſe, and made plain to the meaneſt Capacity,
Seventhly, The Seventh Book is the Art of Dialling by the Gnomonical
Scale,
ܪ
1
+
E
*
To the Reader.
Soale, with the Diagram and making of the ſaid Scales, with Tables alſo
in the Second Book deſcribed; with the Fundamental Diagram of all scales
on the Ruler, as alſo by Calculation, ſhowing the making of all ſorts of
Dials, both within doors and without, upon any Wall, Ceiling, or Floor,
be they never ſo irregular, whereſoever the direct or reflected Beams of
the Sun may come, for any Latitude ; and how to find the true Hour of
the Night by the Moon and Stars ; and how to Colour, Guild, and Paint
Dials; and how to faften the Gnomen in Stone or Wood. We have in-
filted the more upon it, and by our Explanation have endeavoured to
piake it plain and eaſie, it being all our own Practical Arts; fo chat nothing
may be wanting, which either fornier Ages or our own (by Gods Blel
ſing and their Induſtry) have afforded to us. We have to the Artificial
Canon added out of Mr. wing's Harmonicon Cæleſte, page 263. the Rules to
be taken and obſerved in ufc of Mr. Gunter's Canon of Artificial Sines and
Tangents, and Mr. Brigs his Canon of Logarithme Numbers, as in chac
Form, and in this work, we have made uſe of his Directions in the
Aſtronomical Calculation, and the Demonſtration by our own Rules ;
and of Mr. Norwood's Advice in Navigation, and by Demonſtration our
own (the way of our uſual practice at Sea in keeping our Journal ) and
for the Longitude and Latitude of Places, we have had the beſt Experi-
ence we could procure from the ableſt Pilots and Maſters that have been
in the ſeveral Places of the World, and likewiſe of our own Obſervation
of ſeveral Places in the weſt-Indies, and other Parts of the World; to-
gether comparing of them with ſeveral Tables, formerly'made and lately
corrected, and fitted for a Meridian of our own Country, and the prin-
cipal Cape of this Land, for thy caſe; the Lizard being the farewel Cape
to moſt Ships that ſail out of the Britiſh Seas, any way to the South or
weft, and likewiſe the firſt Land made at their return home; and therefore
it muſt needs be very uſeful for all Northern and Southern Navigators in
their Voyages, with great caſe and exactneſs.
It's nothing new, nor does it corne by chance,
That Art is envy'd ſtill by Ignorance.
For the Art of Gauging, I have conferred with Mr. Philip Staynred, Ma-
thematician and Gager in Briſtol; and all the Rules that have been laid
down in the following Treatiſe, are moſt exact and caſie to the meancit
Capacity of ſuch as are skiiful in Arithmetick ; but with a great deal of
Labour, Study, Care, and Charge, in the Tryal of the Practice of them
by our ſelf: which may be conlidered by the Ingenuous Practicioners,
though much more abuſed by ignorant Momus and his Mates, who make
it their buſineſs to ſcoff, deride, affront, and abuſe all ſuch as are Inge-
nious, and pretend to have any thing more than theniſelves; and
know then by their railing Diſcourſe of any Ingenious Work or Artiſt
.
For ſuch Loiterers there is a pair of Stocks fitted in Hell by the Devil,
where for their malice, abuſes, curſed railings, and villainous revilings
of thoſe that Study the Honour of their Country to Poſterity, the harm-
leſs ſtudy of Virtue, and praiſe-worthy commendation of all good honeft-
minded men ; I ſay, ſuch Momuſſes will have their Heads in ſuch Scocks,
and their Tails laſh'd by the Devils for ever, for their malice and envy, if
they
#
1
you ſhall
a 2
1
1
To the Reader.
(
they give it not over, and repent of it in time. But for all honeft-minded
men that love Arts and Sciences, Theorical and Practical, God doth give
them his Spirit to guide them in all Lawful Arts, to the knowledge there-
of, according to their deſire of him.
Others that have either ſpent more time, or made a farther progreſs in
theſe raviſhing Studies, might (if they would have taken the pains) have
haply preſented thee with more in a leſs room; but the moſt of this was
at the firſt collected for our private uſe, and direction of our three Bro-
thers and Son; but now publiſhed for the good of others. Nevertheleſs;
I am not ignorant, how that never any man living, in his writing, could
pleaſe the phanſie of all men, neitber do I expect to be the firſt. To
pleaſe the envious, I cannot ; for they are reſolute: To content the
fcornful, I will not attempt it : To flatter the baughty, were much folly:
To dillwade the capricious, were needleſs; and to perſwade the courte-
ous, were unnecellary. Let every one do as his Genius doth beſt diſpoſe
him, take where he pleaſeth, read whar he liketh, and leave what he
likech not. For my own part, I have with much diligence and induſtry
waded through many ænigmatical Difficulties, and have removed and
drawn back the Curtain of Darkneſs from off our Engliſla Horizon, iti
our Mathematical and Practical Arts following.
Laſtly, I deſire the Judicious Reader, if he chance to meet with any
Errata (as ſome may happen in a Work of this Nature) that he would
courteouſly amend them, and not with cavillation ungratefully requite
my painful Labours. Haply, if this find acceptance, it may cncourage
me to publiſh ſome other thing, which perhaps may give thee much fa-
tisfaction, and be commodious to my Country-men of England. Vale.
#
Yours, and Urania's Servant,
St. Georges Pill,
10 Novemb.
1667.
SAMUEL STURMY.
To
-
.
T
To his Ingenious and Induſtrious Friend
Capt. SAMUEL ŠTUR MY.
A double Acroſtick.
t
S-uch Noble Captains Honour well deſerve,
A-s to their utmoſt, King and Country ferue;
M-aking their skill and Practice freely known
U-nto all others who will Learning own.
E-aft, Weſt, North, South, thy Compaſs will them guide,
L-eft they should wander with each wind and Tide.
ì
S-tir up thy Noble Genius, and go or ;
5: T-hy Book Shall live, when thou art dead and
gon,
U-nto thy Glory: and proud Braggards will
R-epine they could not like thee hew their skill.
M-omus may carp, but from all honeſt Hearts
Y-ou'l Thanks have for your MAGAZINE of Arts.
Ś -ome men, when they this MAGAZINE ſhall spy's
Arnd note the Author, preſently will cry,
M-axy fuch Captains will undo the Trade:
U-nlock theſe Secrets, all are Captains made.
E-nvy thus, Devil-like, would keep men blind,
L-et Noble Sons of Art be free and kind.
!
S-o well ford, Captain, is thy MAGAZINE,
T-hat 'twill invite all ſorts : This Buſh of mine
U-nto thy wine is needleſs
. All men fall
R-eap Profit by thy Labours : and if all
M-en thus would add their Talents unto thine,
Y-od foon compleat a famous MAGAZINE.
Non nobis folum nati ſumus.
1
HENRY PHILLIPPES.
*
1
1
To :
1
To His Worthy Friend the Author
Captain S AMUEL STURMY.
On his BOOK Entituled,
The MARINER'S MAGAZINE.
1
'Weet gentle Reader, ſearch this: Magazin,
And tell me then what thou haſt found therein :
Thou canſt not chuſe, but ſay, There is ſuch Store
of Mathematick Arts, there needs no more.
Well, to be brief, jould I extol thy Fame,
'Twould be in vain : Thy Book Shall do the ſame.
PHILIP STAINRED.
܀
1
To his Judicious Friend the Author
Capt. S AMUEL STUR MY,
ON HIS
MAGAZINE of ARTS.
R
EADE R, Survey, with an impartial Eye,
The Care, the Pains, the Art, the Induſtry,
And Charge, at which the Author, long, hath been,
To Store (with Plenty) this his MAGAZEEN;
Or, rather, MART OF ARTS; where you may buy
(For little Money) INGENUITY:
And be partaker of thoſe Arts we call
Sciences Liberal, MATHEMATICAL:
He having taken Pains on th' Deep and Shore,
Graſping and Grap'ling to increaſe his Store;
And after all his Toil and Pains (thus ſpent)
Gives it his Country for an Ornament.
Ranſack this CASCATE (therefore) wbere goil find
Plenty of JEWELS to adorn the Mind.
I
Gcometry.
And firſt, is repreſented to your Eye
Selected Problems in GEOMETRY.
Navigation.
In NAVIGATION Rules it doth afford,
will make Men Sca-men e're they go aboard:
And when Embarked on the Ocean far,.
May Sreer from th Artick to th' Antartick Star;
1
And
1
H
+
1
܀
+
And so, with Prudeucc, may & Voyage make
Over the Occan's Univerſal Lake :
No Places diſtance hindring of Commerce,
Having free Traffick through the Univerſe.
The GIO DECIAN, in this Book, may have
Rules to Survey his Land, and then his Grave....
VINERIUS now will find it no hard Task
To know he hath his Dwe, and Gauge his Cask.
Surveying.
+
Gauging?
Here MURIFRA GUş certain Skill may gain
To reach his Mark, making no Shot in vain.
Gunnery
Aſtronomy
And fair URANIA leads you by the Hand,
Deſcrying how the Spheres to underſtand ;
Unlocking all the Hidden Treaſury,
And Secret Mylt’rics in ASTRONOMY.
:
In HOROMETRIA Skill
To trace Sol's Courſe out on a Dial Plain.
you may attain,
Dialling.
Fortification,
1
And the Munitor hither may reſort
For Rules whereby to Fabricate his Forr,
To Spring his Myne, and alſo Sconces raiſe
Againſt his Foes, to his Renown and Praiſe.
.
And to the Trader it will be a Treaſure,
Yielding him Knowledge both in Weights and Meaſure.
With this, and ſuch like beneficial Skill,
Our Author This his MAGAZINE did fill;
And that for th Good and Benefit of thoſe
who honour Vertue, and to Vice are. Foes.
Conſider, then, th elaborate Pains he took,
And thank him as thou "profit' Jt by. kis BOOK.
WILL, LEY B O URN.
i
+
To
1
!
1
In Praiſe of his Dear Friend the Author, for his ART,
Capt. SAMUEL STURMY.
9
Or
Vr Artiſt here, I am ſo bold to tell
Je,
Treats of no Toys, nor does be Baubles ſell ge;
Though fuch like ftuff indeed, it well may be,
to have met i tħ Mathematicks Theory:
where he is for the Practick, and does Aight
All fisch as but Theorically write.
'Tis much and high, you'l ſay; but 'tis as fore,
Sutch Practick Works for ever will endure.
And as your fluent Naſo dropp'd before,
Theſe will be read where e're olly Cannons rore.
Here is no trivial Traſh, or frothy Rbimes,
To Drunkards grateful at their tippling times. .
Here's no Acroſticks, nor no Anagrams,
In Combs and Sciſſors for th' Barbarians :
Much leſs your Epigrams, and ſuch like Toys,
For Children fit, or at the beſt for Boys:
No novel Romance, nor no paultry Plays,
To wear out Time with, and miſ-Spend our Days.
The Matter here is all ſublime and high,
That bear: bis Name unto the Starry Sky.
But if you neerly mark him, and his End,
You muſt confeſs, he doth them all tranſcend.
!
1
1
+
j
WILLIAM BERWICK, jun.
!
1
-
1
The
--
**
4 h
Tv
S.
1
1
The AUTHOR to His BOOK.
Gº
O forth, thou ſhapeleſs Embryon of my Brain, ,
Unfaſhion'd as thou art ; expreſs the Atrain
And Language of thy: diſcontented-Sire,
who hardly ransom'd his poor :Babe from fire,
To offer to the world, and careleſs Mena
The timeleſs Fruits of his officious Pen.
Thou art no lovely Darling, ſtamp:d to pleaſe
The Looks of Greatneſs; no Delight to eaſe
Their melancholy, Temper, who reject,
As idle Toys, but what themſelves affet.
No lucky Planet darted forth his Rays,
To promile love unto thy Infant Days.
Thou maiſt, perhaps, bé Merchandiſe for Slaves,
who ſell their Asthors wits, and buy their Graves.
Thou maiſt be cenfur’d guilty of that blame,
which is the Midwifes fault, the Parents ſhame :
Th014 maiſt be Talk for Tables, us'd for sport
At Tavern-meetings, Paſtime for the Court.
Thon mailt be torn by their malicious Phang's
who ne're were taught to know & Parents Pangs:
How cas’ly can proud Ignorance out-fare.
The comlieft weeds thy. Poverty can wear?
when all the siſters on our Iſis ſide
Are oft livorn Servants to aſpiring Pride;
And our renowned Mother Athens groans,
To ſee her Garden ſet with Cadmus Sons,
whoſe Birth is mutual Strife, whoſe Deſtiny
Is only to be born, to fight, and die.
Prometheus is chain’d faſt, and cannot move
To steal a little Fire from mighty Jove,
To People new the world, that we may ſee
Our Mother teem with a new Progeny:
And therefore with thy hapleſs Father prove
To place thy Duty, where thor findeſt Love.
when thou arrivſt at Court, thou long mayſt ſtay
Some Friends alliſtance, to prepare thee way;
As in a cloudy morning i have done,
when envious Vapours Tout me from the Sun.
--when all elſe enter, ſee thou humbly ftand,
To beg a Kiſs from thy Mæcenas Hand:
if Fre vouchlafe a Look to guild thy State,
Proclaime Hins Noble, thy ſelf. Fortunate.
1
1
Y
pre
1
.
1
A
S. S.
b
Thc
1.
។
1
1
Τ Η Ε
AUTHOR'S COMPLAINT.
on in theſe days are Artifs now regarded!
No, not ſo much as oil or Ink rewarded.
Tet shall a Sycophant, or Rhiming Knave,
If be but Impudence and gay Clothes have,
Car harp upon some ſcurr loses Jeft or Tale,
(Though fifteen times told, and i' th city ſtale)
Command a Great Mans Ear, perhaps be able
To prefer Suitsy and elbow at his Table,
Wear Speaking Pockets, boast whom he doth ſerve;
when meriting Mex may either beg or ſtarve.
Mean time we spend our fruitleſs Hours in vain.
and Age of want and hunger doth complain.
It grieves us now, although too late at laft,
Our time in painful Studies to have past;
and what a Folly 'tis we now have found,
To caſt our Seed in an unfruitful Ground;
That in time paßt we have laid xp no store,
which might maintain us when our Heads be hoar ;
And that our fbaken Veſſels, torn and thin,
Can find no eafie Port to harbour in.
Ther, barres Arts, seek out ſome other Friend;
For i henceforth a thriving Courſe intend.
None will with Violets my aſbes grace,
Or ſtrew ſweet fragrant Rofes in the place.
if any loves me, and intends to give,
I wiſh to taſte his Bounty whilſt i live.
what do I care, when Fates my Thred have fpur,
Though Briars and Thorns may Grave ſball over-run,
Impiger extremos currit Mercator ad Indos,
Per marc paupcricm fugiens, pcr ſaxa, per ignes.
1
1
AN
!
AN
:
I N D E X
,
SHE WING;
The CONTENTS of the SEVEN BOOKS
.
1
OF THE
MARINER'S MAGAZINE,
BOOK I.
TH
He Deſcription of Navigation in general.
Page 1
Of what is needfub
firft to be known in the Practick Part : And of the
Compaſs; and bow to divide the Circles and Parts,
3
The Moon's Motion, and the Ebbing and Flowing of the Sea.
6
ibid.
The making a moſt uſeful Inſtrument for the Moon.
ibid.
The Variation-Compaſs, and the uſe thereof, in 10 Propoſitions.
The finding the Golden Number, or Prime, and Epact, according to the Engliſh
Accompt, and all other things relating to the Moon and Tides, Arithmetia
cally.
9
The Practick Part of Navigation, in working a ship in all weathers and
Conditions at Sea.
Geometrical Definitions.
Geometrical Problems.
28, to 43
Is, to 22
22
BOOK II.
{
TH
He Argument, and Deſcription of Inſtruments in general., Page 45
Deſcription of the Fundamental Diagram, and Tables for the
making of the Lines of Chords and Rumbs, as alſo of Sines, Tangents, and
Secants Natural, on the Scale ; and of what Inſtruments you muſt be pro-
vided with before you can make Inſtruments for Mathematical vſes.
47
The Explanation of the other half of the former Semicircle, being a Deſcri-
ption of the Fundamental Diagram of the Dialling Scale on the Mathema-
tical Ruler ; with a Table for the dividing of the Hours and Minutes on the
fame; 4 Táble for the dividing of the Gnomon-line on the scale, called by
fome a Line of Latitudes, as alſo a Table of Tangents for five Hours, to
every five Minutes of an Hour, for the inlarging the Hour-line Scale. 55
The Scales or Lines on the back-ſide of the Mathematical Ruler, viz. A Line
of Artificial Numbers, with a Table how to make the ſame. A Table of
Artificial Tangents, and how to make the Line ; as alſo a Táble of Artifi-
b 2
ciał
The Contents
!
71
cial Sines; The making of Mercator's Meridian Line, by our Table of
Meridional Parts, in Leagues and 10 parts of a League: And the Æquia
noćtial is the Line of Equal parts, by which the Table of Numbers were
taken out, and the Lines made by.
Page 58
How to calculate and make & Table for the Diviſion of the scale of Redučtion,
and the uſe thereof.
62
A Täble for the diviſion and making of the Artificial Rumbs, or Points, Halfs,
and Quarters, on the Traverſe Scale.
63
How to make a Quadrant, which will reſolve many Queſtions in Aſtronomy by
the help of an Index, and alſo very uſeful in Navigation ; with the uſe
thereof in Aſtronomy and Navigation, in ſeven sections.
64
To find how many Leagnes do anſwer to every Rumb and Quarter, in fix Pro-
poſitions in Navigation,
70
How to make a uſeful Protractor,
The Projection of the Nocturnal, and the uſe thereof by the North Star. 73
How to uſe the Pole Stars Declination, and thereby to get the Latitude, with
the Table.
74
How to make a moſt uſefal Inſtrument of the Stars on the back ſide of the No-
Eturnal, and by it to know moſt readily when any of 31 of the moſt notable
Stars will come to the Meridian, what Hour of the Night at anj time of the
rear, at the first ſight; with a Table of the Longitude and Latitude from
the beginning of the Year 1671; with the Right Aſcenſion and Declina-
tion of 31 of the moſt notable fixed Stars, Calculated from Tycho his Ta-
bles, Rectified from the Year of our Lord 1671.
76
The uſe of the moſt uſeful Inftrument of the Stars, how to know the Hour of
the Night any Star comes to the Meridian in any Latitude ; and how to know
what Stars are in Courſe at any Time or Day of the rear.
77
The Deſcription and uſe of 3 Stars.called the Croſiers.
78
A Deſcription of the making of the Croſs-ſtaff, and how to uſe the ſame fully.
79, to 85
A Deſcription and uſe of the Quadrant or Back-ſtaff, in ſix Propoſitions, de-
claring the uſe thereof in all obſervations.
The Deſcription and uſe of the moſt uſeful Quadrant for the taking of Alti-
tudes of the Sun or Stars, on Land or Sea, backwards or forwards, or any
other Altitude of Hills, Trees, Caſtles, or Things whatſoever.
92
A Conftant Kalendar or Almanack for 300 Years; but more exactly ſerving
for 19 Years, being the Circle of the Moon, or the Golden Number; with
new exact Tables of the Suns Declination, retified by the beſt Hypotheſis
until the Leap-years, and the uſe thereof.
IOI, to 122
1
85
}
BOOK
III.
CHAP. I. F the Nature and Quality of Triangles. Page 123
CHAP. II. Containing the Doctrine of the Dimenſions
of Right-lined Triangles, whether Right-angled cr oblique-angled; and
the ſeveral Caſes therein reſolved, both by Tables, and alſo by the Lines of
Artificial Numbers, Sines, and Tangents.
125
CASE I. In a Righe-angled Plain Triangle
, the Baſe and the Angle at the
126
Bafe being given, to find the perpendicular,
CASE II.
1
.
1
The Contents.
Page 127
Case II. The Baſe aad the Angle at the Bafe being given, to find the Hy-
pothennfal.
Case III. The Hypothennfal and Angle at the Baſe being given, to find
the Perpendicular.
128
CASE IV. The Hypothenuſal and Angle at the Baſé being given, to find
the Bale.
129
CASE V. Let the perpendicular be the Difference of Latitude 253 Leagues,
and the Angle at C béS WW1 deg. 45 min. Weſterly, or 58 deg. ler
it be given to find the Hypothenuſal.
129
CASE VI. The Hypothenuſal or Diſtance failed, the perpendicular of Diffe-
rence of Latitude
given, to find the Kumb.
130
CASE VII. The Hypothennfal, and the Parallel of Longitude, and the Ra-
dins given, to find the Rumb or Courſe failed.
ibid.
Case Vill. Having two Angles and a side oppoſite to one of them given,
to find the side oppoſite to the other.
131
CASE IX. Two Sides and an Angle oppoſite to one of them being given, to
find the Angle oppoſite unto the other.
132
CASE X. Having two sides and the Angle contained by them given, to
find either of the other Angles.
ibid.
CAS E XI. Two sides and their contained Angle given, to find the third
Side,
134
CASE XII. Three Sides of an Oblique Triangle being given, to find the
Angles.
ibid.
Β ο ο κ ΙV.
CHAP I. F Sailing by the Plain Chart, and the Uncertainties there-
of ; and of Navigation, with its parts. Page 137
Queſtions of Sailing by the ordinary Sea-Chart.
140
CHAP. II. Declaring what muſt be obſerved by all that keep Accompt of
a Ships way; and to find the true Point of the ship at any time, according
to the Plain Chayt.
144
Directions how wę do keep our Reckoning's at Sea by the Log-board, and alſo by
our Journal Book.
145
CHAP. III. A formal and exact way of ſetting down and perfekting a
Sea-Reckoning:
1:47
A Traverſe-Table for every Point, Half-Point, and Oilarter-Point of the
Compaſs, to the hundredth part of a League or Mile, which gives the
Difference of Latitude and Departure from the Meridian. 149
Examples and uſe of the Tables, with a Journal from Lundy to Barbadoes
by the Plain Chart.
153
The Plain Sea. Chart, and how to make it, and the uſe thereof.
156
CHA P. IV. How to correct the Accompt when the Dead Latitude différs
from the Latitude by obſervation.
157
CHAP. V. Hopp to allow for known Currents , in estimating the ships
Corrſe and Diſtance.
CHAP. VI. Curious Queſtions in Navigation, and how to reſolve them Ge-
ometrically and by Calculation.
161 .
CHAP. VII. The diſagreement betwixt the ordinary Sea-Chart and the
Globe
1
159, 160
1
1
A
The Contents
Globe; and the agreement betwixt the Globe and the True Sea-Chart, made
after Mercator's way, or Mt. Edward Wright’s Projection ; with the uſe
thereof
Page 166
A Table of Meridional Parts to the tenth part of a League, and for every
10 Minutes of Latitude, from the Aquinoctial to the Poles, with the uſe
thereof in Mercator's Sailing, Geometrically and by Calculation. 169
HA P. VIII. How to divide a Particular Sea-Chart according to Mercator
and Mr. Wright's Projection.
174
CHAP. IX. The Proječtion of the Meridian-line by Geometry; and homo to
make a Scale of Leagues for to meaſure Diſtances in any Latitude. 184,185
CHAP. X. The way of Sailing by a Great Circle.
176
CHA P. XL. How to find the true diſtance of Places, one of them having no
Latitude, the other having Latitude and Difference of Longitude leſs than
180 deg. To find (1) Their Distance in a Great Circle, (2) The direct
• Poſition of the firſt place from the ſecond, (3) And the ſecond place from
the firſt.
179
GĦAP. XII. The Deſcription of the Globe in Plano, and the ſeveral com
cluſions wrought thereby
189
CHAP. XIII. To Calculate the Arch of a Great Circle for every fifth or
tenth Degree of Latitude or Longitude.
CHAP. XIV. How by the scale of Tangents to make a part of the Globe in
Plano, whereby goi may trace out the Latitudes to every Degree of Longi-
tude, or every 5 or 10 Degrees, as neer as you will deſire, without Calcu-
lation.
193
CHAP. XV. By the Latitude, and Difference of Longitude from the obli-
quity, to find the true Great Circles Diſtance.
196
CHA P. XŅI. How to make the Trueſt sen-Chart, and the uſe thereof in
Mercator's and Great Circle Sailing, called a General Chart.
CHA P. XVII. How to keep a true and perfect Ses-Fournal by Plain Sailing
and the True Sea-Chart, together with the Explanation thereof.
CHAP. XVIII. A Deſcription of the Table of the Latitude and Longitude
of Places, and the way how to find both.
192
200
202
3
206
7
BOOK V.
1
1
6
+
CHAP. I. He Art of Surveying Land by the Azimuth or Amplitude
Compaſs, with the Deſcription thereof; as alſo the Staff
and chain, with the uſe thereof.
Page 2
How to meaſure a Square piece of Ground.
4
To meaſure a Long Square piece of Ground by the Line of Numbers and Arith-
metick,
5
Hew to meaſure a Triangular piece of Ground.
How to meaſure a piece of Ground of four unequal Sides, called a Trapezia.
ibid.
How to meaſure a piece of Ground being a perfe&t Circle.
7
How to meaſure an Oval piece of Ground.
8
How to meaſure a piece of Ground lying in form of a Se&tor.
ibid.
To meaſure a piece of Ground being a Segment or part of a Circle. 9
Having the content of a piece of Groundin Acres, to find how many Perch of
that Scale was concained in one Inch whereby. it was plotted. ibid.
CHAP. II.
!
1
I
1
13.
your
I
26
27
29
The Contents .
CHAP. II. How to take the plot of a Field at one ſtation, taken in the mid-
dle thereof, by the Compaſs.
Page 1
Chap. III. How to take the Plot of a Field at one ſtation, taken at any
Angle thereof.
CụAP. IV. How to meaſure an Irregular piece of Ground, by reducing the
Sides into Triangles and Trapezia's, and how to lay is down in
Field-
Book.
14
CHAP. V. How to take the Height of Tenariffe, or any other iſland or
Mountain,
18
CHAP. VI. How to find the diſtance of a Fort or Caſtle, or the breadth
of a River; by two stations, with the Quantity of the Angle at each fta-
tion,
CHAP. VII. How to take the diſtance of divers Places one from another,
and to protrakt as it were a Map thereof by the Compaſs and Plain Scale, 24
CHAP. VIII. The Art of Gauging of veſſels by the Line of Numbers, and
the Lines on the Gauging Rod or Staff, and by Arithmetick.
The true Content of a foli: Meafure being known; to find the Gauge-point of
the ſame Meaſure.
ibid.
The Deſcription of the Gauging Rod or Staff.
The Deſcriptiox of Symbols of words for brevity in Arithmetick. ibid.
How to meaſure a Cubical Veſſel.
28
How to meaſure any Square Veſſel.
ibid.
How to meaſure a Cylinder Veſſel.
How to meaſure a Veſſel in form of a Globe.
ibid.
How to meaſure a Barrel
, Pipe, But, Puncheon, Hogſhead, or ſmall Cask. ibid.
How to find the Quantity of Liquor in a Cask that is part full.
31
How to meaſure a Brewers Tun or Maſh-var.
How to meaſure a Cone Veſſel.
33
How to meaſure a Segment of a Globe or Sphere.
34
How to reduce Ale-meaſure into wine, and likewiſe to reduce wine-gallons in-
to Ale.
ibid.
How to meaſure a Brewers Oval Thn.
ibid.
Chap. IX. wherein is ſhewed, How to meaſure exactly all kinds of Plain
Superficies, both by Arithmetick and Inſtrumentally.
How to meaſure a wall of an Houſe in form of a long Square.
ibid,
How to meaſure Boards, Glaſs, Pavement, wainſcot, and the like. 39
How
to meaſure folid Bodies, as Timber and Stone.
ibid.
To find how many Inches in-lexgth will make one foot of Timber, being alike
39
How to meaſure a Cylinder or a Tree whoſe Diameters at the ends be equal. 40
How to meaſure a round piece of taper Timber.
41
How to meaſure a Pyramidal piece of Timber.
ibid.
How to meaſure a Conical piece of Timber.
42
How to find the Burden of a ship.
The Uſe of the
Line of Numbers in Reduktion and the Rule of Three.
44
CHAP. X. Sect. 1. The Art of Gunnery, by a New-invented, uſefuls.
and Portable Scale.
45
Scēt. 2. The Qualifications each Gunner ought to have, and his Duty and of-
fice.
ibid:
Scct. 3. The Deſcription and uſe of the Gunners Scale on both sides. 47
Sect
.
32
1
1
A
30
in the Squares.
43
1
1
The Contents
Sect. 4. The Uſe of the Line of Numbers on the edge of the Scale, for the help
of ſuch as cannot extract the Square and Cube Rööts.
Page 49
Sećt.
5:
As likewiſe how by the
Logarithm Tables and Addition and Subjèracti-
on, to Reſolve with wonderful eaſe all Concluſions in the Art of Gunnery: 50
Sca.6. The Geometrical finding the Diameter for the weight of any shot af-
Signed.
:51
Sect. 7. How to find what Proportion is between Bullets of Iron, Lead, and
Stone ; by knowing the weight of one shot of Iron, to find the weighi of
another Shot of Lead, Braſs, or Stone, of the like Diameter.
:53
Sect, 8. How by knowing the weight of one Piece or Ordnance, to find the
weight of another piece of the ſame ſhape, and the Same Metal, or any
other Metal.
54
Sect. 9. How to make a shot of Lead and Stone in the ſame Mold, of the same
Diameter as the Iron Shot is of.
55
Sect. 10. Hon by knowing what Quantity of Powder will load one Piece of
Ordnance, to know how much will load any other Piece whatſoever. 56
How to make the true Difpart of any true boared Piece of Ordnance, or other.
wiſe, to know whether the Piece be Chamber-boared.
57
To know what Diameters every shot meiſt be.of to fit any Piece of Ordnance. 58
To find what Flaws, Cracks, or Honey-combs are in any piece of Ordnance;
and likewiſe to find whether a Piece of Ordnance be true boared, or no. ibid.
of Iron Ordnance, what Quantity of Powder to allow for their Loading ; and
what Powder to allow for Ordnance not true boared.
61
How Molds, Forms, and Cartrages are to be made for any ſort of Ordnance. 63
How to make Ladles, Rammers; Sponges, for all ſorts of Ordnance and how
the Carriage of a piece ſhould be made.
ibid.
How much Rope will make Britching and Tackle for any Piece.
what Powder is allowed for Proof, and what for Astion.
ibid.
The difference between common Legitimate Pieces, and Bastard Pieces. ibid.
How Powder is made, and the ſerveral ways to know when it is decaying. 65
How to make excellent good Match; and how to make Powder that it shall
not waſte with Time, and how to make good that which is bad, and how to
make Powder of divers Colours.
66
Several ſorts of Saltpetre, and how to make an excellent fort, very eaſie, and
leſs Charge; and how to load and fire a piece of Ordnance like an Artiſt.67
The difference of shooting by the metal
, and by n Diſpart, by Right Ranges,
and at Random; with the Figures thereof,
68
How to make a good shot to any place aſſigned; out of any Gun.
70
How to make an effectral Shot out of a piece of Ordnance at Random. 72
How to find the Right Line or Range of any shot diſcharged out of any Piece,
for every Elevation, by one Right or Dead Range given for the
Piece allign-
ed: And to know how much of the Horizontal Line is contained under the
Right Line of any shot made out of any Piecè, at any Elevation. 74
of the violent, crooked, and natural Motion or Courſe of a Shot, diſcharged
011t of any piece of ordnance aſſigned.
75
How to make a Gunners Ruler, and how to divide the ſame, by the help of
Table, fitting it for any Piece ; and how to give Level with the Gunners
Kuler at any Degree of Random
How to give Level to a piece of Ordnance without the Gunners Rule.
How to make a shot at the Enemies Light inshe Night:
79
64
.
+
76
1
How
t
1
The Contents,
How to ſhoot perfeitly at a Company of Foot or Horſe, or a ship under fail. P.79
How the Same Powder in weight ſhall carry the shot more cloſe or ſcattering:
And how a shot that ſticketb falt within the Concavity of the Piece, that
cannot be driven home, may be ſhot out without any harm to the Gunner ;
and what difference there is in ſhooting out of one piece ſeveral shots tóa
gether.
ibid.
Sect
. 46. How to weigh ships that are funk, or Ordnance isnder Water ; or to
know what
empty'Cask will carry any ſort of Ordnance over a River. 80
How many Oxen, Horſes, or Men, will ſerve to draw a piece of Ordnance. 81
How Gunners may take a plot of their Garriſon, and every Object therein, 'or
ncer it.
ibid.
ARTIFICIAL FIR E-WORKS,
-U Deſcription of the Mortar-piece: How to make one of wood and Paſte-board:
ü
How to fit and prepare Granadoes for the Mortar-piece: How to make
83
How to make Granadoes of Canvas for the Mortar-piece ; and how Granadoes
are to be charged in a Mortar-piece, and fired.
84
How to make Hand. granadoes, to heave by Hand.
How to make Fiery Arrows or Darts, like Death-Arrows Heads. ibid.
How to make Fiery Pots of clay, and Powder Cheſts.
How to make Artificial Fire-works for Recreation and Delight. ibid.
How to make Compoſition for Rockets of any ſize, and how to fire them. 87
How to make Fiery Serpents and Rockets that will run upon a Line, and return
again; and how to make Fire-wheels, as ſome call them; Girondels. 88
How to make divers Compoſitions for Stars, and the uſe of them. 89
How to repreſent divers forts of Figures in the Air with Rockets. ibid.
How to make silver and Golden Rain, Fire-Lances, and Balloons for the Mor.
tar-piece; and the Figures of the moſt uſeful forts of fire-works, and the
Explanation thereof.
90
Moſt Precious Salves for Burning by Fire.
91
Fuces.
1
85
86
}
5
BOOK VI.
1
97
101
TH
He Projection of the sphere by Tangents and half Tangents. 94
The Rudiments of Aſtronomy put into plain Rimes.
95
The Definitions of the Circles of the sphere, and Imaginary Circles, which are
not deſcribed in a Material sphere or Globe. '
The Projection of the sphere in Plano, repreſented by the Analemına; and the
Points and Circles before deſcribed in a Convex and a Concave Sphere, by
Chords and Sines, and likewiſe reſolved by Chords and Tangents.
How to Calculate the suns true place, and the Table of his mean Motion. 105
Probl. 2. The Suns diſtance from the next Æquinoctial Point, and his great-
eft Declinatiombeing given, to find the Declination of any point required.
107
Probl. 3. Having the Suns greateſt Declination, and his diſtance from the
next Aquinoctial Point, to find his Right Aſcenſion.
108
Probl.
4.
The Elevation of the Pole and Declination of the Sun being given,
to find the Aſcenſional Difference.
109
C
Probl. 5.
1
…
The Contents
III
IIZ
Probl. 5. The Suns Righi Aſcenſion, and his Aſcenſional Difference being
given, to find his oblique Aſcenſion and Deſcenſion. Page 110
Probl. 6. To find the time of Sun-riſing and ſetting, with the length of the
Day and Night.
ibid.
Probl. 7. The elevation of the Pole and the Declination of the Sun being gi-
ven, to find his Amplitude, and by it to know the Variation.
Probl. 8. Having the Latitude of the place and the Suns Declination, to find
when the sun comes to the due
Eaſt and weſt.
Probl. 9. The Elevation of the Pole and the Declination of the Sun being gi-
ven, to find the Suns Altitude when he comes due Eaſt and weſt.
ibid.
Probl. io. The ſame being given, to find his Altitude at the hour of fix.113
Probl. 11. The Same being given, to find his Azimuth at the hour of fix. ib.
Probl. 12. Having the Latitude of the place, and the Suns Declination, and
his diſtance from the Meridian being given, to find the Suns Altitude at
anytime aligned.
114
Probl. 13. The Latitude of the place, and the Suns Altitude and Declination
being given, to find the Suns Azimuth, and by it hope to find the Varia
ation.
I 18
Probl. 15. How to find the Altitude of the Sun by the shadow of a Gnomon
ſet perpendicular to the Horizon,by Scale and compaſs
, as alſo by Calculatiox.
Probl. 16. Having the Latitude of the place, the Declination of the Sun, and
the Suns Altitude, to find the hour of the day,
Probl. 17. Having the Azimuth of the Sun, and his Altitude, to find the
hour of the day.
I24
Probl. 18. Homo to find the Right Aſcenſion of a Star, and the Declination of
4 Star, having the Longitude and Latitude of the Star given. ibid.
Probl. 19. Having the Declination and Right Aſcenſion of a Star, to find the
Longitude and Latitude thereof.
Probl, 20. Having the Meridian Altitude of an unknown Star, and the di-
ſtance thereof from a known Star, to find the Longitude and Latitude of
the unknown Star.
128
Probl. 21. To find the Parallax of Altitude of the Sun, Moon, and Stars. 131
S
I22
123
126
1
BOOK
VII.
T
2
5
1
He Fundamental Diagram of the Dialling Scale, and the Argument. P.1
The Preface of the kinds of Dials and Theorems premiſed.
How to make the Polar or Æquinoctial Dial, and how to place it.
How to make the Æquinoctial Diab, or Polar Plane, Geometrically, and by cal-
culation.
8
How to make the Eaſt Æquinoctial Dial, or the Weſt, Lat. 51 d. 30 m. 9
How to make a Vertical Horizontal Dial.
How to make a South inclining 23 deg. in Latitude 5ı d. 30 m.
14
How to obſerve the Declination of any Declining Plane.
15
How to take the Declination of any wall or Plane, without the help of a Needle
or Load-ſtone.
How to make a Declining Horizontal Dial, or South erect declining from the
South Eaſtward.
17
II
..
16
TO
1
The Contents.
21
22
1
38
To find how much time the Subſtiler is diſtant from the Meridian, or Inclina-
tion of Meridians, Geometrically and by Calculation.
18
How to draw the Hour lines in a Declining Horizontal Plane, or South Erect
declining 32 d. 30 m, from the South Eastward.
19
How to obſerve the Reclinationer Inclinatioñijf any Plane:
How to draw Hour-lines in all Declining Reclining Inclining Planes. ibid.
How to deſcribe the sphere or Diagram.
How to make a North or South Reclining Dial
23
How to make an Eaſt and weſt Reclining or. Inclining Dial.
25.
How to find the Arches and
Angles that are requiſite for the making of the Re-
clining Declining Dials.
27
How to draw the Reclining Declining Dial.
30
How to find the Horary diſtance of a Reclining Declining Dial.
31
How to know in what country any Declining Dial ſhall ſerve for a Vertical.33
How to find the Arches and Angles which are requiſite in a North Decliner Re-
cliner, and a South Decliner
Incliner.
ibid.
How to draw the Declining Inclining Dial.
36
How to know the fundry forts of Dials in the Fundamental Diagram of the
Sphere.
37
How other Circles upon the Sphere may be deſcribed upon Dials, beſides the Me-
ridians.
How to deſcribe on any Dial the proper Azimuths and Almicantars of the plane.
How to deal with Declining, Reclining, or Inclining Planes, where the Pole.is
but of Small Elevation.
ibid.
How to inlarge the Hours of any Plane.
40
How to make a Vertical Dialupon the Cieling of 1 Floor within doors, where the
direct Beams of the Sun never come.
42
u Table for the Altitude of the Sun in the beginning of each sign, for all
the Hours of the Day, for the Latitude of sid. 30 m.
44
How to make an Univerſal Dial on a Globe, and how to cover it if need re-
quires.
45
How to make a North Dial for the Cape of Good Hope, in South Latitude
35 deg. and Longitude 32 deg. 54 min. to the Eaſtward of the Meridian of
the Lizard.
How to find the Hour of the Night by the Moon Shining upon a Sun-dial. 48
How to find the Hour of the Day or Night by a Gold Ring and a silver Bowl,
or Braſs, Glaſs, or Iron veſſel.
How to Paint the Dials that you make, and faſter the Gnomons in wood or
49
The uſe of the Tables of Artificial Sines and Tangents,
The Uſe of the Logarithme Numbers.
Next follow the Tables.
After them is an Abridgment of Caſtom-Laws in Navigation.
And laſt of all is annexed, A Compendium of Fortification both Geometrically
and Inſtrumentally.
39'
1
46
ibid.
Stone.
go
51
1
1
t
5
с 2
To
1
6
!
+
---
1
TO THE
1
Truly Induſtrious, and Highly Deſerving of Engliſh-men
Captain SAMUEL
STURM Y.
On his Excellent and Elaborate Treatiſe, Entituled,
THE MARINERS MAGAZINE, GC c.
IF
F'Earth and Water make one Globe, then he
Had brave Columbus worn lo peor a Soul,
Muſt like a Stranger to his Country be, Or bold Americus a Brain ſo Foul,
That's ignorant in Navigation;
Or Noble Cabot of that Temper been,
And is indeed the out-ſide of a Man.
The Indies to this Day had not been ſeen
For he that's truly Microcoſmical,
By brisk Europeans! whereas nore the Name
Omns Noah's Ships, as well as Adam's Fall. Of Worthy Columb, gives the Spaniards Fame;
The Sun and Man (they say) beget a Man: Americus the Portugals; and Cabot pields
A Menſal-Line appends to David's Span: Stout Albion Honour, all its Glories guilds.
who thinks by Rctail.pow'r his Kind to keep, who breaks the Ice in any Great Deſign,
And merrily expoſe the Sun to Sleep,
And happy Actions unto Judgement joyn)
May of a Kingdom soon a Cortage make, Deferu's to wear their Countries brighter Bays!.
And Fences build to't, without Wall or Scake. Who Perfects it, merits Immortal Praiſe.
Nature requires no Miracles, but proceeds
Then thank our painful Author, thu imparts
In Order, by Fit Cauſes, as the needs.
Unto the World his choice and dear-bought Arts;
And he's the true Philoſopher, who views Whereby our ruder Underſtandings may
Earth, Sea, dnd Sky; not he on Earth doth Muſe. Gain Knowledge in the Mathematick way:
Can any by one Seaſon underſtand
No Spot on Earth, no part of Sea or Sky,
That this onr Earth, Four diff'rent ones command ? | But by this Book Men may learn to Survey.
Or that Phyſician reach Anatomy,
Horologie (an Art fo highly priz'd,
who knows nor Bladder or Emunctory?
And by Sumse Nacions almoſt Idoliz'd!)
Or can tbe Ableft-Theoriſt deny
Is here so plainly taught, that Vulgar Men
That Juſt Experience's witneſs’d by the Eye? May Dials make, although they ſcarce know Pen.
who to such Sopliſtry their Reaſon bow,
Now Hogen-Friar's Skill muft be laid by,
Lift Failax
up,
and ſet the Truth lelow.
li's far out-done by th' Author's Gunnery.
In Schools, I hold Theorick Knowledge good; Merchants and Sea-men here their Cuſtoms learn,
l'th' World, it leads our Senſes in a Wood. Their Juſter Laws from Tyranny diſcorn.
The ačtive Soul txkes nothing upon Truſt, N. Book ſo comprehnlive in our Tongse
But proves its Trutlı: while duller Wights in Ruſt As This. But left my ruder Lines do wrong,
And Sloth conſume their Time:Their empty Faith I will retire, and leave Thar to its Fare,
Believes all true, whatever any faith.
which I fore-ſec will be moft Fortunate,
Thus Three parts of the World (in Error grown!)
'Gainſt Practick-knowledge vouch Opinion.
John Gadbury, Φλόμαθηματικός.
$
1
A
1
t
1
2
1
A
FRIENDLY ADVERTISEMENT
TO THE
Navigators and Mariners of ENGLAND,
BRETHREN,
He Duties of a Friend and the Properties of a Flatterer do dif-
fer ſo greatly, that a Man cannot perform the Office of the one,
but he muſt renounce the Practice of the other; And a very
Fountain it is, from whence many Miſchiefs do ſpring and
overflow the wretched Life of Mankind, that the true dealing
of Friends is moſt commonly unpleaſant and hateful; but the
ſoothing of Flatterers is become plauſible, and much ſer by:
In reſemblance they bear many times like ſhow; but in purpoſes
they always differ. A true friend will ſometimes commend
and praiſc divers things in his Friend; and ſo will alſo che Flaccerer, in thoſe whom he
followeth. The one commendech that which in Judgment lie thinketh commenda-
ble, to the end that his Friend ſhould ſtill proceed in Actions worthy of Commenda-
tions; the other commendeth even choſe things many times which in his heart he dotli,
deteſt, to the cnd that he may ſooth up the Humour of the Party. A faithful Friend,
what he diſallows in his Judgment of his Friend, he will be earneſt with him to ſee
the fault, to the end the Party may amend, and give no advantage to his Enemy: The
Flatcerer' ſometimes, though ſeldom, will alſo diſcommend, but evermore crifling
matters ; fearing ty offend the Party, if he ſhould touch him ; fo counterfeiting in
cere Love (the Badge only of true Friendſhip) and ſo leavechi che Party, thus abuſed,
to the ſcorn and reproach of the Adverſary, reaping the Commodity wnich he looked
for, as clie only end of his deſire.
I do not think that there is any Man, that either regardech Gods Glory, or cſteem-
echi of Hamane Society, but holdeth our Art worthy to be numbred with the moſt
excellent that are exerciſed among men; and clierefore ic is great reaſon the Practicers
of ic thould be had in greater reputation than they be now adays. Neither is chere
211y other Art wherein God Mhewech his Divine Power ſo manifeſtly, as in ours; pet-
miccing unto uis certain Rules to work by, and increaſing of them from time to ciine,
growing ſtill onwards towards perfection, as the World doth towards its end; and
reſerveth ſtill unco himſelf the managing of the whole, that when we have done
what we can, according to the Skill we have already, or may have by any thing that
we may learn hereafter, yer always will God make it manifeft, That'he alone is Lord
- and Ruler of Sca and Land; That all Storms and Tempeſts do but fulfil his will and
pleaſure, who oftentimes adminiſtrech many helps, beyond all expectation, when the
Art of Man utterly faileth ; which is lively expreſſed in Pſalm 107. where is nothing
omitted which is neceſſary, nor any thing affirmed but that which the continual ex-
perience of aur daily dangers do proclaim to be true.
Oʻihat ive that see his wondrous Works in the Diep, would therefore praiſe the Lord for
his Mercies, and ſhew forth his Wonders beforc the Children of Men; that we might
once learn, That the Pear of the Lord is the beginning of wiſdom. Moſt undoubtedly
then would our Art flouriſh, our Voyages proſper, and have better ſucceſs ; yea, our
felves would be more cſteemed and honoured of all men. Whercas now the profane
Lives, and brutiſh Behaviour of too too many of our Trade, doth ſomewhat eclipſe
the Glory of the Profeſſion it felf. Beſides other manifold puniſhments
, God ſtrikech
ſome of us with the Spiric of Blindneſs, as no mon living, of any Trade whatſoever,
are to be found ſo ignorant as many of us are: lo ſenſeleſs are we in our own defects,
fo
V
yer
1
1
A
A Friendly Advertiſement, cc.
ful unto him for it. Farewel.
folitde deſirous to amend them , Yea, and ſome of us of the greateſt Skill and Pra-
ctice are ſo loch to give God his due Glory, that many times labouring to ſuppreſs it,
we make Shipwreck of our own Credits and Repucations, which otherwiſe of right
might accrew unco us. When we have performed a long Voyage, of great difficulties,
wherein many a time and oft we have been at our wits end, and knew not which
way
in the World to turn our ſelves, God delivering us beyond our cxpectation, as our
Conſciences can witneſs; yet when the danger is once paſt, and that liome we be come,
We take it as a blemiſh of our Eſtimations, and a great Impeachiment to our Credits,
to give God the Praiſe, and yield him Thanksi imagining that would derogate too
much from the Admiracion which we ſo greedily hunt after among Men. But let me
give you one Example of this Ingratitude to God, on à Voyage from the Weſt-Indies
,
in the Society of Zepham, a Ship that I had command of
. It pleaſed God by a vio-
lone Storm and Sea,' 500 Leagues from England, we loſt all our Maſts, and were le-
veral ciines like to founder our Ship: It pleaſed God that little Proviſion we made for
Sail, and the miſchievous Storm continuing, turned to our good; for the Wind
was fair, buc the Sea ſo dangerous and gronc, that we could noć Scud or Sail but ſome-
times; hur in good time it brought us fafe into the foreſaid Harbour of Toplam,
And in our diſtreſs our Men were very mindful of Prayer, as all are; but coming to
our deſired Port, I deſired them to return our gracious God Thanks, wich
mc, for
our great Deliverance : Some were willing, but two rcfuſed; whereupon I told
them, That when they were next in diſtreſs, it may be God would refuſe them helpor den
liverance : And ſo it fell out; for William Witheridge of Kenton in Devonſhire was
drowned at Billoa the next Voyage following, and the other was drowned at London.
Therefore let me adviſe all, to have a care not to be ungrateful to God.
tve of this Na. There are many men that perform long Voyages God knoweth how, but not they
tion are toomuch themſelves; yet will ſwear and ſtarc, crack and boaſt, That they have done all things
given to admire according to Art; and tell a Tale to Strangers at home, of ſuch Ġulfs and ſwift Cur-
God made, to ſhadow their Ignorance, and rob God of his
coatema our own rents, more than ever God made, to thadow
Countrg.meile
Praiſe : But yet for the Navigators and Mariners of England, I do hope and verily
believe in my Conſcience, That divers of them do fear God unfeignedly, and do as
much diſlike the diffolute courſe of the common fort, as any men can: And I do
nothing doubt, although the number of ſuch are too few in our Nation, yer are they
more than any Nation in the World can ſhew beſides. However, two things arc
greatly wiſh'd by all our Well-willers ; An Increaſe in us of the true Fear of God,
and a careful Diligence in us in things belonging to our Art. Where che Fear of God
is not, no Art can ſerve the curn; for that were to make of Art an Idol: And yet all
thoſe that fear God, muft take heed that they do not tempt God ; and therefore ought
they to uſc Art, as the means that God hath ordained for their benefit
, and be thank-
1
+
1
Yeurs,
From my Houſe and Study at
St. Georges, or Pill, neer
Briſtol, November 12.
1 667.
SAMUEL STURMI.
Errata.
+
1
1
ERRA TA;
".
Courteous Reader,
I
NÅwork of this Nature it is impoſſible to eſcape Miſtakes, the which Man-kind could
nevër totally evade fince the first lapſe. of his
Great grandfather Adam. I hope I ſhall
obtain your pardon therefore, though this preſent Tract bath some few Typographical Errors,
which yer are not many, nor conſiderable, though the Ampbor were far remote from the Preſs
all the while. I have bere given thee notice of as many as I could here readily efpy; If those
findeſt any other, I deſire thy favourable Cenſure, and that thou wouldeſt correct theſe in wanner
Following.
In the Firſt, Second, Third, and Fourth Books.
Page 1 6. line 3. for when I al Sea, read mpērc I. et Sex. 135.for baft of r. bamplaft, p.19.1.22.for all the amo!
1.35. for therefore's the fore, 1.41, for 2.1454 muſt, p.18.1.5. for laugbtr.caughi, 1.45. for Laught r. taught, p.19.
13.for Private r. Priuaicer, p.20.1.17.for Guns 6. Graner, P-34.1.6.for flate flat.p541.10.for inexed r. invet.
ed, p.61.1.10. in the Table of Sincs againft 40 m. puc in 1121 P.640.114 for
firfc. fth, p.145. the laſt Courſe
but one upon the Log-board, for S E, B r. S , L; chc laſt Courſe on thc log board, for S'Eg knots r. S EO
Inols, P.146.1.47.for the wind at IS and E S E s. the wind at SSE and S S W the ship made Leeward way,
P.147.in the Table at the fifth Courſe, for N by w r. NN 450.160.1.5 3.for N Sr.NE, p.162.1.3.for Arabicala
ig c, analytically, p.174.1 40.for su deg.r.si parts, p.117.1.4.for legree s. degrees pigs.1.46,for 20 deg. of Lor.
gitude s. 30 deg.30 m. of Latitude, p.200.1.38. for letr. fet.
In the Fifth and Sixch Books.
1
.
1
P.2.1.44. for Verafter r.Peraator, p.4. the Square ABCD ſhould be a Geometrical Square of equal fides, pa
7,8,9. Tone figures in the Sims of Diviſion are ſet our of order,.p.24.11 1. for rotract r.protraft, p.8o.l.az.in
marg.for having done rohanging down, p.ioz.1.1.for cóxvex r.concave.
In the Seventh Book.
In cic.p.for Gomical r,Gromorical, 9.1.1.5: for Gromicaļ r. Gnomonical. p.8.1.23. for C H 1.CG, p.12. in the
Dial chere waars the letter E toward the top of the Scile; p.13ichis ſhould be added to Chap. 8. for the North
and Southerca Dial.
Stile 38 deg.. 30 min.
To Calculate the Hour-Lines.
Angle of the Arch 0x the
Hour,
Plaxe
Hohys.
d. m.
d.
m
:
12
O
O
1
As the Radius,
To the Stiles Height 38 d. 30,
So the Tangent of the Hour,
To the diſtance from the Meridian.
9 28
/
IS
30
45
60
75"
90
19 54
31 54
9
3
4
S
6
66 42
90
O
P.18. in the Figure, C is wanting in the Center, p. 19.1.7. for Br H, and ſome ſmall Lercers are wanting in
the Hour Scale, p.23.1.9.for Fir, EW, 1.8.for continue r. contain, 1.2 1. for tbird Poirit to three points, p 286
1.1. for Fler. F Lé, p. 30 Pis wanting at the edge of the Srile, P.31
. 1.34. for os r. as; for FORC gr.
FORel. p.34. in the Dial, for er. CP.40. in the Dial, che Letter 1 is wanting at the Interfe&tion of the
Line F B A with the Hour-line of 5. P 40.1.14. for Rr, little i. 1. 16. for ERC. É r. p.41. in the Dial, for K
make B. p.43. in the Dial, for 4 make q.
4
1
I thought Good to advertiſe thoſe that have occaſion for
any
Inſtruments
mentioned in this Book, or any other for the Mathematical Practice, either in
Silver, Braſs, or Wood, they may be exactly farniſhed by Mr. Walter Hayes at
the Croſs-daggers in Moor-fields, next door to the Popes Head Tavern, where
they may be furniſhed with all ſorts of Carpenters Rules, Poft and Pocket
Dials for any Latitude, at reaſonable Rates.
THE
។
1
1
-
.
• Τ
THE
AUTHOR
1
4
Implores Aid of
GOD'S EVERLASTING FOUNTAIN .
:
C
1
OH
H! All-fufficient Fountain, Lord of Light,
Without whoſe gracious Aid and conſtant Sprite
NO Labours proſper, homſoe're begun,
But fly like Mifts before the Morning-Sun.
0, raiſe my Thoughts, and cleer my Apprehenfion;
Pour down thy Spirit on my weak Invention :
Be thouthe Load-ftar to my wandring Mind,
New rigg'd, and bound upon a new Deſign ;
O, fill my-Canvas with a proſperous Wind,
Grant that of thee I may Aſsiſtance find:
So bleſs my Talent with a fruitful Loan
That it at leaſt may render Two for One.
THE
1
.
!
ነ 1
1
CHAP. I.
The Compleat
MARINER,
OR
+
NAVIGATOR
I
The Firſt Book.
CHAP. I.
The Argument by Deſcription of the Art of Navigation in general.
AVIGATION of all Arts and Sciences (ſetting Divinity
aſide) hath much reaſon to have the preheminence, it being
of ſuch neceſſary and publick Concernment; and what uſe
there is made of ic by Seamen at this preſent, as well as hach
been in times paſt, All men know, to whom the Countries
are beholden for their good Service, whoſe Courage hath kept
Great Britain, Queen and Regent of the Sea, and deſerves it
well, in reſpect of the skill and Valour of her Mariners, and
Goodneſs and Number of her Ships. I wiſh as long as the
Sun and Moon endures, That they may maintain their Courage, and improve their
Art, as they ever have, againſt all Nations that have been England's Enemies; and
ever may they crown their Undertakings with everlaſting Credit.
The Art of Navigation being ſuch, I think I may be bold to affirm without pre-
fumption, This Art is more neceſſary for the well-being and honour of our Nation,
than any other Art or Science Mathematical, which is inore carefully kept in the Uni-
verſities. Look upon Grammar, Rhetorick, and Logick; theſe are but Introductions
to other Arts; Mufick is but of little uſe.
The chief Profeſſions now in the Univerſities are Phyfick and Law. Without en-
vy be it ſpoken, we may as well live as the ancient Romans without Phyſicians, and
as honeſt Neighbours without Lawyers, better than without skilful Seamen, which
are che chief Importers of our Wealth, and Supporters of our Warfare.
Beſides that, of all Mathematical Sciences and Arts profeſſed in the Univerſities, of
this Art of Navigation is inade the moſt gencral and proficable uſe; for what can the
Scholar make of his Geometry, with all the nice and notional Problems thereof; of
Aftronory, with all his curious Speculations about the motion of the Planets
, wịchouc
they be applied to ſome more Mechanical and Practical Arts, as Coſmography, Geograa
phy, Surveying, Dyalling, Architetture, Military Employments, which ſhall in ſome
meaſure (ſufficient for the help of Mariners) bc ſhewn in the following Treatiſe,
wherein it will appear, That the Art of Navigation comprehends them all in the uſe
thereof
And thoſe that will be comipleat Sea-Ariiſts, had need to endeavour to have ſome
skill and underſtanding in moſt of theſe Arts, namely, the Theorick and Practick
parts, whereby they may be fully informed of the Compoſition of the Sphere in ge-
neral; and in particular for the Figure, Number, and Mution made in the Heavens by
cho
or
14
B
!
1
The Compleat Mariner.
2
BOOK I.
the higheſt Movcable called Prime Mobile, and likewiſe of the firſt, fourth, eighth,
and ninth Heavens. It will alſo inform them how the Elements are diſpoſed, with
their quantities and ſcituations, cſpecially in the Compoſition of the Sphere of the
World, which is cominonly underfood to be the whole Globe of the Hcavcns; with
all that therein is contained; which is divided into two parts; Elemental and Çæbeſti-
al. The Elemental liath again four parts
, viz. The firſt is the*Earth, which ebgether
with Elie Water, as the ſecond, maketh a perfe&t Round Globe, thereupon we dwell;
therefore the Nature and Circles which are ſuppoſed to be contained in that Sphere,
are fit to be known. The next is the Air, comprehending the Earth; and the fourth
the Fire, which according to the opinion of Philoſophers containcth the ſpace which
is between the.Aizand the Heavens, or Circle of the Moon. Out of cliefe Elements;
which are the beginning of all things that are ſubject to change, together with the
warinth of the Heavens, all things do come forth; and decay, as we ſee and find
upon the Earth, by the continual Change and Motion of the One into the
other.
The Cæleftial part ( containing within the concavity thercof the Elementaltis
tranſparent and perſpicuous, ſhining, ſevered, and free from all mutability; and is
divided into cight Spheres, of round hollow Globes, which are called Heavens,
vlereof the greateſt doch contain tie next unto it Globe-like; the ſeven Inferior
ljave in cach of them but one Star or Planet only, thereof the firſt (the next to the
Earth) is the Heaven of the Moon ; The ſecond of Mercury; The third of Venus;
The fourth of the Sun; The fifth of Mars; The ſixch of Jupiter ; The ſeventh of
Saturn; And the cight of all the Fixed Stars, The number of thele Hermens are
kuown by their Courſes round about the Poles of the Zodiack. The Moon rünnctlı
through her Heaven by her Natural Courſe froin the Weſt to the Eaſt in 27 days 8 ho.
Mercury, Venus, and the Sun, their courſe in a year, and ſome leſs than a ycar ;
Mars his courſe in two years, Jupists in 12, and Saturn in 30 years; The Eiglith
Heaven, according to the Obſervation of Tycho Brahe, in 25400 years.
Theſe Heavens are carried all together in 24 Hours upon tlie Poles, about the Axle-
tree of the World thorow the ninth Heavcıl, by vcrtuc of the Primum Mobile; that
is, the Firſt Moveable; by which Morion to 'our appcarance is cauſed the Day and
Night, and the daily Riſing and Setting of the Celeſtial Lights; But more of this in
another place, for here I have made a Digreſſion. So that 110 Art is more capacious;
and were the Excellency well underſtood, and put in practice, as it might be ( as
Mr. Philips ſajthi in the like caſe) 110 Employment would be more honourable and
advantageous for the moſt generous Gentleman, and Learned Student, than this of
Navigation ; thus it was in eſteem in the days of Qucen Elizabeth,
T
When Drake and Candiſh Sayld the World about,
And many Hero's fossnd new Countries out,
To Britain's Glory, and their lasting Fame;
Were me like-minded, we might do the ſame.
;
1
The Practick part of Navigation is properly placed in making and uſing of Inſtru-
ments, which is thewn in the ſecond Book. Yet there is a certain Compoſition in
the Practick, more rare than all the reſt, in the compleat Sca-Artiſt; and that is the
right Words and Phraſes uſed in guiding, governing, and conſtraining, to perform
the expert Navigator's pleaſure in the Sca; In ruling the unparallelld Fabrick of a
gallant Ship, which hath been omitted by moſt men that have writ of this Art;
therefore I ſhall explain it with my Pen, becauſe I know with proper Phraſes how to
pçrforin it, not hindring any other, as they not me, to ſhew truly and lively their
skill in controlling, guiding, and working a Ship, according to all Weachers at Sea;
although it be of no uſe to Sea-men, that have been all their lifc-time at Sea: but for
Gentlemen on Sliore to read for their Recreation, the Words of Command ac Sea,
which may be delightful unto them. But for experienced Sca-men , they have all
thoſe things imprinted in them, and make uſe thereof according as their buſineſs
Thall fall out at Sea, but after the ſame manner.
In regard all Arts and Sciences are divided into two principal parts, that is, the
Theorick and Pračtick, I tooke upon me to demonſtrate according to my ability,
which
CHAP. II. The Compleat Mariner,
3
which will give the moſt reaſonable men facisfaction; for the unreaſonable, I care
not a fig for them; for I know it to be impoſſible for any man to be a compleat Sea-
man, wherein this Knowledge is wanting, they being both inſeparable Companions
which always wait upon Perfection. I ſhall draw out the Deſcription in as ſmall a
compaſs as it can be, and haſten to the moſt material Practice.
CHAP. II.
of what is needful firſt to be known in the Praktick Part of the Compaſs,
and how to divide the Circles and Parts.
T
1
rence.
HE principal Hand-maids that expert Sca-men are furniſhed with, that their
Undertakings may be crowned with everlaſting Credit are theſe, viz. Arith-
metick, Aſtronomy, Geometry, and the Mathematiques. By the operation
of theſe loving Siſters, and excellent Arts, as hath been ſaid, Navigation is daily pra-
cticed by expert Sea-men : but much abuſed by hundreds of ignorant Affes, that know
nothing what belongs to them, yet do undertake Voyages, to direct a ship naviga-
ble upon the Terreſtrial Globe, reſting wholly upon favourable Fortune, which hath
made ſome of them famous ;- but many times diſaſterous Periods have ended their
Undertakings, with the loſs of many mens Goods and Lives; which yet I muſt
confeſs have and do happen to the beſt, bue not ſo often as to them by great diffe-
But to come to the Subſtance of what is here intended, I would have it to
be underſtood, That he that intendcth the Art of Navigation, háth Arithmetick in
readineſs. If he want it, he may be inſtructed by divers Books now excant, as Rea
cord, Blandevill, and Mr. William Leybourn's Arithmeticks. As for the Mathemacical
and Aſtronomical Knowledge, ſo much as is uſeful for Sca-inen, will be ſhown in the
Projection and uſe.of divers Inſtruments, which will after follow in its due place. In
ahis Treatiſe we will come to the Sea-Compaſs, chat we may proceed in a regular form.
The knowledge of it is the root of that famous Art we chiefly treat of, and preſents
himſelf as the firſt Principle framed bý: God in the
Operation and Nature of the
Magnet, which being in its quality beyond our capacities
, yet it is the firſt thing to
be learned and underſtood, it being the foundation to all the following Concluſions,
and is firſt caught to our Youchs ånd Boys which are intended for Navigators. They
are caught firſt to know the Point on the Card, and by what Name it is called, and
to lay it perfectly backwards and forwards; and to know that to every Point of the
Compaſs there is allowed for Time of an Hour, which is 11 Degrees Is Minutes ;
and how to number the Hours from the North and South, either Eaſtward or Weſt-
ward readily to anſwer as ſoon as demanded: As alſo to know how the Ship Capes;
that'is, to know the Point of the Compaſs chat looks ſtraight forwards to the Head of
che Ship: As likewiſe to know upon what point of the Compaſs the Wisid blows
overs that is, if the Wind be at North, it blows over the Flower.de-Lusce toward the
South; and ſo of the reſt. So we teach them to know what Point the Sun is on;
That in England a South-eaſt Sun on the Aguator "makes 9.24' of the Clock; and
when he is South, makes 12 of the Clock; and South-weſt, 2. 36 of the Clock. As
alſo they learn to ſet the Moon in the ſame manner on the Full and Change-days, to
know the Tides by, as ſhall be ſhewed.
The Compals we Secer our Ships by, is only a Circle of ſome 8 inches diameter ;
and is divided into 32 Points, which have ſeveral denominacions, as you may ſee ex-
preſſed in the Figure. The whole Circle is divided into 360 degrces, and 24 liours:
The Compaſs contains alſo 16 diſtinct Rhombs or Courſes; for each ſeveral Courſe
hath two Points of the Compaſs, by which it is expreſſed. As for example, Where
there is any place ſcituated South-eaſt, in reſpect of another place, we ſay the Rhomb
or Courſe that runneth berwixt them; is South-caſt and North-weſt: or if ir bear
South or North, ſo we call it: or if Weſt, we ſay Weſt and Eaſt: The Compaſs
ſwings in the Boxes, the Wyers firſt well touched with a good Load-ſtone, and the
Chard ſuvimming well on the Pin perpendicular in the middle of the Box; it repre-
B2
fenes
1
1
!
1
4
BobK I
The Compleat Mariner.
40.000 170
091OL 018
Ble 1719 GOSO 40
Sea?
{cuts the plain Superficies or Horizon as we call it, when looking round about us at
and ſee the Heavens make interſection with the Waters, theweth that
you
are in the Center, and that every place in the Horizon is cqually diſtant from you: So
that wlien you eſpy any Iſland, Rocks, Ships, or Cape-Lands, by looking ſtraight up-
on the Coinpaſs,you thall know upon what Point of the Compaſs the object bearech
from you. But we will hatte to thew the young Practitioners the Sca-Compaſs, with
the 32 Points, cxpreſſed by the Letters
, upon each Line, and alſo how to make it, as
followcth.
The COMPASS
4
XII
I
XI
10
llo
B
X
EL
310
26
20
IX
N
Intw
N 6E
ODOSTO
NNW
wwén
NW
NNE
TITA
NEEN
NWbvy
ITI
WNW
Fr.wb
ILA
w
slo
W
VI
W
90
VI
wis
SIT
уц
W ŚW
08 OZ02
agas
swbw
E
SW W
SJAS
VIL
MSS
SE
ot
MgS
JUL
gs
ISS
gas
016
TI
017
os
X
이다​.
이다​.
D
I
TIX
IX
How to divide the Circles of the Mariners Compaſs.
FIA
Irſt draw a Line at pleaſure, and croſs it in the midſt with another Line at righe
Angles; Then in the croſſing of theſe two Lines fer-one foot of your Conipals
,
and open the other to what diſtance you pleaſe, and with that diſtance draw the Cir-
cle, which by the croſs Lines of Eaſt and welt North and South, are divided into
four Quadrants and equal parts, cach of them containing, 6 hours apicce; ſet VI at
Eaſt and VI at Weſt, XII ar Nirth, and XII at Sosth, fo have you the four firſt
Diviſions of your Figure:
Then keeping your Compaſs at the ſame diſtance as you
draw'd the Circle, ſet one foot in the croſſing of the Line and the Circle at Eaſt 6,
with the other make two marks, one of II, and X. Then ſet one foot iar the
Weft at 6, on the other ſide mark out the hours of II and X, as before ; keeping the
Compafies ſtill at the fame diſtance, ſet one Foct at South XII, and with the other
you thall inark out the Hours of VIII and IIII. Then ſet one foot of your Compal-
fes at North at XII, and in the ſame manuer mark out the Hours of VÍII and IIII.
Thus the Circle is divided into 12 cqual parts, and cach of them contajuis 2 hour
apiece ;
08*01608oL
CHAP. II. The Compleat Mariner.
5
done, you
ſo
piece
j
ſo that it will be caſie for you to divide each of theſe into two parts; which
have the 24 hours. Laſtly, you may divide each liour into 4 cqual parts,
which will be quarters of an Hourc, as you may ſee in the Figures.
To divide a Circle into 360 cqual parts, is a thing very neceſſary; for in all Quc-
ſtions in Aſtronomy, and in the Calculation of all Triengles
, theſe parts are the mea-
ſure of clie' Angles : ſo that in reſpect of this
, every Arch is ſuppoſed to be divided
inco 360 cqual parts or Degrees; and every Degrce is ſuppoſed to be divided into 60
lcfler
parts,
called Minutes. To divide a Circle after this manner, draw a Line at
pleaſure, and croſs it at right Angles with another Line, and draw a Circle as befcrc.
Kcep your Compaſſes at the ſame diſtance, and divide the Circle from the 4 Quar-
eers into 12 equal parts. Then cloſing your Companies, divide cach of theſe into 3;
you have in all 36 parts. Then you inay caſily divide with your Pen cach of theſe
parts into 10 little parts, as you may loc in the middle Circle of the Figure, whiclı are
Degrces.
For the 32 Points of the Compaſs, draw the Line of North and Soseth, and croſs
it at right Angles with the Line of Eaſt and West, and draw the Circle, as before ;
and with the ſame diſtance, ſet one foot of your Compaſſes, at Eaſt, and with the
other draw a ſinall Arch at A and B, and croſs it from North to South with the ſame,
diſtance ; tlic like do from the Weſt Point to C and D. Then laying your Rule crofl-
ways to theſe Crofics, draw the Line B D and A C; ſo is your Circle divided in 8
equal parts. Then cloſing your Compaſſes, you may caſily divide chefe 8 parts into
4; divide one and that diſtance which will divide all the reſt into cqual parts, if you
have followed Directions. And ſo you have tlie 32 Rhcombs or Points of the Com-
paſs; and ſo you may ſubdivide theſe Points into halves and quarters, as you may
fec in the Figure. So have you made the Mariner's Sca-Compals. The Uſe thall de
ſhew'd in its place.
The Figure of the Compaſs, and the Traverſe Quadrat,
joyned both together.
B
The weſt Longitude:
7 The Eaſt Longitude. с
Globocze gazob
ob 17.210904/SG0.29%
4
*.
4
.
110
20:30
1
Northerly Latitude.
N
103107
MINI
HAN
EINO
0
og of Ag aso alcol com
NG IN
NNW a
NW
INWON BY
AN
80%O BO SO4 0.3 0 210 10
Northerly Latitude
NW6W4
60,70
WNW
1
NEBE
NE
WON
9 gbni
810
W.
W
I 표
​ESS
9° 810
A ESE
WS
위​이이
​9
“
SELE
WSW
SE:
60
SWÓW
I
SWT
Southerly. Latitude.
SWOS
U M5S
SEESH
ocol
ole
ASSES
1o do go40 S06 Zo 80.919
ags.no
AIJS 5W
80 70 80
Southerly Latitude.
CE
Holzkole
& duzlog103,04% 30 2010
510
:
Ilo 20:30 Logo 50 780 9
A
The weſt Longitude. The Eaſt Longitude.
D
The Traverſe Quadrat (hewch tlie making of the Traverſe Table, in Chap. 3. OF
Sayling by th' plain Sca-Clart.
The
e
1
+
6
The Compleat Mariner. Book I.
1
.
2
.
.
the true Point of the Compaſs the Moon will be, when ſhe is 16 days old
The Moons Motion, and the Ebbing and Flowing of the Sea.
THE
"He Praditioner in Navigation, is next to Icarn to know the certain time of the
Flowing and Ebbing of the Sea ; In all Ports called by Sea-men the fhifting of
Tydes, which is governed by the Moon's Motion, as it is found by experience.
Wherefore I will firſt ſhew the uſe of a ſmall Inſtrument, which is licrc framed,
whereby the meaneſt Capacity (which is void of Arithmetick) ſhall be able to know
the Age of the Moon, with what Flood and Ebb it maketh in all Channels, and in
every Port and Creek at High-water ; and
alſo be able to know what a Clock it is ac
any:i.ne of the Night; and divers other Queſtions, only by moving the Inſtrument,
according as ſhall be directed.
I thall alſo thew your, how you may do all theſe Queſtions of the Tyde by the
Moon Arithmetically: But firſt by your Inſtrument it muſt be projected according
to the following Figure, which'you may make of three pieces of Board, well planed,
and cxactly divided, according as you ſec it formed in tlie Figure. The outward
Circle, being the biggeſt Board, hath 32 Points of the Compaſs ; The inward Cir-
clccn the fame Board is divided into 24 Hours, being chic chickeſt Board. The fc-
conc! Circle muſt be divided into 30 equal parts, repreſenting the diſtance 30 times
24 Hours, or 30 natural days, attributed to the Sun. The uppermoſt Circle of the
thrce, is attributed to the Moon, with an Index as that of the Sun, and to be turned
and applied to cithcr thc 30 days, containing the Computation of the Moon betwixo
Change and Change, or the 24 hours; as likewile to the Points of the Compaſsi
And ſo may the Index of the Sun be applied either to Time, or the points of the Com-
paſs, which ſhall be made plain by the following Queſtions ; which will
appear de-
lightful and eaſie; and the illiterate man will find in moſt uſeful; and he that hath
better knowledge, will ſometime uſe the Inſtrument for‘variety fake. Firſt, for the
Figure of thc Inſtrument.
u uſeful Variation-Compaſs.
U
Poli che two upþér Circles of the Inſtrumont I haye ſet ajhof uſeful Variation-
Compals; calie to be underſtood, and as exact as'any Inſtrument whatſoever for
that
purpoſe. You ſhall havefüll
direction how to uſe them
in the following Diſcourſe,
who we come to treat of the Variation of the Compaſs. But this obſerve, The mid-
le Compaſs repreſentech the Compaſs you ſteer your Ship by, which is ſubject to Va-
riation; but the upper Compaſs and Circle repreſentech the true Compaſs, that never
varieth; and by it you may very readily know the Variation of the Steering Compaſs
,
how much it varieth from the true Point. The inward Circle of the middle Com-
paſs is divided into the 32 Points, with their halves and quarters: and likewiſe the
outward Circle of the ſmaller or upper Compafs. This is too hard for Practitioners
at firſt to know how to uſe this Inſtrument, to rectifie the variation of the Compaſs;
therefore I ſhall be no longer on this Diſcourſe, and proceed to what was promifed,
and ſhew the farther uſe of it afterward.
PROPOSITION I.
Here the The Moon being 16 days old, 1 demand upon what Point of the Compaſs ſke will
Variarion
be at 8 of the Clock at night.
Compaſs
is to be In all Queſtions of this nature you have the Hour and Time given, and the
placed. Moons Age, to find the Point of the Compaſs. To anſwer theſe Queſtions, place
the middle"Index of the Sun on 8 of the Clock at night; then bring the upper In-
dex of the Mcon right over the 16th. day of her Age, in the middle Circle of the Sun,
and the Index of the Moon or upper Circle points to E. b. S. half a point Southerly,
the Clock at night.
J
2
1
be
1
?
*
-
ar 8 of
1
PRO-
L
1
1
INEBEENEN
NWBW NW NWBNWNWNBW North NBE INNE INEBN NE
MMMMMS STASIMSSS unos
XI
An Inſtrument Shewing ý Changing of
ý Tides and y Variation
ofÝCompas
TI
12
13
al
lo
14
E
1
SW
SWBS
1
15
SWBW
SSW
SBW
WSW
Sout?
WBS
aas
HWest
ISS
sebe
ESE
Semin
SEBS
Vio
IWBN
Els
SSE
SaaS
2193
elos
INW
qua
South
sbw
5
2924
7507
SSW
NWBW
US
swos
Tu
SW
NW
4
OS
+49au
swber
EBS
gun
NWBN
wbs
3
353
j9u
[1127
NNW
Hiztowy
NBW
nn um
Inwbr
west
nu
mumn
uqm
hwbwl
S807
NBW
7
E
set
ourt
C*
VEBE
NE
NEBN
NNE
NBE
North
SEBELSE
shez
1
أعلم
ar
i
Soll
}
leta
81
le
I
X
www
DR
ISIS as
betwvere fol:6 & 7.
TAS
1
中
​CHAP. II. The Compleat Mariner.
1
PROPOSITION II.
4
The Moon being 16 days old, I demand, what is the clock when ſhe is upon the
S. E. Point ?
In Queſtions of this nature, you have the Point of the Compaſs, as well as the
Moons Age: therefore turn the Index of the Moon to the Souch-Eaſt Point; keeping
the Index faſt to the Point, bring the 16th. day of her Age in the middle Circle, righe
under Luna's Index, and the Index of the Sun points and ſhews you 10 a Clock at
night, that the Moon will be South-Eaſt.
I
PROPOSITION III.
The Moon being S. E. b. S. at 8 of the Clock at night, I demand, How many days
old ſhe is
In ſuch like Queſtions, you muſt firſt put the Index of che-Moon to the point of
the Compafs South-Eaſt by South, and the Index of the Sun ſet to clie Hour of the
Night; then look where the Index of the Moon cuts thc Circle of Days, and you
Thall find it cut the 13th. day of the Moons Age required.
1
PROPOSITION IV.',
The Moon being Eaft-South-East, and the Sun Weft-North-1Veſt,
' , demand the
Moons Age.
In all Queſtions of this nacure, obſerve what you have given ; then put the Index
of the Moon to the E.S. E. Point, and hold it ſteady whilſt you put the Index of
the Sun to Weſt-North-Weſt, and then look where the Index of the Moon curs is
the Circle of Days, and you ſhall find it cut the 15th day of her Age, thiar'is the
Day ſhe was at the Full, and anſwerech the Queſtion deſired.
PROPOSITION V.
1
The Moon being 17 days old, I demand the time of the Day ſhe will be Eaſt
North-Eaſt.
In all theſe forts of Queſtions, Nore, You muſt put the Index of the Moon to chę
Eaſt-North-Eaſt Point, ſtop it faſt there, and turn the Index of the Sun. about inil
the 17th day of her Age is under the Index of the Moon; then look wliat hour che
Index of the Sun is upon, and you will find it 6 a Clock and 5 Minutes paſt, which
anſwers you the time of the Day required.
This is worth your Obſervation, That if the Index points to the Eaſtward of the
North and South, it thews the morning 12 hours; if it points to the Weſtward of
the North and South, it ſhewech the Evening 12 hours.
Thus you ſee how ready
the Practitioner in Navigation may have the uſe of this Inſtrument, and by- often
practicing it, may be in a ſhort time lo imprinted in his mind, that as ſoon as he
hath occaſion to uſe it, he will be able to reſolve it by memory, which is an excellent
Drnament to a young Mariner; Therefore let ſo much fatisfie for the Uſc of the In-
ſtrument in ſuch like Queſtions,
}
PROPOSITION VI.
5
!
By the Inſtrument to find the time of Elling and Flowing, in any Port, River, os
Creek.
Note. You are to obſerve juſt at the time of High-Tyde, what Point of the Com-
paſs the Moon is upon that Day which ſhe changeth, in any Port, River, or Creek,
where you
' would know the Ebbing and Flowing of the Týdes, and time of High-
Water or Full-Sea. This being found, you may know what Hour belongech to
that
8
!
The Compleat Mariner,
Воок І.
1
that Point of the Compaſs, by turning the Index of the Moon, as b. fore was ſhewed:
So you may
be ſure to have the Hour always under the Index, on the
Change-day,
throughout all the Points of the Compaſs; and ſo we ſhall proceed to Examples.
PROPOSITION VII.
The Moon being 16 days old, I demand, What a Clock it will be Full-Sea at Briſtol,
Start-point, Waterford, where an Eaſt-by-South Moon on the Change-day
makes the Full-Sea ?
You are to conſider the Point of the Compaſs the Moon is upon in theſe Ports,
when it is Full-Sea on the Change-day (as in all other Ports) which in theſe Ports
is found by obſervation to be always Eaſt-by-South Moon (which is 6 hours 45 mi-
nutes) Then conſider whether it be the Day or che Night-Tyde you would know the
Time of Full-Sea; if it be the Morning-Tyde, bring the Index of the Moon to the
Weft-by-North Point, ſtaying it there; bring the 16th. day of her Age under thic
Index of the Moon, and the Index of the Sun will point you to 7 of the Clock and
33 Mínures, the time of Full-Sca in the Morning. If it be the Evening Tyde, bring
the Index of Luna to the Eaſt-by-Scuth, and ſtay it there, until you have brought
76 Days and half under the Index of Luna, and the Index of Sol will point directly
upon 8 of the Clock at Night, the time of Full-Sea in the aforeſaid Ports: Thus you
ſce there is 27 Mimites difference in every Tyde in theſe Ports. So you may know
in every other Port in the ſame manner, if you do as before-directed, and allowing
half a day more for the Night-Tyde, by turning of it half a day further. And take
this for a Rule, That the Moon betwixt Change and Full is ever to the Eaſtward of
the Sun,and riſeth by day, ſtill ſeparating it ſelf from the Sun until ſhe be at the Fall:
Then after the Full, in regard the hath gone more Degrees in her feparation than is
contained in a Semicircle, ſhe is gotten to the Weſtward of the Sun (riſing by night)
and applieth towards the Sun again until the Change-day, which you may ſee plain-
ly demonſtrated by the Inſtrument.
PROPOSITION VIII.
The Moon being 16 days old, I de fire to know at what hour it will be FullSea af
London, Tinmouth, Amſterdam, and Rotterdam, where a S.W. And N.E.
Moon makes a Full-Sea xpon the Change-day.
It is found by obſervation, That the S. W. and N. E. Moon makes Full-Sea in all
the aforeſaid Ports. You may know the Moon is to the Eaſtward of the Sun that is
before him; Bring the Index of the Moon to the S. W. Point; then turn the 16 day
of her Age under the Moons Index, and the Index of the Sun anſwereth the Que-
ftion, That it is Full-Sea at all the aforeſaid Ports at 3 of the clock and 48 minutes
in the morning, the Moon being 16 days old.
PROPOSITION IX.
At Yarmouthi
, Dover, and Harwich, where a S.S.E. Moon maketh Full-Sea on
the Change-day, the Moon being 9 days old, I demand the time or hour of
Full-Sea that day in the aforeſaid Places.
Here it hath been found by experience
, That a S.S. E. Mcon makes Full-sca on the
Change-day, in the aforeſaid places; therefore bring the Index of the Moon to the
S.S. E. Point, keep it there faſt directly on the Point, and bring the Moons Age to
cut the edge of the Moons Index, and the Index of the Suu will ſhew you, That the
time of Full-Sea in the aforeſaid Ports, will be at 5 a Clock and 42 minutes in the
morning.
PROPOSITION X.
At St. Andrews, Dundee, Lisbone, and St. Lucas, where a South-Weft-and-lig,
South Moon makes High-Water or Full-Sea on the Change-day; The Moon
being 28 days old, I demand the time of Full-Sea that day in theſe Places.
You
1
A
1
1
1
ret
}
5
Chap. II. The Compleat Mariner,
9
You have here given you the S.W.b. S. Point of the Compaſs; therefore bring the
Index of the Moon, and ſtay ic on chat Point, and bring the 28 day of her Age un-
der the edge of the Index of the Moon, and the Index of the Sun will point you
out the time of Full-Sea, which is at 39 minutes paſt 12 of the clock at noon, in
the aforeſaid places. And ſo are all Queſtions of this nature anſwered. And ſo I
will conclude the uſe of this Inſtrument, for finding the Ebbing and Flowing of the
Tyde, and ſo will proceed to thew you Arithmetically how to find the Golden Num.
ber or Prime without a Table, the Epact, and Full, Chang', and Quriers of the
Moon, and how to know her Age for ever; and what Sign and Degree ſhe is in the
Zodiack, how long the Moon thineth, and whac time of the day or night it is High-
Water or Full-Sea in any Port; and alſo the Moon's Motion, as far as it is uſeful for
Mariners.
t
!
How to find the Golden Number or Prime, according to the Juliari;
Engliſh, or Old Account.
I.
Y:
Ou may obſerve this, That the Prime or Golden Number is the ſpace of 19 years,
in which the Moon performeth all her Motions with the Sun: At the expulaticu
of which Term, ſhe beginnech again in the ſame sign and Degree of the Zodiecky
that the was 19 years before; and always finiſheth her Courſe with the Sun, and
never excecdeth that Term. To find this uſeful Number, you muſt do tlus; Al-
ways in what year you would know what is the Prime Number, add 1 to the date
thereof, and then divide it by yo, and that which remaineth upon the Diviſion,
and cometh not into the Quotight, is the Number required. As for example,-- In
thie
year of our Lord 1665. I demand what is the Prime Number. Add to the
year of our Lord always i, which makes in this Queſtion 1666. Then divide that
fum bý, 19, the remain is the Prime or Golden Number , as you may ſee bý tlic
Work, which anſwerech the Prime or Golden Number for this preſent ,.
year to be 13, it being left out of the Diviſion that comech not into
(1
ile Quotient. Thus you ſee it is very caſie to do it for any other year. 27
Obnič, That when you find uothing remaining upon the Diviſion,
that is Jelaſt year of the Moon's Revolution, and may conclude, *$66 (87
that' 10 is ihe Prime for that year. Note, The Prime beginnech always
in Fanuary, and the Epact in March.
84.3,
Y
IL
-
How to find the Epact, according to the Julian, Engliſh, or old Account,
and what is proceedeth from
He Epax is a Number that proceedeth from the difference which is made in the
{pace of one whole year, between the Sun and the Moon. Note, The Solar
year doth contain 365 days, s hours, 48 minutes; and the Lunar year, allowing is
Moons, there being 29 days, 12 h. 44 min. between Change and Change, doch .con-
tainbit 354 days
, 8 hours
, 48 min. So that there is almoſt ir days difference between
the Revolution of the Sun and Moon, at every years end; which difference makes the
put. Therefore to find the Epact for any year, fült you muſt know the Prime
Number for that year, which we found before for the year 1665. Tobe I'3. Then you
muft multiply this
. Prime Number 13. by the difference II.
and it will make 143: which divide by 30. and there re-.
II.
mainech of the Diviſion, that comech not into the Quotient, 13 (2
23 which is the Epoct for the year required. So I make' no *4'3 (4
Queſtion.but that you underſtand host to find the
Prime and
33
the Epit for any year paſt, preſent, of to come. Therefore
I hold this ſufficient to expreſs to facilc a thing as this is. I 143
have told you already, That the Epact always beginnech in
Märch; but I ſhall make a ſmall Table for thoſe that are ignorant in Arithmetick;
and cannot find theſe two Golden Numbers, as I may call chem, for 45 years to
с
II
come,
10
Book I.
The Compleat Mariner.
come, where any one may find the Prime and Epact moſt readily in any year you ſhall
deſire.
III.
1
A Rule to find the Change, Full, and Quarters of the Moon.
A year
Dd unto the Epact propoſed all the Months from March, including
the Month of March, and ſubftract that ſum from 30. the remain ſheweth
the day of the Change : But if the Epact be above 26. there this Rule faileth a day
at the leaſt; but at other times it will be no great difference: Therefore it
may
ſerve
for the following Concluſions.
As for Example, I deſire to know the New-Moon in October, 1665. The Epaxt is
23. the Months from March are 8. which added makes 31. from it 30 ſubftracted,
remainis 1. which taken from 30. one whole Moon, there remains 29. So that the
29th. day of Otober is the day of her Change, or New-Moon, which by exact Calcu-
latiòn it is at 58 min. paft 4 in the morning,
Having thus found the time of the New-Moon, you may from thence reckon the
Age of the Moon, and ſo find the Quarters, or Full-Moon.
;
Thus the Moon's Age is
At the Firſt Quarter.
At the Full Moon
At the Laſt Quarter
At the whole Moon
Days Hours Min.
07 09--II
-18
18-22
-22 03-33
-29-2--
I2-44
-14
J
IV.
1
L
V.
Hors to finde the Age of the Moon at any time for ever.
"
A
Dd to the days of the Month you are in, the Epact, and as many days more as
are Months from March, including March for one ; and if theſe 3 Numbers
ac'ded together excecd 30, rake 30 from it as often as you can, and the remain is her
Age :'But if the Numbers added be under 30, that's her Age : As' for Example,
166 5. the Epact is 23. I demand, What Age the Moon is the zith. day of September
From March có, September is 7 Months, the Epact 23, and the day of the Month is
21. Added together, makes 51. From it fubftract 29, becauſe the Monch hath but 30
daysin it, and the remain is 22, the Age of the Moon that day. Had it been the
2ath. of Auguſt, and added them together, it would have made $1. Then to have
taken 30 out, there had remained 21 for the Moon's Age the 22 day of Auguft
:
{":,:
To finde what Sign the Moon is in, by which is gathered, what the
Moon differeth from the Sun.
Ultiply the Age of the Moon by 4. divide the Augment or Sum by 10, the
Oxotlent ſheweth the sign the Moon differech from the Sun; the Remain
-milciplied by 4, giveth the Degrees to be added. As for ExampleThe Moon 22
days old, I demand what ſhe differcth from the Sun? Multiply
22 by 4, and the Product is 88. That divide by 10, and in the (8.
Quotient is 8, and 8 remaincth upon the Diviſion : Thar multi 88 (8
plied by 4, iś 32; from which take 30, the number of Degrees to
in a Sign, and add the 8 Signs in the Quotient; it makes 9 Signs.
88
The odd 2, mulciplied by 4, make 8 Degrees; to which add clie
San's Motion from his entrance into the Sign in which was the 14 day,to the 21, makė
7 days or Degrees to be added to the 8 Degrees make 9 Signs 15; which counted, after
this manner, from , ſaying, m 1, ** 2, VP 3, * 4, * 5, 76, 87, II 8,5 9 Sign,
and the odd 25 Degrees is i's Degrees of Cancer: So the Queſtion is anſwered, That
the Moon is 9 Signs Is Degrees from the Sun at 22 days old; which note, She
differeth but 4 Degrees from her true Motion by the Tables, which is near enougli
for the Mariner to anſwer any Man.:
M
22
4
12
How
.
CHAP, II. The Compleat Mariner,
11
Hope to find what Sign the Moon is in more exact ; with the Moon's
Motion for every day of her Age.
100
II
21
02
28 35
II
1205 08
Stronomers divide the Compaſs of the Heavens into 12 Signs, which they ſer
forth by chefe Names and Characters, which you muſt be a little acquainted
with, and the place of the Sun in the Zodiack. Each of theſc Signes you have them
as followeth.
Firſt know, That the Sun entrech the firſt Sign r the
i Ith of March, 8 the 11th of April, II che 12th of
D. Age.ls. D. M.
May, So the 12th of June, ol the 14th of Faly, m the
13
14th of Auguſt, - the 14th of September, m the 14th
200 26
of O&ober, the 13th of November, 2 the 12th of 3 or 09
0932
December, the oth of January, * the Toth of Fe 401 22 42
5 02 05
05 53
bruary. This known, the place of the Sun is well found,
602
adding for every day paſt any of theſe, 1 Degree.
19 04
Thus you ſec, the Sun runs thrcugh theſe 12 Signs but
703 041
once in a year; The Moon in leſs than a Month, viz. in 803 15 26
27 days, 7 hours, 43 minutes. Note, That every New-
903 26
Moon, the Sun and Moon are in onc Sign and Degree; but
1004
46
the Moon hath a Motion of about 13 Degrees every day, as
I104 24 56
is ſhewed in this Table. Therefore according to the Age
07
of the Moon, add the signs and Degrees of the Moon's
1305 21 18
Motion, to the place of the Sein at the New Moon, and ſo
1406 04 28
you ſhall have the sign and Degree which the Moon is in
1506 17 39
at any time defircd.
1607 00 49
Thus for Example, A New-Moon 1665.che 26th. No-
1707 14
vember, and the Sun and Moon are both in 14 Degrees of
1807 27
ixNow upon the 17th of December, the Moon being
14 days old, I would know what' Sign the Moon is in.
2008 23 32
This Table thews, for the 14 days of the Moon's Motion,
21 09 06 42
you muſt add 6 S, 4 D, 28 Min. to the ſaid 14 Degrees
22,09 19:53
of mi
2310 03 03
Now couạting thplc 6 Signs upon your fingers, recko-
2410 16 14
ing the Namçs of the Signe in order from Sagittarius, 18.1,
2510 29 251
* 2, 43, ņ 4, 85, II 6, it falls upon the sign Gemini.
26 II
Laſtly, adding the odd 14 Degrees unto thc 4 Deg. of the
2711 25
25.46
2813 08
Hon's Motion together, ſhews the place of the Moon to
be in 18 Degrees of Gemini.
29 I 2 22 07
There is much uſe made of the Moon's being in ſuch
3013 05 17
and ſuch Sigris, in Phyſick and Hasbandry, of which
I Thall ſay nothing ; but give you one Conclulion which much depends liercon ;
A Table ſewing the Moon's Motion according to her Age.
oo
II
19.08
10
21
1 2 35
:56
that is;
F
)
i ?
i
,
r
int-
To know the time of the Moon's Riſing, Southing, and Setting.
Or her Riſing (know tħiis) having found the place, or wliat Sign ſhe is in,
ſeek out in the following Kalendar toliat time'the Sun is in this sign and De-
gree, and there you ſhall find the tric'time of the Sun-Setting, being in that place:
Thịş is half the continuance of the Sun above the Horizon in that sign and Degree.
Add this
to the time of the cell points coming to the south, it ſhevxs the time of her
Setting; and fubftracted from it; "heivs the time of her Riſing.
Thus upon the ruth of September, 'as before, the Moon being 14 days old, aırd'in?
the 18 Degree of Gemini, 'I deſire to know the time of the Moon's Riſing and
Setting
C 2
Tirt
ܪ
i
12
The Compleat Mariner. Book I.
$o. 11
I2
10
( Rif. 3
02
Added 19
22
(Set. 22
Firſt multiply 14, the Moon's Age, by 4. Divide the Product
H. M.
by 5. In the Quotient will be li a Clock, and the one Unite
€
upon the Diviſion is Min. 12, that the Moon will be South that
Ö Šet. 8
night. Secondly, The Sun is in this Sign and Degree about the
firſt day of June, and then ſecs at 8 a Clock 10 minutes paft.
This ſubſtračted, ſhews the Riſing of the Moon to be at 3 of
the Clock 2 minutes in the afternoon. The ſaid 8 hours, 10 being
added, makes 19 hours 22 min. which by caſting away 12, the remain ſhews the
Moon's Setting to be ar 7 of the Clock, and 22 min. paſt in the morning, which
anſwers the Qucftion deſired; which is as neer as can be for your uſc.
PROP. I. How to find when it is Full-Sea in any Port, Rode, Creek, or
River.
I have ſhewed you already how to find the Prime, Epact, and Age of the Moon,
at any time deſired. Now we will proceed to thew you the finding of Full-Sea in
any Place; as in manner following.--- Firſt, Carefully watch the time of High-Water,
and what Point of the Compaſs the Moon is upon, on her Change-day, in that Port or
Place where you would know the time of the Full-Sea, or find by the Table what
Moon makes a Full-Sea in the ſaid Port. Secondly, Conſider the Age of the Moon;
then by Arithmetick reſolve it in this manner. Multiply the Moon's Age by 4, di-
vide the Product by 5, the Quotient ſhews the Moon's being South. If any thing
remaineth
upon
the Diviſion, for every Unite you muſt add 1 2 Min.. If it was 4 rc-
maining, it would be 48 Minutes to be added. Then add the hour that it Flows
on the Change-day to it, and the Total is the hour of. Full-Sea. If it exceed 12, fub-
Itract 12 from it, the remain is the hour of the day or night of Full-Sea, in any
Port, River, or Creek. Which I will make plain by ſome Examples, (viz.)
PROP. II. The Moon 16 days old, I demand, what a Clock it will be Full-
Sea at Briſtol, Start-point, and Waterford, where E. b. S.
Moon maketh Full-Sea on Change-day ?
¢ Age
16
Conſider herc an E. b. S. Moon maketh 6 hours 45 min. and
the Age of the Moon is 16 days old: Therefore multiply the
-64
Age by 4, and it makes 64; divide that by 5, and it is 12, 4
and 4 reinainech, which is 48 min. To it add 6 hours 45 min.
24 (i2 48
E.b.S. it makes 19 ho. 33 min. Therefore ſubſtract 12 hour's
55
from it, there remainech 7 a Clock 33 minutes, the time of E.b.S.
Full-Sea in the morning at the aforeſaid Ports which
you Total
may compare with your Inſtrument, and find it very well Subſt. 12
agrec.
j
::
4
6 45
3
19 33
00
F. Sea. 733
PROP. III. The Moon being 25 days old, I demand, what a Clock it will be
Full-Sea at London, Tinmouth, Amſterdam, and Rot-
terdam, where it flows S. W?
.
Age ( 25
4
100
-
pe
Conſider that at theſe Places on the Change-days a s.w.?
Moon makech Full-Sea, which is 3 hours. Therefore multi-
*ply 25, the Moon's Age, by 4, it makes 100. Thàt divide
by 5; in thc Quocient will be 20, and nothing remain. To
it add 3 ho. S.W.and it makes 23 hours. From it ſubſtract
12, and the Remainder ſhews you, That it will be Fxl-Sea
at all the aforeſaid Places, at 11 of the Clock in the mor-
ning. So you will find it agree with your Inſtrument.
2$ (20
55
2 S.W.
12
II Full Sea.
PROP.
1
CHAP. II.
The Compleat Mariner. .
13
.
9
lölt
PROP. IV. The Moon being 9 days old, I deſire to know the hour of Full-Sea
at Quinborough, Southam. And Portſmouth.
Note, That a, South-Moon on the Change-day, maketh Full-Sea at theſe Places.
Therefore multiply the Moon's Age by 4, it makes 36. That divide by
5, and the Quotient is 7 of the Clock; and I remaineth, which is
12 minutes, the time of Full-Sea at the aforeſaid Places, the Moon's
4
Age being nine days. Note, If a North or South Moon makes Full-
36
Sea on the Change-day, there is nothing to be added to the Quotient; (1
but the Quotient is the hour of the day, and the Remainder is the 36 (7 18
min. as before directed. One Example more ſhall ſuffice.
5
PROP. V. The Moon 5 days old, I demand the time of Full-Sea at Rocheſter,
.
Malden, Blacktail, where. S. b. W, Moon is Full-Sea.
Here you may note, That on the Change-day at theſe Places it flows S.b.W. which
is but one point from the South, being but 1 of an hour, or 45 min. And it had been
all one if it had been North-by-Eaſt. Multiply by 4, divide by 5, and the Quotient
will be 4; to it add 45 min. s. b. w. ſhews you it will be Full-Sea at the aforeſaid
Places at 4 a Clok and 45 min. in the morning. But note, Had it been S.b. E. or
N.6.W. it had been 11 ho. 15 min. By this time I hope I have made the Practitio-
ner able to know the time of Full-Sea in any Port, by Inſtrument and Arithmetick:
Therefore I will leave him a ſmall Table for his uſe.
4
A TIDE-TABLE.
i
8 15
* *
H. M.
Rye, Winchelſey, Cullhet, a S. b. E. Moon.
II 15
Rocheſter, Malden, Blacktaile, S. b. W.
O 45
Yarmouth, Dover, Harwich, S.S. E.-
10 30
Graveſend, Downs, Blackneſs, Silly, S.S.W.
I 30
Needles, Orford, South and North Fore-land, S.E.B.S. 9 45
Dundee, St. Andrews, Lisbone, St. Lucas, S.W.b. S. 2 15
Poole, Iſles of Man, Dunbar, Diepe, S.E.
9 oo
London, Tinmouth, Amſterdam, Rotterdam, S.W.- 3
oo
Portland, Hartflew, Dublin, S. E. b. E.
Barwick, Fluſhing, Hamborough, S.W.b. W 3 45
Milford, Bridgewater, Lands-end, E.S. E....
7 30
Baltimore, Corke, Severn, Calice, W.S. W.
4 30
Briſtol, Start-point, Waterford, E. b.S.
6 45
Falmouth, Humber, Newcaſtle, W.b.S.
5 is
Plimsouth, Hull, Lyn, St. Davids, W.CE.
6 00
Quinborough, Southam. Portſmouth, N. & S.. O oo
Add any two Numbers together of the foregoing Table, and they ſhall be I?
hours; Except the two laſt
, N, S. and E. w. So that you may perceive, what hath
been ſaid from the South, cicher Eaſtward or Weſtward, the ſame anſwereth to che
North, either Weſtward or Eaſtward. And ſo much for the Tydes
. But we will know
the Moon's Motion, and the Proportion between Tyde and Tyde.
PROP. VI. The Motion of the Moon, and the Proportion of Time betwixt
Tyde and Tyde.
After all this
, I will thew you in brief che Motion of the Moon, and the reaſon of
the difference between Tyde and Tyde.
You muſt note, the Motion of the Moon is twofold. Firſt, A violent Motion,
which is from Eaſt to Weſt, cauſed by the diurnal ſwiftneſs of the Primum Mobile.
Secondly, A natural Motion from Weſt to Eaſt, which is the reaſon the Moon requirech
27 days and 8 hours 8 min. to come into the ſame minute of the Zodiack from whence
Thc
14
The Compleat Mariner. Book I.
tinent in this placa. 10
the departed. But coming to the ſame Point and Degree where ſhe was in Conjuncti-
on with the Sun.laſt, ſhe is ſhort of the Sun, by reaſon the Sun's Motion every day
is natural Eaſt, Degree, or 60 Minutes, which maketh ſo much difference, thác
the Moon muſt go longer 2 days, 4 ho. 36 min. neareſt, more than her natural
Motion, before ſhe can fetch up the Sun, to come into Conjunction with her: So that
betwixt Change and Change is 29 days, 12 hours, 44 min. by my account. The
Marinors always allow juſt 30 days between the changes, by reaſon he will not be
troubled with ſmall Frations of Time, in this Account of Tydes, which breedech
no great error : Expérience therefore muſt needs ſhew me this, That I muſt allow
the ſome Proportion to the Moon in cvery 24 hours, to depart from the Sun 12 De-
grees, which is 48 min. of time, untill her full Eaft; but then having performed
her Natural Motion above half the Globe, ſhe is to the VVeſt, as we may know by
Reaſon. Now if the Moon move in 24 hours, 48 min. then in 12 hours ſhe muſt
move 24 min. and in fix hours, 12 min. By this proportion cach hour ſhe moveth 2
min. So the Tydes differeth as the time differech.
I will add one old approyed Experience for the Mariners uſe, though it is imper-
clipping in the Wane, cauſech baldneſs; but the beſt time in the Wane, is in s, m,
or *. "So I hope I have ſatisfied the Learner concerning the Moon.
2
17
t
t
2.
ܨ ܇
1.
1
1
IK
12
1
7
?
1
+
1
3
ใ ?
THE
1...,
!
.::
1
1
ज
1
1
+
I
4
4
i
}
LE
(5)
ily
1
11
1
.
4
1
:.******
1
Mariners Magazine;
OR,
STURMY's Mathematical and Practical
ARTS
+
Mariners and more ſufficient Men before the Maſt, which are firſt to hawl a bowl
CHAP. II.
15
THE
1
The Practick Part of Navigation, in working of a Ship in all
Weathers at Sea.
E have been ſhewing the Practitioner all this while, the Course
and Motion of cle Moon, and ſo by it to know how to ſhift
the Tydes, or time of High-Water, in any Port, Road, Har-
bour, or Creek, Inſtrumentally and Arithmetically. The next
thing to be obſerved by a Learnir, is the Words of Command,
with readineſs to anſwer and obey, which is the moſt excel-
lent Ornament that can be in a Complear Navigator, ur Maria
ner. And as Captains Exerciſe their Men on Shore, that their
Souldiers may underſtand the Poſtures of War, and to execute
it when the Word of Command is givca by their Commander; In like manner are
Seamen ond Mariners brought up in Practical Knowledge of Navigation at Sea, in
working a Ship in all Weathers. Although the Rules here demonſtrated, are but of
liccle benefit to him, that hath been brought up all his Life-time at Sea; and leſs to
thoſe that be altogether ignorant in Marine Affairs : But that the Pra&tick may be de-
livered in proper Sea-Phrafes, according to cach ſeveral Marerial that belongs to a
Ship compleatly rigg'd, with the Uſe of the ſeveral Ropes in working and crimming
of Sails at Sea on all Occaſions, cannot be denied by thoſe that know theſe things
perfectly : Therefore it is impoſſible for any Man to be a Compleat Mariner or Navi-
gator, without he hath attained to the true Knowledge of Theorickand Practick, be-
ing both Siſters and inſeparable Companions, that makes them perfect Navigators;
Therefore I could not let this fcape my Pen.
And to explain myſelf
, that I may prevent the Cenſures of all ſuch chat will be
curious; înquiring whether I am not lame, or incapable of that, and like themſelves
appear imperfect; I may ſpeak it with trouble to my ſelf, and ſhame to others, That
there was never more lame and decrepit Fellows preferred by Favour and Fortune, as
alſo by Kindred, and by Serving for Under-Wages (which a deſerving Man might
and would have) as is now adays. "Let a man go aboard the beſt Ship at Sea, and it
will be very rare to find Ignorance out of the Officers Cabinsand commonly able
ing, through the averſeneſs of their Fares, which is great picy. I ſhould be glad
to live to ſee a more equaller Balance among Seamang, and their Imployers, to further
the induſtrious, and encourage the deſerving Men; for if this partiality ſhould con-
çinue long, it is to be feared, in ſome ſhort time, the Compleat Mariner wil be hard-
ly found aboard any Ship, to the great diſparagement of our Englifh Nation, which
liach from time to time so long deſervingly had the Superiority over all other parts of
the World, for breeding the moſt famous Navigators. The Hollander to his Loſs
knows it right well
, that there are none like Engliſh for Courage at Sea; but that ma-
of them out-ſtrip us in the Art of Navigation, which proceeds from the former
unequal
1
ny
.
16
BOOK I.
The Compleat Mariner.
uncquial Balance, which makes our expert Saylers to ſeek if Fortune will be favoura-
ble amongſt them. They had not at this day been High and Mighty, and in ſuch a
flouriſhing Condition as now they are. Therefore I hope to ſee and hear, That the
Eng!iſk Mariner will make berter uſe of ſwift-ſtealing Time, that he may redcem vybac
isletty and atcali coſuch perfection, as that he may parallel his Art with his Vialpur
and Courage; And thač Imployers will uſe more Equity , in placing de øving
meu according to their merit. 'I ſhall not draw duriny Digreffior to any longerdi-
ſcourſe; for I know my plain Rhetorick will not relliſh in ſome mens ears, chcuch it
may in others : Therefore I thall draw to a Concluſion, deliring that no man will
cenſure me, before he knows what is in me, cr is able to mend this. For ſome there
are, Feing a little couched (as the commou Saying is) that if they had me at Sea, they
would put me to ſeek all my preſcribed Rules; but I would have ſuch to kiciu,
That when I at Sca, I ſhall work the Ship in all Aſſays as well as ever they did, aud
can as often as I ſhall be called thereunto, after this manner, (viz.)
PROP. I. The wind is fair.
The Wind is fair, though but little; he comes well, as if he would ſtand; there-
fore up a hand and looſe fore Top-fail in the Top, that the Ships may
ſee we w ill Sail;
Bring Cable to the Capſton, have up your Anchor, looſe your Forc- fail in the
Crailes; pur abroad our Colours, looſe the Milne in the Brailes. Is all our mon on
board ? Thoſe that be on Shore may have a Towe, and be bleſt with a Ruchers for
we will ſtay for 110 man. Come my Hearts, have up your Anchor, that we may
have
a good Prize. Come, Who ſay Amen? One and all
. Oh brave Hearts; the Anchorys
a Peck; have our fore Top-fail
, have our main Top-fajl, hawl hầme the Top-lail
Shcers. The Anchor is away, let fall your Forc-fail, hoiſe up your fore Top-fail, hoire
up your main Top-fail; upan: looſe the Main-fail
, and ſet him;
looſe Sprit-ſail, and
Sprit-fail Top-fail. A brave Gale. Bring the fore-Tack to the Cat-head, and trim
our Sails quartering ;-hoiſe up our ſmall Sails
have out the Milne Top-fail and ſet him.
Now we are clear, and she Wind like to ſtand; hoiſe in our Boats before it is too
inuch Sea ; aboard Main-tack, aboard Fore-tack, a Lee the Helmne handſomly, and
bring her to eaſily, that ſhe may hot ſtay. Breace the Fore-lail and Fore-top-fail co
the Maſt, and hawl up the Lec-Bowlings, that the Ship may not ſtay,;, paſs Ropes
for the Boor on the Lee-fide, and be ready to clap on your Tackle, and hoiſe them
in; ſtow them faſt. Let go the Lec-Bowling of Fore-Sail
, and Weather-Braces. Right
your Helmnes, hale aft the Fore-ſheet, trim che Sails quartering as before : Let go the
Sprit la 1 Breales, and haft of the Sheets; and hoiſe up the Sprir-fail Top-fail, and
other ſmall Sails. See the main Stay-ſail, and fore Top-ſail
, Stay-ſail and Milne Staya
fail, and main Top-fail, Stay-fail and leeſc in your Boonets, chat we may make moſt
of our way. To our Station, and clear your Ropes. Comc, ger up our ſteering
Cails. The Lee ſteering Sails of Main-lail, and Main-top fail, Fore-ſail and Fore-
top-fail only;, for they will ſet faixeſt
, and draw moſt away. I have on purpoſe
omiited leveral Words, by reaſon I would nor trouble the Reader with ſuch indiffe-
rene things as is conceived by all Mariners to be done; as Cooning the Ship, Breaſing,
Vereing, and haleing aft, and hoiling, looring, and the like; but it is to be fuppa-
ſed all to be done at the ſame time. Thus have you a brave Ship under all her Sails
and Canvas, in her ſwirteſt way of Sailing upon the Sca. Now let us have her righe
before the Wind.
1
7
Right afore the wind, and a freſlo Gale.
::
The Wind is vered right aft, take in your Fore and Fore-top-fail
, Steering=fail,
and Fore-top-ſail
, and Main and Main-top-lail
, Stay-fails; for they are becalmed by
the after Sails, and will only beat or rub out. The Wind blows a freth Gale, rrund
zfr the Main-ſheets, and Forc-sheets, braſſe. Square your Yards,'take in your Main
and Ma'nı-top-fail,Steering-fails. Unleaſe your Bonnets. Take in your Main and Fore-
top-gallant-fails
, In the Sprit-fail
, and Milne Top-ſail; let go the Sheets, hale from
che
chu
Book I.
ر
The Compleat Mariner,
17
1
che Cholyens, caſt off Top-gallant Bowlings. Thus you have all the ſmall Sails in,
and furled, when it blows too hard a Wind to bear them.
The wind vereth forward, and ſcanteth.
.
The Wind ſcanteth, vere out ſome of your Fore and Main-ſheets, and Sprifle-
ſheets, and let go your Weather Braces; tope your Sprit-ſail Yard. The Wind ſtill
verech forward; Get aboard the Fore and Main-Tack; caſt off your Weather-
ſheets and Braces: The Sails are in che Wind, hawle off Main and Fore-ſheers; the
Wind is tharp, hawl forward the main Bowline, aud hawl up the Main top-fail,
and Fore top-lail Bowline, and ſer in your Lee-braces, and keep lier as near as ſhe
will lie. Thus have you all the Sails trimm'd harp, full, and by a Wind.
The wind blows Frisking.
The Wind blows hard; fecule our fore and main Top-fails two thirds of the Maſt
down. It is more Wind, come, hawl down both Top-fails cloſe. Come, ſtand by,
take in our Top-ſails; Let go the Top-Sail Bowlines, and Lee-Braces ; let go the
Lcc-Sheets, ſet in your Weather Braces, ſpill the Sails, hawl home the Top-fail Clue-
lines, ſquare the Yeard. Now the Sail is furled, and you have the Ship in all her
low Sails, or Courſes at ſuch a time.
1
It bloweth a Storm.
It is like to over-blow; Take in your Sprit-ſail, ſtand by to hand the Fore-fail. .
Caſt off the Top-ſail Sheets
, Clue-garnets, Leech-lines, Bunt-lines ; ſtand by the
Sheer, and brace ; loure the Ycard, and furl the Sail, (here is like to be very much
Wind) See that your main Hallyards be clear, and the reſt of your Geer clear and
caſt off. (It is all clear.) Loure the main Yard. All down upon your doone hall;
now the Yard is down, hawl up the Clue-garnets, Lifts, Leach-lines, and Bunt-
lines, and furle the Sail faſt, and Fasten the Yards, 'that they may not trayers and
gall. Thus have you the Ship a trije under a Mizen. .
i very hollow grown Sea.
+
.
We make foul weather, look the Guns be all falt, come hand the Mizen. The
Ship lies very broad off; it is bercer {pooning before the Sea, than trying or hulling:
go reefe the Fore-fail, and ſet him; hawl afc the Fore-ſheet ; The Helmne is hard a
weather, mind at Helmne what is ſaid to you carefully. The Ship wears bravely ftu-
dy, ſhe is before it, and the Sheets are afle and braces ; belay the fore doon hall,
that the Yard may not turn up; it is done. The Sail is ſplit; go hawl down the
Yeard, and get the Sail into the ship, and unbind all things clear of it, and bring
too the Fore-bonnet ; make all clear, and hoiſe up the Fore-yard; hawl aft che
Sheets, get aft on the Quarter-Deck, the fore Braces.
Starboard; hard up, right your Helmne Port. Port hard, more hands, he cannot
put up, the Helmne. A very fierce Storm. The Sea breaks ſtrange and dangerous;
Nandby to hawl off above the Lennerd of the Whipſtaff, and help the man ac
Helinne, and mind what is ſaid to you. Shall we get down our Top.maſts? No, let
all ſtand; yer we may have occaſion to ſpoon before the Sea with our Powles. As we
maſt, get down the Fore-yeard- She cuds before the Sea very well; the Top.maft
being aloft the Ship is the hollomeſt, and maketh better way through the Sea, ſeeing
we have Sea-Room. I would adviſe none in our condition to ſtrike their Top-maſts,
before the Sea or under. Thus you ſee the ship handled in fair weather and foul,,
by and learge. Now let us ſee how we can turn to windward. .
i
1
D
The
1
18
Book T
The Compleat Mariner.
The Storm is over, let us turn to Windward.
V.
The Storm is over, ſet Fore-fail and Main ſail; bring our Ship too; fet che Milue
and Main Top-fail, and Fore Top-fail. Our Courſe is E. S. E. the Wind is at
-South: Get the Starboard-Tacks abcard, caſt off our Weather Braces and Lifts; Set
fuithe Lec-Braces, and hawl forward by the Weather Bowlines, and hawl them laught
and belaye them, and hawlover the Mizen Tack to Winerd; keep her full, and by
as- near as fhe will lye. Hon Wind 2014? Eaſt. A bad quade Wind. (No near) hard,
so 'near. The Wind verech to the Eaſtward ſtill. How Wind you ? N. E. hard, no near.
The Wind is right in our teeth; no near ſtill. How wind you? N.W.b. N. The
Wind will be Northerly, make ready to go about; we ſhall lay our Courſe another
way, no near, give che Ship way, that ſhe may ſtay: ready, ready a Lee the Helmuc.
Vare out there Fore-hect, caſt off your Lec-Braces, brace in upon your Weather Bra-
ces. The Fore-fails is a back ſtays, hawl Main-fail, hawl, about, let riſe the main
Tack, caſt off your Larboard-Braces, let go main Bowline, and main Top-fail Bow-
line; brace about the Yard, hawl forward by the Larboard-Bowlines; gee the main
Tack cloſe down, in the Cheeſe-trce : hawl up the Weather Bowline, and let the Lee
Brace of Main and Main Top-fail Yards, and the Sheets is cloſe aft; hawl, of all;
hawi; get to fore-Tack, let go fore-Bowline, and fore-Top-fail Bowline; hpwl afte
the fore-Shect, hawl taught, the main Bowline, and main Top-fail Bowliite; ſhift
the Mizeni, tack, hawl bout forc Bowline, and fore Top-fail Bowline; fet in the
Lec-Braces taught, fore and aft, kecp her as ncar as ſhe will lie. No near, Hox
Wind you? N.N.E. thus werr no more ; no ncar,keep her full. The Wind is at N.N.E.
thus werr no more. (Hom Wind yox?) E. N. E. The Wind is at N. keep her away, her
Courſe E.S.E.Caſt off the Lee-Braces and Weacher-Bowlines and ſee in your Weather-
Braces. Vere out the main Sheet,and fore Shect,looſe the Spric-fail, and Spit-fail, Top-
fail,and Mizen Top-ſail,and Top-gallant-fails;hoife chem up, the Wind vears aflc ftill;
ler riſe the fore-Tack:the Wind's quartering,hawl aft the fore-Shect,bring him dovan
to the Cat-licad with a paſs-a-rec; ſtuddy in your Weather Braces; the Wind ſtands,
here. Thus you have the Ship as at firſt, ſteering under all her Canvas, quarter Wind,
las the did at firſt, ſetting Sail. She hath been wrought with all manner of Weather,
and all ſorts of Winds. Therefore we will draw to the laſt with a Man of War im
Chaſc and taking of her Prize, and ſo leave this Practick Parç to your Cenſure.
ܪ
1
The Man of War in her Station.
T.
ril
Now we are in our Station, and a good Latitude, hand your Top-fails, and furle
your Main-lail and Fore-fail, and brail up the Mizen, and let her lie at Hull, until
Fortune appear within our Horizon. up alaft to the Top-maſt-head, and look
abroad, young-men ; look well to the Westward, if you can ſee any Ships that
have been nipt with the laſt Eaſterly Winds. A Sail, a Sall. Where? Fair by us
How ſtands ſhe? To the Eaſtward, and is cwo Points upon her Weather Bow, and
hách her Larboard-Tacks aboard. “O then ſhe lies cloſe by a wind; we ſee her izpon
the Decks plainly. A good mani to Helmnc. Up young-men, and looſe the Fore-fail
,
Main-fail
, and Mizen. Get the Larboard-Tacks aboardžhave out the Main top-lailand
Porc-top-fail, and fooſe the Spritzfail, aloofc. Keep her as neer as ſhe will tic; hawi
afé tle, sheets
, and hawl up your Bowlincs laught. Do you ſee your Chaſe? How
Wind' jouÉN. E. Then thę Wind is at N. Hoiſe up your Top-Sails as high as
you can have quit Sprit-ſail, Top-ſail, and Mizen Top-fail; hawl home the Sheets,
and hoiſe themfüp: A young man and looſe the Main-top-gallant-fail, and Fore-top-
gállant-ſail; hawl home the Sheets, and hoiſe them up; hoiſe up Main Stay-fail,
and Mizen Stay-ſail, and looſe the Main-top-fail, and Fore-top-fail; Stay-fail
, and
ſet them. It blows a brave chaſeing Gale of Wind; The Ship makes brave way
through the Sea; we riſe her apace; if ſhe keep her Courſe, we ihall be up with lier
in three Glaſſes. No ncar, keep the Chaſc open with little of the Fore-fail
. So, thus,
Keep her thus. Come ofle all hands; the ship will Stear the better when you ſit all
quices
1
+
Book I. The Compleat Mariner.
t
1
She ſettles her Top-fails, we are within ſhot; lec all our Guns be looſe, In the
19
quier ; by, by her ſmall Sails, for ſhe is too inuch by the Head. The Chaſe is a
luſty
brave Ship. So much the better, the hath the more Goods in her Hold. The Ship
hath a great many Guns (no force) it may be ſhe is a Private
Port, the
Chaſe is about, come ferch her wack, and we will be about after her. We fayl far
better than the; we have her Wack; a Lee the Helmu,about Ship, vere our Fore-
ſheet. Every man ſtand handſomly to his buſineſs, and mind the Bowlines and Bra-
ces, Tacks and Sheets; hawl Main-lail, hawl about. Let go Main Bowline, Top
Bowlinc, Top-gallant Bowline; Hawl off all, hawl Fore-fail, about, ſhifts the
Helmi ; bring her too, Hawl the Main-ſheets cloſe aflc,and fore-thect. Set in che Leen
Braces, hawl too the Bowlines. Thus the Chaſe keeps cloſe to the Wind; keep her
open under our Lee. Gunner, ſee that you have all things in readineſs, and that
the Guns be clear; and that nothing peſter our decks.
our decks.-Down with all Hammocks
and Cabins that may hinder and hurt us. Gunner, is all our Geer ready? Is the ſtore
of Cartrages ready fillid, all manner of Shot at the Main-maſt? Is there Rammers,
Sponges, Ladles, Priming-Irons, and Horns, Lyncſtocks, Wads, and Water at their
ſeveral
quarters ſufficient for them? Be ſure that none of our Guns be cloy’d; and
when we are in Fight, be ſure to load our Guns with Croſs-bar and Langrel. Always
cbſerve to give Fire when the Word is given. See that there be Half-Pikes and Jave-
lings in a readineſs, and all our Small-ſhot well furniſhed, and all their Bandaliers
fill'd with Powder, and Shot in their Pooches.
See that our Murtherers and Stockfowlers have their Chambers fill'd with good
Powder, and Bags of ſmall Shoc to load them, that if we ſhould be laid aboard, we
might clear our Decks.Starboard, the Chaſe pays away more room, Starboard
hard; Vere out ſome of the Main-lhece and Fore-lhect; Caft of the Larboard-Bra-
ces, (Steady) Kecp her thus: Well Steerd; the Chaſe goes away room, her Sheets
are both aft, ſhe is right before the Wind: Stereboard hard ; Let riſe main Tack, let
riſe forc-Tack; Hawl afle 'Main-lhect, hawl afle fore-Sheet. We have a ſtearn-Chaſe,
but we ſhall be up with her preſently, for’we fetch upon her hand-going. The Chaſe
hawls
up his Majn-fail and furles it; ſhe puts aboard her Waſte-clothes; ſhe will
from alow young mcır
, and furle our Main-lail; ſling our Main
Yard, with the Chains in the Main-top; lling our Fore-yard, put aboard our Waſte-
Cloaths; he will fighe us before the Winde I lce; she
is full of Men; It is a hor
Ship, but deep and foul. Come clearly my Hearts, It is a Prize worth fighting for;
The Chaſe takes in her ſmall Sails; up aloft and take in our Top-gallant-ſails, Sprit-
fails, Top-fails, Mizen Top-fail, and furle the Sprit-ſail, and get the Yard alongſt
under the Bowſprit. She puts abroad her Colours, It is Red, White, and Blew;
they are Dutch Colours; no force, the worſt of Enemies. Boy, up and put abroad
St. George's Colours in our Main-top; ſtep oft a hand, and put abroad our bloody
Ancient ; Call all hands aloft; Coine up aloft all hands. They are all up Captain.
Gentlemen, We are here employed and maintained by his Majeſty King CHARLES
and our Corniry, to do cur Endeavours to keep this Coaſt from Pyracy and Robbers, and
his Majeſties Enemies;, whereof it is our Fortune to meet this ship at this time :
Therefore I de fire you in his Majesties Name, and for the sake of our country, and the
Honour of our Engliſh Nation, and our felves, for every man to behave himſelf coura-
geous like Engliſhmen; and not to havi the least skem of a Coward: but to obſerve the
Words of Command, and do his utmoſt endeavour againſt this barbaroris and inhumane
Enensy the Durch, who have treacherouſly and inhumanely murthered fo meny of our
Engliſh Nation, in the Exlt-Indies and other parts, whose Blood
tries for vengeance.
Therefore our Quarrel is jaft, and into Gods hands we.commit our Cauſes and ourſelves
.
every man to his Quarters, and flew his Courage, and God be with you.
Tackles and the Ports, all knockt opein
, that wc.inay be ready to sui out our
Guus
when the Word is given." Up noiſe of Trumpets, and hail our Prize; ſhe anſwer-
eth again with hçe - Trumper: Hold faft Gunner, do not fire till we hail them with
our Voices. (Haye, Hool) Amain for King CHARLES. (Port) edge towards
him, he fires his Broad-lide upon us. (What chear my Hearts?) Is all well betwixo
Docks?
fight us.
Come up
1
1
1
1
So
D%
-
20
The Compleat Mariner.
Воок 1:
Decks? Tea, Tea, only he rackt us through and through. No force, it is his turn
next; but give not Filc until we are within Piſtol-thot. Port, edge towards hiin. He
plies his ſmall Shot; hold faſt Gunner. Port, right your Helmuc. We will run up
his side. Starboard a little ; Give fire, Gunner. (That was well donc.) This Broad
fide hath made their Deck thin, but the ſmall Shor at firſt did gaul us. Clap in
fome Cale-lhot in the Guns you are now a loading; Bracc tco'the Fore-top-fail,that we
may imt thoot a head; He lies broad off to the Southward, to bring his orier Broad-
fide to bear upon iis. (Starboard hard.) Get to Larboard Forc-rack"; trim your Top-
fails, run out your Larboard Guns. He fires his Sterboard Broad-lide upon us; He
pours in his ſmall Shot. Starboard give not fircuntil he fall off, that the Prize may re-
cive our full Broad-ſide. Steady: Port a little ; give fire Gunner: His Fore-maſt
is by the board. This laſt Broad-lide hath done great Execution. Cheerly my Mates,
the day will be ours; He is thot a Head; He bears up before the Wind to ſtop his
Leaks : Kecp her chus; Well Steercd. Port, Port bard; Bear up before the Wind,
chiát we may give him our Scarboard Broad-ſide. Gunner, Is there great ſtore of Caſes
thot and Langrel in cur Guns? rea, yea. Port, make ready to board him ; Have your
Laſhers clear, and alle men with them. Edge towards him Guns wlien you give fires
Bring your Guns to bear amongſt his Men with the Caſe-lhor. Well ſteered, we are
cloſe on boord. Give fire Starboard; Well done Gunner; They lie Heads and Points
aboard the Chaſe. Come, Aboard him bravely; Enter, Enter. Are you lached falt?
Tea, yea, We will have hin before we go here-hence. Cur up the Decks; Ply your
Hånd-Granadoes and Srink-Pors. He cries out Quarter ; Quarter for orir Lives, and
we will yield up Ship and Goods. Good Quarter is granted, Provided you will
lay down all your Arms, open the Hatches; lawł down all your Sails and furle them;
looſe the Lachings, we will ſhcer off our Ship, and hoiſc out our Shallop. If you of
för to make any Sail, expect no Quarter for your Lives. Go with clie Shallor, and
fend aboard the Captain, Liçutenant, and Maſter and Mates, with as many more as
the Shallop will carry. So vre will leave the Man of War to puc his Priſoners down
into thc Hold, and ſecure. And fo likewiſe I have hewn you thus much of the Pra
Otick
part of Navigation, in which yoti may perceive that I have wrought the Ship in
all Eſſays, in Words and proper Sca-Phraſes; and if I was at Sea, I ſhould perform
it both in Word and Deed: therefore I leave it all to your Judicious Cenſures. And
Jer not Ignorance, the Arch-enemy of Arts deceive you, and cauſe you to think that I
have writ what I cannor do; but that I can as caſily turn him in the Theorick, which
Tray I liſt, as I càu the ship in the Practick. And ſo I will conclude with Ovid, whca
ve failed in the Straighit Tonian,
1
+
.
+
و :
1
no
+
Nothing but Ilaves we view in Sea where Ships do float,
And Dangers lie, huge IVhales, and all Fiſh play:
Above our Heads, Heaven's Star-embroidered Coat,
Whole Vault contains two Eyes, for Night and D.ly;
Far from the Main, or any Marine Coaſt,
'Twixt Borean Blafts, and Billomosy ipe are toff.
If Ovid in that. ſtraight Ionian Deep
Was toſt. So hard, mach more are we on Seas
Of larger Bounds; where Staff and Compaſs keep
Their ſtrict obfervance Yet in this uneale
of Tackling Boards, we fo the way make short,
That Still osr:Courſe.draws nearer to our Porta:
Bezbeen the Stream and filver-ſpanigled 'Skicsson)
We rolling climle, theus hurling fall beneath;
Our way is Serpent-like; in Meads which lie;
That boss the Grojis, but never makes it Path :
· But fitter, like young Maids and Youths together,
Run here and there; all sphere, andinone know whether:
1
hu
3
?
1
t.
d
ini
)
You...
OKT
1
1
21,2%
Book I.
The Compleat Mariner.
:21
Our way, we know, and yet unknown to other;
And whiles misknown to is; before we dive :
The Hand and Compaſs that governs the Ruther
Do often erre, although the Pilots ftrive
iyith Card and Compaſs
, get oxr*Reck'nings fall
Too narror, ſort, too high, too wide, too ſmall
.
To diſcon this, remark when we fet Land,
Some this, some that do gueſs, this Hill
, that Cape.
For some howers our Skill in ſuſpence ſtand,
Terming this shire, that Head-land Points the Map;
which when miftook, this forg'd Excesfe goes clear,
O ſuch and ſuch a Land it firſt did 'pear.
In all which ſtrife, Streſs’d Sailers have the pain,
By drudging, palling, hawling, ſtanding to it
In Cold and Rain, Loth dry and wet, they ſtrain
Themſelves, and toyl; wone elſe but they muſt do it.
Both Prow and Poop do anſwer to the Helm;
The Stcarſman fings, no Grief bis Foy can whelm.
By Night our Watch we ſet, by day our Sight,
And furle our Sails : If Piratcs do appear,
We reft refolv’d; 'Tis Force make's Cowards fight,
Though none more dare, than they that have most fear.
It's Courage makes us rath, and Wiſdom cold;
Yet Wiſemen ſtort, and ftung, grom
Lion-bold.
Sapientiam Sapiens dirigit, Artes Coartifex, &c.
The Wiſe-man knorys biřs Wiſdom how to uſe,
Tib? Artificer, what Art 'sis beſt to chuſe.
...Tiskrue Saying, and approved long?
The Wiſe-man is more worthy than the Strong
The Fields he tills, the City he can guide,
And” for the Ships in Tempeſts well provide.
Pbocilides.
THE
*
22
Book I.
THE
GEOMETRICAL
DEFINITION S.
T
He ARTS, faith Arnobius, are not together with our Minds
ſent out of the Heavenly Places; but all are found out on Earth,
and are in proceſs of time ſought and fairly forged by a conti-
nual Meditation. Our poor and nccdy Lives perceiving foine
caſual Things to happen prepoſterouſly, while it doch imitate,
attempt, and cry, while it doth ſlip, reform, and change, hath
out of theſe, ſome Fiduous Apprehenſion made by ſmall sci-
ences of Art, the which afterwards by Study are brought to
ſome perfection.
Yer the Practice of Art is not manifeſt but by Speculative Illuſtration ; becauſe by
Speculation we know that we may the better know. And for this cauſe I choſe a Spe-
culative Part; And firſt of Geometry, that you may the better know the Practice.
To begin then.
1
A
D
I. u Point is that which hath no Parts.
A Point is ſuppoſed to be a Thing indiviſible, or that cannot be divided into
parts; yet it is the firſt of all Dimenſions. It is the Philoſopher's Atome. Such
a Nothing, as that it is thre very Energie of All Things. In God it carrieth its Extremes
from Eternity to Eternity; which proceeds from the leaſt imaginable ching, as the
Point or Prick noted with the Letter A;
and is but only the Terms or Ends of Quan-
A
tity.
II. A Line is a ſuppoſed Length, with-
out Breadth or Thickneſs.
A Lines Extremes or Bounds are two 4
Points, as you may ſee the Line a ; b is made
古
​古
​G
by moving of a Point from a to b. A Lime is
either ſtraight or crooked; and in Geometry
of three kinds of Magnitudes, viz. Length,
Breadth, and Thickneſs
. A Line is capable
of Diviſion in Length only, and may be di-
vided equally in the Point C, -or-unequally A
B
in D, and the like.
III. The Ends or Bounds of . Line are Points.
You are to underſtand, the Ends or Bounds of a finite Line is A, B, as before: but
in a Circular Line it is otherwiſe ; for there the point in its Motion returnech again to
the Place where it firſt began, and ſo makech the Line infinite.
*IV: A Right Line is the ſhorteſt of all Lines, drawn from any two of the ſaid
Points,
As you may ſee the Right Line A B ſtraight,
and equal between the points A and B, with A-
3
-B
out bowing, which are the Bounds thereof.
V. A
1
07.
Book I. Geometrical Definitions.
23
i
1
V. A Superficies is a Longitude, having only Latitude.
Superficies is That which hath only
A
B
tr length and breadth , whoſe Terms and
Limits are two Lines. In the firſt kind
of Magnitude the Motion of a Point pro-
4
ducech a Line : So in the ſecond kind of
Magnitude, thc Motion of a Linc produ-
D cech a Superficies
. This is alſo capable of
с
two dimenſions, as the length A B and
CD, and the breadth A C and B D; and may be divided into any kind or number
of Parts,
VI. The Extremes of a Superficies are Lines.
.
As die Ends of a Line are Points, ſo the Bounds or Extremes of a Superficies are
Lines; as before, you may ſee the Ends of the Lires A B, and BD, and 'D'c, and
CA,
VII. A Plain Superficies lieth equally between his Lines.
Șo the Superficies A B C D is that which liech direct and equally bericen his Lines.
And whatſoever is ſaid of a Right Line, the ſame is meant of a Plain Superficies.
VIII. An Angle is when two Lines are extended upon the Same Superficies, Sex
so that they touch one another in a Point, int not dire&tly.
B
As you may ſee the ord Lines A B and B C incline
one towards the other, and touch one the other, in the
Point B. In which point;' by reaſon of the bowing in-
clination of the ſaid Lines, is made the Angle A BC.
And here note, That an Angle is moſt commonly
ſigned by three Letters, the middlemoſt whereof
ſhewech the Angular Point, as iu this Figure, when
we ſay Anglė A B C, you are to underſtand the very
A
C
Point ar B.
;**
1
IX. A Right Angle is that which is produced of a Right Line, falling upora
Right Line, and making two equal Angles on each ſide the point where
they touch each other,
+
1
-
19
A
As upon the Right Line CD fup-
poſe there dotlı ſtand another Right
Line A B, in ſuch fort char it ma-
kech the Angles on cither ſide there-
of; namely, the Angle A BD on
the one fide: equal to the Angle
ABC on the other ſide, then are
either of the two Angles Right An-
; and the Right Line A B, which
D D
Itaudeth erected
upon the Right Line
B
CD, without bowing or inclining
to either part thereof, -isa Perpendicular to the Line CD.
6
J.
glesi
X An
24
Book I.
Geometrical Definitions.
A
E
X. An Obruſe Angle is that
which is greater than a
Right Angle.
7
D
So the Angle CBE is an Obture An-
gle, becauſe it is greater than the Angle
A B C, which is a Right Angle; For
it doth not only contain that Right An-
gle, but the Angle A B E allo, and
therefore is Obtwiſe.
B
XI. Ax Acute Angle is leſs than a Right Angle.
Therefore you may ſec the Angle EBD is an Acute Angle; for it is leſs than the
Right Angle A B D, in which it is contained by the other Acute Angle AB E.
XII. A Limit or Term is the End of every Thing.
As a Point is the Limit or Term of a Line, becauſe it is the End thereof; fo a Line
Likewiſe is the Limit and Term of a Superficles, and a Superficies is the Limit and Terms
of a Budy.
XIII. A Figure is that which is contained under one Limit or Term, or many.
As thc Figure A is contained under one Limit or Term, which is the round Line;
alſo the Figures B and C
are contained under four
Right Lines: likewiſe the
Figure E is contained un-
B
А.
der three Right Lines,
C
which are the Limits or
Terms thereof; and the
Figure F under five Right
Lines: And ſo of all ocher
Figures.
And here note, We call
F
any plain Superficies, whoſe
E
sides are unequal (as the
Figure F) a Plot, as of a
Field, Wood, Park, Forest,
and the like.
T
XIV. A Circle is a plain Figure contained under oxe Line, which is called a
Circumference; unto which all Lines drawn from one point within the
Figure, and falling xpon the
A
Circumference thereof, are egaal
one to the other.
As the Figure A ECF is a Circle contained
under the Crooked Line A ECD, which Line
is called the circumference. In the middle of
F
E
this Figure is the Center or Point B, from which
Point áll Lines drawn from the Circumference
are equal, as the Lines BA, BE, BD, BC;
D
and this Point B is called the Center of the Cir-
23
cle.
C
9
ch
XV. A A
t
1
1
BOOK I.
Geometrical Definitions.
25
XV. A Diameter of a Circle is a Right Line drawn by the Center thereof, and
ending at the Circumference, on either fide dividing the Circle into two
equal Parts.
So the Line A B C in the former Figure, is the Diameter thereof, becauſe it paflech
from the Point A on the one ſide, and pafſeth alſo by the Point B, which is the Center
of the Circle ; and moreover, it divideth the Circle into two cqual parts, namely,
AEC being on one ſide of the Dinmeter, equal to A F C on the other. And this
Obſervation was firſt made by Thales Mileſius; For, faith he, if a Line drawn by the
Center of any Circle do not divide ic equally, all the Lines drawn from the Center of
that Circle, from the Circumference, cannot be equal.
XVI. A Semicircle is a Figure contained under the Diameter, and that part of
the Circumference cut off by the Diamecer.
As in the former Circle, the Figure A F C is a Semicircle, becauſe it is contained of
the Right Line A B C which is the Diameter, and of the crooked Line A FC, being
that part of the Circumference which is cut off by the Diameter: Alſo che parc A EC
is a Semicircle.
XVII. A Section or Portion of a Circle, is a Figure contained under a Right.
Line, and a part of the Circumference, greater or leſs than a Semicircle.
B
So che Figure A B C, which conſiſtech of the
part
CA
thie Circüimference AB C, and the Right Line A C, is
a Sektion of Portion of a Circle, greater than a Semi-
circle.
Alſo the other Figure A CD, which is contained
E
under the Right Line A C, and the parts of the Cir.
cumference ADC, is a Section of a Circle leſs than a
Semicircle. And here note, That by a Section, Sega
ment, Portion, or part of a Circle, is meant the ſame
thing, and ſignifieth ſuch part as is greater or lefler
than a Semicircle: So that a Semicircle cannor properly
D
be called a Section, Segment, or part of a Circle.
XVIII. Right-lined Figures are ſuch as are contained under Right Lincs.
XIX. Three-ſided Figures are ſuch as are contained under three Right Lines.
XX. Four-ſided Figures are ſuch as are contained under four Right Lines.
XXI. Many-ſided Figures are ſuch as have more Sides than four.
XXII. Al Three-ſided Figures are called Triangles.
And ſuch are the Triangles A B C.
.
B
A
C
:....
E
XXIII. Of
;
26
Book I.
Geometrical Definitions.
XXIII. Of Four-ſided Figures, A Quadrat or Square
is that whole Sides are equal
, and his An-
gles right, as the Figure A.
3
1
XXIV. A Long Square is that which hath right An-
gles, but unequal Sides, as the Figure B.
B
1
+
1
XXV. A Rhombus is a Figure Quadrangular, having
equal Sides, but not equal or right Angles,
as the Figure C.
C
XXVI. A Rhomboides is a Figure whoſe oppoſite sides
are equal, and whoſe oppoſite Angles are
alſo equal: but it hath neither equal Sides,
nor equal Angles, as the Figure D.
D
1
2
1
XXVII. All other Figures of Four Sides are called
Trapezia's.
i
:
XXVIII. Such are all of Four Sides, in which is ob-
ſerved no equality of Sides or Angles, as
the Figures L and M, which have neither
equal Sides nor Angles, but are deſcribed
by all Adventures, without the obſerva-
tion of any Order.
M
XXIX. Parallel
+
Book I.
Geometrical Theoremes.
27
XXIX. Parallel or Aqui-diſtant Right Lines are ſuch which being in one and the ſame
A-
-B Superficies, and produced infinitely on both ſides, do
never in any part concur; as you may ſee by the two
C
D Lines A B, CD.
XXX. A Solid Body is that which hath Length, and Breadth, and
[
Thickneſs, as a Cube or Die; and the Limits and Extremes
of it are Superficies, as the Figure I.
XXXI. Axis is the Diameter about which the Sphere or Globe is
turiicd.
K
XXXII. The Poles of a Sphere are the Extremes or Ends of the Diamen
ter, and are terminated in the Superficies of the Sphere.
XXXIII. A Sphere is defined by Euclid to be made, when the Diameter of a Semi-
circle remaining fixed, che Semicircle is turned about, till it be returned
to the place whence it began to move at firſt.
1
Geometrical Theoremes.
I.
NY cwo Right Lines croſſing one another, make the contrary
or vertical Angles cqual. Euclid. 15. I.
SU
II. If any Right Lines fall upon'two parallel Right Lines,
it makech the outward Angles of the one, equal to
the inward Angles of the other; and the cwo inward
oppoſice Angles, on contrary ſides of the falling Line,
allo cqual. Euclid 29. I.
1
III. If any Side of a Triangle be produced, the outward Angle is equal to the in-
ward oppoſite Ingles
, and all the chrce Angles of any Triangle are equal to
two Right Angles. Euclid 32, 1.
IV. In Æqui-angled Triangles all their Sides are proportioned, as well ſuch as con-
tain the equal Angles, as alſo the ſubtendent Sides.
V. If any four Quantities be proportional, the firſt multiplied in the fourthi,
produceck a Quantity equal to that which is made by multiplication of the
fccond in the third.
VI. In all Righe-angled Triangles, the Square of the Side ſubrending the Right
Angle, is equal to both the Squares of the containing fides. Euclid 47.1.
VII. All Parallellograms are double to che Triangles that are deſcribed upon their
Baſis, their Altitudes being equal. Euclid 41. I.
VIII. All Triangles that have one and the ſame Baſe, and lie between two Parallel
Lines, are equal one to the other. Euclid 37. I.
E 2
GEO
28
Воок І
1
Geometrical Problems
G
PROBLEM I.
Upon a Right Line given, how to erect another Right Linc which ſhall be
perpendicular to the Right Line given.
HE Right Line given is A B, upon which from the Point E it
is required to creet the perpendicular E H. Opening your com-
paſs at any convenient diſtance, place one Foot in che aſſigned
Point E, and with
the other make the
HF
two Marks C and
D, cqual on cach
ſide the Point E;
G
then opening your compaſſes again to any
other convenient diſtance wider than the
former , place one Foot in C, and with the
other deſcribe the Arch GG; alſo (the
Compaſſes remaining at the ſame diſtance)
Isten
place one Foot in the Point D, and with
the other deſcribe the Archo FF. Then
from the point where thoſe two Arches in-A. C
E
B
terſect or cut each other (which is at H)
draw the Right Line HÉ, which thall be
Perpendicular to the given Right Line A B, which was the thing required to be done.
F...
ఆ
D
1
PROBL. II.
How to erect a Perpendicular on the end of a Right Line given.
Firſt upon the Line A B, with your compaſſes opened to any ſmall diſtance,
Ba , required
make five ſmall Diviſions, beginning at
A, noted with 1, 2, 3, 4, 5. Then take
D
hi
with your Comp-les the diſtance from A
to 4, and place one Foot in A, and
with the other deſcribe the 'Arch će:
Then take the diſtance from A to 5, and
7
placing one Foot of the compaſſes in 3,
with the other foot deſcribe the Arch
16
h h, cutting the former Arch in the
Point D: Laſtly, from D draw the Line
DA, which ſhall be perpendicular to
the given Line A B.
This operation is grounded upon this B
4 3 2
Concluſion, viz. Theſe three Numbers
3, 4, and 5, make a Right-angled Tri-
angle
, which is very neceſſary in many Mechanical Operations, and eaſie to be re-
membred.
PROBL.
for compare......
6
o Š Á.
2
*
BOOK I.
Geometrical Problems.
29
PROBL. III.
How to let fall a Perpendicular upon any Point aſſigned, upon a Right
Line given.
T
He Point given is C, from which Point
it is required how to draw a Right
Line which ſhall be perpendicolar to the gi-
17
ven Right Line A B.
Firſt from the given Point C, to the Line
A B, draw a Line by chance, as CE, which
divide into two cqual parts in the Point D.
D
Then placing one foot of the Compaſſes
in the Point D, with that diſtance DC
deſcribe the Semicircle CFE, cutting the
given Line A B in the Point F. Laſtly,
If from the Point C you draw the Right
Line CF, it ſhall be a Perpendicular to
h B the given Line A B, which was required.
►
L
PROBL. IV.
How to make an Angle equal to an Angle given.
Et the Angle given be AC B, and let it be required to make another Angle
cqual chercunto.
Firſt draw thc Line E F at plea-
ſure; then upon the given Angle ac
C (the Compaſſes opened to any di-
ſtance) deſcribe the Arch AB; and
allo upon the Point F, the Compaſſes
unaltered, deſcribe che Arch D E;
B
Then take the diſtance A B, and ſec
the ſame from E to D; Laſtly draw
the Line DF: So Thall the Angle
DFE be equal to the given Angle A C B.
A
here to
18
1
+
PROBL. V.
Right Line being given, how to draw another Right Line which shall
be parallel to the former, åt any diſtance required.
He Line given is A B, unto which it is required to draw another Righe Line „ ра-
rallel thereunto, at the diſtance A Cor D B. Firſt open your Compaſſes to the
diſtance A Cor BD; then placing
C
D
one Foot in A , with the other.de
ſcribe the Arch C; alſo(at that diſtance
place onc Foot in B, and with the
Othier deſcribe the Arch D. Laſtly,
10
draw the Line CD, that it may only
touch the Arches C and D: So ſhall
А.
B
the Line CD be parallel to che Line
A B, and at the diſtance required.
4
PROBL.
1
1
30
Book 1.
Geometrical Problems
4
D
PROBL. VI.
To divide a Right Line into any number of equal Parts.
Ec A B be a Right Line given, rand
let it be required to divide the ſame
into five equal Parts.
Firſt, From the given Line A, draw
the Line AC, making any Angle from
the end of the given Line which is ac
che Point B. Then draw the Line BD
cqual to the Angle CA B. Then from
В
the Points A and B, ſer off
upon
cheſe
(wo Lines any Number of equal parts,
bcing leſs by one than the Paris into
which the Line A B is to be divided,
which in this Example muſt be 4.
Then
draw ſmall L ne rrom 1 to 4, from 2
to 3 twici, and from 1 to 4, etc. which
Lines crofing the given lin.' A B, thall divide it into five equal Parts, as was required.
1. TITIII
2
1
2 그
​UVIIIII
B
4
F
PROBL. VII.
u Right Line being given, how to draw another Right Line parallel
thereunto, which ſhall alſo paſs through a Point aligned.
L
Ec A B be a Line given, and let it be required to draw another Line paralle
thereunico, which ſhall paſs through the
given Paint C. Firſt, Take with your Com-
C
paſſes the diſtance from A to C, and place-
ing one Foot thereof at B, with the other
D'
deſcribe the Arc DE; then take in your
Compaffes the whole Line A B, and place one
Foot in C, and with the other deſcribe the
B
A.
Arch FG, cro.ling the former Arch in the
Point H: Then if you draw the Line CH,
it
Thall be parallel to A B, the thing required.
H
>
PROBL. VIII.
Having any three Points given which are not ſcituated in a Right Line,
How to find the Center of a Circle which ſhall paſs directly through
the three Points given.
He three points given are A, B, and C;
R
T
now it is required to finde the Center of
:Circle whoſe circumferenice Šliall paſsthrough
the three Points given..!
Firſt open your compasſes to any diſtance
greaser" elan half the diſtance between B
and C; then place one Foor in the Point
B, and with the other deſcribe the Arch
F.G;. then the Compaffes remaining at the
ſamc diſtance, place one Foot in the Point C,
G
and with the other turn'd about make the
marks F and G in the former Arch, and draw
the Line F O G at length, if need be.
Ir like manner open your compaſs at a di-
ſtance greater than half A B; Place one Foot
in the Point A, with the other deſcribe the
22
Arch HK: Then the compaſſes remaining ac
the ſame diſtance, place onc Foot in the Point
C,
ic
-
1
-
Book I.
Geometrical Problems.
31
C, and turning the other about, make the marks H K in che former Arch. Laſtly,
draw the Right Line H K, cutting the Line F G in O, fo ſhall o be the Center, upon
which you may deſcribe a Circle at the diſtance of O A, and it ſhall paſs directly
through the three given Points A B C, which was required.
•
PROBL. IX.
A
How to deſcribe a Circle in a Triangle,
that Mall only touch the three sides;
and to find the Centre.
1
E
C
A
Ay down che Triangle A B C, the three
Sides equal; then divide the sides of
the Triangle A B in two cqual parts, as at B,
and draw the Line CE and likewiſe di
vide B C, and draw the Line A D; and
where they croſs one che other, as at O,
that is the Center: Therefore put onc Point
of the Compulles in the Cenier O, and extend
the other to cither ſide, and deſcribe the Cir.
cle GF, which will only couch the sides A
B C of the Triangle.
D
23
В.
1
1
PROBL. X.
How to lay down a Triangle in a Circle, and to find the Center of the
Circle in the Triangle.
D
C
not
F
H
Raw the three sides of a Triangle AB
..t C, it is no matter if they be equal or
then
put one foot of
your Compales
in the Point B, open the other to more than
half the length of the greateſt fide, as to C;
and with that diſtance deſcribe the Arch F
HDG; and ſo removing the Compaſſes to
C, croſs the former Arch at F and D, and
draw the Line DF. Again, the Compaſſes
at the ſame diſtance, pur one Foot in A,
and deſcribing a ſmall Arch, croſs the for-
mer Arch at H and G; and laying a Ruler
B.over the Interſections of theſe two Arches at
Hand G, draw the Line GH; and where
theſe two Lines croſs áne che other, as at K,
that is the Center of the Triangular Points.
From it exrend the compaſſes to either of the
Points, and deſcribe the Circle A B C; and.
the Triangle will be within the Circle.
А
SD
1 1 1
24
;
I
PROBLE
32
Geometrical Problems.
Воок І.
}
:
PROBL. X I.
Any three. Right Lines being given, ſo that the two shorteſt together be
longer than the third; To make thereof a Triangle.
Ee it be required to make a Triangle
L of theſe three Lines A, B, and C,
YA
the two ſhorteſt whereof, viz. A and B
together, are longer than the third-C.
Firſt diay the Lire D. E cqual-to che
given Line B ; then take with
F
poiler che Line C, and ſetting one Foot in
E, with the other deſcribe the Arch FF: TE
alio take with your compaſſes the given
25
Line A, and placing one foot in D, with
the odlier deſcribe the Arch G G, curcing
the former Arch in the Point K. Laſtly,
from the. Painit K, if you draw the Lines
KE and KD, you ſhall conſtitute the
KEM
Triangle KD E,whoſe Sides ſhall be cqual
to the three given Sides A B C.
***
1
تفنني
-41
PROBL. XII.
Having a Right Line given, How to make a Geometrical Square, whole
Sides Mall be equal to the Right Line given.
He Line given is RI, and it is required
an inake a Geometrical Square whoſe :D
C
Sides ſhall be equal to the Line R I. Firſt draw
shiegiken, Line Ja then.(by the firſt and ſem
::
cond Wrobleme upon the Point B raife the per-
pendicular BC, making the Line B C equal to
the given Line R I alſo: Then taking the ſaid
R. I iy your compaſſes, place one Foot in C,
dith the ocher deſcribe an Arch at D, The
26
Campafs at the ſame diſtance, ſet one Foor in
1, and trofs the former Arch at D; then draw
the Lines DC and D A, which ſhall conclude A
B
the Geometrical Square ABCD, which was
itquued.
RU
I
.
1
ܫܐܵ
+
in
PROBL. XIII.
Two Right Lines being given, How to find a third which ſhall be in prog
portion unto them.
A.
8
L
Èt the given Lines be a
B В
12
and B; and it is required
to find a third Line which
ſhall be in proportion unto
niem.
Firſt draw two Right Lines,
F
making any Angle at pleaſure,
as the Lines L M and MN,
E
making the Angle LMN:
Then take the Line A in your
Compaſſes, and ſee the length
thereof from Mto E ; alſo take
the line B, and ſet the Length M
N
H
chereof
than,
12
1
BOOK I,
Geometrical Problems
33
thereof from M to F, and alſo from Mto H: Then draw the Right Line E H, and
then from the Point F draw the Line F G parallel to EH. So ſhall MG be the third
Proporcional required: For Arithmctically ſay, ,
.
As ME to MH: Sos MF to MG 18.
8
12
I 2
12
+
3
24
I 2
844 (1-8
88
144
PROBL. XIV.
Three Right Lines being given, To find a fourth in proportion to them.
-36
42
ES
He three Lines given are A B C, unto which it is required to find a fourth Pro-
portional Line. This
is to perform the Rule of
A.
24
Three. As in the laſt
B
29
Problem, you muſt draw
two Right Lines, making
any Angle at pleaſure, as
the Angle E FG; then
take the Line A in your
Compaſs, and ſee it from
F to 1; then take the
Line B in your compas-
fes, and let that from F
28
to K; then take the third
given Line in your Con-
pafles, and ſet that from
G
F to H, and from thar
K
L
Point H draw the Line H
L, parallel to I K; So ſhall the Line F L be the third Proportional required.
Note, Thar theſe Lines are taken off a Scale, that is divided into 20 parts to an
Inch : To do it Arithmetically ſay,
A FI is to FK: Sois F H to F L.
24
28
36 42
28
24
288
2$$8 (42 72
244
1008
Here 110te,
That in performance of the laſt Problem, That the firſt and third
Terms namely the Lines A and C muſt be ſet upon one and the fanie Line, as
here upon the Line F E, and the ſecond Line B muſt be ſet upon the other
Line F*G, upon which Line alſo the fourth Proportion will be found.
1
T
PROBL:
!
34
Geometrical Problems:
Воок І.
9
PROBL. XV.
How to work the Rule of Proportion by a Scale of equal Parts, and ſuch
other Concluſions as are uſually wrought in Lincs and Numbers, as
in Mr. Gunter's 10 Prob. 2 Chap.
He Scale of Inches is a Scale of equal Parts, and will perform (by protraction
upon a Flat or Paper) ſuch Concluſions as are uſually wrought in Lines and
Numbers, as in Mr.Gunter's 10 Prob. 2 Chap. Sector, may be ſeen, and in ochers that
have writ in the ſame kind. This way Mr. Samuel Foſter hath directed in the I Chap.
of his Posthumus Fosteri.
T
An Example in Numbers like his Tenth Probl.
As 16 to 7: So is 8 to what?
Here becauſe the ſecond Term is leſs than the firſt, upon the Line A B, I ſet AC
the firſt Term 16, and the ſecond Term AD 7, both taken out of the Scale of equal
parts: thence alſo the third Number 8 being taken, with it upon the Center C, Ide-
fcribe the Arke E F, and from A draw the Line A E, which may only touch the
ſame Arke ; then from D, I take DG, the leaſt diſtance from the Line Á E, and the
fame meaſured in the ſame Scale of cqual parts, gives 3 , the fourth Term required.
A
4
6
29
re
rut
But if the ſecond Term (hall be greater than the firſt, then the form of working
muſt be changed, as in the following Example.
EXAMPLE.
As 7 to 16: So 21 to what ?-48.
Upon the Line A B, I ſet the ſecond Term 16, which is here ſuppoſed to be A D;
chen with the firſt Term 7 upon the Center D, I deſcribe the Arke GH, and draw
A G chat may juft touch it : Again, having caken 31 out of the ſame Scale, I ſer one
Foot of that Extent upon the Line A B, removing it until it fall into ſuch a place, as
that the other Foot being turned about, will.juſtly touch the Line A G before-drawn;
and where (upon ſuch Conditions) it reſtech, I make the Point C; then meaſuring
A Cupon your Scale, you ſhall find it to reach 48 Parts, which is the fourth Num-
ber required.
The form of Works (although not ſo. Geometrical) is here given, becauſe it is here
more expedite than the other by drawing Parallel Lines; but in ſome Practice the
othier may be uſed. I have been the more large upon this, becauſe in the following
Treatifc
Book I.
Geometrical Problems
35
Treatiſe I ſhall quote ſome more remarkable Places in Poſthuma Foſteri: and the So-
lution of Proportions muſt be referred thither, the form of their Operations being the
fame with this. In them therefore {hall only be intimated what muſt be done in ge-
neral, the particular way of working being here directed.
30
PROBL. XVI.
To divide a Right Line given, into two parts, which ſhall have ſuch pro-
portion one to the other as two given Right Lines.
He Line given is A E, and it is required to divide the ſame into two parts,
which ſhall have ſuch proportion one to the other, as the Line C hath to the
Line D.
Firſt, From the Point A
draw a Right Line at pleaſure,
B C D making the Angle 'BAE;
30
then take in your compaſſes the
Line C, and ſet it from A to
F; and alſo take the Line D
and ſet it from F to B, and
draw thc Line B E: Then
F
from the Point F draw the
Line F G, parallel to B E,
10
cutting the given Line A E in
the Point G: So is the Line A
A
A L16
2.4
E
in
B divided into two parts
30 G
the Paint G, in proportion to
icone
the other, as the-Lise C is to
thc Line D:
Arithmetically, ler che Line A E contain 40 Perches or Foot, and let the Line C be
20, and the Line D 30 Perches; and let it be required to divide the Line A E inco
two parts, being in proportion one to the other, as the Line C is to the Line D.
Firſt
, Add the Lines C and D together, their Sum is 50: Then ſay by the Rule of
Proportion, If so (which is the Sum of the two given Terms) give 40, the whole
Line A E; What Thall 30 the greater given Term give? Multiply and divide, and
you ſhall have in the Quotient 24 for thic grcater part of the Line A E; which being
taken from 40, there remains 26 for che other part AG: For
AE
16
G 40
As A B is to AE: So.is BF to EG.
50
30 24
40
40
2z$ (24
00
120
880
I 200
! :
PROBL. XVII.
"How to divide a Triangle into two parts, according to any proportion af
figned, by a Line drawn from any Angle thereof; and to lay the les-
ſer part unto any side aſſigned.
L
Ec A B C be a Triangle giveri, and let it be required to divide the ſame by a
Line drawn from the Angle A, into two parts, the one bearing proportion to
the other, Asche Line F to the Line G; And that the leſſer part may be towards the
Side A B..
F. 2
By:
1
+
BOOK I.
36
.
Geometrical Problems.
A
GTF
31
der
*****
C
B
45
D
By the laſt Problem divide the Baſe of the Triangle B C in the Point I, in
propor-
tion as the Line F is to the Line G (the leſſer part being ſet from B to D.) Laſtly, draw
the Line A D, which ſhall divide the Triangle A B C in proportion as F to G.
As the Line F, is to the Line G:
So is the Triangle A DC, to the Triangle A BD. ·
PROBL. XVIII.
The Baſe of the Triangle being known, To perform the foregoing Problem
Arithmetically.
Uppoſe the Baſe of the Triangle BC be 45, and let the Proportion into which the
Triangle A B C is to be divided, be as 2 to 4. Firſt add the two proportional
Terms together, 2 and 4, which makes 6; then ſay by thc Rule of Proportion, If 6,
the Sum of the Proportional Term, give 45 (the whole Baſe BC) What ſhall 4 ché
greater Term given ? Multiply and divide, and the Quotient will give you 30, for
the greater Segment of the Baſe DC, which being deducted from the wholc Bafé 45,
there will remain 15 for the leffer Segment B D.
ܪ
1
As 16 is to 45: So is 4 DC 30.
4
180
x86 (30
66
PROBL. XIX.
How to divide a Triangle (whoſe Area or Content is known) into two
Parts, by a Line drawn from an Angle afligned, according to any
Proportion required.
L
Et the Triangle A B C contain 9 Acres, and let it be required to divide the ſame
into two Parts, by a Line drawn from the Angle A, the one to contain 5 Acres,
and the othei 4 Acres. First, meaſure the whole length of tlie Baſe, which ſuppoſe
45 ; Then fay, If 9 Acres the quantity of the whole Triangle, give 45 the whole
Bafe, What parts of the Baſe shall 4 Acres give ? Multiply and divide, the Quotient
will
1
.
Воок І.
Geometrical Problems.
37
Triangle
will be 20 for the lcfier Segment of the Baſe BD; which being deducted from 45
the whole Baſe DC, then draw the Line AD, which ſhall divide the
A B C according to the proportion required.
If 9 Acres give 45, What fall 4 Acres give ?
Anſwer 20
{
180
288 (20
99
2
PROBL. XX.
How to divide a Triangle given into two parts, according to any Propozi
tion aligned, by a Line drawn from a Point limited in any of the
Sides thereof; and to lay the greater or leſſer part towards any Angle
aſſigned.
TH
'He Triangle given is
Á BC, and it is
N-
required from the Point
А.
M to draw a Line that
O
Thall divide the Triangle
into two parts, being in
proportion one to the
other, as the Line N is
to the Line O ; and to
lay the leſſer part to-
wards B.
Firſt, from the limited
Point M draw a Line to
B.
the oppoſite Angle at A;
M
then divide the Baſe BC
in proportion as O to N,
which Point of Diviſion
will be at É; then draw E D parallel to A M : Laftly, from D draw the Line D M,
which will divide the Triangle into two parts, being in Proportion one to the other,
as the Line O is to the Line N.
در
1
H RE
I
PROBL. XXI.
To perform the foregoing Problem Arithmetically.
T is required to divide the Triangle A B C, from the Point M, into tivo parts in
proportion as 5 to 2.
Firſt divide the Baſe B Caccording to the given Proportion; then becauſe the leſſer
Part is to be laid towards B, mcäfui'c the diſtance from M to B, which ſuppoſe 32:
Then ſay by the Rule of Proportion, IF M B 32, give E B 16, what ſhall A R 28
(Perpendicular of the Triangle) give Multiply and divide, the Quotient will be 14,
at which diſtance draw a Parallel Line tò Bíc, namely D; then from D draw the
Line D M, which thall divide the Triangle according to the reqùired Proportion.
+
F
1
*
PROBL:
38
Воок І.
Geometrical Problems.
PROBL. XXII.
How to divide a Triangle (whoſe Arca or Content is known) into two
Parts, by a Line drawn from a Point limited, into any Side thereof,
according to any number of Acres, Roods, and Perches.
1
N the foregoing Triangle A B C, whoſe Area or Content is 5 Acres 1 Rood, let the
I limited Point be M in the Bare thereof; and let it be required from the Point M
to draw a Line which ſhall divide the Triangle into two parts becween Johnſon and
Porell
, ſo as Johnſon may have 3 Acres 3 Roods thereof, and Powell may have Acre
and . Roods thereof.
Firſi, Reduce the quantity of Poppellºs, being the leſſer, into Perches (Obſerve, 160
ſquare Poles contaius i Acre, half an Acre contains 80 Perck, a quarter or one Rood 40
Perch.) which makes 240. Then conſidering on which ſide of the limited Point M
this
part is to be laid, as towards B, meaſuring the part of the Baſe from M to B 32
Perch, whereof take the half, which is 16, and thereby divide 240, the Parcs be-
longing to Powell, the Quotient will be 15, the length of the Perpendicular D H, ac
which Parallel-diſtance from the Baſe B C, cut the Side A B in D, from whence draw
the Line D M, which ſhall cut off the Triangle DBM, containing i Acre 2 Roods,
the parë belonging to Powell: Then the Trapezia A DMC (which is the part be-
longing to Johnſon) contains the reſiduc, namely, 3 Acres 3 Roods.
:
3
28
160
80
246 (15
240
288
Z
>
1
PROBL. XXIII.
Horý to divide a Triangle according to any Proportion given, by & Lins
drawn parallel to one of the sides given.
He following Triangle A B C is given, and it is required to divide the ſame into
two Parts -by a Line drawn parallel to the Side A C, which ſhall be in pro-.
portion one to the other, as the Line I is to the Line K.
Firſt (by the 16th Problem) divide the Line B-C in E, in proportion as I to K;
then (by the 27th Arpblem following) find a mean Proportional between B E and BC,
which ler be BF, from which Point F draw che Line FH, parallel to A C, which
Line ſhall divide the Triangle into two parts, viz. the Trapezia AHFC, and the
Triangle H F B,which are in proportion one to the other, as the Line I is to the Line K.
1
PROBL, XXIV.
To perform the foregoing Problem Arithmetically.
L
Er the Triangle be A B C, and let it be required to divide the ſame into two
parts,
which ſhall be in proportion one to che other, as 4 to 5, by a Line drawn
Parallel to one of the Sides.
Firſ
i
.
Воок І.
Geometrical Problems.
39
1
H
Firſt let the Baſe BC,
А A K.
4.
containing 54, be divided
according to the
propor-
I-
tion given; ſo ſħall the
leſler Segment B E con-
tain 24, and the greater
EC 30; Then find out a
33
mcan Proportional be-
tween B E 24, and the
whole Baſe BC 54; by
multiplying 54 by 24,
whoſe Product will be
B
1296; the Square Root
thereof is 36, the man
G D
E F F
Proportional ſought, sch
is BF. Now if BF 36
give BE 24, what ſhall
AD 36? The Anſwer is HG 24, at which diſtance draw a Parallel Line to the Baſe,
to cut the Side A B in H, from whence draw the Line H F, Parallel to A C, whichi
ſhall divide the Triangle as was required.
1 -
54
24
216
To8
1296
1296
PROBL. XXV.
To divide a Triangle of any known Quantity into two parts, by a Line
Parallel to one of the Sides, according to any Number of Acres,
Roods, and Perches.
He Triangle given is A B C, wlioſe Quantity is 8 Acres, o Roods, and 16 Peň-
ches; and it is deſired to divide the fame (by a Line drawn up parallel to the
Side A C) into two parts, vič. 4 Acres, 2 Roods, O Perches; and 3 Acres, 2 Roods,
and 16 Perches.
Firſt, Reduce boch Quantities into Perches (as it is hereafter caught) and they will
be 720, and 576; then reduce both theſe Numbers by abbreviation into the leaſt
proportional Term, viz. 5 and 4; and according to that proportion, divide the Baſe
B C 54 of the given Triangle in E: then ſeek chc mcan Proportion between BE and
BC, which Proportion is B F 36, of which 36 take the half, and thereby divide
-576, the leſſer Quantity of Perches, the Quotient will be HG 32, ac which Paw
rallel-diſtance from the Baſc, cut off the Line A B in H, from whence draw the Line
HF parallel to the Side A C, which ſhall divide the Triangle given, according as it
was required.
23.0
876 (32
288
1
2
"
PROBL
40
Book.
Geometrical Problems.
1
Į
PROBL. XXVI.
From a Line given, To cut off any Parts required.
r
Tom
7
34
He Line give is AB,
from which it is rés
quired to cut off"; Parts.
i Fird, draw the Line A
C, making any Angle, as
GAB is ellen from a ſet
off any 7 cqual Parts, as
10,7, 3, 4, 5,6,7; an
trony 7 draw the Line 7 B.
Now becaute 1 is to be cut A
oH fiqi dhe Lin: B, chere-
--B
Fore from the Poin: 3, draw
dic Line 3 D, parallel m 7 B, cutting the Line A B in D; So thall A D be che! of
the Line A B, and D.B thall be of die ſame Line.
1
1.
3
ܝܬ
i.!
As 7 is to A B: So is A i to A D.
IN
PROBL. XXVII.
To find a Mean Proportional between two Lines given.
N the following Figure, let the two Lines given be A and B, between which it is re-
quired to find a Mec:n Proportional
. Let the two Lines A and B be joyncd roge-
ther in the Point E, making one Right Line as C D, which divided into two equal
Parts in the Point G; upon whichi Point G, with the diſtance G C or GD, deſcribe
skis semicircle us Di Then from the Point E, where the two Lițies arc joyned moge-
mile'tlfé perpendicular E F : So ſhall the Line E F be a Mean Proportional becwveen
ofte ato giveri Dine's A and Bi For; As E D is to E F: So E F to CF.
16
tier,
9
12
12
OI
68
1
PROBL. XXVIII.
How to finde two Lines,which together ſhall be equal in Power to any Line
given; And in Power the one to the other, according to any propor.
tion aſſigned.
A-
B
-16
C
25
F F
N this Figure let
CD bc a Line
givçın, to be divided
in Poruci, as the Line
A is to the Line B.
Firſt, divide the
Line CD in the Point
E, in proportion as A
to B (by the 16th
Probl.) Then divide
the Line CD into
two equal Parts in the
Point G, and on G,
at the diſtance G D
orGC, deſcribe the
3
12
15
20
16
9
256 H 9
Semicircle
1
.
Book II. Geometrical Problems.
41
Semicircle CFD, and upon the Point E raiſe the Perpendicular EF, cutting the Se-
micircle in F. Liftly, draw the Line C F and D F, which together in Power will be
equal to the Power of the given Line CD; and yet in Pover one to the other, as
A to B.
**
PROBL. XXIX.
A B A la ng AN 。
F.
DITUIT
How to divide a Line in Power according to any Proportion given.
Fi: Divide the Line C
Irſt, Divide the Line C
D in the Point E,in pro-
portion as A to B : Then di- A
vide the Line CD in two e-
B
qual Parts in the Point G,and
upon G as a Center, at the di-
ſtance CD, deſcribe the Sem
micircle CFD, and on E 3
raiſe the perpendicular of EF,
cutting the Scmicircle in F:
Then draw the Line CF and
DF, and produce the Line
CF to H, till FH be equal
to FD, and draw the Line
12
DH. Laſtly, draw the Line
FK, parallel to DH: Then
D
Thall the Line CD be divi-
14.
10
ded in K; ſo charche Square
of C K hall be to the Square of K D, as C E to E D, or as B to A.
VITIT
PROBL. XXX.
How to enlarge a Linc in Power, according to any Proportion aſſigned
N the Diagram of the 28th Problem, Ice CE be a Line given, to be enlarged in
a
Power as the Line B to the Line C.
Firſt (by the I6th Problem) find a Line in proportion to the given Line CE, as B
is to C, which will be CD; upon which Line deſcribe the Semicircle CFD, and
on the Point E erect the Perpendicular E F: Then draw the Line C F, which ſhall be
in power to CE, as C to B.
PROBL, XXXI.
To enlarge or diminiſh a Plot given, according to any Proportion required.
L Et ABCD E be a plot given, to be diminiſhed in Power as L to K.
Divide one of the sides, as A B in Power as L to K, in ſuch fort chat the
Power of A F may be to the Power of A B, as I to K; then from the Angle A draw
Lines to the Point' C and D. That donc, by F draw a Parallel to B C, to cut A C
in G, as FG: again, from G draw a Parallel to D C, to cut A D in H. Laſtly,
from H draw a Parallel to D E, to cut A E in I: So ſhall the Plet A FGHI be
like ABCDE, and in proportion to it, as the Line L to the Line K, which iras
required.
G
Allo
42
Geometrical Problems.
Book I.
€
G
D
F
H
a
A
be the length of the Perpen-
Alſo if the leſſer Plot was
given, and it was required to
make it in proportion to it as K-
K to L; then from the Point
L
A draw the Lines A Cand
A D at leisgtli; alfo increaſe
AF and A I: That done,
B
enlarge A F in Power as K to
L, which ſer from A to B ;
then by. B draw a Parallel to
FG, to cut A Cin C, as B
C: likcwiſc from C draw
Parallel to GH, to cut A D
in D: Laſtly, a Parallel from
37
D to H I, as D E, to cat A I
being increaſed in E; fo ſhall
you include the Plot A BC
DE, like A FGHI, and
in proportion thereunto, as the Line K is to the Line L, which was required.
PROBL. XXXII.
How to make a Triangle which ſhall contain any Number of Acres, Roods,
and Perches, and whoſe Baſe ſhall be equal to any poſſible) Num-
ber given.
Er it be required to make a Triangle which ſhall contain 6 Acres, 2 Roeds, 25
L
Perches, whole Baſe ſhall contain 50 Perches. You muſt firſt reduce your 6 Acres
2 Roods, and 25 Perches, all into Perches, after this manner.
Firſt, Becauſe 4 Roeds makes 1 Acre, multiply your 6 Acres by 4. makes 24; to
wliich add the 2 odd Roods, ſo have you 26 Roods in 6 Acres 2 Roods; then becauſe 40
Perches makes i Rood, multiply your 26 by 40, which makcs 1040, to which add
the 25 Perches, and you ſhall have 1065, and ſo many Perches are contained in 6
Acres, 2 Roods, and 25 Perches.Now to make a Triangle that ſhall contain 1065
Perches, and whoſe Baſe ſhall be 50 Perches, do thus ; double the number of Per-
ches given, namely 1065, and
they make 2130, then be A
D
cauſe the baſe of the Trian-
gle muſt contain 50 Perches,
divide 2130 by so, the Q:10-
tient will be 42. which will
+
1
3
125
dicular of the Triangle. This
done, from any Scale of equal
Parts, lay down the Line B C
equal to 50 Perches; then
38
upon C raiſe the Perpendicular
C E, cqual to 42 Perches,
and draw the Line A E, pa-
rallel to B C; then from any
Point in the Line A E, as from
G, draw che Line BG, and
G'c, including the Triangle
BGC, which ſhall contain
6 Acres, 2 Roods, 25 Perches, which was required.
1
50
BL
ول
t
PROBL
1
3
Book I.
Geometrical Problems.
43
PROBL. XXXIII.
How to reduce a Trapezia into a Triangle, by a Line drawn from any
Angle thereof.
33
f
А.
He Trapezia given is
TH
ABCD, and it is
required to reduce the ſame
39
into a Triangle.
Firſt, cxtcnd the Line D
C, and draw the Diagonal
BC; then from the Point
A draw the Line A F, pa-
rallel to CB, extending it
D till it cut the Side DC in
F
C
the Point F. Laſtly, from
the Point B draw the Line
BF, confticuting the Triangle F BD, which ſhall be equal to the Trapezia A BDC.
And ſo I have concluded what I did intend of Geometrical Problems : Neither had
I gone ſo far as I have, in regard Mr. William Leybourn hath ingeniouſly and very
fully demonſtrated them in his Firſt Book of his Compleat Surveyer. But 110 Book (as
I remember) now extant of Navigation, hath the foregoing Problems ſo large. Be-
ſides, I ſhall direct (in tlic following Treatiſe) the Mariner to Survey any Plantation
or Parcel of Land very exactly and caſily, by his Sea-Compaſs.
+
The End of the Firſt Boo
1
G2
-
}
1
1
#
1
1
1
1
生
​中​,
4
:
P
生​。
5
45
THE
1
val
V
in
>
OR,
STURMY's Mathematical and Practical
ARTS.
1
1
The Second Book.
1
b.
The ARGUMENT.
Yon
On're come to ſee a Sight, the World's the Stages
Perhaps you'll ſay, 'Tis a Scar-gazing Age.
Cume out and ſee the uſe of Inſtrument,
Can Speculation field for ſuch Content ?
That
you can reſt in Learning; But the Name
Offijing Pegalus, or Swift, Gharles-Wain.
And muuld you learn. to know how bé dóth movie
About his Axis, fet at work" by Jove ?
If you would learn the Practice, rçady and then
I need not thus intreat you by my Pen,
To tread in Arts fair Steps, or gain the way;
Go on, make haſte, Delinquent, do net Stay.
Or will you ſcale Olympick Hills so high?
Be Care take faſt hold on Aſtronomy :
Then in that fáir-ſpread Canopy no Way
From thee is hid, no not Galaxia.
They that deſcend the Waters deep, do fee
Our great God's Wonders there, and what they ber
They that contemplate on the Starry Sky,
Do see the Works that he hath fram'd so high.
Then learn the Worlds Diviſion, and that Art
which I ſhall ſhew you in this Second Part.
N this Book is contained both a general and particular Deſcription, Making, and
Uſe of all the moſt neceſſary inſtruments belonging to the Art of Navigation ;
As the Mathematical Ruler, on which are theſe Scales following; viz. The
Line of Chords, Points, Leagues, Longitude, Natural Sines, Tangents
, Secants, at one
End; at the other is Dialing Scales, viz. The Art of Dialling of all forts
, reſolved
by the Chords and Gnomon Line, and Scale of Six Hours; Scale of Inclination of Me-
ridians, and two Scales inlarging Hours; Lines upon any reclining, inclining, or de-
clining, Plain without a Center, called the greater and leſſer Pole : On the other ſide
is a Line of Artificial Signs, Tangents, and Numbers; A Meridim Line, according
to Mercator's or Mr. Edward Wright's Projection ; And Tables for the making of theſe
Scales, with a Line of Longitude and Reduction, which are the Lines on the Mathea
matical
I
3
}
Paces point-blank
46
The Argument.
BOOK II.
matical Scale; alſo, A Portable moſt uſeful Travis-Scale, with a Table for to make
it, with artificial Rhombs, Points, 1 Quarters, and Tangent-Rhombs, and the making
of the Sinical Quadrant, and ſo ordered, that by the help of an Index, and Lines
thereon, it ſhall anſwer moſt of the uſeful Queſtions in Affronomy and Navigation.
Alſo the making the plain Sea-Chard, and the true Sea-Chard, and particular Chards
for any Place; with the moſt uſeful and neceſſary Semicircle, that will protract ariy
Angle, or run upon any Chard, without drawing Rhomb-lines to fill the Chard; that
fo, by help of this Inſtrument, the Chard may ſerve for many Voyages. Alſo the Ma-
king and ilſc of a Compleat infrument, made in the manner and on the back-ſide of
a Noturnal, wich 31 of the moſt uſeful and caſieſt Stars to be known in the North
and South Hemiſphere, of the firſt, fecond, and third Magnitude ; which in a Mo-
meilt, the Inſtrument being rectified, thewcth the Hour of the Night thaç any Star
comcth to the Meridian, with his Declination N. and S. Alſo a Table of thic De-
clination, Right Afcenfion, Latitude, and Longitude calculated from Tycho's Tables, re-
stified for the year 1671. On the other ſide a Nocturnal fo ordered, that it ſhall give
you the Hour of the Night by the Nurth-Star, and the brighteſt Guard, and his
bearing every Point of the Compaſs from the Pole, whereby you may take the true De-
clination; and alſo being ſo rectified, ſhoweth the Suns place in cach Sign and De-
gree in the Ecliptick every day in the year. The Making and the uſe of the Croſs-
Staff, Back-staff, Quadrant; The Making and Uſe of the ſmall Pucket-Inſtrument, on
which is contained the moſt uſeful Lines, Scales, and Proportions, that in an Inſtant
will ſhow the Diameter of any ſort of Ordnance at the Bore, and the length and
weight of the Gun, and Shot, and Powder, in Brafs or iron; and the Diameter and
Names of caclı Piece, Diameter of the show to cach Pičce, and the weight of any Iron
Skot, the Diameter being given in Inches, with the breadth and length of the Ladle;
Point of the Quadrant, which may by this Inſtrument be anſwered near enough for fo
ſhort a time, to give any reaſonable Man an anſwer to any uſeful Queſtion in the Art
of Gunnery. Allothe Deſcription of the Mariners Azimuth-Compaſs, ſo ordered that
it ſhall meaſure all kind of Grounds whatſoever, whether Wood-land or other; and
for taking of Heights and Diſtances, wheeltér acceſſible or inacceſſible : And by the
help of the aforeſaid Semicircle, to protract any Plot of a Field or Plantation whatſo-
ever, as ſoon as any Inſtrument, as the Plain Table; the Theodolit or Circumfcrenter,
with much delight and pleaſure to the Ingenious Mariner, it agrecing ſo well with
his Traviſſes at Sen. All which ſhall be thewn in the following Treatiſe in its duc
Place.
4
1
}
А
1
Book II.
06:01
cos
C
47
А
DESCRIPTION
OF
INSTRUMENTS.
*
CHAP. I.
Of Inſtruments in general,
He particular Deſcription of the ſeveral Inſtruments that have
from time to time been invented for Mathematical Practice,
would make a Treatiſe of it felf'; and in this place is not fo
neceffary to be inſiſted on cvcry of the Inventors in their
Conſtruction. To omit therefore the Deſcription and Super-
fluity of unneceſſary Inſtruments; I ſhall immediately begin
with the Deſcription of thoſe which are the Grounds and
Foundation of all the reſt, and are now the only Inſtruments
in cſtcem amongſt Navigators and Mariners at Sea, which are chiefly theſc; viz.
The Mathematical Ruler, chc Plain Scale, che Sinical Quadrant, the Plain Sea-Chard,
and die True Sca-Chard, the particular Chard, the Semicircle or Protraktor, the
Notarnal, thic Croſs-ſtaff, Back-staff, and Quadrant; the Gunter's Scale, and
the Mariner's Azimuth-Compaſs. Now as I would not confine any Man -to the
Uſe of any particular Inſtrument for all Imployments; ſo I would adviſe any Man not
to incumber himſelf with Multiplicity, ſince theſe aforeſaid are fufficient for all Occa-
fions. Theſe ſpecial Inſtruments have been largely deſcribed already by divers; As
namely, by Mr. Blundevil, Mr. Wright, Mr. Gunter, and others : but not fitted wich
Tables for the making of them, or demonſtrated ſo plain to the Capacities of Sea-
men, as they are here. Therefore in this place it will be very neceſſary to give a parti-
cular Deſcription of them, becauſe that if any Man hath a deſire to any particular In-
ftrument, he may give the better direction for the making thereof, or making of it
himſelf.
Foraſmuch as there is a continual uſe both of Scales and Chords, which are on the
Mathematical Soale, in drawing of Schemes in the Art of Navigation, and all other
forts in this Treatiſé ; Therefore we will demonſtrate the fundamental Diagram of
the Mathematical Scale, that all Mariners
may underſtand ( that have not the know-
ledge already) the making of them, which is a moſt commendable Verrue in an expera
Mariner. I could wiſh that all Maſters and Mates were able to make their own
Infruments, that if they ſhould be long at Sea, and by diſaſter break or loſe their In-
fruments; or if any in the Ship diſcovers the Practice, he may be able to make more
for himſelf and others, without the help of the Artificer's Labour, and ſupply that
This
Diagram plainly ſheveth the making of the scale of Degrees or Chords, and
Points of the Mariner's Compaſs; in a Right Line B 8, being the
Degrees, containing
in all 90; and F & is the Scale for the Points of the Compaſs, being in all 8 Points for
the part of the whole Circle.
Now for the Sines, Tangents, and Secants, you ſhall note, That the Semi-dia-
meter A B muſt be divided into a Radical Number, for the more eaſe in Calculation
as into 100, or 1000, 10000, 100000; and that by the Table of Natural Sines,
Tangents
defect by their own pains.
;
48
ba
co
I
1
o
i
40
800
to
mom
O
1
50
Tangent Line
3
ini
40
2
Poynts
2
Tangent
sployh
10
181 313 331 5681
A Deſcription of the Fundamental Book I.
Tangents, Secants, Chords, and Points, which I have fitted on purpoſe for this
Work. You may take off ſo many Nambers as the Table directs you, as ſhall be
ſhewn.
Miles
Equall
parts.
10 20 30 40'slo do zlé sloblo
70 80
fol:48.
Here followeth a Table of 90 Degrees of the Quadrant. He chat deſires it larger,
may make it to the Parts of a Degree. I have joyned the Chord proper to it, which is
the Natural Sine of half the Arch doubled.
Fot; Example, :If you double the Natural Line of 6.15.25. 30 Deg. you ſhall
produce the chords of 12. 30.45.60 Degrees; thus 10453 is the Sine of 6 Degrees,
being doubled, the Sum will be 20906 the Chord of 12 Degrees; and ſo of the reſt,
as in the Table following.
The Table of Degrees and Chords.
De Chord De Choral D Chord 1 DelChord De Chord De Chora
17 16 278 311 5341 46 7811 101 10151701231
25 17 290 32 5511477971 162 1039 771245
03 52
04 7019 330 34 585 49 830 641060791373
87 20 347 351 601 150 845 65 1074 801 286
06 105 21 364 36 61851 861 66 1089 811299
07 132 22 382
22 382 1371 6351 152 870 167 1104 821312
os 139 23 398 138] 651 153 892 1681118 831325
09 151 24' 416 139 668 154 908 169 11331 184.1338
10 1751 251 432 406841155923170 1147 85/1351
II 19226450 141 700 56 9391711161' 861364
12 2091 271 466 42 717 57 954 721176 8711377
13 228 28 4841 1431 733158 9701731190 881389
14' 244) 29 Soil 144 749 59 9841741204 89.1402
15 2611301 581.145176 60 roooi 175 1217 Jool1414
F
A
O
so
30
610
50
D
1
OI
02
a
!
This
܊ ܐ
Book II. Diagram of Scales and Tables, .
8
26 15
49
This done, Proportion the Radius of a Circle co what extent you pleaſe; make A B
equal
. thereto, which muſt be divided into equal Parts, as before-directed, by half
thereof, and this Table, the Chord of any Arch proportionable to this Radius, may
ſpeedily be obtained. As for Example, Ler thicre be required the Chord of Thirty De
grces, the Number in the Table is si8; or in proportion to this Scale of 100 cqual
Parts, A B is 52 almoſt; I cake chcrefore 52 from the Scale of equal Parts, and ſet
them from B towards 8 to hand o, and draw chc Line ho, which is the Chord delired
30 Degrees : Thus may you find the Churd of any other Arch agrccable to this Radira.
Or if your Radius be of a greater or leſſer extent, if you make the baſe of your Right
Angle A B equal thereunto, You may in like manner find the Cherd of any arch,
agreeable to any Radius given. Only remember, That if the Chord of the Arch deſi-
red exceed 60 Deg. A B which is divided into ioo equal parts, you muſt continue thic
Baſe A B in the diviſion of ſuch parts, as nccd ihall require.
In this manner is made the Line of Chords in the Fundamental Diagram anſwerable
to cliat Radius.
And in this manner you may find the Cherd of the Rhomb, Pointszhalfs,and quarters,
and the Sines, Tangents, and Secants of any Arch proportionable to any Radius, by
help of theſe Tables following (which is an abbreviation of the Canon of Natural Sines,
Tangents, and Secants] and proportioning the Baſe A B thereunto, which is the Scale
of equal pares; as by Example may more plainly appear.
A Table for the Angles which every Rhomb makroh, with the Meri-
dian, and the Chords of every Quarter and Point of the Compaſs.
North, South. deg. mi. ſec. Chor South.
North.
2 48 45 49
5 37 30 98
15 147
N. b. E. s. b. E.
IS
00 1955. b. W. N. b. W.
4.3 451 244
52 30 29
1941 151 333
N.N.E.S.S.E.
22 30 oo 39 S.S. W. N. N. W. 2
25 18 45 427
28
30 56 15 533
N.E.b.N.S.E.B.S. 33
33 45
od 580 5. W.b. S./V.W.6.N.2.
33 451 627
39 22
39 673
42 is 720
N. E. s. E.
45 00 oo 76715. W. N. W. 4
47 48 45) Ön
50 37 30 855
53
26 151 899
N.E.b.E.S. E. b. F.56 15
00 942)S.W.b.W. N.W.6.Ws
59 3
451 985
61
52
64 41 151069
E.N.E.E.S.F. 67
30 001111 W.S.W.W.N.W. 6
70 18 451151
73 7 301190
75 56 151230
001268w. b. S. W.b. N.
7
81
33 45, 1305
301 343
11 151378
Eaſt. Eaſt.
90 001414 wcft. Welt. i 8
H
Les
.)
T
16
7 30 485
36 33
II
30.1028
+
E. b. N. E. b.S. 78 45
184
2.
1
87
O
1
.
50
The Tables for the Making the Book II.
}
Let there be required the Chord of the firſt Point of the Scale, 11 Deg. 15, in this
Table, as I have ficted for every Point, Half, and Quarter, for of the Compaſs.
The Numbers anſwering to 11 Deg. 15 Min. is 195. I take therefore with my
Compaſſes 19, or reckon ſo many on the scale of Equal parts, which is joyned with a
Scale intended to be made; and ſo with a Square for that purpoſe, as thall be (hewed,
mark from F towards 8 the firſt Point 11 Deg. 15, where the Redius of the Circle is
AB; and ſo of the reſt.
The Scale of Longitude.
TH
His Scale is made allo by the Table of Degrees and Chords, as beforc.
E X A M P L E. ·
It is required to know how many Miles make a Degree in the Parallel of 10 Deg.
If
you cxtcnd thic Compaſſes. from A, to the Complement of the Latitude 80 Deg. in
the Line of Sines, and ſetting one Foot in F, turn chat diſtance from F toward A,
you will find it reach 59 Miles scarcſt, in the former Diagram.
Another EXAMPLE:
It is required in thic' Latitude of 60 Degrees to know the Miles anſwering to a De-
grec. In chat Parallel cxtend the Compaſſes from A to the Complement of the Latitude
3), in the Line of Sines; and ſetting one foot of the Compaſſes in F, turn that di-
ftance towards A, and you will find it rcach 30 Miles, that makes a Degree in that
Parallel; and ſo of the reſt.
But if it be required how to make a Scale of Longitude in Miles anſwerable to the
Radius of the fame Scheme, for the Parallei of 10 Degrees, you will find in the Ta-
lle, thc Chord for so Degrees is 17.5 for the firſt Mile, and for 60 Degrees 1000,take
Ico from A to B, as yol was before-clirected, and ſo do with the reft, until you
have made the whole Scalc. Remember, chat 60 Miles muſt begin where the firft
Degree of the Chords aloci on the Scale, and ſo diminiſh towards the Pole 90 Degrees
of the Scale, as rcaſon will give you,
1
2
SIN E S.
N
Ore, That a
Sine falls al-
A Table of Natural Sines to the Radius of 1000.
ways within the
Qandrant of a De Sines. De Sines. TD: Sines. De Sines. De Sines. |De Sines.
Circle, as CD, 17 16 275|311 515 40 719 61 874 70 970
whicli is the Sine
of the Arch BC
34 171 292 32 529147 731 62 8881 771 974
3
52181.309 33 544 48 7431 163 891 179 978
60 Degrees; and
4 69 19 325 34 5591 491 754 64 898 179 981
by thc Table of
5 871 20 343 35 573 50 766 65 906 80 984
Natural Sines, to
6
1041 21
21 358 36 5871 151 777 160 913 81 987
7 134 22 374 137 603|152 788 67 9201 82 990
8 139 231 390138 015153 798 68 927 834 992
I have fitted
9 150 24' 400 351 629 54 809169 933 84 994
for this purpoſc, 10 -1781251 422 40 642 155 819 70 939 85, 996
whoſc Radins is II 1901 126 438 41 656|156 829 71 945 ' 997
1000, you thall 12 207271 4531 42 669 57 838 72 951 871 998
find the Sine of 13 224 28 469 43 682 58 848 731 956 88 999
60 deg.to be 86.6. 14 241 291 4841 44 694 59 857,174 961 99 999
I take therefore 15 2581'c sool 1451 7071 160! 866 75 905 19011000
with iny Compaſes
86 from my scale of Equal Parts, and ſee them froin A towards 8 in the Line of
Sines for 60 Degrees, where the Radics of the Circle is A B, and CE is the Comple-
ment thercof, or Sine of 30 Degrees of the Arch CS, the Number in the Table an-
fwering 30 Degrees is 500; take therefore with your Compalles so equal Parts of A
B, and lay it from A upon the Line of Sines for 30 towards 8 ; and ſo of the reſty
TAN
every Degr. of the
Quadrant which
F
*
**
BOOK II. Chords, Rhombs, Longitudes, &c.
51
T A‘N GENT S.
A
De Tan.
I
2
37321
Tangent Line is
A Table of Natural Tangents to every Degree of the
always falling
Quadrant.
without the Quadrant,
and is drawn at the end
De Tan. De Tan. | Del Tan. Di Tangents.
of a Semidiameter at
17 19 344 371 7531155/1428 73 3270
Right Angles, as B6
34 20 363 38 7811 501482/ 174 3487! in the Fundamental Di-
3 52 21 383 39 809 57115591 75
agrani,
which is the
4 691.221 404 40 839 58116001 176
40101 Tangent of the Arch
s 87 23 424 41 869 59 1664 177 4331
BC 60 Degrees, as in
6 105 24 445 421 900 160 1732 178 4704
the Table of Tangents
7 122 25 456 43 932 61 1804179 5144 you ſhall find it to be
80
81 140 26 487 44 905 6211880 5671 1732 cqual parts,
9 15 27 509 45 1000 63 1962 181 6313 which take with your
10 170 128 5311 46 1035 64 2650 82
711S Compaffes from
from A,
II 194 295541 47 10726521441 83
whcı
you
lave conci-
12 212 30 577 481111066122401184 9514 nued thc Line beyond
13.230 131 600 4911150 16713355 11430 B, take 173 parrs, and
14 249 32 6241501191 6812475 86 14300
that will reach from B
IS 367 33 649 S1|1234.69 2601 87 19081 to G, the Tangent of
16 286 34 674 152 1279 70127471 188 28336 60 Degrees in the
17 305 35 700
35 700 15:1327 51 2904 1891 57289 Scale, and 8 H is the
18 3241 36 720 541370723177 90 Coocooo Complement Tangent 30
Infinite. Degrees 577 parts;
cherefore takc 57-pares,
ic will rcách from B to the length of 30 Degrees ; 'and Co. of che iota ...
8144
85
I SE C A N T.
A
7185!
Secant Line is
A Table of Secants to every Degree of the Quadrant.
drawn always
from the Center of the
Del. Sec. Del Sec. Del Scc. De Sec. Del Secants.
Circle, until it cut thic
I 1000
191057 381269561788 74 3627 Tangent Line ; as A
21000 2010641 39 1286 157 1836|175
3863 G in the foregoing
311001121110711 40113051 1581887/ 176
4133
Diagram cuts the
410021 122 1078 4113251 59 1941 177
771
4445 Tangent of the Arch
51003 23 1086 142134516020001 78 4809 B C 60 Degrees in G:
61005
24 1094 43 1367|61 2062 179 52401 fo is A G thic Secant
71007 251103 1441139011622130 180 5758 of 60 Degrees, which
811009112611112 4514141631220281 6392 in this Table of Se-
9 TOI227 1122
27 1122 46 143916412281 82
cants is found 2000
1010152811132 471466165236683 8205
equal parts; therefore
1111018 29'1434814941 166 2458 84 9566 take off ſuch
akc oft ſuch parts as
0111541 149 15241 67125591 185 1473 are in proportion to
131026 131066 150 1555
5011555 68 2669 186 14335 A B 200, it fall
141030321179 157 158269 2790 189
19107 reach from A to G
151 1035 33 1192 15216241 70292383
28653 for the Secant of 60
161040 341206, 531661 4130711 89-57208) Degrees, and A H
17110451 35 1220
54 1701 1723236100 000000
18.1051
is the complemeni-Se-
361228
Infinite.
551743733420
371252
pos
cant, or Secant of the
Arch 8C,30 Degreos,
which in this Table of Secants is found to be 1154; therefore take with your Compal-
ſes, or other Inſtruments, 115 equal parts, and it ſhall reach from A towards Gº for
ché Secant of 30 Degrees, as you may find by the Scale in the Diagram.
H 2
Morfeld
I21022
52 A Deſcription of what Inſtruments Book II
A
Verſed Sines.
Verſed Sine is found by ſubftracting his complement-Sine out of the Redimu.
Example. For to know the Verſed Sine of 60 Degrees, you muſt ſubſtract EC
or A D, which is the Complement or Sine of 30 Degrees, viz. soo out of the Redins
1000, or Sine of 90, A B, the remain will be DB soo, for the Verſed Sine of the
Arch B C 60 Degrees. In like manner F 8 will be found 134 for the Verfed Sine
of the Arch C 8, being 30 Degrees; and ſo work in like manner for any other De
gree. The Word Perſed is a fufficient Direction, to let them underſtand, that do
not, That the Degrees of this Scale, or ſort of reckoning, begins at B or F, and con-
tinues to 180 Degrees, the Diameter of the Circle; or the Line of Sines Reverſed, by
putting the two beginnings of Degrees together of the Quadrant or Scale, and lobe
gin to count at cue End; for 80 Degrees muſt be placed 10, for 70 Degrees 20 Deg.
and fo to 180 ; and of the firſt 90 or middle of the Scale, count the Sun's greatest
Declination 23 Degrees 30 Min. towards both ends, that is
, 47 Degrees alurideti in
that diſtance; by the ſide thereof muſt be placed the Reverſed lix Northern Signes; ac-
cording to the Sun's Declination, and place in the Ecliptick at ſuch Declination : And
likewiſe 23 Degrees 30 Min. the ſpace for dividing the Reverſed Southern Sigres'-ad-
ward 180; and are reckoned double, as occaſion requireth.
Either of the Semidiameters A Bor A F, the sides of the Quadrant, you may take
the equal diviſions thercof, and make a Scale of Leagues or Miles, or Equal Parts,
for the demonſtration of all plain Triangles, which you cannot be without, having it
ز
upon the Ruler.
CHAP. II.
u Deſcription of what Inſtruments of Braſs, Steel, Iron, ard Wood
you muſt be provided with before you can make Inſtruments for Mathe-
matical ulcs.
B
Efore we explain the other half of the Fundamental Diagram and Semicircle,
it will be neceſſary for to give a Deſcription of what Inſtruments in Braſsi
Steel,, Iron, or Wood, you muſt have by you in readineſs, before you can
make a Mathematical Inſtrument; Thac Men that are ingenuous may be provided
in ſome meaſure with ſuch, before they go to Sea, in ſpending their ſpare-time on
this Practice. In brief, they are theſe. Firſt, For Inſtruments of wood, you' muſt
Scales of
be provided with ſeveral scales of Equal Parts, of ſeveral lengths, which muſt be
equal parts exactly and carefully divided, the length you intend to make the Radius of the In-
ſtruments by it. Firſt, divide this Line into 10 cqual parts, and each 10 into 10
more; ſo is your Line divided into 100; and ſo you may continue it into 200,
300, 400, ſo much as you
pleaſe, as the Inſtrument
you are making will re-
quirc;which you may quick-
ly ſee by the Table. You
Will
VIIMTIWA
muſt be firred with ſome
100 gog
20.10
ina
Scales of pieces of Box (dry, clean
Box of 6 from Knots, ſtraight, and
Wher lood. ſmooch planed ) or other
I Food, on which you may
When
inake what Scale you pleaſe.
You muſt have by you a true
2 Squares. Sgenre of Rrafs and Wood,
ſuch as you may ſee in this
2 Cramps. Figure with a pair of Cramps
made of Iron, with Scrers to
faften the scale of Equal
when
manninnian
UND
TUTTO
V
Parts,
!!
on Deal
ner.
BOOK II. or Tools muſt be provided. 53
Parts, and the Scale to be made together, ſo as they may not flip, whereby may
be
made no miſtake in Graduating. Or for ſmall Scales, you may faſten the scale of Scales
Equal Parts, and the Scale to be inade by it, on a piece of Deal Board, with the Heads saftened
with Nails
of Scuper Nails, ſo as they will not ſtir; but for greater Inſtruments, and Croſs-ſtaves,
and Gauging-flaves, you muſt do by them as in this Figure. You muſt have a Gauge Boards.
made of Brals, with a good Steel Pin, for the drawing of ſtraight Lines on your Bials a
Scale, for the diviſion of the Columnes for Graduation. You inuſt have two or three Gange.
Sorts and Sets of Steel Letters and Figures, and Figures for Ornament, with a neac Steel Let-
Hammer to uſe with them: And the Figures and Letters and Ornament-Figures, fet ters and
in an Alphabet-Box:, with written Letters and Figures before them, for the ready find- Figures.
ing of them; wich Characters of the Signes, and Planets, and Stars, in like man-
The Inſtrument that you graduate with, the Edge muſt be very thin and ſharp, Gauging
and you may have ſeveral of them; or the end of a Pen-knife may do for a ſhift. Inftrx-
You muſt have a Braſs pair of Compaſſes to go with an Arch and Screws; to faſten at Braſs come
any diftance; and four Steel Points to take in and out; two long Points for to rcach a paſles, to-
great diſtance. I have a pair by me will extend 3 Foot,; on a large scale of Artificial gether with
Sines, Tangents, and Numbers, they are to be uſed. The other are thort Points. One an Arch &
is to be made round for a Center-Point, that it may not go too far into the Wood; and 4 Points,
the other pointed like a Dutch Knife, and the Shoulder fitted ſquare as che other
Points, to be faſtned, and taken in and out at pleaſurc. The Uſe of theſe Points is to be taken
to draw Circles on round Inſtruments, as Notturnals
, and the like. You may have in and out
two pair of Dividers, the leaſt 3 Inches and, and the biggeſt 7.5 Inches long. I hold
A pair of
them beſt chat go with a Bow at the Head, and to be ſet together by a Screw in che Dividers.
midſt. Be ſure they be made of good Steel. Theſe are to divide equal Parts, and
any ceher equal Diviſion. You muſt have for great Inſtruments, as Bows, Quadrants,
and the like, a pair of Beam-Compalles, for to ſweep the Arches of them. You Beam-com-
- Thould have a Hand-Vice, ſo made as to ſcrew into the edge of a Board for your uſe, paſſes,
and to take out again with three or four forts of ſmall Files, for to file and make Hand-vice.
Pins, which you will have occaſion for. Theſe Braſs and Iron Inftruments or Scales
ġou inay now give direction to an ingenuous Smith (Thomas Moore) in Briſtol (if you where they
cannot liave them before of theſe forts) and he will fit you with them : Or you may may be
have them ready made of Walter Hayes, at the Croſs-Daggers in Moor-Fields, wich made or
many uſeful instruments in Braſs; Or of Andrew Wakely, Mathematician, at his bought.
Shop on Redriff-Wall, near the Cherry-Garden Stairs; Or in Briſtal of Philip Staynred,
Math. And now I have thewn the Practitioner what Inſtruments he muſt be fur-
niſhed with, I will return to the Explanation of the other half of the Fundamental
Diagram of the Mathematical Ræler. I had almoſt forgot a Receit for ſetting off che
Graduation, when it is newly done on Box-Inſtruments, which is this. Take Cbarcoal, To ſet off
and beat it to a fine Powder, and temper it with Linſeed-031; and let it be rubb'd on the Laftrise
the Inſtrument newly made, and lie fo on it for a time, untill it be pretty dry; and ment,
then with ſome Sellet-Oyl rub the Inſtrument, and make it clean: So will you have
che Gradaation and Figures ſet off very neatly on Box Inſtruments, with Blaćk.
The uſe.
Filos.
Tbe
1
The Figure of the Foure Poyntcd. Compafés
The
Short
Doints
M
1
LTD
The Long Points
6
De
L
"
IIII.
Green
F
1
The figuer
of the Zivide
The Figure ofÝ Gaudge
f
sphelin
Kalousell
2
ܕܳܕ݂
Book II,
Of the Dialling-Scale.
55
CHAP. III.
The Explanation of the other half of the former Semicircle; being a De
fcription of the Fundamental Diagram, of the Dialling-Scale on the Ma-
thematical Ruler.
T
"His annexed Diagram ſheweth plainly the Deſcription of the Dialling-Scales
on the Mathematical Ruler; It being the moſt caſic and exact Inſtrument
uſed in that Art, as by the uſe will be manifeſt in the Seventh Books
IH
3
C
80
90
O
::o
1
6lo
+
510
40
Sines
D
os
80
bo'slo 40
oc
***
60
O 이
​gla
Gnomen Line
4050
Chords
10
04
10
---
Inclination
of Aleridians
2/0
370 40
50
610
BI
A
5
21 3R
Hour Line
fol:55 and
ME
1
4
!
How to make the Diagrain.
FM
Irft, Make a Semicircle by a leſs Radius, as A D B, and upon the midſt of the
Arch at D, with the diſtance D A deſcribe the Quadrant- Arch, as A EB,
which muſt be divided into fix equal Parts, for the Hours in the of the Sphores
which is ſufficient to reſolve the whole; and from cach Point draw Lines to the Center
ar D; So will it cut the Line A B in 1, 2, 3, 4, 5, 6, for the Hour-Lines
upon
the ſaid Scale for Dialing: and thus you ſee it is a Tangent Line, for which uſe it is
more certainly done by this Table of Naturel Tangents for three Hours, if
you
do but
obſerve where the Right Line D E cuts the Tangent Line A B, which you ſee in the
middle or Center of the Semicircle at R; therefore you muſt begin to make this Scale
in the midſt, and lay the diſtance of parts anſwering the Hoxers both ways from R to-
wards
1
15.6
Book 1
The Tables for the Making
IO
2
20
5
00
00
10117 30
+
20 20
3022
1
oo
00
40/40 or
wards B and A: As by Example, To Graduare 2
Hours and 4 Hours, you ſee in the Table, che Wisimbers
A Table for the dividing of
tbe Hours and Minutes
anſwering to 2 Hours and 4 Hours in the first Column
so cheletc hand; is in the ſecond: 60 Minutes, or is the
upon the Dialling-Scales
shirdi. 15 Degrees; and fix the fourth, Column the Tann : HO.MD. M. l'an.par. .
gent-parts 267; therefore if you take 267 fuch Parts
3
30 43
whereof the Semidiameter R B is divided into 1000, as
87
was ihowed in the former Diagram, and put one Foer
01 7 30
131
of thie Compaſſes with that extent ac R che midſt or 3
40110 00 1761
Hours, and turn the other toward B, it will make the
5012 30 221
diſtance of 4 Hours; and turn chat diſtance towards A,
2 4160115
267
it will be 2 Hours of the Scale: And ſo do with the
rcſt of clie Hours, and diſtance of the Minutes.
315
20 00
In like' manier for the Scale of Inclination of Meria
363
30 414
dians, you miift-cakë put the Tangent-parts out of the
40125 0
466
Table of Tangents to every Degree, and graduate in the
ſame manner as before, from the Center which is the
5027 30 520
I 511030
577
midft of the Scale 45 Degrees, as is hewn plain in the
Diagr.m.
0132 30
637
For the Gnimen-Line, as others call it the Line of
20135
700
Latitud:, Let B A be the Semidiameter; fo on B de 301?7 30 767
fcribe the Quadrant ABC, whoſe Arch AC divide
839
into 90 Degrees, from whence you may project the Line 50142 30 916
of Sines BC.
10
16014? Of
1000
Now from each. Drgree of thoſe Sines, draw Lines
toward the Center of thein at A, and note where they cut the Arch of the Quadrant
BD: Thien from B as a Center, take the diſtance of cach of theſe Intersections, and
lay them on the Line B D; ſo ſhall you liave che Diviſion of the Gnonson-Line, or
Line of Latigde
For the more ready
A Table of Latitudes for Dialling.
making of this Scale,
here is a Table of Láti.
tudes calculated to the
90 Degrees of che. Quan
80992160 926 45 817130 63205 354
drant, and the way to
78 989 59 920 44.80729 617 14 332
calculate': it your ſelf.
76, 985158 915 43 797 281 60113, 310
As for Eximple, To
74 98057 909 42-787 27/ 585 12 288
find the Latitude parts
72 975 56 903.411 77626 568 11 265
for 30 Degrees of Lati. 190 1000 70 969155 896 40 765251 55910 242
tude,
39 69 965154 890 391 753124) 533 9 219
Firſt, Find the Sine 88 68 96253 88338 741 23 515 8 195
thereof in the Natural 87 671 958152 8751371 729 22 496 71 171
36
Table of Sines , which
66 954151 868 36 717 21 477 6 147
will be found to be 85 998 65 950 50 86035 704 201 4581 5 123
Joooo; which ſought 84
64 945 49 852134 690 19 438 41 98
for in the Table of 183 63 241 48 844 33 676 18 419 3
82 62 936 47 835 32 662 171 3972
Tangents, giveth an
49
Arch of 26 Deg. 34
81 611 9311461 8261311 648116 376 il
Min. Then the Pro-
portion will hold, As the Radius.
-100000
To the Secant 45 Deg-
141421
So is the Sine of 26 Deg. 34 Min.-- 44724
Unto the Latitude-parts-
Whici anſwers to the Radius 100000 : But in iny Table the Parts 632 anſwer to the
Radius 1000, which will be ſufficient for the Graduating the Line of Gasmons or
Latitude.
But obſerve, To make 30 Degrees of Latitude on your Scale, you muſt take off
632
1
Degalo
Paris
D:81
74
25
63249
!
of the Dialling-Scale.
57
I
21
5
Io
871
25 6
fame, as you will find in the
Book II.
57
632 ſuch Parts as the Line is divided into 100, or 1000, as you have been ſhewn in
the former Diagram.
How to inake the Line of Chords, you have been fully inſtructed already in the for-
mer Figure; which is only by dividing the Arch of the Quadrant A D inco 90 equal
parts; And from A as a Center, take the diſtance, and lay thein down in a ſtraighé
Line AD: So ſhall you have the Line of Chords or Sublemes. Or you may do ic by
the Table of Chords, as before-directed.
How to make chie two Lines or Scales of Inlarging Hour-Lines upon any reclining
Plain, wichout a Center, called by me the greater and the leſſer Pole.
Invexed, you have a Table ready fitted for the making thereof.
Firſt, You inuſt make choice of the length of this Scale, that is in Proportion to
the former Lines of the Scale.
The firſt 3 Hours muſt be di-
vided into 10 parts, and each of
A Table of Tangents for 5 Ho. to every 5 Min. of them into no more, which ſtånd
an Hour, for inlarging the Hour-Line Scale.
for 100, or as you have been
thewd for Icoo. You muſt have
Hours. Mi.Deg. Min l'an.pa Hours Mi Deg. Min. Tanpa two of thele Lines of Equal
IS
546
15 1044 parts
parts , of two proportionable
2 30 43 10 47 30 1091 Lengtlis, for the greater and lef-
15 3 45165 1548 45 1140 fer Pole; And ſo take of the
2015 00
20150 0011191 Tangent-parts anſwerable to eve-
151 100
25 51 Is 1245
ry 5 Minutes of an Hosr: As
301 7 301 131 30152 30 1303 you ſee the firſt and ſecond Co-
35 8 451 153 35153 451363
451363 lumns of the Table are Hours and
40110 001 1761 40155 0011428 Minutes, the third Degrees, and
45II IS 198
45 156 1511496 the fourth Tangent-parts.
So
50I2
30 22)
50 57 30 1569
30 1969 the Tangent of the firſt 2 Hours
5513 451 2441
55 58 45 1647
451647 of the Scale or 30 Degrees, is
oo 267; 4 60160 oo 1732 577 Parts ; take of your two
IS! 291
1822
Scales 57 Parts; Firſt of the
10 17 301 315
10162
30 1920 largeſt Radius for 2 Hours
2 on the
15 18 451 339 15163 45 2027 greater Scale, and the like num-
2012000 363 20165 0012044
002044! ber of the ſmaller Radiw, or
2521 15:388
25166 1512272
152272 Line of Equal Parts for 2 Hours.
30 22 30 414
30167 302414
of the lefler Pole-Scale. And ſo
3523 451 440 35 68 45 2571
52571 in the ſame manner, you muſt
40125 00 466
40170
00:2747
work to finiſh the whole Scales.
4526 15.493
of what Radiss you pleaſe, by
50127 301 520
theſe Tables, as hachbecn di-
55 28 45
451 548
5
55173 45 3430
45 3430 rected.
ooi 577
003732 The Ulle liereof is fully ſhewsa
in the Seventh Book, 29th and.
301 637
30 Chap. of the Art of Dialling.
33 45 668
Theíc Scales are ſufficient to
20135
700
make any ſort of Dials, in
any
25136 15
151 733
Latitude (as is there ſhewn) with
3037 301 767
caſc and cxactneſs.
35 38 45 802
There are two Lines called by
oo 839
the Names of Style and Subſtyle-
Scale; but is only for this Lati-
tude, but may be found for any,
30 916
.
5543
But the Scales before-explained
451 957
3 160'45 oo' 1000
are moſt uſeful, and do the
Seventh Book and Twelfth Chapter of the Art of Dialling. And theſe are the Scales
of one side of the Ruler.
I
CHAP
1
1
1 60115
5/16
5161
15
15]
457
5072
IS12945
3013171
hi
60175
2 16030
5131
10132
is 606
j
40140
4541
50142
15 876
]
។ 1
1
58
A Deſcription of the other Side Book II,
1
H Н
1
Itance will reach from 10 in
manner do with the reſt; for
CHAP. IV.
The Scales or Lines on the Back-ſide of the Mathematical Ruler, are theſe:
A Line of Numbers, A Line of Artificial Tangents, A Line of Sines,
A Meridian Line according to Mercator's or Mr. Wright's Projection;
and the Scale of Equal Parts, by which the Numbers were taken off for
the Graduating theſe Scales; and a Line of Longitude or Equinoctial,
with a Scale of Reductior., as followeth.
I. "Ow to divide the Line of Numbers is thus. You muſt prepare a Ruler of
what length you pleaſe, and alſo a Scale of Equal Parts, divided into
2
100 or 1000 : You muſt count them. But if you divide the Artifici-
al Tangents and Sines with the
Line of Number, you were beſt
to divide the Line into 2030 A Table for the Diviſion of the Line of
Parts; ſo will you have 100
Artificial Numbers.
on the Line of Numbers. This
Table is taken out of the Lo-
garithms, by rejecting the In-
dex or firſt Figure. It is beſt to
omit the firſt Number, by rca-
0021 322411 61261 785 811 908
fon they will take up ſo much
30122 3421425623162 7921 821 913.
room; and begin at 1 or 11,
31 47 23 361 431 63363) 799 831 919
and take the Logarithm-part at
4 60 241 380 44 643 64 806 841.934
41 for the firft ioth or Integer.
5
69.25 397 4565365! 813 85929
But if you intend to make 100
77126 414 46 662166 819 8611934)
on your Line of Numbers, firſt
84/27 431147) 672167) 826 871.939
take 100, which is réckoned
9028 447 48 681 68 832 88 944
1000, as you ſee in 'the forca 95 291 462 49 690 691 838 89 949
going Table, Parts of the Scale 100 30 47750698708451 901954
of Equal Parts, for the firſt 41 311 4911517071711 8511 911950
10 or middle of the Scale: 1932 505 52 71672 857 92 1963
Then ſuppoſe you were to
13 113 33 518531 72473 863 93 968
make the fift 2 or 20, take
14! 14634! 5311541 7321741 869 94 973
with your Inſtrument or com 15! 170135 54415517401751 875 95 977
paſs 301 equal Parts, and lay 16 20436 556 56 74870 880190982
it froin I to 2, and the ſame di 171 230 37 568 57 755177 886 97 986
18 255381 579158763 78 8921 98 99
the middle to 20. In the like 191 2781 391 5911591 770 791 897) 991 995
20 30140 60260 778180 903/100/1000
3 or 30 the cqual parts is 477,
and for 4 or 40, the Log. parts is 602: So you may eaſily perceive haw to do it, by
what is written.
Num.
(parts.
Num
parts.
Log.
Num
parts.
Lag.
Num.
parts:
Log.
Num.lo
parts.
Log.
2.
i
!
6
9
II
I 2
20
)
1
II. Hon
$
>
G
+
BOOK II.
of the Scale of Scales.
59
II. How to make the Line of Artificial Tangents on the
Ruler.
THE
*Hc Artificial Tangents are made in the ſame
manner as before directed, beginning upon
a Right Line of Numliers, omitting the firſt 30
Minutes, and beginning at 40 Minutes. The
Tangent-parts are 106, taken off the former
Scale , and applied as beforc-directed uptvard,
will make 40 Minutes on your Scale: So the
firſt and 89 Degree, the Tangent-part anſwer-
ing thereunto is 241; with them do in like
manner, and ſo of the reſt, until you have fi-
niſhed the wholc Line or Scale, as you may ſee
in the Figure.
1
1
+
1
2
i
+
The following Table is ſo plain to be undera
ſtood, that I need write no more, buc, Thac che
firft Column to the left hand is Minutes, The fe-
cond Tangent-parts anſwering to the Minutes and
Degrees over cach Column to 30 Degrees, and af-
ter to every 20 Minutes, as you may ſee in the
Table
:
2
:
1
1
T
W!
1
t
1
1
1
;
.
+
7
1
t
I 2
Tablo
1
7
The Tables for the Making the Book II.
Minutes.
851 84
81
80
o
I
2
10
1
. . . . . : .
60
III. A Table for the Diviſion of the Line of Artificial Tangents to
45 Deg. and the Minutes fit to be ſet thereon.
Deg. Dig: Pego Deg. Deg. Deg. Dig. Dug. D-g. Deg. Dil. Dep.
Tang. Tang. Tang. Tang. Tang. Trang Trang. 1/ang. Tang. aag. Tang.
Tag: 1 dig.
is.
parts. parts. parts. parts. paris. Paris. parts paris. parts. parts. parts,
89 88 87
!
86
84 83
82
79
6
13
8
4 5
7
୨
co 241 546 719 844 941 1021 1089 1147 1199 12461288
IO 46 308 577 742 862) 956 10331099 1150 1207 12531295
20 76 366/ 610 765 879 97010451100116511215126011301
301 96 4101 6401 786 895) 9831105611191117412231126711308
40 100 4631 668 800 911 996 106711291183123112741314
50 162 595 694 826 227 10090781139 1191 1238 1281 1321
70
12 | 13 | 14
| 22 | 23
013271363 1395 142811457 148515111536156111584 16061627
1013331369 1402 143311462 148915161541|1564158711610 1631
:013391 3741407 143814661494) 1230154515681591101311634
301345 1380 1412 144211471 14981152415491572 159511617|1638
401351 1385 1417 1447 14761503 15281553 1576 1599 1620 1641
5013571191_14221452 1480 150715321557 1580 16031624 1645
601
24
25 | 26 | 27 28 29 130
35
0164816687688 1707 1725 174 1761 1738 17951812 1828 1845
10 1651 1671 16911710 1728 1746
2011655 1675 1694 171357341749 176711784 1801181818341850
30 1658 167811697j17161735-1752
40 166216811700 1719 1737|175517731790 1806 18231839 1855
501665 1684 1704 1722 1742 1758
|
SOM
45
36 37
38
41
43 44 45
0 1861 1877 1892 1908 1923 1939 1954 1969 1984/2000
201866 1882 1898 1913 1928|1944 1959 1974 1989
401187111887 1963 19181934' 1949 1964 1979|1994
Min.
IS
16 | 17
18
19 120
21
1
Min |
31
32 33
24
39 40
42
1
IV. How to make the Scale or Line of Artificial Sines 80 90 Deg. and
Minutes fit to be ſet thereon.
Ow to make this Line, was ſhewn by making the laſt; only that is ſufficient to
HOME
45 Degrees, and this muſt be to 90 Degrees
. And if your Line of Equal
Parts be divided into 100, or as they be reckoned 1000, you may omic clie laſt Fi-
gure
of the Number : But if you number the Scale to 2000, as the Tables are made
to, if
you
would fer off the Sine of 30 Degrees, the Parts anſwering chereunto is
1698; therefore take off your Scale of Equal Parts with your compaſſes 169, and
it will reach from the beginning, to 30 Degrees on the Line of Sines.
So I hope you underſtand how to do the reſt, it being made fo plain and caſie for
che mcanert Capacity, by what hath been writ already.
i Table
::
*
ܟܪܟ
ya
Book II. Artificial Lines on the Scale.
61
7
4 Table for the Diviſion of the Artificial Sines on the Ruler.
Minutes. 10
Deg. Deg. Drg. Dig. Deg. Deg. Dog. Deg. Deg: Deg. Deg.
Sine
Sine
Sine Siat Sine
Sine
Sine
Sine Sine Sine Sine
parts. parts. Parts. parts. parts. parts. I paris. parts. I paris, parts. parts.
I
3 4 5
6
7
8 9 IO
O
2
IO
II
I2
16
20
21
| 26
22
1
1
00241 542 718 843 940 1019 10851143 1194 1239
46 308 5771 743) 361 95410311096 1152 1202 1 246
201 76 366 609 7641 8781 9681042 110511611209 1253
30* 94 4171 639 7851 894 9811053111151169 1317 1260
40 463 667 805 910 994 106411251178 12351267
50 362 505 697| 825 925/1007 1075 1134 11861232 1 274
13 14 15
17 18 19
0 1280 13171352138311412 1440 1465 1489151215341554
10 1287 13231357 188 1417 1444 1470 1493151615321557
2012931329 13621393|1422 1449 1474 14971519 1540 1560
30129991335 136811398,1426 14531478115011523-15441564
4013051340 1373 14031431 1457 1482 1505 1527 1547 1567
5013121346.1378 1408 435 1461 1486 1508 1530 1551 1570
23 | 24
1
25
27
28
29
30 31 32
01573.15911609|1625|16411657 1677|1685|1698 171111724
10 15761594161216281644116591673 1687
201579 1597 1614 163, 1646 16611670 1690 170311716 1728
30 1582 1600 1617 1633-1649 1664 1678 1692
40 15851603162616361652|16661680/16941707|1720 1732
5011588160511623116391654 1669 1683 1696
-33 34 35 36 37
38
39
40 41 42 43
5 1736 1747|1758 1769 1779 1789 179818081816 1825| 18 33
20 1739 1751 1762 1772 1782 1792 1801
401743 1754 1765 1706 1786 179511805
|
44
45 | 46 47 48
49 50 | 51 | 52 | 53 | 54
1841 1849 8561864 1871 1881 18841789011896 1902 1907
55 156 157 58
59 60
61 | 62
62 63
64 | 65
1913 1918 1022 19281637 1937 1941 1945194011953 1957
71 72 73 74 75 76
0 1960 1964|1967 1970 1972107511978 1980 1981 1984 1986
86
77 78
80
81 82 83 84
84 85
90
87
1988 1990 1991 1993119941995119901997119901199-2000
DI Min.)
0
Min. Min./ ° | Min.
66
68 69 70
67
Min.
|
og
79
}
1
1
V. Hon
+
.'
1
62 Of making Mercator's Meridian Line, Book II.
V. How to make a Meridian Line according to the true Sea-Chard, or
Mercator and Mr. Wright's Projection.
to
His Line is made out of the Table of Meridian parts, called alſo tlic Diviſion of
the Meridian Line. To every 10 Minutes of Latitude, nearer we have no
Chards or Plots made, which I have as yet fecn; but they may be made by Mr.
Wright's Tables to every Minute, if any perſon will be ſo curious.
For the Graduating this Line in the scale, you muſt note the Number anſwering
the firſt Degree is 200; therefore divide the Degrees of the Aquinoctial into 20
eqızal Parts, which ſtand for 200 of the Numbers of your Table.
As by example,
Suppoſe you would make the firſt 10 Degrees from the Aquator, towards either of
the Poles, on the Scale; the number anſwering 10 Degrees is 201, omitting the laſt
Figure o: Therefore you may take out of the Line of Longitude (which is Equal
Parts
, or the ſame Line by which you made all the reſt), 201 or 20 Parts, and lay that
diſtance for the beginning of the firſt 10 Degrees; and for 20 Degrees 40,8; and
for 30 Degrees 62,9; and fo of the reſt.
But if you are to make a particular Line, you muſt take the difference of the De
grees and Minutes, as thall be fully ſhewn in the Treatiſe of making a general and
particular Sea-Chard, according to Truth, and Mr. Wright's Projection ; but what
hrath been done alrcady will ſerve for both, if you follow direction.
There is demonſtrated and ſhewn tlic making of Mercator's Scale, to meaſure Dia
ſtance in any Parallel of Latitude in any truc Sea-Chart.
+
VI. How to Calculate a Table, and by it how to take out the Numbers,
and make a Scale of Reduction, to be uſed in Surveying of Land.
Slalute
Acres.
25
00
and Artificial Sines on the Ruler, for the more ready
uſe thereof, as will be thewed : I ſhall firſt ſhew the Cal-
06 Qulation and Proportion uſed in making the Table, which is
thus as followeth.
12
272
255
189
161
138
I 21
106
13
14
15
16
Іо
91
oo
35
Example, For a Perch whoſe Meaſure is 21 Foot (whichi
is the Iriſh Chain) this muſt be done by the back Rule
16
2 1
100
of Three.
17
18
19
00
094 21
84 03
75 42
68
7
As 16 Squared, to 100 Acres:
So is 21 Squared, so 61,73 Acres.
20
21
61 73
22
A Table for the Diviſion of the Scale of Reduction.
23
24
25
126
27
28
29
30
31
56
25
SI
47
47
27
43 56
40 271
37 35
34 73
32 37
30 25
So a piece of Ground being meaſured by the Statute-
Chain of 16 Foot to che Perch, Ihould contain 100 Acres.
Then the ſame piece of Ground being meaſured by the
Iriſh Chain of 21 Foot, will contain but 61-7. Acres, as
you may ſee in the Table, which is near 61 Acres 2 Quarters
38.
28 33
By che Line of Artificial Numbers extend the compaſſes
from 16 to 21, the fame will reach twice repeated from
100 unto 61,73 in the ſame Line of Numbers.
32
33
34
35
36
37
38
26
59
25
23 55
211
21 I
1989
18 85
17 90
17 02)
To make this Live on the Scale, Take the Numbers off
the Line of Numbers of the ſame Scale you make this Line
upon.
39
40
EXAM-
+
Book II. and the Scale of Reduction.
63
EXAMPLE.
1
+
I ſhall place the firſt to on the Scale, to tliis Number anſwers 272. 25"; there-
fore extend the Compaſſes from 103 to 272, or from 274, and lay one Foot of
the Compaſſes at A, and the other will reach to B the diſtance to 16 From 165
you must ſay all your other Numbers. As ſuppoſe you would fet down 14 on the
Scale, the Stars:e Numbers anſwering thereunto is 138 and 9r". Extend the com-
paſſes from 100 to 138 and 9r", and that diſtance will reach from 16, at B, to 14
of thc Scale
. The like if you would ſet off 20, the Numbers to chat is 68.7;
and that diſtance will reach from 16 at B, to 20: and ſo do with the reſt. Thus
have I done with this Scale, being ſufficient to reſolve all manner of Mathematical
Concluſions whiatſoever. The Uſe follows in the ſucceeding Treatiſe.
1
1
$
FO
of
A
B
1
CHAP. V.
2
I
A Table for the Diviſion of the
Artificial Rhomb, or Points,
Halfs, and Quarters on the
Travis-Scale.
1
1811630
1673
T:
36
1870
22 1802
Poinis.Nor. Sourb. Deg. Mir. Sine Tang. (Tang.
parts Rbomb. 191400
N. DET 481 688
689
Sb. E.
5
371 990
992
S. 5, W.
3
8
.2611160 I 1177
N. b. w.411 15 1290
71299
N. N. E. 14
3/1385
1398
S.S.E 16
52] 1462
1481
). S: W. 19
41/15 271 2
1553
NN. W. :2
30, 1582
6 1617
N.E, b, N. 25
S. E. b. S. 28
7 1673 1727
S. W. b. S 30 56 17101 3 1777
3
N. W.b.8.133
45/1744
5 11824
N. E.
3311774
S. E.
39
1914
N.W.
42 IT 182714 1957
4
45 00 1849 4 2000
N. E. b. E.47
¡
26 1904
S 2
N.W.b.W.. 56
031933
E, SE.
161
52:1945
W. S. W. 164
411956
6
W.N. W 167
301965
t, b. N. 70
E.b. S.
73
7|1989
75 5611986
1
78 45|1991
81
33.1995
84 4.522|1997
wet.
00 2000
S. W.
He Uſe of this Tabe is eaſily
underſtood : The firſt con
lumn is the Number of
Points in one quarter of the
Compaſs, and the ſecond
their Names
in the whole; The third the Degrees
anſwering to each quarter of a Point
in the Quadrant ; The fourth the
Sines and Equal Parts anſwering
thereunto 3
The fifth the Tangent-
Rhombs; The ſixth the Tangent-parts
anſwering to each Quarter and Point
to 45 Degrees s, which is ſufficient,
S. E. b. E. 150
s.Web.w. 53
48.1869
37 1890
1
15 1919
E. N.E. 159
18|1973
W, b.S.
Wib. N.
Egg
&
87
120
II 1999
The
1
64 A Deſcription of the Travis-Scale. Book II.
!
The Deſcription of the Travis-Scale.
Thngents.
$94 Eat Weft.
North South
Numbers.
Equall parts.
1100
200
1190
ti
8.
190
70
3
60
180
50
170
The making of this Scale is all one in a man-
ner as you made the former ; only the Line of
Sines is there but once made, and here the Parts
anſwering each Quarter are twice put down, or in
two Lines marked with N.S. which ſtands for to
ſhew the Line to be Northing, Southing; and E. W.
ſignifies Eafting and Weſting. The firſt is the Sine,
the ſecond is the Complement that any point or Quar-
ter makech an Angle with the Meridian. The Line
marked with T. is the Tangent-Rhomb and Quar-
ters, and the firſt Line is a Line of Numlers, which
you have been already ſhewn to make.
One Example I will give the Learner, notwith-
ſtanding it is ſo eaſie; for ſome there are that will not.
underſtand, though they ſee it often done; yet (to
my knowledge) are Mates to good Ships.
160
30
150
140
10
A
130
EXAMPLE.
170
110
10
The Traverſe Scale.
100
9
8
1
Suppoſe you was to ſet ile firſt and fevener Rhomi
or Artificial Point on the Scale, which is 11 Degrees
15 Minutes, the Equal Parts anſwering chereunto is
1290; therefore cake 129 of your Scale of Equal
Parts, and lay it from the beginning upwards, and
you have by that diſtance the firſt and ſeventh Rhomb
of your Scale:
In.like manner do: for any other of the Points
and: Quarters by thcle Numbers, until you have fi-
niſhed the Scale; and when you have done, you liave
an Inſtrument the moſt caſie' ,--çady, and neceſſary
that I know of, for the working of Traviſés, and
correcting your dead Reckoning, which ſhall be
ſhewn in the Part of Sailing by the Plain Chard, in
the Fourth Book. On the back ſide of this Scale you
may ſet a Line of Chords and Egual Parts, and Points,
for the ready protraction of Angles..
7
180
:
5.
70
4
60
.
3
59
1
ET
2
30
1
20
10
fol:64
CHA P. VI.
How to make a Quadrant which will reſolve many Pullions in Affort
my, by the help of an Index; and alſo very useful in Navigation.
Fter you have made choice of the Radius of your Quadrant CD, draw Pai
rabel Circles chercunto, to hold the Degrees of the Quadrant and Columns,
for the Figures, Points, and Quarters, as P 8. Then divide che Semi-
diameter or Side of the Quadrant CD and C Minta 60 equal parts, and draw Pa-
A
rallels
1
CHAP.VI. Of the Sinical Quadrant.
65
rallel to each of the Diviſions, as Sides, C D and CM. Firſt divide them into 6,
and then each of them into io more, as you ſee in the Figure ; at every Parcs
make a Point, for the ready numbring of the Diviſions.
P
90M
810
{
710
5
!
a
che
210
oll
en
ho
:blar
osipto
o1 Hot
mm
jot
+
T
OS
O E
oc
olt
1109
ols
015
OE
ot
ol1
I
1
pood
Sorg/ot of
이
​cata
fol. 65:
Make an Index anſwerable to the Radius CD, with a Line of Sines on the firſt
Side, and a Center-Ear to put over the Center-Brals-Pin of the Quadrant C; as oc-
cafion ſhall require: And on the other Edge make a Line of Equal Parts, equal to the
60 Diviſions of the Side CD, with an Ear in like manner to remove at pleaſure.
Make on the Edge a Tangent-Line; from it you muſt take the Sun's Declination,
as you ſhall be fully ſhewn in the Uſe thereof.
This. Quadrant will ſerve excellent well for a Protractor, with a long Index divided
into 100
or 200 Equal Parts, with an Ear as the former, and a Needle put into a
Stick, to put through the Center of the Index and Quadrant on any Point, in a Plain
or Mercator's Chard, by which you may Protract any Rhomb without drawing Lines
upon the ſaid Chard; as likewiſe the Protractor or Semicircle which follows may do
the fame, being inade in the ſame manner. On the back ſide of the Quadrant you
may puc Mr. Gunter's or Mr. Samuel Foſter's Quadrant, or any other as you ſhall
think fit.
1
1
'A
11
지
​The
1
66
Book.II.
The Uſe of the Quadrant
The Uſe of the Quadrant in Aſtronomy.
SECT. I.
Having the Latitude of the Place of the Sun's Declination, It is required
to find the Time of the Sun-Riſing and Setting.
The Latitude 51 Deg. 30 Min. Northward, and the Diclinatiin 20 Degrees, the
difference of Aſcenſion will be found thus.
Fiſt, Lay clic Center Ear at E of the Index, over the Brafsu Pin in the Center at A
of the Qualrant, and lay the Edge of the Index to E L; to the Latitute of the Place
on the Arch D M; and take of the Tangent-Line on the Edge of the Quadrant 20
Digrees the Sun's Declination ; and lay that diſtance from the Center at A cowards
D, at that diſtance run with your Eye al ng the Parallel-Lines, and mark where it
couclicth the Edge of the Index; there follow that Parall:I-Line to the Arch, and
reckon the Degrees from B to that Parallel-Line will be 27. Deg. :4 Min. chre dife-
rence of Aſcenſion between the Sun's Riſing and Setting, and hour or 6, according to
thic cime of the Year.
The Degrees reſolved into Hours and Minutes, is 1 Hour 49 Min. ivliich is 4 of the
Clock' and about 11 Min, for the San Riſing in the Morin, and 7 of the Clock 49
Min. his Setting in the Evening. In the fame manner you muft vork for all La-
titudes,
SECT. I
Having the Latitude of the Place, and the Diſtance of the Sun from the
next Æquinoctial Point, To find the Amplitudeš
So the Latitude being 51 Deg: 30 Min. and the place of the Sun in one Degree of
Aquarius, that is 59 Degrees from the next Aquinoéti.el Point; therefore let the
Ear ats of the Line of Sines of the Index on thg. Pin it A, and the Edge chercof
to the Latitude, and reckon 59 Degrees the Sun's diſtanc from the first Aquino&i-
al Point, from the Genter to C along the Line of Sinos of the Index; there noce che
Line that cuts the 59 Degrees following with your Eye, to tic Degrees in the Arch,
and reckon the Minutes of Degrees from M to the Edge of che Index, and you will
find it about 33 Deg. 20 Min. the Amplitude required.
+
SECT. III.
Having the diſtance of the Sun from the next Aquinoctial Point, To
find his Declination.
The Sun being either in 29 Degrees of Taurus; or I Deg. of Aquarius, or I Deg.
of Leo, or 29 Deg. of Scorpio, that is 59 Degrees from the next et quinettial Point
To find his Declination do clius: Par the Ear of the Line of Sines on the Pin and Edge
of the Index; pue to 23 Deg: 30 plin. in the Sun's greateſt Declination, reckoned
from M on the Arch; then count the Sun's diſtance 52 Deg. on the Deg. of Sines of
the Index : From the Center put one Foor of your compactes by the Degree, with the
other take the neareſt diſtance to the Line or Side CM; apply thac diſtance in the Line
of Sines of the Index, from S along, and the other Foot will reach to, 20 Degrees, the
Declination required when the sun is in the aforeſaid Degrees and Sines. In like
manner you muſt do for any other Degrees of the Sun's Place.
1
SECT.
BOOK II.
In Aſtronomy.
67
+
r
SECT. IV.
** Having the Latitude of the Place, and the Declination of the Sun, Ta
find the Sun's trne Amplitude from the true Eaſt and Weſt.
This is a moſt excellent ready way by chis Quadrant, and as icar the Truth as
any Man can make any rational uſe of this problem at Sea: It is thus. Suppoſe the
Latitude to be 13 Degrees, and the Sun's Declination 20 Degrees Northward, the Sun's
true Amplitude of Riſing and Setting is required, from the true Eaſt and Weft.
Set the Ear of the side of the Index on which is the Line of Sines on the Center,
and Edge to the Latitude 13 ; then count from M 20 Degrees of Declination, and
carry your Eye upon the Parallel-Line from that Degree of the Arch, and mark what
Degree it curs of the Index and Line of Sines, as in chis Queſtion it doch 20 De-
grees 25 Minutes, and that is the true Amplitude required.
Secondly, Suppoſe you was about the Cape of Virginia, in Latitude 37 Degrees and
30 Min. and Declination 10 Deg: If you work as before-directed, you may find the
true Amplitude to be 12 Deg. and about. 36 Min. You may eſtimate the Min. but
you cannot Steer by a whole Deg: when you have retified your compaſs by this;
therefore this is ſufficient for that uſe, to ſhow you the difference between the true
Compaſs and the Steering Compaſs, if you obſerve his Riſing and Setting by it.
Note, The Amplitude is the diſtance of Riſing or Setting of the Sun or Stars from
che true Eaft and Weſt Points upon the Horizon.
As for the foregoing Work In dhe Latitade of 13 Deg. the Sun or Star having
North-Declination 20 Deg. therefore they will riſe 20 Deg: 33 Min. to the Northward
of the Eaſt, and ſet 20 Deg.33 Min. to the Northward of che Weſt. But if the De-
clination had been 20 Deg. South, then they would have riſen 20 Deg. 33 Min.
Southward of the Eaſt, and ſet 20 Deg. 33 Min. to the Southward of the Weft.
And ſo if you bring theſe Degrees and Minstes into Points and Quarters, and uſe
the Variation-Compaſs upon the Inſtrument of the Moon in the Firſt Book, you may
readily rectifie the Compaſs you Steep by.
r
SECT. V.
The uſe of the Quadrant and Variation-Compaſs in the Firſt Books on
the Inſtrument of the Moon for Shifting of Tydes.
This Inſtrument contains two Parts or Rundles, which are the two uppermoſt in the
aforeſaid Inſtrument made of wood or Braſs, moving one upon the other, as there
you may fee. The biggeſt of the tivo uppermoſt Rundles repreſents the Compaſs you
Sreer the ship by, which is ſubject to Variation : but the upper Compaſs doth repre-
ſent the true Compaſs that never varieth, whereby you have a moſt neceſſary Inftru-
ment to rectifie che compaſs, as Mr. Wakely hath made Tables to be uſed with it;
buc
this will ſerve for uſe as near by the Quadrant.
Admit I am in the Latitude of 27 Deg. and Declination 20 Deg. Northward, and
I obſerve the Sun's Rifing and Setting to be due Eaſt and by Vorsh, and Weſt by
South Point of my Steering or Varixtion-Compaſs; the Variation in that Latitude is
required.
The Sun having North-Declination, and in that Latitude of 27 Deg. if there be
no Variation the Sun will riſe (as you may preſently find his Amplitude by the Qua-
drant and Index, 22 Deg. 34 Min. which is buit 4 Min. not to be taken notice of, 11 d. 1 sito
above) E. N. E. and fees w. N.W. But according to the foregoing Propoſitions
, the cach Point.
Sun did riſe at 2.b. N. and ſet at W.b. N. Therefore it plainly appeareth that there
is a full Point Variation.
Therefore on the Variation-Compaſs on the Inſtrument of the Moon, you muſt al-
ways bring the true Point of Riſing and Setting on the upper Compaſs, to touch the
falſe Peine or Riſing and Setting, found by Osſervation and Steering-Compaſs
, on the
K 2
middle
68
The Uſe of the Quadrant, and Book II.
1
1
thoſe few Rules following.
middle Rundle,being ſet in this poſition. You will find the Enb. N. to be the true
E.N.E. and the W. and by N. to be the true W.N.W. and the N. 6. E. to be che
true N. and the S.b. E. to be the true S. and the S. E. Point Southerly, to be the
crue S. E.b. S. Sontherly; and the South Eaft, to be che South Weſt: And ſo
you may do with caſe in all other Obſervations, in like manner as you have been
ſhewn, by Points, Halfs, and Quarters, which is on the two Trundles; and be fure
ncarer than of a Point I never did ſee any man Steer or 1heape a Courſe:
SECT. VI.
To know the Variation by the Quadrant.
You may do the ſame thing by the Quadrant, without the help of the Rectifier
before ſpoken of, if you will remember, That this Quadrant hath cight Points, or
of the whole Compaſs, by which you may orderly reckon the whole, and let the
Index to the greatest difference either from the Eaſt, Southward, or Northward, or
West. In like manner as in the foregoing Propofition, the true Amplitude of Riſing
and Setting was 2 Points, or 22 D.g. 34 Min. E. N. E. Set the Index to the Degrees
and Points, reckoning the Deg. from D on the Arch of the Osadrant, toward the
Scale of Leagues of the Index ; then reckon the Point and Degree taken by Obſerva-
tion, which is 11 Deg. and 15 Min. a juft Point of the Compaſs: therefore it being
but E. by N. ſhort a Point of the true Amplitsde, therefore the E.V. N. of the Steer-
ing-Compaſs, reſpecteth the true E. N. E. and the N.b. E. reſpectech the truc N. and
fo account all round the Compaſs a Point more than the Steering Compaſs ſhewetli:
And if you would kæow whiclı way the Variation is, you ſee it is a Point inore from
che E. than your Compaſs thewech Northerly.
• But if the Steering or Azimuth-Compaſs, had ſhewn a Point more than the true
Amplitude found by the Quadrant and true Polnt, the Variation had been Weſterly.
Bur ſuppoſe the Amplitude found had been a Point Southerly, E. b. S. and the
Sæn's Riſing and Setting had been a Point Northerly; by Obſervation of the ship-com-
paſs, you ſee there is cwo Points difference: therefore ſet the Index to two Points,
from M the Eaſt or Weft ſide of the Quadrant, as in this Propoſition you muſt reckon
it
, and you may ſee plainly that the Eaſt Point by the Steering Compaſs is the true
Eaſt-South-Eaſt Point, and the South Point is the true South-Sosth-Weff Point; and
the North is the trųe North-North-Weſt Point ; and ſo of all che reſt: And the Va-
riation is Southerly. So that you ſee how readily this Quadrant doth theſe things, when
the Points of the Compaſs is imprinted in a Mans mind, which muſt be, and is in all
Maſters and Mates.
Suppoſe I would know by the Quadrant the true Point of the Compaſs, when
Bootes Ar&turus riſeth and fetrech: In the Latitude of 40 Deg. Bootes Arcturus De-
clination is 20 Deg. 58 Min.
Set the side of the Index and Sine to the Latitude of 40, and count the Declinga
tion 21 Degrees almoſt, from Mon the Arch; and run your Eye up the Parallel, and
it will cut the Index about 27 Deg. so Mix. which is reduced into Roints and Oxar-
ters by allowing, Gr. 15 Min. to a Point, his Rifing will be almoſt E. N. E. & a
Point Northerly, his Setting W. N. W. Weſterly
. But if the Declination of a Star of
the South 'ſide the Aquino&tial, the Riſing had been E.S. E. Southerly, and his
Setting W. S.w. Southerly.
In the like manner you may know the Riſing and Setting of any Star in an inſtant,
by this Quadrant and Index, which I hold to be as neceſſary an Inſtrument as Seamen
can uſe, in reſpect of its plainneſs, and brevity, and portability, ſo made as you ſec
the Figure, the larger the better : And on it you may work all manner of Traviſſes
to the diſtance of 60 Leagues or Miles which is on the ſide of the Index. It being ſo
plain and caſie, I need not write any thing thereof; but for the Learner's ſake, take
1
SECT
f
(
.
7
7.
Book II. Variation-Compaſs, in the Firſt Book. 69
SECT. VII.
To finde the Number of Miles anſwering to one Degree of Longitude, in
each ſeveral Degree of Latitude.
4
60 00
66
In Sailing by the Compaſs, the Courſe ſometimes holds upon a great Circle, fome-
time upon a Parallel to the Aguator ; but moſt commonly upon a crooked Line,
winding towards one of the Poles, which Lines are well known by the Name of
Rhombs.
If the Courſe hold upon a great Circle, it is either North or South under fome
Meridian; or Eaft or Weft under the Agxator.
In theſe Caſes every Degree requires an allowance of 60 Miles,
or 20 Leagues, every 60 Miles or 20 Leagues will make a Degree
Deg. Min. Miles,
difference in the Sailing; therefore as was ſhewn in the firſt Di-
oo od
00 60
agram, and uſe of the Line of Sines, may be ſufficient here, which
18
I2 57
is the Rule of Proportion.
25 15 54 But if the Courſe hold Eaſt or Weſt, on any of the Parallels
31 48 51
to the Aquator,
36 52 48
41
251 45
As the Radius is to 60 Miles, or 20 Leagues, the Meaſure of
45 341 42
one Degree of the Æquator :
49 281 39 So is the Sine-Compl. of the Latitude, to the Meaſure of
53 08 36 Miles or Leagues to one Degree in that Latitude.
56 381 33
001 30
But if you would know by the foregoing Quadrant thc Miles
63
01 27
anſwering to a Degree in each Parallel of Latitude, it is thus.
off
251 24
Set the Ear E on the Center-Pin, and reckon the Degrees
69 30 21
Latitude from D: to which ſet the Edge of the Index, and note
72 321 18 the Parallel-Line that is at the Degree; carry your Eye on ic to
75 31 IS the Side CD, and from the Center to that Line you have the
Number of Miles anſwering a Degree in that Latitude,
78 28 12
81 231 09
EXAMPLE.
84 15! 06
87 08 03
In the Latitude of 18 degrees 12 min. ſet the Index 18 gr. 12
min. from D, and the Parallel-Line riſing with char Degree,
wich your Eye or a Pin follow to the Edge, and you will find it to be 57 Miles,
the Miles anſwering one Degree of Longitude and sr Miles, in the Latitude of 31
gr. 48 min. as in the foregoing Table ; and ſo work for any other Latitude in like .
manner.
But if the Courſe hold upon any of the Rhombs becween the Parallel of the
Æquator and the Meridian, we are to conſider beſides the Aquator of the World
to which we Land, which muſt be always known.
Firſt, The difference of Longitude, at leaſt in general.
2. The difference of Latitude, and that in particular.
3. The Rhomb whereon the Courſe holds.
4. The diſtance
upon
the Rhomb, which is the diſtance we are here to conſider,
and is always ſomewhat greater than the like diſtance upon a great Circle. The firſt
follows in the next Propoſition,
..
I. To
1
1
70
The Uſe of the Inſtruments. Book II.
1. To find how many Leagues do anſwer to one Degree of Latitude,
in every ſeveral Rhomb.
Rhombs.
49120
3720
III
39
16
19
15/20
412062
52 20
4121
2 22
I 2
7122
68
II 26
In this Table is the Degrees of every quarter Point, , and
whole Point in the Quadrant ; as the firſt quarter is 2 gr. Iaclinati. Number of
49 m. ſo the half Rhomb is s gr. 37 m. the third is 8 gr. on 10 the Leagues,
26 m. and the firſt point from the Meridian is II gr. 15 ms.
Meridian.
and ſo you may plainly ſee the reſt.
Gr. Min. Leag. par.
2
2
As the Sine-Complement of the Rhomb from the Meri-
5
IO
dian, is to 20 Leagues or 60 Miles, the Meaſure of
8 26/22 22
1 Degree at the Meridian :
So is the Radius or Sine of 90, to the Leagues of Miles
14
anſwering to one Degree supon the Rhomb.
go
24
Suppoſe by the Quadrant it were required to anſwer this
3021_65
O Heſtion,
Sailing N.N.E. from 40 Degrees of North-Latitude, How
25 19/22
28
many Leagues ſhall the Shiprun before it can come to 41? By
30 56123
56123 32
reaſon this is the ſecond Rhomb from the Meridian, and the
Inclination thereof is 22 deg. 30 m.
3133 45 24 05
Therefore ſet the ſide of the Index EL to the ſecond Point
36 34124 90
from thic Meridian N. N. E. 22 d. 30 m. and reckon from
39 2225 87
C 20 Leagues towards D, and with your Eye or a Pin fol 42 26 991
low the Parallel- Line to the Index, and you will find it cut
4145 0028 08
21 Leagues 65 parts (or better than more) the number of 47 4929 78
Leagues you muſt Sail before you can reach 1 Degree.
50 37131 52
You may do the ſame by the Travis-Scale thus. Extend 53
2633 57
the Compalles from 2 Points neareſt the end of the Scale, and SI56 15/36
greateſt Number of the Line of Numbers that is N. N. E. 2, 152
4138 90
and E. N. E. 6 Points, unto 20 Leagues on the Line of
542 43
Numbers; remove the compaſſes to 100 in the Line of Num 64 41 36 76
bers, and the other Point of the Compaſs will reachi to 21
667 3052 36
Leagues or 6s parts, as before in the Line of Numbers.
70 19 59 37
73 7168 90
This
may be found alſo by a Line of Chords and Equal
75 5682 31
Parts, if
you draw a Right Line, and take with your com-
2178 45) 102 521
pafles 20 parts, and lay it from one end on the Line ; then
34 136 30
take 60 deg. and ſweep an Arch, and take 2 Points with
84 22 205 14
your Compaſſes, and lay from the Meridian on that Arch
87
11 407 60
from N.N. E. and draw the Secant or Rhomb-Line, at 20 890 oolad infinit.
Leagues draw a Perpendicular or Line at Right Angles chere
to the former, and meaſure the diſtance from the Center, to
the Interſection of the Line drawn from 20, with the Rhomb-Line on the scale of
Leagues or Egnal Parts, and you will find it the ſame as before. And ſo the Qua-
drant thews you all at one fight, if you underſtand without more words. By the
Artificial Sine and Number, Extend your Compaſſes from the Sine of the Rhomb 67
deg: 30 to 20 in the Line of Numbers, the ſame Extent will reach from 8 Points, or
go deg. or 100 in the Line of Numbers, to 2 1 Leagues 65 parts, as before. This
conſider in general; I ſhall thew you more particularly in 12 Proofs (how of theſe
four, any two being given, the other two may be found, both by Mercator's
Chart, and all other ways, as is uſual) when I come to treat more particularly of Na-
vigation.
00
161
81
i
II. By
1
+
thing
CHAP. VI. The Uſe of the Quadrant.
71
i
II. By one Latitude, Rhomb, and Diſtance, To find the difference of
Latitudes.
Ler the place given be in the Latitude of 40 Degrees, that is on the Center of the
Os idrant, the ſecond Latitude unknown; The diſtance upon the Rhomb 2 1 Leagues
65 parts of a League ; the Rhomb N. N. E. the ſecond from the Meridian: Therefore
fee thic Index ro che point, and count 21 Luogues 10 parts, and run your Eye up
the
Parallel-Line you there meet with, and reckon the Leagues from the Center Ċ to that
Line, ard you will find it 20 Leagues; and ſuch is the difference of Latitude required.
It is cafic to be underſtood how to lay it down by the Plain Scale; therefore I
Thall farbear to write any more of that Way.
As the Radiis, to the Co-fine of the Rhomb from the Meridian :
Su the diſtance upon the Rhomb, to the difference of the Latitudes.
Extend the Compaſſes form the Sine of 90, to the Co-line of the Rhamb 67 deg.
30 m. the fainc distance will reach from 21-65 Leagues in the Line of Numbers,
torche clifference of Eatitade 20 tegurs. In like manner you muſt work for all ſuch
Propofi:ions, ler chic Number be greater or leſs, by citcr Inſtrument.
The Travis Scale is clic fame manger of Work, as the Artificial Sines, Tangents,
and Numbers; For at find the compaſſes from 8 Points to 2 Points, the fame diſtance
will reach from 21:65 in the Line of Numbers, .to 20 the difference required.
"UL. By the Rhomb and both Latitudes, To find the Diſtance upon the
Birt'. Rhomb:
As'ſuppoſe'che one place given wore C che Center of the Quadrant, in the Latitude
40 deg. the ſecond place in the Latitude 41 deg. and the Courſe the ſecond from the
Meridian.
Set the Index to the R3,5ml, and account 20 Leagues, which is 41 deg. the ſecond
Littitude, and carry your Eye on that Parallel chat leads to. che Iridex; and there iç
will cut the diſtance upon the Rhimiz jhich in this Queſtion is 2 1 Leagues 65 parts:
Extend the Compafis from the co-line of the Rhomb froin the Meridian, to the
Ruzdins er Sine ni 90
The ſame Extcur tvill rein!: from 20 Leagues, the difference of Latitude, to 21:05
in the Lize of Number's, chc diſtance upon the Courſe required.
IV. By the diſtance and both Latitudes, To find the Rhomb.
Suppoſe the Place given was at C, in Latitude 49 deg. and the ſecond Place a
Degree or 20 Le.gues further Nurihivard, and the diſtance was 21~-65 Leagues
upon the Code.
From the Center C reckon 20 Levignes towards D, follow that Parallel, and ſet
the Index, and count the diſtance until it touch the Parallel, and look in Arch of
the Quadrant, and you will find the Rhomb 22 gr. 30 m. or N. N. E. 2 Points from
the Meridian.
Or, Extend the compaſſes from the diſtance upon the Rhomb
21:65, to the diſtance of Latitudes 20 Leagues; The ſame Extent will reach from
the Radius or Sine of 90, to the Sine-Complement of the Rhomb 67 deg. 30, which
was required.
V. By the difference of Meridians, and Latitude of both places, To find
the Rhomb.
As if the Place given was the Center of the Quadrant, 40 deg: and 20 Leagues
was the difference of Latitude Northward, that is 41 deg. and the difference of Longia
tude 8 Leagues 45 parts of a League.
Firſt, count from the difference of Latitude 20 Leagues, on that Parallel coudt
8 Leagues 45 parts ; to chat put the Index ; and in the Arch you will find the Converse
22 gr. 30. from the Meridian.
Extend the Compaſſes from 20 Leagues to 8:45, the fame Extent will reach from
90 to the Tangent of the Rhonib 22 gr. 30 min. as before.
VI. By
72
The Deſcription of the Protractor. Book II.
31012015
)
*
807016050 401 30
LA -
VI. By the Rhomb and both Latitudes, To find the difference of Longi-
tude, or departure from the Meridian.
Let the Rhomb be 2 Points from the Meridian, the one Latitude giveni 40 deg. the
other Laritude 41 deg. the difference 20 Leagues.
Sct the Index to the point and Rhomb 20 gr. from the Meridian, and count 20
Leagues the difference ; on that Parallel reckon the Leagues between the side and the
Index, and you will find it in this Queſtion 8 Leagues 45 parts, the Meridians-diſtance
required. - Or, Extend the Compaſſes from the Tangent of the Rhomb 22 gr. 30,
to Radius 90, the faine Extent will reach from the difference of Latitude 20 Leagues,
to the departure from the Meridian 8 Leagues 45 parts.
Theſe ſix laſt Propoſitions depend one upon the other, as you may plainly ſee;
which may be ſufficient for the Explanation of the Owadrant, by wirich may be un-
derſtood much more.
+
CHAP. VII.
How to make a moſt uſeful Protractor.
His Inſtrument is always to be made in Rrafs or Copper, but beſt in Braſs.
On the Center C draw che Semicircle BO, and divide it into two go, or
180 Degrees, as you may ſee the Figure ; and let the fixteen Points
of the Compaſs P P be ſet in she inward Circle, with the Quarter-Points. And let the
T
1
2
3
5
6
7
8
40
TAITITI
910
80
30
6020
30
A
10 610
20
20
eller
.
LO
10
시​의
​www
w
www
22
B
IA
C
+
to
*
210
18
310
4/0
51
019
710
80
E
fol: 72
.
1
Index
1
1.
|
-
1
1
14
A : Nocturnal:
he
NY
September
003
ningu
20
01
18 Gemini
01 €
OLE
.
April.
olz
TOW
20
011
2010
10
| Aries
0
31
Cancer
March
I
e
i
XII
310
II
une
Ilo
01
IX!
20
201
ILL.
281
02:
30
O Leo
10
Iuly
1 Pise
Februa
ols
10
VIII IX
-
20
1
131
+
VIT
20
:..
II. VIII
9
1IUHN
January
wer
1 Aquarius
Auguſt
son
VI
www.
Virgo
310
20
K
M
VI
011
20
0 %
11
20
nus
을
​12
Dereinber
XII
Th.
mi Libra
Capricorn
310
교
​:30
310
20
10
vetober.
20
* 10
012
et
LOUITTROSELIS
210
Nouember
yo 이
​3 lo
101
20
Scorpio
m Sagitarius
CHAP. VIII. A Deſcription of the Nocturnal.
You may
73
Index A E be two Diameters and long, and ſo fitced as the Semidiameter of the Cir.
cle may be the diſtance froin the Center, for the ready ſetting to any number of De-
grees, or Points and Quarters che Edge thereof, and divide from the Center to the end
of the Index into 100 cqual Parts, which are accounted ſometime Leagues, and rec-
koned by the Surveyors of Land Perches, or any other Denomination of Numbers:
call it for protraction according as you have occaſion to uſe it.
Let the Index be faſtned to the Center with a Braſs Rivet, and through the midſt
of the Rivet there muſt be a Hole drillid; you inuſt put the Pin or Needle ſpoken of
in the laſt Chapter, upon any Point aſſigned, in any Chart or Protrałžion whatſoever.
You may divide the Edges into equal parts,' by which you may make a Meridian-Line
on the blauik Charts, according to Mercator's or wright's Projection.
And 110w you have a neceſſary Inſtrument, which will protract any Travis or piece
of Land upon Paper, with as much ſpced as any Inſtrument I ever yer knew, and
readier by much, the uſe whereof ihall be thewn in this Treatiſe.
1
CHAP. VIII.
The Projection of the Nocturnal.
T
T consiſts, as you ſec, of three Parts. The greateſt or handle-part liach on it two
Circles divided : On the firſt or outmoſt is the Ecliptick, divided into 12 cqual
Parts, in which is put thic 1 2 Signes; and cach of thoſe I 2 Parts is divided into
30 equal Parts or Degrees, in each Signe, numbered 10, 20, 30, as you ſce tire Figure
plainly theweth. The inward Circle is the 12 Months of the Year, ſet in by a Table
of the Sun's place every day of the Year, accounting the Degrees of the outward Cir-
cle, and the number of Days in cach Month equally divided and ſet down 10, 20,
30.
Note, Where to begin to divide the Months and Days is thus. Obſerve the bright-
eſt Guarde, or by Calculation or thic Globes find when he comes to the Meridian juſt
at 12 a Clock at Night. In the following Tables the Right Aſcenſion of the brighteſt
Gusard is 223 Degrees 31 Minutes; from it fubftract 12 Hours, or 180 Degrees
, the
Remainder is 43 deg. 31 min. che Right Aſcenſion of the Soin the 26th day of April
,
in 16 deg. of Tauries, which muſt be uppermoſt next the Zenith in the middle.
Thic other Part or middle Rundle equally divideth the outward Circle into twice Iz
Hours; and within that is a Circle cqually divided into 32 Parts, or Points of the
Mariner's Compaſs projceted thercori. The upper part is an Index, the length is from
thic Center to the foot of the Inſtrument ; all three being faſtned with a piece of
Braſs, lo Riverted that the Center is an Holc through which you may ſee the North-
Star. You inay make it in Braſs, or good dry Box.
The uſe of the Nocturnal.
TH
He Llíc of the Nocturnal is caſie and ready. Let the Tooth or Index of the mid-
dle Circle be ſet to clic Day of the Month, and it will cut in tlie outward Circle
the Sun's Place in thc Ecliptique. Then hold thc Inſtrument on high, a precey diſtance
from you, with the Poor A B right with the Horizon level : Then look through the
Holc of the Center, and ſec the North-Star, turning the long Index or Pointer up-
wards or downwards, untill you ſee the brighteſt of the Guards over or under the
Edge that comes ſtraight to the Center. Then look on the Hour-Circle by the Edge of
chc Pointer, and it ſhows the Hour of the Night, and likewiſe thc Point of the com
paſs the Guard beareth from the Pole; by tlic avhich you may have his Declination by
the following Tables cxactly.
Thic Hour of the Night may be alſo found by the Right Aſcenſion of the Sun and
Stars. Thus, When that you ſee any Stars in the South, whoſe Right Aſcenſion is
known, and alſo the Right Aſcenſion of the Sun for that day, you ſhall ſubſtract the
Sha's Aſcenſion from the Star's ; that which remaineth divide by 15, to bring it inco
L
Hours
+
;
74
The Uſe of the Table of the Book II.
Here place Hours; for 15 Degrees makes an Horr, and 4 Minutes make a Degree; thereby you
the North have the Hour of the Night. If the Sun's Aſcenſion be more than the Star's
, in fuch
caſe you ſhall add 360 Degrees to the Aſcenſion of the Stars, and then ſubſtract the
Pole Star Sun's Aſcenſion, as before directed, you have the time alſo.
No&urnal
for the
andGuards
CHA P. IX.
How to uſe the Pole-Star's Declination and Table, and thereby to get
the Latitude.
T
year
Hc Pole-Star, being ſo very well known to all Sea-men, is thicrcforc made the
moſt uſe of by them: Therefore know, That this Table is made for the
1660. the true Declination being 2 deg. 30 min. but will ſerve for many
years after. This Table is made contrary to the two former Tables; for whereas the
North Point of the Nocturnal is the fiſt Point you reckon from, and was on the for-
mer Nocturnal reckoned from Senth: ſo of this Nostwrnal you muſt take the Point at
the Toth for North, and ſo reckon forward North and by Eaſt, and ſo on to Eaſt and
South-East, South and Weſt, to North again.
And likewiſe in the Table, you muſt begin in like manner at that part of the Ta-
lle that lies directly under the Pole ; which, as beforc-ſaid, is properly called the North,
and ſo procecd about the Pole, aſcending from this loweſt or North Point of the Me-
ridian, as was ſaid before, to the North-Enft, Eaſt, and South-Eaſt, ſo to the South or
higheſt Point of aſcending, being directly over the Pole : From the Soutb they deſcend
again by the Weſt, and ſo return to the North again.
Obferve chis, That the brighteſt of the Guards is the firſt of the little Bear, which
is the Star you are to obſerve, and is almoſt in oppoſition to the Pole-Star.
Note, That when the guard-Star is under the Pole, then the Pole-Star is above the
Pole ; and when the Guard-Star is above the Pele, then the Pole-Star is under the
Pole ſo many Degrees and Minutes as the Table ſhows you.
The Uſe of this Table and Inſtrument is this : Look with the NoEtarnal, and fee
what Point the Guards bears from the Pole, as before-directed ; and if you find the
Gward is not on a full Point, ſtay a little longer until he is juſt, and then obſerve the
height of the Pole-Star exactly as you can; then knowing by your Dead Reckoning
within a Degree or two what Latitude you are in, look for the ncareſt to that Lati-
tude in the top of the Table; and if you find the Point of the Compaſs, which the
Guard-Star is upon, in the firſt Column of the Table, and in that Line under the Com
Inmon of your Latitude, you ſhall find the number of Degrees and Ministes the Polen
Star is cither above or below the Pole, according to the direction of the laſt Column of
the Table, which you muſt thus make uſe of ; If the Star be any clıing above the
Pole
, lubſtract the Number in the Table from che height of the Star obſerved : but
if the Star be under the Pole, then add the Nismber found in the Table to the height
obſerved, by wliich you !hall have the height of the Pole.
For Example
. Eſtimate the Latitude to be near 40 Degrees
, and obſerve the Pole-
Star. Suppoſe you find the Altitude 40 Degrees, and the Grard-Star bears N.N.E.
from the Pole; therefore look for N. N. E. in che firſt Column, and right under
your
eſtimated Latitude 40, in the ſame Line with N. N. E. you will find the Declination
to be i Degree 30 Minutes; ſubſtract that from 40 Deg: the Altitude obſerved
leaves the true Latitude 38 Degrees 30 Minutes.
But if the Guard-
Star had been S.S.W. then the Pole-Star had been i Degree 33 Minstes d. mo
under the Pole; which being added to the Altitude obſerved 40 Deg. 40
the Latitude would have been exactly 41 Degrees 30 Minutes by the Star.
So the Star’s. Altitude by obſervation being ss Deg. the Guard bears from 38 30
the Pole-Star S. E. b. S. the Declination againſt char Point is 2 Degrees
30 Mindtes, added to 55. had been 57 Degrees 30 Minutes for the Latitude: but if
your eſtimated Latitude had been near 50, and the Guard bear from the Pole North-
}
+
V
00
OI
30
Weft
CHAP.IX. North-Star, to get the. Latitude.
75
Weſt by North, the Declination againſt that point is 2 deg. 30 min. ſubſtracted from
55 Degrees, the Altitude obſerved, chere remains g2 deg. 30 min. the Latitude of the
place by the Star.
A Table of the North-Star's Declination in theſe several Latitudes.
O
20
so
•
I
I
2811
Above the Pole.
T
021
OIO
0810
0410
2210
4910
ISI
21
IS12
152
Under the Pole.
S. E.
2
302
The True
30 40
60
70
Point of the
Compaſs. D. M. D. M. D. M. D. M. D. M.D. M D. M
North. iz
1012 1012 1012 092 0912 0812 07
N. b. E. 5311 53 I 5311 52 1 521 Sili 49
N. N.E.
311 3111 3011 301
291
25
IN.E.B.N. 061 051 041 031 02
58
N. E. lo 3910 3810 3710 3610 3500 3310 30
N. E. b. E.O
100 09
070 060
01
E.N.E. 180 190
2010
210
2310
26
E. b. N.
10 500 solo
Solo si lo 5210
5210 5310
5310 56
Eaſt.
1511 1011 17 18 1
1811 1910
E.b.S.
3811 391 391 401 411
411421
421 44
E.S. E. 2 001 2 00
002
002 0012 002 OI2 02
S.E. b. E.
2 1512
152 162 1612 16
2 252 252 2512 2512 2512 2512 25
S.E.B.S.
3012 2012 303 302 302 30
S.S. E. 292 2912 292
292 292 2912 29
S. b. E.
South.
IO 2 1012
Sob. W.
580
5411 531
57
S.S. W.
32 I
3311 341 350 38
S.W.b. S. li 0711
071 0711 08
III
13
S. W. 0 390 400 4110 4010 4310 4410 47
S.W.b.W.O,100
S.W.b.w.
Illo
I 210
130 14
19
W. S. W. 190 1910 170
1610 150 1310
W.b. S. 480 470 460 450 440 4310 42
Weft.
isl. 141 1311 1211 1111 101
08
W.b. N.
397 391 381
371 3611
35 I 33
W.N. W. 12 0011 591
5911 5811 580
5711 56
N.W.b.W.12 15.2
1512 1512
1412 132 12
7512
2512 25 2 242 242 24
N.W.b. N.12 3012 302
3012 3012 3012
30.
N.N. W. 292 2912
29/2
292 2912
292
29
N. b. W.
2212
222 222 222 2212 2112
2
2
2212
222
2212
2211
2
222
the North or lower part of the Meridian,
If the former of the Guards be afcending from If the former of the Guards be deſcending from
the South or upper part of the Meridian.
22/2
22
z
102
N
IIZ
I I2
II 2
12
1.
531
321
55/1
551
I
31 I
I
I
IQI
131
I
160
O
IO
I
I
Above the Pole.
1412
1412
N. W.
2 2512
3012
2
2
21
I hope the young Seamen are pleaſed for Examples, it being made ſo plain to their
Capacity, and as profitable for their Liſe as any Kule whatſoever.
3
L 2
CHAP.
1
1
76
A Deſcription of the moſt Uſeful Book II.
CHAP. X.
How to make a moſt uſeful Inſtrument of the Stars, and by it to know
moſt readily when any of 31 of the moſt notable Stars will come to the
Meridian, what Hour of the Night, at any time of the Year, at the firſt
fight.
1
T
His Inftrument conſiſtech of two parts, which is two Rundles; on the back ſide
of the foregoing Nolturnal it inay be very fitly made : On the matter or
greater Rundle are three Circles divided; the outermoſt is the 12 Months
of the Year, begun the 10th of March, the day the Sun enters inco Aries ; and the
Days equally divided to the Number of Days in each Month. The ſecond Circle re-
preſenteth the 24 Hour Circle, divided equally into 24 Hours, { and “, beginning the
joth of March at 12 a Clock at Noon, the time the Sun comes to the Meridian, and
the firſt Degree of Aries. The third and inward Circle is the Aquinoctial, divided
into 360 Degrees; by them is accounted the Righi Aſcenſion of tlicíc 31 Stars in the
Table following
A Table of the Longitude, Latitude, Right Aſcenſions, and Declinations,
of 31 of the moſt notable Fixed Stars : Calculated from Tycho his
Tables, rectified for the rear of our Lord, 1671.
!
2
2
1
1
I
2
1
.
Longitude. Latitude. Afcenfi-|Decli- Nor.
nation. Sou.
The Whale's Tail
27 56 * 20 47. So6 45 19 48 S
The Bright Star in the South Foot of Androm. 09 39 8127 46 N 25 57140 44 N
The Bright Star in the Right Side of Perſeus. 27 17 &?o os N144 16 48 30 N
The Bright Star of the 7 Scars or Pleiades.
24 24 804
ON 53 00123 03 3
The South Eye of the Bull Aldebaran.
05 [2 IOS 31 564 17 15 48 N
The Bright Star in the left foot of Orion Riges. 12 17 I 31 ir 574 44 8 37 N
Orion's right Shoulder towards the Eaſt.
124 12 16 6 S 84
6 S 84 23/07 181 N
The glittering Star in the Mouth of the great Dog. 09 35 5 29 30 597 4216 141S
The Little Dog's Thigb Procyon.
21 18.15 57 STIO 34 06 03
In the South Arm of the Crab.
09 03 02 05 08 S125.22 20 48
3
The Bright Star called the Heart of the Hydra. 22 45 or j22 24 N 137 5407 15
The Lion's Heart Regulus.
225 17 Stoo 26 N 147 4313 33 N 1
The lower of the Pointers.
14 43.02 45:03 N 160 1858 081 N
The white or North Pointter.
10 34 2 49.40 N 160 4863 32
The Lion's Tail.
17 03 12 18 N 173 04 16 25
The Firſt between the Tail and the Body. 04 10 me 54 18 N 189 03157 47 N
The ſecond of the T'ail of the Horſes.
10 56 7h56 14 N197 37 56 41 N
The Fore-Horſe, or lat in the Tail
.
2 2 2 254 25 N203 3715 00
In the Skirt of his Garment Arcturus.
19 39 31 2 N 210 13:20. 58! N
The South Balance of Libra.
10 31. m00 26 N 218 13.14 371 S
The Brighteſt of the Guards.
o8 16 z2 SI N 323 37|75 38 N
The Scorpion Heart Antares.
08 13 04 27 S 242 2325 37 S
Engonaſis Head Hercules.
It 31 x 37 23 N 254 12114 sol N
The Bright Star of the Harp Lyra.
10 43 761 47 N 276 2738 30
The Swan's Bill.
26 44 249 02 N 289 23 27 18 N
The Bright Star the Eagle's Heart,
27 09 229 21 N 293 41108 03 N
The Dolphin's Tail.
og 32 29 08 N 304 24 10 14
The Mouth of Pegaſus the winged Horſe.
27 22 22 7N 323 03 08 24
The Bright Star of. Pegaſus Neck.
I 397 17 41 N 336 21 09 08
The Southern Star in the Wing of Pegaſus Macrobe 18 56*119 26 N 342 07113 28
Andromeda her Head.
09 47 7 25 42 N 357 54127 18 N
On
2
:.
2
2
2 고
​1
3
2
2
1
...340 -331060 i 10
1111:11 II III
:. 3 210 TL
icorpion
Heart
25
37
Douth
Ballanc
A4.37
Bullseye Aldebre5 48||
acatwaing
Point
63
32M
220vinteren
OG ab 091 01
1
28 March
flo
210
28
31 February
32 April
312
IX
olz
ola
20
ci 30
30 40
IT
30 May
of oz
IK
La
114 50NI
81
ole
8 28M
011
S
VIII
con
has8s Loj san
3001110
Heart
Dolphins taile
210
31. Ianuary
Pagales Mouth
Neck
312
200
VIE
Ils
Zorth 10.37
311 June
G74
210
...
270
thes are
Arydromeda head 27.18W
hale tolle South 1948
{
Petrus Boates
Andromedis foote 40474
to
270
Béarstailegi.com
Beers
Penkeus his side
es his side 48 30M
go oli 1go
Wit 98 puosal 119
30 December
20
vo
GBears taile
firſt 57'47MB
VI
이는
​250 1290
Lions taile
30 31
30 Iuly
Hydrae heart
Cancery Crabb
Wir gil
LittleDoggs thigh 16.3
Greate Doggsmouths
Orions left foote
ons Right shouldedy
VII
Hear Point" | 58 59
v 33
30
8.37
17 25
18
371
D
31. November
thing
20
48
210
N
0172 000
31 Auguſt
I
tlo
180
210
210
MIX
30
1
2
1
110
October
31 September
30
1
O
។
។
19
}
CHAP. X.
Inſtrument of the Stars.
77
1
On the other Rundle or upper part, is placed all theſe aforeſaid Stars; and
any other
you may ſet thercon, if you follow this Rele.
For example, Firſt ſet the Intex to eliet och day of March of the upper Circle ;
on the under Circle, which will be at. 1 2 2 Clạck at Noon, or 360 Degrees, ſtop it faſt
there, that it may not move, until you have placed the Stars on it as you intend to fer
thereon. As ſuppoſe you would let the whale's Tail on the Rundle in his place; look
in the foregoing Table
, and you will find his Right Aſcenſion 6 deg. 45 mins-account
that from the noch of March, and on the Aquinoctial Circle, and lay a Ruler from
the Center over the 6 deg. 45 min in the Matter and Aquator or inward Circle, and
draw the Line from the Center to the outward Edge of the upper Circle
, and thereon
fet the Name of the Star, next to that the Declination of the Star, and the Letter S
or N. repreſencing South or North Declination :: on the inward Circle, fet before cach
Star the Magnitude of the Star, wvliereby you may know the better, as the Figure
following thews you all plain.
Take this Exemple morc. Suppoſe you would ſet the Lion's Heart in his place; In
the. T'ebles I find his Right Aſcenſion is 147 deg. 43 min. reckon that Number on the
Aquinoctial Circle next the Rundle, and draw the Line as before-directed, i fignify-
ing the Firſt Magnitude, ſecondly his Name, thirdly 13 deg. 33 N, for his North
Declination,
The Inſtrument in this poſture, you will find the Lion's Heart will come to the
Meridian at almoſt 10 a Clock in the Evening the roch day of March in
tude.
any Latie
*
How to know the Hour of the Night any Stár comes to the Meridian
Latitude.
in any
re с
)
Eye
Ou
the om
or Hand of the upper Rundle to the Day of the Month, and right againſt the
Star is the Hour of che Night, in the Matter the Star will be on the Meridian.
For example, Suppoſe you would know che Hour of the Night the. Bull's
comes to the Meridian chc 20ch Day of Otober ; Sec the Index to the Day, and right
againſt the Bull's Eye is of an Hour paſt 1, che cime in the Morning that Star will
be on the Meridian South. And in the ſame manner you may ſee the Stars, and
Hours they come to the Meridian that Night and Day. For note the upper half of
the Circle, and 12 Hours is the Day-hours, and the lower and Handle-half is the
Night-hours. You begin to reckon the Day-hours on the left ſide of the Inftrument,
and the Night-hours on the right ſide; ſo round with the Sun.
How to know what Scars are in Courſe
' at any time or Day of the Year.
THe Courſe and ſeaſonable coming to the Meridian of the Stars, and what are
fit to be obſerved, is Thewn you ar once, the Inſtrument in the former polture,
if yo:1 took againſt each Star, you have the Hour of the Night and Day, being the
whole 24 Hors. This is ſo plain, you need no further Precept.
L
How to know the Hour of the Night, by the Stars being on the Meridian.
Suppoſe it were required to know the Hour of the Night the 10:h of December,
the brighteſt of the 7 Stars being on che Meridian South; Set the Inies to the
day of the Month, and right againſt the brighteſt of the 7 Stars, is half an Hour
paſt 9 at Night, which is the Hour required,
1
CHAP.
!
"
1
78
Of the Croſiers and Croſs-Staff. Book:II.
!
CH AP. XI.
of the Crofiers.
W
Hen the Mariners paſs the Aquinoctial Line towards the South, ſo that
they cannot ſee the North-Star, they make uſe of another Star, which
is the Conſtellation called the Centaur ; which Star, with three other no-
table Stars which are in the ſame Conſtellation, makech the Figure of a Crofs (be-
twixt his Legs) for which cauſe they call it the Crofiers. And it is holden for certain,
That when the Star A (which of all four cometh neareſt the South-Pole) is
1
I
Cox fout
North and South by the Star B, then it is rightly ſcituated to take the Height by:
And becauſe this Star A, which is called the Cocks-foot, is 30 Degrees from the
South-Pole, ic comech to paſs, it being ſcituared as before-faid, we take the
Height thereof (which is then the greateſt that it can have) chis Heighe will truly
Thew how far we are diſtant from the Aquinoctial: For if the ſaid Height be 36
Degrees , then we are under the very Aquino&tial : But if it be more than 30
deg. then are we by ſo much paſt the Aquinoctial
, toward the South: And if it
be leſs than 30 deg. ſo much as it wanteth, we are to the North of the Aquinoctial.
And here it is to be noted, That when the Guards are to the North-Eaſt, then is
tho Star in the Crofiers' fitly ſcituated for obſervation, becauſe then they are in
the Meridian.
.
CHAP. XII.
1
How to make the Croſs-Staff.
T
Hc Mariner's Croſs-Staff is that which by the Aſtronomers is called Radius
Aftronomicou, by which we obſerve the Celeſtial Lights above the Horizon.
The Mathematicians have invented many kinds of Inſtruments, whereof
the Croſs-Staff and Quadrant are the
moſt uſeful above all che reſt
. At Sea it is not
every Mans Work to make and mark a Croſs-Staff and other Inſtruments, for want of
Practice needful thereunto; yet norwithſtanding it is fic and neceſſary that a Mafter,
his Mates, and Pilot, who are to have the uſe of it, ſhould at leaſt know when it is
well made.
For to mark well a Croſs-Staff, you ſhall make a plain Alat Board of good dry
Wood,
1
CHAP. XII. A Deſcription of the Croſs-Staff,
part of a
79
Wood, fifteen or fixteen Inches broad, and about four Foot or three Fuot long : Paſtes
it well with good Paper; draw along the one side a Right Line, as in the next fol-
lowing Figure CAD; out of the Line C draw a Square Line upon A C, as CB,
and upon the Center C draw the Arch A E B, being a Quadrant or fourth
Circle; divide that into two parts; the one half thereof, as A E, divide into 90
Equal Parts or Degrees, thus; firſt into three Parts, and cach of the ſame again into
three; theſe Parts each into two, and each of the laſt Parts into five; ſo the Arch
A E ſhall then be divided into 30 Parts. Then take a right Røler, lay the onc end on
the Point or Center C, the other upcn cach Point of the foreſaid ſeveral Diviſions,
and draw ſmall Lines out of C, through each of the foreſaid Points or Degrees of the
Quadrant, ſo long as they can ſtand upon the Board, as you may ſee it plain in the
Figure. Then take with a pair of Compaſſes, juſt the half length of the Croſs that
you would mark the Staff after; prick it from the Point C cowards B; as by Exam-
ple, from Cto F, and from D to G; joyn theſe two Points with a Line to one ano-
ther; and cven into ſuch Parts as that Line is cut, and divided by the aforeſaid Lines
coming out of the Center of the Quadrant, muſt your Staff be marked, whether the
Croſs be long or ſhort, as appcareth by the Lines H I and KL, which are drawn for
Cruſſes: the half thercof is ſo long as CH, or CK, or CF. If the aforeſaid Quadrant,
for want of good Skill.or Practice, be not well divided, or Lines not well drawn, the
Staves being marked thereafter will alſo be faulty. Therefore they may be marked
more exactly by Points equally divided, in manner as followeth.
I
39
C D
49
12
F
W
t
20
70
24236
42222lo
E.
|
.
1
B Ε Η K C
F
-10-20.7450ma
Prepare
1
86
The Uſe of the Croſs-Staff. Book II.
A Table for the Diviſion of the
Crols-Scaff.
7
Prepare you a Staff; draw thereon a
Right Line ſo long as your Staff, and take
with a tharp pair of Compaſſes the half
Length of the Croſs after which
you
defire
to mark your Staff: prick it ſo often along
the aforeſaid Line , as it can ſtand upon
the ſame. Divide cach of the Lengths of
the half Cross into 1000 Equal Parts.
Then prick upon the Staff you will mark
from the Center-end, juſt half the Length
of the Croſs; and mark there a ſinall
thwart Stroke. Off from thence prick for
cach Degree ſo many of the ſame Parts
as the Cross is divided in his half Length,
like as is marked in the Table liere annexed
for every Degree. For the firſt Degree
you ſhall mark off from go the aforeſaid
thwart Stroke 176, for the fourth Degree
724, for the io Degree 1918 of thoſe
Parts, and ſo of the reſt. If you cannot
divide the half Crefs, by reaſon he is ſo
little, into 1000, divide him into 100,
and ſcave out the cwo laſt Figures, and
that ſhall ſatisfie your deſire: For 30 De-
grees take 73, and for 40 Degrees 114,
and for 10 Degrees 19 Parts, and ſo of
the reſt.
1
D. Parts D.Parts. DI Parts.
1 176
176 31 76751161 28667
2 355 32 80401162 30108
31 53811331 841811631 316531
41 7241 1341 88071 641 33315
51 9131 351 9210165 35107
6 11061 1361 262666
37046
71303 37 1005767 39152
811504 13811050368 41445
9 1708 139 109651 169
43955
10 1918 40 114441 70 46713
11121314111943117 4975 8
121234914211246072 53197
1312572 143 12998 173 56912
1412799 44 13558 174 61154
15.303214514142175 65958
16132704014751176 71445
173514 47 15386177 77769
18 3764 48156051 178 85144
194019 49 167467995854
2014281 50 174751 180 104301
21 4550 51 18239 81 117062
22/4826 52 19042 82 133007
23 s108153198871 83 153499
245399 5420777 841 180811
2515697155121716 851 219038
26.600356227081 186276332
276318) 57 237591 187 371885
2866431 1581248741 881 561810
29 69761 15926059891135891
2017320 1601272211 190 Endleſs.
1
:
CHAP. XIII.
A
How to uſe the Croſs-Staff.
Et the end of the Croſs-Staff to the outſide of the Eye, ſo that the end of the
Staff come to ſtand right with the Center of the Motion of the sights. Then
move the Croſs ſo long off from you or towards you, holding it right up and
down, and winking with your other Eye, till that the upper end come upon the
middle or Center of the Sun or Star, and the lower end right with the Horizon,
the Croſs then ſhall ſhow upon the ſide of the Staff belonging chereunto, the Degrees
of chc Altitude of the Sun or Star. Note, The Staff is marked with two Lines of
Numbers, with 90 Degrees next the Eye, and dininishing from 90 to 80 and 70,
60 towards the outmoſt end: The Complement-Sine beginneth towards the Eye-end,
ni cncreaſech contrarywiſe towards the outinoſt end, as from 10, to 20, 30. The
Firſt Number ſheweth' you the Altitude, the ſecond Number is the Sun's diſtance
from the Zenith.
The Sünor Stars being liigh clevated from the Horizonthe Cross cometh nearer the
Ege than when they are but a little clevated, and do ſtand neer the Horizonzthereby the
eye makchi(ſeeing oow to the lower, and then again to the upper end of thecroſs)greater
motion
1
CHAP.XIII,I he Uſe of the Croſs-Staff.
81
.
1
motion in looking upand down, than when the Sun or Star doth ſtand low.And info-
much cheCenter of the Sight by ſuch looking up and down together with the end chc
Staff, a Man ſeech then Imaller Angles then if it did remain Stedfaſt, in regard where-
of the Croſs comech nearer to the Ege than it ſhould, and there is found too much
Altitude. This being found by many, beſides my ſelf, by experience, they were there-
fore wont to cut off a piece of che end of their staff
, or fet the Croſſes a Degree and
or cwo Degrees nearer the Eye; but it isnot the right means for to amend the afore-
ſaid Errors. The beſt means of all in mný. opinion is this; That upon each ſeveral
Height which men will obſerve, they do try with two Groffes fer upon the like De-
grees, how the Staff muſt be ſet
, chat they may ſee the end
of the ſame ewo.
Crolles
right one with the other.
1
+
1
+
i.
31
cu
.
Long
+
)
Ł
&
Bunum
17
:
B
1
+
1
lozi
80
8
"
MUI.
D
mais
с
nanin
TININ
- +
ii
is
Having found that
, and then taking off one of the Crofes
, and ſetting the Staff
again, in the ſame manner as before, all Errowrs will be ſo prevented, which by the
lifting up or caſting down of the Eye, might any 'manner of way happen.
F
1
1
EXAMPLE
I deGre to obſerve the Sun or any Star in the South : I make my Eſtiination, as
neer as I' can, how high that ſhall ſtand, or take the Height of them a little before
they ſhall come to the South, which I take to be 50 Degrees. I ſet therefore the two
Creffes
in.
M
ra
+
!
The Uſe of the Croſs-Staff
. Book II
.
2
But if the Sum hach North Declination, and in the Zenith, then look how many
82
Croſſes each upon their so Degrees, and the end of the Staff in the hollow of the
Eye-bone, on the outſide of the Eye, and bow the Head forward or backward, or
over the one ſide or che olier, till I ſee the utmoſt end of both the Croſſes right one
with the other, according as is ſhewn by theſe Lines A B and CD, as is apparent
enough by the foregoing Figure ; That the Sight-beams over the ends of the Croſses
Mall their agree with the Lines which might be drawn over the end of the Croſſes, to
the Point or Center at the end of the Staff, which doch agree with the Center of the
Quadrant, or the beginning of the Equal Points upon which the staff is marked.
Keeping in memory lich ſtanding of the Staff, I take off the one Croſsy and fer the
Staff again in the aforeſaid manner to tlic Eye, and obſerve without any errour of
the Eye.
In taking the Height of the Sun with the Croſs-Staff, Men do uſe red or blew
Glaſſes, for the ſaving and preſerving the Eyes; yet it is notwithſtanding a grcat let,
and very troubleſom for the Sight, eſpecially if it be high : therefore the Quadrant
and Back-Staff is much better, as will be thewed in the next Chapter.
Thus I have ſhewed you how to take an Obſervation by the Fore-Staff. The nexo
thing that followeth in courſe will be to ſhew you how to work your Obſervation;
which to do, take notice of theſe following Rules.
To Work your Obſervation.
F the Sun liath North Declination, and be on the Meridian to the Southwards of
you then you muſt ſubſtract the Sun's Declination from your Meridian Altitude,
and that Remainder is the Height of the Aquino&ial, or the Complement of the Liana
titude North. But if the Sun hath South Declination, you muſt add the Sun's De-
clination to your Meridian Altitude, and the sum is the Height of the Aqualor, or the
Complement of the Latitude North. If the Sun hath
North Declination, and be on
Meridian to the Northwards, then add the Sun's Declination to his Meridian Al-
titude, and the Sum is the Height of the Aquator, or the Complement of the Latitude
South, if the ſaid Sum doth not exceed 90 deg. but if it doch exceed 90. deg, you
muſt ſubſtract go deg. from the ſaid Sum, and the Remainder is your Latitude North
If the Sun hath South Declination, and be to the Northwards at Noon, you muſt
then ſubſtract the Sun's Declination from his Meridian Altitude, and the Remainder is
the Complement of your Latitude South. When the sun hach no Declination, then
the Meridian-Altitude is the Complement of the Latitude. If the Sun be in the Ze-,
nith, and if at the ſame tince the sun hachi no Declination, then you are under the
Aquineltial.
Degrees and Minutes the Declination is, and that is the Latitude you be in North.
But if your Declination be South, then you are in South Latitude. If you obſerve
the Sun or Star upon the Meridian beneath the Pole, then add your Meridian Alti-
tude to the Complement of the Sun or Stars Declination, and the Sum is the Height of
the Pole,
If
Rules for Obſervation in North Latitude.
+
1
Uppoſe I am at Sea, and I obſerve the Sans Meridian Altitude to be 39 deg. 32
min. and the ſame time the Sun's. Declination is 15 deg. 20 min. North; I doo
mand the Latitude I am in.
deg. min.
The Meridian Altitude
39 32
The Declinacion North, fubft.-
15 20
The Complement of the Lati 24
90 00
65 48 North..
o
Suppoſe
I2
The Latitude I am in
i
.
؛ ܢܼܲ ܢܼܲ.ܨ ܂
Se
CHAP. XIII. The Uſe of the Croſs-Staff. .
83
Suppoſe I were in a Ship at Sea the 18th of April, Anno 1667. and by Obſervati-
on I find the Sun's Meridian Altitude to be 62 deg. 15 min. The Latitude is required.
deg. min.
The Meridian Altitude-
62 is
The Declination North, fubft.
14 18
The Complement of the Latitude
47 57
90
The Latitude I am in, required. -42 03
00
1
Admit you were in a Ship at Sea the sth of November, Anno 1679. and I find
the Sun's Meridian Altitude to be 24 deg. 56 min. The Latitude is required I am in.
deg. min.
The Meridian Altitude-
24 56
The Declinacion South, add---
Tbc Complement of the Latitude
43. 33
90 00
The Latitude 1 an in
46 27
18 37
1
oo
Suppoſe a ship at Sea the 28th of May, Anno 1666. and I find the Sun's Me-
ridian Altitude by Obſervation 56 deg. 45 min. The Latitude is required I ain in.
deg. min.
The Meridian Altitude
"56 45
The Declinacion North, Subst. 22 46
The Complement of the Latitude 33 59
go
The Latitude required I go in-
a56
Admit a Ship at Sea the I th of June 1668. and find the Sun's Meridian Al-
titude by Obſervation 79 deg. 30 min. North, It is required the Latitude I am in.
deg. min.
The Meridian Altitude
79 30
The Declination North-
-23 3l add.
103 01
90 00
The Latitude I am in
13 OI required.
Suppoſe I were at Sea the 14th of May 1693. and the Meridian Altitude of the
Sun was 69 deg.07 min. North, I demand the Latitude the Ship is in at that time.
deg. min.
The Meridian Altitude
69 07
The Declination North-
។
1
-20
53 add.
oo
90
go
OQ
The Ship is under
-00
oo the Aquinoctial.
A+
;
M2
Rules
!
84
.
The Uſe of the Croſs-Staff. Book. II
.
Rules for obſervation in South Latitude.
E
SY
Uppoſe I at Sea in a Ship the ſecond of Fune, Anno 1666. and I find the Sun's
Meridian Altitude by Obſervation to be 64 deg. 45 min. The Latitude the Ship
is in, is required.
deg. min.
The Meridian Altitude North-
64 45
The Declination North, add.
23 15
The Complement of the Latitude-88
88 00
90 00
The Latitude the ship is in
-02 oo South,
Suppoſe a ship at Sea the 28th of December, Anno 1695. and in Longitude 169
deg. Eaſt, and I find the Meridian Altitude by Obſervation to be 59 deg. 52 min. The
Latitude the Shipis in,is required. The Declination in the Meridian of Briſtol for the
28th of December, is 22 deg. 25 min, and the daily difference of Declination is at this
time 8 min. Therefore if you look in the Table of Proportion following, you will find
the Proportional Minutes to be about 4, which you muſt add to the Declination of the
Meridian of Briſtol, and the Sum will be the true Declination for the Longitude 169
deg. Eaſt, which is 22 deg. 29 min.
deg. min.
The Meridian Altitude North--
59 5?
The Declination South, Subſtr.-
The Complement of the Latitude---37 23
-22
29
A
90
00
1
62 23
22
26
84 49
/
The Latitude the Ship is in, which was-52 37 required.
Suppoſe I were at Sea in a Ship the 29th of June, 1679. and I find the Sun's Me-
ridian Alsitude to be 62 deg. 30 min. North, The Latitude is required.
deg. min.
The Meridian Altitude North
The Declination North, add
The Complement of the Laricude
90 00
The Latitude the Ship is in
n!
OS
11 South.
Admit I am in a Ship at Sea the 20th of Fanuary 1667. the Sun's Declination 20
deg. 4 min. and the Sun's Meridian Altitude 79 deg. 36 min. South, I require the
Latitude che ship is in.
Anſwer, 9 deg. 30 min. South.
Admit a Ship were at Sea, the Sun's. Declination 13 deg: 5.3 min. Sowth, and the
Sun's Meridian Altitude 80 deg. 43 min. South; The Latitude is required.
The Declination South
53
The Meridian Altitude-
80 43 add.
94 36 the Sum:
Sulſtr.-9000
The Latitude the Ship is in----04 36 Soutb.
If ycu obſerve the upper part of the Sun, you muſt ſubſtrazt 16 min. But to the
contrary, if you obſerve the lower part of the Sun, you muſt add 16 min. for the
Sun's Diameter, and the Sum will be the true Altitude of the Sun's Center.
Rules
}
deg. min.
I3
I
:
1
CHAP. XIV. A Deſcription of the Quadrant. 85
.
Rules for obſerving the Stais.
Uppoſe I am at Sea, and obſerve the Brighteſt of the 7 Stars upon the Meridia
an, and find his Meridian Altitude to bc 47 deg. 20 min, and the Latitude were
required.
The Declination of this Star is 23
The Meridian Altitude-
47
Subſtract the North Declination----23
The Complement of the Latitude----24 17
go
03 North
20.
23 03
OO
The Latitude I am in-
65 43
Admit I were at Sea, and obſerve Hydra's Heart on the Meridian, his Altitude
is 36 deg. 15 min. and luis Declination is 7 deg. 15 min. Sowth, The Latitude of the
Place is dcnianded.
deg. min.
The Meridian Alcitude is
36 IS
The Declination is South-.
-07
The Height of the Æquinoctial above-43 30 the Horizon.
go
The Latitude the Ship is in -46. 30 required.
.
15 add.
oo
This you ſee is plain, and needs no further Precept but what is already ſaid.
CHAP. XIV.
A Deſcription of the Back-Staff er Quadrant.
.1
./
"He Back-Staff or Quadrant is a double Triangle, as this Figure following
ſhewech; whereof the Triangle A B C the Arch is equally divided into 60
deg. and the other Triangle is divided equally into 30 deg. A DF the Vanes
are ficted ntar. In proportion to him, the Uſe followeth.
T
The uſe of the Back-Staff or Quadrant.
Et the Vane Ġ to a certain number of Degrees, as the Altitude of the Sun re-
quircth; and looking through the Vane F, to the upper Edge of the Slic of the
Sight of the Horizon, if you ſee all Skie and no Water, then draw your Sight-Vane
a lictle lower towards E:' but if you ſee all Water and no Skie, then put your Eye-
Vane up higher towards F; aid when you have done ſo, obſerve again; and then if
you ſee the shade lie upon the upper part of the Slit, on the Horizon-V'ane, and you
at the ſame time do fcc che Horizon through the Sight-Vane, then
that is all you can
do untill the Sun be riſen higher; and tending the Sun until he be. upon the Meridi-
out, you will perceive he is deſcending, or as we commonly
, fay he is fallen, you will
ſee nothing but Water ; your Vanes faſt in this poſture, you
have done obſerving the
Sun 1pon che Meridian chat day: Therefore reckon thc Degrees froin B to the up-
pèr ſide of the Vane Gy to it add the number of Degrees from E to the Eye-Sight,
and their Susm is the Distance of the Sun from the Zenith to the upper Edge of the
Sup; to which Sum if you add 16 minutes, which is the Sun's Semidiameter, you
will haye the truc diſtance of the Sun's Center from the Zenith or Complement of the
Meridian Altitude. Nore this, If you obſerve the upper Limb of the Sun by the up-
per
1
CIM
86 A Deſcription of the Quadrant. Book II.
you work
per part of the Shade, then it is the upper Limb that gives the Shade ; but if
you
obſerve the lower part of your Shade, then it is the lower ſide of the Sun that
gives
the Shade : Therefore you muſt ſubſtract 16 min. from whiar your Back-Staff gives
you, and the sum or Difference gives you the right Diſtance of the Sun from the
Zenith. You may have the Altitude of the Sun from your Quadrant, if
thus; from Cto G is 40 deg. for the Vane ſtands at 20 deg: from D to Fis 16 deg.
being added cogether makes 56, the Altitude or Height of the Sun above the Hori-
zon, which you may uſe as you were ſhewn by the Fore-Staff : Bue in regard che
Engliſh Navigators work their Obſervation by the Complement of the Sun's Altitude,
when he is upon the Meridian, being ſo ready to be counted by their Osadrant ;
Therefore we will direct you in general, and after in particular Rules.
F
+
A HALILI
un
The Figureof the
Quadrant
Ашдципшивашхиції.
պայմաննես
ITาพ NTTI
IM
II XUT*11
B
M
D
C.
Go
25
FE
A
$
2
Firſt, If the Sun hath North Declination, and you in North Latitude, and the Sun
upon the Meridián, South of you; then if you add the san's Declination to his Žer
mith-Diſtance, that is the Complement of the Sun's Aseridian Altitude, the Sum will
lac the Latitude you are in.
But if the Sun hath
South Declination, you muſt ſubſtract the Complement of the
Meridian Altitude, and the Remainder will be the Latitude the ship is in.
If
you be ro the:solithward of the Aguinoctial, and the Sun to the Northwards
of the Aquinoctialy in ſuch caſe you muſt add the sum of the Declination to the
Zenith-diſtance, and the Sum will be your Latitude South.
But if the Sun be to the Northwards of che Aquinoctial (that is, have North De-
clination)
!
.
th
CHAP. XIV, The Uſe of the Quadrant.
87
clination) you muſt ſubſtract the Declination from the Zenith-diſtance, and the Re-
mainder will be the Latitude South.
If you underſtand the forc-going Rules given of the uſe of the Fore-Stuff
you cannot miſtake the Uſe of the Quadrant or Back Staff.
We will now come to Examples what are needful.
obſerve theſe Rules for North Latitude.
1
the
00
.-56 54
1
IO
16 07
A Dmit we were in a Ship at Sea the fifth of May, Anno 1694. and by Obſervation
I find the upper ſide of the Sun to be diſtant from the Zenith 37 deg. 36 min..
Sun being upon the Meridian, I require the Latitude the Ship is in.
deg: min.
The Sun's Diſtancc from the Zenith-
37 36 his upper Edge.
The Sun's Semidiameter, add-
16
The Center of the Sun from the Zenith 37 52
North Declination.
-19 02
The Latitude reguired, the ship is in
Suppoſe a Ship at Sea the 29th of July, Anno 1682. and I find the Complement
of the Sun's Meridian Altitude by obſervacion to be 32 deg. 54 min. The Latitude of
that Place the ship is in, is required.
deg. min.
The Complement of the Altitude ist
32 54
The Sun's Semidiameter add so it
do 16
The Diſtance of the Sun's Center from Zenith-33
North Declination, add-o
The Latitude the Ship is in
42
49 17
Suppoſe a Ship were at Sea the 13th of Sept. 1683. and I find the Complement of
the Sun's Meridian Altitudė, or Diſtance from the Zenith, 45 deg. 42 min. I demand
what Latitude the ship is in.
deg. min.
The Complement of the Altitude is
45 42
The Sun's Semidiameter add to it
2016
The Diſtance of the Center of the Sun 45 58
The Declination South, fubftract
The Latitude the Ship is in
45. SE
Admit a Ship were at Sea the fourth of December, Anno 1690. and the Comple-
ment of the Sun's Meridian Altitude chat day were 49 deg. The Latitude the ship is
in, is required.
deg. min.
The Complement of the Meridian Altitude-49 07
The Sun's Semidiameter
Add-
The Center of the Sun diſtant from the Zenith—49 23
The Sun's Declinación South, ſubſtract-
30
The Latitude the Ship is in North
- Suppoſe I were in a ship at Sea the 23d of May, Anne 1695. and I am allo in
Longitude to the Eaſt of the Meridian of London 135 deg.. and I find the Comple-
ment of the Meridian Altitude by Obſervation to be 13 deg. 12 min. The Latitude is
„required.
1
00 07
00
16
23
25 53
T
.
Declination
.
88
The Uſe of the Quadrant Book II.
IZ
OO
deg. min.
Declination in the Meridian of London- -22
17
The Proporcional Minutes, Sabſtract--
00 03
The Sun's Declination in the Meridian given-- 32 14
The Complement of the Sun's Altitude I3
The Sun's Semidiameter, added.
16
The True: Zenith Diſtance of the Sunë
-1.3
28
Tihe True Latitude the ship is in North-
That is, by rcaſon the Sin is to the Northward of my Zenith, and the Declination
more than the true Diſtance from the Zenith or Complement of the Meridian Altitude;
therefore ſubſtract 13 deg. 28 min. from 22 deg.20 min. and the Remainder is the true
Latitude 8 deg. 52 min. Norch.
-08 46
EXAMPLE.
d. m.
Let the Complement of the Sun's Altitude bez S,che Altitude 76 deg. 32 min. in the
North BS, the Declination North E S 22 deg. 14 min. if you
add the
Altitude SB 76 deg: 32 min. to the Declination SE 22 deg. 14 min.
the Sum is B E 98 deg. 46 min. the Diſtance of the Aquinoctial BS 76 32
from the Horizon in dic North B Z 90'deg. being ſubſtracted from SE:22 14
it, remaineth for Z E, the Diſtance of the Aquinoctial from the BE 98 46
Zenith towards the South, 8 deg. 46 min. juſt B P the Latitude of the
Place, and Altitude of the Pele above che Horizon.
90 00
B
08 46
is
i
S
1
N
>
}
4.1.
.
North
Q.
.
ü
B
i
South
4
1
Let the Complement of the Sun's Altitude be Z S 13 deg. 28 min. the North Decli-
nation ÉS 22 deg. 14 min. being more than the Diſtance of the Sun from the Zenith,
ſubſtract ZS 13 deg. 28 min. the Complement of the Altitude from S E the Declinati-
on, there remaineth deg: 46 min. the Diftance of the Aquinoctial from the Zenith
Z'E or Latitude of the Place BP, as before. I have been the more large on this,
by reaſon I would have Learners perfect in it, it being moſt uſeful Questions,
When
.
CHAP. XIV. In South Latitude,
89
When that you Sail far Northward or Southward, that the Sun goeth not down, as
they find that Sail about the North Cape, and to Spitsberghen, or Greenland ; and that
you would obſerve the Altitude by the Sun, alſo when he is in the North at the loweſt,
Firſt, There muſt be added to the Altitude of the
Sun taken above the Horizon, the
Complement of the Sun's Declination ; that is, the Diſtance betwixt the Sux and the
Pole, that Number ſheweth the Altitude of the Pole.
Secondly, or elſe che obſerved Altitude moſt be ſubſtracted from the Declination,
that which remainech is the depreſſion or depth of the Aquinitial under the Hori-
zon in the North, juſt to the Altitude of the ſame in the South, the Complement
thereof is the Altitude or Height of the Pole.
Thirdly, If you take the Complement of the Sun's Altitude, and ſubſtract from it
the Complement of the Sun's Declination, there remainech che Diſtance of the Pole
from the Zenith, or thé Altitude of the AgninoEtial in the South; the Complement
thereof is the Altitude or Height of the Pole.
Fourthly, or elſe if you add the Declination to the Complement of the Altitude, and
you ſubſtract 90 Degrees out of that Nämber, chere remaineth the Depth of the
Aiguinoftial in the North under the Horizon; that being ſubſtracted out
of 90, there
remainech the Altitude of the Pole.
A
Dire&tions for obſervation in South Latitude.
Dmit a ship at Sea the 7th of July, Anno 1695. and I am in Longitude 135
deg. Eaſt, and the Sun being upon the Meridian, I find the Complement of
I his Meridian Altitude by Obſervation to be 42 deg. 34 min. North'; The
Latitude is demanded, the ship is in.
The Complement of the Meridian Altitude-42 34
The Sun's Semidiameter, addin
The Sun's Center diftant from the Zenith.
The Declination North, Subfrad-
The Latitude the Ship is in
deg. min.
00
16
42 50
21 16
-21 34
00
16
T
18. 37
-00
42
Admit I were in a Ship the fifth of November, Anno 1687, and in Longitude
122 deg. Weft, and the Complement of the Sun's Meridian Altitade by Obſervation is
31 deg. 37 min. North; The Latitude is required, the ship is in.
deg. min.
The Complement of the Meridian Alcicude----31 37
The Sun's Semidiameter, add
The Sun's Center diſtant from the Zenith
31
53
The Declination South
The Proporcional Minutes-.
05 Added.
The Sun's Declination in the Meridian given---18
which add to the Zenith-diſtance
-31 53
The Latitude the Ship is in
50 35
Suppoſe I were in a Ship at Sea to the Southwards of the Aquinoctial, the third of
Fansary, Anno 1683. and I find the Sun upon the South part of the Meridian, and
by Obſervation his Meridian Altitude is 75
deg. 38 min. The Latitude the Skip is in,
is required.
deg. min.
The Complement of the Meridian Altitude14 22
The Sun's Semidiameter, adla
The Sun's Center from the Zenith
The Declination South
14 38
28
The Latitude the ship is in South-
N
EXAM-
1
00
16
21
06 50
90
The Uſe of the Fore-Staff, cc.: Book II.
.
EXAMPLE.
F
South
TIRTILA NI
B
!
In this Figure let C
bc the South, and .P
the North Pole, DE
the Æguino&tial, AB
F
the Horizon, Z thc Ze-
D
nith. Let A F be clic
Altitude of the Sun a-
bove the Horizon, in
the North 58 deg. DF
,
Sosth Declination 8 deg.
If you ſubftract the
Declination D F 8 dege.
from F A the Altitude, A
thicre remains so deg.
the Height of the X-
quinoctial above the
Horizon in thc North
that being deducted
out of 90 deg. there :
remaincth AP 40 deg.
for the Depth of the
North Pole under the
Horizon, juſt to E C.
the Elevation or Alti-
tude of the South-Pole above the Horizon in the South.
deg. min.
Sun's Altitude-
58 00
Souch Declinacion-
-08 00
Height of the Equator-
-50
5
P
North
1
1
00
90.00
er
The Latitude isomea
40
00
2
7
..
1
I
23 15
90
00
The uſe of the Fore-Staff in South Latitude for the Sun and Stars.
Su
Uppoſe I were at Sea in a ship the ſecond of Fune, Anno 1694. and I find the
Sun's Meridian Altitude by Obſervation to be so deg. I demand the Latitude the
Ship is in.
deg. min.
The Meridian Altitude North-
-59 'oo .
The Declination North, add-
The Complement of the Latitude-in82 15
The Latitude regwired
207 45
Admit I were 'at Sealin a Ship to the Sesstbivard of the Aquinoétial, the 12th of
January, Anno 1682. and in Longitude 135 Eaſt; and I find by Obſervation the Me-
ridian Altitude 63 deg: 34 min. North: There is required the Latitude the Ship is ini
.
The Declination for chis Meridian, the Lands-end-of England; is about 19 deg.
33 min. the daily sifference in Declinating at this time is 14 min. Therefore if you look
in the Table of Proportion, you will find the Proportional Minutes to be 5, whiclı yoil
muſt add to the Declination of the former Meridian, and the Sum-will be the truc
Declination for the Longitude of 135 deg. Eaſt, which is 19 deg: 38 min.
The
1
.
::
I
CHAP.XV. Directions for obſerving the Stars.
91
deg. min.
63 34
The Meridian Altitude
The Declinacion South--
The Complement of the Latitude
-19 38 folftract.
43 56
90
00
The Latitude required-
46 04
+
Admit a Ship were at Sea the chird of Axguft, Anno 1675. and I find the Sun's
Meridian Altitude to be 59 deg. 36 min. North, The Latitude is required.
deg. min.
The Meridian Afritudė North
59:36
The Declination North, 'addomion
The Coinplement of the Latitude 74.18
34 42
90
00
The Latitude the Ship is in-
15 42
:
:
Suppoſc, a Ship at Sea, the Sun's Declination being 21 deg. 42 min. Sosih, and the
Sun's Meridian Altitude 74 deg. 23 min. South, The Latitude is required the Ship is
in.
deg. min.,
The Complement of the Sun's Meridiari Altitude-15. 37
Subftracted from the Sun's Declinacion
42
The Latitude the Ship is in
، ، ;ܛ ܝ ܕ ' ܙ
21
206 OS
This being made ſo plain and eaſie to be underſtood, need 110 more Precedent:
But obſerve this, If you obſerve the upper Edge or part of the Sun, you muſt ſub-
ftract 16 minutes; if the lower part, add 16 minsites for the Semidiameter of the
Sun, and the Sun (hewech the true Altitude of the Center of the Sun.
Chie.' XV.
Directions for obſerving the Stars.
S
Uppoſe I am at Sea in a ship, and I obſerve the bright Star in the left foot of
Orion Rigel, upori the Meridian, and find his Altitude 44 deg, 32 min
, his
Declination iş. 8 deg. 37 min. North; The Latitude is required, the ship is in.
deg. min.
The Meridian Altitude
44 32
The Declination North -08 37 Substratti
The Complement of the Latitude-
35...55
.90 00
The Latitude the Ship is in
54 05
deg. min.
Suppoſe I am at Sea, and I obſerve the South Balance of Libra, and by Obſervati-
on of the Star upon the Meridian, I find his Altitude 39 deg. 27 min. and his
Declination South 14 deg. 27 min. I require the Latitude I am in.
The Meridian Altitudemmm 39 27
The Declination of the Star -14 37 South.
The Height of the Æquinoctial
90 00
The Latitude the Ship is in
35 56
N2
I
54 04
1
1
92
The Deſcription of the Quadrant. Book II.
I have furniſhed the Practitioner with all uſeful and needful Examples, which I
thought neceſſary for direction, which explains the following Tabies, and ſhews the
moſt eaſie and perfect way of Obſervation, and how to work them on either ſide the
Agrater. Others I confeſs have been larger, but none more plain: for he that can-
not underſtand theſe Rules and Directions, is not fit to be a Mate of any Ship or
Veſſel, nor fit to be ranked among the Ingenious Mariners.
i
CH A P. XVI.
The Deſcription and uſe of the most uſeful Quadrant for the taking Alti-
tudes on Land or Sea, of the Sun or Stars, backwards or forwards, or any
other Altitude of Hills, Trees, Steeples, or Caſtles, or any thing what-
ever,
T
His Quadrant is made of well-ſeaſoned and ſmooth dry Box Wood or Pear-
tree. The Sides or Semidiameter of the Circle is about 19 or 20 Inches. C
V and CH the Arch of the Quadrant is divided into 90 Degrees firſt, and
cach Degree into 6 Equal Paris, each Part being 10 Minutes, which is near enough
for Sea or Land Observations, and numbred as you ſee from 10 to 90 deg. The two
Sides next thc Center, E F and GF, are divided each of them into 100 Equal Parts:
1
ŽA
.
iro
...
inde
Blo
E
WHO
stade math
+
400 01 02 OZ de
K
wadrats
hall
30
Luadrat
2 31
2 ROAN.10007
sight uane
20
4
nieppe
1
G
Eige
Pendriant
1
1
H
That
:
CHAP. XVI. The Uſe of the Quadrant.
93
That which is next the Horizon GFH, are called the parts of Right Shadow : che
other Side E F V, is the parts of contrary Shadow. In the Center at C there is a Braſs
Pin, and on it hang the Thred and Plummet; and on the Side there is a Sight made
of Braſs at E. There is alſo an Horizon-Vane, lec in upon the Center C, with two
Laggs that the Braſs-Pin comes upon in the middle of the Slit; and a Shade-Vane
and Sight-V ane, for Back-Obſervation. The Uſe of the Quadrant is,
EXAMPLE.
.
Admit I am aſhore upon any Land or Iſland, and would know the Sun's Men
ridian Altitude, and true Latitude of the place. Take the Altitude chus; The
String and Pluimmet being hanged on the Center C, turn the Braſs-Pin to the Sun, and
held up the Center until the Shade of the Brafs-Pin ſtrikes on the Sight and Line of
E, the Thred and Plummet playing eaſily by the Side: mark where it cuts the Arch
of the Quadrant, as at F, that is the Sun's Altitude, and reckoned from H; and
the Latitude is found by the ſame Rules as you have been given in the Uſe of the Fore-
Staff. The beſt way to hold the Quadrant ſteady, is to skrew it with a Braſs-Pin
through at K, to a Staff ſet perpendicular, and then you may raiſe it by degrees, as
the sun riſes.
PROPOSITION I.
For Back-Obſervation at Sea.
Take the Handle of the Quadrant at H in your Hand, after the Vanes are
ſer
011,
and fix the Shade. Vane; then hold your Quadrant as upright as you can; then bring
your Sight-Vane to your Eye, and look through your Sighe upon the Horizon-Vane.
You muſt be ſure to hold your Quadrant, lo chiar che
upper part of your Shade-V
ane,
may be upon the upper part of the Slit on your Horizon-V ane, and look through
the slit for the Horizon: But if you cannot ſee the Horizon, but all skie and no
Wador, you muſt draw your Sighi-Vane a little lower down towards H; but if, on
the contrary, you do ſee all Water and no Skie,then ſlide your Sight-Vane a little higher
towards V, and then make 'Obſervation again ; and then if the upper part of the
Shade do lie upon the upper part of the Slit, and you ſee the Horizon at the ſame
time, then it is well, and you muſt wait a little longer as your Judgment thinks fit,
till the Sun is upon the Meridian, and ſo do as you did before; and if the Sun be
to the Weſtward of the Meridian, and falling, you will ſee all.Water and no Skie,
the Work is done for that time and day. Then look what Degrees the Shade-Vane is
put at, which in the Figure is at 70 deg. which note. Look alſo whar Degrees and
Minutes do ſtand againſt your sight, which ſubſtra& from the former Degree's by the
Shade Vane, and the Remainder is the San's Meridian Altitude. As in the Figure, .
The Sight-ý ane is at 25 deg. 30 min. which taken out of 70 deg: the Remainder is
44 deg. 30 min. the Sun's Altitude, or the diſtance of the upper part of the Srin
from the Horizon; from which if you ſubftract 16 min. which is the Sun's Semidi-
ameter, che Remainder will be the Diſtance of the Sun's Center from the Horizon,
or the true Meridian Altitude. · And the way of working your obfervation, is the
very ſame as you have been given in che Uſe of the Fore-Staff.
PROP. II.
Any Point being given, To find whether it be level with the Eye,
or not..
Take the Quadrant and look through the Sight at E and Center-pin C, unto the
Point given, or the Place you would know whecher it be level, or not. If the Thred
fall on ch the Horizontal Line, then is the place level with the Eye: But if it
ſhould fall within, upon any of the Diviſions, then it is higher ; if without the
Quadrant, then it is lower than the level of the Eye.
PROP
94
BOOK IT
The uſe of the Quadrat
PROP. III.
To find the Height of an Houſe, Steeple, Tower, or Trce, from the
Ground, at one obſervation ; and the length of the Ladder which
will scale it.
TE
you
can approach the bottom or foot of the Thing whoſe Height you deſire,
the thing is eaſily performed by this Quadrant or Croſs-Staff, holding up your Qua-
drant to the Place whoſe Height you would know, and looking through the Sight on
the Side E C, going nearer or further from it, till the Thred cut 45 neg. or fall uponi
100 Parts in the Qyadrat : So ſhall the Height of the Thing above the level of your
Eye, be equal to the Diſtance between the Place and your Eye.
If che Thred fall on so parts of a right Shadow, or 26 deg. or Vanes on the
Croſs-Staff, ſet to the Number of Deg. the Height is buc half che Diſtance -
If the Thred cut 25 Parts in the Quadrat, or 13 deg. 55 min. in che Arch of the
Quadrant, it is but a quarter of the Diſtance : But if it fall on 75 Paris, or 36 deg.
53, it is three quarters of the Diſtance. The Rule is,
As 100, to the Parts on which the Thred falleth:
So is the Diſtance, to the Height required.
And on the contrary,
As the Parts cut by the Thred, are to 100:
Soss the Height, to the Diſtance.
E
F
)
1
集
​46
a
Wan
E
( K KEC
ANONİKumNumTMIINI
IIKKIKI
10
KLIK
214.
*t
21
135.13
ILIKI
LGL
P
1.29
ܣܛܘ
A 64
of 192.
B 3.0
C
upon
the contrary
But when the Thred ſhall fall on the parts of the contrary Shadow, if it fall on
50 Parts, or 63 deg: 30 mix. as it doth at C, the Height is double unto the Diſtance
CD. If on 25, it is four times the Diſtance. If the Thred fall
Shadow, this is the Rule,
As the Parts cut by the Thred, are unto too:
So is the Diſtance, unto the Heighr.
.
On the contrary,
As 100, are unto the Parts cut by the Thred :
So is the Height, to the Diſtance.
Theſe are the Rules Mr. Gunter ſhews by the Quadrat , And what hatlı been ſaid
of
7
.
!
95
CHAP.XVI. The Uſe of the Quadrant.
of Height and Diſtance, the fame inay be underſtood of Height and shadow; but
here follows more uſclul Rules than theſe before going.
PROP. IV.
The Diſtance being given, To find the Altitude.
+
Suppoſe E F G D were a Tower, or Steeple, or Tree, or Houſe, whoſe Altitude you
would know, and you cannot come ſo nicer as to meaſure between your Station of 45
deg. and the Befe of the Thing, by reaſon of ſome Wall, or Moat ; yer by the Pro-
portion of the Line of Quadrature, you may help your ſelf by going backwards.
Thus if you could not meaſure the Diſtance from B to D, then go backward from
B to A, until the Thred cut the 26 deg. 30 min. of your Quadrant ; and meaſure
the Diſtance between B and A, as ſuppoſe it to be 32 Foot or Yards, equal to the
Height D E 32 ; The whole Line D A being 64 Feet or Yards, which is double to
the Height. By the Tables,
Suppoſe then the Angle made by tlıc Thred on the Ouadrant A D B, be equal
to the Angle E A D 26:30 min. and the Diſtance A D bc 64 Yards,
or 192 Foot, to fund die Height D Е, I ſay,
ti?
1
,1
As the Radius 90 Degrecs.
100000
To the Tangent of the Angle E AD 26 deg. 30 min.969773
So is AD 64 Yards-
180618
To the Height required, DE 32 Yards 4,50391
1
PROP. V.
The Diſtance being given, To find the Diſtance from the Eye to the
top of the Tower.
+
Let the Diſtance A D be 192 Feet, the Angle at the Eye A 26 deg. 30 min. and the
Diſtance from the Eye or. Hypotenuſa A-E is required.
sveikatos
1
fe
As the Sine of the Angle A ED 63 deg. 30 min.
Is to the Radius 90 deg.
So is AD 192 Fect--
TO A E 214 4. Feet
995179
-IO
I 228330
233151
PRO P. VI.
Some part of the Diſtance being given, To find the Diſtance from the
Eye or Hypoténuſc.
}
1
Let the Part of the Distance given be A B'96 Fiet, and it is required to find tlie
Diſtance from the Eye or Hypothenuſe. E B," which is che' Length or Hypotherufe to the
Triangle D B E. Firſt find the other Angles thus:
A E is 214 4. Fect:2 The Suni is 310,
349206
AB 13.90 . Feet The Difference i 18 1
307371
The Angle at B is 153 deg: 30.7. Tö Tång. : Sum of
1062808
The Half-Tangent is 76 deg. 45.5 oppoſite Angle 76 deg. 45.---1370177
1020971
The
96
The Uſe of the Quadrant. Book II.
+
.
I
deg. min.
The Half-Tang. difference is
58 20.
The Half-Tangent Sum
-76 45
Sum
135 os
Taken out of
180
со
whoſe Complement remains - 44 55 Angle E B D.
Then to find the Length of B E,
As the Sine of the Angle A B E 135 deg. 5 min. or 44 deg. 35 min.-984885
To A E 214 Feet
333142
So is the Sinc of the Angle E A B 26 deg. 30 min.
964952
To the Diſtance from the Eye B E 135 1. Feet or Hypotcnuſe--1298094
3t3209
PROP. VII.
Some part of the Diſtance being given, To find the Altitude,
IO
Kcep the Angle and Diſtance from the Eye found by the former Propoſition.
As the Radius 90 deg.--
Is to the Sine of the Angle E B D 44 deg. 55 min.- 984885
So is EB 135
313225
TO E D the Height required, 95 4. Feet1298100
which is the ſame as before found, without ſesſible difference. By the ſame Rule
you may find the neareſt Obſervacion in the Figure to che Tower.
1
1
PROP. VIII.
To do the ſame thing by the Quadrant, and Scale of Ëqual Parrs,
another way.
Without Calculation, by your Quadrant or Scale of Equal Paris, you may be re-
ſolved of all the foreſaid Propoſitions, by the help of a Line of Chords; you may lay
it down and demonſtrate it, as you ſee the Figure , by the fame Scale of Equal
Parts as you meaſured, the firſt Diſtance, will anſwer all the reſt. This is ſo plain,
it needs no other Precept.
Here is another way to find the Length of the Scaling-Ladder without Calculation,
which in mauy Caſes is the chief thing looked after; which cannot be ſo well done by
the Quadrat, as by obſerving the Angles of the Quadrants and this ische beſt way
I know.
Let your Station be any where at random, or as neer as you can come to the Food
of the Tower or Wall, for the Ditch, or Móat, or Cannon Moe: As ſuppoſe at B, aud
obſerve there the Angle of the Helghi of the Thing, which let be any Degrees what-
foever, as here is 45 deg. I ſay, If you go ſo far backwards from this Station of B,
toward H, till you make the thing appear juft at half the aforeſaid Angle, which is
here 22 deg. 30 min. the half of 45 deg. That then this Diſtance from B to H is the
true Length of the Sloap-ſide BE, without farther trouble; and a Ladder of that
Length will Scale the ſaid Myat or Wall, allowing only the Height of your Eye from
the Ground.
F
.
PROP.
1
1
1
CHAP. XVI. The Uſe of the Quadrant.
97
1
A
PROF. IX.
Part of the Diſtance being given, To find the Remainder of the
Diſtance.
Let
part of the Diſtance given be A B 96 Feet, and the Remainder of the Diſtance
cannot be meaſured, by rcalon of danger of Shos, or Moat of Water, or ſome other
Impediments; Therefore by the 7 Rule I found the Angle at B to be 44 deg. 55
So that thic Angle BED 45 deg. 05 min. is the Complement thereof: Which knowing
I ſay,
As the Radius
To the Sine of the Angle B.ED 45 deg. 5 min. 985011
So is the Diſtance from the Eye BE 135. Fecting 313235
To the Renainder of the Diſtance B D 96 Foot-298236
min.
1
TO
PROP. X.
1
5
By the Height of the Sun, and the Length of the Shadow, To find the
Height of any Tree, Tower, or Steeple,
This Conc leſion may be tried by a little Quadrant or Pocket- Inſtrument, by which
you may take the Sun's Altitude to a Degree, or $, which is ncar enough for theſe
Concluſions.
+
E
47
VIA
C
MY
A.
Sony
А 4 В
D
IO
Suppoſe D E to be a Turret, Tree, or Steeple, whoſe Height is required to be found
by the Sbadow it makes on Level Ground, the Rule is thus, viz. Let the Height of
the Sun be 37 deg.00 wir. and the Length of the Sbadow 40 Foot, the Rule is,
As the Radius 90 deg.
160306
To the Length of ibe Shadow 40 Foot
So is the Tangent of the Sun's Height 37
deg.
-987711
To tbe Height of the Thing defiredo
147917
which is found to be 30-14 Parts, which ſhews the Height to be a little above
30 Foot.
Here is another way to do the ſame without the help of a Quadrant and Sun's
Altitude, viz, Set up a Staff of any Length, ſuppoſe 3 Feet in Length, as C B, and
tha
o
1
98
The Vje of the Quadrant. Book II.
the Shadora which it makes from B to A is 4 Feet; Becauſe the Shadow of the Toms
froin the Baſe thereof to B is 40 Feet, I ſay,
As the Shadow of the Staff, is ta the Height of the Staff:
So is the Shadow of the Stceple, to the Height of the Steeple.
The Operation may be performed by Natural Numbers, or by Logarithms, thus,
viz.
As 4 Feet, to 3 Feet: So 40 Feet, to 30 Feet.
3
228 (30
AA
As the Sliadow of the Staff 4 Feet A B, Log.
4.0, 60206
To the Length of the Staff 3 Feet B C--
3.0, 47712
So is the Shadow of the Srecple 40 Feet DB-
-40, I, 60206
2, 47918
To the Height of the Tower 30 Feet D Emmen 1,47712
*
1
:
1
:
;.
A
1
:
A
I
Conſtant Kalendar;
OR AN
ALMANACK
For Three Hundred Years.
But more exactly ſerving for Nineteen Years,
BEING THE
CIRCLE of the MOON,
OR THE
GOLDEN NUMBER.
With New Exact
1
TABLES
+
OF THE
Suns Declination 1,
Rectified by the beſt Hypothefis, until the
L'E A P-YEARS 1695.
BY
Capt. S A MUEL STUR MY.
London, Printed Anno Domini 1669.
1
1
1
1
1
I
1
1
1
4
1
Book II.
Ιοι
AN
AL MANACK
For XXXII, Years
According to the Engliſh and Foreign Accounts.
Anno
Dors.
Prime.
Epact. 1
Sund. . Shrove
Letter Sunday.
Eafter
White
Sunday, Sunday.
Diff.
The uſe of the Almaa
nack for 32 Years.
be
1410
T*
1666 1404
G
17
131
300
22 1
CON an
22
1 A
1664 1212 CB Feb. 21 10 May 29 1
His Table The
1665131231 A
5 March 26
eth firſt the
251 April 15 June 030
Date of the
1667 15 15 F
7 May 26 I
Tears ; ſecondly, the
1668 16 26 ED
2 March 22
1010
Prime, or Golden Numa
1669171 C
21 April 11 30
ber ; thirdly, the Epact ;
1670 18/18 B
3
fourchly, the Dominical
1670.19129) A
March os
23 June 15
Letter for theſe Tears ;
1672 01 11 GF Feb. 18
7 May 2610 and then in their Or-
1673 2221 E
9 March 30
18! I der the Chief Movable
1674 3 3 D March 1 April 19 June 07 Feafts (vizi) Shrove-
1675 4140
Feb.
141
4 May 239
Sunday, Eaſter.day, and
1676 525 BA
6 March 26
1410 Whitſunday, upon which
1677! 06G
25 April is June 03 1 all the reſt depend. The
1678) 717 F 10 March 31 May 1
190 Foreign Account is com-
1679 828 E March 2 April 20 June 08 monly ten days before
1680 91 9 DCFeb. 22
II May 2010 us; but their Mova-
1681110 20 B
13
31
Ble Feafs fall ſometimes
1682 II
26
16 June @4
at the ſame time with
1683) 12/12 G
18
8 May 27 outs, ſometimes 1, 2, 3,
16847323 FE
10 March 30 18
43 or 5 Weeks before
1685 1404 D March April 19 June 07
ours, as you ſee in the
1686 1515 C - Feb. 14
4 May 23 laſt Column of the Ta
1687 1626 B
6 March 27
ble.
1688171 7 AG 26 April 15 June 03
168911818 F
10 March 31 May
05
1690 191,20 E Marcli 2 April 20 June 08
of the Terms.
Feb.
22 April 17 May 31
1692) 2122 CB
7 March 27 May 15
Here are four times
of the rear apo
1693! 3 3. A Feb. 26 April 16 June
4
16941 414 G
18
pointed for the Deter-
May 27
16951 51251F Feb.
3 March 24!
mining of Cauſes; theſc
are called Terims. Two
of theſe Terms (viz. )
Hillary Term, and Michaelmas Term, are at a conſtant time of the Year: but
xafter Term and Trinity Term are ſooner or later, as thoſe Feafts happen. Each of
theſe Terms liath ſeveral Returns, and each Return hath four Days belonging to it.
The firſt is the Day of Return or Eloin, for the Defendant in a Perſonal Action, or
the Tenant in a Real Action. The ſecond is the Day of Exception, for the Plaintiff or
Defendant to lay an Exception. The third is the Day whereon the Sheriff mult rs-
turn che Writ. The fourth is the Day of Appearance in the Court. "Theſe four
Days follow each other in order, except a Sunday or Holyday take
Holyday take up any of them,
and then clic Day following ſerves for both Occaſions.
The Beginning and End of Hillary Term and Michaelmas Térm, with all their
Returns, you ſhall find in this following Kalendar, which are conſtant if no Sunday
hinder them.
Eaſter
IS
1691 1111
III D
1
The che rear ap-
12
IO2
To. Rectifie the Sun's Declination, Book II.
2 1
121 36147140141
Eaſter Term begins Wedneſday, Fortnight or 17 Days after Eaſter, and ends thg
Munday after Holy Thurſday, or the Munday before VV hitſunday: It hath chieſc five
Returns.
Quind. Paſc. A Fornight)
Trel. Pafc. Three Weekš
after Eaſter.
Menf. Pafc. A Month
Quing. Pafc. Five Weeks)
Craft. Afcen. Thc Day aftcr Holy-Thurſday.
Trinity Tirhi begins the Friday aftci Trinity-Sunday, which is next Sunday to Whit-
Sunday, and hath cheſe four Returns.
Craft. Trin. The Munday after Trinity-Sunday:
OEZ. in. A Week after Trinity-Sunday.
Quind. Trin. A Fortnight after Trinity-Sunday,
Tref. Trin. Three Weeks after Trinity-Sunday.
The Exchequer opciis four Days before Trinity-Term ; but eight Days before the
otlier Terms.
Lo! here a Trade ſurpaſſeth all the reſt;
Nº Change annoys the Lawyer's Iutereſt.
His Tongue besyes Lands, bæilds Houſes, without Toil;
The Pen's his Plough, the Parchment is his Soil;
Him Storms diſturb not, nor Militia Bands,
The Tree roots beſt, that in the Weacher ſtands.
!
Sept.
Dec.
fan.
Feb.
Mar.
Apr.
May.
June.
July.
Aug.
Day Monih.
Sec.
Sec.
Sec.
Sec.
Sec.
Sec.
Nou.
O&ob. Sec.
Scc.
Sec.
Sec.
Sec.
*
4
2
1
.
How to Rectifie the Tables of the Proſtaphărcſes of the Suns Declination
Sun's Declination at any Time by
Proſtaphareſes.
The uſe of the Table.
111713614240128 8115133 42141130
218 37 43 3928 715 34 43 42129 8
When tbe
N huis Kalendar, Printed for the rear
31191374340 27 01635431421291 7
Declination 1665, 66, 67, and 1668. the Sun's 41203814440127 51173543 42 28 6
increaſeth, Declination to be truſted fufficient ; but
5120137144139 26
4118341431411285
then add; for ayy rear after 1668. thc Rule is thus.
61211371431391251 4119135143|41|28
and for de-
7 21 38 431381251 31191354441127 3
creafing,
8 22139143|38|24 2 1935 43 40 26
Subfiraft. For Example. I vould know the Sun's
924 38 44138123
Declination for the Year 1689. you
muſt 10124139 45137123
always lubftract 1668. from the rear gi- 1|25|39|44|37|22|21|37|43|40|24
yen, which is here 1689. the Remainder is 1225140144 37 22
1 2213843139-3
21 Years; which being divided by 4, the
13126 41 43 37 21
1/22 38/4339122
1426 4044 3720 2123138/44/38 21
Ossotient is 5, Leap-rears, and 1 remains,
15127 41 44135 201 3123138 43138/21
which ſheweth it is the firſt Year.
161281421433519!
35119! 412
Now I deſire to rectifie the Table for the
24139 44 37 20
17128 414213519 4.253943138 20
firſt day of April, which in the Kalendar 1829141 43 35 18 5126 39 43 37 19
8 d. 39 m
you have 8 deg. 36 min. and in this Table 19:30141 43 34117 626 39143 37 1817
Declinations
you have 40 Seconds; which multiplied by 2013114242 3417) 727 40|43|36|178
I April
2131142143133116 8127|40|43|361619
5 Leap-years, give 200 Seconds, that is 3m.
22132142143 33 15812841 44/35/1610
20 ſec. to be added to 8 deg. 36 min. So
23|32|4314313914 9129401431351511
have
24/32/43142132114 10129141142134/14/12
you 8 deg. 39 min. for the Sun's Decli-
nation in 1689.
25133144|42|31|12|11|3014143134113112
26/33144141131112/12/30/41/42133113113
27/34 43 4130111131301411423411214
28/35 4341,291 9141314143132 11 16
29 35 43 41 29 91413141 43 32 11 16
30135 1411291 9114132142142132/10116
311361 14111 8113311 1311 117
2
3
4
5
6
1689.
::
A Table
1
Book II. A. Conftant Kalendar. .
103
A Table of the Sun's Declination.
JANUARY XXX.I.
}
Leap je.
Firſt.
Second.
Third.
1
1664
1668
1672
1665
1669
1673
1677
1681
1685
1689
1693
1
1676
1680
1684
1688
1692
1666
1670
1674
1678
1682
1686
1690
1694
1667
1671
1075
1679
1683
1687
1691
1695
1
Day Mo.1
( Epal
Hour.
CMin.
Differ.
europe
Min.
Differ. 1
Min.
|Differ. 1
Min.
Differ. I
21
21
2.B 28
2
2010 2 1
21
21
1014
8120
8120
9
20
20
II
820
South Declination.
0314
II
19
O Rif.
Setting.
A 29
Curcumcif. 21
50 21 43
45
47
4 I
8
21 40110121 3310 21 35110 21 3819
310 120 1214 3 21 3011021 231121 25/10 21 18116
41025
08
4 3
8
21.
II 1121
14 11 21 17111
SE 23
4 4
8
OD 11/21
0011121
031121
6 II
6F *
Twelf day. 20 57.12 20 48112 20 511220
120 54121
ZG 22
7
45:112 20 36/12/20
361220 39. 12j20 42112
8A21
8
33/12 20 24/12/20' 27 12 20 30112
B
06
21/12/20 II 13 20 14 13 20 1713
ICICIS 18
14
812 19 5811320 11320
413
UD
*
14 12
8119 55113119 44114119 49114119 51113
12 E 17 154 14 819 411419 301419
41 14 19 30114119 33114 19 37 14
1 3F 16 Hilary. 19 27 1419 16 14 19 19/14/19 23
23 14
54/G 15
8
12 15 19 01 1519 04 1519 815
15
ISA i*
14 18
8 818 57 15.18 46,15,18 soj 14118 54114
16 B 14 4
8118
42115118 31115118
31115118 35115118 39115
171C 12
124 21
8118 271518 15116118 1910 18 23 16
*
4 23 818 1916 17 59110118 311618
1116
19 E
0 19 de 17 55 16 17 42 17 17 47 16 17 S1,16
9
Oēt. Hilar. 17 38 17 12 2611617 303717 3417
7
17
22 16 17
1717 13117117
13 1717 17117
23 A* Ret. Brev. 17 511716 51 1816 56 1757
0017
23
B 6 6 Term beg. 116 4718 16 34.17.16 38 1816 41
24 C 4 18 4 33 816 301716 16 1816 20.1816
20/18/16° 25117
25 D * Conv.Paul 16 1218115 5818 16
2/18/16
26 E
3 74...37., 815 541815 40 18115 441815 4918
Qu. Hilar. 15 35 19115 2119115
- 3 Except. is
2/10/15
718115 11/19
Ret. Brev. 14 57 19 14 43.11915 481914 52119
29 16 Appear.
14. 38/19/14
22 21 14
C 28 12/4" 46 8114
814 18120114
4/28/14
13120
South Declisation,
20
1819
3
20F
3116
10 Except.
471181
25/17
718
27F
2
25119115
30119
128G
29 A*
T
16119115
POB
28/2014 3319
8120114
1
A Tables
+
1
1
--
104
A Conſtant Kalendar
,
Book II.
A Table of the Sun's Declination.
FEBRUA ë ï XXVIII.
Leap-je.
Firft.
Second.
Third.
1667
1671
1675
1679
1664.
1668
1672
1670
1680
1684
1688
1692
1665 1
1669
1673
1677
1681
1685
1689
1693
1666
16go
1674
1678
1682
1686
1699
1694
1683
1687
1691
1695
Hour.
k Epact.
Soria
1 Da. Mo.
D.Week.ee
Min.
(Differ.
Deg.
Min.
Differ.
Deg.
Min.
Differ.
Deg.
Min.
Differ. I
1
1,53
+
13 24/2013
SA 22
૨૦
1721 12
12112
1121
1
12
! I
IT
19/21/11
9E18
8121
36/2010
2122
South Declination.
O Rif. &
Setting.
37 4
18. 8
13 591
13 44
13 48
2 E 26
Purif. Mar. 13 39|20 24/2013 28 2013 33120
3 F 125
Craft. Pur. 13. 10/2013
32113
8120113 13 30
4G 23 10 Except. 12 5812213 43120 12
48120112 5212
22 Ret. Brev. Iz
IZ 3.9
3/2012 2212312. 27/21112 3212
6 B 21
Appear.
6211129
71C120
1814 59
8
50121111 4012117 451
451?1111
50121
8D19
O in *
35/2011
24121/11 29121
8 0#. Par. 111
142110 57 2211 22TL
Io F 18
8 18 Except. 10 522210 36 24 4022110.47 21
111G117 3 Ret. Brev. 10 3022;to 1412?
192 2110 242
12A s'16 Term'ends. 10. 81221 9 52122 9 57722 10
13B * 5 1 7 9 46 32.9 30122 9 35122 9 40/13
9 24122
2412219
7123 9 1322 9
15 D 113
is 14 719 2122 45433 8
50
8
55423
16 E IZ
IS 718
16 71 8 392; 8 221228 27 23
8
33122
17 F: 111 145 18 7 8
18 7 8 17/22 8 0/23
8
si
I122
18 G
9
100 10 * 7
*7 5423 7 37 23 7: 4223 7.4423
8
15. 22
71 7 312317 14:23 7 20 221 7
7
ols 24 17
81236 5723 6 57231 7 325
6
1915 26 7 6 45 236 28 24 6 34123 6 39 24
5
5 28.
28. 716
221236 41231 6 10 241 6 16 23
23 E4
475, 30 715 59235
30 7 5 59231 5 41123! 5 4712315 53122
24 F
3
20 Matchew.
5 36 23 5 18/23 5 24 23 5 30123
25
5 34 715 12 24 4 55124 5
24 5
261A I
165 36 7 4 4923 4
4 49231 4 37 23 4
4 37231 4 43/22
27 B *
538 7 4 25 24! 4
13241 4 19124
510 20 #4
3 50123 3 5023
33923
C14
+
14 12 Valentine.
IE 122
South Dec inution.
9A
20 B
21 C
22 D
2
5125
81244
2231 3 44
28/C29
{
U Tablc
i
11
.
..
ke
1
Book II. A Conſtant Kalendar. .
dos
A Table of the Sun's Declination.
MARGH XXXI.
Leap-ye.
Firſt.
Second.
Thörd.
:
Hour.
Deg.
( Epactos
Min.
|Differ. I
Som
Deg,
Min.
Differ. I
.
Min.
Differ..
\Deg.
IS
20
Day Mo.
South Declination.
Min..
Differ. I
South Declination
2812312
292412
16 13:12
212
.:58
58143
162411
11123
Il 24
1240
1
1664 1665.
1666
1667
1668 1669 1670
1671
1672 '1673
1674
1675
1676
1677
1678
1679
1680 1681 1682
1683
1684 1685
1686
1687
1688
1689 169а 1691
1692 1693
1694 1695
O Rif.
Setting.
28 1 David.
3.
3
3 26
3 32
JE 26 1415 44 7 2 512412 5712313
31233 9123
31F *
is 46 712
3311412
45:4
4G 25
25
105 48 72
412412 92412
PA
SA 24
5
50 71 40241 46123 1
52 241
6 B 123
05.
52
7
22241
28 245534134
71C 21 115 54 710 $312310 5912311 5231
8D 21 5 56 70
292410 35 2410
412410.46 24
9 E 20 75 58 70
5 2410
17 2410 12321
IF 18
2010 in Nor. 18/24 Nur. 1324 North 611 North i|221 AguinoEtial.
UG IK 6
6io
42 2410
307410:25124
12 A 17 1016
4 61
24.
4924
6
29241
I8
241
516
5312311
47124
I
42241
ISID 114
10 612
12412
Q0131
16 E 12
12 1416
1416 12 612
40/24/2
35 23, 2
2913412 2323
F Flak
14 63
412312 5812312
532312 40123
181G 11
G
216 18
212413
1512313 ! 10
19 A 10 16 16
3 50243
4512313
33128
9 olo 10 714
1412314 812314
56123
7 11
116 614
3712314
312314 251234
1933
22 DI* 6 24 65
oo 114
542314
49|244:43 24
6 76 26 6
6'5
2312315 1712315
I 2723 5
623
2016 28
65
46/2315
40 2315
351235
25 G *
An. Mar. 16
3225
52123
26 A
3 86 32 66
312216
2012316
15123
6 34 616
5412316 482316 42 2216
2810
716 36 67
11/2017
0123
29 D 29 1716 38 6
38227
2222
30 E 28 14 0 20 r18 1/237
502217
45/23
211F 27
16
42
618
1212218
7122
2
3712310
6 2310
00124
542410
13 B 16
61
8 61
24/23/1
* 12123
141
C15
36/24
116
161242
II
512312
North Declination.
North Declination...
6 14
3
2712313
1024
20 B
3912413
212313
211C
22
37
23 E
24 F
4 2016
29123
9/2316
57 225
25 2216
27 B
2
3722
I
16 2217
512317
281237
3312217
552217
1712218
23'22/8
1
P
A Table
!
:.
PO6
Book II.
A Conſtant Kalendar.
A Table of the Sun's Déclination:
-A PRIE XXX.
Leap-je.
Firſt.
Second.
Tbird.
1
1664
1675
1679
1683
.
Y
Dig
Min.
(Differ.
SPA
151
Da: Moi
Hour:
ke Epalt.
D.Veekil A
Min.
Differ. 1
Serie
Wins.
Deg.
Differ. ::
Min.
Differ. |
6 8 45
8 39
8 34
8
Di
B! So
20
A 18
1665 1666 1667
1668 1669 16701 1671
1672 1673 1674
1676
1677
1678
1680
1681 1682
1684 1685 1686 1687
1688 1689 1690
1691
1692 1603
1694
1695
O-Riſi o
Setting.
26 216 44 6 8
29
2 A 25 :326 46 61 9 7122! 9
122 8 562-28
5122
3 B 124
01648
48. 69 29221 g 23 22 9
18122 9 1312?
4.C 23
I 111
69 5021 9 4421
2i 239 23 9 34 21
DI* 6:52'610 II 21/10 6/22 10
122 9 56 32
22
6 54
54 6/10
610 32 2110
32 21/10 27121
27|2110 2221110 1721
boje. 566|10: 53/2010
56.616: 53121110 48121110 43121110' 38137
G 19 18:57
57 611
14121 II 92111 4 21 10 5921
18 1916
35|2I|II
2521lus
20121
B: * Tilo in
55120 11 5020 TI
4621111 401:20
517 5112 75120112
Iof20 12
6120112 I121
D$15 o 67.: 4 S12
26/2012
12 3512012
2120
E !
114 1417 6 512 55 20 1 2 50/2012 46 20/12
41 20
13 17 8 5/13
5113 15.2013 IO/19 13
6.2013
I20
15 G 12
5,217
217: 10
5113 34/19113 29 19 13 2519 13 2019
16 A TI
ISTII 5113: 53.19113' 481913' 441913 40 20
27-13
13 5114 12 1914
71914
3119113 59119
9 117
15 514 37119 14 26 19 14 22 19 14. 1819
8 17 17.5114
17.5 14 5014914 45118 14 41 19 14 3618
10 E7 pto 10:8 15
81815
311814 59:18 14 5519
6 207 20 515 361815 21 1815 1711815 13
18
22,
Gis
7. 22 : 515 44 1715 391715
39 17 15 35 1815 31 18
16
4
: 9 George.
1117,15 5611815 53118115 48 17
B
5116 181716. 1447 16
:21 7 25
24
17/16
1017
3
618
Mark.
-16 3517|16 31 0616 27 17 16 27/17
26 DI 1817 49
49 516 521716 4718 16 44716 40117
271E
*
7 31 5 17 59 16/17 5 16 17
117116 57 17
617
7 32
32
slis 25|16|12 311617 17116117 1316
219 34
34 517 41 15 17
3711617 3316 17
3316 17 29/10
20 A 27
© 20 81756.1517 52/1517
S
521517 4916 17 .4415
ObCNQuito
6.500 Tyto
59
6 IT
in II
30/20111
C17
2
3020112
12 2614
Nurth Dedinations.
North Declination.
10
18.C
19 D
21F
231A
?SIC
2
29
2817
79 G 28
*7.50
17
W Table
P
Book II. A Conftant Kalendar.
167
A Table of the Sun's Declination.
MAY XXXI.
A
Leap-ye.
Firſt.
Second.
Third.
mar
1664
1668
1672
1676
1610
1684
1688
1692
1665
1669
1673
1677
1681
1685
1689
1693
1666
1670
1674
1678
1682
1686
1690
1694
1667
1671
1675
1679
1683
1687
1691
1695
Honr.
Rif. 6
Deg.
Deg
Day Mo.
DaWeeklogo
а Ерасі! *
Min.
1 Differ. 1
Deg.
Min.
\Differ. 1
Deg.
alin.,
| Differ. 1
Min,
Differ. I
QO
55 1418
6.1419
2117
520
I
1112120
8 1220
North Declination.
14A 12
2/20
Setting
B
Ph. Ja. 18 11 18
8 18
4 18
21C 125 II 7 39
Sl18 261518 23115118 19115118
18 151.5
3 D 24 7 40 5118
5 18 4115118 37114118 341518 30 15
41E 23 7 42 5 18
52/15 18 48 14 18 4515
5 F 22 1317 44
slig 91419
2 1418 5914
61G 21 7 45 sig 2511419 19113 19 161419 13 31 141
A 120 917 46 5119 36114119 33114119 3011419 271141
8 B 118
47 519
51194911319 4011319 4311319 40113
91C* 7 49
2113119 591319
56 12 1953 13
10 D 17 1810 in II 20
1412120
512
E 116
7 so 5120 26112/20 24113120 20112120 8113
12F 15 67 53
5 20 38 12 20 30/12/20 3211220
30/12
131G 14 27 54
5120 49
4911 20 471120
43122041
157 55 521 00111/20 57 1220
II
15B * 17 56 52 I11021 9110121
9
3111
16C 11 47 57 5121 2110121
1910121
17D 10 7 58
31 9121 29 9121
18E
9 017 59
381 921
21
36 921
191F
4121
soj 921
921 47 9:21 451 921 431
439
2016 * 8
4121
59 21 56 9 21 54
8
21 529
21 A
6
910 IO II 22
7
58122
4
118
3 412
15 7 22
7 22 131 8/22
91 8
23 Ci* 8
22 7:22 21 7 22 19 7 22
171 8
5 4122 291
241 7/22 31
25E 2
8 6
4/22
361 7122 351 6122
38
7
26 F
I
618 7 4122
43
622 411 622
40
4517
27G
8
491 6132
6/82 471 6/22 461 622
8 4 22
5/22 sol
27 8
9 4123 001 5:22
) S
30026
418
1 423
? 3 5123 2
512} 6 S
13110 18
14122 81 5123
4123
5511/20 52!
611 21
16l1clai
2610121
North Diclipation.
1
I
5121
52: 401021
1411
14/10
34/10
7 138
O
I
81
8122
1
22 B
2 932
11 922
4 4132
19 7:22 17
24D3 1018
722 28
28 722
33 7/22
6122
4122 · 49
281A 28
1918
1518
51! 6
512
2,
291B
55
5/22
53 5122
522 581 5/22
4.23
52 522
571 523
*
IO
P 2
A Table
1
A Conftant Kalendar.
BOOK II.
A Table of the Sun's Declination.
menang
1675
1683
1684
1687
Da.M10.
Hour.
30+ 125
108
JUNE ***
12
Leap-ye.
Firft.
Second, Third.
1664 1665 1666 1667
1668 1669 1670 1671
1672 1673
1674
1676
1677
1678
1679
1680 1681 1682
1685 1686
1688
1689 1690 1691
1692 1693
1694 1695
Rif.
Setting.
25 이이
​13. 23
23
23
23 1318
II 423 17! 4:23 161 4 23 15! 4.23
3/G * * 18
4/23 20
20 323 19 3/23 19 4123 18 4
118
231 223 22 323 22 323
5
B 20
12 4123 25 2123
23 251 3123 241 223 241 3
6C19
8 I2
4123 271 2123
26 2123 262
D 118 1018
13 4123 29 123
28! 1123 28) 2123 281 2
8 E * 8 13 4/23 30 1123
423 30 1123 29 1123
291 123
123 29 1
9
17 6 Days at a 23 311 0 23 301 123 30 123 30
10G|15 191 stand. 23 311 0123
31 1/23 31
123 311
IA 114 IS10
31 1 23
311 0,23 311
12 B 13
Dags
23 30 I 23
31 023 31
13/C 12
4 Morten. 23 291
291 023 30 0123 301 123 30
DII
108 13
13 412}
2} 291 223
29 223 291 1123
291 123 291 123 29
13 4123
4123: 271 2123 271 2;23
281 123
281
16
9 I 3
8
4123
25 3123 251 2123 26
26 2123 26
17
G: 8
8
12 4123 221 3123
23) 2123 23 3123
2241 2
7
8
4/23 19 323 20 3123 20 323
201 323 21 3
19
B! 6
21
II
4123 16, 4123
4123 171 423 171 3,23 19
20 5
8
4123
1 2 4123
13 4 23 13 4123 15
15! 4
21
4
4123 5123 9 $123 91 4/23
II 4
22
E
O
3 5123
5 4123
615
23
F 3
58 5123
O; 5:23
Ii 4!23 24
24 Gli
19 John Bapt.22
531 622 6 22
56 5/22 5715
*.
8 8. 4122 471 6122 49 4/22
26 B 29
6/8 8 422 41 7 2243 222 45
271C28
34, 7 22 30 722 381 7122 39
61
6 4/22 271 7122
24/ 7/22
31 7122
7 22 331
29
E
Peter Ap. 122
2017122 22
8122 341 712?
26
4 4/22
12/8/22 14
14 822
161 812 18
de Epalt.
1D.Weck.lu
Min.
(Differ. 1
Deg.
Min.
| Differ. I
Deg.
Min.
\Differ.
Copy
Min.
(Differ. I
20 I 23
I 2
II
10
141 4
II 4123
12 423
41A 122
21
21/8
271 2/23
018 13
F 17
I
in $123
31 o;23
311 0123
North Declination.
14
North Declination.
I
E 1o
I 2
O CO O CO
18 A
I
I 2
192
Olc
10/8
II
II
IO $123
8 IO
8
*
41 4/23
422
55
251A
Sul 5/22
451 6/22
51 6
451 6
$18
7. 4122
28 D 126 17
*
17/8
13/8
16 이
​A Table
-"
i
i
Book II.
A Conftant Kalendar.
109
A Table of the Sun's Declination.
J u L ♡ XXXI.
Leap-je.
Firſt.
Second
Third.
+
Meeting
1665
1669
1673
1677
1666
1670
1674
1664
1668
1672
1676
1680
1684
1688
1692
1681
1667
1671
1675
1679
1683
1687
1691
1695
1678
1682
1686
1690
1694
1685
1689
1693
Hour.
Deg.
Min.
IDiffer. 1
Shop
Min.
Differ. 1
Suome
Hlin.
| Differ.
Deg.
Min.
Differ. 1
Day Mo.)
( Epalt
Da Weeklo ou
22
22
22
14
21 أو 51
arاو او 21 أو أ47
421
2
429
4519
1018
26 9
57 sle
31021
A16
B15
812
North Declination.
O Rif. ajo
Setting
8
24
3 4 22
4
6
8
IO
21A 23
Ilo 20
2012-
SO! 8.21 58 8122 00! 822 21 8
3 B 122 1418
541 9
410121
8
1
421
38921
921 40.921
21
SD20
421
28 1021 31
11021
33 9/21
35110
61
19 7 58 5121 1811021 21 1021 2311021
7.
F 18
o]7
이​7
21
810121
II II 21
13 10:0 16
81G 17 1917
197 56
56 5720
57 11 21 00111121
STI
91
7 55 520
520 4611120 49|1220 521120
5411
ICB 817 54 520 34/12 20 37 21 20 40 12 20
43 II
uC 14 417 53 520 2311120 26/12/20 29111120 31112
120 12 1717
1717 52 5120 II1220 14/12 20 17 12 20 2011
13 E *
O
in th 19 59 12 20
21330 5 1220
14F II 517 49 519 46 1319. 42319 521319 5513
ISG 10
•
Swithin. 19 33113119 361319 39113119 4213
16A 9 117 46 519 2013119 23 1319 261319 34.13
7 1417 45 519
614.19 1014.19 1311319
18/C * 7 43 518.52 14 18
5214118 56 14/18 59114119
1910
6 10 Dog d. beg.18 3014118 4245,18 451141
8
48.14
5
7
40' 518 2311518 271518
3015 18
34 14
4
017 39 518
81518 121518 151151
8
3 II Magdalen. 17 5311517 57 16 18
415
10 $217 371617 411161745115 17 49 TSI
817 34 4 17 617 2511611 2916 ir 2316
25 C 29 20 S.Jam.Ap. 17 511617 91617 131017 1716
0 516 42 16 16 5311616
116
7 29 S116 33116.16
3311616 3711716 4116,16 45115
517 27 516 (6/17/16 20/17/16
2017 16 24/17/16
2 417116 28
2817
291G*
7 25 555
59117116
7 17/16
301 A 25
117
501715 5417
24 5155 411815 45 18 25
3
515
North Diclination,
17B
16/13
2141
1
19/15
2015
21F
23/G
23 A
24B
2
O
T
21 1617
57/16/17
26 D:28 14,7
27 E 27
28 F 26
IT17
73.23 1414
3/18/16
23/18/1s 2118li's
22
3211815
3618
!
A Table
1
1
{
Sunt maintenant
1fΟ
A Conſtant Kalendar
Book II.
1:
.
1664
1675
1679
1683
-
1694
Hour.
porno
NI Min.
Shop
(Differ. 1
Min.
Du Mo.
1D.Weck.lu
ka Epact. *
Shop
(Diffr. 1
Min.
Differ.!
Deg.
221.7 55122
A
A u Gus I XXXI.
Leap-ye.
Firſt:
Second.
Third.
1665 1606 1 1067
1668 1669 1670 1671
1672 1673
1674
1676 1677
1678
1680
168
1683
1684 1685
1686
1. 1687
1688
1689 1690 1691
1692 1693
1695
O Rif. 6
Setting.
CIK .
Lamas,
5 IS
IS
IS
19
2D 22
317 19 5114 47118114 5211814
52118114 5618115
I 18
3 E 21 00 29
20 '14 19 18 14 34 18 14 38 18 14 42 19
4 F 20
olz 16 514 1019114 15 1914 19914
1217 14 513 5119113
5111913 56194 00 1914 S119
6 A* 17 I 2
5113 32/19/13 37 19/13
37119113 4119,13 4619
71B17 817
10 5/13 1311 1913 1811913 5211913 2719
8C
2017
8 512
54 1912
31913 0819
OD 14 1717 7
SI2 34120112
3912
2012 43 2012
43 20 12 48120
Laurence. 112 14/20/12 19/2012 33 2012
F (12 517 3 Siri 54120111
5912012
312012
8 20
2G11
Sir
33121 II 38:111 43 2011
4
320
O in M11
131201 I 821 II
II 232011 28120
4
B 9 14
58 610
57121 II
320 II
8
6
56 6110 32;2010
42121110 472
16D
7
6
o 11/21 10
2 1/2 TO 26121
*
52
9 5021 9 54121110 00|21|10
21
6 016
6 619
9
9 38 22 9 44 21
19 G4 126 48 69
9 I2;221 9
1712112
46 08 44211 8
55122 9
21
B 3 016
44 618
23 22 8 20/22 8 33/221 8
39122
618
1122 8
I 221 8
24 E 29 9 Barthol.
7 171227
217. 231331 7 28/22
516 36 6 6
562317 001231 7
51231 7 II22
266 26 186 34 0
6 32221 6 38/22
38 22 6 6 43122
6
1012316
27 A*
6 1523 6. 21 22 6 20 22
14 Dog d. endal s 4722
47/22 5 5312
531226 582316
2910 24
66 29 6 S 2512315 30 231 5
2523 5 30231 5 35231 5 4112?
30 D 23 316 27
6) 5
S
Ś 1322 5
1823
21 E 122. 1516 25 6 4 391271 4 441244 50123' 4
Min.
12 ffer. 1
his
IO
14
24118
SG 18
8
Ciis
59119,13
E13
28120
12
187
I
13 A110
Nurth Declination.
522110
721
North Dec.ination.
361211M
1512110
3
54 0 0
lo
619
181
so 6
28122
6122
33/2119
33/21
30%
22122
1 21
20 A *
6. 46
50219
21
39
I
2210
2016 42
O'10
722) 8
1732
23D i*
Ź 33/22
25 F 28
45/23.
6 32 66
28 B 25
4/22
2
8221 5
55123
u Taple
2
1.
e
Book II.
A Conftant Kalendar
III
A Table of the Sun's Declination
.
SEP.TE MB E R XXX.
Leap-ye.
Firſt.
Second.
Third.
1
rum
1
+1664
, 1668
1672
1676.
1680
1684
1688
1692
1665
1669
1673
1677
1681
1685
1689
1693
1666
1670
1674
1678
1682
1686
1690
1694
1667
1671
1675
1679
1683
1687
1691
1695
L
Hour.
Rif. &
Deg:
Day Mo.]
DaWeeklizo
( Epactant
Min.
I Differ. !
Supe
Min.
Differ. 1
Deg.
Mir.
Differ: 1
Differ. I
16
4
C 17
2
North Declination,
EIS
916
I 2
I
13124
North Declisation,
OG
161231
2411
2311
olo
IO2410
1624
13 D
O
Setting.
F 21
Giles.
4
4
22
4 27 4
32
6
2 G 20 1216 21 3 532313
592314 412314 9123
31A 19 io 2013 30233 352413. 412313 4623
B 18
016 17 3 71233 1 2 2313 171243 23123
2016
15
43 142, 492312: 54233
0023
D16
6. 13
6 2023 2
262312
31/232
37123
5712312 3 23 2 81232
8 F 14
6 Lady Fair.t
3312411 391241 451231 50123
GIZ 1816
7 61 ID 2311
21
2723
IOA * 6 5 60 4612410 52 2410 58
3124
пjВ II
76
3 60
23/330 2912310 302410
4024
12C 10 6. I
I 2210
512410
9 .310
in Scuth24 23 South18 24 South 13 7 South 7 9 Aquino&ial.
141
E
7
57 710 4812410 42 2310
3124
151 i*
is ss
55 71 II231 512411
0012610 541231
16 G
6 1215 53 710 35/2411
29241 232311
5 IS SI 71 582311
521231 47 24 1 41123
18 B
4 5 49 72 22 2412
102312
24
is 47
47 72
451232
401241
2 3412412 28i
23
5 45
45 73
3232 5712312 5224
I 9 March.Ap.13 322 3 3 27243
15 23
22 F 29 2215 42 73. 56124 3. 5.01233 441233
3823
O
104
19234. 13;23 4
72314
24/A 27 s 38 714 42 244 362314
301234
25 B 26 1715 36 75
592314
40123
261C
5 34 75 2912315 23/245 171235
3.5 32.75 52 2315 361235
41;24,5 35123
2815
23 16 30 716
6
152316
F
4275
*
Michael 6 38 2316
55.
6
Or 236
492316
44123
165 57
3612310
1824
17A
South Declination.
1612412
1910
2010
211E
South Declination,
2
9 243
21|2413
231G 28 18
2124
25123
36
062314
54/2414
12124
27 D25
29
09/23
3.2/2316
1612216
5823
2123
OG 122
Als
415 26 77
77
A Table
1
112
BOOK II.
A Conſtant Kalendar.
Å Table of the Sun's Declination.
OCTOBER
XXXI.
Lear-ye.
Firſt.
Second.
Tbird.
1664
1667
1668
1672
:
1665
1669
1673
1677
1681
1685
1689
1693
1671
1675
1679
1683
1676
1606
1670
1674
1678
1683
1686
1690
1694
11680
1684
1688
1692
1687
1691
1695
Hour.
O Rif.
6
Se
care
Min.
(Differ. 1
Min.
| Differ.
Deg.
Setting.
(Da.Mo.
D.Werk.lcea
ke Epact.
Mix.
Differ.
I
Deg.
Min.
Differ. 1
I 2
ܪ
1310 20
w 13
8 31122
252318
915 16
16 7
9/22/
2219
37/22
59122
21122
4312
2
D10
71
I
1022
13 F
4112111
3121
South Declination.
IA 21 5 24 7 7 24 7 17 7
7 06
2B20 ols 22 7! 7 401221 7 40231 7
9 40231 7 35 2213
35 22 7 29123
31C 118
8
923/ 8
21221.7 58231 7
581231 7 52123
521231
18
S
8
8
41D *
20221 8
Isi23
SIE117
8 53228 47/22
22 8
42221 8
6F 22 2215 14 71 9 151221 9
4 22 8:59:22
71G114 1815
12 71 9 371221 9 321231.9 271231 9
SA 13
5.10 719 59 32 9 54122 9 49 22 9
9 B 12 :75 8 7110 21 22 10
162210 102110
2
101C II 195 6:7110' 4312210' 37121110 322210
27122
5 4 711 04121110
04121110 59122 10 53 2110
53121110 48121
12/E
9 165 2
25211
20 21 II 15122 11
8 O in M11 46/21/11
36 21 II
141G 7
4/4 59 812 721 12 22I|II
57 21 11 5221
ISA * 4 57 8112 38 21412 2312112 18/2012
16B 6 04 55 812 48 2012 4412112. 39121 12
19C14
4 134 53 813 92113 4
2012
59 2012
18 D * Luke. 13 29/2013 34/2013 20 21 13 14120
19E 3
14 49 813 49.20413 44 20 23 40 20 13 34 20
Treſ. Mic. 14 9/2014
4/2013 591913. 5420
14 1914 24/2014 19/2014 14
22 A 129
11 Ret. Brev. 14 48/2014 4319114 381914 3420
7
7!19115
2/19/14 57119/1 53119!
3410 26 194 40 815 26 19 15: 271915 16.1915
12 191
25|Dl* Criſpin. 115 441875' 40 19 15 35 19115 31119
E 25 16 0 13 m 16
211815 5811815 53 18 15 4918
27°F 24
Menf. Mi. 16 2018 16 16 1816 II!ı8116
181
4 Sim. Jude. 16 38 17 16 33117116 291816
16
29 A 22 17 Ret. Brev. 16 5517 5111816 4718116
3217
Appear. 107
12/17/17
41717
110 120
C120 1314".2 8117 29117'17 25117117 21117117...17117
Sonth Declination.
13 21
34/21
54/20
Nw
2017
211G
o Except.
28
14/20
Term beg. ins
23.B 28
A
1
26
28 G 23
25/18
301B 21
81717
00/18
A Table
!
Book II.
A Conſtant Kalendar .
113
A Table of the Sun's Declination.
7
*
NOVEMBER XXX.
Leap-ye.
Firſ.
Second.
Third.
1678
k
Da.Mo.
Your.
Shop
Min.
(Differ. 1
Epati.
Min.
I Differ, i
\Deg.
Minto
Differ.
Gore
Min.
Differ.
46
17 38
3D*
II
ro
1664 1665 1666 1667
1668
1669
1670 1671
1672 1673 1674
1675
1676
1677
1679
1680 1681 1682 1683
1684
1685 1686 1687
1688 1689 1690 1691
1692 •1693 1694
1695
O Rif.
Setting.
D19
All Saints. 1746
17 41
17 32
2 E 18 . 2 2 All Souls. 18 2116117 58117 11 54116 17 50
3 F * Craft. An. 18 1816118
1816 18 14 10 8 10 1618.
6116
41G |17: 0 Except: 18 3341518 30
16 18 26 1618 22116
A 15 In Powder Tr. 184815118
4875 18 451518 41115118' 37115
B 14
71 Appear. -
19 311519 0011518. 56:51:8 53116
71C 12 1914 16 8119 18115119 14115119 '11115119 8
15
O 25 M119 32 1419 29 1419
25.14 i9 22.14
୨
84 15 819 46 1419 43 14 19 9914 19 36 14
8 4 13 819 5913 19 5013119 $31419 $014
II G 19
4 Martin.
120 13114120 1014120
6113,20 : 313
12 AT 17
7 17 Craft. Ma. 20
2313110 19113 20
1913.20 16.10
13 B* Except. 20 38/12 20
3511 2 20
3112 20: 291
14
13
Ret. Brev. 20
501220 47 12 20 43/12 20 41
15
Appear. 2 10 20 59,12 20 551220 53
S3112
4
210 4
131121
IOI121 7|12] 21
17
8/21 23.1021
21 TI 22
18 11 21
Ox. Mar.) 21
34 II 21 31/10/23 29 1421 216
266
19 A
i
ri Except. 21 441031 41 102T 39110 21 36110
201B
B 29
29 23 Ret. Brev. 2. 5410|21 5
SI 1021 49 10 21. 4610
20 Appear.
31 01:22
ool 921
21 81 9 21 5 ID
27
*22 IL 8122
7 922 59
23 E 26 83
$6,922 2011
22
18
I6; 9122. '13! 8
8/22
3 55 9122 28/8/22 20 8 22 24
4 Qui Mar. 22 35 722 341 8 22
22 301 8
26 A
23 17| Excépt. 421 782 41 722 39
37
27 B *
Ret. Breü. 122 48622 481.722
481 322 45 622 44-7
281C 22
6 Term ends. 22 54
5426]22 541 632
52 7.2.2:5040
299
D121
3.51 9123. 001 o 22 52.5 22 58
522532 58 622:56
41 5123 031 523
216
23 os
03
1320 2311320
South Declination.
South Deciination.
20
1
16
XT21
5112
14/4 3
BI
181G
2
i
21C 28
23
+
22 D 37
10
at او اوه 22
N N N
9.22
22!
24 F
25/G
37 7122
8122
22
30/E 20
2 Andrew. 23
os! S/23
t
Q
Table
}
+
1
1
114
Воок II,
A Conftant Kalendar.
}
A Table of the Sun's Declination.
DE CE M B E R XXXI.
Leap ye.
Firft.
Second.
Third.
1
1
chamane
1664
1668
1672
1665
1669
1673
1677
1681
1666
1670
1674
1678
1682
1686
1690
1694
1667
1671
1675
1679
1683
1676
1680
1684
1688
1692
1685
1687
1689
1693
1691
1695
Hour.
Deg.
Shop
Day Mom
Min.
| Differ: 1
DaWeeklarar
( Epact *
SHIP
Min. 10
\Differ. I
Min.
Differ. 1
Min.
Differ. 1
7123
223
28 I
31) 1/23
1
+
Sonth Declination.
O Rifs
Setting
ISO 20
20 X
23
23 og 23 08
23 06
2/G * 3 SO 9123 14! 623 131 4123 124123 II
5
A 117 1113 49 9/23 18
18 4 23 17 423
4123
10 4123 15 4
4B 15 233 48 923 21 3 33 20 3123
201 4123 191 4
SIC 14 20 3 48 923 241 3123 23 3123 23 323 22
6D 13 3 48 9.23 26 2123 26 3123
26 3123 25! 323
25! 31 23 25 31
7/ E 112 813 47 923 281 2123 28 2123 27) 2123 271 2
8/F II
21 Days are 23 30: 223 29
29 1 22 291 2123
9
GIO
at a ſtand. 23 311 1123
23 301
301 123 30.1123
1123 30
301 2
IOA 9 17 기
​23 1 023 31 123 23 31
311
IIB 8
o in ve
23 311 0123 311 023
31
@j23 31
31 O
1210 7
6 Days in- 123
23 301.123 31 023 31 012331
13 D * creaſe.
23 291 123
301 133 30 1123 30
E 6
23 47 923 28 123 291 2123 291 123 29
4 1513 47 923 26 2123
2123 271 2123 271 223 271
Ꮐ
3 48 9123 24
251 21 23 251.2123 25
IFA :3 313 48: 923 211 3123
221 3123
22 3123 23
3 49 - 9/23 18 3/23 191 323 191 3123 20
19°C
0 3 49 9/23 141 423 15 4123
IS, 4123 16, 4123 16.
20 D 29
I 9 1823
25/23 Io 5123 121 4123
8 Thomas.
23 51 4123
71 5123 8
F 27 3 50: 9123
oo 5123
o
Il 623 21 5133. 315
3
55 623 57 522 541 5
24
À 25 173, 52 922 48 622 49 723
491 7/23 sil
51) 622 526
Chriſtmas. 22
41 7 22 43822 45) 6 22! 46
26 C 23
6 Stephen. 22 34): 7122 36 8122 381 7
22 22
39
22
127D 22 78 John.
27! 7122
211 8122 311 722 327
28 E 21
Innocents. 22 191 8122
23 8
II 8/22 12 8/22
29 F 20 1513 56 9122
IS 8122 17
8
4 9122
61 9122
8 9
1311A118
3'3 58 9121 531 621
South Declination.
2/23
18/B
Mango
I
i
124
6 623
21 E 28
22
23 G 26
SI: 9.122 541
6123
25
B 24
201 822
22 251 ?
301G 19
3 57 9/22
91.8 22
او اور 21 او 57
31 او 5
TO
Book II, To find the Sun's Declination.
115
T
To find the Sun's Declination upon every Day of the Year.
"He Sun's Tear (that is, the time that the sun goeth out of a certain Point of
the Ecliptick, and returnech again to the fame) is not of 365 days juſt;
but about 5 Hours and 49 Minutes inore (char is, little leſs than 6 Hours ;)
Wherefore after three rears, there is always added to the fourth four times 6 Hours,
that is, a Day more in February, for to count the rear or the Repolution of the Sun ini
even Days therefore chat fourth rear is called Leap-year: Therefore when we de-
ſcribe the Sun's Declination in Tables, we always uſe to make four ſeveral Tables, for
four ſuch Years following one the other; and yet by reaſon of the foreſaid difference,
that four Revolutions of the Sun do not juſtly make up one Day, but wants about
48 min. bringeth in proceſs of time ſo great a difference in the Declination, that it is
needful every twenty Years to renew fuch Tables.
How to find the Leap-years, it is thus: Divide the rear of our Lord above 1600.
by 4 ; If the Diviſion doch fall out even, without any over-pless, that rear then is a
Leap-year of 366 Days: But if one of the Diviſion there remain any Nismeber, that
Remainder theweclı how many rears that rear propounded, is after the Leap-year.
For EXAMPLE.
I deſire to know what rear the Year 1666. is. Leaving 1600, I divide 66 by 4,
and find there remains 2 ; for 16 times 4, or 4 times 16, is 64; that taken from 66,
there remains 2; whereby I find the rear 1666. to be the ſecond rear after the
Leap-year. In the like manner you muſt work for
any other Years: Only note this,
If nothing remainech upon the Diviſion out of the Quotient, then it is a Leap-year if
ic be even.
As for EXAMPLE.
• It is required to know what Year 1692 is. Leaving the 1600, divide the 92 by
4, and nothing remains upon the Diviſion, but is even 23 in the Quntient; where-
by I find that rear 1692. is a Leap-year.
For to know the ſame by the foregoing Tables, it is thus. Each Month hath 12 Co-
lumns; The firſt chereof thews che Days of che Month; The ſecond Column, having
the Dominical Letters, ſhews che Days of the Week; The chird Column having cwo
Rows of Figures, the firſt of them ſhews the Epact of the Moon, and
the other the
Hour of the Day, reckoning che ſaid Hours always from Noon; the fourch Column
Thews the Chief Days of the Year, and the Terms and their Returns which are fixed
and certain ; and in the void places it ſhews the Riſing and Setting of the Sun in this
Latitude, and the Place of the Sun every 10 Day or Degree. Theſe four Columns of
themſelves are fit for Mens ordinary uſe, and may be made with a little Art and Pains
to perform all the Concluſions which the yearly Almanacks ſhew and teach, as you
ſhall ſee by the following Rules and Obſervations.
The fifth Column of the foregoing Tables thews the Sun's Declination for every Day
of the Year, for all theſe years in the firſt Column under-wfitten, which are all Leap-
years. The ſixth Column ſhews the Daily Difference of the Sun's Declination. The
ſeventh Column ſhews the firſt Tear from the Leap-year : The eighth, the Daily
Difference of the Sun's Declination in that year. The ninth ſhews the ſecond Year
from the Leap-jear ; The tenth, the Difference; The eleventh, the third Tear from
the Leap-year; The twelfth, the Difference every Day of the Sun's Declination,
you ſee in the Tables. This Table following thews the Leap years, Pirſt, Second,
and Third Tears, as they are plainly expreſſed in the Head of each Table.
as
1
R 2
Leapa.
1
1
1
To find the Sun's Declination Book II.
I 6 6 8
I 67
I 674
g
I 679
I 6 8
I 690
I 694
I
I 692
of the Line, as well between the
116
Leap-years.
Firſt. Second.
Third,
I 6 6 4
I 665
I 6 6 6 I 6 6 7
I 669
I 6 7 I
I 6 7 2
I 673
I 6 7 5
I 676 I 677
1678
80 I 6 8 I
I 6 8 2 1 683
I 6 8
1 685 [ 686
4
I 687
I 6 8 8 I 689
I 69
I 6 9 3
1 6*25
For to find the Sun's Declination, Look for the Day of the Month in the left hand
of the Table, and in the common Angle of meeting you will find the Declination
which you ſeek after.
I: EXAMPLE.
I deſire to know the Sun's Declination for the 22 of May, in the rear 1693. being
the firſt year after the Leap-jear. In the Head of the Table I find the Month and
Year ; on the left hand of the Table I find the Day; and in the Common Angle or Line
of Meeting, I find the Declination I look for to be North 22 deg. 13 min.
II. EXAMPLE,
Upon the 5th of November in the Leap-year 1692. I deſire to know the Declina-
tion of the Sun. In the Head of the Table I find the Month and Day, and in the firſt
Column to the left hand I find the Day of the Month, and in the Common Line of
Meeting, under the rear, I find the Sun's Declination required, to be 18 deg. 37 m.
South Declination; and His Difference in -24 Hoitrs, 15 min.
The foregoing Tables of the Sun's Declination is rectified properly for the Meridian'
of the moſt famous and Metropolitan Cicy of London. The conſtant Kalendar I bor-
rowed out of Ingenious Mr. Philips's Purchaſer's Pattern; at the end of page 247
With ſome addicion it is very afeful with the foregoing Tables..
of the Difference and Æqua-
A Table by which you may proportion the Sun's tion of Declination in dia
Declination to any other Meridian.
vers Places of the Earth.
The Difference in MMMMMMMMM
Declination daily. 00 03'06091215 1831/24
more
Deg. 150
dian of London, have the Ducis-
3010
nation leſs when the Sun de-
4510
oli
clinch from the Line, an 11-
Degrees of 6000
3
creaſcch in Declination eithi
31 4
Difference of 75 o
Northward or Southward, as well
Longitude
gololo
2344
between the soch of March and
either Eaſt or 10s
che i 2ch of June, as between the
O I
2 2 3 4 5
7
Weft.
I 3th of September and the 12th
2'31415 7 8
of December, and more wlich the
135OI 21314 5171 8 9
Sun returned again totvards the
15010
315 618910
16510
Line, whiccher it be North or South
2141568 1011
1801
314161719'II'12
I 2ch of December and the roch
of March, as between the 12th of June and the 13th of September.
Oo
Nether, They come are more
1
I
I
I
I
I
I
2
I
I
2
2
Nmm
w
I,I
22
O
I'2
23.14
1
I 2010
I
2
I
I
'
HIEM
BOOK II.
By the Tables.
!!7.
On the contrary, They that are more Weſterly from the veridian cf London, when
the Declination increaſcth North or Sopil, have more Declination, and leſs when the
Declination decreaſeth ; that is, when the Sin is going towards the Aquinoctial, ci-
ther on the North op Sarth ſide of the Lince the reaſon is, becauſe the Sun çoraçth to
the Meridian Eaſtward, to them that live there, always before it doth to us; and
them thát live mórc Weſterly, have him later, to-their Meridian.
L::
EX A.MA LE. I.
24
440
220
of thoſe that are more Eaſterly, which increaſe in Declination:
On the 26th of March, the firſt rear after the Leng-wear, I deſire to know the
Declination of the Sun at Noon at Bantam in the Eaſt-Indies. I
find by Globes, ör the Plat of Mercator, -that Bantam is to the - 360.24. 110
Eaſtward of the Meridian of London about tio Degrees; we do not
cſteem of a Degree or two, becauſe ic' ambunteth to nothing in this
Praćtice.: 'The Sun for his Courſe round the Heavens and Earth,
which is 360 Degrees
, háth 'need of 24 Hours; What time tvill
(1
2640
110 Degrees have * Fácit:7 Hours, and ſomething more not avoreh
362
the noting; whicreby the Sun comes to the Méridian: 7. Huller's
ſooner, at Bantan, than it doth at London That it is 12 a Olockmat
26109 62.
Noon at Bankam, when it is 4 of the.Clock in the Morning with us
36.
at Londen. The San’s Decliation for the 26th of March, iso deg. ibi
25 min.: The Difference of the Declination of the Day following,
you find is 2 3 min. which it is increaſed; Therefore I ſay, If in 24
24., 23:7
Hours the Declination increaſeth 23 Minutes; How much ilien in
ven
7 Hours ? Facit almoſt .iq Minutes, that the Declination is letto
than it is at London. So that the Declination at Bantam thar Day,
1
is but.6 deg. 18 min. North:"And on the contrary, when the De-
467
ISI
clination decreafeth, work, and you will have the Declination South,
Eaſtward, or Weſtward.
1
16
24
EXAMPLE II.
The uſe of the Table.
On the 17th of September in the ſame Year, I defire to know the Declination thac -
day at Noon at,Bantam." The Declination for the Meridian of London is cha Day
1 deg. 52 min. and the Difference of the Declination of the Day following is 24 min.
decreaſed; and, as was ſaid before in the laft Example
, the difference of Longitude
is rro deg. Therefore I look in the Hcad of the foregoing Table, for the neareſt Inm-
ber to the Difference 24, and find it to fall juſt even
on the Head of
, the laſt Columns
then look on the left hand of the Table for the Difference of Longitude, and I find
105 deg. ncareſt, and in the common Angle of Meeting I find 7, which is to be
ſubſtracted from the Declination in the Azeridian of London abovelaid, i deg. 52 min.
and the Remainder will be the Declination for the. Meridian gr Longitude
, I am in,
which is i deg. 45 min. South : But if the Declinafion decreaſech, as it dotlı here in-
cicaſe, then you muſt have added.
deg. min.:
the Meridian of London the Dçclinación
co2 52
The Minutes Proportional ſubftracted.
The Dçclination for 11o deg. Lo:gitude of Bantam, Eator 45
The Declination of iro deg. Weft of the Meridian of London-00_07
1
+
1
1
7
00 07:1
1
.
'Weftmoisa
EXAMPLE
i
A
118 The difference of Equation of Declinat. Book II.
EXAMPLE III.
19
A Ship coming on tlic ſeventh of November, in the third Year after the Leap gear,
into the great South-Sen, thwart of the Coaſt of Peru, in Longitude 76 deg. The Pi-
lot defircth the Declination there at Noon in that Meridian.
deg. min.
In the Meridian of London the Declination is
19 08. Suuth..
The Minutes Proporcional anddelen
00 03
In the Longitude 76 deg. the Declination.
11 Weft.
In the Longitude of 76 deg. Eaſt, the Declinacion ism 19 os
Two Ships being in Company, they parted at the Lands-end of England: The one
Sails Eaſtwards, and comethi upon his Reckoning upon the 28th of September 180
Degrees on the other ſide thc Globe of the Earth (being the firſt Year after the
Leap-year) and by the foregoing Tables finds the Sun's Declination 5 deg. 57 min.,
The other Ship Sails weſtwards, and meecech the firſt ship at the aforeſaid place, by
his Reckoning not the 28th, bue cn the 27th of September, and findech the Declt-
nation in theſe Tables for that Day;, ſo that they differ in the Time one Day, and
in Declination 24 min. the which proceedeth from this caliſe : The firſt having Sailed
againſt the Riſing of elie Sun 180 Degrees, hach ſhortned his time 12-Hökrs; the
other hach Sailed with the Sun 180 Degrees, hath lengthned his time 12 Hours,
and thereby hath one Night leſs than the firſt. Seeing then in 24 Hours increaſeth
34 Minutes, he that Sailed Eaſtward muſt reckon 12 Minutes Declination leſs, and
he datesailed weſtward 12 Minutes more than the Table doth ſhew; and ſo both of
them all keep one' manner of Declination, to wit, 6 deg. 9 min.
7
may be had.
18
>
21
3 | 17
4 | 15
net ir no no
He ,
A Table of the Refractions of the Sun, Moon, Moon and Stars, cauferli
and Stars, according to the Obſervation of them to appear higher above the
thrice Noble Tycho Brahe.
Horizon chan they are: Therefore
the Refraction is always to be ſub-
Sun Moon Stars
ſtrasted from the Altitude ob-
Alti-
alti-
s'an. Moon
ſeryed, that the true Altitude
tudes.
tudes.
min, min. min.
min, min.
0 34 33 30
06 06
As for Examplc.
26
25 21. 19 os 06
2 20
20
15 20 04 05
The Sun's Meridian Altitude
17 I 2
04 og
by Obſervation being 9 Degrees,
15 II
03 04
I require the true Altitude.
14
14 I
23 03 04
6
13
13 00 24 | 03
Altitude by Obſervation--900
I 2
13 08
25
02 03
II I 2
Refraction ſubſtract -- IO
07 26
02 03
9
IO II 06
27 02 | 03
The true Meridian Alci-
8
10 IO 11
28
O2
tude-
II 09
1 og 29 02
02
I 2-1 09
0909
04 30 OI
02 of the Refraction of the Sun,
13
08
04 31 OI
02 A Dutch ship being the
08 08
03 32 QI OL Diſcovery of a North-Eaſt Paſſage
IS 07 08 03 33
OI
to the East-India, was forced to
16
07 07 02 34 OI 01
Winter in Nova Zembla : the
17 06
07 02
35
OI Mariners beheld the San 14 days
ſooner than he ſhould by his De-
clination, and by Computation 5 Degrees under the Horizon; which is cauſed by the
grols Vapours, and thickneſs of the Air neer the Horizon.
22
OA
deg. mi.
O
1
so'
02
the
+
01
OT
THE
1
1
T
t
the filling up of the Column.
BOOK II.
119
The U SE of the
CONSTANT KALENDAR
I. To know the Day of the Month.
His is the Chief and moſt uſeful Olfervation of any Almanack, and may as
well be performed by this, as by any other. To this purpoſe, you muſt by
the general Kalendar at the beginning hereof, know the Dominical or Svina
day Letter for the Year; then conſidering with your ſelf, whether it be the beginning,
midſt, or end of the Month (as you muſt do in any Almanack) find this Letter
in the beginning, midſt, or end of the Monih, and reckoning from it to the Day
of the Week, either Munday, or Tueſday, or whatſoever other Day it.is, right
againſt the Day of the Weck, you ſhall find what Day of the Month it is. Here is
110 difficulty in this ; only when it is Leap-year, you ſee there is two Sunday Letters,
the firſt of theſe you may uſe only to the 24th of February, and the other all the
rear after.
For Example. In the 1668. the Dominical Letter E D the firſt Sunday in faniary,
is at the firſt E, which is at the fifth Day of the Month; the firſt Sunday in February
is at the ſecond Day of the Month; but the firſt Sunday in Márch is at the firſt D,
which is at the firſt Day of the Month, and ſo all the Year after.
II. To know what Day of the Week any Notable Day will fall
upon, in any Year.
's Firſt find the Dominical Letter in the former Table; then find your Letter in your
Month next before che Day you deſire, and ſo from thence count the Days of the
Week, till you come to the Day deſired. Thus if you would know what D of
the Week Lady-day, or the Annunciation of the Lady Maryfalls upon this rear
: 1668.
the Dominical Letter is D; this is three Days before the ſaid Day, therefore that falls
upon a Wedneſday.
But now in the rear 1669. when the Dominical Letter is C, Lady-day will be upon
the Tharfday. This will be in a ſhort time as ready to you, as if theſe Letters were
painted out for you in Vermilion.
III. To find the Time of Sun Riſing and Setting.
This is ſet down for moſt of the Days in the whole Year, for London; and may
ſerve for all the Eaſt, South, and Weft Parts of England: And this is done after
ſomewhat a briefer manner than is uſual, making the Minutes which are placed in the
midſt
, to ſerve both che Hours of Setting and Riſing ;-which you muſt underſtand
thus: The 7th of February you ſhall find theſe Figures, 4. 59. 8. that is
, the Sin
that Day ſets at 4 h.59 m. that is 59 m. after 4. and riſech at 59 m, 8 h.that is 59 m.
before 8. or almoſt 7 a Clock. And ſo you muſt account them always remembring,
That as the Minutes follow the firſt Figure, ſo they muſt be reckoned in Time after:
as they ſtand before the laſt Figure, ſo they muſt be reckoned in :Time before it.
And think it not prepoſterous that the time of the Sun's Setting is ſet down before
the Riſing; for the Sun's Setting is of moſt uſe, and the other ſerves in a maitner for
ho. min,
If you double this time of Sun Setting
04 59
You have the Length of the Day.
09 58
If you fabſtract it from
-I2
00
You have the time of Riſing, differing in bero from the Kalendar-
07 OI
But all one in effects and this doubled, ſhews the Length of the Night-14 02
IV. TO
1
+
.
1
1
1
P3
1
L
1
A
i
120
The Uſe of the Conſtant Kalendar. Book II.
IV. To find the Place of the Sun.
His is ſet down in the Kalendar, about every tenth Day, to every tenth Degree;
TH
ſo that reckoning a Degree for each Day becween, you ſhall have the place of
the Sun exact enough for moſt ordinary Uſes. Thus the s oth of March the Sun
cnters into Aries; therefore the 15th Day, or five Days after the Sun is in five De-
grees of Aries.
V. To find the Day and Hour of the Change or New Moon, and
thereby the Full and Quarters.
0
F
Irſt you muſt find the Moon's Epa£t for the preſent rear you are in : This Name
ber is found out in the Firſt Book, Page 12. and alſo in the Table before at the
beginning of the Kalendar. The Change alſo may be found out by the Golden Nume
ber; yet that would ſtand fo fcateering and without form, that it is much hand-
ſomer and readier to find out by this Epalt, which runs for the moſt part in a
Conſtant Order, only here and there skipping a Day or a Number, which is marked
with this *.
Having found out the Epalt for this preſent Year, turn to the Month you deſire,
and there find out the ſaid Number of the Epact in the third Column of the Months,
and mark what Day of the Month it ſtands againſt ; for that is the Day of the Change
or New Moon. Likewiſe if you have reſpect unto the Dominical Letter, which is
by it, you ſhall ſee what Day of the Week it is.
Now here in this Column there are two Rows of Figures; The firſt ſhews die
Epact-Number, and the next the Time of the Day reckoned by the Hours from
Noon, which are plain to underſtand till you come to 12 Honrs after Noon, which
is Midnight; but then the Numbers above 12, you muſt reckon to the Morning
of the next Day.
So that theſe Hours after Noon,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
are all one with theſe,
1, 2, 3, 4o 5, 6, 7, 8, 9, 10, 11, 12,
the next Day in the Morning.
Thus in the rear 1666. the Epact being 4, and the Dominical Letter G, you ſhall
find this Epact-Number 4 againſt the ar of July being Saturday; and the Figure o
ſtanding by it, ſhews that the New Moon is juſt at Noon.
Again, You ſhall find the Epact-Number againſt the 16th of November, being
Friday; and the Figure of 2 ſtanding by it, thews char it is about 2 Hours after
Noon the Moon changeth. Now this is the true cime of the New Moon, according
to the Moon's mean Motion ; which though it may differ half a day from the true
Change, yet it ſeldom differs ſo much, and is better for the following Concluſion than
the true time.
Having firſt found out the time of the New Moon, you may from thence reckon
the Age of the Moon, and find the Quarters and Full Moon.
Thus the Moon's Age is Days Hours Min.
At the Firſt Quarter
mm 7mm 09
At the Full Moon
At the Laft Quarter-
22-03 33
An whole Moon
39. I 2 44
Or elſe obſerve the Dominical Letter that is againſt the Epact, or Day of the New
and where you find that Letter again, that is the Firſt Qxarter; for the Full
Moon take two weeks and one Day,which will fall upon the Letter next to it; for the
Laſt Quarter take one Week more, which will fall upon the Letter of the Full Moon.
Thus
II
-14-18
22
Moon;
Book II, The Uſe of the Conſtant Kalendar.
IZI
1100 upon
Thus is the New Moon fali aron A, the Firſt Qurter falls upon the next A, and
the Full Moon on the next B a wrek after, and thc Laft wrter on the next B. And
thus you have this brick Kalend.ir or Conſtant Almanack for many rears; only for the
more cxa&tuſs in the Huur of the Moon's Change and Age, it is reſtrained to 19
Tears: För chough the change of the Moon (for the moſt part) hapnech again upon
the ſame Days, fór ſeveral Revolutions of the Prime or Golden Number XIX ; yet
the ſame Hour of the Day, but alters every Revolution 7 Hours, 27 Minutes,
30 Seconds, procceling forward for the moſt part;, but the Leap-years coming in
with a Duty more than ordinary, kceps this Motion ſo much backward, that in 300
Tears it neither gains nor loſeth a Day, only differeth in the Hear of the Day ; yet
for the more exactneſs, it will be bercer to renew this every 19 Years. All theſe
things this brief Kalendar 1hcws plainly, with little or no trouble more than in an
yearly Almanack.. I thall nowy proceed to ſome ocher Concluſions. I have been very
large already, in the Firſt Book, of Things concerning the uſe of the Moon in other
Concluſions ; to which I refer you for any thing of the Tides, or the Southing of the
Moon, or the Riſing or Setting of the Moon, or what elſe is neceſſary in Navigation.
I thought to have entred my Figure of the Sea-Compaſs, for the Surveying of
Land, which was promiſed in the Argument; As likewiſe the Gunner's Scale and
Gauging :
Rod: But I refer you to the ſeveral Books in the following Treatiſe, where
· the figure and the Life of it, is together for
your
ſatisfaction.
The End of the Second Book.
R
1
زی
}
:
په
:
4
2.
ا
:
أ
IF
1
TE
f
و
.
+
.
123
1. A Triangle is a Figure conſiſting
of three sides and three Angless
1
Mathematical and Practical
THE
IVI iners
IV.
Zini
OR,
STURM Y’S
܀
5
I
;
ARTS
The Third Booki
men
CHARI
of the Nature and Quality of Triangles.
1
His Third Book is as it were a Key to theſe that follow, the Sub-
ject whereof is Trigonometry; therefore I hold it convenient
before I come to the following Praćtice, to ſay ſomething con-
cerning Plain Triangles at leaſt, although that Subject be
handled by divers more able Mathematicians already, whoſe
Works are extant, (viz.) Pitiſcus, Snellius, che Lord Napier,
Maſter Gunter, Maſter Norwood, Maſter Gellibrand: fo that
thoſe who deſire to make a farther ſcrutiny into Trigonometry,
may peruſe the forementioned Authors.
Before we come to thew
how the Quantity of the Sides and Angles of any Triangle
may be found by the Tables of Artificial Sines and Logarithms, and by the Lines of
Artificial Sines, Tangents, and Numbers, cake theſe following Theorems, as Neceſſa-
ries thereunto.
as in the Figure DBC.
i
D
B
II. Any two sides of a Triangles
are called the sides of the Angle
comprehended by chem ; asche
Sides C B and'D B, are che
Sides containing the Angle
CBD.
R 2
III. The
11
124
Trigonometry.
Book II.
(
1
III, The Meaſure of an Angle is the Quantity of an Arch of a Circle, deſcribed
on the Angular Point, and cutting both the containing Sides of the ſame Angle; As
in the Triangle following, the Arch CB 'is the Meaſure of the Angle at A, the arch
KD is the Meaſure of the Angle at E, and the Arch F G is the Meaſure of the Angle
at H. Each of theſe Arches are deſcribed
on the Angular Points A, H, El and
cus
clre contaiging Sider
contain
A
SOM
ol
7181
1
wir Thors
6+
he
E
B
F
tril on
478
414
yo
E
1
Elly
1
+
***
H
D
pole i
Tiltak
:RSV.?
IV. A Degree is the 360 part of a Circle.
V: A Semicircle containech 180 Degrees.
VI: A Quadrant containeth's0 Degrees.
VII: The Complement of an Angle leſs than a Quadrant, is to much as chat Angle
wanteth of 90 Degrées, as if che Angle A HE ſhould contain 41 deg. the Conspie-
ment thereof would be 4ô deg: Forif you taikė 41 from go, there will remain 49 deg.
VIII. The Complement of an Angle to a Servicircle, is the Remainder chercof to
180 Degrees.
· IX. At Angle is' eittier Right; Achte; or 'Oltiife.
X. A Rigbi Angle is that whoſe Meaſure is a Quadrant.
XI. An Acute Angle is leſs than a Right Angle.
XII. An Obtul Angle is greater than a Quadránt.
XIII. A Triangle is either Righi- Angled; or Obrufe-Angled,
XIV: A Right-Angled Triangle is that which hath one Right Angle as the Tri-
angle A HE is Right-Angled at E.
xy. In every Right-Angled Triangle, that Side which ſubtendeth or lieth oppoſite
to the Right Angle, is called the Hypothenufa ; and of the other two Sides, the one
is called the Perpendicular, and the other the Baſe, at pleaſure : Bue moſt commonly
the ſhorteſt is called the perpendicular, and the longer is called the Bafe
. So in the
former Triangle, the Side A His the Hypothenuſa, HE the Baſe, and a Ethe Per-
XVI: In every Righi- Angled Triangle, if you have one of the Acute Angles given,
the other is alſo given, it being the Complement thereof to 90 d:g. As in the Triangle
A HE, ſuppoſe there were given the Angle H A E 49 deg. then by conſequence the
Angle -AH E muſt-- Be 4 Degrees, which is the Complement of the other to go
XVII. The three Angles of any right-Lined Triangle whatſoever, are equal to
LI
pendicular.
1)."
Degres.
1
two
CHAP. II.
Iriganometry.
125
two Right Angles, oy to 180 Degrees: Sæthat if of aný Right-Lined Triangle, you
have any two of the Angles given, you liave thc chird Angle alſo given, it being clic
Complement of the other to 180 D grees.' I.
Lisac
org
1
i Luxos
c
8',
1
k126 :
+
::D!
1
CA
&T
6 5
B
**
که م
L
so
"
.
**
+
I
So in this Triangle A B C, if there were given the Angle B A C 36 Degrees, and
the Angle A CB 126 Degrees, I ſay by the conſequence, there is alſo given the third
Angle. For if you add tlacitivo given Jalgtskotagèrbertzange fubftrax it from 180
Degrees, there will remain A B C 18. Degrees. The tw@given Angles being taken
from 180, the Remainder is the Angle required.,
In all plain Triangles whatſoever, the IsFoletare'in proportion one to the other, as
dic Sives of the Anglès arperipe, sp thựſe Sidese : So in the Triangle A B C, the Sine
of the Angle AC Bºis in fuch proportion cpo flic
Side AB, as the Sine of "cht Angle
CAB is to the Side B C. And 10 of any other.
mesne
CHAP, IN
Containing the Do&rine of the Dimenſions of Right-Lined Triangles,
whether Right-Angled or Oblique-Angled, and the ſeveral Caſes
therein refolved, both by Tables, and alſo by the Lines of Artificial Num-
bers, Sines, and Tangents.
Come now to Thew you how a Plain Triangle may be reſolved; that is, "bý hia-
ving any three of the ſix Parts of a Plain Triangle
, ta find a fourth by the inftrue
ments before-inentioned.
In all the Caſes following I have made uſe of but two Triangles for Examples; one
Right-Angled, the other Oblique-Angled : Bur in either of them I have expreſſed all
the Variccies that are neceſſary; lo that any threc Parts being given in any of them, a
fourth may be found at pleaſure.
The Sides of any Plain Triangle may be meaſured by any Meaſure of Scale of
Equal Parts; as an Inch divided into io Parts, or 20, 30 Parts; or likeiviſe into
Inches, Feet, Tards, Poles, Miles, or Leagues,
Draw a Line at pleaſure, as Ą B; How to las
and from the Peint A leçit be requi- dowa di
red to protract an Angle of 41 deg. Angle by
TL 134 min. Firſt extend the companies the Line of
upon the Line of Chords, from the
beginning thereof to 60 deg. alvvays,
and with this diſtance ſetting one
riFoot upon the Point A, with the
other deſcribe the pricked ArchB.C:
Theo with your Compaffes take 44
deg. 24 min. (which is cliç Quantity
A
6 В. of the inquired. Angle) out of the
Linc of Chords, from the beginning
thereof
I
t
!
9
Chords.
1
1
+
,
B
i.
126
-Trigonometry. BOOK III.
In the reſolving of Plain Triangles there are ſeveral Cafes, of which I will only
thereof co 41 deg. 24 min. Keeping the Compaſſes at this diſtance, if you ſet one Food
thereof upon B, the other'will reach upon the Arch to C. Laſtly, draw the Line
A C. So the Angle C A B ſhall contain 41 deg. 24 min.
To find the Suppoſe C A B were an Angle given, and that it were required to find the Quan-
Degrees tity thereof. Open your Compaffes
, as before, to 6o deg. of your Chord; and placing
one Foot in A, with the other deſcribe the Arch B C. Then take in your compasſes
gle.
the Diftance C B, and meaſuring that Extent upon the Line of Chords, from the
beginning thereof, you ſhall find it reach 41 deg: 24 min. whích is the Quantity of
the inquired Angle.
If any Angle given or required ſhall contain above 90 Degrees, you muſt then
protract it at twice, by taking fiſt the whole Line, and then the Remainder.
The ſeveral caſes of Right-Angled Triangles,may not only be applied to Navigation,
but alſo in the taking of Heights, as is thewn elſewhere: And the Obligu:- Angles
Triangle, for the taking of Diſtances, taught in this following Treatiſe.
contained
in an An.
1
inſiſt on theſc, which have moſt relation to the Work in hand.
1
of Right-Angled Plain Triangles.
1
CAS E L
In a Right-Angled Plain Triangle, The Baſe and the Angle at the Baſe
being given, To find the Perpendicular.
Uppoſe that the Line C A (in the following Figure) in the Right-Angled Trian-
Sun
gle, were a Tree, Tower, or Steeple, and that you would know the Height
thereof; you muſt obſerve with your inſtrument the Angle CBA, and meaſure the
Diſtance BA.
So have you in the Right-Angled Triangle A B C, the Baſe 405 Poot (Miles or
Leagues the denomination might have been as well) and the Angle at the Baſe 32 deg.
and it is required co find the perpendicular A C...
Now becauſe the Angle C B A is given, the Angle BCA is alſo given, it being
the Complement of the other to 90 deg. and therefore the Angle BCA is 38 Degrees:
Then to find the perpendicular CA, the Proportion is,
As the Sine of the Angle B C A 58 deg. (which is) 9928420
Is to the Logarithm of the Side B A 405 Foot
2607455
So is the Sine of the Angle CBA 32 deg. (which is) 9724210
The Sum of the Second and Third added.
12331665
The firſt Number fubftracted from the Sum-
9928420
To the Logarithm of the Side CA
1403245
The neareſt Abſolute Number anſwering to this Logarithm 2403245, is 253 fert;
and that is the Length of the Side CA in Miles or Leagues, or the Height of the
Tree, Tower, or Steeple
, which was required.
U GENERAL Ruir.
ING
N all Proportions wrought by Sines and Logarithms, you muſt obſerve this for a
General Rule
, (viz.) To add the ſecond and third Numbers together, and from
the sum of them to fubftract the firſt Number ; ſo ſhall the Remainder anſwer your
Queftion demanded, As by the former Work you may perceive, where the Loga-
rithm of the Side B A 2607455 (which is the ſecond Term) is added to the Size of
the Angle CB A 9724210 (which is the third Term) and from the sum of them,
namely from 12331665, is ſubſtracted 9928420, the Sine of the Angle B C A,
which
t
0
A
1
CHAP. II.
Trigonometry.
127
which is the firſt Number, and there remaineth 2, 403245, which is the Logarithm
of 253 almoſt, and that is the Length of the Side required.
L
A
283
ferie
478
32
go
A
B
405
To reſolve the ſame Work by the Line of Sines and Numbers.
Ou may work theſe Proportions more eaſily by help of the Line of Sines, Tan- Place here
gents, and Numbers, on your Scale, the Proportion being as before.
the Line of
Therefore if you ſet one Font of your compaſſes at 58 deg. in the Line of Sines, Numbers,
and extend the other Foot to 405 in the Line of Numbers, the fame will reach from tlre Sines, and
Sine of 32 drg.co 253 in the Line of Numbers, which is the Length of the Side A Tangents.
C, which was required.
Or otherwiſe
, Extend the compaſſes from the Şine of 32 deg. to the Sine of 58
deg. in the Line of Sines; the ſame Extent will reach from 405 in the Line of Nam-
bers, to 253, as before, the work is much abbreviated, there being no need of Pen,
Ink, nor Paper, or Tables; but only of your compaſſes.
CA S II.
The Baſe and the Angle at the Baſe being głven, To find the
Hypothenuſa.
N the ſame Triangle A B C, Ler there be given (as before) the Baſe A B 405 Foor,
Miles, Leagues, or Perches, and the Angle A B C 32 deg. and let it be required
to find the Hypothenuſa B C. Now becauſe the Angle CB A is given, the other
Angle BCA is alſo given; and the Proportion is,
As the Sine of the Angle BCA 58 deg.
9, 928420
To the Logarithm of the Side 405 Foot
2,607455
So is the Sine of the Angle C A B 90 deg.-
ļo, nooooo
The Sun of the ſecond and third Number
12, 667455 added
To the Logarithm of the Side B C, which is 2,679033
Tlie Abſolute Nymaber anſwering to this Logarithm is 478; and ſo many Feet,
Atiles, Leagues
, Plichek. is_che Hypothenifa, according to the denomination of the
Queſtion; that is, whether it be Fees, Perches, Miles, or Leagues. By either of theſe
.
By the Line of Numbers and Sincs..
As was ſaid before, the work is altogeclitr, che-fame with the Tables; For che
Proportion being
As the Sine of the Angle BCA 58 degrees
Is to the Length of the Side B A 405 Foor:
So is the Sine of the Angle CA B 90 Degrees,
To the Length of the Side CB 478.
Extend
the work is the ſame way.
-
+
128
Trigonometry. BOOK III.
Extend the compaſſes from the Sine of 58 deg, to 405 in the Line of Numbers;
the ſame Extent will reach from the Sine of 90 deg. to 478 in the Line of Num'ers,
and that is the Length of the Side B C. Or you inay cxtend the Compaſſes from the
Sing of 58 deg. te go deg.chę ſame Extent will reach 405 to 478, as before.
CAS E III.
Miles, Leagues,
The Hypothenuſal and Angle, at the Baſe being given, To find the
Perpendicular.
N the ſame Triangle let there be given the Hipothenuſal B C 473 Fies, Poles,
and the Angle at the Baſe C B A 32.deg. To find the Perpen-
dicular CA.
The Angle CAB is a Right Angle, or 90 Degrees ; Thereforeihe Fruportion is,
As the Sine of the Angle CAB 90 deg
10,000000
Is to the Logarithm of the Side B C 478 2,679428
So is the Sine of the Angle C B A 32 deg.--
9,724210
Tothe Logarithm of the Side A C 25392, 403638
The Number anſwering to this Logarithm'is 253 fere; and that is the Length of
the Side CA in Feet, Poles, Miles, or Leagnes.
Here the work is ſomething abbreviated; for the Angle CA B being a Right Angle,
and being the firſt Term, when the ſecond and third Terms are added together, the
firſt is cafily ſubſtracted from it, by cancelling the Figure next your left hand, as you
fee in the Example; and fo the reſt of that Number is the Logarithm of the Numza
ber ſought.
By the Line of Sines and Numbers.
Xtend the Compaffes from the Sine of 90 Degrees, to 478; the ſame Extent
will reach from che Sine of 32 Degrees, to 253.
Or, Extend the Compaſſes from the Sine of 90 Degrees, to the Sine of 32 Degrees;
the ſame Extent will reach from 478, to 253; and that is the Side C A.
T
CAS E IV.
The Hypothenuſal and Angle at the Baſe being given, To find the
Baſe.
Et there be given in the Triangle the Hypothenuſal B C, and the Angle at the Baſe
CBA, and by conſequence the Angle BCA the Complement of the other to
go degrees : Then to find B Å, the Proportion is,
rot
응 ​&
1
OMFEST
827
+
11,1
с
As
.
m
HAL
1
CHAP. II.
Trigonometry.
129
As the Sine of 90 deg, CAB.
-10, OCO000
To the Hypothenuſal C B 478-
2, 679428
So is the Sine of the Angle ACB 58
9,928420
To the Logarithm of the Bale A B-
22, 607848
The neareſt Number anſwering to 2,607848, is the Logarithm of 405: And ſo
many Foot or Poies, or if the Queſtion be Miles or Leagues, is the Baſe or Parallel of
Longitude AB.
Now you ſee the former Figure is turned, and therefore very fitly may have
other Denominations (or Nim's) So that in the Art of Navigation, it will not be unfic
to call one of theſe Sides the Parallel-Side, as A B, or Side of Longitude, or Meridian
Diſtance; the other the Perpendicus!ar-Side, or the side of Latitude, asĆ A; and the
Hypothenufa!, the side of Diſtance C B, and che Arches to lay down from the Chords,
as before-directed.
By the Line of Sines and Numbers.
He Angle given, as before, Extend the compaſſes from the Sine of 90 deg. unto
478. the ſame Extent will reach from the Sine of 58 deg. to 40s in the Line
of Numbers.
Or, Extend the Compaſſes from the Sine of 90 deg. co the Sine of 58 deg. the fame
Excent will reach from 478 to 405, which is the Length of the Baſe turned up, or
Parallel-Line of Longitude, as before laid, A B.
Case V.
Let the perpendicular be the Difference of Latitude 253 Leagues,
and the Angle at Css. Web. W. 1 deg. 45 min. Weſterly, or
58 deg. Let it be given to find the Hypothenuſal or Diſtance upon
the Rhomb.
7
F the Perpendicular or Difference of Latitude 253 Leagues AC be given, and the
I
Anglè ar A CB, S.W.6.W. I deg. 45 Wefterly, or $8 deg. Then by conſequence
the Angle A B C, or Complement of the Rhomb is allo given; taking the firſt out of
90 deg. then the Hypothenaſal may be found thus.
As the Complement Sine of the Rhomb 32 deg. at B---- 9,724210
Is to the Logarithm of the Difference of Latitude 253---12,403121
So is the Sine of the Angle or Radius 90 deg.
.
To the Logarithm of the Hypochenuſal, or Diſtance upon?
the Rhomb or Courſc failed! 478
2,678911
Here becauſe the Angle CAB is a Right Angle, or go Degrees the Radius, and
comes in the third place, I cherefore only put an Unity before the ſecond Term, and
ſo ſubſtract the firit Term, andittie Rennainder is 2,678911s the Abſolute Number
anſwering thercunto is 478, the Side required.
10, 000000
1
By the Line of Numbers.
Xtend the Compaſses from the Sine of 32 deg. to 253 deg. the ſamie Diffànce. will
reach from the Sine of go deg. to 478, the Side required.
Or, The Diſtance between che Sine of 32 deg. and 90 deg. will be equal to the
Diſtance between 25 3 and 478, and givech the Side required
;
S
CASE
1
1
1
130
Trigonométrý.
Book III:
9
CASE VI
The Hypothenuſal or Diſtance Sailed, and the Perpendicular or Diffe-
rence of Latitude given, To find the Rhomb or Angle A B C.
IN
N the foregoing Triangle, there is given the Hypothenuſal or Diſtance failed, CB
478 Leaguis, and the Perpendicular or 253 Leagues difference of Latitude, and
it is required to find the Angle A B C, and by it the Rhomb.
As the Logarithm of the Hypochenuſal C B 478 Leagues - 2,679428
Is to th: Right Angle or Radius 90 deg. C A B-
10,000000
Só is the Logarithm of the Perpendicular 253 CA---
2, 403121
To the Complement Sine of the Rhomb, or Sine of the Aa?
gle A BC 32 deg.--
$ 9,723693
The ncareſt Nem!er anfwering to 9; 723693, is the Sine of 32 deg. which de-
dufted from 90 deg. cherc remains the Angle of the Rhomb 58 deg. or J.W.b.w.
I deg. 45 V Veſterly.
E from 253, 40 32 deg
1
By the Line of Numbers.
Xtend the Compaſſes from 478, to the Sine of 90 ; the ſame Diſtance will reach
from
to
Or, Extend the compaſſes from 478, to 253; the ſame Extent will reach from the
Sine of 90, co the Sine of 32 deg, whiclı: is che inquired Angle A B C, and the con-
plement of the Rhomb..
9
C A'S E' VII.
The Hypothenuſal, and the Parallel of Longitude, and the Radius
given, To find the Rhonb or Courſe sailed.
N the forcgoing Triangle there is given the. Hypothenuſal or Diſtance Sailed, CB
478. Leagues, and the Right Angle C AB.90 deg. the Radiisand the Parallel of
Longitude or Baſe 405 Leagues, to find the courſe or Rhomb: failed, or the Angle
A CB.
91:
Hii haitotine?
عم :'(
ر.ن.
4
mai 2; 679428
As the Hypothenuſal or Diſtance Sdiled 498 CB
To the Right Angle CA B Radius or Sine of go deg. TO,000000
So is the Parallel of Longitude; or Baſe A B 405 Leaguesit 12, 607455
To the Sine of the Angle of the Rhombor Courſe Sailed 58
or S.W.B.W, i deg. 45 weſterly
9,928027
ki
By the Lines of Sines and Numbers.
EX
Xtend the Compaſſes from 478 in the Line of Numbers, to the Sine of 90 deg.
the ſame Extent will reach from 405, to the Sine of 32 deg.
Or, Extend the Compaffes from 478 to 405; the ſame Extent evill reach from the
Sine of 90, to the Sine of 37 deg, A C.B, the Angle of the Rhomb or Courſe failed,
which was required.
+
ܪܢ
4
of
1 2 A
ܐ ܀
f
CHAP. II.
Trigonometry.
131
Of Oblique-Angled Plain Triangles:
CASE VIII.
Having two Angles, and a Side oppoſite to one of them, To find the
Side oppoſite to the other.
IN
N the Triangle QR 5, is given the Angle QSR 25 deg. 30 min. and the Angle
QRS 45 deg, 20 min. and the Side RS 305 Feet ; And it is required.co find the
Side QR.
Here note, That in Oblique-Angled, Plain Triangles, as well as in Right-Angled,
the sides are in proportion one to the other, as the Sines of the Angles oppoſite to
thoſe Sides: Therefore,
As the Sine of the Angle QRS 45 deg. 20 min.-
9,851997
Is to the Logarithm of the Side QS žoš
2, 484299
: So is the Şine of the Angle QSR 25 deg: 39 min. 2.6339.84
The Sum of the ſecond andi tbird Terms-
I 2, 118283
The firſt Term ſubſtractedom
9, 851997
To the Logarithm of the Side QR--
The ncareſt Abſolute Number anſwering to this Logarithm is 185 ; änd ſo many
Feet is the Sido QR.
1
2, 266286
1
By the Line of Sines and Numbers.
He Line of Sines.and Numbers will reſolve the Triangle by the ſame manner.of
Work, as in the other before. For if you excend the Compasſes from the Sine of
45 deg. 20 min. to 305 Foot, the ſame Diſtance will reach from 25 deg. 30 min. to
185 Foot, and ſo much is the Side QR.
Or, Extend the Compaſſes from the Sine af 45 deg. 20 min. co 25 deg. 30 min. the
ſame Diſtance will reach from 305 to 185, the Length of the Side inquired.
$
i
1
1
309
3
Logoro
+
T8S
230
45.20
1
R
40 g
s
HT
In like manner if the Angle R RS, 10g deg. To min, and the Angle QR $ 45 deg.
20 min. and the Side OS 305 Foot, had been given, and the side Ř S required, the
manner of Work had been the fame: For,
As the Sine of the Angle QR S 45 deg, 20. min.
9, 851997
Is to tbe Logarithm of the Side R$ 305
2, 484299
So is the Sinc of R OS 109 deg. io min. (or go deg. so min.) 9,975233
The Sum of the second and third Terms
12,459532
The firſt Term fubftrated-
9,851997
To the Logarithm of the Side R S 495->
2, 697535
S2
The.
12
132
Trigonometry Book III.
RS.
-
til
E
::
T
TO
..
together wich thc Angle QSR 25 deg. 30 min. and let it be required to find
The Abſolute Number anſwering to this Logarithm is 406, and ſo much is the Side
In this Cafe
, becaületbe'küçlerosis hable than yopilegres
, you mult chere-
fote take the complement thereof
' to 180 deg. ſo 109 deg. 10 min. being taken from
180 deg. there remains 70 deg. so min. whoſe Sine is the ſame with 109 deg. 10 min.
And fo you muſt work with all Angles above, 90 Degrees; and" ſo will the Geosple-
ment to look as befcte directed; effect the Dame thing.
By the Line of Numbers and Sines.
Xtend the Compaſſes from the Sine bf 45'deg: 20 min.' to 305 Peets the ſame Di-
ſtance will reach from 70 deg. 50 min. to 405.
Or, Thie Compaſſes exrčnded from the 'Sing of 45deg. 20 min. to 70 dég: 50 min.
the fame Excent will reach from 305, to 406 in the Line of Numbers, which is the
Side RS required.
.
CA SE IX.
Two Sides, and an Angle oppoſite to one of them being given; To find
G... the Angle oppoſite to the other.
N the fàlme Trianglez-let-chere be given the Şide: 098395,' and Q R 185 Feet,
IN
the Angle Q R S. The Proportioh is,
As the Logarithm of the side Q R 185
2, 267172
Is to the side of the Nagle OSR"zigidega za minket 93633984
So is the Logarithm of the Side R$ 305-
2,484299
The Sum of the ferond and third Numbers--
12,118283
The firſt Number ſubftracted from the Sum
2,267172
To the Sine of the Angle QRS 45 deg. 20 miu.
i 97851111
The neareſt Degree anſwering to this'sine is 45 dég: 20 min. which is the · Angle re-
quired Q R S.
By the Line of Sives and Numbers.
E
Xtend the compaſſes from 185, to 25 deg: 30 *} the ſame Diſtance will reach
from 305, to 45 deg: 20 min. che Angle QR %.
Or, Extend the Compalles from 185, to 305; the ſame Extent will reach from 25
deg. 39 min. to 45 deg. 20 min. as before.
ins
CASEX
Having two Sides, and the Angle contained between them given , To
find either of the other Angles. : 3)
Or the performance of this problem, Suppoſe there were given the Side RS 406,
and the Side R Q 185, and the Angle comprehended by them, namely the An ?
gle at Rx 45 deg. 10.min. and it were required to find either of che other Anglese
Firſt; Take the Sum and Difference of the two sides given; their sum is 591,
and their Difference is 291. Then knowing, that the three Angles of all Right-lined
Triangles, are
equal to two Right Angles
, or i80 Degrees (by the 17th Theor,
of Chap:
3.) Thicrefore the Angle ÖR being. 45'deg. 20°min. if you fübftract this Angle
from 180 deg. the Remainder will be 134 deg: 40 min: which is the sum of the two
unknowa: Angles ar Qand S;. the half chereof is 67.deg. 20 min.
The
S.
"L
inni
rii
1
2
OS
2
i
i
1
T
“
+
0:*:
A
CHAP. II.
.
Trigonometry:
133
2
1
1
" Ia ao
deg. mi.
The Side R SW-406 Paces,
Two Right Angles-180 00
The Side RQ_185-paces. The Aliglasat ibis 20
TheSum of the two
134:40
The Sum-----591 of the Sides given, RS and R:Q.. joppoſite Angles-
The Difference-221 of the sides.
The Half-Süm-28
The Sun and Difference of cheesides being alias found, and alſo the Half-sism of
the two unknown e Angles: The Proportion by
" włticle you muſt find the Angles ſe-
verally is,
As the Logarithm of the Sum of the Sides 591
,2771587
Is to the Logarithm of the Difference of the Sides 22.1. 344392
So is the Tangent of the Half-Sum of the two unknown Angle 10379313
67 deg. 20 min.
The Sum of the ſecond and third Number--
-12723605
The firſt Number fubftracted from the Sum
2771587.
The Tángent of 4r deg. 50 min. (is this )--
9952018
which added to the Half Sum, makesam 199, dega 10 min. Greater Angle.
The greater Of tbe Angles required, 7
Sulfrat 41 deg. so min. from the ze deg. 30 min
. Leſſer Angle:
Half-Sum,leaves the leffer Angle at S
:::I::
“21iIslt (e”
.
.
Sum 67 20
Tang 41 50
109 TO
Leffer}30
Angle
7
+
13
3
6:11
1
}
2
30
1
(togdro
I83
i
ܝ ܀
2.30
45.20
R
40% I?
S
ON
ir vi HI
By, the Line of Sines and Numbers.
I
F
Xtend the Compaſſes from the sum of the Sides 591, to the Difference of the
Sides 221; the fame Extent upon the Line of Tangents, will reach from chei
Half-Sum, to the Tangent of the found Angle 41 deg: sa min.
Or elle extend the Compaſſes from
the Difference 231,
to-the Tangent
of the Half-
Sum of the unknown Angles; the ſame Dinânce will reach from the Half-Sum 67
20 m. in the ſame Line-toʻthe Tangent of 45 deg: 50 min. which added to, or
fubftracted from the Half-Sum, as before is thetřn, will give che Quantity of either
of the two unknown Angles.
deg.
-
:)
i
i
A
.
'CZ
j
1
.
134
Book III.
Trigonometry.
CA S XI.
Two Sides and their Containing Angle given, To find the third Side.
Here is given R S 406 Pades, fand RQ 185 Paces, and the Angle at R 45 deg:
T 20 min. which is by the 10 Cafe,
As the Sum of the sides given RS+RR 594P. 02771587
Is in proportion to their Difference R Safe R Q 221 -2344392
So is the Tangent of the Half Sum of the 67 deg. 20 min. -1079213
Two oppoſite Angles Q and Sanknown, 2, 3 Numb. 1272 3605
To the Tangent of the Angle
41d.som.--~-9952018
which added to the Half Sum--- -67 20
Leaves the greater Angle at a required-109
' whose Complement to 180 deg. is-70*50
IO
1
Then ſay,
1
Asshe Sine of the Angle found 109, ar 70 deg. 50 min.-
Is in proportion to his oppoſite Side R S 406 Paces.
So'tke Sine of the Angle given at R45 deg: 20 min.-
To his oppoſite. Side required. R s žos Paces!
The Logarithm of the Side required
By the Line of Sines and Numbers.
9,975233
mi 2, 608526
9,851997
12,464523
- 29485290
L
E
Xtend the Compaffes from the Sine of 70 deg. 50 min. to che Logarithm-Side
RS 406 Paces; the ſame Extent will reach from the Sine of 45 deg. 20 min.
to the Side 305
.
Or, Extend the Compales from the Sise of 70 deg. so'mln. to the Sine of 45 deg:
20 min. the ſame Diſtance will rcach from 406, to 305 Paces, which is the Length
of the Side Q S, which is required.
A
1
+
CASE XII.
Three Sides of an Oblique Triangle being given, To find the Angles.
best!.
:
7
!
**
Indom
30
581
3
406
1
1
B
R
!
N this Triangle SQR, Lee the three Sides known,
The Side SR
406
The Side SQ
-305
The Side QR
185
And it is required to find the three Angles.
To perform this, you muſt firſt let fall a Perpendicular from the Point Q, upon
the Side's R, which you may do by ſetting one Foot of your Compaſſes in the Point
Q, and open the other to the Point R, draw the Arch RE, and divide che Space
E Ř into two equal parts; and ſo the Perpendicular will fall upon the Point B.
То
.
.
Chap. II.
Trigonometry.
1
135
To perform this more exactly by Numbers,
As the greater Side or Baſe S R, 406
To the Sum of the two leffer Sides 490.
So is the Difference of theſe two Sides 820-
To the Part SE (cut off by the Arch RBE) 145--
2,608526
-2, 690196
2, 079181
-4, 769377
2,608526
2,161851
This ſabſtracted from the whole Line 406, leaves for tlne pārt wițhin the Arch
261; the half thereof is 130, which is the Place B where the perpendicular will
fall, icckoned from the Angle R; and by this perpendicular you have divided the
Triangle into two Right Angles, whoſe Sides are known: For R B being 103, ſub-
ſtracted froin the whole Line SR-406, leaves for the remaining Part 275 Now
having thoſe two sides of theſe cwo Right-Angled Triangles, and the two fiift given
Sides, :05 and 185, being the two Hypothenuſals thercot, you may bay the oppoſition
of Sides to their Angkes, as in the 6 caſe; or by the Sides, and Hypothens/al, as in the
7 Caſc, find the Angles.
By the Line of Sines and Numbers.
Xtend the Companies from 406, to 490°; the" (aře Diſtance will teach from 120,
to145 SE Leagues, the Side required.
EX
Theſe are the moſt necdful Caſes in the Reſolution of Plain Triangles, which inghe
have been ſet forth with much Variety and Inlargement; but I rather ſtrive to thew
the beſt and plainelt way. The Practitioner bcing perfect in what hath beenraid, be;
fore, we will proceed to our interided Diſcourse of NAVIGATION..
The End of the Third Book
t
1
+
THE
}
A U THOR
TO HIS
Fourth Book.
Let ſlip
$
M
Y Nem-riggd Ship, Strike now thy Sails,
thine Anchor, the Wind fails;
And Sea-men oft in Calms do fear
That foul and boiſťrous Weather's near.
If a robuſtious Storm ſhould riſe,
And bluſter from cenforious Eyes;
Although the ſwelling Waves be rough,
And proud, thy Harbour's ſafe enough.
Reft, reſt a while, till Ebbing Tides
Shail make thee ſtanch, and breme thy Sides :
When VĚinds ſhall ſerve, hoiſe up thy Sail,
And fly before a Proſprous Gale;
That all the Coaſters may reſort,
And bid thee Welcome to thy Port.
1
11
--
Compleat Sea-Artiſt;
A R T
1
NAVIGATION.
ſtructed in che principal Points of the foreſaid Arts ; that is, that he know the Order
137
THE
OR THE
OF
gi
The Fourth Book.
CHAP. I.
Of Sailing by the Plain Chard, and the Uncertainties thereof ";
And of Navigation.
HE Art of Navigation, is a Knowledge by cercain Rules for
to Sreer a Ship through the Sea, from the one place to the
other; and may not improperly be divided into two parts,
namely, the common, and alſo the Great Navigation.
The Common Navigation requireth the Uíc of no Inſtre-
ments but the Compaſs and Sounding-Lead, as chiefly confift-
ing in Practice and Experience, in Knowledge of Lands and
Points how they lie in Diſtance and Courſe one from the other,
and how they are kilown at Sea, in knowledge of Depths
and Shoulds and varieties of Grounds, the Courſe and Setting of Tides, upon
what
Point of the Compaſs the Mron maketh High-water in cach ſeveral.place, and the
like ; which muſt be reckoned partly by the Information of skilful Pilots
, but far
better by a Man's own Practice and Experience.
The Great Navigation uſeth, beſides the foreſaid common Practice, divers other
Artificial Inftruments and Rules, which they muſt take out of Aſtronomy and Coſmo-
graphy. It is therefore needful, that every Pilot and Officer, that takes
charge of any
or riffel in the Practice of the Great Navigation, be firſt and chiefly well in-
and underſtand che Diviſion of the Sphere of the World, and the Mocions of the
Heavens, eſpecially the Eighth, Fourth, and Firſt; Together with the contriving or
Making and uſe of Inſtruments, as I have ſhown briefly in the Second Book. Know
this
, Without this Knowledge it is impoſſible to perform great Voyages (not before 'ar-
over the Sea. In regard ſuch Knowledge may be attained to, by good In-
Struction, we liave ſet forth the ſame in this Treatiſe,for the benefit of all ſuch young
T.
1
f
Ship
tempted)
Sea
A
138
Uncertainties in the Plain Chart. Book IV.
!
Sea-faring Men, as are deſirous to be Sea-Artiſts or Navigators, fo clearly and plainly
as the brevity of the ſame could ſuffer to be done.
The Defeats and Imperfections of this Art are many; partly in the Skill or Theo-
rick, partly in the Practick.
After a long Koyages the ship ſuppoſed to be nçar the Šhore, the Commander or
Mafter rệquires from their Mates an Account of their Judgement how the Land or
Cape bears from them, the Courſe and Diſtanceof it when they ſee it : Hechat comes
neareſt the shore, is ſuppoſed to have kept the beſt Reckoning. I have known ſome
shat have not been ſcarce able to nuinber and make five Figures, have gone neereſt the
Shore than the beſt Artiſt in the Ship; bur they have been wonderfully miſtaken, to
my knowledge, in other Voyages
. I went a Voyage to Barbadees in the Rainbow, and
took our Reckoning from Lundy, in the Mouth of Severn; and in the Ship were 12
Practitioners and Keepers of Accoant ; eleven of them kept it by the Plain Chart, and
my ſelf made uſe of Mercator and Mr. Wright's Projection. When we came in the
Latitude (which was 400 Leagues from the Shore) every Man was ready to give his
beſt Judgement of his Diſtance off the shore: But they all fell wonderfully ſhort
of the truth; for he that ſhould have had che beſt Reckoning, was 300 Leagues
ſhort, and moſt of all the reſt was 268 and 250 Leagues, and he that was account-
ed ați excellenc“
Artiſt aboard the Ship, was 340. But by the Reckoning kepa by:
Mercator's Chart, which wanced but three Lengnes ſhort of the Iſland. In the ſame
Ship going from ebence to Virginia, they alſo fell ſhort, by the ſame way of Account
by the Plain Chart, 90 Leagnes the ncareſt; and thoſe that were adviſed to keep it
by Mercator, found it come but 4 or 5 Leagues ſhort of the Cape of Virginia: But
coining froni thence home, they gor their Credit mcyded; they came all within 30,
20, and 10 Leagues of the Shore.
So I ſay, If the Courſe and Diſtance had been firſt agreed upon from the Place
they were bound to, to be juſt the fame, unto the Cape or Land they fift deſcried;
If men differ then, there is ſomething in that, in refpect of the uncertainty of the
Longitude: A bad Reckoning may prove betrer than a good.
Buc we find that there is near 180 Leagues difference Error, between the Meridian
of Barbadoes and Lundy, and mucli more in the Diſtance; and in ſome Charts about
620 Leagues Eirour, in the Diſtance between Cape Fortuna, the South Cape of Anian
Fretum, to Cape Hondo by the River Depiſcadores; and theſe Errors may be aſcribed
partly to the uncertainty of the Longitude, and partly unto the Plain Chart, and
Sailing by it, which makes ſome places nearer than they are, and other places far
more diſtanc chan they are, and ſcituated much out of their true Courſe or Rhomb.
Secondly, Men many times commit great Errors in bad Sreerage, and careleſs look-
ing to the compoſis; for I have known many Seamen wlien their crike or turn have
beca.cat, and the Log hove, chey have told the Miſter or his Mate, they have
Steered a Point a Weather the courſe; beſides, the Points of the Needle or Wyres
beiug opuchcd, by elle Lead-ſtone, are ſubject to be driwn aſide by the Gans in the
Stecrage, or any Iron neer it, and liable to Variation, and doth not ſhew the true
Northand South, which ought continually to be obſerved by a good. Meridian, or
as Tomac.call it an azimuth:Compaſs, which is the proper Name. Such a one you have
deſcribed, by which I Survey Land with, as is ſhewn in the following Treatiſe; lo;
the
Variation ought to be carefully allowed.
I found 11 Beſides, on Land there is great difference in the ſame Country and Places, as Dial-
degr, in a lifts well know, by taking often the Declination of ſeveral Walls; as alfo Mi. Gunter's
George's Obſervations at Limehouſe, for the finding of the Variation, found it a Degree more;
and Briſtol and other places of the ſame Ground leſs; and Mætius faith, hic hach found a Deo
being four free or two.difference. This difference at Land muſt needs (hew the uncertainty we
miles dio hayo at sem. ' Beſides, many times the ship is carried away by unkuown Currentsa
,
frant ; and which when they be diſcovered by their Ripplings, as alſo ſome by reaſon of Trade-
Obertatio Winds, we ſet chem in our Journal; as alſo if we meet with any Soundings
,
ons, as in p. is in divers Places 1.00 Leagues off the Land or Ipands, to my knowledge, I would
330. and" adviſe all Learners to be careful to put down all ſuch remarkable things as neer as lic
differed can, their Latitude and Longituds
. So I believe did Mætius, to remember the Current
of a degr. chat ſet becueen Braſilia and Angola, in the oppoſite
Coafts of Africa, where he in-
only.
Itanceth,
1
1
CHAP. I. Uncertainties in Sailing.
ac
fee
139
ſtanceth, That an able Maſter bound to St. Helen’s,in 16 Degrees of South-Latitude,
in the mid-way becwixt boch Coafts, and being in the Parallel of Laticude thereof,
ſteered Eaſt, was notwithſtanding carried by the unknown Motion of an unknown
Current 800 Miles Weſtward, and yet ſtemmed the Current with a fair Wind, and
laſt made the Coaſts of Brafilia.
From chic 10 of April to the 15 of July, the Current fets near North-West. From
the
15 of July to the 12 of O&tober, there is no Current perceivable. From the -12 of
Otober to the 13 of January, it ſets South-Weſt; And from the 13 of January to the
12 of April, it ſeems to have no Motion perceivable
. Again,
Currents is a means of great miſtake in keeping of a Reckoning; for Captain Luke
Fox in his North-Weſt Diſcoveries, and the reſt, complained tearfully of the falt
Lands of Ice upon thoſe Coafts, that ſo alters the current, that in ſome places they
cannot make good their Courſe they ſteer upon, by three Points; eſpecially in Davis his
Streights, where ſteering East-by-South, they ſcarce could make good South Eaſt-
by-South, which is four Points of the Compaſs, and the Error at leaſt 70 Leagues.
I have alſo perceived a good Current to let to the Eaſtward, E. S. E. about the.
Weſtern Iſlands, and the Madera's, in ſeveral Voyages I have made to the West-Indies
but more eſpecially I have obſerved it in my laſt Voyage to Barbadors. I went out of
England in Company with Captain Jeremy Blackman, in the Eagle bound to the
Eat-Indies, and a Dutch Ship in his Company, and one of Phimouth for the Iſle of
May: So we kepe company together as far as the Madera's, but intended never to
it thac Voyage ; for we reckoned our felves 25 Leagues, and ſoine more, to the Weſt-
ward of the Meridian of the Mideras: But being in the Lalitede near about we had
eſpied the Land; and being becalmed, drove with the Current by the Eaſtern end
of the Iſland, betwixt Porto Sancto, and the Departs or Rocks that lic off from chat
end. I compared Reckoning with moſt aboard each Ship that kept Account, and found
ſome 39 Leagues to the West ward of the Iſland and thereby in five Voyages made
before that way, kịcw by Experience there is a Current ſets ſtrongly nicar about it
E, S. E. Beſides, ſeveral 'Ships of London and the Weſt-Country have miſtir, afrer
much labour and trouble to find it. Snellises inſtancech, That one of good repure,
failing out of Holland twice, milt it and came home. I thall not here trouble
you
with more Inſtances, nor mulciply needleſs Queſtions, nor ſtrive to branch chem out
in their ſeveral Varieties; but give you thoſe which are moſt uſeful and neceſſary :
And then if my time will permit, I will ſhew you ſome Arts which will as much
delight you to learn, and this as briefly as I can.
As for the firſt and moſt uſeful Queſtions in Navigation, is 'this; By the know-How we
ledge of the Rhomb or Courſe you failed upon, and the diſtance of Miles or Leagues keep our
that you ſailed thereon, to know your difference of Latitude and Longitude (that is
, Reckoning.
how much you are Northerly or Southerly in reſpect of Latitude, or Eaſterly or Weſter-
ly in reſpect of Longitude.) This is the moſt ordinary manner of keeping of Account
by moſt Maſters and Mátes, of the Ships Way, which is called the Dead Reckoning
And to keep this Account, firſt you ſee, That the knowledge of the Rhomb they fail-
cd is always ſuppoſed to be had of the Log-board, ſuppoſing the Compaſs by which
we ſteer, either doch or ſhould ſhew the ſame exactly; and (o you have the Diſtan-
ces in Miles and Leaguesa, put
down
every
half Watch upon the Lng-board, with
the Courſe failed, and Winds By or Large: 'Therefore we will come to the firſt Que-
ſtion, and Reſolve it by the Traverſe-Table following, and allo by the Traverſe-Scale
in the Fifth Chapter of the Second Book. I have ſhowed by the Sinical Quadrant alrea-
dy, in the Sixth Chapter of the Second Book: And we will reſolve it allo by the Arti-
ficial Sines and Tangents on the Ruler, and the Tables.
But know this, I never knew any Courſe ſteered at Sea, nearer than go half a Point;
for there is no Halfs nor Quarters marked on the Compaſs.
.
T2
The
-- -
140
Book IV.
Plain Sailing.
The Firſt Propoſition.
Queſtions of Sailing by the Plain, Ordinary
Sea-Chart.
fifth
1. Sailing 57 Leagues tipon the fear Rhomb, How much ſhall I alter
my Parallel of Latitude?
TH
He Angle that any point makes with the Meridian, we call che Rhomb; but the
Angle that it makes with any Parallel, is called the Complensent of the Rhomb.
I. lnto cvery Point of the Conspaſs there anſwers 11 deg. 15 min. thercfore the fifth
Rhomb from the Meridian makes Argles cherewith of 56 deg. 15 min. namely, S.W.
b.11. S.E. b. E. N.W.b.w. N.E.1. E. whoſe Complement 33 deg. 45 min. is the
Angle of the ſame Rhomb with every Paralel.
Now adinit I fail from A to D, S.W.b.W.57 Leagues, I demand cl.e
difference of Latitude E A.
3345
1
+
.
31 100
HII IIIIIIIII
Firſt, by the following
Traverſe-Table, at the Head
of the Table, over every Co-
47. 200
D
lumn, is put the Figure of
E
Halfs, Quarters, and whole
Rbombs; and in cn: of the
Columns cver liead is N. S.
an. I at the foot'E.W. and ſo
is numbred at the Head, from
the left hand to the right.
N. S. ſtands for Northing.
Thien the Rhumus are reckon-
ed at the bottom, froin the
right hand back again; The
A
Margent of the. Tables thews
the Leagues failed; and over
E.W. or under E. W. Thews how much you have failed Eaft or Weft from the Meridi-
an. N. S. fhews North or South from the Latitude. As in this Example, The di-
ſtanice failed is 57 Leagues on the fifth Rhomb; therefore under
Diſtance Sailed, in the Side, I enter with 57 Leagues, and in the
? Rhimb.
Common Angle or. Lize of Meeting, I find 31. over N. S.
N SW
in the Foot; and ju the next Column, over E.W. is 47.39, as
you
ſee in the Tallein the Side : So thac che Difference of Lati-
47 3931 Blog
tude is 31 Leagues and ... Paris of a League. "And if it were
E ;WIN
required to find the Departure, you ſee it to be 47 Leagues and
.
parts. This is very plain and calis, yqu need ng, falther
sastow buzzle
i
.
1
E
S
:5 Rhomb.
A
Preceptai
L
;
con
seisma . By the Traverſe-Scale
.
E
Xtend the compaſſes in the line of Numbers from 100 to 57, the fame Diſtance
will reach from5 Point
and about ' in the Line of Numbers.
1
IT
A
6.
.BY
11
1
0
CHAP. I.
Plain Sailing
141
By
the Artificial Sincs and Numbers on the Ruler.
E
Xtend the Compaſſes from ico in the Line of Numbers, 'to 57, as before; the
fame Diſtance wi'l reach from the Sine-Complement of the Romb, to the Diffi-
rence of Latitude, which is the ſame way as by the Traverſe-Scale.
By the Tables of Artificial Sines und Numbers, by the Fourth Caſe
of Plain Triangles.
TO00000
As the Radius, which is the Sine of go deg. or Angle at Ę
Is to the Diſtance run 57 Leagues A D
175587
So is the Sine Complement of the Rhomb at D 33 deg. 45 min., 974473
To the Difference of Latitude required A E 31 Leag. 1 150060
In like manner you may find the Difference of Latitude for any Diſtance run
upon any point of the Compaſs : But remember to add the ſecond and third Numers
10 zether, and from it to ſubſtra&t the firſt or upperinoſt.
c
II, Sailing, 57 Leagues upon the firſt Rhomb , How far am I de-
parted from the Meridian of the place from whence I came?
By the Traverſe-Table.
1
His Qu•ſtion was anſwered in the laſt Example, and found over E.W. to be 47
Leagues and ..., as you may ſee in the ſmall Table in the foregoing Side. In
the like manner, you may find the Difference of Latitude and departure from the Me
ridian, for any Diſtance run upon any Point of the Compaſs; which is the life of elfs
Table
By the Traverſe-Scale.
ELeagues ;
1
1
Xtend the Comp-les from 100 in the Line of Numbers, to che Diſtance run 57
ſo is the Sine of the Rhomb; that is, pụt cne, Point of the Compaſs
On 5 Points, in tlie Line of Eaſt and Weſt of the Scale, and the other will reach
to the Departure from the Meridian 47 Leagues i Paris.
By the Tables of Sines and Numbers, by the Fourth Caſe of
Plain Triangles.
As the Radius'or Sine of go deg. at E-
1000000
Is to the Diſtance run 57 Leágues ADL
So is the Siric of the Rhomb-56 deg. 15 miñ ALI 1991084
To the Departure from the Meridian to 4771. ED.
?! ED.citesti 166671
175587
By the Artificial Lines on the Ruler.
E Send the Compaſſes from 99-dog.tq 575 the fame Diſtance will reach from
from $7 Leagues, to 47 , as before
Or, Extend the comparies from 90, to 56 deg: 15 ming the ſame Diſtance will reach
III. Sailing
1
:
.
1
142
Plain Sailing.
BOOK IV.
III. Sailing upon the fifth Rhomb, until 1 alter my Latitude 1 deg.
35 min. I demand how far I have sailed?
AS
S failing from A to C, S.W.b.w. till the Difference of Latitude be 31 Leagues
6. I demand the Diſtance run A C.
47.000
A
Aam
3345
INRINNWI
68
a
1
1
156wowbizzle
Annunun
A
Firſt, By the Traverſe-Table, Look in the Foot of the Table for the fifth Rhomb,
and over N. S. in that column, look for 31 Leagues, and in the Common Angle
of Meeting, to the left hand, under Diſtance Sailed, you will find Diſtance Sailed
57 Leagues A C required.
By the Line of Sines and Numbers.
57 the Diffe-
E
Xtend the compasſes from the Complement-Sine 33 deg. 450 to 31.10.
rence of Latitude; the ſame Extent will reach from go deg. to 57 Leagues.
Or, Extend the Compaſſes from 33 deg. 45 min. to 90;
the ſame Diſtance will
reach from 310 Leagues, to 57 Leagues, the Diſtance AC, as before.
Say by the ſecond Cafe in Plain Triangles,
As the Sine-Complement of the Rhomb, 33 deg. 45
9,744739
Is to the Difference of Latitude 31.01. Leagues-
3500648
So is the Sine of 90 deg: "Radius
10000000
To the Diſtance run AC 57Leagues-
3755909
1
61
IV. Sailing upon the fifth Rhomb, until I have altered my Latitude
31.7 or deg. 35 min. How much am i departed from my
firf Meridian ?
AS
S failing from A to C, S.W.b. w. till the Difference of Latitude A B be 31
Leagues, I require BĈ my departure from my Meridian.
By the Traverſe-Table.
AS
S in the laſt caſe, find 31:17. Leagues over the fifth Rhomb, in the Foot, and in
the next Column to the left hand, over E. w. is 47 1. Leagues, the Departure
required.
)
1
4
B)
1
Chap.I. Sailing by the Plain Chart.
143
56-15
By the Line of Sines and Numbers.
Xtend the Compaſſes from the complement-Sine.of the Rhomb, to 33 deg. 45, to
31. Leagues; the ſame Diſtance will reach from 56 deg. 15 min. the Sine of
the Rhomb, to 47 1. Leagues, che Departure from the Meridian.
By the fourth Caſe of Plain Triangles,
As the Sine of go deg.--
-10000000
To the Difference of Latitude A B 315
2501059
So is the Tangent of the Rhomb 56 deg. 15
10175107
To the Departure from the Meridian 47 4. Leaguesa 2676166
In the like manner, by the Departure from the Meridian, you may find the Diffe-
rence of Latitude.
V. Sailing upon forme Rhomb between the South and the Welt 57
Leagues, and finding I have altered my Latitude i deg. 35 min..
I demand upon what Point I have failed.
Uppoſe I had ſailed from A to C (being a Rhomb bectveen the weſt and South)
57 Leagues, and then find the Difference of Latitude 310 Leagues, I demand
the Angle B AC.
By the Traverſe-Table.
N VY
Umber 57 Leagues in the Column of Diſtance Sailed, and in that Line or com-
mon Angle of Meeting, you muſt find the Difference of Latitude 31 Leagues,
at the Foot of the Table in the fifth Rhomb, which was required.
*
1
By the Line of Sines and Numbers on the Scale.
1
Xtend the Compaſſes from the Diſtance run 57 Leagues, to the Sise of 90; the
fame Diſtance will reach from the Difference of Latitude, to the Sine-Comple-
nicat of the Rhomb 33 deg: 45 min.
By the fifth Caſe of Plain Triangles.
R, Open the Compaſſes from 37 Leagnes the Diſtance, to 31. the Difference
of Latitude; the fame Diſtance will reach from the Sine of 903 to the Sine
of 33 deg. 45 min. the Sine-Compl. Rhomeb.
As the Diſtance on the Rhomb AC 57 Leagues--
Is to the Difference of Latitude 311. Leagues AB-
2501059
So is the Sine of 90 deg. B-
rod00000
To the Compl. Sine of the Rhomb at C 33 d. 45 m. the Sum-12501059
The firſt Number Subſtract-
2755874
The Sine of the Anglemm
9745185
The Sine-Complement of the Rhomb is C 33 degi 45, ſübſtracted from 90 degrees,
there remains the Angle of the Rhomb at A 56 deg. 15 min. which is five Points
namely, s.Web.w. We neglect ſome part of a Minute, which is not co:be regarded.
2755874
VI. Sailing
+
144
1
Sailing by the Plain Chart. Book IV.
VI. Sailing upon fome Rhomb between the South and the Weſt 57
Leagues, and finding I have altered my Latitude i deg. 35 min.
I demand my Departure from my firſt Meridian.
"
By the Traverſe-Table.
Umber 57 Leagues in the Column of Diſtance Salled, and in that Line or Angle
of Meeting find 31 7. Leagues, and in the Column to the left hand you will
have 471.. the Departure from the Meridian.
By
the Sixth Caſe of Plain Triangles.
Diſtance run A C 57 Leagues? Sum 88 Leagues
2947923
Diff. of Lat. A B 316. Leagues SRemain 261. Leagues 2424881
5372804
Departure from the Meridian B C 47. Leagues--
This is thus done. To the Diſtance run, add the Difference of Latitude, and alſo
ſubſtract it from the ſame, noring the Sum and Remainder; then add together the Lo-
garithm of this Sum and Remain, and half that is the Logarithm of the Diſtance froin
the firſt Meridian.
-2681402
titude;
By the Line of Numbers.
EXtend the Compaſſes from the Diffance 57 Leagues, to-31. die Diference of La-
the ſame Diſtance will reach from 88 the Sum, to the Departure, as be-
fore, 47 Leagues.
Or, Extend the Compaſſes from 57, to the Sum 88 Leagues; the fame Diſtance
will reach from 31, 1047}, as before, which is the Departure required.
All things that have been done by the Artificial Sines and ii umbers are done by thie
Traverſe Scale, or Artificial Points, Halfs, and Quarters, and Tangent-Rhombs, with
the Line of Numbers in the Traverſe-Table; and chis agreeing very well in Leagues
and 100 Part of a League.
1
i..
CHAP. II.
what muſt be obſerved by all that keep Account of a Ship’s Way at Sea;
And to find the true Point of the Ship at any time, according to the
Plain Chart.
I
Might have further inlarged and mulųplied Queſtions, but that, I think theſe
fufficient for any uſe at preſent'; and therefore † will be brief, and come to the
moſt inaterial Buſineſs, (viz)
The whole Practice of the Art of Navigation, in keeping of a right Reckoning,
conſiſts chiefly of three Members or Branches.
Firſt, Well cxperienced in Judgment, in eſtimating clio Ship's Way in her Courſe
upon every ſhift of Wind; allowing for Leeward-way, and Currents.
Secondly, In duly eſtimating the Corse or Point of the Compaſs on which the
Ship hath made lier way good; allowing for Currents, and the Variation of the com-
paſs.
Thirdly, Thc diligent taking all'Opportunities of due obſerving the Latitude:
The Reckoning ariſing out of the two firſt Branches, we call our Dead Reckoning;
and of theſe Branches there ought to be ſuch an Harmony and Concent, that any
TIVO
+4
ou
--
CHAP. II. Sailing by the Plain Chart, .
145
two being given, as you ſee by the Work before-going, a third Concluſion may thence
be raiſed with Truth.
As, Having the Courſe and Diſtance, to find the Latitude of the ship's Place.
Or, By the courſe and Difference of Latitude, to find the Diſtance:
Or, By the Difference of Latitade and Diſtance; to find the courſe.
But in the midſt of ſo many Uncertainties chat daily occur in the Practice of N14-
vigation, a joynit Conſent in the three Particulars, is hardly to be expected ; and
when an Error ariſeth, the ſole Remedy to be truſted to, is the Obſervation of the
Latitude, or the known Soundings when a Ship is near Land: and how to rectifie the
Reckoning by the obſerved Latitude, we ſhall ſhew.
I would adviſe all Sea-men to yield unto Truth in this particular, That about:24
of the common Engliſh Sea-Leagues, are to be allowed to vary a Degree of Latitude,
Sailing due North or South, under the Meridian; otherwiſe they put themſelves to
many Uncertaintics in their Accounts.
Firſt, In Sailing directly Nereb of South, where there is n10 Current, finding their
Reckoning to fall ſhort of the obſerved Latitude, they take it to be an Errour in cheir
Judgement, in concluding the ship's Way by cſtimation or gueſs to be too little.
and ſecondly, If there be a Current that helps ſer chem forward, that there is a
ncer agreement between the obſerved and the Deal Latitude, they conclude there is
no luach Current.
Or laſtly, If they ſtem che Current, they conclude it to be much ſwifter than in
truth ic is: And thus one Error commonly begets another. But luppoſing a Confor-
mity to the Truth, we ſhall preſcribe four Rules for corre&ting a Single Courſe.
THE
LOG-BO A Ř D.
But firſt of all it is moſt
neceſſary. co ſhew how we do
keep our Reckonings at Seag
by the Lg-board, and allo by
our Journal-Book.
i
Half Fathom. Large.
2
1
9
8
L
The firſt Column is for
Time.
SS
The ſecond for the ship's
I 6 6 6.
Courſe.
By or
The chird for the Knots. ;
Hoxrs.
Courſe.
Knots.
.
The fourth for the Half
2 S.E.b.S.
L
7
Knots.
4 S.S.W.
6
L
6 E. b.N.
L The fifth for the Fathoms.
8 N.b. E. E.
L
IO N.N.W.W.
7
L
The fixth is to put down
che Sailing Large; that is, co
12 W.N.W. 9
L
make her way good on the
2 2 S. E. 6. S. 8
11 / 을
​L
Point ſhe Sails, ſignified by L;
4 S.S. W.
7
L and Sailing By the Wind, ſigni-
6. S.W.b. S.
L
fied by B; that is, to give al-
8 S. W.
lowance to your Courſe ac-
L
cording to the Lee-way you
S. E.
3
B have made (by taking in or
having out more Sail, or by
Currents or Variation) choſe
ſeveral Diſtances
Our Engliſh or Italian Mile by which we reckon at Sea, contains 1000 Paces,
and cach Pace ş Foot, and every Foot 12 Inches; the 120 part of that Mile is
41 Fers, and ſo much is the ſpace between the Kness upon the Log-line : ſo many
V
Knots
IH
Holalalalal-10
2
5
ܚܙܐ ܚܐ ܝ ܝ ܝ ܝ ܝ ܝ ܕ
6
10
8
I
1
1
12 S. E.
9
B
:
1
ma
146 Sailing by stihle Plain Chart Book IV.
Knots as the Ship runs in half a Minute, ſo many Miles She Saileth in an Hour ; or
ſo many Leagues and ſo many Miles she runnerh in a Watch, which is four Hours, the
time in which half the Company belonging to the Ship watcheth ar orice by turns.
EX Á M P L E.
!!!
of the Compass.
1
>
Nine Knots in half a Minutes is 'inine Miles in an Hour, which is nine Leggises
and nine Miles in a Watch, which is 12 Leagwes or 36 Miles in all. Every. Noon,
after the Maſter Mates having obſerved the Sun's Altitude, for every Day at Nooni
they take the Reckoning from the Lag-board, and double the Knots run, and élien di
vide the product, which is the number of Miles run, by three; the Quotient is the
Leagues run ſince the former Noon. Or eļſe add up the Knots, and multiply them
by 2, and divide by 3, you have the ſame': But be ſure it is all upon Ccurſes Wé
throw the Lng every two Hours, and we neveſ expreſs the Conife nearer than is a Point
Mr. Norwood gives full ſatisfaction in liis Seaman's Practice, by his own experience;
That in our ordinary Practice at Sea, we cannot, if we will yield Truth the Con
queſt, allow leſs than 360000 of our Engliſh Feet to vary one-Degree of Latitude
upon the Earth, in failing North or South under any Meridian. According to this
Meaſure, there will be in a Degree 68. of Miles of our Statute-me difiere, cach Mile
5280 Feet; and by the common Sea-meaſure, 5000 Feet to a Mile, there will be
72 Miles or 24 Leagues in a Degree, which we svill take for truth.
Now if you would liave ſhown the Miles of a true Degree, allowing 60 to a De-
gree, the Miles muſt be enlarged proportionally, and the diſtance between every one
of the Knots muſt be so Foot; as many of theſe as run out ih half a Minute, ſo
many Miles or Minutes the Ship faileth in an Howr; and for every Foot more, you
muſt allow the 10 part of a Mile
. And ſo if you will work the old way by Leagues,
you muſt reduce them by Arithmetick into a Degree, and 100 parts of a Degree;
or Miles or Minutes may ſervc. For I have ſeen no Chart that the Meridiart is divided
into more parts chan 6 cimęs 10, which is 60 Minutes; or Mercator's Chart 20 times
3; which is 60: So che ſmall Divifions on the Dutch Mercator's Charts, every 3 is
a Mile or Minute, which is ncar enough for any uſe at Sea and theſe Degrees are
not above . an Inch upon the Aquator.
Sailing Eaſt or Welt berwen any cwo Places, and uſing a Log-line that hath a
Knot at every 7 Fathoms, and to reduce it into ſach Miles 60 to a Degree, each con-
taining 6000 Feet, the Proportion in Number of theſe two is chis, As 6 to 5; for
6 Knots of 7 Fathoms makes's of 87 Fathom, or so:Feet. Admit 2. Man keeps à
Reckoning of his Ship by a Log line of 7 Fathoms, and by it find the diſtance of tivo
Places 1524'Milestor 508°Leagues
, and would know the diſtance by a bog-line
of 50 Feet to a Knot, or 6000 feet to a Mile : Say then by the Rule of Proportion,
As 6 i 70,5: Sois 1924 to 1720 Miles, whereof 60 Miles make a Degree of 10
Leagues.
Next we will work the Courſes of the Log-board, and_by_it find the difference of
Latitude, and departure from the firſt Meridian.
A Ship being in the Latitude of 47. deg. 30 min. North, and Longitude oo degrees,
the firft. Coarse of the Log-board is S. 2.6. Six6 Miles, and s: S.W.13 Miles
;
2.6. N:18 Miles, and N.b, E. E. 16 Miles, N. N.W. W.18 Miles
, and w.
N.W:18 Miles, and S. E. b. S. 18 Miles, and S. s.w.is Miles, S.W:b, S: 12
Miles, and S.W.18 Miles, 3. E. 18 Miles by the Wind, the wind at W. S.W. and
E.S. &. The Ship made two Points Leard-way on the two laſt Courſesa.
.. . I demand the Difference of Latitude, and departure from the Meridiani che
laſt 24 Horrs, and the Latitude I am in.
There are ſeveral ways to work Traverſes; but the moſt neceſſary and readieſt is
by the Traverſe-Scale, and the following Table; the firſt is to the 10 part of a
League, and the Table to thic 100 part of a League or Mile. We ſhall work clic
for-
mer Traverſe by the Tables following, and you at leiſure may work it by the Tra-
verſe-Scale, and find the necr agreement of both
without any ſenſible Error.
You
+
*
1
CHAP. III, Sailing by the Plain Chart ,
1
1
Miles.
100.
Miles.
100.
I 3
I 2
OT
17 65
04 641
12
Leeward-way, it is but E. S. E, that is, 6 from the South; therefore I reckon them
1
ard
147
You muſt
put
North. South Eaſt. Weft.
down the Courſes
made good upon
Courſe by com- Miles
pafs. failed.
cach Point of thie
Compaſs, and the
number of Miles or
S. E. b. S.
16
13 30108 89
S.S. W.
04 97
Leogues you find
E. b. N.
failed on them by
18
03 51
N. b. E. E. 16
the Lng-board; in
15 31
ſuch manner as I
N.b. W. W. IS 13 231
07 07
have done in this
W. N.W. 18 06 89
16 63
Table: then accord-
S. E. b. S.
18
14 97110 00
ing to the Rbombs,
S. S. W.
IS
13 86
os 74
look in the Table
S. W.b. S.
0998
106 67
S.W:W.S.W.
following for the
18
06 89
16 63
S.E: E. SE.
Print, Half, or
18
06.89116 63
Quarter failed, and
Sum
of all
38 94177. 90157 81.57 61 the Diſtance in
Subftract Icaft
38 9457 61
Miles or Leagues, in
Remains differ. Latitude.
38 96.00 30
the right hand or
II Differ.
The Laticude the Ship is in is 46 d. 49 m.
lefc hand Column ;
Sonth and
The Courſe made is South almoſt.
and count the four
Eaſt.
Points and Quarters
in tiic head of the
Table, and the four next the Eaſt and West, from the left hand to the right hand, in
the foot of the Table.
Put down in four Columns N. S.E.VV. and under put what anſwers cach Point.
As for Example. The firſt Courſe failed is three points from the Meridian;
namely, S.E. b. S. under that Column I count 16 Miles in the ſide, and find
againſt it 13 * Miles Southing, and 87. Miles Eafting. I put it down in tlieta-
ble isi its placc, 13. 30 under South, 8.89 under Eaſt. In the like manner you
muſt do by the reſt
. Likewiſe the laſt courſe ſailed s. e. but by reaſon of 2 Points
iu che foot of the Table, and right againſt 18 I find 06.89 Southward, and 16.63
Eaſtward, which you may pur down as I have done in the Table
. In the like
manner
you muſt do if your Courſe were North or Vvesting. This is ſo plain it needs no far-
Then add up the Sums in cach Column, and ſubſtrast the leffer out of the greater,
the Remainer is the Difference of Latitude and Departure : As I find that the Ship
hath gone but 38.6Miles to the Southward, and the Latitude ſhe now is in is 46
deg. 49 min. and the Eaſtward but 20 parts of 100 of a Mile : Therefore her Courſe
is neer South ſhe made good the laſt 24 Hours.
I
ther Precept:
CHA P. III.
1
F
T
Formal and Exact Way of Setting down and perfecting a The Rule of
Sea-Reckoning.
keeping a
perfc&sea:
Reckoning
His being the moſt neceſſary Rule in this Art of Navigation, How to keep an is beft fer
Exact Reckoning; Although the Courſe and Diſtance cannot be ſo truly and down in
certainly known, as the Latitude may be ; yet we muſt endeavour in theſe
particular
alſo to come as neer the truth as may be, the rather, for chat ſome Reckonings muſt
geacral
neceſſarily depend wholly upon them. Therefore we come 11ow to thew an Orderly true Sca-
and Exact way of Framing and Keeping a Reckoning at Sea'; for which purpoſe 1 Chart, ir
have inſerted this Table following, which ihewch how much a ship is more Nor- Chap. 17
the fly or Southerly, and how much Eaſterly or VW efterly, by ſailing upon any Point Circle Sail-
after the
or ing.
148
Sailing by the Plain Chart. BOOK IJ.
ti:
or Qstarter-point of the Compaſs, any diſtance or number of Miles or Leagues propo-
fed.
Mr. Norwood many years ſince laid the ground of inaking this Table, after this
Proportion, [ As Radiasis in Proportion to Diſtance run: So is the Sine-Complement
of the Rhomb, to the Diſtance of North or South : And ſo is the Sjne of the Rhou:b
to the Diſtance of Eeft or Weſt) as you may ſee by the firſt and ſecond Caſe of
Plain Triangles.
Therefore for every Point and Quarter-point from the ileridian, chere are four
Columns: In the firſt chereof is ſet down the number of Leagues or Miles run or failed
upon that point or Quarter-point of the Compaſs ; The ſecond thewech how much you
have altered the Latitssde, that is, how much you are more Southerly or Nurtherly, by
- running ſo far
upon that point or Quart:r-point ;: The third theweth how much yon
are morc Eaſterly or Weſterly, by running or ſailing chat Courſe and Diſtance, as your
have been bcfore directed.
Nore this, Thc Numbers ſet in the firſt Column from I to 10, are alſo to be un-
derſtood from 10 to 100, or from 100 to 1000 : and the Figure of the fourth place
anſwers to the Figure in the firſt.
As, Suppoſe a ship ſails away Soxibi a Point Weſterly 173 Leagues or Miles; we
ſet down this Number thus. Look into the
firſt Column for the Point, or the firſt
Courſe Diſtance. Southing. Weſting.
Rhomb from che Meridian againſt To is South
98
ſometimes made uſc of, and underſtood to
Po. W. 70 697 68
be 100. I find in the ſecond Column againſt
it 995 (or you may have the ſame Number
Leagues. 173 | 1721 168
at 100 towards the foot of the Table,
omitting the laſt Figure) and then in the
third Column you may ſee 98; alſo againſt 70, or 7, there is 697, and in the ſecond
68; and in the third againſt three in the firſt Column is 29, in the ſecond is
2 and, which is almoſt 4:; therefore I put down 3.
Theſe ſummed up as in the Table, ſhews that the Ship failing upon the firſt half
Point from the Meridian, as namely, S. W. is to the Scuthwards of the Place the de-
parted 1727. Leagues or Miles, and to the Weſtward 16 Leagues and... If you
deſire more Exactuels, you may uſe all the Places for the greateſt Number, which is
100, (viz.)
.
100
995
3
29
3
r
Color:
2
०६.
*
ol
ملز
30
3
Traverſe-Table
21
S); 1)
;
1
T
1
A Traverte-Table for every Point, Half-Point, and Quarter-Point of the Compals, to the
100 part of a League or Milc; which gives the Difference of Lac. and Departure from tbe Meridian.
The Traverſe. Table, The r Points Quarter.
49 m. is deg.
26 m. 111 deg. 15 m
.
149
2 dig.
38m.
18 deg.
nr Milis faild.
Dift.in Leagues
1
o Point
SIE W
00100 os
ooloo
IS
for Miles failid.
Dift.in Leagues - wmtu
OI
98100
N
1 i lot
202
3 03
4 04
§ 104
6 Jos
IO
99100
OI
oooo
9600
58
oooo
ogloo
94/00
9200
9၁ဝဝ
104
78
98
20. اوه
94100
9300
92/01
8 107
56 ,
9 08
109
08
90101
09
98
811101
10
1
II
76
IS II
34 12
10 109
II lio
12
11
13 12
14
13
15 14
16 IS
17 16
18
17
13
37
9301
93121
14
84 02
20
15
82102
80102
6703
64
SI 18
79
20
lis
62103
20 쩨
​20
20
76103
ܐܐ
22
22
681 24
8810m
88102
87.14
32
96
49105
4605
851.30
30
091 25
29
7 106
09.00
34
09100
39
0900
44
09100 49
93103
$4
98100
59
98 00 63
9800
68
98105
731
98100 78
9800 83
9700 88
19 118 9700
93
19
9700
21
97101 031
II
97 01 08
23
9701
13
24
23.
9701
17
25 24
97/01
22
26 125
961or 27
27 26
96101
.28
96101
32
29 28
9601
42
30-129 9610i 47
31
30
96101
521
32
960I
57
33. 132 96101 61
34 33
65
35 134
36 135
95 lor
75
37
36 95 10T 88
38
37
95101 86
9501
91
9501
961
41.140
95 oz OI
43341 '9502 06
43
95 02
4
94103 IS
94102 420
9402
25
.
9402
30
47
94 OZ
35.
94102
40
941 45
93/02 50
12
55
93102 60
93102
65
55154
93102 70
55
93102
751
57 156
93102
59. 158
31
14
هاها به به به با
o Point
o Point 1
I Point.
N
SIE W IN
SIE W
N
SIE W
0000 101 100
141 100
98100 20
I
OY
20
01
97100
29
9609 39
98100 29
44 OZ
3
03 9800 39 03 95 oo
58 03
4
9700
491 04
731 04
los 971no 59 los
88
25
6
83101 171
06
97100
69 06
O2
v6 86101
37 7
07 9600 78 07
OI
17 07
85101
8
9600 88 OS
32 108
83101
Oi 76 9
95/00
109 8901 46
09
95 10
10 95101 08
8801 61
IO
79102
9401 18
87 01
II
77102
I2
94101 27
12 86101
91 I 2
75102 541 13
13 85102 os 13 73102
73 14
14
47
114
7102 19 is
91101
57 IS 83102
34 105
69103
I 2 16
16
92101 67
16
49
16
32 17
17
I 76 17
17
65103
18
911or
86 18
79102
18
64103
71119
19
90101 96
19
78 oz 93 19
901.20
20
90103 06
77103
08
20 60104 101 21
ZI 98102 16 21
22
21 $804 291 22
22 89102 25
75103 37
56104 49 23
23
35
23
74 03
$2 23 5404
24
45 24 73103 66 2 52104 $8) 25
25
55 25
85 125 50105 07 26
26 97102 65 26
26
27 22
27
8603 75 27
IO 27
46 28
28 86102
84
28
25 28
44 05
66 29
29
86102 94
29
40 29
42 05
04
55 30
40106 05 1 31
31
69
31
23
32
84
32
441 33
33 33
98
33
06 63
34
13 34 33106 831 35
.83103 53
31 135
28
31/07: 021 36
3682103 63 36
42
29/07
37
73
37
5905
37
27107 41 38
82
53105
25107
39
92 39 57105
39 23107 801 40
40 80104 021
40
02 40
21/08
41
IZ
41 $4/06
16
1942
42 79104 21
42
15106 31
17108 39 43
43 79104. 31 43 2106
45 43
44
78104 411 44
5106
60이 ​44 14/08 78 45
45
51
145. 60106
75
12 08.971 46
46 77104
61
49/06
89
171 47
47
7704
47
48 07
04
08/09 361 48
80
47107 20
06109 49
149
90
49
33 49 04109_751 só
so 75los
so 44127
48 so : 02/09
SI 7505
10
51
43107 63
OO IO
7410S 20
52
4207 17
9810
7405 29 53 41107 22
9610
54 73105
391 54
40108 07
53
94/10
731 SS
73125
491 55
92110 921 56
56 72105 59 56
36
90 11 I 2
57
57 7210s 68
57
SI
56
89111 311 58
71 OS 78
57
87111 511 59
7110S 88 59
158 85u
701 60
86
24110
68
65/13 651 70
79
13/11
78
82
601 80
46115
88 27/17
99
80
551 १०
silog
98
1999 0119
08/19, Sol 100
60
197
34 196
1639 001200
It
S! E
HIN
SI:
IVIN S
7 Pomnt
7 Point
4
7Point.
84 drg. 22 m 81dpg.
74 m.
178 deg
45.m
The 7 Points Quarters.
3806 241 32
71103
7003
69104
68104
67 04
66104
65104
6404
63104
62105
61/05
60105
85103
85103
9403
84103
83103 431
95101
37106
35106
9510i 75
63 34
135
30
36
221 37
57
39 38
40, 39
38
82103
8103
81103
38
72
38
61 39
87
55106
001 41
30104
19108
berperanto
41
42
#443
15 08 58 44
7804
45.744
46 145
4&. 147
49.: 148
145
46
10109
70
46
47
48
SO 149
in 604
7604
48
48
56!
4607
00
si so
Sim SI
53 5.2
54 53
51
SI
951 51
14) $2
341 53
531 54
53
۶۶
154
93102 79
58 157
155
84
ܐܘܪܐ9
30408
38 108
3708
3608
35108
89
92102 94
60 159
58
58
ES
80
70 169
59
169
1
80 79
66106
di 07
1
.
84
271
73
89
56108
89
69
79
02/13
98 91114
82129
too. 199
200199
20
91 PR
43
90103 92
90 89 8904 41
go
76109 80
S
7 Point
8704
67
E
WIN
WIN
1
.
87dre.
150
A Traverſe-Table for every Pomc, Halt-Point, and Quarter-Point of the Compals, to the
100 part of a League or Milc; which gives the Difference of Lar, and Depareure from ihr Meridian.
122 deg.
18 Miles fail'd.
Diſt.iis Leagues
28
1 Point
or Miles failed
Diji.in Le.g!!!
100
I
OI
94100
91 00
02
3 Oz
O2
܀
O
7701
7710I
21
ܐܪܪ
7 106
)ن
54102 3016
47 02
59 02
53/02
૧
7302
03
0
ola 의
​16104
10
IZ
59 12
13
(2
IZ
IZ
OI 04
61103
58103
18 04
20
14
Izlos
16 115
39
71106
46104
18105
95106
8906
40
55107
86
20
22
21
05/06
20
20
32/08
25/08
22
801 23
1824
280S
22
22
25106
26 [25
24
24
19/06
56
79108
26.
29 28
13.07
7911
1
The Traverſe. Table, The 2 Points
Quarter.
14 deg. 04m. 16 deg. 52 m. 119 deg. 41m.
30 m.
1 Point
I Point
2 Point!
N SE W
pl
IN SE 11 IN
SIE V N SIE
I loo
9700
24
96100 291 oo
94/00 331 00
9100
381
12 lor
48
ΟΙ 91100
58 OI
8800 67
5109 76
72
87120 87
82101 OI
75 3
4 103
8800
97 03
83101
16 03
34 03
70101
53 4
5 04 8 SOI
104 7301
451 04
71101
68
04
6:3I 9! S
6 105
82101 45 os
74101 741 105 65102
los
7910T 70 06 70102 03 06
35
a
7
8 07 7601
94 07
66102
32 07
70 07 3903 16 8
9 08
18 08 6110Z
61 los
4703
08 31103 44!
9
10 109
7002 43 09 57 22 90
41103 37 09 21103 831 10
11 110
67102 67
53103 19
3003 71 jo
21
II 64102 21 11 48103 48
II 3004 04
09104
IS 12 •
44103 77
24/04 36
971 13
14 13
40 13
06
40104
12
13
72
930S
36 14
I5 114 55103
64 14
35104 35
351
051 13 86105 14). 15
52103 88 15 31104 641 15 06/05
14 73116
12 16
17
16
49104
13 16
27104 93
16
00105
73 15
SI 17
18 17
37 17 2210S 22 16
06 16 63106 891.18
19 18
43104
61 18
SI 17
17
27 19
20 19 40104
19 14/05 81 18 83106
18
74
48107 651 20
21
37 05 101 20 1006 IO 19 77107 08
19
40108 041 21
340S
34
39
71107 41
42 22
23
34105 48
or|06 68
.64107
75 21
24 23
83
97106 97
60198
08
17 99
25 24
07 23
91107 26 123
54108
42 23 10109 57 25
22106 311 124 83107 55 24
4808 76
02/09 951 26
27 26
25 84107 84 25
42109 IO
9410
331 27
28 27
16106 80 26
13
26 3609 43 25
87110
71 28
04 27 7508 42 27 30/09 77
20 129
1007 .28 28
U
71108
71 28
27
7: II
3.1 130
07/07
29 66109 ou 29
441 28 64 II
32 131 04/07
77 30
29 30
741
56|12 251 32
33 132
01108
01
31
58109 58 31
I 2
30
631:33.
34 32
98108
32 5409 87
OLII 45 31 4413
35 133
sol 133
981
791 32 34113 391 35
36134
92108
74. 34
33
89/12 131 33
26113 781 36
3735
8008
99 35
74
34 84.13 47 3-4
16
38 36
03
23
35 78/12
80
35
54138
39 37
83.09
47
32
72113
03/14
4.0 138
80109
28/11
31
61
66113 48
96115 3.1140
81
96
37
39
38
23!1.90
88115 691 41
42 40
7410
1912
20
39
40
138
IS
34/14
8016 o
0; 4.7
43 14!
:7IIO 41
41 15/12 48 40 44114
73116 451 43
44 47
69
42
IzI2 77
41
72
4)
65116 8.4
4.5 43
43
06/13
061 42 37/15 16 141
57117 45
46 44
17
44
3.5 43
50
50117 601 46-
47 45
42 44
64 44
83 43 4917 991: 47
48 46
66
45.
9513 93 45
17 44
37|| 48
49 47
90
46
8914
46
37 45 29 18
75! 49.
50 148
147
51 47 OS 16
146 19119
131 go
SI 149
39
80
01117 18
47
I 19 saj so
52 150
63
49
09
9617
31
204|19 Salsa
72115
50
53 SI
49 9017 85
9720
67
54 32
67115
12
so 84 18 19
661 54
55 353
961
63115
51 7818
15$ 01
35113 361
56 154
87
32/13 60 53
52
91
:73121
-6719
5755
20
29113 85
55 53
54
66/21
58 156
09: 55.
84 54
54 53 58/22 811. 58
2314
4617
56
5519
23
33
55
87
21:2
54
2015
60 158
571 157
4217
156
66 90 20 311
165 90133 581 64
781 70
80 71
6011943
55123
75
95 73.
9130
611 so
86 86
101:6
7330 39
14134
z'9
0024
95
190 197
O2
94 15133 68 92
16 100
00148
58 TOI 38158
2,00 1194
04
188 30167 36 184
7676
SI:
E IN
WN
E
NIN S
S?
MN
6. Points ]
6 Points
6 Points
6 Points.
75 dec.
***6 m.
173 deg.
70 deg. 19 m. 67 deg.
30m.
The 6 Points Quarters.
10 29
48 30
861 31
6
53
25 10
1910
13/10
07 11
-62109
29.
4912
26
al 34
32
32
95108
16}
45
49/10
45110
41 to
36111
32
2
1814
II 14
37
26109
1o
36
37
14
36
92: 39
36
37
1.36
38
AI 39
7709
60113
H #4
19
49
39
43/14
1
: 6810
6511o: 63
62111
224
6213
98 13
4115
2515
59111
5611
3518
+
1916
13/16
22
33/11
P/12
14
85
85114
Sol14
48
76115
47112
4412
4112
48
48
48
87
38
48,
2853
ol
3813
SI
851:0
49.
661
53
26
73118
196
(
59116
55/16
5016
15! 5G
431 57
.26114
61 19
S157
20114
42
49/20
23
Iss
581 60
43/22
67:6
70 167
90117 00
76
22
32/26
30121
90 87
84
83
441 go
69129
38 38
041
5 2 200
S
>
***
4
7 m
.
151
.
133 deg. 45 m
lor Miles faitd
Dijl in Leagues
1
of Miles fail'd.
wift in League
I
00
OO
00
1
goloo
8110
2
Or
ol
OI
2
or
02
ܐܘ
89
49 01
32102
22
>
4 03
5 104
14
$3/01
41
:)
104
4252
33102
99
1703
82103
.8 07
8604
II
.9 108
oo
2
31155 55
94194 701
IO
44105
29/06
15/06
09
13
10
81107
.64 07
4606
3506
ES
6625
IZ
20
14 IZ
IS 13
78:14
( 2
ܐ11
4916
37137
14.
14
99198
OD)
14
14
15
441 17
DO
90
18
21
98108
.
4010
22
22
ปี à des
84.
781 23
20
.7311
:59 12
-44/12
19
691
LI
05/11
193,482
2
22
mܐ
.62|14
441 26
27 24
59111
41 IT
31|LI
73
22
24
ܘ
,1613
foil14
87114
23
2815
25.
IBIZ
14
9416
A Traverle-Table for every Point, Half-Point, and Quarter-Point of the Compals, ta the
100 part of a League or Mile; which gives the Difference of Lac. and Deparsore from the Meridian.
The Trauerre. Table. The 3 Points Quarter.
250 g., 19;m 28 deg. 07m. 2o deg 56 m.
.
2 Paints
2 Points
2 Points 1
3 Points,
N
SIE W
Wil N SIE W
N
SIE
W
N SIE W
431
8810)
47
86100 511 83100 561
85
7600 94
7101 03
6601 SS
3 102 7101 28
65121
41 OZ 5701 54
67 3
6:lol 71 03
03 43102
06
4
52102
za los
04
29102
571
기 ​04 · 180 781
5
6 25
56 los 29 02 831
us
is 03 os
col
99103 331 6
06
30
36
00103 60
us
897
23 03 42 27 0503 77 06
06 65104
44
8
14103 85
07 94194 24
07 72104
63
07 48 los 19
jo 109
04/04 108
82104 21
23 58105
14
os
ss 10
il" 109
c9' 70105 18
109
66
109
IS196
II II
I2 I 8505
13
58105 66
17
98/06 67| 12
755 $6
68
II
13
:22 13
60
99
0007
II
56106
41 13 27107
에
​07
85101 711
47108 331 15
16
14
3,4
I1107 5.4 13
72108
23
13
30108 891 16
17 IS
27
58108 74
13,09
18 16
27 7
ZQ
87108 48 15 44109 25 14 97 10
18
19 17
18198 1.2 16 76108
I -30199 77 IS
8010 56 19
08138 551 17
64139
431 17 16.15
28
16 6311
18
18 52409 gol
13
0110 801 17
4611
6.71 zi
19 89109
41
19
37
13 8711 31
18
29112
zo 79109 83
28.10
19
82 19 1212
24
2 1 62110
26 21 1711
31
34
95,13 33. 24
25 22
6010
78
85 20 : 7913 891 25.
26
23
26
.30113 521 zi
54 ? 8112
23
88
45 15
15.2.7
28
25
97
69113
25+
39
561.28
29 26 22 IZ
40
5.8 13
67 24
91 24 II16
129
10 127
831
26
401.24
25
731.25 421
24
671 30
31 28
0213
25
94 125 70 17
31
93113 68 28 115 08 27
45
78
33 29
83114
29
56 28
97 27
331 33
34 130 74114 54 29 9916 031 29
48 2S
35 31
14
96 १० 87116 sol 3.2 02117 99
441 35
36 32
5415 391 31 75/56 971 30
88118
SI 29
37 33
4515 82
631!7 44 31 74119 02 30
7620
3516 25 33 $717 91 32 5919 54
31
6021 II 38
26116
68
34
38
33
OS 32
40 36 16/17
135
28 18
34
56 33 2622
221
40
017
36
34 35
17:1
08
34 0922
78141
42 37 6717
96 37 0419
81
36
59 34
92123 3442
43 133 87118
38 37
27
36
II
75123 891 43
44 139
78/18
81
8020 74 37 74122 62
36
58124 441 44
45
24
40
139
69125
38
14 37 4225
09:45
46' 141 51/19 67
140
68 39* 46/23 65 [38 25125 -561.46
47 42 4920
09
16
40 3124 16 39 0826
IT-47
391:0
52
63 41 1724
68
39
30 20
95 +3
lo
22 49
03125
19 40 74127
50 las
2011
44
57
8925 71 141
51-146 1021 61
44
98124. 04
40128 331 SI
01/22 23 45
51 44
6026 73 43
'53 147
66 46
7414 98 45 4627 25 44
0729
54 48
08
46
3217 76
44
9030
5549 72123
521
28
13130 56133
45
156 150
46
49
40
79
57 151
53 24
371
47
so
89129
58 152
4324
79
82
SI 15127
34 49
75129
221 5.8
59 53
33125
-23 52
8)
33
50
49 06132 78 59
60 154 14:5 65
91128 281 st
46130 84
52
49
33
70 163
27129
GI
60 04135
73137.99
98 58 22138 881:70
31134. 20
63
70 5537
66 5144 44180
47 79
77 19146
74 8315000 901
1,09 190
75 88
13 85
41 83
1455 551100
78185 50 176
38194
26
171 54/102, 87 166, 281111
HIN SI
IN S E
HN SI E MIN SI
Ś Points ?
5 Points +
5.Points,
téa deg
Godee.
59 06.
156 deg. IS m
The s Pointe N70Pys,
126]
34114 61
22
TIS
4516
3116
26
3228
61 17
II
10155
15 17
44/18
27/18
891 34
129
10119
441
COW
93/20
ool 36
561 37
32
38 34
39 135
4018
4321
67 39
45 20
3120
101
10
Ou :0w
41 137
531
16/19
0221
9220
88122
35
38
6919
21
21
60123
5721
ܐܐ]41
41
42
4322
48 43
49 44
9126 67 48
21123
10123
42
42
38
571-4 _-78156
43
74/26
22
42
$2 147
86.4
24/28
891.52
44 53
91122
82123
62125
47
001 54
46
$1125_93
148
6123
17/28
03/28
47
48
43
94
56131
39/20
IL1:56
671 57
2725
87
30
39131
48
22/32
,61.30
3:27
89133
331,6
80 72
6141
I 2
71
43
90 81
26
ool.90
35138
39/42
j
3742
19347
7751
200 180
111700
LT
s Prints
S2
4 m
152
A Traverſe-Table for every Point, Half-Point, and Quarter-Point of the Cuinpals, to the
100 part of a League or Mile; which gives the Difference of Lat. and Departure from ihr Meridian.
39 deg.
1
jur Miles ſaild.
Dif in Leagues
or Miles failid.
Diff.in Leagues
80100
77100 63
00
I
2
4101
2
4|oi
02
OZ
69
Z
83102
83
981. 103
70103
82103
62104
104
18104
78
1805
6605
g 107
34
107
IZ
28107
.
06
89108
63/08
49108
491 It
1
24/08
10
IZ
VIO
6110
IZ
85129
53
IS
II
II
31/11
17 13
OZI
OL
34 12
73112
14
82113
20
67113
11
17
Q113
VO
261 23
22
17
16
97 16
20
891
86
79
08 14
88115
68117
63125
20
20
09 18
ܐ
75 18
8019
80 28
49/16
29/17
43/18
27
87
21
The Traverſe. Table, The 4 Points
Quarter,
36 deg. 34 .
22 m.
42 deg.
II. 45 deg. com.
3 Points
3 Points
3 Points of
4 Points,
N SIE
w
IN SIE W
N
SIE W N SIE
W
I loo
601 foo
loo
74 09 671
7100 71
01 bilan 19 OI
55 JOI
27
OI 4801 34
4.1
3 02
41 02
32101
90
oi 22/02 OI
1202 3
4 03
31 oz 38 03
0902
54
96102
4
5 04
02102
86103 17 103
361 123 54103 54 S
6 104
571
64103 81 04 44104 031 04
2404 24 6
7 l'os
17
4104 44 OS
04 9504
95 7
8 Job 43104
76 06
07 OS 9305 37
66 8
2310s
36
106 960s 71 06 67106
04
06 36106 361 9
10 J08
03los 96. 107 73106 07 4+1 06 72
07 07
071 10
11 138
83106 55
los
SO106
981 108 15107 39 07 78/07
781 ir
ng
64107 IS 09
61 08
08
13 fo
44127
74 10
0508
09
73 09 19109 191 13
14
34
10
82108 88
37109 47
90109
9014
IS
05 08
94 lin_601095 52
II
ILO 07
611 15
16
IZ
37110
85110 74
31| 16
66110 13 13 14 10 78 2
70 II 42
12
17.
18
14
46 10 72 13
9111
42 13
09
12
73 18
19 15 2611
32 14
69112 041
0812
76 13
4413 44 19
16 06/11 911 46112
691 14
43 14
14 14 141
ZI
16 87|12 SI
116 23.13
32 IS 5614
IO
14 8514 851 21
1.7
96 16
30114 77 15 '9615
56 22
23 78
47 13 70 17
78 14
59 17 0445
45
16 26/16
24 1.9
28 14 30
18 SSIS
78 16
12
971 24
25
19 32/15
18
5116
17
26 20
49 2) 10/16 49 19
26117 461 118 38118
381 26
27 21
69116
08
87117
13
00:18'13 19
09 27
28
68 I 64/17 76 10
77 19
29 23
22
40
4919 44
20
SI:0
SI 29
30 (14 10/17
23 19119 03
23/2 ! I: 21
2121
30
31 124
47
23 96119 67
972
82
92121
92 며 ​31
32 125 70119 06 24 74/20 30
49 23 63122
119 66
25
5120
93 24
16 23 33:3, 331 33
34 27
15 26 28121
25
83 24 04124
11 20 85 127 06 22
125
93123 50 24. 75124
91121 46 27 83122 84
126 67|24
17 25
46125
37 129 72/22 04 28
47 27 41 24 85
26
1626
5 2 12
64
29 3724
28
lib
52
87126 87
39 31 33/23 23 30 IS 24 74
28
19 27
56127
40 132
130 9212 38 29 6426. 861 28
28128
93124 42
69126
30
28
38127 531
991:9
9941
7325
32
64
31 12/28
29
43134 54125
33
28
86128 88
30
41 43
4435 34126 ZI
0127 91 37
SS 31
44
14/26 81
34 78 28.551
621
33 3430
31
46,136 94/27
401 35 56129
18 34 08130 89 32 53132
36
33129
82 34
56
33 2333
23 47
37
45 35 5732 23 33 9433
49 39 36/29 19
37
31 32 91
34
65! 49
so 140
17129_781 38
721
37
os
35/35
351 sa
SI 140
39
351 37 79134 35 36
06136 06
SI
5241
99
53134 92
77136
5342
57
5731
40
62
9733
39
2735
59
37
26
7434
17
37 32
40
26
14138 14 $4
5234
86
75136 94
89155
98 33 36 43
29135
53 41
49 37
601 so
57 145
16
96
06 36
44
42 23138
10
30140
30] 57
58 46
59134 55 44
83136 79 43 0738 95
01 58
59 147
39135 IS 45
43 43
62
7239
41
72/41 72] 59
60 48 19135 741 146
3838 06
44
45140
142
4314
22|41 691
41 51
85147
49149 491 70
80,64
25147 65 61 84150
75
59
72 56 56136 56 80
61 69
5757
09 66 67160
44 63 63163 631 90
77
30163 43 74
IS 70 7170
71 roa
200'|160641119 12
154 60 126 86
148 161134 30
141 41141 41200
WIN SI IE WIN S E WIN S
HN SI
4 Points
4
4 Points
4 Points.
5.3 deg. 26 m.
27 7 147 deg.
49 m. 45 deg.
The 4 Points Quarters.
90/18
22
631 33
33 26
7121
45 22
1922
31/20
20
35 28
36 123
04 34
751 35
461 36
1637
60123
38 30
1
1625
38
90126
561 39
2840
13123
83
4132
437133
31
01
02
ZI
IO) 42
61
47/26
24127
31
10129
4130
II31
8231
34
60129
II
45 i 26
22
821
821 45
53 46
OO
37
1828
55/28
mmmmmmmm
82131
59
38-
941 48
03
36
65134
10130
8831
65131
42132
2013
05133_58 lei
96130 38
77130 98
40
38
36
771 52
48 53
4837
ao
33
0136
54 43
55 144
56 144
18132
76)
41
42
38
38
40
611
28
89138
60139
39
78133
01 41
6137
29
431 60
70 156
44
11144
49
26153
28153
90 172
Too 80
3259
08167
200
E
2
I
4 Points
so deg.
QO m.
CHAP.III. Sailing by the Plain Chart.
153
As for Example.
100
3
29
Beforegoing, if
you take all the Numbers
9951 980
in the Table, they will ſtand as here appear-
S. Weft.
70 6966 686
eth, where the Southerly Diſtance is 172***
299
Leagues
, and the Weſterly is 16,25. Leagues
,
Leaguos. 117: 3172:10 16:95 But I hold it more convenient to omit the
laſt Figure to the right hand, and ſo take the
Tenths, as in the ſecond Example; and then in all things it will agree with the Tra-
verſe-Scale, on which if you extend the Compaffes from 100 to 73, the ſame Diſtance
will reach from the firſt's Point 11ext che Line of Numbers, to 172.o and from the
half Puint of the Weſting, to 16. Leagues, as before.
As alſo, If you extend the compafles from 172.-- the Difference of Latitude, or
72.000, which ſtands for 16., on this or the like occaſion, and apply
this Distance from 4 Points on the Tangent-Line of the Scale, and the other Point of
the compaſſes will reach co ; Point, which is from the South Wefterly, as before.
Now for the point and Points reckoned at the Bottom, it is thw.
Admit a Ship fails 57 Leagues or Miles North-Weſt and by Wift, or che sth Rbomb
from the Meridian ; I would know how much I am to che VVeftward, and
how much to the Southward.
from
1
3
B
-474
S
5
{
Therefore look in the bottom of the Tain
Diſtance 3 Rhomb. I
ble for the sch Rhomb, and in the Side for
IN SIE
57 Leagues or Miles; and in the Line of
Meeting over the fifth Rhomb you have.
57
47. 39 or 47 6. for the Weſting, and 31.67
317
or 31almoſt for the Northing.
Now had
you
been to find the Northing
Sailed E
MIN
and Weſting of the third Rhomb" from the
5 Rbomli. 1
Meridian, as N.w.beN. to 57 Leagues di-
ſtance, the Northing would be 47, and
the Wiſting 31.7, as you ſee ſignified by the Letters N. S. and E.W. at the head of
the Table, and North N. S. under E.W. at the foot of the Table. This is ſo plain, it
needs no further Precept.
Or by the Traverſe-Scale, Extend the compaſſes in the Line of Numbers from 10
of foc, to 57 Lergues; the ſame Diſtance will reach from 3 Points in the next Line,
with s Points of the Easting and Westing, to 47.4 Leagues or Miles; that Diſtance
will reach from s Points in the Line of N. and S. to 31.7 Leagues, as before.
And the Compaſſes extended from 47c, to 31. on the Line of Numbers; che
fame Diſtance will reach from 4 Points in the Tangent-Line, to s Points from the
Meridian, or 3 Points it the Caſe ſo required, as if it had been N.W.6. N. The
like do in all ſuch lHeftions.
Likewiſe by the Traverſe-Scale, Let the Courſe be given N.W.1.W. and Departure
47*., To find the Diſtance and Difference of Latitnde.
Extend the compaſſes from s Points in the Line of e.w.of the Scale, to the De-
Parture; clie ſame Diſtance will reach from 10 or 100 in the Line of Numbers, to
57 the Diſtance ; And alſo from 5 Points in the Line of N. S. to 31, the Diffe-
rence of Latitude
. I make this plain by the Scale, by reaſon the compaſſes and the
Scale, are more portable than the Book and Table.
A larger Example I will give you of the Tables and Traverſe-Scale together,
whereby you inay perceive, That the Artificial Numbers, Points, and Quarters
agree in all things with the Table; nay, I hold the Scale the beſt of the two, for the
ready allowing for Variation, and for Currents, which is done by removing the
Compaſſes froin one Point or Diſtance to another. Now let the Queſtion be this
,
X
Suppoſe
rumenti
154 Sdiling by the Plain Chart. Book IV.
+
Suppoſe a ship fail from the IP and of Lundy, iu Latitude 51 deg. 22 min. Nurth,
and Longitude 35 deg. 52 min. to che Iſland of Barbadoes, in Latitude 13 deg. 10 min.
North, and Longitude 332 deg. 57 min. By the Plain-Chart, Difference of Latitisde
is 764 Leagues, and Longitude 1059 Leagues
, and I fail cheſc ſeveral Courſes, (viz.)
S: S.W. W. from A to B 400 Leagues, S.W.b.S. W. 125 Leagues, and s..
180 Eeagues, and S.W.b: w., Wefterly 190 Leagues, W. S. W. 146 Leagues, and
Wib.S. 159 Leagtes, and South 8 Leagues 776: All theſe Courſes and Diſtances I
ſce down as followech. In the firſt Column is expreſſed the Days of the Month, and
Diſtance failing upon cach Courſe ; The ſecond, the D.ry of the Veek: The third,
clie Courſe failed; The fourth, the Diſtance from the Meridian ; The fifeli, che
Place and Point of each courſe by Letters; The ſixth, the Diſtance ſailed; The fe-
venth, eighth, nich, and renth, the Northing, Southing, Eaſting, and teſting,
which is the Difference of Latitude and Departure from the Meridian in Leagues
Parts; The eleventh Column is the Latitude; The twelfth, the Longitude; The thir-
tecnich, the Variation of the Compaſs.
Diſtance
Latitud. Longit.
Courſe from the The Dift.
Dift. Norib- South-Eaſt-Weſt-
Sailed.
Meridi- Places. Sailed. ing. ing. ing. ing.
Variatic
Dige
D1. Month. The
D... Week. 14
Min.
Degr.
ON.
Min.
lan.
2,212 5
10
Cur.fets
3
с
2001
Variat.
Ho
S. W. 4 From C
Variat.
Points.
);
S. W. S.W.54 Prom A 1306 764 704 1058105851
52 Erfterly
Apr. 21 f
2.12 m. K. Lcagu. Leagues Leagu. Leag. Leag. 13 3 32 57 Sm.
Current
S. S. W. S. W. From A 200
176,4 94
May 21
Sets
W.
33 44 16 26 by eſtim.
2 Po.ro B.
176 41
E.S. E.
194 3
E.S.E.
100
773 634
S. W.b.S. S. W. From B
G
20
ISS 1 27/28 2812 38
W: .113 Ponto C.
00 n.
5
39
32
100
707 707
S. W
rold
80
566
S.W. W.
566 32 06
ro D.
42
00 m.
1 2
87
S. W.by S. W. From D 100
471
882
Weſterly
W.W. 5. Poso. E.
424
2 degr.
100
383 9241
S. W.6 From E
Weſterly
19 f W. S. W.
153
40
Points. co F.
370115 50/350 24
4 degr.
6
23
55
S. W. 7 From F
1001
195 981
Hefterly
Points. co G.
50
490 13 11 342 46
5 degr.
92
The Courſe S.W.48
862
made good. 14. 34 m.
$W.3
8c
151 b
90
o
79457 38/35 19
24d W. by S.
981
38
7637
12121
13 17|342 46
2
220
This done, add up the South Column, which Sum is 763. Leagues; which redu-
7834. (38: 11 çed into Di grees, by dividing by 20 and multiplying the odd Leagues under 20
by 3, and adding the Minutes in the Tenths, you will find the Difference of Lati-
tude in Degrees to be 38 deg. 11 min. which ſubſtracted from si deg. 22 min. there se-
deg. min. mains 13 deg. Il min. the Latitude of Barbadoes.
"Add
up
elle Soomes of the Weff Column, which is 862 Leagues ; that converted in-
to Degrees
, is 43 dig. 6 min. Subſtract that from the Longitude of the iſland of Lun-
dy; it, you cannot, add to it 360 deg. So, 25 deg: 52 min. added to 360 deg. makes
Latitude of Bar-
385 deg. 52 min." Then the Difference of Longitude ſubſtracted from it, 43 deg. 6 m.
badoes. there remaius 342 deg. 46 min. the Longitude the ship is in.
You muſt note, The Degrees are ſuch that 60 Miles or min. makes a Degree
Longitude or Latitude, or of a Great Circle.
Nore, The day we ſet ſail, we put down the day of the Montband Veek, the di-
rcet Courſe to the Port we are bound to, and the Place marked with two Letters, as
* 22
-IT
SI-
38.
13-
11
of
in
CHAP. IV. Sailing by the Plain Charti ! 155
t
1
in this Table A for Lundy and K for Barbadoes; and alſo uncler Diſtince, the number
of Leagues upon a ſtraight Courſe; and under Northing and Southing, the Difference
of Latitude in Leagues and Tenth Parts; and ander Latitude, the Latitude of the
two Places; and under Longitude, the Longitude of the two Places, and alſo the Va-
riation of the Compaſs from whence we ſet our firſt : which you may ſee all plain in
the head of the Table, in the Common Angle of Mciting with the 21 of April
.
And remember,Yon have the Latitude and Longitude given you; therefore by it you
muſt find the direct Rhomb and Diſtance, as you have becn (hewed by the ſecond and
ſixth Caſe of Plain Triangles.
Now if you would ſct down this Reckoning on the Plain Chart ſeverally, you muſt
extend your Conopalles from one of the Paralleis of Longitude, to the Latitude
you
are in; as alſo take off ſo many Leagues of the Meridian Line, as your Depariure
hath been, reckoning s Degrees for 100 Leagues, and every Degree for 20 Leagues.
As for Example.
Suppoſe we would ſet down the firſt Diſtance of Soush and Weſt, Extend
your
Compaſſes from the Parallel of 40 deg. to your Latitude you are in 33 deg. 44 min. And
alſo extend another pair of Coropales on the Aquinoctial, if chere is one divided ; if
not, on the Meridian, which is all one; and take off 16 d:g26 min. by one of the
Parallels of the Meridian: or take off 188 Leagues which is 9 deg. 26 min. the
Difference to the Weſtward from your firſt Meridian; and fo let the compaſſes of the
Difference of Latitude run upon the Parallel of 40 deg. and the other Compaſſes with
188.or 9 deg. 26 min. on one of the Parallels of North and South, until they meer
in the Point B: (And ſo add the Meridian-difference of the ſecond Place to the firſt;
and the Difference of Latitude of the ſecond Place, fubftract from the firſt, by reaſon But add the
you are going from the North Pole toward the South or Æquator.) As for
your
De- difference
grees of Longitude, you muſt know where you begin the firſt Meridian; and as you of Lati:
go to the Weſtward ſubſtract the Difference of Degrees of Meridians, and as fail
to the Eaſtward add the Difference of Degrees, and you have the Longitude in Degrees sporto melard
where you are.
So that this may ſuffice for a Preſident, to lay down on your Draught or Blank
Chart che Point of the Place of the ship, by the Meridian-diſtance and Difference of
have been directed , fo are che Points C, D, E, F, G let down,
So that you need not peſter
che Chart with Rbomb-iines, as formerly; but take the
Difference between the Latitude and Meridian-diſtance off the Line of Numbers, and
apply that Diſtance to the Tangent-line of Rbombs on the Traverſe-Scale, and that
will preſently ſhew you the point or Rbomb between any two Places aſſigned.
The drawing of the Plain Sea-Chast, and the way of ſailing thereby, is the moſt
eaſie and plaineſt of all others: And though it be fit to uſe only in Places neer the
Aquinoctial, or in ſhort Voyages, yet it will ſerve for a good Introduction to the
other kinds of Sailing. Therefore we ſhall not loſe our labour ; for in all kinds of
Sailing the ſame Work muſt be obſerved with ſome caution.
11
tude as jou
you
mi
Firſt,
1
1
B
156 The Plain Sea-Chart, and how to make it, B.IV.
The Plaine Sea Chart:
SI
10
20
30
50
T
lo
ot
The
Equingtial
10
10
G
K.
F
E
C.Deiverd
St Antonio
Barbados non
$
20
D
E
W
.
с
Palero
3016
30
B
Madara
P.Santo. !
ws: Michal 0
401
40
is
+
ibo
507
AU Londy
125
15
folio 156.
5
35 5
3415
335
in Bib
ܪܝܢ
folio
+
1.
Firſt make the Square ASTB, of what length and breadth you pleaſe and divide
each Side into as many equal Parts as your occaſion requires; and theh draw ſtraighc
Lines through theſe Paris, croſſing one the other at Right Angles, ſo making many lit-
cle Geometrical Squares, each of which may contain one Degree: but I have made
this, by reaſon of its largeneſs; to contain 10 Degrees. Note, That the Degrees of
the Meridian at the Aquinoētial are all of equal diſtance to the Poles, which is a
groſs Error, which ſhall be ſhewn in the following Diſcourſe. So that you may make
the Meridian-Line in your Chart 25 deg. 52 min. to the Weftward of the Meridian
of Lundy: Or you may divide the two Sides into Degrees as far as you think fit,
and every Degree into 66 Parts, which is the old way; and I know molt Mariners
will not be directed a new way of dividing the Degrees cach of them into 10 Parts;
ſo each Part will contain about 2 Leagues; and that diviſion of double Leagues is
near enough for the Mariners uſe. You may ſuppoſe each of theſe Parts to be ſub-
divided into 10 ; ſo every Degree will contain 100 Parts, which is a very ready way
if you keep your Account by Arithmetick, by Decimals or 10 Parts. This is fo
plain, it needs no further Precept; therefore we will proceed to the uſe of it.
Now
your ſeveral Courſes and Traverſe-Points are laid down on your Chart, from
Lundy
20)
Ch.IV.The Plain Sea-Chart,and how to ufë-it, ist
000 02
The Errors
Lundy at A, to B the firſt,ſecond to C, third to D, fourth to E, fifth to F, the ſixth to
to G, which is the point the ship is in when you
764 00 caſt up this Reckoning. Now to know how far 1058
763 08 you are ſhort of the ſand by the Plain Chart, 862
fubftract the sum of the South Column 7634 796 Leagues
from the Difference of Latitude 764 Leagues, sort of Barbadoes at G.
and
you will find you are bur ia parts of a League to the
Northward of your given Latitude, which is not to be regarded ; and alſo ſubſtract
the Sum of the Weſt Column, from the Difference of Longitude, and the Remainer is
197 Leagues, which you are ſhort at G of the Barbadoes; aná being in the Latitude
of 13 deg. 11 min. the Iſland bears off you due Weſt at K : So char you ſhould fail
197 Leagues on chat Point Weſt, before you ſhould be arrived at your Port by che
Plain Chart.
But by the true Sea-Chart you are arrived at G, which is the Iſland of Barbadses :
For the true Meridian-diſtance is buc 865 Leagues between Lundy, and Barbadoes, and of the Plais
the Plain Chart makes it 1059 Leagues; and the true Courſe from Lundy to Barbas sea-Cbart.
does is but 48 deg. 34 min. which is S.W.a little above a quarter of a Print Weſterly;
and the Plain Chart makes it 54 deg. 12 min. S.W. which is above of a Point; And
che true Diſtance is but 1152 Leagues, and by the Plain Chart it is 1366 Lergues.
By this you may plainly perceive, that no Ifand, nor Cape, or Head-Land, can be
truly laid down in the Plain Chart in its true ſcituation, but near the Aguine£tial
only, and near about the ſame may be uſed without ſenſible Erior, becauſe there
only the Meridians and Parallels are cqual; but on this
ſide or beyond the Aquino-
Etial there is Error committed proportionally to the Difference of the Meridian and
Parallel, that is.-Thetrue Difference of Longitude found out by the Plain Chart, hath
the fame. proportiointa the-true-Difference of Longitude, chat the Parallel hath to the
Meridian. " But moſt Mariners will not be drawn from this plain caſie way of
Sailing, notwithſtanding they have it plailly demonſtrated to them by us: Buc
thoſe that take the true way of Sailing, find the Credit and Benefic of it, to the
Ihare of thoſe that are ſo obftinats, conccited, and grounded in Ignorance.
Bus in the following Diſcourſe I will uſe my endeavour to make chings ſo plain,
tliat-if the Ingenious Mariner will but ſpend half an hours time at the ferting forth
of liis Voyage, to find, by direction his
true Courſe and Diſtance, and Meridian-di-
ftances and put it at the Head of his Journal, as you ſee in the Table, he ſhall ale
his plain Sailing all the reſt of his Voyage ; and he ihall have direction how to uſe le
by the Chart made according
to the Globe. But ſomething more of this way, accord-
ing to my Promiſe.
they
CHAP.: IV.
How to Correct the Account, when the Dead Latitude differs from the
Obſerved Latitude.
E are come now to make good what was promiſed in the ſecond Chapter,
to preſcribe four Precepts for correcting a Single Courſe.
I ſhall be brief, in regard Mr. Collins, in pag. 22. of his Mariner's
Scale new Plained, hath imitated Mætiu a Hollander, a Latin Author, in theſe Examin
ples; but good Rules, che oftner writ, che mose they get.
W
The Firſt E XAMPI E.
F a Ship fail under the Meridian, if the Difference of Latitude be leſs by Eſti-
mation, than it is by Obſervation, the ship's Place muſt be corrected and enlara
ged under the Meridian; and the Error is to be imputed either to the Judgment in
eſtimating the Diſtance run, in making it too little; or if the ſaid Diſtance bc eſtima-
ted by a ſound experienced Judgment, it is to be ſuppoſed you ſtem ſome Current,
Adinic
1
158 How to Correct the Dead Reckoning. Book IV? :
Admit a Ship fail from A, in the Latitude of -36 deg. dirc&tly Souch, 70 L'aguses,
or 3 deg. 30 min. and by Eſtimation is at B,but by Obſervation lie is in Latitud
32 deg:
The Reckoning rcctified, the ship's Place is in the Point C; b.ie if the Difference of
Latitude be more by Eſtimation, than it is by Oblervation, the Judgment may crri
in ſuppoſing the Diſtance run to be too much. In this Cale, che Diſtance is to be
ſhortned, and the Correction muſt be made according to the Latitude obſerved ander
the Meridian.
Admit a ship ſail South from A, in the Latitude 36 deg. untill hac have altered lier
Latitude 3 deg. 30 min. by Eſtimation being at B, in Latitude 32 deg. zo mir. and if
che obſerved Latitude be 33 deg. oo mino the ship's Place corrected is at C, and 100
at B.
no
W
MIINIIIIIIIIINNI*
RULE II.
Uppoſing no Current, If the Dead Latitude differ from the Olſerved Latitude, the
Error is in misjudging the Diſtance run, which is to be made longer or ſhorter,
as the Caſe requires.
.. Admit a ship ſail from A, S.S. E. Eaſterly 70 Leagues, and is by Eſtimation
at P in the Latitude of 33 deg. but if the obſerved Latitude be 32 deg. 30 min. ad-
mit at B, then a Line drawn through B, parallel to NA, crofleth de Lin. of the
Ship’s Courſe at, which is che corrected Point where the ship is: So that the Di.
ſtance is inlarged 10 Leagues , the whole Diſtance AQ_is 82 Leagues 1..
The ſame manner, If the Ship
liad failed 94 Leagues on the
C
ſaine Courſe, and by Eſtimation
were at the point R, in the Pas-
R
rakel of 32 deg. and by Obſerva-
32
BI
tion the Latitude were found to
D
be 32 deg. 30 min. In this Caſe
the ship's Diſance is to be ſhort-
ged, by drawing the foreſaid
33
Line B parallel to N A; and
it will croſs the Line of the Ships
Courſe at , the Corrected Point
where the Ship is.
134
By the Traverſe-Scale. M
V135
Xtend the Compaſſes from
10o, to 94; the ſame Di-
S
ſtance will reach from 2 Points,
to 824 Leagues in the Line of NK
Numbers.
I
A
101
0
பா
F
1
136
데
​K
1
RULE
III.
i
Suppoſe there is ſome
Current, and
you can depend upon
the Olferved Difference
of Latitude, and Lng-diſtance, as both true; then the Error
may be impuced
to the Rhomb, which alters by reaſon of the ſuppoſed Current.
Eſpecially when you fail in Rhombs near the Eaſt and Weſt ; for then if the Dead
Latitude differ from the Obſerved Latitude, the Error is to be im puted citlier wholly
to the Rhomb, or partly to the Rbomb, partly to the Diſtance.
If wholly to the Rhomb, then retain the obſerved Difference of Latitude, and Di-
ftance by Obſervation, and thereby find the D.parture from the Meridian, by draw-
ing a new Rbomb-line
But if your Judgment would allot the Error partly to the Rhomb, partly to the
Diſtance,
CHAP.V. How to Córrect the Dead Reckoning. 159,
Diſtance, keep the obſerved Difference of Latitude : And for the Departure from the
Meridian, let it be the ſame as was by the Dead Reckoning.
Suppoſe a ship fail Eaſt by Sostb. a Point Southerly 72 Leagues, from the Latitude
of 36 deg. from A to M, and by Dead Reckoning ſhould be in the Latitude of 35 deg.
If the Obſerved Latitude be 35 deg. 20 min. which is at S; In this Caſe, if the Error
be wholly imputed to the Diſtance, the Line S X being drawn parallel to N A, would
cut off or ſhorten the Diſtance as much as the Meaſure MX, which is 26 Leagues;
which becauſe it ſeems abſurd and improbable, is not to be admitted of: Whicrefore
imputing the Error to the Rhomb only, place one foot of the Excent AM in S, and
with the other croſs the Line N A ac L; and ſo is A L the Departure from the Meri-
dian required ; whereby the Rhomb-line, if it were drawn, will be ordered to paſs
through F the Crofs.
By the Traverſe-Scale.
F
I
you extend the Compaſſes from 100, to 72 the Diſtance; the ſame Extent will
reach from the Difference of Latitude by Obſervation, to the true Rhomb, which
is almoſt Eaſt by South: and if you apply that Diſtance to one point on the Line N. S.
of the Scale, the other will reach to the Departure required 70. Leagues.- Which
is far better than the other way.
1
The Fourth PRECEPT, CASE, or EXAMPLE.
1
If
a Ship fail Eaft or Weſt, and the Dead and Obſerved Latitude doth agree, the
Reckoning cannot be corrected; but if they differ, the Error will be partly in the
Rhumb, and
partly in the Diſtance : In ſuch a Caſe keep the Meridian-diſtance
, and
the Difference of Latitude is the Diſtance you are gone to the Northward or South,
Dard of the Eaft and Weft.
1
4
By the Traverſe-Scale.
Xtend the Compaffes from 100, to the Diſtance failed; the fame Extent will
reach from the Difference of Latitude by Obſervation, to the true Courſe: So
that
you may in a moment do all theſe Queſtions and Cafes by chc Traverſe-Scale,
and Line of Numbers and Artificial Points and Quarters thercon. If you have buc
the
perfect Uſe of it, I know there is no Inſtrument whatſoever more ready to reſolve
any uſeful Queſtion, and correct your Reckoning.
Laſtly, If by frequent Obſervation you find the Ship is ſtill carried from the Eaſt
or Weſt, cither Northward or Soutbward, you may conclude ſome Current to be the
cauſe thereof : Keep the Diſtance by Dead Reskoning and Obſervation, and the Diffe-
rence is the Diſtance from the Parallel.
We will not multiply too many Examples, but rather adviſe the Ingenious to
make uſe of ſuch as his need ſhall require; for underſtanding what hach been ſaid,
will be advantageous to the Practitioner.
CHAP. V.
How to allow for known Currents, in Eſtimating the Ship’s Courſe
and Diſtance.
His Subject hath been largely handled by Mr. Norwood, at the end of his
Sea-mans Praćtice ; and by Mr. Philips, in his Advancement of Navigation,
page 54, to 64. As alſo how to find them out by comparing the Reckoning
homeward with the Reckoning outward, which was kept betwixt two Places: There-
fore
T
160 ; Hom to allow for kroppn Currents, Book IV.
fore I ſhall be brief, and demonſtrace by Soale and compaſs, what they have done by
Tables.
Firft, This is caſie to be underſtood, If you fail againſt a Corrent, if it be ſwifter
than the Ship’s way, you fall a Stern; but if it be flower, you get on head ſo much as
is the Difference between the Way of the ship, and the Race of the C#rrent.
V
EXAMPLE.
.
Soulb
Current
mil,
S
3
1
Current
8
D
1
4
5
LEPTH 2
2
.
(
C
If a ship fail 8 Miles South in an Hour, by Log or Eftimation, againſt a Current
that fets North 3 Miles in an Hour, that fubftracted from 8, leaves s Mile an Hour
the Ship goes a head Sonh: But if the ship's way were 3 Mile an Hour South, againſt
Gres a head 5S a Current that ſees 8 Mile an Hone North, the ship would fall s Miles an Hour a
Stern.
mil
Admit a Ship runs East 4 Miles an Hour, and the Current runs alſo 3 Miles an
Ships Itay 13
Hour, What is the true Motion of the Ship? Anſwer, 7 Miles an Hour a Head.
Falls afterns
Admit a Ship croſs a Current thac fers North Eaſt-by-North 4 Miles an Hoxr; the
Ship fails in a Watch, or 4 Hoxrs, 9 Leagues East-by
. North, and in two Watches more
ſhe ſails 13 Leagues. E. N. E. by the Compaſs
.
Now it is required what Courſe and Diſtance the Ship hath made good from the
firſt place of fecting out from A.
Firſt draw the Right Line AL,
IT
H 411
their with the Chord of 60 Deg.
Z
deſcribe the Quadrant ön it; to 2
be ſure take 90 deg. off the Line
3
of Chords, and lay it from N to
R
O;, then draw the North Line
AP; then ſec off the ship's firſt :
Courſe one point from thic Eaft
from N to G, and draw the Line
G
N
AG, and from A to B lay off the
firſt Dilance 9 Leagues : Then
E
prick off the courſe of the Cura
B
xent, being s Points from N to F,
and draw the Line A F, being
the Courſe N. E. b. N. of the
Current. And becauſe the Cure
Tent in 4 Hours ſets 5 Leagues
forward in its own Race, there P
fore draw the Line B C, parallel
to AF, that is, take the neareſt
North24
2 LA.
Diſtance from B to AF, and .
ſweep a ſmall Arch, and from B to the upper Edge of the Arch, draw the Line BC
thereon, put from B to Cs the Currents Motion, and draw the Line A C, which
thews the Courſe the ship hath made good the firſt Watch.
Now for the ſecond Courſe, draw CH parallel to the Line A L, and with the Raa
dims or Chord of 60 deg: upon C asa
Center, draw the Arch Hz, whereon prick 22
deg. 30 min. or 2 Points for E. N. E. for the ship's ſecond Courſe from the Eaft; and
draw cz, whereon prick down the Diſtance failed 13 Leagues from C to D; then
draw D W parallel to AF, as you did B.C; then becauſe the Current ſets 10 in
two Watches, therefore prick down 10, Leagues from D to W, and draw the line
A W; which being meaſured upon the ſame scale of an Inch divided into 10 parts,
ſhews the ship's direct Diſtance is 35 - Leagues ; whereas if there had been no Curs
rent, the direct Diſtance had been A R 22 F Leagnes : Then meaſure the Arch NE,
and you will find it 35 deg. which is a little above 3 Points from the Eaft. So the
Point the ship hath made good is North-Eaſt-by-Eaſt a little Northerly; whereas if
there had been uo Current, the Courſe had been N s, that is, Eaſt and by North
of a Point Northerly, and had been at R, but now the Ship is ac W, therefore di-
ftant from it equal to RW 15 to Leagues. The prick'd Lines are the Courſes and
Arches without a Current.
1
E
3
1
1
وا
This
r
CHAP.VI. Queſtions in Navigation:
!"
A
1
ther Number which here I
* ; I ?
161
This is a good way to work theſe Queſtions: If you have no Compallis, draw on a
Slat or Quadrant to work Traverſes by; if you have, that way is the ſooneſt donc
by them after the ſame manner. Some will expect, that knows me, ſome other fort
of Queſtions, (beſides theſe moſt uſeful beforegoing:) For them, and their leiſure-
cime, I have inſerted theſe fix Queſtions following.
QUESTION I.
A Ship Sails 40 Leagues more than her Difference of Latitude, and
is departed from the Meridian 80 Leagues, I demand her Diffe-
rence of Latitude.
Ake a Right-Angled Trian-
H
Make
gle, to that the Bafe FG
be equal to her Difference 40
Leagues, and the Perpendicular
GĦ equal to her Departure 80
Leagues: Then continue che Baſe
FG, and find the Center point E
unto H and F, ſo it will be E,
and G 60 Leagues for the Diffe-
80
rence of Latitude ſought.
Atohmetically
.
40.
Square GH 80,you have 6400,
which divide by GF 40, the
R Oslotient is 160 ; from whence 6400
ſubftract G F 40, there remains 2
120; che half is 60, for the Difference of Latitude fought.
6400 (160
40
Questi II.
A
60
A Ship Sails 20 Leagues more than her Difference of Latitude, and
but 10 Leagues more than her Departure from the Meridian, I
demand her Diſtance Sailech
B
IN, the Trlangle A BC,
you.haye E B 20. Leagues
more than the Difference of
T12Datitude Aic'; and A D, 10
mit L'agnesimore than theildar
parture from the Meridian
BC.
IT
Firſt, with the double of ei-
A
ji take, the double of E-B 20,
??
wch is 40 Leagu.and
lay from
A ana
add it thereunto, as Ġ H.
Now on the midſt of FH,
210
as at K, making it the Cen-
ter, I deſcribe the Semicircle
HIF: Then on G crect the
2 TO
43
F
Perpendicular which cuts the
G
K
Yrch in I ; then pleaſuring
GI,
8
80
80
1
E
G
640
44 lo
I20
Slulinis
IN
20
E
ID
D
f!
1
+
.T 210
E
1
t
Queſtions in Navigation. Book IV.
162
200
200
.
GI, it will be equal to D E 20 Leagaes, which added to the two former Numbers
10 20 and 10, you have in all so Leagues for the Diſtance failed, required.
Anabically: 2 AD:X:EBSDE2400, whoſe V q is 20, 4$$ (20 che Root.
The Square 490
Ques T. III.
Two Ships Sail from one Port; The firſt Ship Sails dixeEtly South, the
Second Ship Sails W.S. W. more than the firſt by 35 Leagues,
and then were aſunder 76 Leagues; The Queſtion is, How ma-
ny Leagues each Ship Sailed.
Fire
Irſt draw the Meridian-line A B, and from A draw a W.S.W. Courſe as AC
continued, and from C lay down the 35 Leagues unto D. Now draw the
Chord-line of 6 Points, as B C; then take 76 Leagues, and lay it from D to cut the
Chord-line in E. Laſtly, from E you muſt draw a Parallel Meridian, which will
cut the Rhomb-line in F'; fo meaſuring E F, you ſhall have 45 To Leagues,
that the firſt ship failed directly Somsh : So the ſecond ship ſailed 35 Leagues more,
therefore muſt Sail in all 80Leagues, which is the Diſtance required.
}
B
A
)
56.15
li
:.
>D
pa
1
35
! ۲: دلي.
1
A 16130167.39
Til
f
Al WSW
1
А
13 viis vir
It gigante unit:
91657cuIII: los client
By the Artificial Tables of sines and Numbers
i ci
WA?
aub
!!)
Asthe Side E D 76 Leagues co:ar.
8119.19
Ta the Sine of the Angle EČ D 56 deg. 15 min
991985
So is the side CD 35 Leagues
154407
To the Sing of the Angle CED 22 deg. 31 min.- 9583111
which ſubſtract from. 56 deg: 15 min. you have the Angle.at D 33 degi 44 min.
Then,
As the Sine of the Angle at F 67 deg. 30 min. co. ar.
99348
Is to his oppoſite Sider
76
188081.
So the Sine of the Angle at D 33 deg. 45 min. S
add
974455
Tio his oppoſite Side FE 45 ** Leaguesa
165974
Şamah Ship Sailęd 45.5 Leagues, and the other W. S. W. 80 m Leagues.
1
irti
niciale
t
:
QUEST.
e che
1
}
1
Chap. VI. Queſtions in Navigation.
163
3
QUE's T. IV.
Two Ships Sailed from one Port: The firſt Sails S. S.W. a certain
Diſtance ; then altering her Courſe, She Sails due Weſt 92
Leagues : The ſecond Ship, Sailing, 120 Leagues, meets with
the firſt Ship. I demand the ſecond Ship's Courſe and Rhomb,
and homo many Leagues the firſt Ship Sailed S. S.W.
D
Raw the firſt Ship's Rhomb from A unco E, being S.S.W.chen lay her Diſtance
failed weſt 92 Leagues from A unto C, and from C draw a s. s. w. Courſes
as CD continued: Next take i 20 Leagues, and lay it from A, fo that it ſhall cut
the continued Line in D: ſo drawing Å D, you ſhall have the ſecond ship's Rhom!
near W. S. W. Laſtly, meaſuring CD equal to A B, you ſhall find it to be 49
Leagues that the firſt Ship failed S.S.W.
F
I 퍼
​D
"
B В
ASS
10
1
A
-
For the Courſe,
As the Side AD-I 20 Leagues,
co:ar.
Is to the Sine of the Angle at B 67 deg. 30 min.-
So is the Sine of the Side BD 92 Leagues-
To the Sine of the Angle BAD 45 deg. 6 min.
792082
996562:
196379
985023
Unto which add the Angle F A B 22 deg. 30 min. you have the ſecond Ships
Rbomb 67 deg. 36 min. being near W. S.W. whoſe Complement is the Angle ADB
21 deg. 24 min.
For the Diffance
As the Angle
BAD. 45 deg. 6 min. co.ar.
Is to the Side
-BD 92 Leagues
So is the Angle ABD 22 deg. 24 min.-
Tatbe Side A B 49. Leaguesrequired.
614976
196379
958101
169456
r
1
:
1
Y 2
Qui ST.
1
1
164
Queſtions in Navigation: Book IV.
Ship.
Now 32 deg. 30 min. added to 78 deg.45min the halfis so deg:37 min. 3.
Quos T, 'V.
Two Ships Sail from one Port 7 Points afunder: The one Sails in the
S.W. Quadrant, and departs from the Meridian 57 Leagues ;
and the other Sailed in the S. E. Quadrant, and was departed
from the Meridian but 25 Leagues, and then are both fallen
into one Latitude; I demand the Rhomb ør Courſes of each
FT
Irſt draw an Eaſt and Weft Line continued ; and making choice of a Point at D,
upon D erecta Perpendicular, which will be a Meridian-line; as D A continued.
Now from D lay down the West Ship’s Departure D B 57 Leagues; alſo the Eaft Ships
Departure 25 Leagues DC: ſo their whole Diſtance will be ČB 82 Leagues. Now
apon the Puint at B, or elſe as here at C, draw an Angle of the Complement of 7
Points, or one point, which is w.b. N. as CF the prickt Line; but if their courſes
had been more than 8 Points, then you muſt lay it to the Southward of che West Line.
C
D
E
LS 32
P
ΤΑ
Now from the midſt between B and C, ar E, draw another Meridian-line, until
it cut the former Rhomb-line C F in the Point G: So taking the Diſtance from the
Point G unto C, lay the ſame from G until ic-cur the Meridian-line in the Point A,
I Courſe which is the place and Port you Sailed from. Laſtly, From A you ſhall draw their
S.W. W. Rhombs or Courſes, as.A B, which is 4 from H to N from the South, Weſtwards;
2 Courſe and the Eastward Ships Courſe is A Cas Point from. Pro N, from the South, Enfamando
S.S.E.SE
As the Sum of their Departures CB 82 Leagues-
191381
To the Difference of their Departure SB 52 Leagues--
150515
So is she Sine of the Sum of their Courſes C AB 78 deg. 45 min.- 999080
To the Sine of the Difference of their Courſes, Sum -1149595
SAB 22 deg. 30 min. the Sum
958214
25
.
22/A
1
+
1)
that is
4 Points or S.w.w.for che one Ships Courſe Sailed from A to B: and 22 deg. 30
min. ſubſtracted from 78 deg. 45 min. the half is 28 deg. o7 min. ;; that is, a Points
and; S.S. E. a Point Efterly, for the other Ship's Courſe
0
QUESTO
.
1
CHAP.VI. Queſtions in Navigation. .
165
1
QUE s T. VI.
From the Port at A I Sail S.S. W. unto B, and from B I Sail N. w.
b. W. unto C, and from C I Sailed unto my firſt Port at A,
E. b. N. Now having sailed in all 120 Leagaes, I would
know how many Leagues I have Sailed upon each Point.
t
F
Irſt draw A B a S.S.VV.Courſe, at any convenient diſtance; then from B dray
a N. VV.b.VV. Courſe, and from A draw the oppoſite Courſe of E, WN.
which is V veſt by Suth, which will cut B Cin C; ſo continue the sides of thciri-
angle A B unto E, and A C unto F. Then lay BC from B unto D, and A C
from D unto E. Then take 120 Leagues, and lay the ſame from A unto F: Next
draw thc Line E F, and from D and B draw Parallels chereunto, which will cut
A Fin G and H. Laſtly, meaſuring A H, you ſhall have 33 } Leagues char you
have Sailed S. S.VV. And meaſuring HG, you ſhall have 39 Leagues parts that
you have Sailed N. vy.b.VV. Allo meaſuring G F, you Thall have 46 Leagues
nicar, that you have Sailed E. b. N. which makes in all near 120 Leagues.
E
3
D
7
2
B В
1
G
Sam
Arithmetically, By the Table of Natural Sines in the
Sea-mans Kalendar.
A
J
8314
+
28
{
Firſt, Add upall che Sines of the Angles together,
deg. min.
45-minoo
007071
Which is
56154
78-459790
25175
Then by the Rule of Three,
SS. S. W. 3310 AB.
$45-00
As 25194, to 120 Leagues: 50756-15
2 To the Diſtan- N.W.b.W.39Leag.
E. b. N. 46. CA.
78-45
ces failed
I might have added ſeveral other Queftions of this nature, but I hold theſe ſufficient;
for choſe that underſtand how theſe are donc, may do any of the like nature: But
way of demonſtrating and laying of them down, as you ſee in the Figures, Ine-
ver ſaw before of any other Mans Work. Therefore now we will come to the true
Way of Sailing, and Uſe of the true Sea-Charf.
CHAP
1
.
"
WL
166 The Difference of Agreement of the Book IV.
+
i
CHAP. VII.
71.
The Diſagreement betwixt the Ordinary Sea-Chart, and the Globe ;
Cand the Agreement betwixt tbe Globe and the True Sea-Chart, made
after Mercator's Way; or Mr. Edward Wright's Projection.
ſtrated.
He Meridians in the ordinary Sen-Chart are Rigbt Lines, all parallel one to
i another, and conſequently do never meet; yet they cut the Aquinoctial,
and all' Circles of Latitude, or Parallels thereunto, at Right Angles, as in
the Terreſtrial Globe: 'Buc herein it differeth from the Globe, for that here all the
Pinabels to the Aquinoctial being lefſer Circles, are made equal to the Aquinoctial is
ſelf, being a great Circle; and conſequently, the Degrees of thoſe Parallels, or leffer
Circles, arc equal to the Degrees of the Aquino&tial, or any other great Circle, which
is mccrly fallë, andicantrary to the nature of the Globe, as ſhall be plainly demon-
The Meridians in the Terreſtrial Globe do all mcer in the Poles of the World, cut-
ting the Aquinoctial, and conſequently all Circles of Latitude, or Parallels to the
Aquater, at right Spherical Angles : ſo that all ſuch Parallels do grow leſſer to-
ward either Pole, decreaſing from the Aquinoctial Line.
As for example, 360 Degrees, or the whole Ciècle in the Parallel of 60 Degrees,
is but 180 Degrees of the diguingEtials and ſo of the reft: Whereas in the ordina-
ry Chart, that Parallel and all others are made cqual ane to another, and to the
Equinoctial Circle, as we have faid before.
The Meridians in a Map of Mercator or MryWright's Projection, are Right
Lines, all Parallel one to another, and croſs the Aquino&ial, and all Circles of La-
titude, at Right Angles, asin the ordinary Sea;
Chart: But in this
, though the Cir-
cles of Latitude are all equal to che Aquinoctiat, and one to another, both wholly,
and in their Parts and Degrees; yet they keep the ſame proportion one to the other,
and to the Meridian it ſelf, by reaſon of the inlarging thercof, as the ſame Parallels
in the Globe do "Wherein it differeth from the ordinary Sea-Chatt, for in that the
Degrees of great and lefſer Circles of Latitude are equal; and, in this, though the
Degrees of the Circles of Latitude arc equal, yer äře the Degrees of the Aleridian un-
equal, being inlarged from the Aquinoctial towards either Pole, to retain the ſame
proportion as they do in the Globe it ſelf; for as two Degrees of the Parallel of 60
Degrees, is but one Degree of the Aguinoctial, or any Great Circle upon the Globe,
ſo here civo Degrees of the Aquinoctial, or of any Circle of Latitude, is but equal
to one Degree of the Meridian; bécwikt the Parallel of 59 and 60; and ſo forth
of the reſt.
Now for clie making of this Táble of Latitudes, or Meridional Parts, ic is by an
addition of Secants; for the Parallels of Latitude are leſs than the Æquator or Me-
Radines ridian, in ſuch proportion as the Radius is to the Secant of the Parallel.
10000, For Example
. The Parallel of 60 Degrees is leſs than the Æquator; and con-
ſequently, cach Degree of chis Parallel of 60 Degrees is leſs than a Degree of die
Secant Aquator or Meridian; in Tüch proportion as 100000 Radius, hath to 200000 the
20000. Secant of 60 Degrees.
Norv how Mr. Gunter and Mr. Norwood's Tables are made, which are true Meridi-
onal Parts, is by the help of Mr. Edward Wright's Tables of Latitude. Mr. Gunter's
is an Abridgment, conſiſting of the Oxotient of cvery ſixth Number, divided by 6,
and
two Figures cur off. :
As for Example. . In the Tables of Latitude for 40 Degrees, the Number is
1242's deg. parts.
3B21739 (43:712 That divided by 6, the Quotient is 43 deg. 712 parts of the
66666
Æquator, to make 40 Degrees of the Meridian. And Mr. Norwood's Tables of
Meridional parts, is an Alridgment of Mr. Wright's Table of Latitudes; namely,
every
}
to
4 g -12
w!
1 L
2
- in
ig
7
2:
-262217559
4
CHAP.VII. Plain Chart, True Chart, and Globe, 167
every
fixth Number cutting off four Figures to the riglır hand, as for 40 Degrees: as
before the Number is
in regard it wants but a little of nodo sue offz. we make the Meridional parts 2623,
as you will find by his Table. So this talle hensch hos-many. Parts every Degree,
and every Tenth part of a Degree of Latitude in this Chart, is from the Aquinoctial
namely, of ſuch Parts, as a Degree of the Aquesor containeth 60, And this
which I here exhibit, and call a Table of Meridional Parts, is alſo ati Alridgment
of the Table of-Larizudes of Mr:Wright's, namely; the fift-Nummitverse Samitting al-
ways the three laſt Figures.
As for example. All thic Numbers are for 40 deg. 26. 227.559; bmit the three
laſt, and divide the reſt by 3;tand in the Quotient is 8742, the Meridional parts
for
40 Degrees; and ſo of the reſt: So that this Table Thewech how many Paris
cvery Degree, and
every
Tenth part of a League, and every Tenth Minute of Lati:
1xde in this Chart. is from the Æquinsetial to the Poles ; namely, of such.Parts as a
Degree of the Aquinoctial contains 20. Leagues. This is large cnough for our Uſes
at Sea, and as ready, being in Leagues, by cutting off the laſt Fignre, which is a
Tenth: For I could never ſee any Draught or Plat made according to Mr. Wright's
Projection, excepting his own in his Book, that is divided into more Parts than 6;
for all the Mercator's or Dutch Charts as I have ſeerty, are divided into 6 times 10,
which is 60 Minutes: But he that deſires a larger Table,max make uſe of Mr.Wright's
Tables of Latitudes.
The Uſe of this Table (hall partly appear in the Problems following, and may
be
illuſtrated after this manner.
.
)
1
4
11:]
+
.
1
.:01 #
PROBLEM I.
How to find by the following Tables what Meridional Parts are cona
tained in any Difference of Latitude.
ү°
Ou muſt take the Meridional Parts anſwering to tachi Lätitude
, and ſübſtract
the leffer from the greater; ſoʻthe Remainer is the Number of Meridional parts
contained in the Difference of Latifnde propoſed.
As, Let one Laticide be gi deg. 20 mini
. -1 2002
And the other Latitude le 13 deg. 10 min.
Meridional Parts:
2657
9345
The Meridional Parts contained in the Difference of Lacitude are 934. Leago
Thie Degrees are over șhe Parts, and the Minutes are ou cach ſide under the De-
grees; and in the common Angle of Meeting or Line with the Minutes, is the Mer
ridional Party you defire
.
là !)?!ic
:.:02 m.
RO. B'Ir? II.
:!:04
The: Latifudes of two Places being given, and Difference of Longi-
itude of both.Places. To find the Rhombiand
Diffance. I
and !
To the intene the Application may be the more:evidenieji bhuri Examples Yhåll be
Suppoſe the Latitude of the Iſland of Lundy in the Mouth of Seavern, to be at A;
18
51 deg: 22 min. and the Latitude of Barbadoes 13 deg. 10 min. ac B, and the Diffe-
1058
rence of Longitude 52 deg. 55 min. CD, chat the Barbadoes is to the VVeſtward of the sid.az m.
Iſland of Lundy; The Course and Diſtance from the one place to the other is de-
manded.
Firſt you may demonſtrate the Queſtion by the Scale. Draw the Right Line A C
for the Meridian ; and in regard the Difference of Latitude is 38 deg. 12 min. con-
vcrt"them into Leagues, by multiplying them by 20, the Number that goes to a De-
gree,
10
sunt
06:5:1
::22:
L
}
و )
wiwa
Til!
It'y?!
*
77
79..
C
ri
Ländi 25 si
360 00
385 52
Barbad.332 57
Differ. 32 55
im
s!
ܢ
3
*
1040
13
10
38 12 (4
20
3
760
764
168
Sailing by the True Sea-Chart. Book IV. .
IOS& Leagires
D
巧
​B
F
2764 bd 9341
2
K
TE
A
}
gree
, ard the odd Minutes diýide:by B, and the Difference in Leagues will be 764;
As Was ſhewed which lay from A to B, for the common Difference of Latitude. Then cake the Diffe-
in ibe lajt Ex- rence of che two Latitudes ¿qlarged, 934 Leagues, and lay from A to C; chen
draya
ample.
the two Parallel Lines, as B'E and C D. Then 52 deg: 55 min. the Difference of Lona
gitude; converted into Leagues, as before-directed, is 10583 which lay from C'to
D, and draw the Line A D, which is the true Courſe from A to D, and the Diſtance
according to the True Chart inlarged: Therefore AE is the true Rhambaline and Di-
Stance found out, produced by elie former Worko, And as D is the true Poins by
Mercator's Chart of Batbadoes, ſo ise the true Point of the ſame Place of Barbadoes by
the Plain Chart; and A E the true Diſtance, B.A E the true Courſe. And as CD
is the true Longitude by the Globe; ſois B E the true Meridian-diſtance between Lun-
Then you muſt dy and Barbadoes. You may work afterwards by the Rules of the Plain Chart ; and
put down the you need not work Mercator's way witymore, without you lave a Mercator's True
Courſe and Di- Chart; and to work by that, you ſhall
be directed'ini che following Discourſ. There
fance and Meo fore to work by the Plain Rules all the Voyage after, meaſure' A E, and you will have
in the Head of 1157 Leagues for the Diſtance; and for the Courſe take GH, and apply it to the
Tour Journal.. Points on the Scale
, and you will findithe tfue-Tobirse S.VV. a litdle more than 2
Quarter Wefterly, which is all onç Courſe, with the True Sea-Chartą but the Di-
Flance Inlarged is 1908 Leàgris
AD: Nów bylihe'rlain Stathaft the Courſe is
BAF, S, W. abovci kofia Point Writerly; and the Diſtance is À Filző6 Leagues :
So that the Plain Chart ſheweth the Diſtance more than it is by 149 Leagues, and the
Courſes more VVeſterly by half a Point, and the Meridián-diſtance, rooimuch bj 194
Leagues, which is a groſs Error; and in ſuch Diffances grolly arc theſë Men miſtaken,
that
uſe a Plain Chart.
::
gabi?
។
܂ ;io، ، :.
2:2 .
10.vi:
1
po
!
cii
.:19T'I
4
Táble
+
10
1
.
3o da
yo
CHAP.VII, Sailing by the True Sea-Chart.
169
Io
or
3 3
7
16
1
Minutes.
2
ba
...
Minutos. Join
OO
200
6001
801
1001
I202
Io
233
267
20
67
100
40
to
M.
II
MI
IO
zo
50
20
21
22
23
29.
IM.la
0127917129130130404132019133739,35881130441141740463871543311
A Table of Meridional Parts to the 10 part of a League, and for every 10 Mi-
nutes of Lacitude from the Aquino&ial to the Poles,
•DEGREES
3 4 5
16
7 8 9
Leon Le... Le, Le. Le... Le Le Le Le Leon
400
1403 1605 1807
33
433 6341 834! 1035 1236 1437! 1639! 1841 10
467
6671 867) 1068 1269 1471 1673 187520
30
300
500 700 901 1102 1393 1504 1706 1909 30
1331. 333
534 1734 934 1935 1336 1538 1740 1943140
1671 367 567 767 968 1109) 13701 1571 1774) 1976
591
IO
12 13 14 15
16
17 18
19
2010 2314 2418 2623 28 28 3035 3242 345 3661 3827
2044 2248 2452 2657 2863) 3069 32771 3436 3696 390710
2078 2282 2486 2691) 2897) 3104) 3312 3521 3731 3942120
301 2112 2316 2520 27251 29311 3138
29311 31381 3347 3556 3766 397.7
40 21641 23501 25541 27601 2966 31731 33811 35911 38011 4013 40
2180 2384 2588 2794 3000_3208 3416 36261 38361 4048 50
24 25
26 27
28
of 4084 4297 45121 4729 4947 5167 5388 5612 58376065
10
10 4119
4119 4333! 4548 4765; 49831 5203 5425 5648 58756103 10
20 41551 4369 4584 4801 5020 52401 5462 5686 5913 6141 20
301 4196 4405 4620 4838 5056 5277 556057241 59511 617930
401 4226 44404656 4874 5090 5314 5537 5761 59891 6218
49
501 4262 _4476.4692 4910 5130 5351 5574 5799 6027 625650
30 i 31 32 33
34
35
36
37 138 1 39
62946527 67616998 7238 7481 7726 79758277 8433 0
10 6333 6565 6800 7038 72781 7521 7768 801782708526 10
3711 6604 6840 7078 73197562 7809 805 9 83128569:20
301 6410 664468791 7118! 7359 7603 7850 8101 8355 3612 30
40 64496683 6919 71581 24001 76441 7892 81431 83978655140
50 64886722 6938 7198 7440 7685 7934) 8185 8440 869950
40
43 44 4 46 47 48
49
Oj 87421 9005| 9272 9543 9819 toloo 103851067510971 11273
10 8786 9049 93171 95891 986510147 1043311072411021111324 10
20 $8291 9093 9363 9635 9912 10194 10481 10773 1107111375 20
308873 9138 9407 9681 995910241 10530108231112111426 30
40 39171 9182 9452 9727 1000516289110578|10872 11172 11478 40
so 8961 9227)_94981 9773 ICO5210337 10626 1092211122211529150
50 51 52 53
54 55. 1 56
57 | 58 59
1158111896 12217125.45 12831132261358113941114313 14696
11633|11949|12271|12600 12938132841364014004 14377 1476110
20.1163512002123251126361299511334213700 1406314440 1482620
30111737 1295511238012713130521340111375914121 145041489230
40|1179312109124351 2768 13110 1346013820 1418814568 1495840
11842113163 1249012825,13168 1352011388114251 14632 15024 50
60 loi 62
64 65 66 67
68,
15091 15497 159171635016798117263.17745 1824618769119315
101515811558611598816433116874 73421178271183321885811940810
2015225|15635 16059|16497116957 1742117920 1844818948 19502 20
30115293 15705 16131 16572 17028 1750117993 18505 19035 i959730
4015300 1577516204 1664717106|17582 18077 18592 19130 19693 40
5011542815846 162771 167221718417662 181611156801922219789 50
71 72 73 74 75
75 76
76 77 78
78 | 79
19836 20485 21116 -1781122455)23234/24033 24890 25815 26819
019984/20588 21224 21896 2260923368 241722503925976 2699510
20120083/20692 21333
23211 -2730123495 24312251901261402717320
301201832079621444-212822854123628 244552534426306 27354
301
40120383 2090120555|2224623979 23762-4593625 492/26474 27530140
50 2038421008 21668 22365 2310623899 247432565626645 27727
so
83
84
85
88 89
있​M.
39 1
TV
ZO
41 14?
I!!
"
.
1.1.1
IO
63
ور)
O ol'w
Il 4 9
170
1090
1
I
ܐܐ
1
lo
333
1293
*O
81
8
82
86 87
1 1 49
1 44
101281101293451307261322961341121362701389294239514738515642010
20/28307 29564 3097432579344453667313943943090 48477 5897720
3012850729787 3122632870 3478837096 39973 43830 4958462274/30
4012871130015131484331681351413752340532 4462113103466920140
5028919/20247131748133474135 505!37973|41120145970 52564174863150
Z
1
1
By
1
170
Sailing by the True Sea-Chart. Book IV.
By the Tables
1
A
Dmit a Ship is at A, in Latitude 55 deg.2 2 min. Northgas is Lundy,and fails,or is
to fail to E, in Latitude 13 deg. 10 min. according to the Plain Chart corrected,
which is Barbadoes; or by Mercator's Chart, Barbadoes is in the Point D, and the
Difference of Longitude is 1058 Leagues, which is 52 deg.55 min.. Firšt find the
Difference of Latitude inlarged, as is before-directed in the firſt Problem, and found
to be 934 . Leagues.
Now you have given A B the Difference of Latitude 38 deg. 12 min. inlarged from
B to C, and CD the Difference of Longitude 52 deg: 55 min. whereby the Angles
and Hypothenuſal ſhall be found by the Fourth and Fifch Caſe of Plain Triangles.
But becauſe in this kind of Proje£tion, the Degrees of Longitude and Latitude are
not equal (except in Places near the Aquinołtiat) the Degrees of Latitude at every
Parallel, exceeding the Degrees of Longitude, in ſuch proportion as the Aquin.Etial
excċeds chat Parallel; therefore theſe Differences of Longitude and Latitude muſt be
expreſſed by ſome one common meaſure; and for that purpoſe ferves the foregoing
Table, which theweth how many Equal Parts are from the Aquinoctial, in every
Digree of Latitude, to the Poles; namely, of ſuch Equal Parts as a Degree of Lon-
gitude contains 20 Leagues,
Wherefore, as before-diłę&ed, multiplying 52 deg. by 20, and dividing the odd
Minutes, being 55, by, 3, it will be 18, Leagues; added to the former Sum, makes
1058; Lengkes, for the Meridional parts contained in the Difference of Longierde.
Allo by the laſt Problem, I find the Méridional parts.contained in the Difference of
Latitude to be 9341. Leagues: So that A C is 934 . Parts, and CD 1058 of
ſucli Parts.
: Therefore, By the Second Caſe of Plain Triangles.
IO
As the Difference of Latitude inlarged AC is 934 1. parts 297057
Is in proportion to the Radius go deg.
Sois the Difference of Longitude in ſuch Parts CD 10581_-1302448
To the Tangent of the Rhoinb at A 48 deg: 33 min.ro 1005391 į
Extend the compaſſes from 934. Leagues the inlarged Latitude , 'to to$85
Leaguess
the fame Diſtance will rcach from the Radius to the Tangent of the Course
48.deg. 33 min. which is the Courſe from Lundy to Barbadoes, S.W. a little above'a
quarter of a Point Wefterly.
.
By the Fifth Caſe of Plain Triangles.
As the Sine+Complement of the Rhomb at D 41 deg. 27 min.- 982083
To the Difference of Latitude A B 764 Leagues
288309
So is the Radius o 90 deg.
To the Diſtance A E 1154 , Leagues
306226
Extend the compaſſes from the Complement-Sine of the Rhomb 41 deg. 27 min. to
the Sine of 90 deg. the ſame Extent will reach from the trne. Difference of Latitude
764 Leagues, to che Diftince A E 1157 Lergues, which is required.
IO
:
4
1
PR Ó BL,
.
4
I
L
CHAP.VII. Sailing by the True Sea-Chart.
171
.
'
-IO
hach failed upon.
So is the Sine of the Rhomb, 48 deg. 33 min., at A-
P R O B L. III.
The Latitude of two places, and their Diſtance given; To find the
true Courſe and Point, or Place you are in, by Mercator's
Chart.
ADmit I fail from the I find of Lundy, in the Latitude si deg
. 22 min. in the
Southweſt Qaarter of the Compaſs, 1154 Leagues; and then find my felf in
the Latitude of 13 deg. io min. I would know what Point of the Compaſs I have
failed upon, and my Difference of Longitude to the Weftward.
The Difference of Latitude A B is 38 deg. i 2 min. which reduced into Leagues
is 764 Leagies.
As the Diſtance failed 1154; Leagues A E
306226
Is in proportion to the Radius go deg.
So is the true Difference of Latitude 764 Leagues. A Bm 288309
To the Sine-Complement of the Rhomb4 i deg, 27 min. at D 982083
that is, S.W. W. or Southweft 3 deg. 33 min. Weftefly, the Courſe that the ship
Extend the Compaſſes from 1154 Leagaes the Diſtance, to the Sine of 90; the
ſame Diſtance will reach from the Difference of Latitude 764 Leagues, to 41 deg:
27 min. die Co-fine of the Rhomb: The Sine is 48 deg. 33 min. that is, 4 Points and
above a Quarter from che South Weſtward from the Merididn.
Secondly, For the Difference of Longitude.
Find by the Firſt Problem the Difference of Latitude inlarged, as is thère dire-
cted, 934 1. Leagues : Then it is,
As the Radius 90 deg.
-IO
To the Difference of Latitude in Parts 934, FAC-~ 297057 Inlarged.
So is the Tangent of the Rhomb 48 deg. 33 min. An-1005395
To the Difference of Longitude in Parts 1058 Leagues-- 303452
Extend the compaſſes from the Sine of 90 deg, to the Difference of Latitude inlar
ged 934 , Leagues; the ſame Extent will reach from the Tangent of the courſe
48 deg. 33 min. to 1058 Leagues : which laid off from C to D, ſhall be the Point or
Place in Mercator's Chart where the ship is.
Or, 1058 Leagues converted into Degrees, by dividing by 20, che bisotient is
52 deg. 55 min. tlic Difference of Longitude required.
PROBL. IV.
Sailing 1154 Leagues upon the 4; Rhomb from the Meridian, or 48
deg. 33 min. from the South Weſterly, I demand the Departure
from the Meridian.
As the Radius 90 deg.
To the Diſtance failed 1154 Leagues A-E
306226
987479
To the Departure from the firji Meridian 865 Leagues 293705
Extend the Compaſſes from the Sine of 90 deg.to 48 deg. 33 min. the fame Excent
will reach from the Difance failed 1154, to the Meridien -Departure.865 Leaguets,
Z 2
BE
!
!
.
,
1
TO
h
1
172
Sailing by the True Sea-Chart. BOOK IV
1
BE is the true Meridian Diſtance, which you may ſet in che Head of yo:ur Journal,
to ſubſtract your Daily Diſtance from your firſt Meridian,
PROBL. V.
Both Latitudes and the Meridian Diſtance of two Places being given,
To find the Difference of Longitude, and Courſe and Diſtance
on the True Sca. Chart.
TH
His is a moſt uſeful Problem, when the Mariner hach caſt up his Traverſe : Sup-
poſe a ship ſail
lipon
the $.w. Quarter of the Compaſs, from Latitude si deg.
22 mir. unto Latitude 13 deg. 10 min. and the Departure from the firſt Meridian to
chie Weſtward 865 Leagues.
You muſt find firſt che Difference of Latitude inlarged, as is before-directed in the
firſt Problem 934 1.
As the tru. Difference of Latitude AB 764 Leagues
288709
Is to the Meridian-diſtance or Deparcure B E 865 Leagues 293701
So is the Difference of Latitude inlarged A C 9341 - Leagues -297057
To the Difference of Longitude in Leagues 1058 CD --590758
302449
By the Line of Numbers.
Xtend the Compasſes from A B the true Difference of Latitude 764 Leaguess.co
BE 865 Leagues Meridian-diſtance, the fame Extent will reach from Ad
934. Leagues che Difference of Latitude inlarged, to the Difference of Longitude
1058 Leagues; which "iaid off upon the Parallel-Line from Cto D, is the point and
place of the ship in Mr. Wright's or Mercator's Chart.
As the true Difference of Latitude 764 Leagues A B-
288209
Is to the Meridian-diſtance 865 Leagues B E
1293701
So is the Radius go deg.
-IO
To the Tangent of the Courſe 48 deg. 33 min. at A 1005392
By the Artificial Lines on the Scale.
j
A
reach from 90 deg. to the Tangent of 48 deg. 33 min. that is, 4 Points and above
a Quarter from the Sousb weffard, that is, š.W. Wefterly, the Courſe the Ship
hach kept.
As the Sine of the Courſe at A, 48 deg. 33 min.-
987472
Is to the Radius 90 deg.
So is the Departure from the Meridian 865 Leagues -1293701
To the Diſtance Salled Ą E 1154 Leagues-
306222
By the Scale.
90 deg.
the fame Extent will reach from 865 Leagues-BE, to 1154 Leagues A É, the
Diſtance failed.
PROBL. VI.
By the Difference of Longitude, and one Latitude, and the Courſe,
To find the other Latitude and Diſtance run,
Uppoſe I ſail from Lundy, in Latitude 51 deg. 22 min. North Latitude, S. W, 3
deg. 33 min. Wefterly, until my Difference of Longitude be 52 deg. 55 min. chac
is,
-10
#
2
t
1
A
CHAP.VII, Sailing by the True Sea-Chart.
173
-10
is, from Cto D, which is the Place of the ship in Mercator's Chart; I demand how
much I have laid the Pole, and how far I am from Lundy?
As the Tangent of the Rhomb 48 deg. 33 min.
1005395
To the Difference of Longitude CD tos8 Leagues
302448
*So is the Radius
To the Differcnce of Lacicude in Leagues 934'. A C-.. 297053
By the Artificial Lines on the Scale.
Xtend the Comp-affes from 48 deg: 33 min. to 1058 in the Line of Numbers; the
fame Extent will reach from 90 deg. to 934 f. Leagues.
Or, Extend the Compaſſes from 48 deg. 33 min. to go dig. the ſame Diſtance will
reach from 105 8 Leagues, to AC 934 f. Leagues, as before.
Now the Meridional parts anſwering the Latitude of si deg: 20 min. is I 2002;
from it ſubſtract 934. here found, and there remains 2657%, which Number I look
for in the Table, and find it under 13 deg, and in the Line of 10 min. which is the
Latitæde of the ſccond place where the Ship is; and the Difference of Latitude is 38
deg. 12 min.
The Diſtance may be found as before, in the ſecond and fifth Problems.
I 2002
9345:
2657
7.
PROBL. VII.
TO000co
1
By the Courſe and Diſtance, and one Latitude, To find the other La-
titude, and Difference of Longitude.
Uppoſe I ſail S. W3 deg. 33 min. Wefterly, 1157 Leagues, and by obſervation
find my ſelf in the Latitude of 13 deg. 10 min. I require the Latitude of the
Place from whence I came, and the Difference of Longitude becsveen the two Places.
For the Difference of Latitude,
As the Radius B 90 deg.
To the Diſtance failed 1154 Leagues A E
306222
So is the Sine-Complement of the Courſe E, 41 deg. 27 min. 982083
To the D'fference of Laticude 764 Leagues---
288315
Extend the compaſſes from the Sine of godeg. to 1154 Leagues; the ſame will
reach from the Sine of 41 deg. 27 min. to 764 Leagues, which converted into Degrees,
is 38 deg. 12 min. the Difference of Latitude ; which added to 13 deg. 12 min. the
Latitude of the laſt Place, the Total is so deg. 22 min. the Latitude of the firſt Place
required.
The Difference of Longitude is found as before in the chird Problem, ſaying,
As the Radius, To the Difference of Latitude inlarged 934 4.:
So is the Tangent of 748 d. 33 m. To the Difference of Longitude in Leagues
1058, which is so deg: 55 min.
Now to convert the Difference of Longitude found in any Latitude inco Leagues,
do it after this Example.
Suppoſe two Places in one Parallel of Latitude, as in the Parallel of 55 deg. 22
min. whoſe Difference of Longitude is 52 deg. 55min. I require che Diſtance of thoſe
two Places.
As the Radius-
Is in proportion to the Compl.Sine of the Latitude 5ı d. 22 m.- 979573
So is the Difference of Longitude 1058 Leagues.
302448
To the Diſtance in that Latitude 661 Leagues
282092
You
10
!
1
1
1
174
Sailing by the True Sea-Chart. Book IV.
You muſt underſtand, That the Leagues of Longitudo in any Parallel of Latitude,
are in proportion to the Diſtance in Leagues, as the Aquinoctial is to chat Parallel; or,
as the Semidiameter of the one, is to thie Simidi.imeter of the ocher, as was faid in the
Seventh Chapter
:
CHA P. VIII.
How to divide a · Particòlar:Sea-Chart , according to. Mercator and
Mr. Wright's Projection.
49.
D:8.
Y
+
55:30
F it be a Particular Chart you would makć, you muſt firſt conſider the cwo
49
L-titsdes you would make the Chart for ; and out of the foregoing Table of
13401 2
Meridional parts, take the Numbers anſwering to cach Latitude, and ſubſtract
:
11427 the lefſer out of the greater
, and the Remain is the Numbers which you muſt take
for the extreme Parallel of Latitude.
1975 As for Example, I would make a Blank Merca-
tor's Chart from the Latitude of 49 deg. 30 min. to
55 deg. 30 min. and for 10 Degrees of Longitude. Look
2975 (9:52 in the Table of Meridional Parts, and for the Latitude
301 13401 1975
of 49 deg. 30 min. you will find the Number anſwer-
2013142159 1921
ing thereunto is 11426, and the Numbers for the Lati-
10 13284158 1863
tude of $5 deg. 30 min. is 13401; the leaſt ſubſtracted
09143226 58 1805
from ihc grcarct;che-Remainer is 1975 Equal Parts for
50113168581747
Divide the Dif- the length of the Meridian-Line: Therefore first diaw
40/1311015811689
ference 1973s the Line A B, DC for the Meridian-Line, and croſs
3013052 581631
by 26,the Quo- it with cwo Perpendiculars, as B C and AD: Then di-
citat will ſheme vide one of the Parallels of Latitude into 1o Equal parts,
2012995 571573
you the oumber
TO 1293815711516
of Degrees of as A D,and ſubdivide eaclíof thoſe Degrees inco zorgwal
54 00112882157 1459
the A quinoai- parts or Leagues that makes a Degree of Longitude and
so 12825571402
al and min. that Latitude at the Aquator; and ſuppoſe each of theſe
makes the length 20 Parts to be divided into 10 parts more, ſo will a
40 12768 571345
30/12712 551288
Line; as for 6 Degree be divided into 200 parts: Then take with your
20112656 5612321
deg. form Aro Compasſes 1975 Equal parts out of the Line A D, and
1012600 56 1176
D, will be 9 d. lay from A to B, and from D to C, for the extreme
53
001254515511120
$5 m. of ibe Parallels of Latitude ; and through cach Degree of
Æquator.
50112490151065
Longitude marked with 1, 2, 3, 4, 5, 6, 7, 8, 9,
40 12435155 IOIO
10, draw Meridian-lines parallel to the firſt Meridian :
301238055) 955
Then out of the Table of Meridional parts collect the
Numbers anſwering to every 10 Minutes of Latitude,
2012325155 900
You may divide as in the firſt Column of this Table annexed, che ſecond
10 122711541 846
Column is the Number anſwering the Minutes of La-
152 00112217 541_7921
nutes into two
Equal parts i fo titude in the Table of Meridional parts, which ſubſtra 50112163 54 738
is every Degree cted the leſſer from the greater, the Remain is the Dif-
4012109541 684
divided into sference, as in the third Coluon si deg. for the D.Fe-
3012055/541 630
min. as A B & rence of the two lowermoſt Numbers: Then add the
20113002153) 5761
DC into som.
Numbers together in the fourth Column in this manner; IO 11949 531 523
si for the firſt 10 minutes, and 52 added to it, $1.00 11896 53 470
makes 102 for 50 Minstes; and 52 added to 102,
5011843 53! 417
From D.
makes 154 for the Number of Equal parts you muſt 40 11790531 364
cake out of the Line A D for 30 min. from A to so 3011737153 311
Deg. of Latitude, and lay ic on both ſides of the 20111685152 258
Chart, and draw the Parallels of so Degrees of Lati . To 11633/521 2061
tude; and ſo do of the reſt, as you ſee in this Table. 50 001115811521 1541
And for 51 Degrees the Number is 470;
take
470
So1153952 103
and lay it upwards from A to 51 Degrees on both 40111479152
fidcs
, and draw the Parallels of s1 Degrees of Lati-
3011426151
tudes and ſo do with all the reſt.
CH A P.
F
every 10 Mi-
1
+
1
51
.
.
CHAP.IX, A Figure of the True Sea-Chart.
175
A Particular Sea Chart
N
35. to
515
!
Ilg
514
1
7
Elg
513
East
West
elg
!
}
19
511
1
olg
이다​.
:
E
2
3
4
5.
6
S
20.
7
8
او
А
in
200
(40a]
-600
800
1000
1200
0077
1600
8 D
Souto
175
1
.
.::::..
CX A P. IX.
The Projection of the Meridian-Line by Geometry, and how to make a
*: , Scale of. Leagues for to meaſure Diſtances in any Latitude
HE Projection is the ground-work of Mr. Wright's Table of Latitudes, in
his Book called,
The Corre&tion of Errers in Navigation, where he ſhewech
how to make it
, and hath alſo made a Table by the continual addition of
the Secants of every minute, which Ihiews how much you are to lengthen the De
grees of Latitude in your chart, thar-ſo there may be a true proportion becween the
Degrees
T
*
.
176 The Projection of.the Meridian-Line. Book IV.
.
1
Degrees of Longitude and Latitude in all Places. Which Table I have abridged, and
made it more plain and eaſic, by reducing it into Leagues and Tenth parts, as hath
been ſhewed before. We will here ſhew you how to do the ſame by Geometry,
and alſo how to make a Meridian-line anſwerable to any Line of Longitudes, and
a Scale of roo Leagues to meaſure any Diſtance in any Latitude.
Firſt, Make the Quadrant A B C, of what largeneſs you pleaſe, and divide the
Limb thereof into 90 Degrees, and number them from B towards'C; Then divide
the side of the Quadrant into s Equal parts, which are five Degrees of the Aqui-
noEtial. Then divide che firſt Degree from the Center, as A D, into 6 Equal parts,
and through them draw Parallel-lines to A C: You may divide cach of the other
, four
Degrees from Dco B into 20 Equal parts, which are 20 Leagues, which makes a Deo
gree Longitude at chc Aquator; and ſo you may number them as you ſee, from 10
to 100: So the whole Line A B will be your Radius, and the length of rio Leagues,
or five Degrees and a half of Longitude of your Chart. "And becauſe the Degrees of
Longitude are to be of one length in all Latitudes, therefore the Degrees of Lati-
tudes mnft encreaſe, as the Secants of the Latitudes increaſe. Therefore if you would
know how long one Degree of Latitude muſt be in the Latitude of 50 Degrees, lay a
Ruler on so Degrees, and on the Conter A, and draw the Line AH. Now the Raa
dims being AD, the length of one Degree of the Aguator, this Line A h, or l Kg
of
1
"
B
3 110
N 110
M
>
110
Ho
100
100
go
30
80
80
70
so
60
60
so
40
K қ
zlo
30
:
20 DE AG
R
O
0
ITA
06
d'e
17.6
being both of one length, is the Sesaxt of 80 Degrees to that Radius, and muſt be the
length of one Degree of Latitude in a Chart from 5o. Degrees ta, fi Degrees, as you
may preſently try by the former Charl; and fo'ché Line A @ which is the secant of
20 Degrees, is the length of one Degree of the Meridian-linelim the Latitude:of 20
Degrees; and ſo for any other Latitude. The
fix Lines divided in the firſt Degree
'AD, are 10 Minutes apiece; and ſo you have the Secant of every 10 Minutes of
Latitude, and their length in every Latitude, for a particular Chart, and for a gene-
ral Chart, which hath in it North and Soxth Latitude.
You may divide the Quadrant's Side A B into 10 Equal parts, and ſubdivide
thidm into io mors; fo will D * bero Degrees of the equator, and ey will be
che
7
Y
N
10
20
20
60
80
go
10
751
coo
30
13 L
1
40
100
जात
>
draw Parallel-lines to MN, through every 10 Leagues in the Side A B, you
drant extend your Compages from the Center A,
CHAP.X. How to make a Scale of Leagues. 177
the length and Secant of 10 Degrees in the Latitude of 20 Degrees, and L r the length
and Secant of 10 Degrees in the Latitude of 60 Degrees, which is twice the length of
one Degree of the Æquator : So that you may preſently try the truth of this proje-
ction, how it agrees with the Globe: Whereas one Legree of Latitude in the Globe, is
equal to two Degrees of Longitude in chat Latitude of 60 Degrees; 1 here A L 'the
Secant of 60 Degrees, is twice the length of A D the meaſure of one Degree of
Longitude in the Blank Chart; and I r is twice the length of 10 Degrees D * : Só
cvery Degree is two of the Æquator in the Latitude of 60 Degrees of a general Charts
and by the Globe, in the Latitude of 75 deg. 30 min. one Degree of Latitude is equal
to four Degrees of Longitude. So in the Quadrant, A R is four times the length of A
D: and ſo the Proportion will hold in any other Latitude.
u Scale of Leagues from the Latitude of 25 Degrees, to the Latitude
of 57 Degrees oo Minutes.
How to make the Scale of Leagues.
HE Quadrant being drawn, as before-directed, take 110 Leagues and lay
from A to B, and draw the Line MNB ar Right Angles thicreunto : and if it be
for a particular
Chart, as that before-going, draw Lines from the Center through every
Particular Latitudes as you ſee in the Quadrant I have done, to make a Scale for the
blank Chart before-coing, from the Latitude of 49 deg: 30 min. to 55 deg: 30 min.
So that if you would know the length of 110 Leagues in the Latitude of so Degrees,
lay a Ruler upon so deg. in the Arch of the Quadrant and the
Center, and
draw the
Line Adh, and that is the length of 110 Leagues in that Latitude. So that if you
will
the Quadrant: and ſo you may do for every League, as you ſee the little Checkers be-
wwixe
the Latitude of 20 deg. and 30 deg. for to Leagues between Latitude of 50
Suppoſe you would know the length of 40 Leagues in the Latitude of so deg.
Extend the Compaſſes from A to K, and that Diſtance is 4. Leagues in that Latitude:
And in like manner work by the reſt in any other Latitude,
you
would make this into a Scale, as the Figure Y MmN; Firſt in the Qua-
to the Interſections of the
Lines drawn through every Degree MN B, and lay them down upon the side of the :
extreme Latitude of your chari, as A, O, P, Q, Y, M, with the ſmall Arthes, as you
sce I have done from M to NY, and that is the length of the Meridian-line of
your
go
819
310
!
1
А
10
30
20
40
50
60
80
70
too
M
90.
177
.
T
deg. and 56.
If
Аа
178 Of Sailing by a Great:Circle, BookIV.
fance of the uppermoſt Line Y N of your Scale; and draw Nm the outſide of
ſelf, by altering your Latitude many Degrees, by which you may often rectific
your Scale or Degree of Latitude MY; therefore draw the Parallel-limes Y N and
Mm for the extreme Parallels of your Scoile : Then extend your comp.rſ/es. from Y
in the Quadrant, to each of the Inter feЕtions of theſe ſmall Arches that are drawn
from the Interſections on the Tangent-line MN;' and from 25 deg. uppermoſt, lay
that Extent downward for the Parallel of 55 deg. of Latitude, as che Line above the
Igwermoſt; and fo lay down all your Latitudes by theſe ſmall Arches, in like mani-
ner; and ſo neatly divide the side of your Scale M Y of Deg. of Latitude: Then
draw Parallel-lines to all theſe Degrees, as you ſcc: Then excend the Compages froin
the Center of the Ouadrant co M, for the length of the lowermoſt Line of your Scale
Mm for 110 Leagues. Then extend the compaſſes from the Center of the Quadrong
to N, which is the length of 110 Leagues in the Latitude of 25 deg. and it is the Di-
your Scale 110 Leagues : So take every to Leagues from the Center A, in the Line
A M, in Latitude 56 Degrees, and divide che lowermoſt Line of the Scale ; and
the like do in the Latitude of e5, for to divide the uppermoſt Line of the Scale;
and draw Lines through each of them, which will divide all the reſt of the Parallel-
lines in cach Latitude into 10 Leagues apicce, and number them as you ſee I
have done ; and divide the firſt 10 Leagues by the Meridian-line of the Scale, into
10 Equal parts below and above, and draw Lines through each of the Diviſions: So
have you ncacly divided your Scale, and cvery Degree of Latitude thereof, iuto
Leagues, to 100 and 10 odd Leagues ; which will meaſure any Diſtance in a Chart,
made according to che Degrees of Latitude and Longitude in the foregoing Chart.
For to know the Rbomb berween any two Places, ſhall be ſhewed in the Uſe of the
gcueral Sea-Chart following, by a Protraćting Quadrant, and alſo how to find the
Place of any Shipin Mercator's Chart, and to lay down any Traverſes.
CMA P. X.
The way of Sailing by a Great Circle.
TE will now (hew you the way of Sailing by the Arch of a Great Circle,
which is the moſt true way of Sailing of all others, if a Man is ſure
to have Winds, that he may keep neer the Arch that goes over any two
Places propoundod. But as there is a great deal of uncertainty in having a conſtant
Wind by the Arch, ſo likewiſe the Trade-winds many times lye wide of this Arch
many Leagues; beſides many dangers of Rocks, and Sands, and great Currents, and
danger of Pirates; which by keeping near the Arch may lead Men into many incon-
veniences; which may be a greater crouble to a Man, more than by ſailing a fer
Leggmes the more, for his beſt advantage and more ſecurity ; beſides the trouble in
that way, of keeping of Accounts, which Men chat svarch every 4 Hassrs, cannot al-
low ſo much time cvery Noon, nor will be perſwaded to do it once in three of
four Days, in regard Mercator's Clart comes neer the very truth. Lec the Wind, and
Sea, and Currents,, or Pirates, drive a ship ever ſo far wide of the Archa yer it is all
one by that True Chart. If you keep a true Account of the Ship’s way, allowing for
Variation of the Compaſs, and Setting of Currents, and heaving of the Sea, you may
at any time have the cruc Point where the ship is, and how to ſhape a Courſe the
moſt ready and convenient way to the Port you are bound to : Yet, I ſay, theſe Meni
that are perfect in this way of ſailing, may ſee by their Mercator's Chart the dan-
gers that lie between any two Places, and ſhun them; and likewiſc make many Vog-
ages, where the Winds may favour them, ſailing by the Arch, and no danger of Racks
or Sands to trouble them, which will prove a great advantage when your Courſe lies
neers Eaft and West, for failing upon theſe two Points, Men truſt altogether upor
Plain Sailles, their Dead Reckaning (by the two former
ways) but by this
way you may help your
Mercator's.
W
your Account
Far Examples Admiç you were to fail from Avere on the coaſt of Portugal, to the
Bay
no.
way he
1
Degrees, To find
Chap.XI. Of Sailing by a Great Circle. 179
Bay on the back ſide of Aguamacke ncer Virginia, which lye both nicer the Latitude
This Arch of 4
of 40 Degrees; and ſuppoſe the Difference of Longitude between theſe two places be Great Circle
70 Degrees : The Diſtance of the two Places Eaſt and Weſt is 53 deg. į and ſome- over two Places
ching more; but the Diſtance in the Arch of a Great Circle is buc 52 deg. and a little in Lacitude 40
more, that is, 1 deg. and about ; leſs, which is but a little benefic to this, which is d.oo m. and
the chiefeſt, That in failing between two ſuch Places by the Arch of a Great Circle, Difference of
you will in the firſt half of the way raiſe the Pole ş deg. 41 min. and then in the deg.is the Line
ocher half depreſs the Pole as much ; ſo that in the whole Voyage you will alter your PW in the Di-
Latitude 11 deg.22 min. by ſeveral courſes; by which you may rectific your Dead agram of the
Reckoning, which you cannot do in failing upon a Parallel of. Eaſt and Weſt; by 14. Chap; of
which you ſee it is the beſt way of ſailing, as well as the neareſt, eſpecially in ſuch Globe iz Pla-
occafions, if the Wind favour you.
Now concerning this way of ſailing, Mr. Edward Wright our Noble Countryman
did firſt lay down a way, in his Book of Correction of Errors in Navigation, pag. 63,
64 and 65, by Geometry, which Captain Santanſtal did comment upon in his Book
called the Navigator, which was only of a Parallel Courſe; for
any
ocher
ſaid but little or nothing.
Mr. Norwood in his Book of Trigonometry hath added many Problems of failing by
the Arch of a Great Circle; for thoſe that will or can, inay by his pains find out all
things in this way of ſailing: But as they are difficult
, and the way unknown to
moſt Sea-men how to calculare; ſo they arc tedious to thoſe of clie beſt skill : There-
fore I commend Mr. Philips his Tables
, in his Book called The Advancement of. Nd-
vigation, and likewiſe his 'Plain Figures in his Book of the Geometrical Sea-nsair. I
Thall likewiſe by Geometry, and by Calculation, give you ſome ſatisfaction. Either
way ſhall be done with ſpeed, and as exactly as need be required.
The true Diſtance between two Places in the Arch of a Great Circle contained be-
twixt chein, is chus to be found out.
If the two Places have no Latitude (being boch under che Aguator, and one of
them allo no Longitude, the Longitude of the other being leſs or not more than 180
Degrees, the Longitude is the Diſtance.
But if the Longitude be greater than 180 Degrees, ſubſtract it out of 360 Degr.
the Remainer is the Diſtance.
If two places be in one Meridian, and have the ſame Longitude both, and but
one hath Latitude North or South, the Latitude is the Diſtance.
But if both Places agreeing in Longitude, lave Latitude alſo of like denomination
Latitude,
(that is, both Latitudes Northerly, or both Sostherly) then ſubſtract che lefſer out of Landle. 50:15
the greater, the Romain is the Diſtance
.
Ribedev.43:30
But if both Places in one Meridian, have one Northerly Latitude, and the other Diſtance 7:45
Soutkerly Latitude, add them together
, for the Sum is the Diſtance in Degrees
.
C H A P. XI.
Horo to find the true Diſtance of Places, one of them having no Latitude:
The other having Latitude and Difference of Longitude leſs than 180
I Their Diſtance in a Great Circle:
2 The Direct Poſition of the Firſt Place from the Second.
3 And the Second place from the Firſt.
The Firſt Scituation.
Irft, If any two Places being propored, the one under the Aquinoctial, the
other may be in any other Latitude given, cither North or South, and the
Difference of Longitude of the "Places being known; you may find the three
things before ſpoken of in any Queſtion, by the following Directions. We call che
Aa 2
Angle
11
20
140
IS
Leagues 155
Mult.by 3
Miles 465
:
)
>
1
:
F
180
Of Sailing by a Great Circle. Book IV:
To draw a
Great Circle
Angle that the Rhomb leading from one place to another, makes with the Meridians,
the Poſition of theſe Places : But in regard the Arch of a Great Circle, drawn berwen
two Places, is the moſt neer diſtance from the one place to the other; therefore the
Angles which that Arch makes with the Meridian of thoſe Places, we call the An-
gles of Direct Poſition: or direct way of cwo Places one from the other.
Now in the following Diagram, let A be the Entrance of the great River of Am.u-
zones, under the Aquator; A Q the Arch of the Aquator, or Difference of Longi-
from the Ama- tude and let C repreſent the Iſland of Lundy, lying in Latitude 55 deg. 22 min.
dy, put ebe Dif- Northerly, and the Meridian thercof: and ſuppoſe the Difference of Longitude
ference of Lon- A Q_to be 41 deg. 22 min.
gitude on the
Weſt Side from A toG; ad draw the prickt Liu NS, N for the North, and S for the Soutb Pole; and through G draw
the Azimuth Circle NGS, and draw the Parrallel of Latitude CZC; and brough z diam tbe Arch A ZRV Q,
for ibe Circle which pafesh from £ Amazones to Z Lundy.
N
C
E
H:
ia
Æ
West
G
A
East
का
S.
។
F
as
Hopp to do theſe Queſtions Geographically.
Irft, With an Arsh of 60 Degrees deſcribe the outward Meridian Æ ECQIF.
Secondly, Draw A Q the Aguinotial Line. Thirdly, Take si deg. 22 min.
of the Line of Chords, and lay it from toc and draw the Line CD through the
Center, and the Line EF at Right Angles chiereunto. Fourchly, Take off your Sarthe
of Tangents, counting from 90,-41 deg. 23 min. and lay. from Q to A; Faria re-
preſents the River of Amazones. Now draw the Circle CAD chrough A, the Hori.
zon thereof is EF; then meaſure.FK, and apply it to the Line of Tangerits,
before directed; and you will find the Angle of Direct Poſition to be 48 deg.
25 min.
Take that Number out of your Line of į Tangents from 1 Degree forwards towards
90, and lay it from H to L for the Pole, and draw a Lins from
L through A, it will
cut the Live in I; ſo meaſuring CI on the Line of Chords, it will be gi deg. 57 min.
for the Distance, which is 1237. Leagues and:3712 Miles.
By' the Tables.
Hen in this Triangle CAQ, Right-Angled at there is required C A the
neareſt Diſtance of the two places in the Arch of a Great Circle ; and the Angle
ACQ_of Direct Poſition from the INland Lundy to the Amazopes: and the Angle
Coo
!
1
termos
1
it
CHAP.XI. Of Sailing by a Great Circle. 181
CRO being the Complement of the Angle of the Direct Poſition of the iſland of
Lundy.
For the neareſt Diſtance CA,
As Radius, is to Co-ſine of Difference of Longitude 41 d. 12 m.-989534
So is Co-fine of the Latitude or Difference si deg: 22 min. 979699
To the Co-line of the Diſtance 61 deg. 57 min.-
-967233
which 6r deg. 57 min, converted into Leagues, is 1237 as before, the ncareſt De
stance between thoſe two Places.
For the Angle of Direct Poſition from che Amazones toward Lundy,
NER,
As the Radius, to the Sine of the Differ. of Longitude 41 d.22 m.-982011
So is the Co-cangent of Difference of Latitude 51 deg. 22 min.-990267
To the Co-tangent of the Angle of Poſition 27 d. som. NÆR-972278
For the Angle of Poſition ACQ,
Meaſure NR
As Radius 90, is to Co-fine of Differ. of Lat. 51 d. 32m. QC-989273
on tbc half Tan-
gents, and the
So is Co-tangent of Differ. of Longitude 41 deg. 22 m. AQ1005522
Angle of Police
To the Co-tang.of the Angle of Direct Poſition 48 d.25 m. ACQ-994795
tion, and tbe
Diſtance is all
The ſame Preportions will hold by the Artificial Lines on the Scale:
ont as before
And thus you ſee, he which will fail the ncareſt way from the Amazones to the
Lizard, ſhall at firſt ſhape his Gouiſe 27.dog, 50 min. from the Meridian to the
Eastward; that is, N. N. E. almoſt a Polnt" Eaſterly. Now if the Wind ſhould
ſerve that you might ſail this Courſe; it is to be underſtood, that in this kind of fail-
ing he is not to continue this Courſe long; but to ſhift it, and incline more and more
to che Eaſtward, as often as occafiou requires : which how it may be dons, ſhall be
ſhowed in the following Diſcourſe.
1
P R O EL II.
How to find the Great Circle's greateb Latitude N.or S. or Obliquity.
Ote, Without the knowledge of the true Quantity of the Obliquity or Latinade
of that Great Circle which will paſs directly over the Places propounded, there
can be no complear Demonſtration, much leſs Arithmetical Calculation of things per-
taining thereunto ; therefore it is needfull that the true Quantity of each Great Cira
cle's Obliquity be diligently found to exact certainty: which to do, in ſome caſes is
very caſic, and in ſome again more difficult. Therefore I will propound Rules for the
ſeveral Scituations following, except thoſe that are ſcituate under the Aquator, or
under the ſame Meridian.
If one place be under the Aguator and hath no Latitude, and the other hath
any Quantity of Latitude, and the Difference of Longitude being leſs chan 90 deg, as
before 41 deg. 32 min. it is eaſily found, thus:
The greateſt Olliquity in the foregoing Diagrain is H RV,
As the Sine of the fourd Diſtance 61 deg. 57 inin.
994573
Is to the Sine of the Latitude si deg. 22 mju.-
1989273
So is the Radius (added to the Loff-Number)
To the Sinc of the greateff Obliqúary 62 d.16 m.)
994706
So 62 deg. 16 min. is the greateſt Obliquity or Latitude from the Æquator, of thac
Great Circle extended over thoſe cwo Places,
Buc
182
Of Sailing by a Great Circle, BOOK IV.
But if the Difference of Longitude be 90 deg. as Æ H, and one of the Places have
n0 Latitude, and the other have any Quanticy of Latitude; then it is evident to
reaſon, as in the forcgoing Diagram may appear, that the fecond place is ſcituate in
the very point of the greateſt Obliguity, which is never above 90 Degrees, as HN;
and the other place is in the very point of Interfection of the ſaid Great Circle with
the Aguator : For note, That every Great Circle that paſſeth over any two Places
propounded, cuts the Æquator in two oppoſite Points 180 deg. from each other, as
the Ecliptick Line doth in the two Points of Aries and Lilra; and the greateſt Olli-
gulty of char Circle is 23 deg. 30 min. the Sun's greateſt Declination, and never any
Now if one Place have no Latitude oo deg. oo min. and che other have any
Quantity of Latitude, the Difference of Longitude being more than godeg. to find
the Obligrity of the Great Circle paſſing over thoſe Places.
As admit one Place Latitude oo deg. 00 min. and the other si deg:22 min. Diffe-
rence of Longitude 138 deg. 38 min. Diſtance betwixe them is near 107 deg. There
fore take che Diſtance 107 deg. out of 180, and there remains 73 deg. Then,
more.
.
As the Sine of the Remainer-73 deg.
Is to the Sine of the Latitude 51 deg. 22 min.
Se is Radius
To the Sine of the greateſt Obliquity 54 deg. 25 min.
998251
989273
IO
1
991022
so that 54 deg. 25 min. is the greateſt Obliquity of the Great Circle extended over
theſe two Places. And ſo you may work for any Queſtions of this nature.
The ſecond Scituation.
Econdly, There may be two Places fcituated in divers Parallels of Latitude, be-
twixt the Artick and Antartick Poles, that may have one Degree and Minute of
Latitude, yet may have ſeveral Degrees of Longitude.
..
!
*NFO
?
8
H
R
o 31 drie
1
K
sadsr
Q
L
1
G
Family
t
B В
IS
th
PROBL
1
CHAP.XI. Of Sailing by a Great Circle.
183
PROBL. III.
1
1
PAD:::...
1
....
(1
Admit there be two places both in the Latitude of 51 deg. 22 min.
and their Difference of Longitude be 52 deg. 55 min.
1. To find the neareſt Diſtance of thoſe two places.
2. The Direct Poſition of the one place from the other.
How to do theſe Queſtions Geographically.
Ake off the Lize of Chords the Latitude of the Place 51 deg: 29 min.and:lay from
Æ to X, and froin Q to C; and take of the Tangents clie ſame Liitude, and
lay from K to ();, and through theſe three points draw the Parallel of Latitude
XOC; che Differesce-of Longitude laid from Q to L, draw the Heridian-Circle
NLS, the ſecond place is at R, and firſt at C the Meridian-circle cuts a R:
Therefore draw thic Circle from C through R to B, and meaſure HI on the Tane
gents, and you will find it 68 deg: 46 min. for the Angle of DireEt Poſition HCI.
Now from the Tangents cake 68 deg. 46 min. and lay it from the Center K to E
and from E draw through che Point of Interfe£tion at R the prickt Line ERF; and
becauſe it cuts the Line in F, therefore mcaſure C F en che Line of Chords, and
you
will find it 32 deg. 18 min. forotlic triteGreat-ircle Diſtance, which is 646 Leagues
,
or 1938 Milos.
In ilt "Sevcuch Problems of falling bý Mercabdras-Chaves yonayote ikeře jare-
quired the Diſtance of theſe two mldas meaſured irr fhe Paralla, and fotirid to be
660,4.. but here is required the-ndareſt Difrocesin the Arch of a Great Circle:
Work chus by the Tablese
For the Dištančejo prima
As the Radius;-Is to the site Comisle of the Lars 384. 38 im RN=779541
So is the Sine of the Differvbf Longitude 367 m.R FN-984876
To the Sine of half the Diſtance 16 deg. 09 min. R F
944415
Which doubled is 32 deg. 18 min. and this converted into Leagues and Miles, as bea
fore, is 646 Leagues, and 1938 Miles; the neareſt Diſtance; and leſs than the
Stance meaſured in a Parallel by Miles 42.
To find the Dire& Poſition,
As Radius 90, Is to the Sine Compl. of the Lat. 58d.38m.RN-989273
So is the Tangent of the Differ. of Longitude 260 d.27 m.RNF-969678
To the Co-rangent of the Angle of Poſition 68 d. 46 m. NRF-958951
Which ſhewech, that if you will go the neareſt way from Cto R, you muſt not go
Weft, though both be under one Parallels but muſt firſt ſhape your Courſe from C
from North 68 deg. 46 min Wefterly, that is almoſt w.N.w. and ſo by little and little
inclining to w. b. N. and then W. and W.b.S.and almoſt W. S.W. as before.
How to find the Obliquity,
Wo Places having Latitude both the ſame, as 51 deg. 22 min: and towards the
famc Pole
, whether North of South, and Difference of Lośgitude sa deg. ss
min, or any number of Degrees under 90: If above 90, take it out of 180, and
work with the Remainer the fame manner of way.
PROBL IV.
As Radius 90, Tá Co-tangengi of the Latitudes 5-1 d. 22 m. RN-990267
So is the Co-fine of the Difter of Longitude 26 d.27m.RNF-995197
To the Compl. Tan. of the greatest Obliquity 54 d. 25 m,NF-985464
DIE
1
+
So
184
Of Sailing by aGreat Circle. Book IV.
So that the greateſt Obliquity is 54 deg. 25 min. And the ſame Proportion will hold
for any Queſtion of this nature.
We mighc proceed to frame many Queſtions touching thoſe two Places ; but theſe
being the moſt material, I leavetle" reſt
to your own Practice, to uſe aš inuch brevity
as I may. I might have ſhewn you the Side and Angles; buc in regard chey are Sphée
rical I omic it, and thall demonſtrate them at laſt in Plano: But you may fo low
theſc Rules, if you cannot appiciend the Diagram; but ſome may deſire the Tri-
angle, therefore I lay it down.
In chis Triangle CRN, let the two Places be C and
CR, and let N be the North Pole; then CN or RN
either of them are 38 deg: 38 min. the Complement of the
Latitude and the Angle CN R is the Difference of Lon-
C
gitude: There is required CFR the neareſt Diſtance and
the Direct Poſition of the one to the other, NCR or
NRC; for in tliis Caſe choſe two Angles are equal:
And ſecing NC and NR are equal, therefore let fall the
Perpendicular N F, the Triangle NCR is divided into
tivo-Right-angled Triangles CNF and RNF, which
are every way cqual.
The Third Scituation.
11.386388
ៗ
R
One Place having North Latitude, and the other Place having South La-
titude, of different Quantities, and the Difference of their Longitudes
:s leſs than 90. As I omit one place having North Latitude, as Lundy,
si deg. 22 min. the other South Latitude, as the Rio de la Plata, 35
deg. oo min. Difference of Longitude betwixt them 45 deg. 55 min.
I demand the Diſtance, the Angleof Poſition, and the greateſt La-
titude or Obliquity of the Great Circle that pafſeth over theſe two
Places,
A Fter you have deſcribed the outward Meridian N ESA, stake from the Line of.
Chords the Latitude si deg. 22 min. and lay it from E to P, and draw the
Line PCO and HC M at Right Angles to P, take off che half Tangent Line the Dif-
+
.:In
N
7
.H
(am
SI22
1
1
LA
1
K
5
Pole,
نورر4
لل
E
A
i
11
+
35
MM
El
:)
V
S
W
ference
1
CHAP.XI. In Sailing by a Great Circle.
185
4
ference of Longitude 45 deg. 55 min. and lay it from E'to F, and draw the Me-
ridian-Circle NFS, whereon lay the Latitude of Rio dela Plata 35 deg. from F to
R, by taking 35 out of the Line of Chords, and laying it from E to 35, and the
Tangent of F E from C to Pole, and draw the prick'd Line Pole 35, which cuts
the Circle N FSi R, the Rio dela Plata: and through R draw the Circle PRO,
and meaſure M N on the half Tangents, you will find the Angle of Poſition to be
RPE 36 deg. 2 min. Then take the half Tangent of 36 deg. 2 min. and lay from C
to K, and draw the prick'd-Line from Kthrough R, and it will cut the Line at T;
therefore meaſure T P on the Line of Chords, and that is the meaſure of RP os deg.
18 min. for the Diſtance, or 1906 Leagues, or 5718
Miles: The greateſt Latitude
or Obliquity is: from A Eco L; and Ý LW is the Parallel of 68 deg. 21 min. che
greateſt obliquity required,
PRO BL. V.
1
901318
Then by the Tables;
"As Radius, To the Co-line of Differ. of Longitude 45 d. 55 m.-984241
So is the Co-tangent of the greater Latitude gi deg. 22 min. 990267
To the Tangent of the firft Arch 29 deg. 5 min.
934509
The loſs Latitude 35 deg. andrea As fakes satrdeg. Take the firſt Arch 29 deg:
5 min, therefrom, and there remains 95 deg. 55 min. Take this out of 180 deg. and
there remains 84 deg: 5.min, che ſecond Arch:-Then
mons
71. As Co-Gide of the fir} AFERdeg. $ moin..
994005
Is to the Co-line of the Second Atch 84. deſ. 15 mín.
So is the Sine of the greater Latitude 58 deg. 22 inin,
989273
Out of
ifsódico
Taken
.
84 42 STo the Co-line of 84 d. 42 m.8905ģt Sum
And there remains 95 18 SThe truc Diſtance 95 d. 18m._-896586.
which was required.
Now to find the Obliguity, Take both their Latitudes as if they were perih, or
both Soxtb, and the Complement of the Difference of Longitude to 1 80 deg. which here
is 1,34 deg. og min. half that is 67 deg. 2 min. 30": both the Latitudes added to-
gether make 86 deg. 22 min. half that is 43 deg. II min. ic being too little, I added
i deg. 20 min. to the half, to find the mean and true Latitude 44 deg. 31 min.
which I find the Obliguity, as I proved by this Operation.
As Radius, Tó Co-tangent of the Latitude 44 deg. 31 mia.--1000732
So is Co-line of half the Difference of Longitude 67 deg, 2 min.- 959158
To the Co-tangent of the Obliquity 68 deg.. 21 min.
959890
Now to find whether 68 deg21 min. be indeed the true Obliquity, make theſe
1
1
about
1
Proofs of it.
P
As Radius, To Co-tangent of Obliquicy 68 deg. 21 min.-
959890
Take from it the Tangent of the Latitude 5ı deg. 22 min.
990267
There remains the Co-line of Differ. of Longitude 60 deg.14 min.-969623
Again,
As Radius, To Co-tangent of Obliquity 68 deg. 21 min. 1959890
Take out the Co-tangent of the other Lacitude 35 deg. oo min.-1015477
There remains Co-line of Differ. of Longitude 73 deg.si min.- 944413
Now both the Longitudes found, 73 deg. 51 mix, and 60 deg. 14 min. added Sr
Вь
cogether, 134 Os
60 14
73
186
Geographical Queſtions Book IV. .
together, makes juſt 134 deg., 05 min. the Difference of Longitude at firſt propounded
berwixt thoſe cwo Places ; which proves, That the greateſt Obliquity of the Great
Circle that paſſech directly over theſe two Places, the Iſland Lundy and Rio dela Pla.
ta, lo ſcituate, is 68 deg. 21 min.
Now if it ſo happen that both the Latitudes be of the ſame Quantity, as one
Place North Latitude 11 deg. 30 min. and the other Place South Latitude 11 deg.30 .
and the Difference of Longitude betwise the two Longitudes 55 deg.
48 min. To find
the truc Great Circles Diſtance betwixe ſuch Places, firſt divide the Difference of Lon-
gitude juto civo equal parts, and then take one Latitude and half.che Difference of
Longitude, and find the Diſtance belonging to one Latitude, which doubled, yields
the whole Diſtance betwixt the Places propounded : As Longitude 55 deg. 48 min.
halfed is 27 deg. 54 min. and Latitude il deg. 54 min. Then worķ thus.
As Radius, To Co-fine of Differ. of Longitude 27 d. 54 m. 994633
So is the Co-line of the Latitude 11 deg: 30 min.
999119
To the Co-finc of 30 deg, half the Diſtance
993752
which doubled is 6o deg. the whole Diſtance betwixt one Place: South Latitude 11 deg.
30 min. and and another North Latitude 11 deg. 30 min. having 55 deg. 48 min.
Difference of Longitude. And ſo work for any two. Places ſo ſcituate.
D
1
*
Geographical Queſtions.
Two Ships being at Sea, their Difference of Longitudczas s3 deg. Now
upon a day they obſerved the Sun being between them; the North Ship
found the Sun's Meridian Height 33 deg. and the South Ship 77 deg.
the middle Latitude between them was 15 deg, North of the Æqui-
no&ial Line : I demand the Angle of Poſition, and Diſtance from
the North Ship to the South ?
Meridian alitude. T Irſt add the two Meridian Altitudes complement together, 33 deg. and 77. deg.
33d.Compl.spdeg. Complement 57 and 13, che Sum is 70, che half sum is 35 deg. the middle La-
77d.Comol izdeg. titude 15; add the middle Latiinde and half sum together, it makes so deg. the
Sam 70
Sun 35
Mid.Lal. 15
P.
Nor.Sbip.golat.
South Ship 20 lar.
N
wi
,
)
A
?
I
bi
33
e
20
.
L
H
+
De
В
2.
.
North
1
d
CHAP.XI. In Sailing by a Great Circle.
187
North Ships Latitude; and ſubſtract the middle Latitude from the half Sum, and
the Remain is 20 deg. clie Latitude: of the South Ship. The North Ships Latitade
is laid from Q to N so deg. the Difference of Longitude QF $3 des. Through F de-
ſcribe the Great Circle Meridian PË B, on which lay down the South Ships Latitude
20 deg. as F C, and ſo draw the Great Circle NCD through chc Interfe£tion of the
prick'd Line IH, with the Meridian ac C; for that is the Latitude of the ſecond
Ship: So the Angle of Poſition is NCR, whoſe meaſure is Gon the half Tangents
48 deg. 58 min. from the South Westwards; and the Diſtance is NH 48 deg. 22 min.
that is
, 1683, Leagues, or 5051 Miles, the neareſt Diſtance of the cwo Ships, which
was required. How to do it by the Tables, you have been shewed in the laſt
Example.
T
►
The Fourth Scituation.
1.
17
P R O B L. ,VII.,
The Latitude of two Places being given, together with the Difference of
Longitude, To find,
Firſt, The neareft Diſtance in the Arch of a Great Circle.
Secondly, The Direct Poſition from the firſt place to the ſe
cond. And,
Thirdly, From the ſecond place to the firſt. And,
Fourthly, The Circles greateſt Obliquity that paſſeth ever
thoſe two Places,
ADmit L be the Latitude of Lundy 51 deg. 22 min. and Longitude 25 deg. 52 min.
and B is the Latitude of Barbadoes 13 deg. 10 min. and Longitude 332 deg. 57 m.
and Difference of Longitude 52 deg. 55 min.
:
:
P
*.
:
1
y
H
•22
Pole
(3
K
$2.5
A
G
F
5
Y
+
N
15
1
S
Lay down firſt the Latitude si deg. 22 min. from Q to L; ſecondly, the Diffe-
rence of Longitude QF 52 deg. 55 min. and draw the Meridian-circle P through F
to S; then lay down the
Latitude of Barbadoes from Q to 13, and take of the half
Tangens
B b 2
1
188
Geographical Queſtions Book IV.
1
!
1157 M.
I140 G
166
Tangent Line QF 57.deg.: 55 min, and lay from C to K, and draw the prick'd Line,
and he will cut the Meridian PFS in B, the Latitude of Barbadoes; 13 deg. 10 min.
through B draw the Circle L BN, ſo the Angle of 'Poſition is BLC, whole meaſure
is RO 67 deg. 5 i min. that is, W. S.W.21 min. Weſterly; which taken off the Line
of half Tangents of your Scale, and laid from C to Pole, and draw the Line from
Pole through B, and it will cut the Limb in G: Therefore meaſure. G L on the Line
of Chords, you have 57.deg. for the Diſtance, or 1140 Leagues, or 3420 Miles.
To find the Diſtance in Queſtions of clis Nature,
1 As the Radius, Isto the Co-line of the Diff. of Longit. 52 d.55 m.-978030
So is the Co-tangent of the greater Latitude si deg. 22 min.
-990267
To the Tangent of the firft Arch 25 deg: 44min
-968297
Take 25 deg. 44 min. from 76 deg, 50 min. the Complement of i 3 deg. 10 min. the leſs
Latitude, and the Remain is si deg. 6 min. the ſecond Arch.
2 As the Sine of the firſt Archi:25 4. 44 m. 9954647
Is to the Co-fine of the ſecond Arch 51d.06 m.- 979797? To find tbe Diſtance.
So is the Sine of the greater Latitude srdizzin- 9892733
1909066
1306 P. To the Co-finc of the Diſtance 57 d. oo m... 973602.
keltich is the neareſt Diftangeln dhorch of a Gront Circles by 17 Leagues leſs than
Mercator's Chart by the Rbomb, and leſs by 166 Leagues than by she Rhomb on the
Leag.17 Plain Chart; which confirms this to be the neareſt of all ways of Sailing becwixt any
two Places
A
To find the Angle of Poſition,
3. As the Sinc of the Diſtance 57 deg. oo min.
992359
Is to the Sine of the Difference of Longitude 52 deg. 55 min. 990187
So is the Co-line of the Latitude 13 deg. 10 min.--. 998843
Add the two laft: Subſtrall the firſ Numbers
1989030
There remains the Sine of the Angle of Direct Poſition-
996681
Which is 67 deg. Fi min. from the South part of the Meridian-weſtward, as name-
ly, W. S. W. 20 m. Wefterly.
Now to know the Diſtance and Angle of. Rofition, you muſt þut Barbadoes at B,
on the Weſt ſide of the Circle iz deg. from £gránd draw the Parallel of Latitnde
si deg: 22 min. and lay off the Difference of-Longitade from 2 to F, and draw the
Meridian-circle. PFs, and it will cuc chei Parakei of-Latitudain. Icherefore from
B draw the Circle from L to K: And if you follow your former Directions, BD
will be the Meaſure of BL the Diftance found, as beforc, 57 deg. co min. and LBP
che Angle of Poſition, whoſc Meaſure is R G 38.deg. 26 min. and the Great Circles
greateſt Obliguity is CO 54 deg. 40 min. For,
4. As the Sine of the ſecond Arch's i deg. 6 min
989111
Is to the Sine of the firſt Arch 25 deg. 44 min.
963767
So is the Tangent of Difference of Longitude 52 deg.-55 min.-1012151
1975924
To the Tangent of the Direct Poſition 36 deg. 26 min.
986813
- Fróm B towards L; which is 3 : Points, 2 deg.41 min. from the Aeridian, namelyz
-N. E.b. N. 2 deg 41 min. Eaſterly you mụſt fail.firſt from B towards L; bur al-
ter your Courſe, ſtill increaſing toward thc Eaftward, as ſhall be thewed.
For
L
1
{
SI
-
22
a
CHAP.XII. In Great Circle Sailing.
189
.
D
G
e
t
YI22
1
L
R
B
1
til
C
.za,
a
Æ
F
K
Roll
S
I
::: 1
2
onl!
Pluton
Ballote
1
: ?
I
A
1
insريد
.1
72
!? Oor
For the Obliquity, to find that,
5 As Radius, To Co-fine of leſs Latitude 13 deg. 10 min. 998843.
So is the Sine of the Angle of Poſition 36 deg. 26 min. -977370
To the Co-fine of the greateſt Obliquity 54 deg. 40 min. -976213
Theſe are the Scituations of all Places upon
the Terreſtrial Globe; ſo that there can-
not be any two Queſtions, but, in reſpect of each other, they will be found in one
of cheſe four kinds; except they fall in one Meridian, or on the Æquator: and theſe
Directions you have in the Tenth Chapter : Therefore if you will ſeriouſly obſerve
theſe ſhort Directions already given, and as follows, you ſhall never have your Expe-
ctation deceived.
CHAP. XII.
0
How to deſcribe the Globe in Plano, by the Mathematical Scale.
T:
Heſe, and all other Queſtions of this nature, concerning the Reſolution of any
Spherical Triangles may very eaſily be performed by the Globe: But be-
cauſe the Globe is a chargeable, Inſtrument, and ſo every one cannot have it,
therefore ſeveral Men, have for ſeveral Uſes, invented ſeveral ways to Project the Globe
upon a Plain, as Mr. Gunter hach them in his Book of the sector. The firceſt for this
purpoſe will be chat of Gemma Friſina, which is moſt uſed in the Great Maps of
the World, the Projection whereof is as followeth.
Firſt, By the Chord of 60 deg. deſcribe the Circle ÆNES, and by the Chord of
20 deg.
divide it into four parts, as £ E a Croſs Diameter for the Aquator, and NS
for the great Meridian: Then by taking off every 10 deg.of the Chord, you may divide
each Quadrant into
90 deg. and number chem as in the Figure : Then if you take off
your Line of Tangents in your Scale every 10 deg. and 5 deg. and lay them from the
Center C on the four sides of the Quadrants
, as you ſee the Figure, and number them
as they are in the Figure ; fo thall you divide the Diameters into his parts Æ E the
AquincEtial, NS the Meridian, which arc half Tangents. You may do it alſo wich-
OUC
M
11
1
190
11
1
Geographical Sailing BOOK IV.
you extend
Meridian-cir-
des.
out the Scale, by your Ruler, if you ſtop one end of your Ru'er ac N, and curn the
other end about to the ſeveral Degrees in the lower Semicircle ES E: And alſo if you
keep one end of your Ruler fixed in the Point £,and lay the other end abour to che le-
veral Degrees in the Semicircle NES; ſo have you the Meridien-diam ter divided
into half Tangents likewiſe.
Now you have divided the Diameters, they muſt guide you in clic drawing of
the Meridians from Pole N to Pola S, which arc- perfect Circles; as likewile are the
How to find the Parallels of Latitude. You may find the Centers in the Diameter £, if
Centers of the the Compasſes from the firſt Degree on the half Tangents, to the Secants of every 10
Degrees, and with chat Diſtance put one point at 10 deg. in che Semidiameter EC,
and in EC will the other Point be the Center of.che- Meridiin of the firſt 10 deg.
from £: and do the like from E in the ſame manner, for
any
other Degree. To
draw the Meridian of so-deg. Longitude, take the Secant of so deg. off the Scale,
and one point will ſtand in the Semidiameter Æ Cy'at ço deg. and the other will ſtand
in the Center at Eåff, and likewiſe at Weft for go 'deg. on the other ſide: And ſo do
for the reſt; and ſo you may find the Center of any Circle whatſoever, upon the
Croſs Semidiameter belonging to it, which you muſt continue beyond the Great Cir-
How to find ebe cle, where the Center will be in many Queſtions. For the Parallels of Latitude, it is
Centers of tbe thús: Take the complemient-number of Degrees. off your Line of Tangents, put onc
Parallels of Point in thc Degree of Latitude, the other will ſtand in the Center.
Latitude
The GLOBE in PLANO.
The Globe in Platno.
XI North,
Vrati
واو
9
80
80
To
zo
60
бо
---provela
L
50
59
H
40
iz
999:
30
sܕ
30
2019
ofic.
510
20
410
39
HS:
10
Why
Mulo
20
E. East.
West.
867666
510
0401 ,35/
glid
F
P
210310 40,58
3to
.
m
do
3
of.
#
TA
AA
Comment
20
I.
got
1
ALTO
sto
30
08
m п
0
sif
Zo
50
fool.
+
N
ogs
09
Hi
04
11.
08
08
06S
T
1
Vt
i
xt South
190.
For
1
Chap.XII
. By an Arch of a Great Circle.
191
For Example. If you will draw the Parallel of Latitude for 60 deg. take off the
Tangent-line of your Scale 30 deg. the Compl. of 60 Latitude, and the other will fall
upon V the Center of the Parallel of 60 deg. in the Semidiameter NS continued be-
yond the Circle.
So, Take the Tangent of 40 deg. and it will draw tlie Paralel of 50 deg. whole
Center is at #: and lo do in drawing all Parallels of Latitude. You may draw them
alſo by making ſeveral Trials, until your three "Points be in a Circle, and alſo draw
the Parallels of Latitude : with the ſame Diſtance find their Centers; but if you can,
by the Scale is tlie ſureſt way.
1
1
1
The Four Scituations that are if the Globe.
t
The Firſt Scituation.
you
A is a Point of Interfe£tion for the mouth of the River of Amazones, 2 Lundy,
CRV the Obliquity 63 deg. 16 min. E the other Point of Interſection with the Aqua-
bor, N RV che incalure of the Angle of Poſition, which applied to the Æquator from
Einwards, thews you 27 deg 50 min. from Amazones to Lundy. Now if will
know the Diſtance in ſuch Quefians, meaſure it in the Meridian that agrees with-
the Angle of Poſition; as namely, for this Diſtance ÆZ, you muſt mcafure from N
in the Meridian-line of 27 deg. 50, min. and you will find it 61 deg. 57 min. And
lo do for to meaſure any other Diſtance
.
The Second Scituation.
I is the firſt Places Latitude, r is the Difference of Longitude şu deg. 55 min. T is
likewiſe the ſecond Places Latitude ; and n H is the meaſure of the Angle of Poſition,
which meaſured in the Semidiameter Æc, will be found 68 deg. 46 min. In that
Meridian meaſurel r the Diſtance, and you will find it from N towards C, to reach
32 deg. 18 min. Remember to meaſure che Diſtances from the Poles in the ſame Meria
dian, of the Number of Degrees of the Angle of Poſition : The greateſt Obliquity of
that Circle Nrl is at W 54 deg. 25 min. Interfe&tion of the Aquator at Y, 'W is a
Meridian of greateſt Obliguity.
The Third Seituation.
Listhe firſt Places Latitude si deg: 22 min. North; Ep is the Difference of Lor-
gitude 45 deg. 55 min. R is the ſecond Places Latitude, or Rio de Plata, a che greaceſt
Olliqalty 68 deg. 22 min. m n the Meaſure of the Angle of Direct Poſition: applied to
E C will be found 36 deg. 02 min. in that Meridian: from the Pole meaſure the Di
france L R, and you will find it 95 deg. 18 min. P the Interfe&tion of the Great Circle,
paſſing over the cwo Places in che Æquator.
The Fourth Scituation.
The firſt Latitude is at L Lundy 51 deg. 22 min. Difference of Longitude counted
from E 52'deg, 55 min. that Meridian will cuc the Latitude of Barbadoes 13 deg. 10
min, at b: M H the Meaſure of the Angle of Direct Pofition 67 deg. 51 min. and bL
meaſured in thar Meridian is 57 deg. min. che Diſtance. Now to know the Angle
of Poſition from Barbadoes, being westward from Lundy, ſet it on the weſt ſide of the
Figure, as at B;
and likewiſe if the firſt place be to the Eaſtward, puc his Latitude
to the Eoft ſide of the Meridian.
Now to know the Angle of Poſition from Barbadoes
, and Diſtance, and Obliguity,
BOO is the Arch of the Great Circle that paſſeth over theſe two Places; ais Lundy,
qh is the Meaſure of the Angle of Poſition 36 deg. 26 min. Bo meaſured in chas
Line,
192
Geographical Sailing Book IV.
Line, you will find the Diſtance 57 deg. O is the greateſt Olliquity 54 deg. 40 min.
NÓ Ó S a Meridian of the greateſt obliquity : d is the interſektion with che
Aguator.
CMAP. XIII.
By Arithmetick how to calculate exaktly for any Degrees and Minutes of
Obliquity; what Degree and Minute of Latitude the Great Circle
Shall paſs through for any. Degree and Minute of Longitude, from the
Point of Obliquity, ör of its Interſection with the Aquinoctial.
Ore theſe Rules well; for tliey ſerve ForaliQueſtions of this nature, what
Difference of Longitude focver any Point or Place hath from the Meridian
of its next Obliquity, which is ever go deg. or leſs ; " the Complement chereof
to 90 deg, is the Difference of Longitude of that point or Place, from the Meridian
of that Great Circle's next Inter fečtion with the Æquator.
The firſt fort of RuL I s are theſc.
By the Latitude of two places, the Difference of Longitude betwist them,
and the Obliquity of the Great Circle paffing directly over both Places
given ; To find the Difference of Longitude of each Place from the
Meridian of the greateſt Obliquity.
C
1
Ler this be our Example. Lundy in North Latitude si deg. 22 min. and Barbadoes
in North Latitude 13 deg. 10 min. and Difference of Longitude 52 deg. 55 min, and
the greateſt Obliqnity 54 deg. 40 min, Work firſt with the leſs Latitude, to find the
Difference of Longitude from the Meridian of obliquity of both Places.
Ru LI I.
.
Thus, As Radius, and Co-rangent of Obliquicy 54 deg. 40 min.-1,985059
Take from it the Co-tangent of lefs Laciqde 13 deg. 10 min.-1,
063090
Tbere remains the Co-ſiné of Differ. of Longitude 80 d. 27 m. 921969
The Difference of Longitude of the ſecond place from the Meridian of the greateſt
Obliquity. And by reaſon the Difference of Longitude from the Obliguity is more
than the Difference of Longitude betwixt the two Places, therefore ſubſtract the Dif-
ference of Longitude 52 deg.: 55 min. from 80 deg. 27 min, and there remains the Dif-
ference of the firſt place from the Meridian of Obliquity 27 deg. 32 min. and the firſt
Place is becwixt the ſecond Place and the Meridian of Obliguity. But if the Difference
of Longitude from Obliguity had been leſs than the Difference of Longitude betwixt the
two Places
, then ſubftract the Longitæde from the Difference of Longitude betwixt the
two Places, and the Remain had been che Difference of Longitude from the Meridian
of Obligxity, to the firſt Place.
The uſe of theſe Rules are,
1. To find where to place the Meridian of the greateſt Obliquity betwixt any two
Places, in a blank Chart, or Mercator's Chars or Plat, to trace out the Voyages as we
find that it is 80 deg. 27 min, the Difference of Longitude of the ſecond place from
che Meridian of Obliguity, and 27 deg. 32 min. from the firſt.
2. You ſee you find the Difference of Longitude berwixt the Obliquity and any LA-
titude propounded, by the laſť Rule.
3. By the Obliguity of the Great Circle, and the Difference of Longitude from the
Obliquity, to find the true Latitude,
RuLa
+
CHAP.XV. By the Arch of a Great Circle.
193
Ru I E II.
1
921986
Y
As Radius, To Co-tangent of Obliquity 54 deg. 40 min. -1985059
Take from it Co-line.of Differ. of Longitude 80 deg. 27 min.
There remains Co-tangent of the Latitude 13 deg. 10 min.--1063073
And ſo likewiſe if you work with 27 deg. 32 min. by the ſame Rule, you will find
the other Latitude 51 deg. 22 min. and by it find from any Place through what Lati-
tude the Great Circle pafleth every 10 deg. or 5 deg. or more or leſs Quantity of Longi-
tude from che Obliquity; and thereby find the Latitude, to make marks in every Merl-
dian ; and ſo trace to our che Great Circle Arch, in your Mercator, or Mr. Wright's plat
or Blank Chart.
As thus, for a deg. 28 min. of Longitude morc, added to 27 deg. 32 min. makes
30 deg. oo min.
Ru LE
III.
As Radius, To Co-tangent of Obliquity 54 deg. 40 min. 1985059
Take from it the Co-line of Differ.of Long. from Obliq. 30 d.00'- 993753
Tbere remains the Co-tangent of the Latitude so deg. 41 min.-- 991306
.
Deg. Min.
220
OO
00
00
42
IZ
And by the ſame Rule I made this Differ. of Longit. Latitude
from
Table, of an Arch of a Great Circle, ex-
from Obliq.Lundy Lundy.
tended from Latitude si deg. 22 min.
Deg. Min.
to Latitude 13 deg. 10 min. Difference 27 32 51
of Longitude 52 deg. 55 min. ſetting the 30
50 41
Point of the greateſt Obliquity upon a
35
49 07
Meridian-line, tliát ſo it might be the
40
47 13
better protracted on Mercator's Chart: 45 00
44 56
This is, for every 5 deg Difference of
50 00
Longitude, theſe are the Latitudes the 55 00 38 58
Great Circle will paſs through; ſo chac
60 00
35 12
you ſee there is 52 deg. 55 min. added
65 00
30 48
by s to the Difference of Longitude of
25 46
the firſt place from the Meridian of Obm 75
03
lignity, which makes 8o deg. 27 min.
13
Barbadoes.
the Difference of Obliguity of the ſecond
Place, which was the Difference of Longitude from Obliquity at firſt found. In like
manner work for any other place.
|
1.
70 00
00
20
80 37
IO
-"
1
CHAP. XIV.
Homo by the Scale of Tangents to make a part of the Globe in Plano, where.
by you may trace out the Laticudes to every Degree of Longitude, or eve-
gry 5 or 10 Degrees, As neer As jou will defire, without Calculation,
Y the Line of Tangents on the ſide of your Mathematical. Scale, you may
make the following Projection, which was made by Mr. Philips in his Geo-
metrical Seaman, pag. s. by Tables and Geometry: But here you may ſave chat
labour, if you have a Scale with a Line of Tangents on it.
Firſt
, Conſider of whac length your Tangent or side of your Quadrant muſt be,
and accordingly fee off your Radius from A towards: D, as I have done, by taking off
77 deg. of the
Tangent-line of my Scale, and ſet it from A the Pole to D, for 13 deg, the
rate numbe
.
ſerves for
on the North ſide of the Æquator, or 1 3 deg. of North Latitude, which is the com- of deg. in Soweb
pleruent Latituds
B
Сс
?
1
!
+
194
Geographical Sailing Book IV.
1
plement of 77 to go dog. Then make the other ſide of the ſame length, and draw che
Quadrant A DÉ, the Radius is always a Tangent of 45 deg. Then with your com-
partes take off the Line of Tangents the ſeveral Degrees, and draw the Arches or
Parallels of Latitude, as you ſee I have done in the Figure. Thirdly, divide the
Limb of the Arch D E into 90 deg. and through every s or 10 deg. draw Lines of
Longitade, or Meridian-lines. The Arches of Latitude muſt be numbred as in thic Fl-
gure; but thc Lixes of Longitssde you may number from either ſide, as occaſion re-
quires.
You may,
if
you will, when occaſion requires, divide a Circle into four Qua-
drants, and draw the Lines of Longitude from the Center ; and you may make this
as large or as little as you will, by the Tables of Natural Tangents in ine Sccond Book,
have been there ſhewed how to lengthen or ſhorten your Radins : You may
number the whole Circle of Longitude inco 360 deg.
as you
!
.
1
D
1
I lo
[21
210
.
1
310
2
1
40
51
8.
R
60
.??
30
bylo
HT
FR
F11
510
00
.
710,66
go
73
8/0
7990. 1.50
40
30
210
T11:
194.
4
CHAP.XII. By an Arch of a Great Circle.
0
Places doth paſs.
Differ.) Longit. Latitude Longit. 57 d. the true
195
The Blank Quadrant, being thus made, will ſerve for many Examples; eſpecially
if
you make it upon a Slat Stane, that you may wipe the Arch, that is lightly drawn
by a Slat Den betwixt any two Places, off at pleaſure.
You may ſee down therein che cwo Places you are to fail becween, according to their
Latitudes and Longitudes ; and then only by your Ruler draw a ſtreight Line from the
one Place to the other, which will repreſent the Great Circle which paſiech between
the cwo Places, and will croſs thoſe Degrees of Longitude and Latitude, whidi you
muſt fail by exactly. You may do ic by che. Difference of Longitøde orily, if you
will, as ſhall be thewed in this Examples for proof thereof.
of a Voyage from Lændy, in Latitude 5 deg. 22 min. and Barbadoes, in Longitude
332 deg. 27 min. Difference 52 deg. 55 min. and Latitude 13 deg. 10 min. To find by
whac Longitude and Latitudes the Arch of a Great Circle drawn becween choſe two
Firſt
, Lcc the Line A D repreſent the Meridian of the land of Lundy, marked you may mesa
out by L for its Latitude si deg. 22 min. and the Longitude thercof 25 deg. 52 min. at ſure the Diften.
D, which is ſet down according to its Longitxde and Latitude. Then from D in the cestbus. Take
Limb or Arch of the Quadrant, counc tlic Difference of Longitude 52 deg. 55 min. B L she Dir
and this is the Meridian of the Ifland of Barbadoes, on which you muſt mark out
Bance of sheim
the Latitude 13 deg. 10 min. at B; lay a Ruler from the firft Latitude L to the ſecond it upon the Me-
Places, and put
at B, and draw chic ſtreight Line LB, which repreſented the Arch of a Great Circle ridian B 2 of
becwcen che two Places; and if you guide your Eye along in this Line, you may rca- Barbadocsibe
dily and truly perceive by what Longitudes
ſecond place,
and Latitudes you ſhould fail : For where
you will find
this Line croſſeth the Arches of Latitude and
the Parallels of
Differ.of Latitude to be
the Lines of Longitude, that ſhows the true .
from
i
Longitude and Latitude of the Arch, accord Sabftract
added.
, as bei
Obligu.
ing to your deſire. Now the truth hereof
fore found.
will inorc evidently appear, if you compare
D. M.D.M.D. M. D, M.
the Latitudes and Longitudes which this Line 5527 3251
interſecteth, with this Table, as before, Cal-
20130oolso
00150 41102
culared by inc for every 5 deg. of Longitude: S 0013:5 00149 07.07 28
You may, ſcc by the Figuri, that the Line
5
00/4000/47
13\12 28
BL in the points a, b, c, d, c, F, g, h, i, k, 00145 00
0044 56117 28
L, doth croſs che Parallel of Latitude, as you
5
12
281
ſce in the fourth Column of this Table, at the
5
0038 88127
58127 28
ſame number of Degrees from the firſt Me 5
12 32 28
ridian, as you ſee in the fifth Column. The 5:00 0030 481 37 28
firſt Column is the Number of Degrees of 5 00170 00125 46 42 28
Longitude, the ſecond is the Difference ſub 5 0075 0020
3047 28
{tracted, the third is the Degrees of Longi 5 2780 2713 10152 55
tude from the Meridian of the grcateſt Ob-
55
152 55
Barbad.
liquity, the fourth and fifth arc, as before,
Latitude and Longitude added, for the Dif-
ference of Longitude from Lundy. For Example: I would know what Degree of
Latitude 47 deg. 28 min. of Longitude from the firſt Meridian doch croſs; and I ſee
by che Figure it is at 20 deg. of Longitude, which the Table ſhewech is 20 deg. 30 m.
of Latitude, and ſo of the reſt, 75 deg. from Meridian of Obliguity, and Longitude
338 deg. 24 min.
And ſo in like manner you may lay down upon the former Quadrant any two
Places
, howſoever fcituated, by their Longitude and Latitude, or Difference of Longi-
tade, in the manner as you have been ſhewed in the laft Erämiple
.
Differ.of]
52
22:00
00
2
28
5
00
0042
00150
00155
I222
!
0060
00135
C C2
Снар.
A
1
196
Geographical Sailing
BOOKIV.
!
CHAP. XV.
1
titude well ob-
titude, Rhomb,
Ru L E IV.
How by the La By
the Latitude, and Difference of Longitude from the Obliquity, to
find the true Great Circle's Diſtance.
farved, and the
Rbomb diſcreet.
ly rectified, to
As Radius
90, To Co-fine of the Latitude si deg. 22 min.
979541
find the Lao So is the Sinc of the Difference of Longitude 27 deg. 32 min. 966489'
and Longitude,
To the Sine of the Diſtance 16 deg. 47 min.
946030
and Diſtance,
baving two of So that 16 deg. 47 min. iš the Great Circle's Diſtance from the Point of the greateſt OL-
them knowi, by liquity of that Great Circle.
är Arithmetical
Rue, and by
them to price
Ru I E V.
the same down
is áBlank Chart
or Mercator's
By the Obliquity of the Great Circle, to find the true Latitude to any
pist.
Quantity of a Great Circle's Diſtance, from the Point of his greateſt
Obliquity.
As Radius, To Sîne of greatest Obliquicy 54 deg. 40 min.-
991158
So Co-line of the Diſtance from Obliquiry 16 deg. 47 min.-998109
To the Sine of the true Laticude si deg. 22 min.
989267
Again,
As Radius, To Sine of greateſt Obliquity 54 deg. 40 min.-
Sots Co-líne of Diſtance from Obliquity 73 deg. 47 min.
To Siue of true Latitude of Barbadoes 13 deg. 10 min..
991158
944602
-935760
Ru L E VI.
By the Great Circle's Diſtance from the Point of Obliquity, and the
Latitude given; To find the Difference of Longitude betwixt the
Place and the Meridian of greateſt Obliquity.
As Radias, To the Sine of Diſtance from Obliquity 16d. 47 m.-.-946052
Take from it the Co-line of the Latitude gi deg. 22 min.
-979541
Remains the Sine of Difference of Longitude 27 deg. 33 min.-----966511
By theſe four laſt Rules you may be confirmed of the truth of the former Work,
of tracing of the Great Circle by Longitade, Latitude, and Diſtance, from che Point
of its greateſt Obliguity.
This is worth your Obſervation, That the Complement to 90 Degrees of the Great
Circle's Diſtance of a Place from the Point of the Great Circle's Obliquity, is always the
Great Circle's Diſtance from the point wherein that Circle interſectech the Agui-
noctial.
1
The
E
V
CHAP.XV. By the Arch of a Great Circle..
197
The ſecond ſort of 'R U LES;
whereby to find what Rhomb you are to fail upon, that you may keep in or
neer the Arch of a Great Circle, extended from one place to another.
Ru LE VII.
1
By the Difference of Longitude from the Obliquity, and Latitude
given; To find the Great Circle's Diffance from the Poins and Me-
ridian of greateſt Obliquity.
As Radius
. To the Sine of Differente of Longitude from. Ob-3999393
liquicy &o deg. 27 min.
So is Co-line of the Latitude 13 deg. 10 min.
998843
To the Sine of the Great Circle's Diſtance from Oblic. 73 d. 47m.-998236
So that if you take out 16 deg. 47 min. in the Diſtances of. Obliquity before found oue
of 73 deg: 47 min. the Remain will be the true Diſtance of the two Places 57 deg. as
was found in the ſecond Rule of the fourth Scituation. But if the Point of greareſt
Obliguity had been betwixt the two Places, you muſt have added them together, and
the ſame had been chc Diſtance of the two Places.
Ru L E
VIII.
.
By the Great Circle's Diſtance from the Obliquity, and the Latitude
given ; To find the Rhomb.
As Radius 90 d. and Tang of Great Circles Diſtance 16 d.47m.-947943
Take out the Co-tangent of the Latitude 51 deg. 22 min. 990267
There remains the Co-finc of the Rhomb 67 deg. 50 min, 957676
Having therefore in the four Scituations, before-going, been directed how to find the
Diſtance betwixc thc evo Places, by the Arch of a Great Circle, and the greaceſt Olli-
quity of any Circle, you may by the firſt Rule of Chap. 13. find the Difference of
Longitude of each Place, from the Point of the Great Circle's greateſt Obliquity; and
then by the fourth and ſeventh Rules of Chap. 15. find the Great Circle's Ditance to
the obliguity, and by the eighth Rule find the Rhomb to be failed on, from either place
towards the other.
Ru i E IX..
.1
To find how far a Man fhould ſail upon « Rhombbefore he change his
Courſe a Point, Half a Point, or a Quarter of a Point, wit'ſ
You
may try
this by your Protracting Quadrant, on your Blank Plat or Chart,
made according to Mr Wright's or Mercator's Projection ;} where the Voyage is truly
and carefully traced our as before.
Or you may arithmetically try every Point, or Quartér of a Point, or Half Point,
as you ſee cauſe, by cheſc Rules.
Having found by thc eighth Rule the Rhomb was 67. deg. so min. add co-ita Quarter
of a Point, or 2 deg: 29 min. it makes 70 deg. 19 min. 1. N.W. W. and Latitude
49 deg. 07 min, and Difference of Latitude 2 deg. 35 min. and it is required the Di-
Stance.
1
;
As
.
HU.
+
198
Geographical Sailing Book IV.
As Radius go deg. To Sine of Difference of Latitude a d.15 m.-859394
Take out the Co-liue of the Rliomb 70 deg. 19 min.-
952739
There remains the Sine of the Diſtance 06 deg. 42 min.
906655
Therëföreno6 deg. 42 min. is 134 Leagues, may be failed W.N.w. Weftirly, from
the Latitude si deg. 22 min. to Latitude 49 deg. 07 min.
1
Ru L & X.
By the Great Circle's Diſtance, and the Difference of Latitude given;
To find the Rhomb.
As Radius, To Sine of Difference of Lacitude 2 deg. 15 min.
Take out tbe Sinc of the Diſtance 6 deg. 42 min.
There remains the Co-fine of the Rhomb 70 deg. 19 min.-
859394
906655
952739
Ru L E XI.
By the Rhomb, and Diſtance upon it given, To find the Difference of
Latitude.
}
As Radius, To Cc-finc of the Rhomb 70 deg. 1.9 min.
952739
So is Sine of the Diſtancc 06 deg. 42 min.
906655
To the Sinc of Difference of Latitude a deg. 15 min.
Which 2 deg.. 15 min. taken froin the firſt Latitude si deg. 22 min. there reinains the
Latitude 49 deg. 07 min. which are good Proofs.
859394
RuL E XII.
By the Obliquity of the Great Circle, and the Latitude given ; To find
the Difference of Longitude from the Meridian of Obliquity.
As Radius 90, To Co-tangent of Obliquity 54 deg. 40 min. -985059
Take out the Co-tangent of the Latitude 49 deg. 07 min.- -993737
There remains Co-line of Difference of Longitude 35 deg. oo in -991322
35 deg. co min. is the Difference of Longitude from the Meridian of Obliquity. You
you may alſo try it by the Difference of Longitude in the fifth Column, 7 deg. 28 min.
in the foregoing Table, added to 27 deg. 32 min. makes 35 deg. as before.
4
981592
Ru L & XIII.
By the Latitude; and Difference of Longitude from the Obliquity given;
To find the Great Circles Diſtance from the Meridian of Obliquity-
As Radius, To Co-line of the Laticude 49 deg.:07 min.-
So is the Sine of the Difference of Longitude 35 deg. oo mini.----975859
To the Sine of the Diſtance 32 deg. 05 min.
-957451
Which is the Dift ange from the Meridian of the greateſt Obliguity: Then you may
proceed to the Rhomb next to be ſailed on.
Now if you add 2 deg. 49 min. to che former Rhomb 70 deg. ig mini ic makes 73
deg. 8 min, that iswis.w. half a Poins Wefterly, and the Difference of Latitude. 4. dega
rimin. which you may find by the foregoing Tables, or thus by the ninch Rule.
RULE
1
XT",
CHAP.XIT. By an Arch of a Great Circle.
199
RULE XIV.
As Radius, Is to Co-line of Difference of Latitude 4 deg. II m.--886301
Take out the Co-line of the Rhoinb 73 deg. o8 min.
946261
There remains the Sine of the Diſtance 14 deg. 34 min.
-940040
Which 14 deg. 34 min. converted into Leagues is 291 Leaguesſ the Diſtance you may
fail w. s.w.1a Point Weſterly, from Latiende 49.deg. 07 min. to Latitude 44 deg. 56
min. and the Difference of Longitude is so deg. as you may ſee by the former Table, by
the Lati:M:le 44 deg. 56 min. is 17 deg. 28 min. and the Difference of Longitade from
Obliguity is 45 deg. oo min. and the Longitudes according to my Globes made by Hon-
dius & drg. 24 min. and ſo work until you have calculated the Diſtance for every
Point, or rather half Point, becauſe of a Point cannot be well ſteered
well ſteered upon. It is
110 matter if you do not exactly keep the Arch of a Great Circle, for the Reaſons be-
fore given : but as neer to it as you may conveniently, as Mr. Norwood hath ſufficiently
anſwered in his ninth Problem of Sailing by a Great Circle.
But let me adviſe you to gee into your Latitude ſhort of the Place you are bound
to, for fear of miſtakes in your Reckoning, and ſo over-ſhoot your Port to a greater
diſgrace, than the credit of Great Circle Sailing will bring you; and then to know
whiar Diſtance you are to fail in that Parallel of Latitude,
Rui . XV.
By the Latitude, and Difference of Longitude given : To find the Di-
ſtance upon Courſe of Eaſt or Weſt,
As Radius, Is to Co-fine of the Latitude 13 deg. Io min. 998843
So is the Sine of Difference of Longitude 5 deg. so min. 894029
To the Sine of the true Diſtance 4 deg, 53 min.
892872
which is 97 Leagues š.
After you have made ſome progreſs in your Voyage, you may make uſe of theſe
moſt excellent Rules, whereby a Mariner may make liis Concluſion moſt certain.
Ru L I XVI.
By the Difference of Latitude, and Rhomb failed on ; To find the
Diſtance. You have it in the Fourteenth Rule before-going.
Ru L E XVIL
By the Latitude, and Diſtance failed spon an Eaſt or Weſt Courſe, To
find the Difference of Longitude ; AS 20 Leagues failed in Latitude
51 deg. 22 min.
As Radius, To Sine of the Diſtance i deg. 00 min.
1824185
Take away Co-line of the Laticude so deg, 22 min.
979541
Remains the Sine of Difference of Longitude 1 deg. 36 min.
844644
Therefore failing 20 Leagues Eaſt cr Welt in the Parallel of si deg. 22 min. the
Difference of Longitude made is I deg. 36 min. which is a good Rule
when you are in
the Latitude of the place you are bound co. In the firſt Rule you have how for to
find the Difference of Longitude from Oblignity, Chap. 13. and likewiſe in Rule 19.
where the Difference of Longitude is 35 deg. ſubſtract it from 27 deg. 32 min. and
the Remain will be the Difference of Longitude of two Places, one in Latitude si do
23 min. the other 49 deg. 7 min, as you have been directed before.
1
So
.
:
.
f
+
*
F
1
1
rallel-Meridians and Parallels of Latitude ; fo laying, che Index over, the ſecond place,
200
Geographical Sailing BOOK IV.
So that what hath been written will ſatisfie any ingenious Spirit, to make uſe of
theſe Rules in theſe four Scithations; and cheſe four will anſwer any thing required,
in all ſorts of Great Circle failing. I ſhall now make the blanis Mercator Plat; and
trace out the Arch of a Great Circle; and likewiſe thew how by Latitude and Longin
tude to find the Place of any Ship in Mercator's Chart.
CMA P. XVI.
How to make the moſt true Sea-Chart, and the uſe thereof in Mercator's
and Great Circle Sailing, called a General Chart.
Or the manner of the Diviſion, Ler the Aquater be drawn and divided, and
crolled with Parallel-Meridians, as before directed; only one Degree of
Longitude in the Particular Chart beforc-going, is 10 deg. of the Æquator of
this General Chart. You have been directed how to make the Meridian-line off che
Scale which is for a General Chart; and the ſame Rule makes this. Look in che Table
of Meridional parts, and you will find the Difference between the Aquator and
40 deg. of Latitude in the Meridian-line to be 874,2 which is 874 Leagues ande;
that divided by 20 is 43 deg. 42 min. of the Aguator , therefore take out of a
Scale of Equal parts, anſwerable to cach Degree of che Æquator, divided into 30s.
which you muſt reckon 200 Parts, take 874 Parts, and that Diſtance will reach from
the Águator to 40 deg. of Latinde. Always remember in a General Chart to omic
the laſt Figure in the Table, which is Tenths, tilin this 17
And for so deg. take 1758 ſuch Equal parts, which is 57 deg. 54 min. will reach
from the Aquator at Æ, to so deg. of Latitude in the Meridian-line: And ſo do
for any other Degree or Minute of Latitude, until you have made che Chart, as I
have done the Figure following.
How to make You may divide the Meridian- line by the Proje£tion in the Quadrant, making the
the Meridiana fide thereof anſwerable to 5 Degrees of the Æquator of the Chart, as in this
.
line by the for-
Suppoſe you would know the Diſtance betwixt 40 deg. and so deg. of Latitude, in
mer Geometri.
cal Proje&isu.
the Meridian-line of a Chart.-Take the middle of 1o deg. which is 45 deg. ouc of
that Latitude in the Quadrant, and it will reach from 40 to 50 deg. of Latitude in the
Chart, which you may ſoon try: And ſo work for any other Latitude.
A Scale of You have alſo there a Scale of Leagues for every. Parallel of Latitude, to meaſure
Lengues. any Diſtances in the Chart; for cycry Degree is 200 Leagues in this Chart, as you may
ícon apprehend, without more words, by the former Directions.
The Protracting The Protracting Quadrant (you may ſee the Figisre following) ſheivs you all at one
Quadrant. fight, without more ivords, how to make it by dividing it into eight Peints, and
cach Point into four Quarters, and an Arch within into 90 deg. anda Libal or Index
to be riverted to the Center, and a Hole drilled through the Rivet, to put a Pin through
the Center of the Quadrant upon any Place aſſigned, and let him, ſquare by the pas
the Limb of the Quadrant will fhew you the point
of the Compaſs
, and
whac Angle it
makes with the Meridian, or bearing of the firſt place from the ſecond, as we have
thewn by divers Arithmetical Rules, for your more certain and exact direction, how
to keep your Reckoning upon your Mercator's Chart, or Blank; and.co know.firſt
and afterwards what Rhomb you are to fail upon, keeping in or.nccr the Arch of a
Great Circle; and to know what Longitude and Latitude you are in, after ſome pro-
greſs madein your Voyage.
You ſhall have herealfo the way how you ſhall crace out the Arch of a Great Cir-
cle betwixt the Places in a Blank or Mercator's Plat, and how to prick down upon?
your Chart any Diſtances of Longitude and Latitude:
Before in the third Rule of Chap. 13. you have the way how to calculate for any
Degree and Minute of Obliquity, and any number of Degrees of Longitude; whác:
Degrees and Minutes of Latitaide the Great Circle ſhall paſs through; by the fames
Rules I have calculated, forcvery 10 Degrees of Longitude, reckoning from the greatce
Obliquity,
1
1
.
1
.
A
i
1
**
49
A Generall sea Chart According to Mercator.
North
HI
29
A
A
60
୧୨
G
Londy
H
Trondy
50
50
50
a
이도
​ot
OTE
Edust
Wert
40
olt
Foften this in with a Rivet to the Centor of the fuadrant A.
that it may turne upon it with a hole through the Rivet
40
The Index
min
130
Cancy
Tropicos
D
30
20
1
1
Barbados B
E
1
Barbada
b
ho
F
10
10
10
20
30
401
50
60
.yo
80
90
801
sol
30
50
20
10
Æ
sto
Æquino Etial Line
01
of
ot
20
t
200
150
100
South
ģ
Å scale of equall posts for the deviding the MeridianLine by the Table of Meridionallºparts
ano
071
AS
1
oopt
2600
gooo
2.800
33.00
3490
Page:200
1
4
4
4
F
1
|
4
4
1
;
:
:
}
上
​i
:
r
4
上
​4
i
1
|
{
1
1
4
ve
CH.XVHI. Sailing by the Arch of a Great Circle. 20
Obliquity,
7
20,
$ 60
í
+
:
!
to the Interfe&tion of the Great Circle with the Aguator, viz.
Greateſt Obliquity G 54 deg! 40 min
S 10-
30
For Deg: 'Eat.
400;
50.
254dos m. 52.458 SØ: 4.1 472 13 -42:12 :
70 80:27 590 deg. Longit.
35:13.25:46 13:10'. For
ço deg. Latitude.
In this following. Mercator's Plat, G ffandede at 54 deg. 40 min. of North Las
titude, and Æ. juſt 90 deg. of Longitude Weſtward, and £ N 90;deg. Eaſtward from
Ga-bcing the two oppoſite Paints in the Aguator, 180 deg. from cach. Point of Inn
terfe£tion ,
In every Meridian betwixt Grand-Æ, on both ſides a Meridian drawn at cvely ro
deg. of Longitude in the Charts make a mark at Latitude found in the Table; by the
Meridian-lirie.of the Chart; and having ſo marked every Meridián betwixt Gand ,
then by chefe marks ye may draw. Arches from one to another : but it will ſuffice
to draw Right Lines from Mark to Mark; as from G the greateſti Obliquity of the
Great Circle, to the next Meridian on cach ſide...and likewiſe to the next, until you
come down to the Æquator a¢ Æ on both ſides ; ſo have you pourtrayed on this
Mercator Plat a Great Circle Arch from Lundy to Barbadoes, one being in North
Latitude 51 deg. 22 min. at Ly the other at Bin North Latitude 13 deg. 10 min. with
Difference of Longitude 52 deg. 55 min. from Cunco. L, to C under B.
Note, To make a perfect Circle: the Latitudes of the Arch are the ſame on the
South ſide of the Aqualor, as you have found them on the North ſide. You
might have marked out only ſo much of the Great Circle from the firſt Latitudes as
you fcc I have done from the ſide of the Meridian-line, and Latitude of Lundy 5r deg.i
32 min. atl, to Latitude 13 deg. 10 min. at B, by the Difference of Longitude in the
laſt Column of the Table in Chap. 13. and Difference. you will find of Longitude is
52 deg: 55 min, by the former Direction: The Figure makes all plain in the Chart or
Blank
Two Places in one Latitude, as in the ſecond Soituation Latitude 5 I deg. 22 min. as
in the following Chari, one at H, and the other Place at Lin the ſame Latitude, and
Difference of Longitude 52 deg. Śs min. the neereſt Diſtance is trot upon a Parallel di-
rectly froin H to L, but from H to fail from 5 i deg. 22 min. by G the greateſt Obli-
quity, in Latitude 54 deg. 40 min. W. N.w. almoſt, then w.b. N. and w.b, S: W.
$.w. the other half from Gto L, which is the neereſt Diſtance by 42 Miles; for
the Diſtance by the Parallel is 660. Leagues, but by the Arch of a Great Circle is
buc 646 Leagues: And one would not think but the Parallel were the neerelt, to look
in thc Plat : but he that knows the Globe, conceives that by the Arch that goes neerer
the Poles, croſs the Meridians, to be the neerer ; therefore the Arch muſt be the neereſt
way. And ſailing into ſeveral Latitudes, you have the benefit to correct
your
Reckoning, which you cannot ſo well do by keeping a Parallel of Eaſt and Weſt.
Theſe Directions may be ſufficient for any Queſtions you will have any way in Great
But he that will take the pains, may find great delight in this ſort of Practice : Yet I
mult conclude, That although it is the necreſt way, it is not the convenientoft svay for
Seamer, for ſeveral Reaſons beſt known to them that keep an Account of the Ships way,
which I could lay down here ; but in regard it is needleſs, I leave every one to liis
mind, and ſhall thew you the way how I did
keep my Account at Sea, by the Plain
Coari and Mercator's Chart; and how to meaſure Diſtances in Mercator's Chart,
in ang Parallel alſo: which, if you have a better way, publiſh it, that others may
gain benefic by it; for you will not hurt me any way; but rather I deſirć, that
allthc Nauigators in England did exceed me (for His Majeſtie's ſake, whoſe Subjects
we are) and hope that the Neighbour-Nations will once know, Thac che Engliſh
Mariners are not leſs known in Art, than by their Courage, which the Dutch kuos
by dear-bought Experience.
Dd
CHAP .
:
Dead
Circle Sailing.
:1
1
:
+
1
2022
How to keep a Sea-Journal, "Book IV:
C H A P XVII.
How to keep a Sea-Journal, that ſo every Sea-man, Navigator, and Mari-
ner, may not be ashamed to Shen their Accoune to any Artiſt, and by is
benefit themſelves and others.
I
A
Would not have any ingenious Sea-Artiſt, that hàth a long time kept Account
of a Ship’s way, and hath been commander or Mate many years, to think I pre-
ſcribe him Rules, and to perſwade him ont of his beaten Rach (No, I think that
a hard matter.) But we preſcribe Rules for thoſe that are but new Learners, that ſo
they may have a perfect Micthod and Way of keeping Account of a Ships way at Sea;
that if the Master Thould perceive an Ingenious Praeticioner aboard, and by exami-
ning his Journal find him able, might at his recurn home give him encouragement,
by ſpeaking in his behalf to other Men to make him a Mate; and that is the way to
encourage Artiſts.: Buc I confeſs the greateſt Dunces have commonly the beſt Imploy-
meiits, and many abler men before the Maſt: which is great pity, that the deſerviag
Men had 110r their right. But what thall I ſay? There is ſú h an averſment in Face.
Therefore I ſhall procced to our fournal. I conceive it will be fit to have a Book in
Folio, that a sheet of paper makes but two Leafs, and to keep the left ſide of your
Book voids that you may write all the Pall ages of the Voyage ; that is to ſay, when
you let Sail, with what wind, and what Ships are in company with you, and how far
you keep company; what Storms, and how the Wind was : and likewiſe pur down the
time that you come by any misfortune, of crackiug or breaking a Maft or Yard, or if
any Men ſhould die; and alſo what Damage you receive by any Storm, and the like
Occurrences, as you ſhall think requiſite ; and what Currents and Variation you
mcee with. But before all this, put down the Title of the Voyage, Gver the left-hand
Page, incheſa or ſuch like Words, viz.
4 JOURNAL of our Intended VOYAGE by God's Aliſtance
from Kingrode-Port Briſtol in Latitude 51 deg. 30 min. to
the Illand of Madara in Latitude 32 deg. 10 min. and from
thence to Barbadoes in Latitude 13 deg. 10 min.
The right ſide of your Book throughout may be divided into 13 Columns, by Lines,
as you may ſee in the following Example.
In the firſt muſt be expreſſed the day of the Month, in the ſecond the Letter of the
Week-day that rear; put it once in the top of the Page : In thie third Column the
Months; make him large enough to put down the Latitudes you make by Obſerva-
tion of the Sun or Stars, and Currents, and how they ſer: In the fourch, the Courſe
ſteered by the Compaſs: In the fifth, the Variation of the Compaſs, if there be any;
or clſo the Variation by Currents, if there be any. Ser down the Angle of the Rhomb, it
made with the Meridian in the ſixth Column; and in the ſeventlı, che Diſtance failed
in Leigues or Miles: In che cighth, Ininch, centh, and elevencí Columns, fer down
the Northing, Somthing, Eaſting, and Westing: In the twelfth, the Latitude by Dead
Reckoning; and in the thirteenth Column, the Difference of Longitude from the firſt
Meridian, according to Mercator's Chart, or the Arch of a Great Circle, or a Polar
Chart or Globe.
.
...
The
*
}
}
{
|
生
​1
1
牛
​}
h
上
​}
d
十
​។
1
1
}
4
中
​1
1
d
1
1
1
*
"
)
25 a Ser fail out of Kingroad, in Company with the Fohn bound to Cales, and Ant
1666.
A Journal of orr Intended. Voyage, by God's Aſsiſtance, in the Good Shi
the Eliz. of B.S.S. Commander, from Kingrode in Latitude 5ı d. 3017
to Madara in Latitude 32 d. 30 m. and from thence to Barbadoes,
Latitude 13 d. 10 m.
March
bound to Virginia ; the Wind at E. N. E. thick rainy Weacher,
Month Days.
Week Days.
I
1
1
1
1
+
1
1
1
1
+
!
|
1
f
I
!
+
1
1
1
.
Diſtance 698 Leagues, Difference of Longitude 41 d. 40 m.
The Journal of our Intended Voyage
, by God's Allistance, in the C. (of B.) 5.$. Comman-
det, from Lundy, in Latitude 514.20 m. to the Iſland of Madara, in Latitude 32 d.30m.
Angle of Politión, or Courſe s. s. w.sd. W. Diſtance 411 Leagues, Meridian Diſtance
167 Leagues, Difference of Longitude 11 d. 16 m. from Madara to Barbadoes, in Latitude
Angle of Poſition or Courſe S. W.61 d. 14 m3. Diſtance 798 Leagues, Meridian
1
13 d. 10 m.
si deg. 20 min.
32 deg. 30 min.
'
5 East
ſervacion.
Deer.
48
28
1
----
283
SI
2910
1136 74
39 e 40 deg. 27 min.
36 03
1117 401
2 70
ΟΙ ΤΟ
I 20 10
April,
ſets Ebs.
I 2
16 0722
191
3 days
3 9 3 66
120 13
8 58
13 dog, io min. SWBW half w
Laricudc by Ob Cnurle by
Set laji March 25. SbW halt Wis deg: 30 min. ES W21 0.30 m.
38 deg. 30 min.
32 deg. 30 mio.
Secco th'Eaſtward Add up the Numbers, the ſun is
land almoſt a halt Madara Ifand bears Weft diftant
:72 .
127868, 43 min.
1c 18 deg. 57 min.
He 16 dek. so min.
14 deg, oz min. WSW hall wis d.30 m. Weft S W 69 d.zom. 46
14 deg. oz min. WSW half wis d.30 m. Wcft: W 69 d.30 m. 30
Ship is in Lat.Bar-Add up the Nambers, the lam is
badocs 6. lca. Ea. Difference of Laricude, Depart. from firft Merid. 1092
The Current ſec S.
1316 13 deg. SI min. Difference of Latitude, Depart. from Lundy
1131 Barbadoes ID and bears Welt diftant of Leag._
Courro
Degr. from the Diltan, Diff. La Dif. La. M. dep./M. dcp. 51 d. 20 Milt.LO.
SSW Variacion. Merid.SW 250. 411 1377 377 167 167 32 0.30 110.161
$ W 61 d.14 m. 998 1387 lea.387 lca. 698 698 13 d.109410.40
Dift.
Variation of Degrees from
Norib. Souch. Eaſt. Weft. Lac. dyp.f. of
Compaſs.
the Meridian. failed. North. Souch. Eaſt.
Compaſs.
dead R. Longit.
Min. Points. Degrees.
Dcgrces.
Leag.10 Lea.joo Lca.1oo Lca.100 L ca.100D. M.D. M.
44 351
18 37 49 07
Sb W halfw s deg.30 min. ES W 22 d.30 m. 49
4527
18 7546 51
(44 deg. 31 min: Sb Whalf w ls dee. 30 min. E'S W 22 d.20 m.
47 12
19 5244 30
Add up the Numbers, chc lum is
148
56 64
04 14!
Sis w 2 deg. 45 min.Els W. 25 d.15 m. 43
38 87
18:38142 34
SS W 1 degree Eaſt S W22 d.zom 46
42 50
17 60/40
26
SSW
o degr. Eaſt W:2d.30 m.
39
14 92 38 38
Add up the Numbers, the ſum is
I28
So go
Correáion by Obſervation
: 9
8
The ſum corre&cd is
1309
52 0038 3007 43
Difference of Latitude, Depart. from firſt Merid. 278
1:56 84
108 64
na Current SWbS fid.30 m. Cur." W 220.30 m. 43
43 42
17 99136 17
SWS 11 d.30 m. Cur. S W22 d.30 m. 45
41 57
17 22:34
SWOS 110,30 m. Cur. S W22 d.zo m. 42
38 30
134
123 79
SI 28
by the Current in Corre&tion by Obſervation
I SO
22 Leagues The fum corrected is
131
49 78132 3010 46
.
8.58
cftimat.
, *from firſt
376 97 167 00132 3011 16
swbs
s d. 30 m. East?
SW 28 d.30 m. 36
31 75
16 97 30
by Current.
55
SWIS 2 d. 45 m. Eaſt SW 300.45
W on
39 46
57
SWS
SW 33 4.45 m.
30
24 94
25 deg. 54 min. SW
SW 45 degr.
51
36 06
36 2625 55
min. SW
s W 45 degr.
60
42 43
42 4323 48
SW
IS W 45 degr. 28
19 80122
47
Numbers added, thc tum is
ISI
194 44
I55 58
Corre&ion by Obſervation
3
I 90 7
Sum corrected is
157 48 22 40
Difference of Latitude, Depart. from firſt Merid, 572
573 71 324 48
47
SWbw
oo degr.
IS W 56 d.15 m. 45
25 oo
37 42 21
SWbw oo dogr. SW 56 d.15 m. 47
26 II
39 0820 C7
Swbw oo degr. sw 56 d.15 m. 43
23 89
35 75 18 50
SwbW oo degr. SW 56 d.15 m. 44
24 44!
36 5817 43
SW6W on degr.
sw $60.15 m.
39
21 67
32 4316 37
nya upine Numbers, the ſum is
218
181 26
Corrc&ion by Obſervation
7
$
4 33
1 650
13
Sum corre&ted is
116 78
Difference of Latitude, Depart. from firſt Merid. 782 2 690 49
31 06
dc8.14 min. Ws w half wis degr, Welt 5 W 670.30 31
II 86
WSW half wisd.jom. WeS W 67d.30 m. 33
I: 63
II 48
Add up the Numbers, the ſum is
140
Correction by Obſervacion
5 2
6
Sim corre&cd is
55 57 7
Difference of Latitude, Depart. from firſt Marid. 921 4
746 6
38 50
13 deg. 48 min. Wbs half wi sdegr. half S W 84 d.zom. 55
54 73113 47
Wb S half W. degr. so min.S W 87 d. 11 m.
60
13 deg. 10 min. Wb S half woodeg. 60 min. S W 78 d.30 m: 51
5992|13 39
so 02/13 09
166
164 67
796 06
Well
go inin,
148 41
50.30'Cur.ſetS W 840.22 m.
39
WbN
380
38 6012
Is. 2 d. 45'Curls W 81d.30 m.
59
26
81
Ιο
1163
1764 -33
5
Set fail April 23.
from Madara.
46
23 65128
16 67127 42
23 d. A7
22deg. 40 min.
19 80
30
ISA
196 74
2018
( May
20
1
1
1
121 II
210 2
Star
174,7616
499 241
7
om.
m
17 Go
22
26 64116 151
42 5015
30 4914 44
27 7214 09
127 35
53 57
ܘܘ ܐ
8
4 80
145 2
132 1514 03.
9
lolo
631 32
5 39
2 94
995
18 28
1764 34
At
13 dog.
25 71113
1860 3713 10152:39
1
1
Place this bocween folio 20.. and folio 203.
L
는
​A
1
}
|
f
i
4
了
​}
}
{
1
1
4
1
1
1
1.
4
1
1
4
{
j
4
一
​4
4
十
​1
1
4
4
{
.
|
t
neer tbe Iribor
Miles in the Eaſt and Weft Column, into Degrees and Minutes of Longitude. I will
CHAP.XVII, How to keep a Sea-Journal. 203
Let this be our Example.
We will frame a Reckoning between che three Places before-mentioned, from Lien-
dy to Madara, from thence to Barbadoes, whoſe Diſtance in their Rhombs, and Diffen
peace of Latitude, and Meridian-diſtance, I have put over in the head of the left-
hand page, as you may ſec, anſwers to the words under. And in truth, I have found
theſe Diſtances very near the crytlı ; In two Voyages I differ but two Leagues, and
that I was ſhort. I worked it firſt out of a Mercator-Chart, and in Plain Sailing
took the Product of that Work for my Diſtance, and Meridian-diſtance, and Courſe,
as you have been alrcady thewn in the firſt Queſtion iu Mercator-Sailing
You ſee by the left-hand Page that we ſet ſail the 25th day; but we entred it
not in the right-hand Page until the 26th day at Noon: for it is to be underſtood, Not that I do
That ſince her ſetting fail March 25. to Noon of the 26th day, the Ship ſteers away affirm the Vari-
and makes her Way good on the S.b.w.w. Point of the Compaſs; but the. Varia- ation to be E.a-
tion being 5 deg. or half a Point to the Eaſtward, as you ſee in the fifth Colmen, Berly.for I know
therefore the point the hath made good upon is only S. W. 22 deg. 30 min. as is ex- weltli but being
preſſed in the ſixth Column: Upon this Rhomb the fails 48 Leagues, as in the ſeventh
Colamn appears : And anſwerable thereunto I find in the Traverſe-Table before-going, not, it ſerves to
the Somthing to be 44.1 Leagues, or by the Traverſe-Scale 444. Leagues; and the exemplife the
Weſting 18,?. Leagues by the Traverſe-Scale 18 4. Leagues, as here in the ninth and Rule, that being
eleventh Column appears by the Figures plainly ſee down. The Figures to the left hand the end for
fignifie Leagues in this Journal
, or Miles; and the two Figures to the right hand ſig- ample is made.
nific the 100 part of a League: Tle Southing being 44. Leagues, which is a deg.
13 min. neareſt; if chat be ſubſtracted from the Latitade from whence you came,
Lundy si deg. 20 min. it makes the Latitude the ship is in at Noon to be 49 deg. 07
miz. as appears in the twelfth Column. In the ſame manner, the ſecond entrance, being
the 27th of March, ſhewech, that from the 26th day at Noon, to che 27th day at
Noon, ſhe made her way good' upon the S.b.w. W. Point of the Compaſs; but the
Variation being 5 deg Eafterly, therefore the Angle of the Rhomb which the true
Meridian was from the South to the Weſtward S.W. 22 deg. 30 min. and failing 49
Leagues, the Southing is 45 ? Leagues, and che Weſting 187. Leagues: So the La-
titude is now 46 deg. 51 min. So the third Entrance is the 28th day, the Courſe and
Variation the ſame as before, and the Diſtance si Leagues ; the Southing 477
Leagues, the Weſting 19 Leagues : So the Latitude now is 44 deg. 30 min. You
muſt underſtand the like manner of working of all the reſt
. What hath been ſaid of
a Reckoning may ſuffice; but it is of very good uſe to ſet down the Longitæde in the laſt
Column, and a Resle how to convert the Eaſting and Weſting, that is, the Leagues or
give you this General Rule, that you inay do it neer enough, without any ſenſible Error,
on your Mercator Chart, or Polar Chart or Globe, provided theſe Rhombs differ noć
much one from another; by which Rule I found the Longitude for every Sum in the
Say then,
As the Difference of Latitude,
To the Deparcure from the Meridian :
So is the Difference of Latitude in Meridional parts,
To the Difference of Longitude in Leagues or Miles.
The Difference of Latitude in the South Column fumm'd up (as you muſt do as
often as you have any notable Difference betwixt your obſerved Latitude and Dead
Latitude) is 136.14. Leagues; omit the laſt Figure to the right hand #r, and then
The Departure from the Meridian in the Weft Column is 56.64; omit the laſt
Figure, it is 556: So you puc then down.
But if you fail
All on oueCostley
The Meridional parts for the Latitude si deg. 22 min. isa
12002
The Meridional parts for the Laticude 44 deg. 30 min. iso 9959
of Mercacor
The Difference of Latitude in Meridional parts is-
2043
Sailing
Dd 2
Say
Journal.
it will be 1361
the Rule is in
the tbird Proble
1
..
:
1
204
How to keep a Sea-Journal. Book IV.
:
Say thien,
90
al
:
As the Sum of the South Column, or Difference of Latitude 1367-313576
ds Radius
deg. To Tang Is to the Sum of the Weſt, vr Departure from the Meridian 566-275281
Rhomb: SO So is the Difference of Latitude in Meridional parts 2043 331026
difference of
Lar, in Merid,
606307
parcs, Te diffe.
To the Difference of Longitude in Leagues 84
292731
nence of Longi.
inde in Leagues which reduced into Degrees is 4 deg. 14 min. for the 28th of March. So ſtill you
or Atiles
muſt remember to take the Sum of Difference of Latitude and Departure from the firſt
Meridian. There are ſeveral other Rules you may ſcc laid down before, for a Paral-
lel-Courſe of Eaſt and Weſt, and other Rules to find the Longitude; as occafion re-
quires, you may make uſe of them : But this Rule ſaves you trouble, and comes neer
enough in ſailing ſeveral Courſes.
8 min. is But let us proceed with our Journal. I obſerved the Meridian Altitude of the Sun
21 Lea.neer the third day at Noon, that is from 30. at Noon co 31. I find my Latitudo by obſer-
1201 1: 520
vation 38 deg. 30 min. which, by Dead Reckoning it, is but 38 deg, 38 mln. ſo the Dif-
so 27:) 27
ference is 8 min. Southerly; but being aſſured of a good Obſervation, I correct the
3640
Dead Reckoning thereby, by this Rule of Proportion, ſaying,
1040
As th: Sum of the North Column corre&ted is 1201-
307954
14040
To the Sum of the Eaſt Column corrected 520-
271600
203
So is the aforeſaid Increaſing Southerly 27 cm
143136
24040 (1 :
414736
To the Inertaling Weſterly 7. Leagnesam
entre
106782
220T
Which is 1 League, and ſomething more, not to be taken notice of. This Rule of
Proportion Mr. Norwood hath laid down in page 111. of his Scaman's Pralkice, in the
Deſcription of his fournal in Miles, from Barmsadoes or Summer Iſlands to the Li-
Zard; which method I do in many things follow, but 110t all : But this Rule that I
propoſe is by the Traverſo-Scale, which I hold beſt, which is thus.
$
1
6
1
By the Traverſe-Scale,
Extend the compaſſes from the point made good in the laſt funining up, to the
number of Leagues or Miles Difference of Latitude by Obſervation, and by Dead Res-
koring, in the Line of Numbers; the famë Diſtance will reach from ſome points from
the Eaſt and Weſt, to the Difference of Eaſt or Weſt.
As for Example.
Excend the Compaſſes from 2 Peints and a little more (which was the sum of the
Courſe made good the 31 of March) unto 2 1. Leagues, which is 8 min. or there-
abouts in the Line of Numbers; the ſame Excene will reach from 6 Poists to i
Leagues and ſomething more in the Line of Numbers and that is the increaſing Weſterly.
You may alſo with the ſame Excent correct the Diſtance, if you put one Poot at W or
100 in the Line of Numbers, the other will reach to the Diſtance 27: Leagues corre-
Eted byObſervation, as you ſee I have done in the faxrnal.So you ſee, That underſtand-
ing perfectly the uſe of the Traverſe-Scale,you may do the ſame, and more readily,
as Mr. Norwood doth with his Table, to every Degree and Minute of the Quadrant,
without ſenſiblc Error.
Now this Difference being fonnd, I add therefore and put down in the South Co-
lumn the Difference 27. Leagues, and the Weſt Column 1 Leagues, and under
Diſtance 2 Leagues: Now the ſame corrected is by obſervation 130. Leagues,
Diſtance 120 **. Leag. Sossthing and Wefing 52 Leasi 8 min. ſubſtrated" from the
Dead Latitude,make 38. deg. 30 min. che truc corrected Latitude according to obſerva-
E
1
cion:
+
et
CHAP.XVII: How to prick down your.Reckonings, 205
I
it must be corre-
tion: Then I ſum up the firſt Sems of the 28 of March, and this Sun corrected 3r
of March together, and you have the Diſtance 278 Leagues, Difference of Latituido This is becauſe
256. Leagues, and Departure 108. Leagues, and by the Rules before-giren find way self so
256 Leagsses Sosthing, and 108 Leagues weffing; and with the Difference in Me: timae in terms to
ridional Leegues 364 %. Leng. I find the Differente of Longitude in Leagues 554 the Souirware
Leagues, converted into deg. and min. is 7 deg 43 min.
by II ms.than by
In like manner, upon the third of April I thould be in Latitude 32 deg. 19 min. Dead Receox-
but by very good obſervation, I find the ship in the Latitude 32 deg: 30 min, that is, 113.6m tiverefore
not ſo much Southerly by 11 minutes : therefore to correct it by Obſervation, I put ward in
under Diſtance 3 ? Leagues
, and in the South Column 3* Leagises, and in the East Leag.orgo mich
To Leagues, and under Dead Latitude 11 min. I ſubſtract the corrested Difference to the latwra,
of Diſtance our of the Sum over it
, and likewiſe the corrected Difference in the North by reason my
Column out of the Sum in the South, and likewiſe the Eaſt out of the West Column, Courſe is in the
and add the 11 min. to the Dead Latitude, and then you have the Sum corrected; but
S.11. Qxarters
if there be any Current, you may ſet it down, and allow for it, and note it down, as sted in the cosa
is that Example following the firſt of April to the third, and by your Traverſe-Scale traiy Quarter.
preſently find how much the Current hath fet you to the Eaſtward.
But if your Courſe be ncer the Eaft and Weſt, it is ſufficient to correct it in Lati- In Sailing East
tude only, as in the Example of the 12ch and 13th of May; for in that Caſe the and West, you
Longitude canaot be corrected but from ſome further ground. Now to ſer down this have a Kule ir
Reckoning upon the Plain Chart, or common Sea-Charš, it is ncedleſs and inncccffary : Probl.7.f fai!
The becter
way is to ſet down every one of the Sumns as they are corrected by Obferva-*3.9Mehr
tion, in the ſame manner as you are directed in the larger end of the third Chapter of C1,23
this Book; and ſo by the coral sums of the Difference of Latitude and Departure from
tlic fuft Meristian, or Latitude and Meridian diſtance, you may ſet it down on your
Draught or Chart as often as you pleaſe with eaſe.
Now to ſet off every Sum corrected in Degrees of Latitude and Leagues of Longi-
tude, you have a Scale of Leagues or Miles for that very purpoſc, and Directions how
to do it, in the ninth chapter of this Book: But if you are deſirous to ſec down your
Reckoning in a Mercator or Mr.Wright's Chars, or in the Polar Chart, you have in the
12th and 13th, or laſt two Columens of your Journal, chc ſubſtance and principal ſcope
of your Reckoning ſet down as often as you ſum up or correct your Reckoning : name-
ly, your Latitude and Longituds; which whenfocver you have a deſire to ſee down in
the foreſaid Chart, or any other graduated Chart, with Degrees of Longitude and La-
titude, you may readily do it.
As for Example. Suppoſe I would ſet down the plat of the aforeſaid Juurnal
from the 25th of March to the 13th of May, I and the Latitude againſt the 25th
of March'sı deg. 30 min. and the Latitude of the Barbadoes 13 deg. 10 min. and
the Difference of Longitude 52 deg. 35 min. Therefore in the Latitude of 13 deg. 10
min. I'draw or point out an occult Parallel, and reckon 52 deg: 35 min. from the
Ifand Lundy towards che Welt: I draw by chat Longitude an occulc Meridian; the I hope this way
Interſection of this Meridian with the foreſaid Parallel is the point repreſenting Bar. will find goo.
badoes, or the Place of the Ship; and the like is to be underſtood of any of the other : acceptative with
And ſo I put down in the General Chare of Mercator the 8 Points of the ship's Place,
a; 2 b, 36, 40, 5c, 6 f, 75, 8h, as there you may ſee. This forin of keep-
ing a Reckoning is the moſt fit and agreeable of all others as I have ſeen or heard of, to
all ſorts of Charts, Maps, or the Globe ic felf, and to all kinds and ways of Sailing
1
}
)
the ingenious
Aldrinei cr
Arift.
whatſoever.
1
1
!
2
1
1
CHAP
:
1
1
1
í
.
206
How to find the Latitude, 6c. Book IV.
1
1
CA A P. XVIII.
A Deſcription of the following Table of the Latitude and Longicude of
Places, and the way how to find both.
T
}
nent
+
Y
HE ancient Geographers, from Ptolomy's time downward, reckoned the
Longitude of Places from the Meridian, which pafleth chrough the Calo
Verde Iſlands; and others have the beginning at the Canary Iſlands ; and
Jodocus Hondius beginneth at the Ife Pico one of the Azores; and Mr. Emery Mulli-
doth account the Longitude from the Weftermoſt parts of St. Michael's, another
Mand of the Azores: who, albeic they differ greatly in reſpect of the beginning of
cach of their ſeveral Longitsides, they come all to a neer agreement for their Difference
of Longitude from any particular Meridian or Place: And for the exact ſetling of La-
titudes, we have many ccrtain helps; but the Longitude of Meridians hath ftill wca-
ried the moſt able Maſters of Geography. By Latitude and Longitude the Geographers
ſtrive to repreſent the Parts of the Earth, that they may keep Symmetry and Harmony
with the Whole.
Now the Longitude of any Place is defined to be an Arch or Portion of the Ægsi-
noctial, incercepced between che Meridian of any Place aſſigned; as the Meridian
that paſſeth through the Lizard, the moſt Southern Land of England, or any other
Place from whence the Longitude of Places is wont to be determined. Many have en-
deavoured to ſcc down divers ways how to find by obſervation the Difference of
Longitude of Places; but the moſt certain way of all for this purpoſe, is confeſſed by
all Lcarned Writers to be by the Eclipſes of the Moon: But now theſe Eclipſes hap-
pen buc feldom, and are yet more ſeldom and in very few places obſerved by the
Skilful Artiſt in this Science; ſo that (ſome there are) but very few Longitudes of Pla-
ces deſigned out by theſc means.
If you would know how to find out the Longitude of any Place by the Eclipſe of the
'Moon, you muſt firſt ger fome Ephemerides, as the Praćtick Tables, or Mr. Vincerit
wing's Directions in his Harmonicon Cælefte, pag. 150. or any other Learned Mathe-
maticians Calculation, and ſee what hour ſuch an Eclipſe of the Moon ſhall happen
at that place for which the faid Tables were made; chen afterwards you muſt ob-
ſerve the ſame Eclipſe in that place whole Longitude you deſire to know. Now
if the time of the Eclipſe agree with that other for which the Tables were made, then
you may conclude, that both Places have the ſame Longitude, and are ſituated under
the ſame Meridian. But if che number of the Hours be more than the Place you are in
is ſcituare Eaſtward, you muſt therefore ſubſtract the leſs Number out of the greater,
and the Remainer muſt be converted into Degrees and Minutes.
2
6
1
+
1
I
+
1
M 톡
​A TABLE
1
1
1
:
CHAP.XVIII, 4K*. .
207
t
:
A
7
im
TL
TAB L E
1
1
}
indi
!
OF THE
LONGITUDE and LATITUDE
Of the moſt Notable Places,
F
t
That is,
+
1
HARBOURS, HE A D-LANDS, and ISLANDS
!
OF THE
1
1
W ORL D.
Newly Corrected, and Compoſed after a new
manner, by beginning the ſaid Longitude at
the Meridian of the moſt Southern Port of
England, the Lizard.
1
.
By Capt. SAMUEL STURMy Math.
)
'
I
21
54 Cape Brittan
42 2161
The Sea-Coast of Newfound-Land, and
North | WA
New-England.
The Places Names. Latitude Longit.
D. M.D. M.
North Weſt Cape Raya
48 05'52 49
The Places Names. Latitude Longit. Cape Deganica
54 01:53
D. M.D. M. New-Eng.Cape S.Charles 52 4852
23
Cape Honblanto
52
11 50 841
46 01 52 57
Belile
SI 0248 44 Cape Salila
43 4655 · 22
Cape Bonaviſta
32
49 1947 42 Capo Codde
5449 04
Boſton
42 3964 36
40146 55 Plymouth
42 07.62 35
Nantucker
21 47 49
41 0860 17
Cape St. Francis
41 1761
Cape Daſpaire
48 0147 27 Martins Tinyard
47 36 46 03
Cape de Raca
46 27/46 30 The Sea-Coafts on the main Continent in A-
Bay Bulls
47 28 47 merica, or Welt-India.
St. John's Harbour
47, 47147
47 41
32147 411 The Places Names.
The Places Names. Latitud. Longit.
Gape St. Larinſo
47 10 48 52
D. MD. M.
47 36150 18 Elizabeth Iland 41 0262 04
Bloik
Trinity Bay
Bacalao Iſland
Conſumption Bay
48
48
!
IZ
II
Plalancia Bay
21
Iſland St. Paly
208- A Table of the Lat, and Long of the Book IV.
II
I 2
Cape Hatcrafs
Cape Fara
12 5054
16 00155
16. 20155.
5054 28
31
8
00:12
SEVAC!
57157 28
12
IL
I 2
II
II
1
12
20
I 2
12
I 2
1
TY
1
North West
North Well
The Places Names. Latitud. Longit.
The Places Names.
Latitud. Longit.
D. M.D. M.
D. M.D. M.
Bloik Iſland
40 $562 36-L Barbadus
13 10152 58
Long Iſland
40 45163 16 Tobago
12153 06
Cape May
5364 45 Point Degällaia
IC 45153 31
Virginia. Cape Charles zqu-43165 26- Gianada
10154 32
Cape Henry
37 0.65 38 St. Vincent
35 49651 46 Guardadupa
34. 0669 56 Monfariat
2015 41
Cape de Catocha 27 2380 37: Mayeś.
1700 36 37
Cape de Camaron
16 05176
05176 19
St. Criſtova
17 30 56 45
Cape de Gracias
I5 1311201.56 Hand Devas
IS 57157
Cattcrgaine
io 24121062 Inand Blanco i
20156 52
Bay Tonto
1062 36 Margaita
28 56 37
Cape St. Roman
55 60 36 Turtuga
3057 40
Cape Dacodara II 08156
08156 38 Iſland Derickilla 12 19158 01
Cape Trag, or 3 Points 1755. 41 Boca
12 1958 53
Cape Brama
09:21 54' 16 Illand Deavos
29159
Cape Dasbalſas 08
20153 ir Bonoga
3260 54
Saramo
06 09 50 56 Quiſlai :
25,60 39
Suranam
05 58149 52 Moagos
20161 55
3
Eaft end of Hiſpaniola : 18 4762 38
Middle of Hiſpaniola 18 30 64 58
The Weft-India Inands. . Weſt end of Hiſpaniola 18 25,68 26
Eaft end of Jamaica 18
20171. 58
North
18
Weft Jamaica Harbour
1572 52
The Places Names Latitud. Longit. Weft end of Jamaicz! 18 38 24 57
D. M.D. M. The Eaſt end of Cuba - 22 0075 56
Defonffcaca Ifand": 3:32 33148, 30.
Caimanis
19 4177 41
La Burmuda
32 25156 00
19 21178 45
Beliama
27 5773 06 Santavillä
172877 50
Tcavis
27 277! 04
14 50176 04
Sigvatro
26 18168
Guanabo
16 3381
3318 19
45
Guacro.no
25 47168, o Guanabimo
Guamjua
25 1516453. Cozumál
19 25 84 56
Tiango
Laſallecranas
.-30
00/87 58
Guanahimo
23 5066 39 The Iſland Delas 23 30191 58
Mayagnana
23 05166 SI
Abraio
· 25 50 94 00
22 05164.31
Labarmaia
Amiana
4064 38 Iſand Dearanas 22 30193 14
Inagua
21
19.69 03 Triango
33193 05
Yamatta!
22 32167 49
Zarka
50193 00
Soamia
24 20168
50 The Ipand of Proudancol3 2781 16
25 10 71
1071 30
St. Andrea
12 42180 57
Yamia
24
St. Jolin
18 30160
42
Santa Cruce
17 4259 18
The Sea-coaſts of Brazilia.
Anguilla
18 4857
St. Martin
18 3556 47
South , Weft
St. Bartolama
18 IS156 33
The Places Names. Latitud. Longit.
Barbada
17 678 55 39
D. M.D. M.
Antego
16 32154 52 The River Amazones
00 00 41. 30
Dallijada
16 0054 36 | The Iſland of St. Paul oo 5514 36
Marigallatita
15. 41155 26 The Iland of Aſcenſion 07 48/2006
Dominica
15
00155 05 | Cape Blanco
02 25 22 29
Matralina
14
20 54 44 Inand Rocas
03 42 17
St. Lucia
13 3054 43 Iſland Farnando 03 folis 16
00 Grand Caiman
A
04 Moſquito
16 10 83 -04
22
24 33166
Caycoſs
22
55/93 16
21
21
20
Javaqua
22170
IO
00
Maria
42]17 16
Abratho
1
1
7
CHAP.XVIII. Chjefeſt Places of the World, I
30119 01
08 0011!444
II
II
I 2
101114_44
Oc1114 41
00 114 39
02
II
21
0905191
53191 07
10
II 12.90043
35
I2
}
IT
I 2
1271 44
I
I
Aduose
56 | Nurth end of Sumatra 05' 281116 3.5
, .
209
South Weft
North Enft
The Places Names. Latitud. Longit. The Places Names. - Latitudi Longit.
D. M.D.. M.
D. M.D. M.
Abracho
05 00117 56
Cape St. Raphall 06 10119 36
36 | Gomafpala
05 401116 29
Cape St. Auguſtin 08.25.18 28 Niobar
07 00-115 04
River St. Mignall
Illand Defombro
og
The River Roall
21 20 41 INand Ruſta
09 50114 33
River Gianda.
14. 49122. 06 | Quiarinibar
Cape.de Abeorho
17
52/21 42 Chitra Andomaio
St. Harbara
18 1121 06 Inand Dandemajo
13
Illand Aſcenſion
17 1917 01 Illand Decocols
1.14 39 115 12
Trinidada
19
50114. 24 Celloan
07.:: 50198 39
St. Maria Dagaſta 19.38 12 14
14 | Doda Safia
og 40193
Iſland de Marcin 19
00:08:03 Andaio
30190 SI
Illand de Pidos
25520s 51. Garine
w 19:59 20:56
Cape-St. Toma
47123 38 Moique
29
Cape Frio
Cuballa
23.5.224.. 43
08
Cape Sc. Mariz
00 37 :II
Illand de Profoll
239056
River de Platta :
35 5045:52 Inand de Zoclia
Port St. Juliano 50 00152 30 Chorebaman
12.321870:55
The Screights of. Ma-?
Sucatra
5:3 3056 30
1874 oz
gellane
Abdelcari
Cape, de Sancto Spirito 52 2058 30 Apoluria .
09 S.20190550
Cape Victoria, West 2
11 os 39188901:50.
end of the Sareights 352 30165 * 4*
Degomo
02 40871:35
12 0080 30 Piedros Blanco
Cape St. Frainico OL: 30180. 30
Ser
03 $279.alias
CapciSt. Frances, : North Weſt
Domes Caicuhas. 03 211 6573124
Poincde bon matre 07 3880.00 Iland Quclalla 03 40/64 bao
Nombre de Dips: W.Sesió 0079:-130 De Almiranta . 03 57163 -28
Nova Albion,or New-
Agnalaga
09 006615
England, in the
Aldore Has
46 00162 30
,:!,09 05165 02
Soxtb Sea, the back
John.de Nava
Orton, 09 163,26
fide of it
Calmobodo
og 40 61:1:5
Gape de Fortuna}s's 30 i 30.00
Donatallo
OS
(Anjar. frais
Aignose!
30158 18
Infulz Salamonis So. Lat.w.Long. John de Comoro
09:00 57..20
Nombre de Jeſus
os 50169 30 | Pemba,
12 : 05 09530530
Tapan Infules 36N.00 153E00 | Zanziba
06 261153 39
Cape de Bucnz Deſco...ols.00155.00 Manfia: ...o
07 501551508
John Demiz Es
10 481549,24
TI
20551448
po
*
Mohalla
12: TII56:25
Foannada
12 09157: 03.
- The Eaft India Inanders
12:49 57-55
St. Chriſtopher, a
14 2089115) 303
۲ P
SPOTI
Southi b: Eaſt
John de Nova co
»7 2015.5362251
c: The Pilates Names
. Latitud. Longit: | Ballas de India:
1.2.
Lau-
25
37160
co
DMD, M.
Calcio
Hipon Inand
06, 1-45/1 29 20
St. A pohima
?
20 50 49:119505
Bantam, Eaſt
India
06:15 115 34 Dom(caicahas
20:25,816 9:54
Janba Iklandsi I
49.12.25 Moroſlars
2001IP 68,344
south enth of Sumatra
Q5:06:52 1:25:48 Dolgarias
,
IS.929.7043
Middlaendsof Şumatra 01,30 1:20:49 | $a Branda :
17 13.1741 44
EC
Englands
Lima Cape
06, 1076 155
08 3017811315
01
o
IT
DI
205916:57
09 30
Comoro.co
II
1
61
Mayacxa 21
.
0!
:
22
10 ss.122
do Co., O
>>
$
OO
!
1
ПЕ
*
mat
210
ATable of the Lat. and Long of the Book IV.
+
4
1
20
1
1
I@
IO
CI
ΟΙ
20
18
15 33172.-39
South Eaſt
The Places Names... Latitud. Longit.
D. M.D. M.
Englands Foreſt
5071 14
Diego Roize
20 05174 54 | The Sea-coaſt from Cape Bone Eſprance
John de Lisbon 25 2468 32 to Guiney.
Romoras
28
21
South | Eaſt
The Places Names. Latitud. Longit.
D. M.D. M.
Illand Defiftian 36 57 11 44
The Sea-coaſt on the Main Continent in the Inand Degiaiatica 37 56114.04
Eaſt India.
Cape Agullas
36
20133 54
Cape Bonee ſprance 35 50132 54
North | Eaft Cape Sacos
29 4030 14
The Places Names. Latitud. Longit.
Aſcention Iſland 07 48105 24
St. Elana
16 0310 68
D. M.D. M
St. Elana Nova 16 03.19. 48
Malacca :
or 41116 14
Baſſas
17 45/27 35
Queda
06 47119 44 Cape Lado
ooj29 13
River de Care
45118 54 | Cape Padron:
06 0029 "04
River Bongale
22 09128 33 Cape Lopas
00 2521
Aicopoir
19|112 39
Anabona Iland
22 22 56
Samnabronza
30108 52 Inand St-Mathaos OI, 40107; 45
Arme GOD
14 3510027 Idand St. Toma
Idand St. Toma SooN.10 23: 34
Naga Patam
Iſland Chocos
21199 59
II
16.00 40123 So
Cape Comorin
07
7 50j97 39 River Gaboan
10127 12:16
Cochin
OI
09 40 97 29 River de Angai
Callant
10 48197.27
Illand de Principas 01
SO125 14
Mongalar
I 40197:19
Iſland Defarnanda 03 10126 06
Dodalle!
17
River Boilin
02 4227 29
Goa
14 40 971 01 River Decainaronas 04.00 27.:09
Chaul)
18
10198 51
Calecatini Eaſt India II
II 30 9258
Macao tn the K:of Pegu 19:30112' 49
Domon;
19
5499 01
Surrati,
21 00 99 36 The Sea-coaſt from Samſons River to the
Dio :
20 4896 57 River of Gambo, Corft Guiney and
River Decinda
24 55195 39
Barbary.
Gudaries
24 5089-28
Cape Muchoaridan
25 32 82: 39
North | Eaſt
Cape Ruflallgat
07 84 39 The Places Names. Latitad. Longit:
18 19 79 09.
D. M.D. M.
Dofari's
17 00175-04 Old Callabar 04 50125..15
Nero Callabar
04 40123 37
Adon?
13 08/66,58 Cape Formoſſus 04 03 22 52
Cape Gaardafuy II. 4077* 24 River Binhin 14 06: S0122
Cape de Baflos
0430165 - 19 River Dallagoa
07 4019 49
Magadox 0
02 3059-124 River de Valta'. 06 05 16 52
Molinda!?!
02 5.42 5277 11 Capo g Points
***04 1013
20152
05
OT
River St. Andraſs
os 23108 06
Cape Fallco
o8 02 52:24 Cape de Palmas
04 40 06 05
Dagnade op
15 17153.26 River de Caſto
os 20103 48
Cape Corintes
23 30 48 st
30 48 st | Cape Mounta ob 23 or. 145
Cape St. Marina
25 4046 59 Cape Roxo
II:381613
River St. Lufſcais
28
254609 River of Gambo
12 :17
33 18:43' 59 | Cape de Verd - 14 2416:57
tion
The
00
00127 30
0198 55
.
(
7
1
(
1
22
Cape de Ponto
; !
Cape de Matriaia
h
12
το
Tangga
I 20' 472
Bay Doliagoa
ܐܰ܂ ܙ܆
1
1
+
7
Chap.XVIII. Chiefeſt Places of the World. I
211
1
00
00
1οΙο
16 Chercune
01 Cape Mizarrara
32/14 56 | Cape de Solli
16 Cape Ruſſurta
07 Cape Roartini
30 40 41
I 2
Varo
Vigia
Abrogo
Brava
Fogo
22
02
A
30:9:26
North Eaft
The Places Names. Latitud. Longit.
D. M.D.M.
Stora
39 0915 44
Bona
The Cape de Verd Iſlands.
37 1916 44
Bozarat
37 30 17 50
North Weſt
36 50117. 24
The Places Names. Latitud. Longit
. | Cape Beun
37 0518.11
Sulía
36 02118 07
D. M.D. M.
Britto
35 23 18 41
Abrollio
00116 16
34 5618 36
Vigia
03
32 18124
St. Paul
or
31 1125 49
Rochaſs
02 30116 16
32 58129 26
1 12121 07
32 1813:2
1 81 3-2 04
2619 48 Alexandria
28
16 36 10 19
Michallat
30 30141 55
Cairo
14 49 15 14
30 35 41 44
14 42 14 56 Joppa
31 42143 39
Lantiago
14 52114
Maya
Is 0014
Bouaniſta
IS 5813 49
Sall
17 00113 26
St. Nicholas
16 30 15 44 The Coaſt from Antiochia to Sagua in the
Sz. Lucia
16 50116 08
Straights.
St. Tincant
16 55116 32
St. Antonio
17 07116 55
North Eaſt
Cape Blanco
20
The Places Names. Latitud. Longit:
Cape Boyadojo 26 5512
52
D. M.D. MA
Marquepana
27
From Antiochia to Sagua 34 54146 , 26
28
46 Cape Pollopolla 35 35144
29 501
5011 29
Cape Seridioni
35 5541 24
30 30. I 25 Cape Decoxman
36 16 36 44
Cape Cancin
32 271 I
271. oo
06 Cape Babarnau
37 5836 .22
Tangiar Enft Longit. 35 36 1 49 | Land Miri
39 12137 05
Incomodio
40 26 40 39
Conſtantinople 40 50 40 33
Gallippollo
40
20137 59
Cape Degriffa
40
12137
The S82-Coaſt on the Main, from Tangiar Cape Pimra 40 26/32 23
to Joppa in the Straights. Cape St. George 39 28/32 19
37 4032 35
North East
37 15 31
T531 52
The Places Names. Latitud. Longit. Cape Macapan
im
36 28 30. 53
Caſtelcornis
37. 45130 32
A
D. M.D. M. Drugromaſtra 38 3830 40
Wadallo in the Straights 34 57 3 13 Cape Linga
40
18:29 38
34 571.3 48 Hirafla
40 57 30. 38
Qran,
35 467 59 Antavara
41 49 30 54
Tanis
36 30 9 18 Caccaro
42
Sally
36 40 9 48 Raguſſa
43
Argier
36 40 1
40110' 54 Stanio
42 5728 :
Tádallis
36 48111 25
Trovor
43 30 27 37
Ragin
36 501 44 | Cape Caſta
43 27 26: 48
Gion
36 50 13
50113 30
Salconico
44
Gigaria
37 03 14
44 05:26:39
Colla
37 oglis - 14 | Sagua
44 47125:49
E e 3
22:2
II
521 1
7
Cape Denao
Cape Gillain
Cape de Garro
!
II
Cape Collo
Cape Sille
Ballis
년
​then
21129: 41
2928 57
II
E
01\27: 24
03 14 26 Zaro
From
1
+
!
212
A Table of the Lat: and Long. of the Book IV
38
Cape Fiſtria
02
Gállipoli
+
1
Sallaurio..
22 29 14
40
41
52116
11.08
42
North | Eaſt
The Places Names. Lasitade Longit.
D. M.D. M.
St. Penaga
38 5233 03
Andrea
12/34 46
From Cape Fiſtria to Giblitore.
Ipſava
38 28135
II
Mortalin
North Eaft
38 54 35 52
The Places Names. Latitud. Longit.
Sravifratta
39 28 35 14
Lamnofs.
394135 49
D. M.D.M.
Embroſs
40 09136
44 4024 IO
09
Venecia
Palamos
14136 04
40 14136
45 37 22
22 45
Tafla.
Gorro
40 0035 19
44 57121 35
Giavocha
44 1921 26
Ancana
Iſlands in the Straighies.
43 25124. 29
Angollo
41 3127
Cape St. Maffa
Sapiencia
36 47130 1.6
39 52 26 53
Scouty
40 0821 04
37 10129 43
Zane :
Cape Callom
38 50 25 40
37 37 ?9 38
Cape Sparta-venta
Cape Şidro
38 15 29.33
37 40124 16
Pollicaſtro
Paxa
38 49/29 30
40 0824 02
Corfu
39 26/29 26
40 S1123 32
Faimo,
Napolis
41 08122
39 4429 19
SI
Sellino
Rome
41 Soj21
501210g
Ciritaclia
Pianaſſa
46 20:24
41 46 20
Trinitc
Lcagueorne
43 2819 03
41. Sol?5 - 53
.
Cape Malle
Pollagafia
43 S115 37
17126
37.
Mallida
42 37/28
Cape Larci
42 58114 38
0:3
Corſella
Tallone
42 35177. 38
43 0014 04
Marſilia
43 12 12 44
Aguſta
42 36/27 13
Catllalla
42 40127 93
Cape Degofrito 41 41 II 18
Catrio
Cape Pallomallo
42 44 76 58
10 8 39
40
Lilla
43
0026
Cape Martin
38 46 820
33
Buzo;
43 02/26
Allagant
38 2017 14
$t, Andrea
43 0725
0712558
Cape Paul
37 28:7 "II
Cape Degac
36 471 5:24 Iſland Groſſo
Poma, la
43., 14125 48
Välis
44
36 491 3 33
Malagolc
Sauſſaga
44
Giblitore
36,401 2 06
35 52/22
IS
Malta
36 00/21
54
Comino
36. 15, 21: :24
Eaft end of Cyprus 34 48144 18
$:
Middle of Cyprus 34
I i
Weſt End of Cyprus 34
The Iflands in the Straights, called the Rhodes
35 4037 22
Archipelago.
an
Sivia
36 0537 17
:
:
34 37 38 39
North Eaſt
Eats Eaft end of Candia
32
The Places Numes. Latitud. Longit? Midąle of Candia 35
D. M. D. Mwest endilo Cajidia
35
Sarfanto
-36 57 33.39 Scarpaataa
35
1036
Sapfo .
37 17133
35 35135
Eamania
37 28!33 18 Langa .
36 3.1!36
Trava
37 49 34 41 Stampalią
341.552
Pipor
39 32133 51 Leyatta n.
36 38 34:54
Laffor:
39 58 33 51 Niza !!
2134 135
Laino....
39 44'33 38 Cavari
:
36 403.2018
Střipo:
39 1633 51
51 | Palla?!!
36 52132 09
!
?
18
002536
2025
36 45 3:07 Piper
03
T
18 43 09
22 41 41
>
1
35:04
0413:5.
mmmmmmm
08133.5
1532, 75
04
22
Carose :
133
27
36 11
37
I.
r
.
Cardinal
1
1
CHAP.XVIII. Chiefeſt Places of the World.
213
1.
North Eaſt
The Places Names. Latitud. Longit.
Cardinals Hats
Forlconari
។
Maſſina
21
09 10
28 58 10 42
38
38
38
38
34/22
38
1
4
Lcja
40 46 21
36
1
5
D. MJD. M.
The Canary Iſlands.
37 2532 31
36 5932 37
North West
Millo
36 4033 19 The Places Names. Latitud. Longit.
Goza
35 4120 SI
Samatto
35 46119 39
D. M.D. M.
Lampadoſla
35 5819 59
59 Forta Ventura 28 12106
Linofía
12106 28
36 20 20 os Sanſlorrotca
28 51/06 08
Pantalaria
36 53119 35
35 | Allegranſſa
29 11106
1110,6.13
Kambro
37 10 18 · 34 Grand Canary
Maritimo
27 43 08 3€
37 5220
02 Tenarife
28
20109
28
38 0723 Gomara
28
IS
Eaſt end of Sicilia 37 07/23. 24 Faro
28
osio 43
Middle of Sicilia
37 4222 09
09 Palma
Welt end of Sicilia 37 52/2007
07 Salvagas
30 05108 57
Uſtica
50125 58
58 Dazarts
32 08 09 46
Allicur
45/21 47
47 Madara
32 27 11 19
FaHicur
43122
Of
Por. Santo
33 14110 09
Liſtallin
32
Lipari
40122 27
Võlcana
38 48 22 30
Stroinballo
39 3123
03123 02
Foldemaſlina
38 2023
20123 29
The Weſtern Iſlands.
Ponuſla
yö 40120
40120 32
Palmarolla
North Weſt
40 50119 59
Ginnute
41 5919 48
The Places Names.
Latitud. Longit.
Gigig
41' 58119 39
Criſta
41 55118 51
D. M.D. M.
Planoſſa
42 07/18 33 Abraoſo
37 55129 46
Lilbo
42 31118 36 Valo
40 3027 28
:
Caprera
42 58 18 20 Corvo
40 09 24 59
Gargona
43 20 18 28 Flauris
39 30124 55
Northend of Corſica
42 551!7 26 Fiall
38 49 22 13
Middle of Corſica 42 05117 07
Pico
38 30 21 37
St. George
Southend of Corſica 41 2017
39 002.1 20
Taloro,
40 5617 19 | Traſſara
39 3110
Azancra
41 08116 22 Gratiaſſa
39 30121
Worth end of Sardinia. 41 10117
10117 25 | Abrajo
39 52 78 5.
diddle of Sardinia.
40
06 16 54 Vajo
38 43118 23
South end of Sardinia. 38 5616 37 St. Michael
The Ifand of St. Pedra39 2016 03 Hornisgo
Pallmade folla. 39 IljL6
05. | St. Maria
37 06/17 56
Sarpentara
39 00117 18 Vejo!
22 28 56
43
Callatta
37 57 16 28 Illánd Varda
48
Minorke
39 55112 16 Maiden Iſland 46 30123 36
Mayorka
39 38'ir 12 l'Old Bražecl
SI 03/09 58
Cabrea
39 0711 05
Collombratta
12
39 soos
Eviffa,
39 05109 57 :
Eormentara ::,,38 44109 154
Vile IT
The
::::::...
.܆
7
.
1
1
OI,
22
II
]
38 00 18 16
37 25117 36
46
1
44
1
1
1
!
3
>
1
1
214
A Table of the Lat, and Long of the Book IV
A
North | Eat
Latitude Longit.
mes
The Places Names.
50
50
50
37 00/1
39 081
002
04
Illes of Boyon
85 29
42
43 10/2
212
44 0810
43 492
43/2
1
North | Eaſt
45 285
D. M.D. M.
The Sea-coaſt of Portugal and France,
Chofol
49 53
from Cales to Callis.
Boffin
49
193
42
Jarze
49 303 24
North Eaſt
Sark
49 373 09
The Places Names. Latitud. Longit.
Garnalle
49 4312 49
D. M.D. M.
Arme
49 483 Os
Cales
Caskats
36 32 1 24
73 09
Cape St. Maria
Arderny
36 520 We.24
23 37
Cape St. Vincent
18 i Cape Hag.
43 52
Lisbone
06 Cape Barſlaw
49 574
32
Rock of Lisbon
Rone
39
49 4615 54
Burlings
Saine Head
39 4312 28
50 415 28
2212 St. Vallari
00
so
Cape Finiſterre
55
Deip
50 1516 39
Cape Corian
Callis
43
56
SI 137
16
Sazarka
43 381
52
Cape Artingal
06
Cape Pinas
44 040 Ea.52
Lyons
06
Sr. Andrea
43
36
The Sea-corſt of England from the Lizard
Bilbo
43 413
to Newcaſtlc.
St. Abaſtian
43 4014
19
Burdeux
45 10 5 En. 44
,
Bloy
19
The Places Names. Latitud. Longit.
Shorant
46 0014
56
Rochel
46 174 54
D. M.D. M.
Toppar
45 3613
29
The Lizard
50
ioloo
Mamoſin
45 4914
34
Falmouth
go 22 00 12
45 584
50 35100
35|0o 34
. St. Martins
46 164
29 Ramhed
5o 34 00.49
Barges :
46 3003
54 Plymouth
so
30100 51
uils
46 443
The Ediftonc
so
Piller
47 043
The Start
50 27 1 19
Nancs
47 454
15 Dartmouth
50 377
Radon
47 5513
SO 4211
50 401
The Coafts of Britanny.
Abſom Bar
5 4710 37
50 5512 Ia
Cardinals
47 2712 24
Chiddock
50 5712 14
Ballile
47 192
14 Portland
50 5012
36
Groy
47 351 54 Weymouth
51
44
Glannats
47 3311 34
Pool
51 413 34
Pennes, pr Pennemark 47 3511
10
Ine of Wight
50 584
08
Paiker
48 000:We;os ("Portſmouth
8
4 24
Seames
48 040 E:23 Shorám
57
Camarica Bay 48 25
250 56 Beache
2
50 5815
15
Brift
3510
51
48 450
19 Dongeners
51 916
uſhant
48 480
St 25
32
Iſland of Baſſe
49 011 27 Ripraps
5D 1316
49
Morlias
48 S41
39
The South Foreland
st2216.744
49
02
The Downs
5t 256 45
St. Mallas
SI
Nortli
00
Olloron
30 Foy
20
23 00 44
22
28
36
3.3 | Torbay
i
The Bary
Limes
1
1
3.2,
5
51
51
74
43
59 Rye
1316
Conquer
05 Dover
2516
C
Satta Inics
712
48 4513
39 | Sandwich
27/6
33
1
]
CHAP.XVIII. Chiefeſt Places of the World.
215
North | Enft
The Places Names. Latitud. Longit.
3016
01
I
IIG
38
SI 4819
819
1
51 3719
46
48
1
North Foreland
Margarec
Quinborow
Rocheſter
London
Graveſend
Tilbury Hope
Colcheſter
Harwich
Ipſwich
Orfordnals
Alborow
Yarmouth
Winterton
Cromar
Blackneſs
Wells
Lin
Bolton
Grimsbe
Hill
The Sporne
Burlington
Flamborough Head
Scarborough
Whitby
Hartlepool
Sunderland
Shelles
Newcaſtle
D. M.D. M.
The Sea coaſt of Flanders and Holland
SI
2816
44
from Callis to the Scaw.
51 2916
34
51
North Eaſt
SI 2815
54 The Places Names. Latitud. Longit.
si 3015 24
SI 3515
44
D. M. D . :
SI 3815
54 Duynkirke
SI 1817 49
52 046 O2 oftend
51 308
29
52
27 Sluice
SI
9
II
52 1416 24 Zealand
OS
52 2016
35 The Brill
52
08
52 24.6.39
Antwerp
52 456
Rotterdam
52
519
52 526 46
Amſterdam
524010
53 206
41 The Tafel
4010 01
53 20, 10:16
53 416
19 | The Uly
53 30.10.12
53
Skelling
53 35110 14
52 5815 33 Amaland
53 40110 16
53
915 02
Embden
53 441106
53 3914
28
Breme
53 50 12 26
53 454 16 Hambrough
54
4113 26
53 455
QI
Holikeland
54 30
54 0014
Stonar-
55 17 12 08
54 814 35 The Scaw
57 52113.SI
54 2014
21
54 3514
10
54 3713
29
54 4213
26
55 0213 24 | The Sea-coaſt from the Lizard to Holy-
54 5813 14
head.
1
1
716
1
28
1
SI
Leithe
North Weſt
The Places Names, Latitud, Langit.
The Coast of Scotland.
D. M. D. M.
Lands end
so 2010 34
North Eaſt Gulfe
SO: ILI
· The Places Namis. Latitud. Longit. Scilly
So 711
7 Stones
50 181
'16
D. M. D. M. Harry Point
51. rojo: Eing
Barwick
55 492
39 Londy
2010. W.3
2 09 Holmes
SI 261 E. 44
Dondce
56 262
17 Briſtol
51
29/2
57 222
29 Glocelter
312
39
57 4811
34 Caldy
SI 5310
Cac. Nafsu
58 371 38 Milford
52 Slo W.: 6
58 592 02 Ramza
52
Fair Eſc
59 303
19 | Studwalls
53
IZ10 04
Shetland
бо 2212
54 Barzs
53 1309.16
Fair Head
58 43/2, W.21
2. W.21 Weſtcheſter
Iſland Lewes
53 37 01E.04
58 3012
48 Holy-head
53 440
26
Skey Inand
57 4012
Iſle of Man
54 .2510 36
The
34
Aberdeen
Bafom Naſs
52
IO
Illes of Orkney
120
38
3
216
A Table of the Lat. and Long of the Book IV.
!
1
1
+
56 175
1715 18
20116 44
52 006
IS/22
56
58
41 Heda
56
57
58
58
58
52/26
The Sea-coaſt of Ireland.
The Sea-coaſt in the Sound.
North
West
North Eaſt
The Places Names. Latitud. Longit. The Places Names. Latitud. Longit.
D. M.D. M.
D. M.D. M.
Lamby
44 4411 46 Lizol
57 35114
Dublin
9
53 32 1 56 | Anall
57
814 41
Wexford
52 3311 44 | Ellen-nore
Waterford
56 40115 21
52 3012
24 Copenhaven
Corke
52 0112 56 Mooan
55 41195 29
Kingfail
52 52/3
08 Witmond
SS
20116 37
Old Head
SI 403
14 | Iſmond
55
Mizand Head
şi 2815
go Burnthom
56 00117 40
Cow and Calf
st 4215
42 Erchholm
56 1017 38
Skillukes
06 Gathe ſand
03
Blaskos
52 1516
II Farro Sound
48/21 53
Limbrick
53 0414 51
Gotland
58 2021 22
Loopas Head
52 4415
55 53121 29
Gally Head
53
2015. 36
36 Dormąmel
55 23 44
43 4015 16
Dines Naſs
58 22 23:55
Ifles of Aion..
53 2116
36 Righa
50 26 25
Slages
54 2716 21 Runen
38125 14
Ifles of Art
55 1815 36 Parun
54
Fore Head
SS 3814
4 56 Shorham
5826 30
Fair Foreland
55 3512 36 | Wile
59
6125
6
Ardenbro
59
6
Dagarąco
59 4472355
Oglholm
59 58 24 32
Norgin
60 1025 14
The Sea-coaſt of Ireland-Iland. Eaſt Rand
60 19 18 41
Wibro
61 1630
Nortle | Weft
Wakaco
61 1629 42
The Places Names Latitud. Longit.
Latitud. Longit. Parting
61 0028
15
Burga
61 2 27 14
D. M.D. M. Roſtbrugh
61
3125 04
61 3.21 2 51
Abbo
61
Merchant Forcland 63 5211 42
Bu hoers
60
Horn
63 42 10
16 Stockholm
58 4920 6
Silly
64 509 56 Froučnboro
48/18 16
Bargafar Point
01
Stickholm
2318 24
66 267 36 Yaffro
58 10-18 58
Griinſa
20
Fuland
57 42!19 IZ
Marza
67 8
9 42 Chiping
56 $3|18 -27
Rage Poinc
66 40 10. 00 Fastinboro
215 49
Fair Foreland
65 40/14 53
53 Scarlet Iland
40116 02
Snow Hill
65 11 14
11/14 Šo Ellinbro
56 46 16:00
64 00 14 69 Cape Cole
57 00 15 36
Weft main Illes
63 17112 53 Nading
Gammat Ifles
57 5315 4
63 48115
48115 06 | Hólm Sound
59 813 44
Grimes Hols.
63 23115 46 Mordo
58 37112
Walle Sound
58 25 11 40
Long Sound
59 71.12 54
From
$134
1
00
Mage Nafs
8123 28
921. 54
A
1
65 2717
58
58
Long Naſs.
66 429
56
56
Rook Point
3
CHAP.XVIII. Hom to find the Latitude, G.
217
+
Hope Iſland
1
1
Cape Blande
Staye Angor
Out Shers
Bomal
Harla IIand
Katts Naſs
Swin
Gallee
Gripo
Rols Illes
60 149
61 54
79 Iglig. 29
North East
From Naze of Norway to Archangel.
The Places Names. Latitud. Longit.
North
Eaft
D. M.D. M.
The Places Names. Latitud. Longit.
76 13123 16
D. M.D. M. Hopeleſs Illes
77 00 22 54
The Naze of Norway 58 0010
26 Nageo Point
77 10 23 38
58 571 9 44
Duckus Cone
77 4522 54
59 7
8
30
78 25 23 28
Helis Sound
59 31 9 4
79 27/24 19
02
Point Lookcut
76 2520 18
8 06
Horn Sound
77
720
00
62 40 9
Ball Sound
77 34120 3
63 529 46
Foe Sound
78 38/20 45
63 4010
26 Beare Sound
79 1519
IS19 ss
67 11330
Black Point
78 32 18 34
67 3813 52 Cape Cold
79 00 17 56
68 30 14 40
Fair Foreland
70 28118 32
Aſſumption
71 7 20 38
71 22 22 6
70 5624
Iſland Kilding
68 54 26 16 The Coaſt of the North-Weſt Diſcovery.
Cape Race
65 49'29 28
Cape Gallant 67 11 28. 56
North | Welt
65 17 28 54 The Places Names: Latitud. Longit.
64 12 26 31
Archangel
63 22/26 46
D. M.D. M.
Cape Farewell
590014: 50
Sir Thomas Smith's Bay79 10179 50
The Coaft of Greenland. Botton's Illes
60 2062 50
Belifle
51
2148 44
North | Eift
where the Table is be-
The Places Names. Latitud. Longit.
gun, on the coaſt of
D.' M. Di M, Terra Nova.
Cherri Iſland.
74 34120 32
IO
i
Werro
Low Fat
Zanham
North Cape
Skitanboro
2
Cape Crace
Fox Naze
។
1.
+
1
By, multiplying the Hours by 15, and dividing the Minutes of Hours, if there be
any, bý' 4, ſo will the number of Degrees ariſe; and if there remain any Minutes
after the Diviſion, they muſt be multiplied again by 15, and ſo will the number of
Minigtes of Degrers ariſe, by which theſe Places are diſtant from each other, which
Điffance is called the
Difference of Longitude of that place for which the Tables were
calculated, if the other place be Eaſtward of the firſt ; but if it be more Weſtwardy
it is to be ſubſtracted from the Longitude of the other.
"And this is the way we have endeavoured to ſettle the Longitude, with as much
neetneſs to the truth as poſſible we could. I have’not only made uſe of my own Cal .
cùlation of the Difference of Meridians of Places, as I have often uſed at Barbadoes and
Virginia, or any other Place, from the Meridian of the Lizard'; but I have alſo ob-
täinėd-them from the beſt Geographical Charts that are yet diſcovered, and the lateſt
Tables made; and ſo by conſulting with the able and skilful Mariners, that have uſed
the East and Weſt India; by the firſt we have informed our felves for the fetling che
Longitude of Places in the Eaſt India, with the beſt approved Authors: as in page 161
of Harmonicon Cælefte we find the Difference of Meridians betwixt Calicut in Eaſt-
India and London to be s hoärs and so min. which being converted into Degrees and
Minutes as before directed, is 37 deg. 30 min. the Difference of the Meridian of
London and Calicut; and the Difference of the Meridian of London and che Lizards
Ff
5 deg.
1
}
1
218
.30 - How to find the Latitude: BookIV.
+
5 deg. 24 min. added to it, gives the Difference of the Meridian of Calickt and the
Lizard, it makes 92 deg. 54 min. thc Differenc: of Longitude to the Eaſtward of the
Meridian of the Lizard.
And Macao in the Kingdom of Pegu, whoſe Difference of Meridians with the
City of London is 7 ha. 9 min. which is 107 deg. 15 min. the former Difference added
inakcsó Mateo to the Eaſtward of che Lizard 11z deg. 39 min. The Difference becwise
thê Tables before-going, and the Eclipſes, in the Difference of Meridians of Calicht
and London, is very ſmall, the Tables 4 min. more; and the Difference between the
Eclipfos and the Tables is 22 min. more. Then the Obſervation of Macao and London,
being ſo ſmall, it may very well be born withal: And we have ſetled the Longitude of
the West India, according to long and approved Experience of Voyages of my
ſelf and
otliers, from the Lizard to Barbadoes, and co.Cape Henry and Charles the Capes of
Virginia.
The Latitude of a Place is the Distance of the Zenith, or the Vertical Point thereof :
from the Agrator, or the Height of the Pole elevated above
the Horizon. You have
been thewed ſeveral ways already, for the finding the Pales Elevation above the Horda,
zon: but this Rule will not be impertinent to this place, being nor named before,
which is by the Stars thus.
You muſt obferve ſome Fixed Star in the Heavent, which is neer the Pole, and chat.:
never fets in that Region : Thus, you muſt obſerve the leaſt and alſo the greateſt Alti-
tude of the ſaid Star, when he doth come to the Meridian under the Pole, and alſo
above the Pole; which done, you muſt add the leaft. Altitude to the greateſt, and fo
the half of the deg. and min. thus nuinbred together, will be che Elevation of the
Pale; or Latitade of the Place.
An Example whereof may be this. The firſt Star of chic chrce in the Tail of the
Great Bear, in his leaft Altisude, obſerved at Briſtol, is about to deg. 59 min. and the * '
greateſt Altitude of the.lame, when hiefs above the Pole, is found to be neereſt gi deg.
59 min, both which Numbers being added together, do make 102 deg. 58 min. the half
of rhat fame is.fi deg. 29 min. the crue Latitude or Elevation of the Pole.
You imày cake notice, I begin to Longitude at the Meridian of the inoſt Soutbern
Parts of Brigland acthé Lizard, and increaſes on each ſide of that Meridias, from
I deg. to 180 deg>both Eaſtwardiand Westward, ; therdfore you muſt note, That by
chiefe gtables all Places, that lic to the Eastbeard of the Meridian of the Lizard, are
called Eaſt Longitude ; and all Places on the Weff fide of the Meridian of the Lizard,
is called Weſt Longitnde ora?
Therefore a Ship being in Eaſt Longitude,' ſailiiig to the Eaſt ward, the increal-
eth her Longitude; but ſailing to the Weſtward, it decreaſcth. And likewiſe if a
Ship be to the Weſtward of the Lizard, that is, in West Longitude, and failech to the
Weſtward, the Longitude increaſeth ; but failing to the Eaſtward, the Longitude de-
creaſeth:
You muſt norėsiche Sun riſeth to the Enftward, therefore all the stars, and are
carried Weft; and that all Places tlaat ase
to the Ealimard of dịe Meridian of the
Lizard, the sun comes to their Meridian fiil, according the time it is to the Emblem
ward of the Meridian of the Lizard : As you may note wher: was bcforç, directed
That every 15 degasis an Hours and 4 min. a Degree i Therefore in the former Expert
ple of Calicus, whole Difference of Meridians is. 5ibo 50 min. that is to ſay, the Suni
is on the Meridian in the Angst Indiaat
: Galicut ar: 19 min. paft 6 a:Clask in the
morn-
ing here at the Lizard, that is, 5 hox:£9 m.foontiitlaan he comes to the Meridian of these
Lizard.co make liere 12 a Clock'at Nool. And:o:on the contrary Icfier to the Hub
by every. 15 deg. As for Example.
The Difference of Longitude bervext the Meridian of che Lizard and Barbudaeis
52 deg. 58 min. that converted into Iijone is 3 hours 42 min. the ciraesche Sancome
to the Meridian of the Lizard, before it comes to the Meridian oh Barbadoes; Erla's
is ta lay, it is our 3 a.Clock 42 mida paſt ar thc Ličara in the afternoong beforegsod
12 at Noon in the Barbadoes.
in..!
You may take notices
. I took my firſt Latitude and Longitude froin tinc Nortkerd
parts of Nin found-land, to the Weſtward-at Cape Homblanto, ucet. Bell-ile
, and (a have
coaſted all round che Bay of Mexicu, and taken the Welt India Ifands in the way;
CI;
HU
and
!
.
1
1
219
+
CHAP.VIII. and Longitude of Places.
and ſo round the Coaſt by Brazil, and through the Straights of Magellane to Nova
Albion, where Sir Francis Drake was on the back ſide of New-England, in the Soxth
Sex; and from thence to the Eaſt India, firſt the Iſlands, and then the Main Land,
and back by Cape Bon Eſprance, and round the Coaſt of Guinney and Barbary down
from Tangire, and upon the Chriſtian Shore to Gillitore and Tocke in the Canary
Ifands and weſtward Ipands, and ſo along the Coaſt from Cales to Callis, and from
thc Lizard to Newcaſtle, and from thence along the Coaſt of Scotland to Skey Iſland,
and along the Coaſt from Calis co the Scaw, and along the Coaſt from the Lizard to
the Iſle of Man, and round the coaſt of Ireland to the Sea-coaſt of Iſeland, and ſo
from the Scaw round the Sound, by the Naſe of Norway to Archangel, and about by
the Sta-coaſt of Greenland by the North-weſt Diſcovery, to the coast of New.found
Land, where firſt I began ; whereby you may ſee I have traced a Path, or coaſted
round to chc inoſt Chief Harbours, Head-lands, and Iſlands in the World, by the
Tables. And ſo I ſhall conclude with cheſe Verſes in Mr. Philips's Preface,
when Drake and Candiſh Saild the World about,
And many Herocs found nen Countries ont
To Britains Glory, and their laſting Fame :
Were we like-minded, we might do the same.
4
1
1
1
1
The End of the Fourth Book.
r
1
1
'
*
}
1
{
i
f
}
4
{
1
F
自
​|
|
i
1
4
{
1
|
:
1
-
|
了
​4
4
MATHEMATICAL
Practical Arts
LE
.
1
A new W
On a new invented Scale, which refolyes moſt Queſtions in a moment
S T O R M Yº's
AND
Il
The Fifth Book.
-S HEWING
ETING
of LAND. by the MARINERS AZIMUTH
or AMPLITUDE.COMPASS; By which you may $ ORVEY
and PLOTT with eaſe and Delight, áll manner of Grounds, either
finall-Incloſures, Champions, Plains, Wood- Lands, or any other Uneven Grounds.
AND ALSO,
How to take the Ploit of a whole Town.; and a moft Excel.
lent way to be ſatisfied whether his Plott will Cloſe, before lie be-
gins to Protract the ſame.
ALSO
The ART of GÀGEING all ſorts of Veffels; as Clbe:
Veſſels, or to Meaſure Square Veſſels, as Cylinder Veſſels, and-Pipes,
Hogſheads, and Barrels and to Meaſure Veſſels that are part out ; and alſo to
Meaſure Brewers Tuns, or Oval Tuns, or Malh-Fats, or Cone-Veſſels, or Brewers
Coppers, or any other Veſels.
AND LIKEWISE,
How to Meaſure exactly allkirid of-Plain Superficies, as Walls, Timber
Work, Roofs of Houſes, Tyling, Board, and Glaſs, and Wainſcot, Pavement,
and the like; As alſo Timber and Stones; And of Meaſuring of SHIPS.:
The ART of GUNNERT,
in that AR Tinä moft Excellent Compendious
Form; never by any
ſet forth in the like manner before in the ART of 'GUNNÉRÍ:
With divers Excellent Concluſions, all reſolved, both Arithmetical and Geometrical
and Inſtrumental and by Tables ; Being framed both with, and without the help of
Arithrizetick;
As alſo divers forts of Artificial FIRE-WORKS, both for Recreation, and Sea
and Land Service.
1
1
1
+
T
*
1
By Cap" SAMUEL STURMI
LONDON,
Printed by Iriſtiáni Godbid, Anno Dom. M.DC.LXIX:
.
1
"
t
D
The Mariners Azimuth
Amplitude Compass
Or an Inſtrument for Surpaing of Land
by a New-way .
to
40
I
510
go
HI
VIT
10
0
20
50
PO
word
10
K к
0
810
V VITA
IX
MIL
R
SI
4141.
X
C
이다​.
V
ol
OZ
이​::
F
o
IX
ան
03
O+
02
1
OL
OK
MIX
On
OT
|
honom
boto 468
The AUT H OR to his Fifth Book.
Erc comes a Critick, cloſe thy Page, Bc open as the Eye of Noon,
Thou
art no Subject for this Age:
And let the Dogs bark 'gainſt the Motor
And Cenſure oftentimes you know
Thou haſt no Luſtre of thy own
Will ſtrike the Dove, and ſpare the Crow. But what's deriv'd from Art alone,
· But hold, thy Guilt does not require Fear not, thy Art inſtructed Page
That thou ſhould't lurk, or yet retire. May either pleaſe, or teach the Age.
H Н
1
To my much Honoured Patron,
Sir FOHN SHAVV Knight and Baronet :
And to the reſt of the Honourable.
F A R M E R S
OF
HIS MAJESTIES CUSTOMS,
Sir Fohn Wolſtenholme, Sir Robert Vyner Knights
and Baronets; Sir Edmond Turner Knight,
Edward Backwell and Francis Millington
Eſquires:
External, Internal, and Eternal Happineſs
be wiſhed,
+
M
HONOUR ABLE SORS,
AN bad at the firſt, and ſo have
all Souls before their entrance into
the Body, an Explicite Methodi-
cal Knowledge ; but they are no
Sooner Veſſeld, but that Liberty
is loft, and nothing remains but a vaſt confuſed
Notion of the Creature. Thus had I only a Ca-
pacity without Power, and a Will to do that
which was far enough above me. In this perplexi-
ty I ſtudied ſeveral Arts, and put them into pra-
čtice: For my own fullen Fate hath forcd me to
ſeveral Courſes of Life; but I find not one hither-
to which ends not in Surfeits or Saciety, and all
the
I
a a a
1
-
The Epiſtle Dedicatory.
the Fortunes of this Life are Follies. Thus I ram-
bled over all theſe Mathematical Inventions or
Sciences
, Wherefore (Honourable Sirs): I ha-
ving Compoſed, out of my poor Studies, this Mi-
ſcellany, and conſidering there was nothing in it
more uſeful for your Service than the Art of Sur-
veying, knowing that jou beſton Surveyors Offices
upon many, but divers may be wanting in the
Knowledge and Labour of the Art; my ſelf once
enjoyed both in your Honours Service, by.the
kind thankful Remeinbrance of my much Honou-
red Patronobit Eivy foon eclipſed my. Office
without defert;
and left me only my Art : There-
fore (as St. Paul faith in another Caſe) I will wait
with Patience until my Change will come; and
in the mean time I am your Honours humble Ser-
vant, and humbly proſtrate theſe my poor Labours
at your Honours Feet, begging your Patronage
thereof; knowing; That there was never any
thing fowell contrived by the Wit of Man, that
hath not been ſubject to the Cenſure and Mif-
conſtruction of the Envions : Nor do I at this time
(in the Production of this) expect Immunity from
the Cenforious Criticks of this preſent Age ; jet
for ſuch was this work never intended, but only
for the Fudicious, whoſe candid Cenſures I dare
abide : But yet not without labour and difficul
tycan Books have paſſage into the World; there-
fore to the end the placid Fruits of theſe my La-
bours (now grown up among the wild Grapes of
the Field) may be cheriſhed and preſerved from
the turbulent Storms of diſcontented Spirits, it
1
1
now
1
The Epiſtle Dedicatory.
nom being come to its Maturity and Perfection,
muſt humbly implore the Protection of frme Ho-
nourable Perſons to defend it. And being well
afJured (Sirs) of your Honours moſt Heroick and
Candid Diſpoſitions, I humbly call this into the
Arms of your Humanity for Shelter and Prote-
€tion; not doubting but that your refulgent Rays
Mining thereon, will be ſufficient to annihilate
and diſpel the moſt dark and miſty Clouds afcendo
ing our Horizon ; which will not a little ſtrengthen
both my preſent and future Undertakings for the
Publick Good, and excite the Author to a grate-
ful Acknowledgment of your Memorable Virtues,
and to echo forth the Praiſes due to your
Names and Éminencies. All that I have endea-
voured, is to profit others, and to make my diligent
and ſtudious Reader and Practicioner able to be-
nefit his Country (which,certes,is no more than the
Common Law of Hunanity requires at all our
Hands) and not, like ſome, to bury their Trea-
ſures in the Aſhes of Oblivion; which puts me in
mind of that excellent Saying of Tullie, Non no-
bis ſolum nati ſumus. Wherefore (Honourable
Sirs ) I have endeavoured, in as plain and Suc-
cinct a Method as I could imagine, to lay down
the Art of Surveying, a new way, bythe Ma-
riners Sea-Compaſs, which is the beſt Inſtrument
for their uſe and purpoſe, it agreeing ſo neer their
Traverſe Rules at Sea, that there is very little
difference. And likewiſe, I have ſhewed the
Sea-men the Land-man's Art of Surveying and
Gauging all ſorts of Veſſels, and plain Superfi-
.
1
H
+
à à a 2
cies
-
The Epiſtle Dedicatory.
cies and Solids ; the Art of Gunnery, Artificial
Fire-works, and Aſtronomy, in the following
Book; and ſo furniſhed the whole Work with
ſuch Theorems and Problems, Geometrically,
Inſtrumentally, and by Calculation, as are moſt ne-
ceſary and ſubſervient thereunto : And there-
fore (Honourable Sirs) you being able to protect
it, I most humbly commit it to your gracious Pro-
te&tion, reſting,
1
1
Sirs,
St. Georges, or the Pill,
neer Briſtol, Mar.25.
Anno 1669.
Your Honours moſt humble
and faithful Servant,
SAMUEL STUR MY.
1
!
1
+
CHAP. I.
1
4.
The ART of
Surveying of Land
By the
SE A-COM PASS:
The DESCRIPTION of the COMPASS
and ST AFF, and CHAIN.
The Fifth Book .
CHAP. I.
ward;
Have been all this while a Niewing the Maria
ner, How to deſcribe and make his own 19
ſtruments, and the uſe thereof in Navigaa
tion; I am alſo willing to ſhew him the
great uſe there may be made of his Sen-
Compass, commonly called the Azimuth,
or Amplitude.Compaſs, which all ingeni-
ouis Mariners carry to Sea.
This Compaſs requires but little deſcrip-
tion, it being ſo well known to all Sea-men;
for it is the ſame in a manner as they Steer
the Ship by: But it is called by the name of
a Meridian Compaſs. The Chart within the
Box is divided as you ſee in this Figure each
quarter into go Degrees, beginning at
North and South, numbred Eaſt and Weſt-
on the Glaſs there is a Braſs Circle, and Diameter, that goes over the Center
of the Compaſs-Chart, the Braſs Circle is about 7 Inches Diameter, and about f. of
an inch broad; The outward Circle is divided into 360 Degrees by 90 Degrees in each
Quarter,
as you ſee the former was; the Figure niakes all plain to the meaneſt capacity
and numbred as you there fee from 1 Degree each way from each oppoſite Point, The
inward Circke is the Hours anſwering each 15 Degrees and Quarters of the Horizen,
and they are nunbred as you ſee in the Figure.
There is a Circle the like divide the high of the Needle and Box-Chart, with lines
drawn up the Box at 90 Degrecs every way, that the Degrees of țhe upper Circle, and
lower Circle, and Chart, may agree.
In the Diameter FGHK there is a right Line drawn in the inidit, as GH, and
at each end is two llits of an inch and an half long, each of them, as FG and HK;
which are cut right in the middle : by which in taking any Angle, you muſt be ſure to
fer the North Point right ander the liſt and line of the one, and the South Point under
the flit and Line of the other : and 10 muſt you always, when you take the Angle ofany
twe
9
A 3 2
2 The Art of Surveying of Land Book. V.
A
The life of the
two ſtations from one place to another, You muſt be ſure to keep the two flits in the
Braſs Diameter over the North and South Point of the Chart , and turn the Index that is
riveted to the Center at C to the Object, and look through the lights that ſtand upon it ;
when
you find you ſee it plainly, and have made a good obſervation
of the Angle, look
what Degrees the edge of the Index cuts, and upon.what Quarter of the Compal and
that number of Degrees is the Angle of the two places from
the Meridian.
The lights that ſtand perpendicular on the Index are 1 inch and ; long"; the further
light hath a wire that goeth through the midſt thereof, by which we cut the Object : that
fight next unto you hath only a lit.
Through which you muſt ſee the Wire and Object you look at in one and the fame
Line, when you make any Obſervation.
Betwixt the two lights is a right line drawn through the midſt, and at the further
ſight is faſtened a perpendicular of Braſs with a right line through the midſt as BD:
this perpendicular is faſtned with two ſmall Braſs ſcrews at M: to the further ſight
with a wire; and at D is a hole where is faſtned a Silk thred twiſted and ſcrewed through
a ſmall hole in the Eye-light at S, and faſtned with a ſmall wooden pin.
And this is for to take the Sun's Azimuth at any time of the day, by turning the
Azimusla-com- Eye-light to the Sun ; and the flics over the North and South point of the Chartº as
pass. before directed) you may ſet the Index to what Degree you pleaſe; and when the ſhadow
made by the Thred D S, comes upon the Line in the midſt of the Index on the Line SA,
and on the perpendicular Line RD; then on that inſtant take the Sun's Altitude by a
Quadrant or Staff, and note it down : and likewiſe the Degrees cut by the Index at the
perpendicular end, and that is the Sun's Magnetical Azimuth at that time. When you
The Amplitude have done, you may unſcrew the perpendicular from the ſight; and then you have the
Compaſs. Compaſs ready to take an Amplitude of the Sun's Riſing or Setting : but more of that
in the following Treatiſe, when we ſhall touch upon Aſtronomy.
When you make uſe of the Compaſs for Surveying of Land, you have a Braſs ſocket
ſcrewed faſt to the bottom of the inward Box that holds the Chard, In that ſocket you
put the head of your three legged Surveying-Staff with a ſmall ſcrew on the ſide to faſten
it to the head, that it may not ſtir when the Compaſs is ſet North and South as before
directed ; then you may turn the Iridex and lights to what obje£t you pleaſe, and be
fure of your Angle from the Meridian if your Chard be good, and the Needle well
couched and placed. Thoſe that make them ſhould have a ſpecial care of that; and that
How th: Nee- the points of the Needle be faſtned and cemented together with a little Tin, ſo that they
dis of the do not ſtir abroad, as I have ſeen many Charts careleſly made, doth; It might be to the
thame of them that make them; and likewiſe the Wires put off one ſide s Degr. more
or leſs, as if in all places there were ſtill a Point-variation, which is a lazy trick as
well as faulty in moſt places. I would adviſe all Ingenious Mariners to make a conſtant
practice of taking obſervation of the Sun's Altitude or Azimath, and Steer a Courſe,
and make allowance accordingly, as hath been ſhewed elſewhere, with the Wires of the
Needle put exactly under the Meridian, as this Compaſs before going the Points are;
and then in all things this Inſtrument will come to the Truth, as well as a Needle of
greater charge, and Plain Table and their appurtenances of 3 l. price : or the Theodolite,
and Circumferenter and Veracter. And yet I cannot but highly commend theſe 11-
Struments as very uſeful for Land-aber which have Money enough. Neither dare I re-
Of the Devices ject as uſeleſs, either the Topographical Inſtrument and Croſs-Staff of Mr. Diggs, the
of Infoments. Familiar Staff of Mr. John Blagrave, the Geodetical Staff, and "Topographical Glaſs
of Mr. Arthur Hopton, the Settor Croſs-Staff, and the Pandoron of Mr. George Al-
wel
, or any other witry Invention which hath been deviſed for the Exact Plotting, and
Speedy Meaſuration of all manner of Superficies, as Landşand the like. But in regard
the Authors have in their own Works to their exceeding Commendation deſcribed the
Making and uſe of the ſaid Inſtruments, I ſhall ſay no more.
And for the Mariners Compaſs in a manner to do the fame things for the Surveying
of Land, or Plantations
, or the like, I hope will be well taken and accepted of all Inge-
nious Mariner's, for whoſe fake I take theſe pains.
Let the Glaſs over the Chard be as clear as poſibly you can get him,
1
be ſet.
The
1
3
The Staff:
Links; and that of Mr. Gunter's which
Links : fo that each Link of Mr. Gunter's
bora's. And this year Mr. Wing hach den
CHAP. I. By the Sea-Compaſs.
The.Figure of the staff is plain, it needs no further deſcription : It is to be had at
any Inſtrumeni-Makers.
Of Chains, the ſeveral ſorts thereof.
Of Chains there are ſeveral forts, as
namely Foot-Chains, each Link containing
a Foot or 12 Inches ; ſo the whole Pole
or Perch will contain 16 Links or Feet
according to the Statute Pole.
The Chains now uſed and in moſt
ekteem among Surveyors are Three. The
Firſt I will name is Mr. Rathborn's,
which had every Perch divided into 16
had 4 Perches or Poles divided into 100
Chain is as long as Four of Mr. Rath.
fcribed a Chain of 20 Links in a Perch
for the more ready uſe thereof in his Art
of Surveying ; Therefore when we have
taken the Angles, and Plotted a piece of
Ground, we will ſhew how to know the
contents thereof in Acres, Roods, and
Perch by the two lalt Chains,
1
Sect.I.
is
25
Mr. Gunter's Chain.
After Gunter's Chain is a Chain moſt uſed amongſt the Surveyors of this Age, and
4
7 Inches
some of an Inch, and each Pole according to the Statute contain 16 Feet, the whole
Chain is 100 Links in the Four Pole or 66 Feet. In meaſuring with this Chain you are
to take notice of only Chains and Links, ſaying, ſuch a line meaſured by the Chain
contains 54 Chairs 45 Links, or thus diſtinguilhed 64. 45. and this is all you take
notice of in Surveying of Land.
Now for the ready counting of the Links; at every Perch let there be two Curtain
Rings faſtned, and one Ring at every s Links : ſo you may readily count the Rings
at either end. If the Ingenious Mariner wants a Chain, he may mark a ſix Thred-line
Or ſmall Belch as before directed with Red Cloth marks and White for diſtinction, or
bitts of Leather as we mark our Dipley line ; and be ſure to ſtretch him well firſt; or if
you can, let it be a Top gallant Brace half worn; then meaſure them exactly : and
mark him as before directed, and you may meaſure any place of Land or Plantation,
or any diſtance, as well in dry weather, as with a Chain, without ſenſible Errour,
1
SECT. II.
Cautions to be uſed, and to be obſerved in the uſe of any Chain.
VV
'Hen you have occaſion to meaſure large diſtances, or otherwiſe, you may by
chance miſtake or miſs a chain or two in keeping your account, which will
breed a conſiderable Errour ; and alſo in meaſuring of diſtances, in going along by a
Hedge lide
you can hardly keep your Inſtrument-Chain & mark in a right line and there-
fore the diſtance will be more than in reality it is. For avoiding theſe miſtakes you ought
to provide tan ſmall ſticks, which let him that leadeth the Chain,carry in his Hand before;
and at the end of every one of thoſe Chains, ſtick one of theſe Sticks or Arrows into the
Ground,
4 Ike Aar of Surveying of Land Воок.
66
Figures to the right Hand, and
Ground, whichler him that followeth take up; ſo going on until the whole oxun. of
Sricks be ſpent, and then you may conclude you have meaſured Ten Chains ui sont
further trouble: and theſe Ten Chains if the diſtance be large, you call a Change and lo
you may denominate every large diſtance by Changes, Chains and Links in a fice of
Paper you keep the account by. If the diſtance be tär, you muſt fet up a Cloih uport
a Stick for a mark betwixt your Inſtrument and the further mark, and ſee through your
Inſtrument both the marks in one ; then you may be ſure to go ſtraight with the
Chain.
Sect. III.
How to reduce any Number of Chains and Links into Feet and Yards.
IN
N taking of Heights and Diſtances hereafter taught, it is neceſſary in the Practice of
my Geometrical Concluſions to give your mtaſure, in ſuch caſes, in Feet and Yards
by reducing of your Chains and Links thus.
Multiply your Numbers of Chains and Links, as one whole Number by 60, cutting
off the Product the two laſt Figures towards the right Hand ; ſo ſhall the figures to the
left Hand be Feet, and the Figures cut off ſhall be 100 parts of a Foot.
Let it be required to know how many
5:32
8:06
Feet are contained in s Chains, 32 Links.
66
Set down the Chains and Links with a
Examples. 4836. Comma (3) thus and theſe Multiplyed
4836
359L117
by 66, the ſum will be 35. Feet and
333
351, I2
-531,96 parts of a Foot as thus you ſee it ſtand
351:12. This is the Rule by Mr. Gunter's
Chains. ,
li you divide the Feet by 3, the Quotient will be 11 7 Yards. Now if you have
leſs than 10 Links as 6, you muſt always remember to put (o) to ſupply before the
6, and Multiply the number as you ſee in the laſt Example.
SECT. IV.
How to caft up the Content of any piece of Land in Acres, Roods and
Perches by air. Gunter's Chain.
BY
Y a Statute made the 33 of EDVVARD the I. an Acre of Ground ought to contain
160 ſquare Perches, and every Rood of Land 40 ſquare Perches, and every ferch
contains 16 | Foot; and 4 Perches, Poles, or Laggs in breadth, and 40 in length
makes an Acre: which multiplyed together is 160, Half an Acre is 80, a Quarter 4
' ſquare Perches.
Suppoſe the Figure ABCD
A
were a ſquare piece of Ground
B
as the Marſh of Briſtol, and
were is Chuin 16 Links
every way: Then to find how
many Acres, Roods and Perches
are in it, do thus. Square the
ſides, that is, Multiply
the other, and cut off the s laſt
3192
3192
To
1
one in
A
C
that before is Acres : what re-
mains Multiply by 4 (for 4
Roods makes an Acre) and cut
off 5 Figures as before, and the
comma:is Reods;and that which
remains
1
+
L
5
IS:16
IS16
?
is
is
S
40
+
produce 22,98256: thes
left hand, remains before
the Comma 22, which is 22
stand thus, and cutting off 2 Figures, and you have for the contents.
of che piece of Ground 11 4 Perches.
(bain; the surveyors all generally
Perch, and his breadth AC 8 Chains),
fore in the lalt Example, directed by
CHAP. I. By the Sea-Compaſs.
remains, maltiply by 40 the number of Perch in a Rood, and cut off 5 Figures to the
right hand of the Produkt, and in like manner you have the odd Pershes. This Exa
ample will make all clear and plain.
So you will find 15 Chans, 16 Links Multiplyed together, as before directed, wil!
r
lalt Figures cut off to the
15:16
By the Line of Numbers.
90 96
Extend the Compaſſes on the line of
Acres, and the 5 Figures
7580
Numbers on the scale of Scales from
isultiplyea by 4, the Product
which is at 160 unto the fade of the
15.16
Square AB 15 Ch: 16 Lin, which is
393024: s Fic? Acres 2 21.98256
60 perch and above, the fame diſtance
gures cut off on the s
will reach from the fame 60 Perch, to
left hand, the comma
Roods
3193024
22 Acres, 3 Roods, and 37 Perch, and
? Roods, and the s
the 20960 part of a Perch.
last Figures multipia; } Perch 37 [20960
Figures, and the reſt will be 37 Perch. .
SECT. V.
How to Meaſure a Long Square Piece of Ground by a Chain of 20 Links
to a Perch, according to Mr. Wing.
M
After Wing in his Art of Surveying, in the 113 Page hath deſcribed a Chain of
23 Links in a Perch, which is ſomewhat more ready, if you will reckon the
Liar in perches for ſmall parcels of Land.
Suppole
a piece of and be in length 36 Perches and 16 Links, and in the breadth
3 Perches 2 Links; By this Chain 1 deſire to know the Contents thereof, having 20
Links in a prich, 1 deſire to perform the operation, in a Decimal
Count by half the number of links, and then the Sums will ...:.:368
3685
Practitioner in greater parcels of Land, to follow Mr. Gunter's
1104
1T14:08
;
Secr. VI.
To Meaſure a Long Square piece of Ground.
Et the long piece of Ground be:
A
ABCD whole length A B is
1106.21.F
II Chams 25 Links, or 45.....
Or 32 Perches.
Multiply one by the other, as be-
way
31
of Surveying
O
*
LI
co
Mr. Gunter's Chain; and you will
find the contents of the Ground to be
9 Agres, no Roods, but a Perch.
!
C
D
By
1
1
+
The Art of Surveying of Land
1
Baſe A B-45, the fame extent
6
Book. V.
By the Line of Numbers.
II2S Extend the Compafles from the Center, which is at 160 Perch unto the length
800 AB 45, and the fame diſtance will reach from the bredth AC 32 l'erch, to 9
9100000 Acres.
Sect. VII.
To Meaſure a Triangular Piece of Ground.
Uppoſe the Baſe of a Triangular
of Ground. A B whole means
jure is in Chains 25 Links, or 45
C
Perch, and the perpendicular CD
32 Perch, or 8 Chains ; Take the
half thereof, and Multiply one in
the other, will produce the Como
tents of the whole Triangle to be
4 Acres, 2 Roods, o Perch.
By the Line of Numbers.
II 2S
Extend the Compaſſes from the
400
Center, which is at 320 - unto the
4150000
B will reach from the perpendicular
4
2 00ooo
CD 32 unto the quantity of Agreso
which is 4 A. as before.
4 Apr 2 Ros
32P
FA2R, 27
45 171
F
SECT, VIII.
To Meaſure a picce of Ground of Four unequal ſides called a Trapezia.
1 Et the Ground given be A,B,C,D; after you have taken the Angles with your
Compaſs, and (noted them down in a piece of Paper or Field-Book, (as
dhewn in the following Diſcourſe) You muſt Protract or lay down the Figure as
I do this, by a Scale of equal parts of 10, or 15, or 20, or 25, or 30 parts divided
L
Ihall be
B
A
&A, ROP
20P
E
60P F
c
1
21:P
into
(.
+
emergen
60
22
I20
CHAP. I. By the Mariners Sea-Compaſs.
7
into an Ir.ch, I have laid down all the following by 20 pares or Perch in an Inch : then
draw the Diagonal Line AC, and with your Compaſſes take the diſtance AC, and ap-
ply it to your Scale of 20 Perch to an Inch; and you will find it 60 Perch, or is
hain, and then if you let fall the perpendicislars BF and D E, and meaſure them
in the like manner, you will find by your Scale BF 20 Perch or 5 Chain, and DE
-4 Perch, or 6 Chain.
In reſpect the Baſe is common to both the Triangles : You may therefore add the
o perpendiculars together 20, and 24, the fun will be 44, the half thereof is 2 2
Perch. This Number being multiplyed by the whole length of the Common Baſe AC,
Go Perch, giveth 1320 Perch, that divided by 160, gives the Contents of the Trape-
1320
zia or piece of Ground to be 8 Acres, 1 Rood, o Perch. You might have multipłyed half
the Baſe AC 30 by the ſum of the two perpendiculars 44, and it gives you the fame 5 (7)
as before.
132(0)(8
By the Line of Numbers.
36(0)
Extend the Compaſſes from the Trine A Center at 329
to half the length of the Diagonal AC
30
The fame will reach from the sum of the perpendicular 44, to the quantity of Agres,
which is 8 : Acres.
After this manner you may meaſure a piece of Ground of 5-6-7-8, or any
gumber of Sides, by bringing it into Triangles and Trapezias, as shall be ſhewn.
I20
(
44
30
1320
SECT. IX.
T
:
that is the area of 7:22:56
22
3
II2
To Meaſure a Piece of Ground which is a perfect Circle.
"He proportion of the fircismference of any Circle, to its Dianseter, is as 7 to 22,
(2)
Example. In this Circle ABCD let the Diameter thereof be 56 Perches, Feet, 0(4)
or Inches, which multiplyed in 4(0)
it ſelf giveth 4136. This Num- Acres as
ii.
B В
ber mulriplyed by 11, gives 5:1:24 Peri
45496, which being divided
by-14, the Quatient will be
3249"
the Circle.
How many Poles and Feet,
II2
ór ſquare Inches, is in any Cir-
1232
A
6P
cle whatſoever, you may know
5:4
better by theſe Rules; Firſt,
82321176
A RP
If you know the Diameter, and
would find the Circumference, 22:7:176
FIT
fay, as 7 to 32, fo the Diamke
ter 56 to the Circumference
176; Or if you know the Cir-
cumference, and would find the xr
Diameter, ſay, as 22 to 7, fo 23
X232
D
is the Circumference 176, to
(s
222
the Diameter 56.
10 24
15
,,
:
1237
56
J
88
28
!
704
176
2464
The Diameter and Circumference being this known, the Rule to find the Content is this
.
The Diameter being 56 Perch, and the Compaſ: 176, the half of both theſe multi-
plyed together, and divided by 160, you have the quantity of Acres 15, Roods 1, Perch
24, which is the Contents of that Circle.
By the Line of Numbers.
Extend the Compaſſes from o Center at (203 1. unto the Diameter AC 56, che
fame diſtance will reach again from 56 to the quantity of Acres 15.4.
x86
I confeſs though the ordinary proporcion of 7 to 22, is ſomewhat too much; yet it 246
is byt about 1 in 2000, which will breed no great difference in theſe Queſtions.
((1S
IGG
Bbb
SECT. X. X
3
8
Book. V.
The Art of Surveying of Land
Secr.X.
IN
30 of
со
To Meaſure a Grcund being a trui Oval.
Nthe Oval ABCD, let the length be given
40 Perch, or Fect, or lacher, and DB
the fame memfure: Then to find the Quantity in
Perch, Feet or Incles; if yo:r work by the faine
Denomination of Ecct and inches, as I do of
Perches.
Multiply AC by DC;C, the Product Mal-
tiply by 491, and from that Produit ciit off's fi-
gures as before directed, and as in the Margin, the
Contents will be s Acres, 3 Rools, 22 Perch,
+
Multiply 40
one by.he io
other.
I 20
Muli:-
491
I 200
IOSCO
17800
A:; 89200
4
8.356800
40
I20
3
2²
parts of a Perch.
Per.22372000
By the Line of Numbers.
Extend the Compaſſes from the o Center in the
Line at 203 ?to the length of the Oval AC 40,
the ſame diſtance will reach from the breadth DB
30 unto the Contents in Acres, s Acres ; fere.
C
SBCT. XI.
To Meaſure a piece of Ground lying in form of a Sector of a Circle.
.
Mult. 48
30
00 B
144
:30
C
1440
Divided by 320
(1)
(0)
¥44(0)(4
320
X610(2914
O.
Et the Se&tor be ABC, whoſe fides is AB,
or A C 48 Perch, and the Arch thereof BC
30 Perch : Then to find the Contents in Acres,
Multiply AB 48, by BC 30, the Product
divide by 320, the Quotient is 4 Acres, and 160,
the Remain divide by 18o the of 320, and the
Quotient is the Roods; if any thing remain, it is
Perch; So the Contents of this piece of Ground is
4 Acres, 2 quar. o Perch.
By the Line of Numbers.
Extend the Compaſſes from the A Center at
320 unto the ſide, or Semidiameter AB or AC
48, the fame diſtance will reach from BC 30,0
the quantity of Acres 4 and to
48
148
** CO(2 94.
+
به7
j
1
Sacr.XII.
.
tar
Chap. I.
By the Mariners Sea-Compaſs.
୨
----
Sacr. XII.
60
18
480
60
To Meaſure a piece of Ground that is a Segment, or part of a Circle.
L
Et the Segment be ABC; AB 60 Perch, DC18 Perch: Multiply the
Chord of the Segment 60 by the perpendicuslar Height, and the Produkt divide by
225 the Gage- vumber, and the Osiotient will be the Acres, the Remain divide by
and it is found 1 Acres, 3 'Roods, 3 Perch, the quancity of Ground in the Segment,
1080
(1) (8)
C
29 ()
x=80(4A.
225
3(2)
18
18% (3 R.
c.
so
1
IO
B
2
1
Q
160
-P.
8 A. R. P.
..
4:3:8
1
By the Line of Numbers.
Extend the Compaſſes always from 22 5 unto the Chord of the Segment AB 60:
the ſame diſtance will reach always from the perpendicular Height, unto the quantity of
Acres 4 and i fercie
Sect. XIII.
+
Having a Plot of Ground with the Content in Acres, To find how many
Perch of that Scale was contained in one ffich, whereby it was Plotted.
Etthe long Square piece of
Ground, be containing 9
Acres ,Firſt, Meaſure the
Plott by a known Scale, which
B
ſuppoſe it be of so Perch in an
Inch; ſo meaſuring AB, you
find it 22 : Perch; alſo the
breadth 16 Perch.
3
-
By the Line of Numbers.
If you work by the Rule of
the laſt Square Blote, you will
find the Contents 2. Acres.
Now on the line of Numbers,
take with your Compaſſes the
diſtance between 27 and 9
Acres, the half diftance thereof
will reach from 1o on the Line
of Numbers unto 20: ſo I con-
clude the Ground was caſt up by a Scale of 20 Perch in an Inch.
9 X 109
2
Bbb 2
A
4
2
4
By Arithmetick
Sacr. XIV.
1
11
IO
The Art of Surveying of Land
Book V.
.
Per.. **?
2 I
Sta.Per:16 Xample. The laſt Piece of Square Ground being found 9 Acres by the Statute-
SECT. XIV.
A Piece of Ground being meaſured by the Statute-Perch of 16; Feet, Ta
know how many Acres it is, it being meaſured by a Perch of 21 Foot,
nhich is the Iriſh Perch.
É
Perch of 16 Foot; and you would know how many it is by the Iriſh Perch of 2 1
Ir, 3311 Foot.
By the Line of Numbers.
Extend the Compaſſes from the Iriſh-Perch of 21 Foot, to the Engliſh Statute-Perch
of 16 Foot, the ſame diſtance will reach from 9 Acres turned twice over unto s.
x-
Acre, fere: Or in the scale of Reduction extend the Compaſſes from 16. Foot to 21,
42114
the ſame diſtance will reach from 9 to, in the Line of Numbers, And so of
33
meaſure. Now by, reducing the 9 Acres into Perches, it makes 1440 Perch ; and be-
The Square of cauſe the greater ndeaſure is to be reduced into the leſſer, Multiply the given Quantity
1440 by 1 2 i the Square of 11, which 11 was found, thus. 10 f being a Frałtion, it
and of 14:196 reduced
into halfs, makes 33 divided by 3 is 1 1 ; So the Iriſh-Perch 21 Hoot in halfs
196:12 1:1440 is 42 divided by 3 is 14, thoſe two Numbers Squared a 1r is 121, the Square of 14 is
196, the Product of two Numbers 1440 multiplyed by 121, the Produt is 174240:
2888 that divide by 196, the Square of 14, and the Ouotient is 888,192 Perch, reduced into
Acres, is 5 Acres, 5 Roods, 8 Perch', almoſt 9 Perch according to the Iriſh-es-
#1174249 Sure,
292(1)
197
2724(9)
Suppoſe you had been to Reduce Iriſh-meaſure into Stateite-meaſure;
9246(2) 192
then multiply 1440 by 196, and the Produkt would have been 282240:
174240 883, 96 that divided by 121, and the Produ£t had been 2332 Perches fere,
Ig666
which makes Acres 14, Roods 2, Perch 12 ; fere, Statute-menfure.
42
any other
a I ( 15 (21.
I 21
1440
1440
کرو'I
1
|
.. در
6
By brot Line of Numbers.
3(8)
881815 Extend the Compaſſes in the Scale of Reduction from the Number of Feet in Cuftomary
2010
meaſure, as I do from 21 Foot to the Perch of Iriſh. meaſure : the fanie will reach
quar.
from 9 Acres Iriſh in the Line of Numbers to 14 Acres, 2 Quarters, and 12 Perch
8(812--& P. Statute-meaſure.
4 o
Or if you extend the Compaſſes from 21 in the Line of Numbers to 16 andį, the
5 dC.2 Q.8 P. extent turned twice over from 9, will fall upon 14 and Acres, and a little more.
Perch So that if you remember in all ſorts of meaſure to reduce your Fraction into the ſame
1440
Denomination, and ſeek out the leaſt proportionable terms : by Dividing by 3 if half
196 Fost, and Squaring theſe terms as before directed, you have a Rule that ſerves for all
8640
forts of Cuſtomary or Iriſh meaſure whatſoever.
I 2960
1440
282240
282 2 40
.
19%
196
1211
}
68
12332
12I
!
79
233/2 (14
XOGO
1
1
VIXT:
CHA P. II.
CHAP. II.
By the Mariners Sea-Compaſs.
11
nologna
chonch
babaya
29
CHAP. II
.
B
How to take the Plott of a Field at one Station taken in the
middle thereof by the Azimuth-Compaſs
.
Efore you go into the Field, you muſt Rule a piece of paper in 8 Colamus
as you ſee the Figure following in this Chapter, makes all plain, without any
more Deſcription, which is called a Field-Book. Secondly, when your
come into the
field, firſt place Marks at the ſeveral Angles of the Field, as
at ABCDEF, in the following Figure ; then make choice of ſome con-
venient place about the middle thereof ato, to fix your Compaſs, if you can ſee all the
Marks, and be ſure the Braſs-Diameter and Slits before deſcriebd, be ſet directly over
the Meridian or North and South Line of the Chard, and there fixed.
This done, direct your Sights to your firſt Marks at A, Marking what Degree the
Index curteth, which let be 36 Degr. 45 Min. you may eſtimate the Minutes; This
you muſt note down in your Field-Book in 1 and 2 Column thercof, as you ſee in the
Book it is plain ſet down, then meaſure the diſtance from o the place of the Compaſs
to A your firſt Mark, which let contain 8 Chains, 10 Links, which muſt be placed in
th 3 and 4 Column of your Field-Book, as you ſee in the Figure of the Book,
-
ésst B
10152
A
8.75
75
.
قازق
36445
2017
1790
Soutfit
-N
S-
8
3:27
9:55
.
F
Size!
W
D.
8.15
10.2.3
ย
West
7
12
are.
The Art of Surveying of Land Book V.
Then direct your sights to B the ſecond Mark, and note the Degrees cut by the Index,
wliich let be South Eaſterly 8. Degrees 45 Minutes, and the diſtance 8 Chains 75 Lix.
You muſt
put down in the Field-Book, as before ; Firſt, the Letter B ; Secondly, the
Inclination to the Meridian cut by the Index South Eaſterly 80. Dezt. 45 Min. in the
third Column; then & Chains 75 Links in the fourth, as you may ſee in the Columns
in the Book, all plain; then direct your sights to C your third Mark, and note the
Degre:s cut by the Index, which let be s. £. 16 degr. 45 min. and the diſtance OC
10 Chains 45 Links, put the fame down in the Field-Rock likewiſe, as before directed;
then direct your Sights to D, and note the Degrees cut by the Index, which let be s.w.
32 degr. oo min. the diſtance OD, 8 Chains's 3 Links, Nore it down in the Book, as
before.
Then direct your Sights to E, the Index cutting 72 degr. 45 min, North Weſterly;
and the diſtance OE 8 Chains 15 Links: They muſt be no:ed in the Book as the rest
Laſtly, direct your Sights to F your laſt Mark, the Edge of the Index cutting in
the upper Braſs Cicle M.W. 18 degr. co min, the diſtance OF 9 Chains ss Links ;
then will the Obſervation ſtand in the Field. Book as in the following Table or Figure;
then by a line of Chords, or by the Protractor you may preſently Protract the exact
Figure of the Field upon Paper thus, By a line of Chords; take 60 degr. ard draw the
Cicle. Secondly, draw the Meridian line of North and South, and parallel-line of
Eaſt and Weſt. Thirdly, your firſt Obſervation and degr. cut by the Index, was 36
degr. 45 min. N. E. Therefore take 36 degr. 45 min. off the line of Chords, and lay
from the Meridian line Mto N. Fourthly, the diſtance O A was found 8 Chains 10
Links, take with your Compasses off any Scale of equal parts, 8 Chains and that
ſtands for 8 Chains 10 Links ; lay this diſtance from 0 to A, and draw the prick's
line O A.
Then ſecondly, take out of the line of Chords the ſecond Angle S. {. 80 degr. 45
min. and lay from the south towards the Eaſt on the blind- Arch, and through it draw
the line OB; then take off the fame Scale of equal parts 8 Chains 75 Links, that is,
8 parts, and 75 parts of divided into 100, and lay it from 0 to B, and prick the line
from O to B, and draw the line AB, which meaſure with your compaſſes, and apply
it to the Scale of equal parts as before, and you will find the fede AB 8 Chains 75 Lin.
The like do by all the other Angles and diſtances in the famie manner as you have been
thewed in the two firſt Angles. The Figure makes it ſo plain, it need no further precept;
and you may put down the Number in the ſide, as I have done.
Now by the Protractor inihe Second Book deſcribed.
You may lay the Diameter-Edge thereof on the North and South line, and through
you have laid the Center, put a Pin on the Center of the Plott at 0, and note the degr, and diſtances
down all the in the field-Book, as before: the firſt was North Eaſterly 36 deg. 45 min. put the
Angles in the Edge of the Index to 36 degr. 45 min. in the Arch of the Protractor , and by the Edge
quarterly
, and account in the Scale thereof 8 Chains 1o Links; and by the ſide thereof draw the line
Jay the Center-A O, prick'd as before, and ſo do by the reſt of the Angles and fides in like manner,
l'is of the Pro- and you may preſently draw a Plott of Ground you have meaſured. This is very plain,
fractor on the and may be underſtood by the meaneſt capacity in this Art. The Obſervations Marked
Point O, as be in the field. Book ſtand as in the following Table,
fore ; and the
the Meridian-
The manner how the Field-Book muſt be Ruled.
Mark.
ID.M.Ch. Lin You are to note, that every Degree in the uppermoſt
N. E.36 450810
Braſs-Circle is ſuppoſed to be divided into 60 parts,
which is called Minutes, which cannot be expreſſed in
S. E. 8045 875
с
S. E.16 4510 45
regard of the ſmalneſs of the Inſtrument or Circle of
Braſs on the Glass of the Compaſs; and therefore the
S. W. 32 00 8153
odd minutes muſt only be eſtimated : fo muſt the odd
N. W.7245 815 Links taken off your Scale of equal parts, and it will
55
breed no ſenſible Errour.
"Turn the Pro-
Tractor after
4 ,
Diameter
011
line:
B
E
F N.w.180019
CHAP. III,
Chap. III.
By the Mariners Sea-Compaſs.
13
fond
on ปัว
จ ricanhattartar
om
hur dhe
សយ
begonnen
momento
CHAP. III.
r
Ilow to take the Plott of a Field at onc Station taken in any
Angle thercof, by the Sea-Compaſs.
the Figesre of the Field following, place your Compaſs.at M, fet your Braſs-
Diameter right over the North and South points or Line ; then firſt direct
your
Sighesto A, which let be ſuppoſed to be 22 degr. 15 min. N. E. which Note in
your Field-Book, as before, then meaſuring the diſtance with your Chain MA
which let be 8 Chains 46 Links, which place in your Field Book according to
former directions.
Secondly, Direct your Sights to B, the degrees cut off by the Index, fuppoſe 42
degr. 45 min. and ſuppoſe the diſtance meaſured to be M B 10 Chains 21 Links,
and put them down alſo in the Field-Book.
Thirdly, Direct your Sights to C, the degr.cut is 66 degr. 30 min. and the diſtance
MC ni Chains 64 Link!, put theſe in your Field-Book alſo, as before, and in the
ſame manner you muſt deal with the other marks DM, and EM, and MĚ, and MG,
and ſo having them all in the Field-Book, they will ſtand as followeth.
a
D
C
AL...
J
11.27
B
11.ca
m
u66
10.21
Fast
20:15..
30
3
8
&
21115
Nore
2
N?
cii
7512
sonih
L-
M.
1
2
The Figure of the Field-Book.
Deg. Min. C. L. North.South. Eat Welt.
158 46
42
45/10 21
30 11 64
E SE
45 II 23
57
30 11 68
FIS E
49
45 TO 15
GIS E 18
72
M
ĀNE 22
BNE
CNE 66
DINE 86
it
t
.
0017
СНАР?
ment
..
i
1
14
Book V.
The Art of Surveying of Land
r
hi
Wo din incinta anche di un dia cura dina din cur
vingine indigcinacinanincipcincioginfiraigsinci paipginais
2
VAC
VALOT
CHAP. IV.
1
1
1
S
How to Meaſure any Piece of Ground be it never ſo Irregular;
And how to reduce the Sides into Triangles or Trapezias,
and to caſt up the Content thereof in Acres and Perches.
Uppoſe you were to Meaſure a piece of Ground, or Woad; or Marſh, or any
place whatſoever, by your Compaſs and a Line inarked as the 4 Pole-Chain
before deſcribed in the firſt Chapter , and if you cannot ſee all the Angles by
reaſon of the bigneſs thereof, then you muſt meaſure round about by the ſides
thereof, as in this Figure following ; and the Obſervations made in the field
are fet down in the Field-Book following, ſo plain, that it need no further precept.
Suppoſe you made your firſt Obſervation at A in the Field in the following Fighie.
(the Compaſs being rectifyed as before directed) you direct your Sighes along the hedge
to the Mark in the corner at B, and the Index cutts 54 degr. from the South iveſtwards,
and the diſtance is 5 Chain 12 Links, which ſer down in your Field-Book, thus, AR
bears SW 54 degr. oo min. diſtance 5 Chain 12 Links; Then make your ſecond
Station at B, and direct your Sights to C, the Index cuts NW 45 deg.co min. diſtance
2 Chain 89, Links, which note down in your field-Book, as you did before in the re-
cond place, and ſo do by all the reſt." From C to D NW 76 degr. diftance 3 Chains
35 Links, from Dto E NE 31 degr. diſtance 4 Chains 55 Links, from E to F NE
56 degr. diſtance 2 Chains 57 Links, from Fro G N E 21 degr. diftance 2 Chains 24
Links, from Gto H SE 31 degr. co min. 2 Chains 95 Links, from HtoK S E-34
degr. 3 Chains 25 Links, from K to A SW
4 degr. 2 Chains 95 Links ; Thus you ſee
all the Obſervations plainly ſet down in the Field-Book, you may proceed to Protracting
How to know your places of Obſervation and Marks in the Field, and your degrees and length of
if the sides, be Lines orderly placed in your Field-Book; We proceed two ways to examine the truth
rightly meafu Thus the Rule is.
Ted before you
Firſt,
As the Radius of Sine of 90 degr. is to the length of the fede of the Field in Chains
tract the Plal- and Links, or Perches and 100 parts ; ſo is the Sine of the degree cut by the Index to
form thereof the length of the Parallel of Eaſt and Weft in Chains and Links, or Perches and 100
Therefore by your Scale extend the Compaſſes from the Sine of 90 to the length of the
ſide of the Field in the Line of Numbers, the ſame diſtance will reach from the degrees
cut by the Index to the length in the Parallel of Eaft or Welt.
Secondly,
As the Radius or Size of 90 degrees to the length of the ſide of the Field in Chains and
Links: or Perch and 100 parts; fo is the Complement Sine of the degrees cut by the
Index to the length of the Meridian of North or South in Chains, or Links, or Perch,
Wherefore Extendthe Compaſſes from the Sine of 90 degrees to the lengtb of the fide
of the field in the line of Numbers; the fame diſtance will reach from the Sine Como
plement of degrees cut by the Index tb the length of North or Somsh in the Meridian.
So that you ſee the laſt Columnt in the Field-Book are noted Northand South, Eaſt
and Weft.
Now to know by the Chains and Links, the firſt Obfervation from Ato B, is S154
degr. and the diſtanc AB is 5 Chains 12 Links ; therefore by the laft Rule extend the
Compaſſes from 90 degr. to. s Chains 12 Links in the Line of Numbers, that diſtance
TANO
go out of the
Pield, and Pro-
parts.
or 1oo parts.
1
i
svill
1
15
CHAP. IV. Bythe Mariners Sea-Compaſs.
will reach from 54 degr. cut by the Index to 4 Chains 14 Links in the line of Num-
bers, which is the diſtance in the Parallel of weſt, and alſo the ſame extent will reach
from the Complement of 54 degr. which is 36 degr. to z chains 97 links in the line of
Numbers, which is the diſtance in the Meridian Sossth, and put it in the South Column
of
your Field-Book, as you did 4 chains 14 links in the Column of weft; and ſo you may
do with the reſt of the Obſervations,
1
1
But the moſt ſure way and leaſt Errour, is, to convert your chain and links into
Perches and 100 parts of a Perch, and then you can Protract in Perch and 100 parts
the better.
5:
I Ź
Thus if you Maltiply the number of chains found in the ſide by 4, by reaſon 4 Perches
are in every chain ; and if there be above 25 links in the place of links, divide by 25,
and the Quotient will ſhew the odd Perch to be added, and what remains is links: that
Multiply by 4 likewiſe, the Produkt will be 100 parts o: a Perch.
As for Example.
The firſt fide AB his diſtance is 5 chains 12 links, multiplyed by 4, makes 20 Perch,
is parts, which put in the next Column to it; Now if you extend the Compaſſes from 96 4 : 4
degr. to 20 Perch Sparts, the fame diſtance will reach from the Sine of $4 degr. cur by 20 :
48
the Index to 16 Perch 56 parts, which is in the Weſt Column, and the fame extent will
reach from the Complement 54-degr. which is 36 degr. to 11 Perch - 88 parts, which I
(1
put in the Soush Column; and by the fame Role I work in like manner by the reſt of the
2(4
Obfervations. The ſecond fide' B C is a chains 89 links, reduced as before, makes it
2-89(3
Percher parts; and ſo work by them, as before directed, you ſhall have all your
4 23
Numbers ſtand as in the following Figure of the Field-Book.
3P: - pts.
(Houſes name.
Angle with Merid. cha. Lin.Pol. roopis: North. Soush. Eaft. Weft.
Id-56
ABS W 540015
12 20 48 [P.
48 P. pos. 11 88
BCNW 45 0012 89 11
GDNW 76 603
3513
32
13003
DEN E 31 004 5518
2015 72
09 40
14
EFN E 56 002. 6710
68 5
88
FGN E 21 0012
2418 96 8 40
GHS E 51 002 95 II 80. 7 3219
HKS E 34 003 25 13:00
10.727
28
KAS W
4 00 2 2 oslit 80
2462
The Sum 41 6041 6037 963796
8
1
16 soli
8 16
100
16
56 18
40, 3
oo
3
mt no
3
20
20
f.
.
I7 68
O
н
.
Under the line are the diſtances in the Eaſt Colomy; and likewiſe the Figures on the
inſide the Meridian, are the points of North, as 1,2,3,4,5,6 Column; and the Figures
on the outſide are 7, 8 to A, and are taken out of the Souch Column, and working as
directed in the two firſt ſides, you may find all the reſt. The Figure makes all, plain to
the meaneſt Capacity, and ſo you will have the trae Plott of your Ground, or Park, or
Wood-land, or i lantation, or place whatſoever, drawn on Paper or Farchment. Now
if there be any Houſes by the hedge.fide, made a mark in your Field-Book in that Angle,
and how many Chains or Persh from the place you obſerve, and by it inſert it into
your Plots.
1
*
!
Ccc
As
+
16
Book V.
The Art of Surveying of Land
1
Souit
B
30,
1
ME
0 1
SHA
2,50
25125
20
EastAT
J
1
41
す​。
D
3
West
61 5
P7
and
12:20
I
88
32.86.
99.4
+
10
OL
K;
3.914
P
El
&
2.3.
-1
S
1310
KT
1
H
Ito
***
Norths -
G
j
1
I
As for Examples
How to draw
There is a Houſe in the firſt fide and Angle AB, SW 54 degr:about 2 Chains 42
4.
the Plott by the Links, or 9 Perch 21 parts, and reckon 1 Chain 2 4 Links, or about 5 Perch; there
Protractor.
fore put it down in your Book, with the Man's Name that ows the Houſe, as I have
done, ( Fohn Cooke) or the like ; and if any Howſe, Charch, or Castle, be in the
middle, take the Angle thereof from any Point, and meaſure the diſtance, and note it
in
your Book, and enter it into your Plort, as I have done this Hoxfe. By theſe rules you
may compleatly take a whole Pariſh, Plantation, or Iſland : Now if you draw a Plott
by the Protractor deſcribed in the Book of Inftruments, You muſt rule your Paper or
Parchment with an obfcure plummet Merid. Lines, and Parallel Lines about 1 inch, and
aſunder; and put the Pin, the Center and Rivet upon any Point, and turn the ſide of the
Protractor on the Meridians, and look in the Field-Book for the Angle, and put che
Edge of the Index to the degrees, and count the Perch on the index-fide, there make a
Mark with your Pin for the ſecond place, and draw a Line from that place by the Edge
of the Ruler to the Centers for the ſide of the Hedge or Field.
As for Example.
Suppoſe you were to draw the ſide A B in the Plott with your Protractor, lay the Pro-
tractor on the akt-fide of the Meridian-Line, and the Diameter-Edge thereof to the
Meridian. Line, then in the oppoſite Degree and Quarter as in this Example is NE:
1 put the foot of the Index to 54 degrees, and from the Center, the Edge points SW 54
degr. Number the Chains, or Links, or 20 Perch 42 parts, and from that Number to
the Center draw a Line by the Edge thereof
, and you have the Side AB; by this Rule
you may gain all the reſt. There is no Man that underſtands any thing in theſe Aris;
but knows readily how to Plett a Field
by the Rule before-going without more Directi-
ons,
for they will be all the fame.
1
+
CHAP
ל
GHAP, V.
By the Mariners Sea-Compaſs.
17
3
Cu o cincin sain arte a la cara do Tradio in curan
Diagociagiogiogiapies
OTAQIPgiagio india
Togiani dieciegienia
CAOVADONGAS DE
AGRODNO
DOOR
.
}
CHAP. V
I
1
1
I
VVS
1
1
How to find the juſt Quantity or Content of any Piece of
Ground in any form.
E have ſhewed in the fore-going Chap. how to Meafure
the Geometrical Square; the Parallelogram, the Triangle,
Trapezium, the Circles and the like. Now we will thew
you
how to caſt up the Contents thereof more fully. Sup-
poſe the fore-going Figure A, B, C, D, E;F, G, H, K,
be a Plott drawn or Protrated by a Scale of 10 Perch in
an Inch, and the exact Contents thereof is required. Now becauſe it is an Angular Plott,
neither in the form of a Square, Parallelogram, Trapezium, nor Triangle, therefore
all ſuch Plotis muſt be reduced into ſome of theſe forms: which to effect, I reduce the
main Body of the
Field into the Trapezium ACE K, and the reſidue of it into s Trio
angles; as ABC, and CD E, and EFK, FHK, and FGH.
Now to know the juſt Quantity of Acres, Roods, and Perches the Field contains
I firſt meaſure the Trapeziun A CEK, I meaſure with my, compaſſes the length of
the perpendicular CO, and apply it tomy Scale of 10 Perch in an luch, and find it
14 Perche parts, and likewiſe the perpendicular KP, and find it 10 Perches, so parts,
which I add together, and they make 25 Perches 38 parts, which I Multiply in half
the Baſe AE 16:43, and the Produłt is 416: 99 : 34: Therefore if you cut off
Figures to the right-hand, you will have the Contents of the Trapezium 416 Perch,
and cut off 2 Figures of the 4; and before the Comma is 99 parts of 100 of a Perch and
The remains, which is not to be taken notice of.
1
.
11
4
Fig. 98 ini
Sllowing
es.
*
.
In like manner for the Triangle ABC, I multiply half the perpendicular :4 75 by
the whole Baſe 25 Perch 26 parts, or the whole perpendicular by half the
Baſe, as be-
fore ; it is all one, and the Produét is the Content of the Triangle ABC 119 Perch
ps parts: and ſo likewiſe for the Triangle CDE; multiply half the baſe 9 Perch se
by iz Perch, and 20 D. Y the perpendicular, and the Product is 1/16 Perch 14: is the
Contents of that “Triangle. Likewiſe in the Triangle EFK, the length of the perpen-
Ricular FR is 7 Perch: oo and the half length of the baſe ÉK is 14 Perch 70, multi-
blyed as one whole Number, the Product is to2 Perch 90 parts 00 : the Contents of that
Triangle EFK: the 4 Triangle FH'K perpendicular:6 P.10:Hq baſe FK 11 2:50
the Coriteres is 70 Perch:15: the firſt'Triangle FGH the perpendicular GS is 7?:92
and the baſe FH, the half thereof iso P. 50: multiplyed together, and the Product is
SI P. 48 parts for the Triangle FGH.
Laſtly, I add the ſeveral fun:s together, and they give the Content of the whole
Figure in Perches and 100 parts.
1 Trapezium ACEK 416:99
2 Triangle
А ВС.
119:98
The Area or. Cox.
3 The Triangle CDF
tent of the T
4 Triangle EFK
Triangle
FHK
Lo Triangle
EGH
51:48
The Area or Content of the whole Field or Wood is 87.7 P.64
Dda2
Whichin
1.
t
- 116:14
1:02:90
70:15
e
ie
/
18 The Art of taking of the Heights Book V:
ice Perches
Which if you will reduce into Acres, Roods and Perches, you muſt note, that 16
Acres, Fost and Square isa Perch, and 160 of theſe Perches makes an Acre; therefore Di-
and vide 877 Perch by .150, the Quotiext fhews the Acres to be 5; that which remains
above 40 divide by 40, and the Quotient will be Roods 1 : and that which remains will.
be 37. Perch. This is a General Rule for reducing of Perches; ſo that the whole Plott
of Ground yieldeth the Content of the faid Field 5 Acres, 1 Rood, 37 Perch and of
a Pereh.
This is the way to caſt up the Content of ariy, Irregular Field, by reducing it into
Trapeziums and Triangles, and adding their ſeveral Products into one sam, which ought
heedfully to be regarded, it being one of the moſt material works belonging to the Pra-
Etice of a Særveyor; for unleſs he be perfect herein, he can never perform any Work of
that nature aright
. Thave been brief and plain in ſhewing the Art of Surveying by the
Sea Compaſs; I might have been longer, but to avoid Prolixity, I think what is right is
fufficient :' Íf any deſire a larger Diſcourſe, he may make uſe of Mr. Leybourn's com-
pleat Surveyor, or Mr. Wing's Art of Sarveying, and others that have writ at large of the
Uſe of the Plain Table, which is the moſt eaſy and uſeful Inſtrument in the Art
of Sør-
vering of ſmall Incloſures. But the Compaſs fitted as before, will with a little labour
do any thing as exact as the Plain Table, Theodolite, Circumferentor, Protractor, or
any other Inſtrument : cſpecially large places, as Woods, Parks, Forreſts, or Plant a-
tions or the like : and what Direction I have given in Uſe of the Sea-Compaſs will ſerve
for an Introduction to all other forts of Inſtruments for Surveying of Land.
1
1
40
Apologia
ល
hobo mababa ang tot
poppable
hotos
o$
CHAP. V.
M
1
How to take the Height of any Illand, or Mountain in the Sea
by an Example made by the Author of the Height of Tenariff.
Any Learned Men have writt of the Great Incredible Height of feve-
ral Mountainous Hils and Iſlands in the World. For taking of the
Height of Iſlands in the sea, none have greater opportunities than
Sea.mer. By then may all Men be informed of the truth of ſuch like
things; and alſo of ſeveral good stars that be to the Soutbward, as
the Crofiers and Cannobas in the Stern of the ship, and any other,
did they but obſerve any ſuch Stars from a known Latitude, and take their Meridiana
Altitude and time of Night; or if they cannot, or will not Calculate, and find by, it
the Stars Altitude, and Longitude, and Declination : yet if they bring it home, and
give it to ſome able Artiſt, he will do it, and all Men ſhall have the benefit of it, and by
it, it would be a great help to Navigators.
It is reported of Ariſtotle, Mela, Pliny, and Solinus of the Invincible Height of
Athos a Hill in Macedonia, and of Caucaſus; and of Caffius in Syria, and many
other places : and among the reſt one of the moſt miraculous
things which they have
obſerved of the Mountain Athos, is, that it being a Hill and Mountain ſituated in Ma-
cedonia, it caſts a Madow into the Market-place in Myrrhina, a Town in the Ifland
Leonos, diſtant from Athos 86 Miles to the Eaſtward. It is no marvel it caſt fo large
a lhadow, feeing by Experience of the ſhadow of a Mans Body, we find it extraordi-
zary long at Sun-Rifing, as well as at Sun-Setting. They report it is Higher than the
Region of the Air. Falins Scaliger writes from other Mens Relations, that the
Iſland
3
::
1
1
CHAP. V. of Iſlands and Mountains at Sea.
19
Inand Pico of Texariff riſeth in height is Leagues, or 60 Milés : Moſt Writers agree
that it is the higheſt Mountain in the World, not excepting the Mountain Slotus
it fell, which I queſtion whether any mortal Man ever fee Slotus, beſides the Mork
of Oxford, who by his skill in Magick conveyed himſelf into the utmoſt parts of the
World, and took-a-view-of all places about the Pole : yer that this IMand cannot be fo
high, ſhall appear by the following Deinonftration and Obſervation made by Me; ali
Sea-men that have uſed to Sail to the Canaries know; that the Sńow is not off it above
2 Months in a Year, that is, June and July.
..
Patricire not content with the former Meaſure of 60 Mile high, reaches to 70 Mile
high ; Now that any Snow is generated 60 or 70 Mile above the pláin ſuperficies of the
Earth and Water, is more then ever they can perſwade any Men that underſtand theſe
things, ſeeing that the higheſt Vapours never ariſe by Prolomy 41 Miles, and by Era-
tofthenes's Meiſure 48 Miles above the Earth; that is, There is never no Rain, Dew,
Hail, Snow, or Wind, but ſtill a clear ſerenity.
I have been and paſſed by Tenariff ſeveral times my ſelf, in the Year 1652 I was
there in the Caſtle-Frigos of London Cap, John Wall Mr. and Loaded our Ship at
Garrachica right under the Pico, and ſince bound to the Weſt-Indies in the Year 1656,
in the Society of Topfam, a Ship I had Command of, was put by Weſterly winds to the
Eaſtwards that we had light of the Pico of Tenariff, it bore off us South; about Noon
I was reſolved to make Obſervations of the height thereof, to try Concluſions with my
Quadrant of 20 Inches Semidiameter, deſcribed in Chap. 16 of the ſecond Book ; and
by the Rules of Ouadrature I made theſe Obſervations following. On the sth. of M&y
1656 I obſerved, and found the Sun's appareiit Meridian- Altitude 81 degr. 44 min.
his Declination 19 degr. o 8 min, the Latitude 27 degr: 20 min, the Latitude of Pico
I found formerly to be 28 degr. 20 min. difference of Latitude 60 min. or miles, which
in the following Figure I make ove half of my Horizontal Bafe AD, then at the ſame
time obſerving the Height or Altitude of Pico, I found it 2.4 degr. 14. mir. Therefore
according to the Sphericality of this Terreftial Globe conſiſting of the Earth and, Sea, I
demonſtrated the following Figure. The Section of the Arch A E FS reprefeuts the
Superficies of the Horizan of Tenariff; or a part of an Arch of a great Circle, the Me-
ridian-AD, the differefice of Latitude 60 mile; DC Pico the perpendicular, E B a fe-
cond Obſervation, N the North.part of the Horizon, S the South part; C the Port of See Fig. 98 in
Garrachica.
the following
Schemes.
Now to find the perpendicslar Height thereof, you have the Rules at large ſet down
in the 16th. Chap. of the Deſcription and uſe of the Quadrate and Quadrant ; and by
the firſt Rule of the Ouadrant I found the perpexdicular Height D Pico 27 min. ormile,
if the Sea were a flat plain as a Tableöboard, as the Right-Line ABD.S repreſents.
But having 3 days of Fair Weather, in ſight of Pica I made a ſecond Obfervation,
and ran to it with my ship until I made an Angle with the Pico of 45 degr. oo min. as
Pico E C:or Pica BD; and had the apparent Meridian-Altitude of the Sun 81 degr.
29 min. the Declination 19 degr. 22 min. Latitude the Ship is in 27 degr. 53. min.
difference of Latitude 27 min. or miles, being equal to the Height, as BD to the per-
pendicular D C Pico, the Angle of 45 degr. being the moſt fure as can be made by
any Inſtrument which confirms the firſt, and the Height of the Pico of Tenariff to be
27 min. or miles High, if the Sen were flat as a Board.
But touching the Hypothefis, that the Earth and Sea makes a - Round Body, It is ge-
nerally agreed upon by all the Philoſophers, Aſtronomers, Geographers, and Naviga-
tors Ancient and Modern; and therefore the diſtance of a degree
60 min. reckoned in
the Heavens by Obſervations of the Sax or Stars is more than o miles upon the ſuper-
ficies of the Earth and Sca, as appears by ſeveral Experiments made by able Arrifts:
but eſpecially by the Labour and Induſtry of our own Countrey-Man Mr. Richard
Norwood, as you may ſee in his ſecond Chap. of his Book the Seamans Practice, made
by hiin berwixt Tork and London. He makes it evident that I degree of a Great Circle
4
2
1
1
I
on
5
1
1
"1.
20 The Art of taking the Diſtance from Book V.
rence of the
How to find
6120
.
22
Therefore if you do
on the Earth, is near 367200 Feet, which in our Statute-Poles of 16 and Feet to the
Pole is 22254 Poles; and about half, and theſe reduced into Furlong's at 40 Poles to a
Furlong, makes 5:54 Furlongs and 14 Perch; and laſtly, theſe reduced into Engliſh-
miles of 8 Furlongs to a mile, makes 69 miles and 4 Furlongs 14 Poles, that is 63 and
miles and 14 Peles to a degree upon the ſuperficies of the Earth and Sea. And fee-
ing a Degree is the 369. part of any Circle,equally divided in the Circumference; There-
fore if we can find how many Feet, Perches, Furlongs, or Pieces, are in a Degree
of known meaſure: then can we preſently reſolve how many of the ſame known mea-
ſure are in the Circumference of any Circle fo divided on the Earth and Sea; for if
How to know here is 367200 Feet in one degree of a Great Circle upon the ſuperficies of the Earth
the circumfe- and Sea, therefore it is evident, that if you multiply: 367200 by 360 degr. the. Pro-
duet, is 1 32192000 Feet, which reduced into Poles, is 8011636, and theſe reduced
Earth,
into. Furlongs, are 200290 Furlongs, and 36 Polesi. And laſtly, theſe reduced into
miles, are 2 5036 Engliſh miles, 36 per.ch for the circumference of the Earth and Sea
And now if you deſire the Diameter and Semidiameter of the Earth, as it is proyed by
the Diameter & Archimedes, That the proportion of the Circumference of a Circle is to the Diameter
drflance to the thereof almoſt as .2 2 to 7; therefore by the Rule of Proportion, Multiply the Circui-
Ciucer of the ference of the Earth; namely, 1 32192000 by 7, and divide the Product 925344000
Earth aud Sea. by :2, the Quotjent is 42061091, which is the Diameter of the Earth in Feet: and
the half thereof, namely, 21030545 is the Semidiameter of the fame, or diſtance of
the ſiperficies of the Earth and Water, to the Center, being 2 1 millions of feet, and a
little more ; and theſe reduced into miles, as we did the Circumference, thews the Dia-
meter of the Earth, to be 79.66 miles, and ſomewhat more: and the diſtance to the
2:45. Center or Seriridiameter 3983 miles'; and thus is found the Circumference, Diameter,
and Semidiameter of the Earth and Sea, and alſo the quantity of a degree of the fame
meaſure in Engliſh meaſures of Feet, Perch, Roods, and miles.
33(3)f. Nill retain a degr. in the Heavens to be 60 minutes
, you may find how many Feet is
*%240(370 in a mile on the Earth and Water, if you divide 367200 feet by 60, the Quotient will
3333
be 6120 feet; which doubled, and divided by 33, and half-feet to a perch, the 140-
33
tient is 370 perch, and 30 foot remains : divide 370 by 40 perch to a furlong, and the
( Quotient is g Roods or furlongs, and 10 perch or poles remains, divided by 8 Roods to
3706 a mile, the Quotient is t, and i remains; ſo that a minute in the Heavens by this Rule
400
and meaſure npon the ſuperficies of the Earth and Water, contains i mile, i roed, 10
R.
perch, and 30 foot; therefore my degr. 60 min. of Latitude at my firſt Obſervation,
(I
is found by theſe Rules to be 69 and į miles, 14 perch my diſtance upon the Arch of a
o(IR
Great Circle from the Latitade of Pico : therefore working by the Rules given of
Osadrature in the 16th. Chapter of the 2 Book, the crie Height of Pico will be found
The true beizhe to be 3 miles, and the diſtance from the Eye to the Top of the Pico A,P, will be
and d. flance found by the Rules in the 16th. Chap. 76 miles.
froin the Eye.
And working the ſecond Obſervation by the fame Rules, your difference of Lati-
tude 27 minutes B:D will be found to be 31 miles, 2 furlongs, 14 perch, 18 foos, which
is 31 miles and a little more ; which is almoſtthe fame Height found by the firſt Ob-
fervation : and the diſtance from the Eye to the Top of the Pico is 44. miles.
By theſe ſeveral Rules you may find the Height of any Mountainous Iſlard at Seks
or High-Laxd on the Main-Land, if you can come bring it North or Somth of you,
and make any Obſervation of the Sun or Stars; or if you will but truſt to a Log-Line
marked after the former Experiment, that a mile doth contain 6120 feet, or 1020 fa-
thamis; and fo 3 miles or a League contains 18 360 feet, or 3060 fathoms: then if you
intend to keep a Log by } minute-Glaſs: and becauſe half a minute is part of an hour,
divide 6120 by 120, the Quotient is si feet... Therefore ſo many si feet or knots
fhe runs out in a minste, ſo many miles ſhe ſails an hour. By this Rule you may keep
your Reckoning exactly; for I had Experience in failing North and South by a Login
Line marked after the rate of 6000 in a mile, that is by the ſame Rule so feet or 8 fa-
choms, and 2 feet to every knot, that I have ran or ſailed almoſt 22 Leagues to raiſe
or depreſs the Pele i degr. on a Great Circle ; and if any have impartially taken the
fans
1
8
5
1
f
21
CH. VI. Forts, City-Walls, or Caſtles which have Rivers betwixt.
fame notice and carc, (or will) they ſhall find the like. But many that follow the old
Rule
300000 feet to a degree, and sooo feet to one mile, and 60 mile to a degree, or
the 120 part of an hour 41 feet, or 7 fathoms to the knots upon the Log : when they
Sail North or South, and find the Log to fall too fhort of their Obſervation, impures
it to ill Steerage, Sometimes to the Variation of the Compaſs, or ſome Errour in their
Plotts, or ſome Current, or other accident, but will not believe the truth a great many
without they had Angels ſhould tell them ſo,, a great many have ſo much Ignorance and
Obſtinacy. But for confutation of ſome of the Antient Authors, eſpecially Cleomedes,
whom I take to be a Man which did never ſee, nor obſerved Texariff Pico; He af-
firms, that there is no Hill found to be above 15 furlings in Height; and of Mr. Hughes,
He faith, that if Mercury hinſelf ſhould affirm a Hill to be above 4 miles in Height, he
will not believe him ; neither will I believe them that are of that Opinion, be they
what they will, without they could prove the foregoing Obſervation not good, and
produce better of their own made by the Pico of Tenariff'; and ſo much for the Alti-
tude of Hills at Sea.
na in cura cincino circa ciocura cuina
piegiaciggiodiniocinioniodi
MOUROS
Naciocinio drastici indian
innocopinioniocietie
Da
PORCON
23.com
CH A P. VI.
How to find the Diſtance of a Fort, or Walls of a City, or
Caſtle, that you dare not approach for fear of Gun-Shot;
Or the Breadth of a River "or Water, that you cannot paſs,
or Meaſure over it , made by 2 Stations, with the Quan:
tity of the Angle at each Station.
Uppoſe from ſome private place as at A, you eſpy a Caſtle, Fort, Tree, or place -
whatſoever, that you dare not approach for fear of Gun-Shot, Marſh-Grounds,
or a River betwixt you, or ſome other Impediments, that you cannot make
your ſecond Station in any open place, but are forc'd to make it in ſome other
ſecure Place at B; therefore plant your Inſtrument or Compaſs at A, and direct
the Sights to Cand B, take the Quantity of the Angle CAB 46 degr. 00 min, and
go to B, and take the Quantity of the Angle ABC 79 degr. o min. then meaſure the
diſtance of the 3 Stations A and B 350 fathoms.
S
See Figure 98
in the follow-
!
܀
Then by a Plair Scale, or by the Line of Sines on the Scale of Scales ; you may ing Schemes:
prefently reſolve the diſtance, as I do by the Tables.
As the Sine of s5 degr. og ACB - 991336
46 d. oo m.
АВ:
79
to 350 Fathoms,
354406
So is the Sine of 79 dsgr. ABC
999194
I 25
to the diſtance
i do o
1353600
AC 419:42
362264
125 0
OO
11
oo
SS O
991336
354406
As the Sine of 55 degr. 00 ACB -
is to AB 350 Fathoms,
So is the Sine of 46 degr. o CAB
to BC 307 Fathoms, the
diſtance required.
1
985693
1340099
ma 348763
Sicr.1.
I
1
22 The Art of taking the Breadth of Rivers. Book V.
Sect. I.
1
How to take the Breadth of a River,
a
runs betwixt Gloceſter-ſhire and Somerſet.shire, and found the breadih of the SPuser
upon a Spring-Tide 40 Perch or a Furlong; you muſt do it thus. Being on the River
fide as in the former Figure at E, there ſet your Compaſs; Ubſerve fome mark on the
other ſide of the Water, as ac D, then ſer a mark at É, and go [quare-wife either to the
right-hand, or to the left from theſe 2 marks, ſo far, until you ipie the mark Don the
other ſide the Water doth juſtly make an Angle of 45 degr. with the mark E; and this
will be when you come to F; then meaſure carefully F E, the diſtance of the 2 Stations,
and that Thall be equal to the breadth of the River : fo that if FE be 10:30:30:40:50:
or 100 Poles, or Tards, or Feet, the breadth is the fanie. The like may be done by at:y
other Angle, as if you go to G, and make an angle of 26 degr. 30 min. in D; then is
the diſtance G E twice the breadth; but ever if you can get an Angle of 45 degr. for
that is the beſt and readieſt Angle to find out ſuch a diſtance ; therefore if you can, uſe
to other.
And the like way of Working you may do.at Sea, if you gain the sight of any Cap,
Head-Land, or iſland, ſet it by your Compaſs when you ſee it, without altering
your Courſe, make an Angle of 45 degr. And by your Plan Scale if you have reçe
a good account of your
Way by the ſame Rules as before, you ſhall have the true diftanse
of your ship from the firſt place, or Cape, or Hend. Lund, or iſland whatſoever : Go
you may get the Slope-ſide DF or DG if you meaſure it with your ' ompoles and
apply it to the fame Scale of equal parts by which you put down the wifi anc. Ė F or E G.
Thus you may find the diſtance from the hip, to any ape ; Theſe are made ſo plain by
the Rules before-going, that it need no further precept.
1
1
SECT. II.
SECT. II.
Being upon
the Top of a Hill, Tower, Steeple, or a Ships Top-Maft-Head,
there obſerving the Anglo of diſtance from you, 10 find the true diſtance
thereof.
Ou
,
Steeple, or Ships Top-Malt-Head be 40 yards, or any other mealøre: and from ic
you ſee an House, Tree, or Place whatſoever, and you deſire the diſtance from you.
You have been ſhewed already to find the height of a Tree, ower. Hill, or Steeple, by
this Rule we will thew you how to ſtand upon them, and take the diſtance from any
1
thing elſe, viz.
LU
1
Let the height of the Tower, or Maſt, or Hill be 40 yards, and let the Angle of
diftance taken with your Quadrant be Ro degrees, being 10 degrees under the Lise of
Level, this is the Rule for all ſuch Queſtions.
As the Tangent Compl. 80 degr. which is 10d.
924631
is to the beight 40 Yards.
260206
So is the Radim
to the diſtance from the Top of the Tower or Hill 226 4; 235575 yards.
IO
...?
Sacr. III.
!
1
Ch. VI. The Art of Plotting a City, Town, or Village. 23
SECT. III.
You
ſoever.
By the way of your Ship, and any 2 Angles of Poſition; to find the Diſtance
of any iſland, Cape, or Head-Land from you.
Ou have been ſhewed how to do it with a right-Angle of 45 degr. already ;
but with a little more trouble, you ſhall learn to do it by any z Angles what-
As for Examples
Suppoſe you were Sailing full South from A towards B, and from A Mould eſpy. Land
at C bearing 2 Points from you to the Weſtward, as S S W or SW 22 deg.30 min. and
Sailing ſtill upon your Courſe until you come to B, you obſerve the place bears from
you juſt 4 Points, or SW 45 degr. which is the doubleof the Angle obſerved at A. If
in this
manner you double any Angle ; that is, let your firſt Angle be what it will, you
muſt Sail until you have doubled that Number, then you may aſſure your ſelf that the
diſtance you have Sailed between A and B, is juſtly, equal to the diſtance between Band
CB being the ſecond place where you made your laft Obſervation, and C běing the
Place obſerved. So that if A B be 12 miles, B C'is likewife 12 miles; and this you
may
do without furthor trouble or Calculation, and may lay 'ît down by your Plain-
Scale, as I have done this following Figure.
إن
1
In all fuch Queſtions remember that the Angles at the ſecond place of Obſervation, see Figure 100
Thall be either
juſt the double, if you go nearer
to the Place, or elfe juſt the half if you in the follow-
go further off than the Angle at the firſt place. Therefore the firſt Angle you may ing Schemes.
take at Random, no matter what it is, ſo you be careful to obſerve when you be juſt a
upon the double, or the half; ſo that by Calculation you may reſolve it almoit with as
little trouble as a Right- angle, which is made plain thus. In the Triangle ABC the
180
acute-Angle being the outward at B, being 45 degr. the obtufe or inward- Angle being
the Complement thereof to 180 degr. muſt be 135 degr, and the Angle at A being - 450
22 degrees 30 msic, being added to this, makes 157 degr. 30 min. which Subſtracted 135_o
from i do degr. there muſt needs reſt for the Angle at C 22 degr: 30 min. Now chis 22 - 30
Angle at being equal to the Angle at A 22 degr. s; therefore the ſide A Boppoſite tờ 157 - 30
the
one Angle, mult needs be equal to the fade B Coppoſite to the other Angle, as you
18000
ſee by this Cafe.
d. 22 - 30
And by theſe
As the Sine of the Angle ACB 22 degr. 30 min. 958283
to the diſtance Sailed 12 milė A B metoda
107918
So is the size of the Angle CAB 22 degr: 30 min
poſite fides.
to the diſtance BC 12 miles. -
1066201
To find the 5 As the Sine of ACB 27 deg. 30 min. 958283
diftance AC2. is to AB 12. miles.in
So is the Co-fone of 135. d. which is 45 d. 984948
to the sliſtarce from the firſt place
of Obſervation AC 23 miles parts. 134583
Rules you may
find all the op-
2.958283
107918
1092866
7
};
1
1
1
1
ī:
1
!
:
Dad
CHAP. VII.
1
24
I be Art of Plotting a City, Town, or Book V.
bono
TO
abocholtenboek opboude oortree
condo
hoofd
ററവും
Lanangalanche
ណ្តុរ
CHAP. VII.
.
moir
ir
1
How to take the Diſtance of divers places one from the other,
remote from you, according their true Situation in Plano, and
to rotra& (as it were) a Mapp thereof by the Compaſs and.
Pplain-Scale.
He Problem ſerveth chiefly to deſcribe upon Paper or Patchinent all the
moſt Eminent and Remarkable places in a Country, Town, or Cicz:
whereby a Mapp. thereof may be exactly made by help of a Table of
Obſervations following, as with a little Practice you may foon per-
T
ceive.
Upon ſome high Piece of Ground make choice of:2 Stations as A and P, from whence
you may plainly difcern all the Principal Places which you intend to deſcribe in your
Mapp; then at A Plant or ſet your Comp.:s fixed, and turn the Index about top and
let A and P bear one of the other North and South, as you ſee marked with the Letters
R and S: and then direct your Sights to the ſeveral Marks from A to B, CDEFG
HIKLM obſerving what degr. the Index cutteth. As ſuppoſe your Inſtrument fixed
at A, and the sights directed to B, the Index cutreth NĖ: 83 degr. so min. and like-
wiſe the Index directed to C, curs 82 degr. 5 min. and ſo in like manner take the reſt of
the Angles, as you ſee them in the Table following, which were noted down by you in
a: Paper-Book when they were taken.
1
i.
1
82 : os
I.
20
45 : 26
A
A
09 :'oc
oo"
1
:
.
Places Angles: deg. min. )
AB
NE
83 : 50
AC SE
AD SE 64 : 50
The Stationary Diſtance
AE
SE
56:
AF .SE
730 Perch; or 2 miles AG SE
41 : 30
go Perch.
AH SE
24 : 40
AI
SE
AK w II
со
.
AL... 1 16 :
00
KAM SW
23:00
::..8:::
Next meaſure the Stationary diſtance A P, which was found 730 Perch, which you
muſt Nore down likewiſe in your Book; then plant your Compaſs, and fix him at P,
that the Chard may ſtand North and South on the Stationary-Line PA, then turn the
Index to your firſt Mark K, the Irdex cuts NW 24 degrees;. Likewiie turn the jights
to I, and mark-the-Inclination to the Meridian, and sut it down-Nw 17 degr and
ſo do by all the reſt of the forme: Marks or Points; and Note them down as you ſee in
this Table PK:PL:PM: P1: PD: PB:PC:PE:PG:PF:P H: and where
the Lines Interſect each other, drawn from the two Places r. and P, there muſt
Sco Figure 1 10 ſcribe the ſeveral Places, to which you made Obſervation, where you may write the
Laſtly,
و را
.م
you de
& Name of the Places.
1
Chap. VII. Village By the Mariners Sea-Compaſs. 25
00
OO
:
00
oo
оо
+
1.
33 :
43 :
so
40
IO
QO
20
73 :
Places Angles Deg. Min.
PK NW
24 :
PL NW
17 :
PM NW
12
PI NE
9 :
PD NE 21
PB
NE
PC NE
PE NE
54:
PG NE
64 :
PF
NE
PH
NE 87 :
IS
See Figure 101
Laſtly, If you would know the Diſtance of any of the Places thus deſcribed, one
from another, you have no more to do, but open your compaſſes to the two Places on the
Paper, and then apply it to the fame Scale, by which you laid down the ſtationary Dift-
ance AP, which in this Figure was laid down by a Scale of 20 Perch to an Inch: the
like is to be underſtood of Fathoms, Yards, or Feet, and ſoʻ applyed, it will without
farther trouble effect your deſire.
And you may Protrait it by help of your line of Chords, and Line of equal parts, as
this
you ſee is done, or by the help of your Protractor,as before directed : and if there
any
other Notable Caſtle, or lower, or Place, lying in a right-Line with your Ob-
ſervation upon any Hill, you muſt remember always in taking of Inacceſſible Heights
and Diſtances; as alſo in Plotting Unpaſſable Diſtances, by reaſon of Water, that you
take theſe two ſtationary diſtances as far alunder as may be.And if at any time you require
the Altitude of a Church, Caſtle, or Tree, ſtanding upon a Hill, you muſt perform it
at two Operations ; firſt by taking the Altitude of the Church, or Caftle, or Tree to-
gether as one Altitude; and ſecondly by taking the Altitude of the Hill alone; then by
ſubſtračting the height of the Hill from the whole height, the remainder ſhall be the
height of the Castle, or the like. .
And here Note alſo, That in the taking of all manner of Altitudes, whether acces-
fible or inacceſſible, you muſt always add the height of your Inftrument from the Ground
to the height found, the total is the true height. And thus much briefly touching this
is
Matter.
"
1
4
!
1
!
Ddd 2
THE
1
1
26
Book V.
Gaging of Veſsels .
1
jn an Ale-Gallon, which I believe is the Truth: But that which is received by Autho-
as Corn 272 Inches. Theſe
The ART of
Chap. VIII.
The Uſe of the Line of Numbers, and the Lines on the Gaging
Rod or Staff, and the Rules in Arithmetick in Gaging of
all ſorts of Veſſels, (viz.) to Gage a Cube-Veſſel, to Mea-
fure any Square-Vefſel, and a Cylinder-Veſlel; Alſo, Bar-
rels, Pipes, or Hogſheads; to Meaſure a Veſſel part out,
to Mealure a Brewers-Tun, or a Maſh-Fat, to Meaſured
Cone-Veſſel, to Meafúre a Riſing or Convex Crown ; and
alſo a Convex or Falling Crown in a Brewers-Copper ;
alſo a Brewers Oval Tun.
PROBL.I.
The true Content of a Solid Meaſure being known, To find the Gage-Point
of the Jame Meaſure.
He Gage-Point of a Solid Meaſure is the Diameter of a Circle
whole ſuperficial Content is equal to the ſolid Content of the fame
meaſure; ſo the ſolid Content of a Wine. Gallon according to win
chefter menfure, being found to be 231 Cube-Inches: if you
conceive a Circle to contain ſo many inches, you ſhall find the
Diameter thereof to be 17:15 by this Rule. Example. A
Wine-Veſſel at London is ſaid being the 66 Inches in length,
and 38 inches in the Diameter, would contain 324 Gallons.
If ſo, by the Line of Numbers we may divide the ſpace be-
tween 3 2 4 and 66 into two equal parts, the middle will fall about 146, and that diſtance
will reach from the Diameter 38 unto 17:15 the Gage-point for a Gallon of Wine or
Oyl after London meaſure : the like reaſon holdeth for the like meaſure in all places.
Thus likewiſe you may diſcover the Gage point for Ale.meaſure, an Ale-Gallon,
hath been of late diſcovered containing 282. Cubique-Irches, for asri is to 1: 2733 fo
is 2 82 to 356, 3 whoſe Square-root is 18:95 the Cage-point for Ale-meaſure, becauſe
of Waft and Soil exceeding that of Wine above two Inches: or you may find it as before
by the Content 256, 3 and the length 56, and the Diameter 38, as before. There are
ſeveral other Rules to find it, but theſe may fatisfy to ſave Prolixity, Mr. Phillips
, and
others, have found and proved by Example
. That there is 288 Cabique leches
rity, are theſe forts of meaſures, the wine-meaſure is 331 (ubique Inches, and
Rules are undeceivable with Authority.
C
i
1
:
1
EU
mi
as
Therefore
+
1
i
1
9
į
CHAP. VIII.
The Art of Gaging of Veſſels.
27
Therefore take notice you muſt be very careful in all your meaſures of all ſorts of
Veſels, their length, breadth, and Jephth, as alſo of the Head and Bong; for all ſmall
Errours in them ntay increaſe too much in the Content: for the miſtake of a quarter of an
Inch in a large Veſſel, may make you miſreckon a Gallon in the Content ; therefore how
to be careful is beſt known to the Practicioner more than I can declare by many words.
PROBL. II.
Bogy
The Defiription of the Gaging-Rod, or Staff.
He moſt uſeful Gaging-Rod is 48 Inches or so in length, upon one ſquare there
T is 2 Lines, a Line of Numbers, and a Line of 48 Inches, every inch divided into
10 parts for the ready meaſuring of any Veffels, length, breadıh, or depth.
But for the meaſuring of Great-Veffels, there is cwo Staffs divided into Inches and
to parts, made to ſide.
On the ſecond ſide is two Lines, the firſt to Gage by the Head, and the ſecond by the
which added together multiplyed in the length, will give the Contents; Ás by
Example in the following Problem, and uſe of a Table of wine-meaſure.
And the third ſquare is two Diagonal Lines, for the Gage of wine the firſt; and for
Ale, the ſecond; which ſhews the Contents to the part of a Gallon according to 282
Cubique-Inches in a Beer or Ale-Gallon, the Uſe in Probl. 7.
On the fourth fidèis a Line of Segments, or 6.3 Gallons, divided into 1000 parts, as
have the Ufe by the following Table in Probl. 8; The making of this staff is
beſt known to the Inſtrument-Maker, by reaſon it muſt be exactly done, and you may
have them of Mr. Philip Standridge in Briſtol
, and by Mr. Hays, and John Brown in
London, Mathematical Inſtrument-Makers,
1
you may
mano
PROBL. III.
х
2
The Deſcription of Symbols of words for Brevity in Arithmeticki.
VV
Here theſe following Characters, are placed, you are to Work by theſe Rules;
and that will reſolve your Queſtion.
Ploss or Addition, which is as much as to ſay.add:
Minus or Subſtraction, then you muſt fubſtrat.
In or Multiplication, now you are to multiply.
To Divide by 2 or any other Number under the line.
Equal to the thing deſired.
9 Square the Number given.
2-9 Twice Squared, when 2 ſtands before the Letters.
C Cubs the Numbers,
Z Sum and Z9 Square of the Sum.
Va To Extract the ſquare Root.
Ž. The Sums of the Squares.
* Difference, and X. difference of the Squares.
X9 Squares of the difference.
£ The Rest angular or Plain of them, which is the Produit of 2 Numbers
multiplyed.
:
11
1
PROBL, IV.
5
}
1
t
?
#
28
The Art of Gaging
1
Book V:
PROBL. IV.
How to Meaſure a Cubical Veſſel.
1
S
Uppoſe we have a Cubical Veſſel to meaſure, whoſe ſides let be ABCDEF, which
let be every way 24 Inches, and I deſire to know how many Gallons of Wine or Ale
Sce Figure 1oz the faine will hold.
in the follow-
ing Scheme.
For Beer or Ale by the Line of Numbers.
Extend the Compalles always from the Gage-Point (which for Ale is 16 };) unto
the ſide of the Cube 24 Inches, the fame extent will reach from the fame 24, turning
ewice over unto 49 Gallons, and better.
For wine.
Extend the Compaſſes always from the Gage. Point, which for wine is 15 i- unto the
fode of the Cube 24 Inches: the fame extent will reach from the fame 24, turning twice
over unto 59. Gallons, which is almoſt 60 Gallons of Wine,
1
The Arithmetical way.
AB, C=Gall.of Ale 49. AB, C=Gallons of Wine 59
282
231
PROBL. V...
1
1
1
How to Meaſure any Square Veſſel.
Uppoſe we have a Square Veſſel to Measure, whoſe fide AB let be 72 Inches, and
breadth AC 32, and the depth CD 8 Inches.
See Figure 103
By the Line of Numbers for Ale,
You muſt firſt find a mean proportion between the length AB 72, and the breadth
AC 32, by multiplying it together, and taking the Square Roos thereof, or taking
the middle way between 72, and 32 on the Line of Numbers , and you will find
it 48 for the mean.
Now Extending the Compaſſes from the Gage point 16 to the mean Number 48
Inches, the ſame extent will reach from the depth CD 8 Inches, turning twice over
unto 65 Gallons.
For Wine.
To find how many Wine-Gallons it is, Work by the Gage-point is to as you did in
the laſt Rule, and you will find near 79. Gallons; or you may find a mean proportion
between the breadth AC 32, and the depth CD 8: which will be 16 Inches, and ſo
Work according to the former Rule.
Hew to Work the ſame without the Gage-Point.
Example for all
. Extend the
Compaſſes from the Ale-Gallon 2 82 unto the length AB
72: the fame diſtance will reach from the breadth AC 32 unto 8:2 Gallons: for an
Ínch depth, ſo for 8 Inches you may preſently find it to be 65 Gallons.
For Wine-Gallons.
Take the Numbers 2 3 I the Gage-point, which by the former work you ſhall find
9. Gallons for i Isch depth.
The Mathematical way.
ABX: ACX CD
Ale-G allows 817
2 S2
ABX AC XCD
Wine Gallons 9.225
231
The Brewer's Coolers are meaſured all one as this Peffelis.
PROLL.VE.
1
1
1
Chap. VIII.
The Art of Gaging of Veſſels.
29
PROBL, VI.
1
How to Meaſure a Cylinder Veſſel.
Suppoſe the Diameter of the Head AB be
. 20 Inches, and the length thereof AC
be 30 leches, To find
the contents in Ale.Gallons. Extend the Compaſſes always See Figure 104
from the Gage-point, which for Ale is 18 ? - Inches unto the diameter 24 Inches; the
ſame difance will reach from the length 30 Inches turned twice over unto 48 Gal-
lons
For Wine.
Extend the Compaſſes from the Gage.poizt 17. unto the diameter 24 ; the fame
diſtance will reach from 30 turned twice to 58; Gallons.
The Arithmetical way.
For Ale. АВХАС
Ale-Gallons (viz.) 48,5
For Wine. 'ABqXAC
-Wire-Gallons, (viz.) 587
294
359
PROBL. VII.
Figure Ios
D
How to Meaſure a Globe-Veffel.
Suppoſe the diameter of height of the Globe be A B 24 Inches: Then to know the
For Ale.
Extend the Compaſſes from the Gage point, which is 23 id unto the diameter AB
24 Inches; the ſame diſtance will reach from the fame 24 carned twice over unto 25
Gallons of me,
For Wine.
Extend the Compaſſes from the Gage-point 2 1 unto the diameter 24 turned twice
over, as before, you ſhall have 31 Gallons.
The Arithmetical way.
For Ale AB.C..
Gallons, (viz.) 25
540
For Wine,
ABC
Gallons; (viz.) 31
440
23
20
1
60
3
7
21
z(1 (2
1
To
१०
PROBL. VIII.
23-0
20-0
How to Meaſure a Barrel, Pipe, Buit, Punching, Hogſhead, or ſmall 43—
ci Cask
21-50
3
100
3
I
vu
3
Suppoſe you have a Cask to meaſure, whoſe length is AB 27 Inches, and depth ar
the Roug CD 23 inches, and breadth at the Heat E F 20 Inches.
you are to find a mean-diameter berween the Head and the Bong by theſe Rules.
Take the difference between 23 and 20, which is 3 ; which being multipłyed always 300
by z the product, here is 21, and divided by 10, the Quotient will be 2 to which added
to the Teffer diameter
20, you have 22 for the mean-diameter.
Another way to find the mean diameter is thus. A Vesſel having 20 Inches diande-
30.016
ter at the Head, ?: Inches at the Boxg, I would know the mean-diameter: 20 and 23 45
makes 43, the half is 21: so the leiler taken out of the greater', the difference is
3, which reduced into 10 is 300; then divide by 45, the Product is so added to 212
makes 2 2 3 Inches the mean-diameter required.
Then
..!
21S
O 6
22
30
2.::, The Art of Gaging
Воок у
ܘܘܕ
2 Gallons (viz.) 36
Gallons 44 566
72
Compares from
231 to 282,
Then for Ale.
Extend the Compaſſes from the Gage-point always 18. unto the Mean-diameter
22.4., the fame will reach from the length 27 Inches turned twice over, to 36;?
Gallons.
For Wine.
Extend the Compaſſes always from the Gage-point 15 unto 22 :, the fame will
reach from 3 7 to 44 and Gallons, as before.
The Arithmetical way.
M ſtands for
For Ale.
CD29+: EFqXAB
Meani, D for
Diameter,
1077
MD for meana
For wine.
CD 29+:EFqXAB
Diameter.
880
There is another way to Work this Veffel or Queſtion, by the Mean-diameter which
See Figure 10k was before found to be 22 1. Inches; and that is after the Cylinder-Vellel, which may
be reſolved by the Line of Numbers, as before, and by
Aritbmetick, thus.
For Ale. MDX AB
Theſe fort of
Gallons, (viz.) 36.
Cask Gaged s
359
ſeveral
ways. For Wine. MDXAB
Galons, (viz.) 447 ferr.
294
By the Diagonal Line on the Rod or Staff.
To know what Take the meaſure with your Kod from the Bong hole at C to the lower part of the
it fhiall hold in He-d at F, as the Line FĆ, which in the Exciple is near 251. inches : ſo if you
all.
would know how much Alc the Cask will hold, you ſhall find the Bong Hole to cut in
the Diagonal Line 36 %. Gall. And for Wine it will cut 444 Gall, the contents required.
Extend the
A Table for the Gaging of Veſſels.
the fane cx-
Head. Bang. Head. Bong.
The Uſe of the two Lines uspon the Rod
"tent will reach
D Gpts. Gois Gps. Guido
from the Cone
marked Head and Bong; and of this
ten! in Irise-
ol 0,001 0.002 3:1,08912,78 Table for Wine-meaſure.
020,004 0.009321,160 2.32 I
03 0,010 0.020 33 1,234/2.468 The Uſe of this Table is the root of
Contents in all
04 0,018 0.636 34/ 1,310 2.620
the uſual Making and Uſe of the Lines
Gallons.
05 0,028 0.056351,3882.776
on the Rod or Staff only. In the Table
06 0,041 0.081 36.1,469 2.938
you have the perfect Number, bur
By the ſame
07 0,0560111 371 1,551 3.102 you muſt number upon the Staff, for
Rule you may 0810,072 0.1453811,636 3.272
what
10 account 100, and every ſmall Di.
Ale-Carte will 0910,092 0.183139 1,724 3.448 viſion is 10 ; and you muſt eſtimate
hold in V Vine. 10 0,113 0.226140 1,8133.625 the parts of theſe ſmall Diviſions: then
II 0,1370.274 411,904 3.809 is the Work all one as with this
12 0,163 0.326 422,000 4.000 Table, (viz.)
130,192 10:383 +3 2,096 4.1.91
40,222 0.444 44 2,194 4:288 You muſt meaſure the Diameter
150,255 0.510 45 2,296 4.588
16 0,290 0.580 1 46 2,398 4.796
firſt at the Head, and find the Numa
17 0,32810:557 47 2,504:5.097 it ; then meaſure the Diameter at the
ber in the Table, or Staff belonging to
180,367 0.734 4812,611 5.222
Bong, and likewiſe in the Table, or op
190,409 0.818 49122721 5.442
the Staff, find the Number belong.
20 0,453 0.90650 2,833 5.665
ing to that, then add thoſe two toge-
21 0,500 1.000 512,948 5.895
ther, and workiply the Sum thereof by:
22 0,548 1.097 52 3,065. 6.129
the inches of the Velleis length, mert
23 0,60011.199 5333184 6.367 ſured from Head to Head in the init
24 0,653 1.30554 3,305 6.609 ſide.
250,703 1.416 55 39428 6.856
2610,766 1.532156 3,554 7.108
27 0,826 1.69257 3,682 7.364
28 0,888 1.777.5813,813: 7.625
29 0,953 1.9065913,945 7.8 90
30:1,02012.040160 4,080 %.1 60
Gallons 44 1.
to 36 1. the
+
The Diameter in Incbes.
:
1
.!
!;
The
31
Ву
371.2
i
6223
a,
2390
61
$976 13387
2
10
7672
737029 4706 8 1846
6944 / 26.433815|1235
140: 681125 4213 4 1038
Chap. VII. Of all ſorts of Veſſels.
The Table and Staff Shews for 20 Inches at the Head.
0, 453
For 20 Inches at the Bong.
1,199
Theſe 2 added together, make
1 652
27
Which being Multiplyed by
U1564
27, the length,
3304
Makes
44604
According to this Operation, it ſhould be 44 G allons nous parts, which difference is
of no Moment in theſe Concluſions.
The Table of Segments.
PROB L. VIII.
Gal. l'arts. Gal. Parts. Gal, Parts
the Line of Segments on the Rod or
63|10000 42 6288
Staff, and alſo by a Table , How to
3647
9705
62
95304161582013582
find the Quantity of Liquor in a Cask
60941 3517
that is part full.
9:80 40 604019,3452
9170
Uppoſe you would know the Quantity of
6019065 39 5973 181332 1 Liquor in a Cask whoſe depth at the Bong
8962 15850
3255
is 23 inches, as before, and let the Liquor be
59 8862 381578717 3189 in beight 16 Inches, and the whole Cask to
8765
See Figure 107
5724) 3123 hold 44 Gallons.
58 8661 37 5662 16 3056
8580 56002986
By the Line of Numbers on the Staff, the
57 8491 3615535 15 2918 proportion will be as the whole depth 23 Inch-
8404
54761.1 2847 es is to the depth in Liquor 16 inches, fo is
5618319355415142775
1000 to 392 parts.
8236 5354 2703
Which being fought for in the Segment-line
551 8154 34 5294 13 2630 on the Staff, you shall have in the Line by it
8072
52341 25501 46 Galonsa
54! 7990 33517412 2481
Now if
7909 151152405
you extend the Compaſſes from 63
53 7829 132, 's057 11 2323 00 467. Gallons; the ſame diſtance will reach
sooo 2250
once from 44sh, the Contents of the whole
fa
6
4943
the Cask:
7595
4885 2091
33-To
SI 7519 30 4826
Then by this Rule always as 63 Galors is to
7444
476050|1928 467. Galons, fo is 44 Gallons to 33
Cask.
fo. parts of a
il 7297
4646|50|1764
49:7225 128:4585,1 7 1681 By the ſame Rule Work for Beer or Ale. Gallon: by
7153
4543-501509 To Work this Arithmetically is fomewhat fame Rules
4.8
2082 27 446276 1420 tedious ; wherefore I have here Caléulated a you may
4400 501329 Table whereby you may perform it very eaſy make him
Rule
年 ​**75 50 1138
Examples
In the laſt Vesel whoſe depth at the Bong is The top or
4150501935
Fong is up-
15.6679 24 4087131035 | 23 Inches, and depth in Liquor 16 Inches
1 4024 50
permoſt in
446548 23 3960
The firſt Rule of Proportion.
the Column
720
682 390650
As : 3 is to 15, ſo is 10000 to 69 21 parts, to the left
43 6.418223842
which fought for in the neareſt Number in the hand 63 Gal.
sable of Segments you fhall have againſt it
the bottom is
353
295
?S_ Gallons neareſt.
to the right
hand Colum.
Eee
Then
1
7758
2
1)
9
20IO
This table is
made to or
2
So
2
7012
Quarts or
Pints.
6877
6679
6613
830
2
602
I
470
13777150
45
is
1
11
vremena
32
The Art of Gaging
Book V
Then again.
23-7: 10000
3043 anſwers
76
As the whole Radizes 63 Gallons is to 407., ſo is the Gage of your Veffel 44
to 3049
Gallons to 33. Gallons near, as before.
After this manner of Working you ſhall have for 7 Inches depth of Liquor In
gallons.
63; 15. 76
And ſo by theſe Rules you may work for any other Cask.
10 =
Il 1. Gall.
PROBL, IX.
IS: jou
1
How to Meaſure a Brewers Tun, or a Maſh-Fat.
L
Et the Tunbe ACDE, whoſe Diameter in the bottom let be ED 98 Inches,
and the Diameter at the top AC let be 90 Inches, add both the Diameters toge
thér, you have 188 Inches; then take the half thereof, and it is 94 Inches,
this is the Mean-diameter FG; then get the height of the Tun, which let be AB 40
Iniches. Now to know how many Barrels of Ale or Beer it will hold according to 36
Gallons to the Barrel, you ſhall Work thus.
1
1
By the Line of Numbers.
Extend the Compaſſes always from 113., which is the Gage-point for a Barrel uinto
the Mean-diameter 94, the
fame diſtance will reach from the height 40 Inches turned
twice over vinto 27 of a Barrel.
The Arithmetical way by the Mean-diameter.
FG, XAB
Gallons 9841
359
Which being divided by 36, you have 27 Barrels 121 Gallons!
5
1..
Or thus for Barrels.
FGqXAB
Barrels 27 km, uear as before
12924
This Arithmetical way by the Mean-diameter is not abſolute true, yet near enough
for Brewers Tuns, by reaſon there is difference of Diameters between the bottom and
the top; yet it is ſeldom above 7 or 8 Ipches : But to have an Exact way which allo
ferverh for Coopers or any, take this way for Working this Tun for an Example.
The trueſt Arithmetical way.
E Dq+AC9; + EDX. ACZ, XAB
==985 Gallo y
1077
Divide 985 Gallons by 36 Gallons in a Barrel, and the Quotient 27 Barrelse
and 13 Gallons remains : ſo the one will hold 27 Barrels 13 Gallons de parts.
1990
Or thus for Barrels
EDq+ AC9, +EDXAC.ZXAB
38772
Barels 378
Too!
j
PROBL.
.
Chap. VIII.
The Art of Gaging
33
PROBL. X.
How to Meaſure
a Cone-Veſſel, frech as is a Spire of a Steeple, or te like,
by having the Height and the Diameter at the Bafe.
Suppoſe the Diameter at the Baſe A B be y8 Inches, and the height DC 492
Then by the Line of Numbers for Barrels of Ale or Beer.
Extend the Compaſſes from the Gage-point 169 iunto the Diameter A B 98, the
ſame will reach from the height of the Cone DC 490, turned twice over unto 1214: See Figure 109
Barrels fere.
This is the beſt Proportion to Work for great Cones to have it in Barrels, but ſmall
Cones have it in Gallons.
Then thus Work,
Extend the Compaſſes from the Gage-point 32 .. unto the Diameter of the Baſe 98,
the ſame will reach from the height of the Cone 490 twice turned unto 436941, Gall
.
The Arithmetical way.
For Gallons.
ABgx DC
4369 37. Gallons.
Which being divided by 36, you have 1211. Barrels.
1
Or thus for Barrels?
AB, XDC
121
347. Barrels.
38772
An Example,
The Brewers Tun before meaſured may be meaſured, after this manner by Cones,
by this Example in this Figure I have proportioned the lame Tøn in this Cone, as you
may
, prove thus by the Rule of Proportion to find thie Diaineter on the top E É
was before.
Work thus,
As CD490 is to AB 98, ſo is CG 450 to EF 90 Inches; and ſo back again,
to find the height
of the greater Cone , fay, as the difference of the Diameters $ inches
is to the height of the Tuin 40 Inches : ſo is the Diameter of the bottom AB 98 Inches
to.the greater height DC 490 Inches, from whence ſubftract 40, there remains the
height of the leſſer Cone G C 450 Inches.
Now Working as before, for the Contents of each Coxe.
The greater Cone will be found to be
4369 Gall.
And the leſſer Cone to contain
3384 Gall.
Which ſubſtracted from the greater Cone, there
remains 985 Gallons to Parts.
985 Gallons
For the Brewers Th11, as before found in the 9 PROBL, which is 27 Barrels
13 Gallans,
90, it
*
547 Parts.
432 Parts.
IIS
1000
I
ܘ1
Tee 2
PROBL. XI.
1
}
34
The Art of Gaging
Book V
PROBL. XI.
How to Meaſure a Segment or portion of a Globe or Sphere, which ſerves
for a Convex Signet or Riſing, or Falling Crown in a Brewers Copper.
Dmit you have the Diameter of the Crown AB 80 Inches, and the height there
A А
of CD 6 Inches.
A Convex Riting Crown.
See Figure 1 10
The Falling Crown is nothing but this Figure, the upper part turned down.
Note that a Crown is feldom leſs then 2 Inches, nor above 12 Inches; for in Briſtol
in all their Crowns belonging to the Breivers Coppers, the leaſt that was found, was in
Inch, and the greateſt height or depth u : Inches.
By the Line of Numbers.
Extend the Compasſes from the Gage-point 18, unto the Dianeter AB 80, the
fame diſtance will reach from half the height CD 6, which is 3 being turned twice unto
53 Gallonsfere.
The Arichnesical way for Ale or Beer-Gallons.
ABg.X;CD
53: Gallons.
359
1
PROBL. XII,
1
How to Reduce Ale-mcaſure into Wine; And likewiſe to Reduce Wine
Gallons into Ale.
For Example.
TH
Here is a veſſel that holds 60 Gallons of Ale; the Queſtion is how many Galloss
of wire it will hold.
The Proportion of the backward Rule of 3.
As 282 Ale is to 231 Wine :: fo 60 Ale to
Orthus. As 94 is to 77 ::fo is 60 Ale to
73; Wine-Gallons.
The reaſon is thus, 231 Ale-Galons is 282 Wine-Gallons, or 77 Ale-Gallons is 94
wine-Gallons:
Or, as 2 82 to 231, fo is 116.410 958 30, and as 231:282, fo is 95:30 to 116
1, or extend the Compaſſes from the Ale-gallon 282 to the tVine-gallon 231, the ſame
diſtance will reach from 60 to 73 i Gallons, or from 77 to 94, or from 9400 77.
!
1
+
PROBL. XIII.
1
How to Meaſure a Brewers Oval Tun.
СА.
Sec Figure in
LEt the length in the bottom be A B 1 20 Inches; and the breadth E F 90; let the
length at the top be CD 112 Inches, and the breadth 84; alſo the depth 40 Incher
A Brewers Oval Tun.
Now to Work this, you muſt find a Mean-proportion between the length in the bat-
toni 123, and the breidth go Inches.
The Arithmetical way.
A Bx EFV9 = 103-92. That is, Multiply the two Nambers together, and
of the Product thereof extract the Square Rent a lo hall you have the Mean-propora
tional Number
By:
1
23.
E
A
27
2
to 6.
F
.
109.
.
20
116
49:6
107.
G
E
9:0
F
*
The Ture
40
B
A
9:8
D
C с
go
40
G
64
94
A-Brewers Ovall Tun.
B
E
98
A
H
108
(!
C с
84
112
D
G
D
40
F
b
А
C
B.
A
120
B
+
110,
go
111,
E
Book.V. p. 34-35
חד
1
-
1
1
1
1
1
3
35
CHAP. VIII. Of all ſorts of Vefſels.
By this Rule you will find a Mean-proportion between the 120 at the bottom, and
the breadth 90, to be 103 and 92.
And likewiſe for the top between 112 and 84, will be 97; then as before, you ſhall.
find the Sun to be 200-92 ; the half thereof is the Mean-diameter 100; 5.5 inches ;
ſo Thall you Work all one, as you did in the Round Tun.
Houd to get the Mean-diameter by the Line of Numbers.
Let the Numbers given be 120, and 20. Extend the compaſſes from 90 to 120:
Divide that in half, the ſame diſtance will reach from 90 to 103 %. almoſt 104 the
Mean Number required; and fó likewiſe between the Number 112 and 84, you will
find it 97 ; then as before, you ſhall find the Sum 200:92 andį 100inches.
The Arithmetical way, as before, is thus.
M. Dianı:X CA
31: Barrels of Ale or Bçer.
12934
219
And in Gall, M. Diam. XAC=1119 Gallons of Beer or Ale.
1000
+
By the Line of Numbers for Barrels.
Extend the Compaſſes always from the Gage - Point 113 is to the Mean-diameter
100 14: the ſame will reach from the height 40 turned twice over, to the Quantity
of Barrels of Ale or Beer 31;
PROBL. XIV.
..
How to Gage a Veſſel by Oughtreds Gage-Rulc.
THis is an Inftrument by taking the length in Inches and 10 parts, and is as Exact as
any way
Inftrumental extant; both the Diameters at the fieat and Bong, with a
Line called Oughtred's Gage-Line.
The Uſe is thus.
Take the Diameter at the Bong with thoſe Diviſions before faid from that end where
the Diviſions begin to be numbred, aud fet that down twice : and on the Diameter of
the inſide the Head in this manner, * and then add them together, as here you +
fee the length in Ixches. Suppoſe to be (30. 82) then fay, as 1, is to :
77, fo is 30: 82 to 54 of a Gallan, being a little more then a Gallon,
0,63
or 54 Gallons the Content of a High Country Hogſhead; and fo you may
do by any other great or ſmall ſort of Cask.
0163 2
[ 77
The end of the 8 Chapter of Gaging Veſſels,
t
CHAP. IX.
1
VV
u
F
the Produit is 1254 Feet; which Divide by 324 (becauſe there is ſo many Square
Feet is contained in a Tard, and the uolient is 31 Tards, and 3 remains which
36 The Art of Meaſuring of VValls, and Book ,
To Sun in our in cur si intended in an inain and are
sinaingiaciociniocianidiaciocianinaingioginionici
VACUNOVODAODACOAS
Chap. IX.
Wherein is ſhewed both Arithmetically and Inſtrumentally
How to Meaſure exa&tly all kind of plain Superficies, as
Walls, Timber-work, Roofs of Houſes, Tyling, Board,
Glaſs, Wainſcot, Pavement, and the like ; as alſo Timber
and Stone.
PROBL. I.
Oraſmuch as it is very requiſite for a Compleat Artiſt to know how to Mea-
fure all manner of Buildings, as Walls, Timber-work, Tyling, and ſuch like;
I Mall in the following Example make Illuſtration thereof.
Note this, that Walls and Tyling are meaſured by the Rod of 18 Fert,
Wainſcot by the Tardor Feet, and Board and Glaſs by the Foot only. There-
fore meaſuring any of theſe things, confideration muſt be had to the juſt
Form and Figure thereof : Then by the following Rules you will ſoon have the Arca
Content thereof.
As for Examples
Suppoſe there be a Wall in the
Form of the Figure, and it is required to know how
many Perch, or Rods, Yards, and Feet are contained therein.
The Arithmetical way for Perch.
ABX AD, or thus, ABXAD, ZX2
16
Extend the Compaſſes always from 16 to the length 66, the fame Extent will
reach from the height 19 Foot unto the true Contents of the Wall 76 Perche
A 66
A
VIVA
Then to bring it into Rods, and Feet, and Yards, Work as before. Multiply 66 by
Feet in a Rod,) and the Oxorient is 3 Rods, and 28zremains
, which divide by 9 (for ſo
many ) ,
is Feet; fo this wall being 66 Foot long, 19 Foot high, there certains 3 Reds Erard, 4
Feet (for as 81 Feet or 9 Tards is a quarter of a Rod.
But
11
76 Percb.
19,
37
1
20
1
t
:
:43560
Chap. IX. Dormant-Pikes, Chimneys, c.
But ſuppoſe ABC be a Govel Dormant Pike,
ſuch you muſt meaſure them as Triangles, to bring
it into Feet. Multiply 16 the Perpendicular by
B
half the Baſe AC, the Produ£t is 166, the Contents
in Feer double it : divide by 33 half Feet, the Quo-
116
fient is 9 Perch, 11 Feet remains. Work by the
Line of Numbers as in the laſt Rule, and it will be
A
9 Perch så, (or : divide i 60 by the Quotient 17
Tards and 7 Feet, remains the true Contents of the
Dormant Pike, that is i Rod, 8 Yards, 7 Feet.
But in Meaſuring of Chimneys which require more Workmanſhip then other ordi-
nary Walls, they are uſually accounted at double meaſure. Firſt mealsire them as firgle-
meaſure. Take the length of the braſt Wall E F, and the 2 ſide Angles DE, and FC,
which Multiplyed into the height C B the Produ&t of that Multiplication doubled,
yieldeth the Content,
According to che Cuſtomary meaſure allowed for
Chimneys that ſtand in a Govel or Side-wall, (but if
I
the Chimney fland by himſelf, the Back is to be
meaſured with the reſt of the Chimney; but the
218
Back. ftanding againſt a Govel or Wall is accounted.
K
part.of the Vall; and muſt not be meaſured with
1024
the Chimney.
Admit this. Figure LIK. GH. ABDC be
á Chimney to be meaſured, and according to double
meaſure, the Content is required,
13
D
C
Firſt meaſure the Baſe CF the Braft-wall D C,
€ 24, f
and FC the ſide Angles
, which together makes24
Feet; next take the height of the Square CB 18
Feet, which Multiplyed together, the Produkt
is 435 ! 60 Fect for the Content of the Figure ABDC Then for theand Brafts
Wall GH, and ſide Angles is 15 Feet, height n H 6:26 Feet : X as before, makes
93: Go Feet, for the Content of the Square GHm: n.
Feet. Parts.
In like manner of Working, you will have the 1
ABDC: :
Contents of the Square IK : RV:92: 16; like- | The Squares. GHmn: 93:90
wiſe the Chimney+SHäfe in compaſs is 9 Feet, and
ZIKRV : 93.: 16
8 Foot high. * together as before, is 72 Feet
The Shrift:IL : 32:06
for the Contents: add theſe 4 Sums together, the
The Sum 693:66.
Som is 693:66 Feetrdoubled is 1 387 Feeti
doubled. 603: 66
Fcat.the Content of the Chimney according to
Crstomary meaſure.
The Toral Sum. 1387: 32
Which reduced into Perches as before, is's Perch 36 Foot ? Parts; or into Rods,
és 4 Röds 10 Cards 1 Foot according to theſe Meaſures : But it is fit the Maſter-
Frorkman Should Meaſure it, and ſhould have ſo much Arithmeticks.as to Multiply and
Divide, or elſe he caññót be a Compléat Workman in every part.
Note that after the fame order Slate-work and Tyling are meaſured either by Perch
or Rods of 1 8 Foot Square. Note that Roofs of Houſes, and Timber-work, Partition-
Floors, and the like, are reckoned by the Square of 10 Foot, but Worked by the fame
Rule, as have been already delivered in this Problem , therefore it needs no other pre-
.
cept..,,
PROBL. II.
G G
ital
1845
1
inte
T
!i;
:
/
38 The Art of Meaſuring Timber and Stone. Book V.
0
Hays in Morp- ber of Inches in a foot of flat-meaſure, and the Quotiext ſhews 10 feet, and
PROB L. II.
How to Meaſure Boards, Glaſs, Pavement, Wainſcot, and the like.
N the laſt Problem we have Mewed that Boards, Glaſs, Pavement and Wainſcor,
I
and the like, they are commonly accounted by the foot or Yards . Therefore to
make this plain, we ſhall inſtance only upon Boards which are cut out in long Squares
commonly.
How to Meaſure them.
Take the length and breadth in Inches and Parts, Multiply one by the other, the
Produit will thew the context in Inches ; (that divide by 144 the Numbers of Inches in
one Foot, the Quocient will tell you the Nunsber of Fees, and the remainder is Inches.
For Example.
Such Rilles are
Admit I have a Board that is 7 Foot long, and 18 Inches broad; Multiply 84 Inchi
made bywalcr which is in 7 Feet by 18 Ixches, the Product is 1512; which divide by 144 the Num-
Ficlds, and in
remains,
Biially by P." which is: 144; therefore the Board contains so foot }; but many times the Board falls
Staynard. out to be broader at one end, then it is at the other, add together the breadth at each end;
then take the for the true breadth. :
And Work as before, But commonly Artificers have a Uſeful Linc put upon their
Rules for their ready Meaſuring of Board and Timber-meaſure ; but this is the Exaételt
way, though that is near; and what have been ſaid of Board-meaſure, only the ſame is
to be underſtood is the way of Meaſuring not only Boards and Gials; but likewiſe all
manner of Wainſcot, Paviment, Floors, and fuch like; they depend upon one and the
fame Geometrical Ground, though they be reckoned by different meafure, as you ſee
by the Perch, Rod, Square, Tard
or foot according to the Cuſtom of the Place, there-
fóre needs no further Example.
Extend the Compaſſes always from 12 Inches unto the breadth 18 Inches, the ſame
extent will reach from the length 7 foot, unto the Number of Square foot in the Board,
which is 10 za foot.
AB 18 Inch x AD 7 foot
=10 feet.
72
1
1
21
1
PROBL. III.
1
The Menſurations of Solid Bodies of Timber and Storie, and firſt of
Squared-Timber
VV b -
ber, and Stone, and the likes which are uſually meaſured by the foot: and
therefore you are to obſerve that ai foos of Timber or Scone is accounted a foot ſquare
every way in the Form of a Die; whereby it plainly appears that a foot of Timber is
12 tiines more than a foot of woard, which it 344 Inches; but a foot of Timber muſt
be 1728 Inches: ''
}
For Timber that is Squared you may find the Contents thereof on this wiſe ; Firſt
To find the
find a Mean betwixt the two Sides at the End. Admit the height at the End be AC
Square-Roor by
the Tables.
16 Inches, the breadth thereof AB 25 Inches ? By the 7ables -- 120412
the half Sum
139794
Sum. 260206
The half Sum.
I30103
is the Square.
fost and Mean-propertion between 16 and 25 which is 20:
Ву
S Add together
1
1
CHA P.
X.
of Timber and Stone.
39
By the Line of Numbers.
Divide, and take the middle between 16 and 25, and you will find the Mean 20
as before; Then to know how many foot of Timber
is in a Square of 16 Inches
in
height, 25 Triches broad, and 14 foot long.
Extend the Compaſſes always from 12 Inches unto the Mean-proportion or ſide of the
Square 20 Inches the ſame will reach from the length 14 foet turned twice over to
387: foot of Timber.
The Arithmetical way.
Thus. ABXAC. XAD
to the Contents in feet. 3878
144
Orthius. 'ABXACAD
reduced into inches 168 in 14 foot
38 feet, as before found.
L
1728
A common Era
Er-
Yet it is common with the Carpenters to add the broad and narrow fide together, and for amongt
to take the halfiberçof-the true Square that wayisvery erroneous, eſpecially when the moſt Carpen-
difference between the ſide is much,
ters,
1
E
36.
Ai
1:5
14.50 168, Anchos
HURUMLARUANGLAHEIDELBERGELAR
in the former Example one fide is 2.5, the other 16, the Sum 41, the half 20%*
inches; that is, half an inch roo‘mach, as was proved by the former Rules; that is
Juſt zo for the Meax or true Square : ſo that by taking the 2 ſides 20, it makes the
piece of Timber 40 footwhen indeed it is bür 38 :feet, which is a foot and half too
much.
Now if a piece of Timber that
, iş tapering, the Cortimon Rule is to take the Mean
betwixt both'endtand fo to Work as in the lalf Form, but it is not abſolute true.
For Examiple.
Admit a Piece of Timber were Square at one end 25 inch. and at the other 1 siaches,
and 14 foot long.
This is the abſolute Arithmetical way,
Z'AB, DEFÆXAD
Contents-39:15
432
1
:
PROBL, IV.
How to find how many Inches in length will make one Foot of Timber, be-
ing alike in the Squares.
So
Uppoſe you have a piece of Timber that is forroSquare 16 inches every way,
and
you would know how many inches in length will make one foot of Timber.
By the Line of Numbers.
Extend the Compaſſes always from 12 inches to the Gode of the Square, which in this
Queſtion is 16 inches, the fanie turned twice over from 12 inches, will reach to 6 inch,
in length for one foot of Timber.
FES
The
1
1
1
i
1
!
*
-
int
1
!
1428
11
N
.
foot, Then
as a Tree whoſe Diameter
40
The Art of Meaſuring of Timber, cc. Book V.
The Arithmetical way.
192 "75
Ef 6:257 or 6 Too Inches as before:
A C.169
Having the fide of a Square piese of Timber, at the end and the length in feet; to
find how many feet is contained therein.
Admit the ſide of the Square at the end be AC 16 Inches, and the length thereof 14
By the Line of Numbers.
Extend the Compaſſes always from 12 Inches unto the ſide of the Square AC 16
inches; the ſame diſtance will reach from the length 14 foot turned twice over unto 2:4
si feet in the Piese of Timber.
The Arithmetical way.
Cvi AC169. * B D 14 foots
The Content in Feet 243 :
144
PROB 1. V.
How to Meaſure Round-Timber five ſeveral ways.
A
of thicknefs at the end is 20 inches; I deſire to know how many inches in lexgth
will måke one foot of Timber.
مرد
1
1
+
By the Line of Numbers.
.::Extend the Compalles from the Diameter AB 20 inches unto the conſtant Number
23. the ſame distance will reach from the fame 13 turned twice over unto s jinches for
one foots, as ADU:
1
220
.
ABY
The Arithmetical way.
AD5 inches for one foot.
Having the Diameter of a Piece of Timber, as admit it to be 20 inches, and the
length ſuppoſe 15 foot; To find the Contents in feet.
By the Line of Numbers.
Extend the Compaſſes always from 13 744 to the Diameter AB 20 inches, the famo
diſtance will reach from 15 the length turned twice over unto the Contents. 32 já feet ig
1
12srce.
1
1
Acid
15 foot "
15 foot
10
tri
22
namanya memandan
B В
G
1
The
1
/
before.
1
Chap. IX.
Of Timber and Stone.
41
The Arithmetical way.
Squara the Diameter A B 20, and it is 400; treble it by 3,, and it is 1200, mul-
tiply itby the length 15, and the Product is 18000, that divide by 550, and the con-
sents is 32 mii? feet of timber in a round piece or tree, which is 32 foot, and about quar.
Here is likewiſe another brief Rule, Arithinetically thus.
Square the Diameter A B 20, and it will be 400; Multiply that by 11, and it is
4400, divide it by 14, and the Quotient is 314, and 4 remains, which multiplyed by
the length 15,' the Product is 4714 ; that divide by 144, and the Quotient is 32 -2
Orelſe you may
find the Contents of the Circle by this Rule, as 7 is to 22, ſo is the
Diameter to the Circumference; or multiply half the Diameter by half the Circumfe-
rence, and the Product is the Content of the Circle, that Multiply by the length, and
divide by 144, gives the content of the iimber or Tree in feet or purts.
Now the common way uſed by Artificers, is to meaſure round a Piece of Timber or
Tree, and to take the one fourth parț for the Square, which is very erroneons and falſe.
For Example.
The meaſure of the Compaſs or Circumference by the Rule before-going is 62 t: irch.
of the round piece of timber or tree the thereof iş is inches, which they take to be
the
Square ;
which Multiplyed into it ſelf, produceth 2+3 : 50 for the area of the
Bale; which Multiplyed by the length is frot, the Product-is 365490, the Contents in
feet and parts; that divided by 144, the Quotient is 25 is that is differing from the
Truth no leſs then 7. 29. that is, 7 foot and about a quarter too much : the Buyer hath
then his due; hut I conceive they agree in the Price to ſtand to that meaſure, by reaſon
of the waſt in Chips before it is bronght into Squares; but the beſt way will be to msea-
Sure the tree right, and afterwards allow for the Waſt; or elſe in time the Error will
be taken for Truth, and Truth will be accounted Error, as it is by too many this day.
How to Meaſure a Round piece of Tapering Timber.
Admit the Diameter of the Great End of a piece of tapering timber be AB 20
Inches, and the Leſſer End CD 16 Inches, and the length' E F 15 foot. To find the
Contents, odd both the Diameters 20 and 16, the Sum is 36, the half is 18 for the
Mean.
Then Extend the Compaſſes always from 13 14 to the Mean-diameter 18, the fame
will reach from the length is foot turned twice over unto 26 foot. Or by Aritha
netick, Squire 18 the Mean-diameter, and it makes 324; that treble by 3, the Prodat
is 972 ; that Multiply by the length 16, the Product is 1.4586, that divide by 550,
and the Quotient is 26 foor, the Contents of the taper. piece of timber is 26 foot and
half.
PROBL. VI.
How to Meaſure a Pyramedal piece of Timber.
This piece of timber is meaſured by this Rule, (viz.)
A right Lined
AA
Dmit you have a piece of timber to meaſure, whoſe length at the Baſe is 2 5 inches marp Piece of
Timber is
A B, and breadth AC i6 inches : and the length of the piece DE is foot.
1
led a Pyramide
1
5 cor
5 rool
25
اور
Islitact
16
D
E
1
1
By the Line of Numbers.
Firſt, by the Line of Numbers find a Mean-proportion between 25 and 16 by di
viding it into 2 paʻts, and the middle will fall upon 20 inches, the Mean-proportion
reqnired.
Then
Fff 2
1
+
i
42
The Art of Meaſuring of Timber, cc. Book V.
Then extend the Compaſſes always from 20. unto the Meandiameter 20 Ixch.
the fame distance will reach from the length 15 foot turned twice over unto 13 #foos
of timber.
The Arithmetical way.
ABX ACX DE
XDE
= 13 foot parts of timber in the Fiece.
1
S
11
11
888
1000
432
A B ACXIDE
=
13 foot
888
parts, that is 13 foot and above
1000
three quarters.
144
PROBL, VII.
How to Meaſure a Conical piece of Timber.
Dmit
A you had a Cone Piece of timber whoſe Baſe or Diameter at the End AB is
28 inches, and the length thereof CD 15 fuot, it is required to know how many
feet of timber is in the Piece.
A
stot
e
Támot
Footer
11
sirnaal
11.777
7 777 ERR
But the Baſe
thereof Round
it is a Cone.
G
1
1
[
1
Extend the Compaſſes always from 23. unto the Diameter A B 28, the fame
distance will reach from the length 15 foos turned twice over unto 21.1
The Arithmetical way.
A BqxCD
S21, foot of timber in the Cone Piece.
550
**. And by the former Rule you may Meaſure any part of a Cone or Pyramide-piece.
Admit you were to cut a Piece of 5 foot at the greater End, and you find the Diameter
EF 18:95 Inch. Firſt, Mean-diameter 18:95 and 2 8 Inch. added is 46; 95 the
half is 23:48 the Mean: then extend the Compaſſes from 13 ik unto the Mean-dia-
meter 23the ſame diſtance twice repeated from the length 5 foot, will reach to 15
the foot in the of the Cone at the great End; And likewiſe to Meaſure EFHG the
Diameter HG is 9.45 added to 1895 EF, the Sum is 2 8:4.
The is 14; 20 Inches the Mean proportion the lengths foot; by the former Rule
you will find in that Piece of timbers foot; and to Meaſure the little Come GH 9
inches diameter and 's foot long.; Work as to Meaſure the whole Core, and you will
find it is parts of a foot.
And ſo you have truly Meaſured the Pieces, as you may find by
foot. parts.
IS
07 adding them up, and they make 2 1 foot is parts, as you found in the
whole Cone at firſt; and to by finding the Area of the Circle and part,
si you may find the Segment of any Cone or Pyramide that is Square in the
fides by the Area thereof; by the fame Rules you Meaſure śrone. It is.
21 mm 38 needleſs to make more Examples in this thing.
1
50
5
1
CHAP. X.
ܢܙܐ
sa
7
7
Chap. X.
43
ఉంట
porn
ook
va
balone sono
CHAP. X.
1
1
For the Burden of a Ship, or her Tunnage, Take theſe Rules
following:
SECT, I.
1
S
Uppoſe you were to Gage a Ship that the length of her Keel is 45 foot, the
breadıb of the Beam 17 foot, the depth of her Hould 9 foot always to find the
11nnage.
Multiply the breadth by the length; and with the Produkt Multiply the
depth in Hold, and divide by 100, and the Quoricat will Mew you the Tuna
nage to be in this Example 68 tun.
Or extend the Compasſes always from 100 to 17 the breadth, the fame diſtance will
reach from 45 the length; to 7
Then extend the Compaſſes from I unto 7 ) and the ſame diſtance will reach from
the depth in Hold 9 foot to the tunnage 68, twn of King's tunnage.
But for Merchants Ships who give no allowance for Ordnance, Mafts; Sails; Cables,
and Anchors, which is all a Burden, and no tunnage.
You muſt ſiork thus for tlie tunnage
1
4
Sect. II.
45 * 17 *9
=72.45 Tun Burden.
95
Or, extend from the Gage-point 95 always to the length of the Keel 45, the fanie
will reach from the breadth 17 of the Beam to a 4 Number, as to 873; then extend
from 1 to 8.; the fame diſtance will reach from the depth in Hold 9 foos to the Burden
72 tun.
1
SECT. III.
L
Having the Proportion of any one Ship in Burden, with the length of her
Keel-Timbers; To Build another of any Burden according to that.
t
PROPORTION.
Dmit I have a ship of 8o Tun, the length of her Keel is 46 foot. Now I am to
Build a Ship whole Keel muſt be 65 foot; I delire to know how many Tan The
muſt be.
Extend the Compaſſes from 46 foot unto ós foot, the ſame extent will reach from
the Burden So Tun, being turned 3 times over unto 225 is cunc.
The Arithmetical way.
8. X:65
225:57 tonsa
46 C
..
SECT.IV.
1
44 The Art of Gaging of Ships and Veſſels, Book V.
SECT, IV:
I
A
1
had a Ship of 226 tuns, and the length of her Keel is 65 font. Now I
would Build a Ship of twice the Burden; that is 452 tuns ; Now I would deſire
to know the length of her Keci.
Extend the Compaſſes from 226 unto 452, thes of that diſtance will reach from the
length of her Keel 65 foot unto the length of the greater Ship’s Keel 81 zá foot ferè.
The Arithmetical way.
65. C.x 452
81 i fooi fere.
226
I could have infifted upon more Examples, but it is to no purpoſe; by reaſon the
Carpenters have theſe Rules in Practice moſt of them, and for jaging and Meaſuring
Ships, the breadth and ſharpness of her bottom is to be conſidered, and to abate fume-
thing of 95 the Divizer, or add ſomething to it according to Judgement and Reaſon;
and To likewiſe to 100, to find the tannage.
5
1
A
CH A P. XI. .
The Application of the Line of Numbers in Common Affairs, as
in Reduction of Weight and Meaſure of Cheeſe, Burter,
and the like.
I
elle;
25
Have added this Chapter , not for that I think it abſolutely neceſſary; but only
becauſe I would have the abſolute applicableneſs of the Rule to any thing be
hinted at, that it may be known, that any thing may be meaſured by Rule, as
well as by neighi, ſo far as there is Proportion conſidering that, and any thing
the Application of which I leave to the Induſtrious Practitioner, only here
I give a hint.
What have been ſaid of other things in Reduction, is general in any other, as from
1 2 to ro either shillings or Inches to tenths, as of a Shiling, or tenths of a root, or
Pence or Farshings, Ounces or Chauldrons, Yundreds, either weight or tale. The Rule
is thus, (viz.) In either one ſhilling, or foot, hundred, or the like. If 100 is 12 d.
what thall 66 be ? facut 8 pence or · Inches; that is, Extend the Compaſſes from 100
1012, the fame will reach from 66 to 8, and ſo of all other.
If icobe 112 P. what ſhall so be? facie 66 Pound, If 100 be 8 Pines, what ſhall
be? facit 2 Pints ; If 100 be 48 farthings
, what ſhall 30 be? facit 14.4, that is
3 d. 2 f. near.
If 100 be 36 bushels, what ſhall 2 4 be? facie S buſhels and better.
If 100 be 60 min. what ſhall so be ? facit 30 min. or an hour.
If 100 beL: 0, what ihall so be? facit 96. The like is for any Line of Reduction.
Now if
you
would know how many there muſt be in any greater Number then one
then fay, By the line of Numbers thus: If 48 farthing's be one ſhilling, how many
ſhillings is 14+ farihings? facit 3 Shillings. For the extent from 48 to 1, will reach
from 1 44 to 3; And again, if a Mark and a half be one Pound, how many Pound
is 12. Mark? the extent from 1 : 50 to 1 ſhall reach from 12 to 8, which reaſon muſt
help you to call it 8 Pound. Again, if : Nobles be one Pound, what is :12
Nobles?
facit. ofl. the extent from 3 N.o 17. will reach from 312 to 104 : Further, if a
Chauldron of Colis coſt 26 jhil. what lhall awal.coſt? facit 1?.' But more to the
If :
: 6 buſhels coſt 30 s. what shall's bujh. coſt? facit 4.5. 2 d. If one week
be 7 days how many days in 3 9 week? as i ro 7, ſo 39 to 273 days in 39 weeks ;
As
8 furlong; make 1 mile, how much is 60 furlongs? facit 7 miles, for the extent from
I to 3, gives from 60 to 7: 50, and the like of all other.
t
j
Illatter;
CH AP. XI.
Chap. XII.
45
1
Singin mo ang din tiro con din strainatunci adinin cincinetto
gingisdiagoaingias
ciosiQgingIONIO
Singiagiogieriggiogiagiogia
WACVAVAGNAGRUPACOSTAONA
LOCACAUSA DAVVAUDACOMODO
!,
CHAP. XII.
1
The moft Excellent
1
Gunners Scale,
L.
9
Which refolves the Chief Principles of the whole Art of Gunnery,
in a very brief and Compendious form, never by any ſet forth
in the like nature before; with divers Excellent Concluſions
both 'Arithmetical, and Geometrical, and Inſtrumental;
and by Tables being framed both with, and without the help
of Arithmetick. As alſo divers Artificial Fire-Works, both
for Recreation, and for Sea and Land-Service.
SACT. I.
The Qualifications every Gunner ought to have, and the Properties, Duty,
and Office of a Gunner.
H
E ought to have skill in Arithmetick, to work any Concluſion by the
ſingle and double Rule of 3, to abſtract both the Square and Cube Roots,
and to be perfect in the Art of Decimal Arithmetick, and to be skilful
in Geometry; to the end he may be able through his knowledge in theſe
Arts, to meaſure heights, deptbes, breadths and lengths; and to draw
the Plot of any Piece of Ground, to make Artificial Fire-Works which
are uſed in the time of War : A Gunner that hath a Charge ought to have in readineſs
all neceſſary things for his Artillery :
Aswheels, Axle-trees, Ladles, Rammers, Sheep-skins to make Sponges, Gun-powder,
Shot, Tampions, Chain-shot, Croſs-bar-ſher, Carvas, or Strong Paper to make Car-
tredges, Fire-works, Artificial Torches, Dark Lanthorns; again, to Mount and Dil-
mount Guns, Hand-ſpikes, Coyns, Budge-Barrels to carry Powder, and Baskets to carry
Shot to your Piece. When leiſure will permit, he is to choole good Match-cords, to
Arm his Linſtocks in readineſs to light, for to give Fire, and alſo a pair of Caloper Com.
Posjes to meaſure the Diameters of Shot, or the Muzzle, or Baſe-ring, or the like ;
and alſo a ſmall Braſs pair of Scales and weights, a Ruler divided into Inches, and 8
Parts in
every
İnch, for the ready meaſuring of Cartredges, how to fill them.
A Gunner ſhould never be without ſuch a Scale as this as I have here deſcribed, and
to know the Uſe thereof perfectly; and thereby be ready to give a reaſonable anſwer
any Man of any Queſtion belonging to any ſort of Ordranse uſed in England in a
moment
, as this Scale will do, as thall be ſhewn : He ſhould always carry a pair of
Compaſſes with him to meaſure the Diameter or Bore of any, Piece ; and alſo the length
of the Cylinder within, the better to fit her with a shot, and proportion a Charge.
to
A
늘
​4
46
Book V.
The Art of Gunnery.
A Gunner ought to know the Names, -Length, weight and Fortification of every
Piece about the Chamber (that is as far as the piece is Laden with Powder;) and be able
to tell readily how much Powder is a due Charge for any Piece, what Shot is fir, how
many Matroſſes multfattend the fame, how many Horſes or Oxen will draw the ſaid
Piece, or Men, if occaſion be ; He muſt be careful in making Choice of a fober honeſt
Man, for the Yeoman of the Powder; and he muſt not beat up the Head of his Powder-
Barrels with an Iron tool, but with a Wooden Mallet, which can never Fire the ſame: A
Gunser ought to trie his Piece, to know whether it be true bored or not, to proportion
his Charge according to the thinneſt ſide of the Meral, and accordingly take his Obſer-
vation at the Britch of the Piece, juſt over, where by his Ari he finds the middle of the
Bore within the Piece is; by which means a good shat may be made out of a bad Piece.
ܪ
t
Before he makes a Shot, he is to conſider, that if the Piece lie point blank, or under
Metal, he ought to putin & fufficient Wadst after the Shot, to keep it cloſe to the Powder;
for if it ſhould not be clofe, but ſome diſtance between the Powder and Shot, the Piece
will break in the vacant place; but in caſe you morint your Piece, put no wadd after the
Shor.
And one chief thing is to know very well how to Diſport his Piece, be it either true
bored, or not true bored, which he may try firſt.
When a fit Mag is entertained, the Mr. Gunner (whom he ſerves,) ſhould bring him to
his Pieces, and give him the Denominations of his piece, and parts thereof; which when
he hath learned, which is the baſe-ring, and trunnion-ring, the muffel-ring, and the like,
(you may fee their names all plain in the Fig.of the Gun without more words) and like-
wiſe the Crows, Handspikes; the Cogn, and the likes and how far in the Bore is called the
Chamber of the Piece: Theſe things, with the Gunner's care well underſtood, he
may
give them further Directions, (viz.) But it is great pity, that the Gunners at Sea did
not exerciſe the Sea-mer in this knowledge, as the Corporal doth in Muſtering of them
with their Mufgaets ; for want of the like knowledge, the greateſt part of common
Sea-men, are as dull and ignorant, when they be required to ſtand by a great Gun in
time of Fight; and therefore it would be much for the Credit and Honour of our En-
glifs Nation, to train up their Sea-men in this knowledge eſpecially ; but it is taken no-
tice of, that if any man have any Art above another, he is afraid to let another fee hini
do any thing, or underſtand from him ſuch knowledge, for fear he will be in a ſhort time
as able as himſelf; which many do attain unto in a ſhort time to be as able as himſelf with,
out their help, therefore it is more for their Credit to teach them what they know.
1
1
1
1
SECT. II.
1
Sam
it ho were the Inventors of Gun-powder, and ſome Principles of Philoſo-
phy fit to be known.
Ome Italians have writ that Archimides the Philoſopher was the firſt Inventor of
Guns and Gun-powder: or whether this be truth or not, Learned Men are of divers
minds; Munſter, and Gilbert Cognot have written, that Guns were deviſed firſt in the year
1370 by a Monk, whom Munſter calls Bertholdus; fichen our Country.man Dr. Dee in
his Mathematical Preface and Diſcourſe of Menader faith, that an Engliſh-man was firſt
Inventor of Gun-powder in another Country, and they firſt made uſe of it from him;
alſo our Englim Chronicles do report, that in the year 1380 a Monk did accidentally
ler fall a ſpark' of Fire upon Brimſtone and Salt peter beaten to Powder in a Morter cove-
red with a Slat-ſtone, he ſeeing this mixture blow off the Stone from the Morter, did
thereupon deviſe a kind of Powder, and taught the Venetians how to uſe the fame in
Pipes of Iron againſt the Garvates.
Every Simple Body is either bright and Light, or elſe Groſs and Dark, and Ponde-
rous, and according to the variety and difference, it is always naturally carryed towards
ſome one or other part, the World hath height as upwards, or deprh as downwards; and
the depıb dependeth upon the Influence of the height.
All
1
1
$
4
47
CHAP. XII
The Art of Gunnery.
All pure and rare bodies aſcend, as the Fire more than the Air ; but the thick and
groſs bodies deſcend, as the Earth more than the Water.
Nothing worketh naturally, but in that which is contrary to it, and more feeble
the form of working, is aided by the Qualities; and the matter fuffering, which ſuf-
fereth by the Quantity.
Nature is extremely curious, as well of her perfection, as her conſervation ; and then
when all things conſpire, as well the Action that cometh from the Agent; as the
Pallion from the Patient hath proportion.
Accident hath its variety from the Subject, and goeth not from one thing unto ang-
ther..
Every Corporal thing! repoſeth in its nåtural place : Motion may be made any
where within the Orb of the Moon, Nature admitteth no Empreſs.
A body rarifying it felf, the place thereof increaſeth as the body increaſeth, the
reſiſtance of the moved proportion to the Mover, furthereth the motion ; the longer
the Chace of a Piece, the louder the Report; alſo the force of the kroke dependeth
on the ſwiftneſs of the Courſe,
, ħ hon
1)
1
---
{
1
!
1
1
ܪ
Se cir. III.
The Deſcription and uſe of the Gunners Scale, upon which is all ſorts of
Ordnance from the Canon, to the baſe of their Weight, Lading, Shot,
I and all other things appertaining to them.
T
His Scale is made according to the Diameter of our Engliſh Ordnance, but 8 inch.
loxg, the Diameter of a Canon-Royals and it may be made of Silver, Braſs, or
Box, or any other, fine grained Wood, that
will not warp. Upon one ſide'l have fee
the Names of all ſorts of Ordnance, and in the Angle of meeting with the Names, is
the diameter of the bore ; and begwixt chat and the next leſs diameter, is firſt the com-
mon.length of. fuch Pieces; and upon the ſtep of breadth, is how many Paces theſe
Pieces thoot,point blank, and right in the Argte of meeting, betwixt the two disise
ters with the Angle of meeting with the Names, is firſt the weight of the Gala, the
breadth of the Ladle ; and thirdly, the length, fourthly, the weighe of the Charge in
Powder, fifthly, the diameter of the Shat, ſixthly, the weight of the Shot , ſeventhly,
a Line of Inches ; eighthly, each. Inch divided into 10 parts, and likewiſe into 8 paris,
which are parts and half quarters, which the Line of Diameters of the bore comes
from. The degrees in the diviſions, and on the thickneſs and lengeb thereof, there is
à Line of Numbers, by which you work all the moſt uſeful Queſtions in Gunnery,
as you will find in the following page..
The Uſe of this fide is thus.
Suppoſe you come to a Piece of Ordnance, and it is deſired to know what Diece it is;
take the scale, and put it into the bore of the Piece, mark the ſtep of a Diameter that
fits it, and tře Angle of Diameter goes down into the Line of Inches, and parts; and
that diameter goes into the ſide in the Angle of meeting, and tells you.the Nanie
of the
Piese' : Betwixt the next leſs Diameter, right under, you have as before, the common
weight of the Piece, the breadth and length of the Ladle, weight of Powder, diameter of
Shos, and weight.
As for Examples
Admit I came to a Gun, and found by the former directions, that her diameter of the
bere is 4 Inches
. And in the Angle of meeting in the ſide, I find her Name is Demi-
culvering, lower then ordinary; at the end thereofIfind 9 or 10 for the uſual length,
and betwixt the next leſs diameter and the ftép is 174 the paces the Piece carries the
Beller in a level-line, point blank, right againſt weight in the next lefs Diameter, which
is 4 Inches, is the uſual weight 2000 l. breadth of the Ladle 8, and length iz Inches
,
the weight of the Powder 6 or 4 ounces; and next the diameter of the Shot 4 Inches;
and next, the weight 91. So that you ſee the next leſs diameter is the diameter of the
Shut, as well as of a leſs Piece of Ordnance. This I have made plain to the meancit ca-
pacity: Here they are ſet down in this Table following,
Ggg
The
(
1
1
1
48
1
:!
you
+
The Art of Gunnery. Book V
The Explanation of the Scale may ſerve likewiſe for the Table; only take notice, that
under Inches and Parts, is to be underſtood the firſt; to the left hand is. Inches
, and
tlie other is ſo many 8.parts of an Inch.
As for Example.
Admit you enter the Table with a Saker of the loweſt fort, the hiight of the lore
is 3: Inshes, 8 foot long, the weight 1400, breadth of the Ladle.olength 9 Ixch,
weight of the Powder
. 3.pornd 6 ounces, diameter of the Shot 35 weight of the shop
4 pound 12 ounces, and the paces the Piece carriės;. by Alex. Bianco's Tables is 1 5o of
juniori
Obferve that the Ladle is but 3 diameters of the Shot in length, and parts
Circuimference from the Canon, to the whole Culuering , I allow the Charge of Pow-
der to be about two diameters of the Piece :. from the Culvering to the Minion the
Charge to fill two diameters and a half; all from the Minion to the Baſe three disa
meters of Poroder.
1
s foot to the Piece.
parts of the
:
'
1
3
The names of the Pieces
of Ordnance.
Bore.
Gun.
Ladle.
Ladle.
Shot.
Length of the
blank.
Gun in pounds.
Diameter of the Parts.
Powder.
Weight of the
Breadth of the
Length of the
the Shot.
Weight of the
Diameter of the Parts. 1o
The weight of
Inches.
Parts.
He ſhoots point
; Paces.
: The
L8 18
>
3 : 07:0 750
1
18 18
A Baſe,
1 : 24:6 200 2:04 :00 : 81:50 : 5 60
A Rabanet.
:45:0 300 2:44 : 10:121 : 30 : 81 70
Fauconets.
2: 26150 40074:04:41:92: 21 : 590
Faucons.
67 of 750 4:48 : 22:42:52 : 8130
Ordinary Minion.
$:08:42 :82:73 : 4120
Minion of the largeſt ſize.
3:28:01 000 5:09:03:43:03:12 125
Saker the loweſt fort.
3:41
8:014006:49:63:63: 24 :121150
Ordinary Sakers.
3:6:01 500 6:51 0:44:03:46 : 0160
Sakers of the oldeſt fort. 4:010:01 800 7:211:05:03: 67 : 5163
Loweſt Demiculvering
4:2 10:0 2000 8:012:06:44:09: 0174
Ordinary Demiculvering. 4:4
:0 2700 8:012:64 :44 : 2 10:11 175
Elder fort of Demiculvering. 4:612 : 3000 8:413:48:84412:11 178
Culverings of the beſt ſize. 5:01:04000 9:0 14:2110:04: 6115: Oli So
Ordinary Culvering.
5: : 14500 9:4 16:0 11:05: 017: 5|181
Culvering of the largeſt ſize. 5:46:04800 10:010:011:&'s : 220: 0183
Loweſt Demicanon.
6: 211:0540011:4 20:0 14:06: 030: 0156
Ordinary Demicanon. - 6:412:0560012: 022:0|17:816 : 432: 0162
Demicanon of great liže. 6: 612:0600012: 022:0|18:016: 5 36: 0180
Canon Royal, or of :
18: 011-2: 18000|14:624:0 32:87: 458: oli 85
7
2
is
+
1
P
The Deſcription of the other ſide of my Gunner's Scale.
Upon the other ſide is a Scale of 8. Inchés divided into four quarters, and betwixe
each quarter above it is three Columns; the Inches News the height of alí ſorts of Iron
Mots from 2 ounces to 72 pound; and of Lead from 3 ounces to 806 pounds, and of
Stone from 1 ounce to 26 pounds ; each diſtinguiſhed from other by their names
,
written in the firſt Inch, the Table is in the ſixth Section, and the weights and meaſures,
accommodated into our Engliſh Averdupoiz weight of 16 ounces to the pound, and to
our Foot of Aſſize of 12 inches to the Foot. The Line of Inches being likewife di-
vided into Io parts, the whole into 8o, may ſerve for 800 ; for Protraction as follows:
There is alſo the Gunners Quadrant divided into go degr. in the outmoſt Limb, and in
the ſecond Limb within, is divided into the 12 points of the Gunner's Quadrant, and
!
+
cach
j
51
1
util.this
Ordnant
$
med
amo
fronion Ring
Cormash hing
4
Casacabell deel uit Ring
Touteh hole
--Chamber orcharged eslinder
Ramforce ring
H
hased night
whole
cies
wi
Eric
... "
:: LIEBE
WWW.1017"
2+timi 7 *****:
Fitilista: LT1*11
ANI!F1-11-
1.1.1.
1. VI.
"..**
Het mit
11IHAN
nder
EP111110 " # vittu,
+++++++++
11:3:19 MM LIGHTIRADIM
Trinions
%os
Muzzle Ring
life
touteh hole
nopous Arique |
10.
Diamitor of. y shot
B
K
The Chaçois
per
fi
Concave
d
B
Conca
of the true
Bored
vride
15 C €
saung foonair
4 9
oft
opzon2
49.99
02
su17 woulou,
น
Dead Raing 1922984045101670 22 32 8193 4. 10441129121419222185 22892283 1792 12141000
Right Raing 102209 2272442627 8. 285 302 320 337 13544246338.352000 114012201300 1350
D.
3 4 5 6
8
30 4015060.140
Stone Pounds i 310.410:30.7 0.910.12
41.82 02.72.133 104.35.916.317.818.14.10.10 12.4 12 12 14.3 15.12.17 10.19.1421.12/24 020.12
Lend:
03.014.315.016 98 212. 14.12.54.15 17.25 21.5 24.12 30.05.2019.945 of 31.057.0 63.072.079.8187-0196.0106.8
Iron
Pound 3 10
21.92.212.143.124.
124.1216 7.5.8.13 10.1012.2014.1417 3112-1|23.-26.630.034 038.0 42.0 48.0 53.0 88.064. 0172.10
A
Peces Point blanck
Teces
Boje. 200
Rabanet
Falkennet
Saucon
The ordinary Minion
inion of y. largeſt size
Sakers lowest sort.
Sakers ordinary
Sakers of y oldelt sort.
Demiculver lower then ordin.
09
ordinary Demiculvering
Taldersoft of Demculver
OST
Cauvering of yleast Size
ordinary whole (ulvering
Culvering of lasgelt size
SIN
Lowet Demicannon
SO
ordinary Demicannon
2.
300
Demicannon of greate size
80o
Weight of y Gunn 30o 4 DO
75 750 Loooloolisoa 1800 26.4 2700 3000 4000 4500 4800 54.00
5600 Copo Cannon Reyall. 8000.
Breadth of Tadle
4.
Lt
6. 16:15 18 8 18:19 Użlio
14 31
Length of Tadle 4 +
7.
8 8 :o 9.410.11: 12:11220:1416 16
20.
22-0224
Weight of Powder
1
2 i 2.33:13.640
05 5:06:17 8: 240..
11:6 12:8
1.8 1-8
32 10: Joun
: 58 2.163 03:43:23
4:44:3 4:35 5 1
6
6 66 4
7 /
Weight of sholt 08 1:5
876 3 13:22146.01745100 10:14 12:10:17: $20
32 136
58
BR FI
F DML SL30 SOL DCO ODCDC CLOC CGLD.C ODC DCG C.R
Will
The Mettall ans
SL
ring
Muzzel
Peren
ito
drat
3%/%
19
°
smo
ole
Ad
Right Shadow
5 /4 13/2
saunog
res
9
sonon
7
0
so
1
P
10
20
1
2
8.0 90
Peg of Mournal
Ot
Cou
450.61
71 attres
8
O,
11
2
'
24
2
1
Inches
2
3
6
7
8
7
2
3/%%
24
3
و و
c sf 6%
Book..V.p.48.49.
وه
7
7
1
1
1
|
1
7
|
49
1
.
1
a
Chap. XII
.
The Art of Gunnery.
each point 4 paris; and in the third Limb is a Geometrical diviſion of right and contrary
Thadows, for the ready taking of heights and diſtances; but there is alſo a Geometrical
Quadrate, with each fide divided into 10 parts, which ſtands for 100, and each 10
parts divided into 10 more, the Uſe thereof in taking of heights and diſtances is in the
16 Chap. of the second Book of the Deſcription of inſtruments : But the life for to
level, or elſe to mount or Imbafe any piece of Ordnance, is in the 34 Sect. of this Book.
To the ſide thereof is fitted a piece of Braſs of the ſame breadth as the Scale in thick-
neſs, with two holes within an Inch of each End, and two Screws fitted to ſerve the
four holes, as you may ſee in the Figure to the ſide of the Scale, that if you would
level or mount any piece of Ordnance, Screw the plate to the end of the fide B, with
both Screws, and put the plate in the bottom of the metal as far as he will go, and
put the tomping in upon him to keep the plate faſt, and then level or mount your Piece,
as in 33 Section directed.
But if you will Imbaſe any piece of Ordnance to any place or point alligned, you
muſt ſcrew the plate to the end 'A, and let the ſide with the Line of Numbers be next
the muzzle, and Itop him with the tomping, as before ; then Imbaſe your Piece, or
put him under the Line of Level as you will
, to what degree you pleaſe ; and when
you have done, Screw the plate to the ſide A B, with a ſcrew at one end, and a ſcrew at
the other, (there is alſo over the weight of the Shot a divifion of the right Ranges
, and
degree to degree; and likewiſe you may put the diviſion of Inches in the 38 Section,
for the number of Inches and parts from 5 foot to 14 foot long, requireth to mount her
to any degree of mounture with great facility and eaſe. There is alſo Triangle.wiſe a
plain Scale, that goes along down by the degrees of diameters, or ſteps, the Line is a
Line of Cbords, with the Gnomor-line, and a Line of ſix hours of the fame Radites, and
a Line of Rhumbs, with the line of Sines; and this is for the making any ſort of Dial
iò any. Latitude by the following directions, and alſo for the Plotting any Triangle, or
reſolving any Queſtion in Navigation, or Affrononey. You muſt remember, there is
a Braſs Pin in the Center at C for to hang the Plummet and String, with the Lope
upon.
Thus 1 hope I have filtei all ingenious Gunners with a Scale ſo uſeful, that I will leave
it to them to give me commendation for my labour and pains. If I might adviſe Gun-
ners of all forts, that are able to have one of theſe Scales of Braſs or Wood, to carry a-
bout him, to reſolve any Queſtion preſently for his own credit, and it is very portable
and fit for his Pocket; but it is beſt to have a caſe of Leather or Cloth to keep it
clean; and you may carry a pair of Compaſſes with him, and by him you may reſolve
moſt of all the Queſtions in this Noble Art of Gunnery.
On the ſide of the Quadrant betwixt the Equinoctial, and the Radiw, or Sans great-
et Declination is a diviſion to every 10 minutes of the Suns Amplitude Riſing and Seta
ting anſwerable to the Ecliptick Line, and the Declination on the other ſide the Figure,
makes all plain to any Inſtrumext-maker, without further precept.
.
1
SECT. IV.
7h: Vſe of the Line of Numbers on the Scale, for the help of ſuch as cannot
Extract the Cube and Square-Root.
How by knowing the weight of one Bullet, to find the weight of another
Bullet, the height being giver.
Buller of Iron of.6 Inches heigbt, weigheth 30 l. what will the like Bullet of 7
Inckes, in Diameter or height weigh; always take theſe Rules.
Extend the Compaſſes from Inches to 7 Inches Diameter, the ſame diſtance will
reach from 30 l. weight, turned 3 times over unto 47 1.10 01xces, the weight of a sbos
7 Inchi high.
:::
Ggg 2
The
A
2
.
50
The Art of Gunnery.
Book V.
The Arithmetical way.
C6C7.£. 30x 343
1
=
47 l. 10 DANC.
216
one pound of Iron.
That is Cube 6 makes 216, and Cube 7 makes 343 ; then by the Rule of Proportions
Multiply 343 by 30, the product is 10240, divide by 216, the Quotient is 47 pound;
for which is 10 ounces as before, there is ſomething leſs then 7 Eube Inches in
By she Tables of Logarithms.
The Logarithm of 6 is
07781512
The Logarithm of 7 is
08450980
Subſtract the appermoſt Number out of the lower, the diff. increaſing. 0669468
- 3
The laſt Number Multiply by 3, and the triple of this difference is --- 2008404
Added to the Logarithm of 30 l. with no ounces, (which is 3000) - 34771212
Gives the Logarithm of the weight 47.
36779616
Now to know how many ownces is, work thus by the Rule of Proportion.
If 100 gives 64, what will 16 ounces give? Anſwer, 10 ounces y ſo the Shot of
7 Inches diameter weighs 47 l. 10 ounces or 47 70 pound, the likc way of work is
with all ſuch Queſtions.
+
SECT. V.
+
34
10
Q0O
100
24
Admit the weight of an Iron Bullet being 30 pound, the Diameter was 6
Inches, the weight being 42 1 what may the Diameter be.?
16:100 10
Irſt I will ſhew you how to turn 10 oances and into 100 parts of a pound; always
fay, If 16 give 100, what fall 10 give ; : (64 as you may ſee the work in the Mar-
gent, where the weight is known, and the Diameter required.
Always Divide the weight 30 l. and 47 l. into 3 equal parts, and that diſtance
1000 will reach from 6 Inches Diameter, to 7 Inches the Diameter required on the Line of
Numbers.
1024
By the Tables the Logarithm of 30 is
34771212
The Logarithm of 47 mil. is
36779616
468
Uppermoſt Subſtracted from it, leaves the difference increaſing, - 2008404
*524 (54 The difference divided by 3, or the third part of this difference. 0669468
x06
added to 6 Inch. Diameter, the Logarithm
7781512
Gives the Logarithm of 7 inch. Diameter required
08450980
This is the moſt
eaſy, ready, and certain way of Arithmetick; and fo work for
all ſuch Queſtions, if three Numbers be given, to find a fourth in a Triplicated Propora
rion.
I
+
SECT. VI.
The Geometrical finding the Diameter for the weight of
any
Shot affigned.
M
R. Gunter in his firſt Book, Section 4, hath ſhewed the Making of the Line of
Solids on his Sector : but this "Rule Thews the proportion of the Diamesets in
weight : having a Shot of one pound 2 pounds or 3 pounds weight of the Meral or Store
aſſigned; if it be of a pound, divide the Diameter into 4 equal parts, and 5 ſuch parts
will make a Diam, for a shot of the ſaid Metal or Store that ſhall weigh juſt two l.
And
1
1
1
+
51
Chap. XII. The Art of Gunnery.
And divide the Diameter of a Shot that weighs juſt 2 l. into 7 equal parts, and 8 ſuch
parts will make a Diam. for a shor of 3 l. weight And divide the Liamet, of a Shot of
3 l. weight into 10 equal parts, and 11 ſuch parts will make a shot for 41. weight.
And divide the diam. for a shot of 41. weight into 13 parts, 14 ſuch parts will make
a diam. of a Shot for 's l. w eight.
And divide the diam.of a Shot of sl. weight into 16 equal parts, 17 ſuch parts will
make a diam. for a ſhot that will weigh 6 l. and fo dividing each next diam. into 3 equal
parts more then the next leſſer was Divided into, and it will with one part added from a
diamet. of a ſhot that will weigh juſt 1 l. more; and ſo you may proceed infinitely, in-
creaſing or decreaſing, by taking one part leſs, as is appointed to be Divided, for one i.
jeſs, and the next into 8 7. leſs, to abate 1 for the Remainder, infinitely decreaſing it.
HA
FitrH
Hry
HHHHHHHH ,
Afacond Geometrical way.
Firft you muſt have exactly the diamet. of a ſhot that weigheth one pound, and thieri
deſcribe a Circle, whoſe diamet. Ihall be juſt cqual thereunto ; and Divide it into 4 Quan
.
1
E
A!
1
2
ماه او ا8
7
6
5
4
3
1
1
1
!
3
an
2rants
1
Į
1
:
2
.
1
52
I be Art of Gunnery.
Book V.
The Diameter
94
13 parts.
drints, with 2 diamet. cutting each other in the Centor orthogonally; ; Then take the
Chord of the whole Quadrant ou degr. B C in your Compaſes, and lay it from the
of a shot of " Centor of the firſt Shot one pound D to 2, and ſo A 2 will be the Diameter of a Shot of
pound is i Inch 2 pocend; Then extend the Compaſſes from 2 to the Chord C, and lay that diſtance from
D'to?,' ſo will A 3 be the Diameter of a Shot of 31. And fo likewiſe extend the
Cumpailes from 3 to C, it will reach from D to 4, and from 4 to C, and it reaches
from Dto s, and from s to C, lay it ſtill always from D to 6; and ſo continuing till you
have proceeded as far as you will: You ſhall find that if AB were the Diameter of one
pound, A 2. is the Diameter of 2 pound, and A 3 is the Diam. of 3 1. and A 4 the
Diam. of 41. and As the Dinm. of 5 1. A 6 the Diam. of 6 l, and laſtly, A8 is the
Diam. of 8l. and ſo you may proceed in like manner infinitely.
Likewiſe having the Diameter of a Shot of any weight, the double of the Diam.
is the 'Diam. of a Shot which weighs & times as much. Thus the double of A1, which
is A 8, makes the Diameter of a Shot of 8 pound ; and ſo the double of A2, which
is the Diameter of a Shot of 2 1. makes A 16, the Diameter of a shor of 16 pounds,
that is 8 times 2 pounds, and ſo the double of A 3 makes the Diameter of a shot of
24 pounds, and the double of A4 makes the Diameter of a Shot of 32 pounds, four
times 8 being 32 ; and ſo you may proceed as you pleaſe, and find the bigneſs of
any
Shot.
i
A third way.
This you may do alfo, having the Diameter of a shot of one pound, double that
diam. it will make a diam. of 8 pound; and treble the diameter of one pound, will make
a diameter of a Shot of 27 pound, and quadruple or 4 times the ſame, will make a diam.
of a Shot of 64 pounds, and s diameters will make a Ball of 125 1. and 6 diameters
of a Shot of one 1, will make a diameter of a Shot that will weigh 2161.
L
А.
215
C
2
B
125
1
64
1
1
a
1
1
277
1
37 19
.
M
1
H
New
1
.
* i
t
+
Shot given
1
>
vifons
Stone ſhot of 10b.
Chap. XII. Tbe Art of Gunnery.
53
Now it is convenient to flew how to find the Mear-diviſions between theſe extremes;
as for the diameter of a shot of 2 l. 37.41.51.61.7 l. or what more you will, fo as
by ſuch progreſſion you may proceed from pound to pound, until you come to the last
term of 218 pound; nevertheleſs the ſame manner of working will proceed infinitely.
Lay the forementioned 6 diameters upon one and the ſame right Line you muſt at the
end of them draw another Right-line orthogonally
, and fit therein the diamet. of 2 ſuch
as at C, and from thence draw another Right-line parallel to the firſt, as GH,
and then draw a Ouadrant as A B, and from the Centre G draw right Lines through all
the divifions of the digm. marked upon the right Line AF which are all equal, fo Shail you
have 6 diviſions to be divided; the firſt being divided already,
and is the diam. of a Shor of
I l. but the ſecond diviſion is to be in the Circumference or Quadrant divided into 7 parts
equally, becauſe it containeth the ſecond diameter unto 8, for adding i to 7 it niakes 8,
the third diviſion is into 19 equal parts
, which being added to 8, makes 27, the fourthi
hall be divided into 3 7 equal parts; which together with 27, makes 64 the fifth thall
be divided into 61 equal parts, which added to 64, makes 125 ; and laſtly, the ſixth
place muſt be divided into 91 equal parts, unto which adding 125, you ſhall make a
diameter of a Shot of 2 16 pound juſtly.
Now foraſmuch as theſe diviſions are difficult to make well, within ſo ſmall a Osa-
drant : you may therefore deſcribe a greater, as the Quadranti L M, and there the di
are more diſtint, and larger than in the leſſer they can be; Further, you may
note, that Fire-balls,Granadoes, and other Globous Artifices, muſt have the ſame pro-
portion to their Grandures from their Ball of one pound, which may be exactly confi-
dered; and ſo by this Method you may make Balls of Lead, Braſs, Stone, and Granadoes
,
Fireballs, and all other Spherical Fire-works, of what weight you will
, having one of
One pound firſt, to lead you accordingly:
SBCT. VII.
To find what proportion is between Bullets of Iron, Lead, and Stone, by
knowing the weight of one Shot of Iron; to find the weight of any
other Shot of Lead, Braſs, or Stone of the like Diameter.
He proportion between Lead and Iron, is as 2 to 3, ſo that a shot of 2 pound of
Iron, is of like diameter or height as 3 l. of Lead.
As for Example.
A ſhot of 6 Inches diameter weighs 30 pound,' to find the weight of a ſhot of Lead of
By the Rule of Proportion.
Firſt, if 2 gives 30, what will 3 give ? multiply and divide, and the Quotient is 450g-
the weight of a shor of Lead.
By the Tables, the Logarithm of 2 is
0301300
The Logarithm of 30 is
14771212
The Logarithm of 3 is
Add the z lowermoſt, the ſum is
19542424
Subiraεt the upper Num. Log.of 451. the weight of the
the Remain is the
Žoi in Lead of the fame diam. 316532124
Extend the Compaſes from 2 to 30, the fame diſtance ſhall reach from 3 to 45 i (In
like manner work by the reſt following.)
8-30 --- 3
The porportion between Iron and Stone, is as 3 to 8 ; ſo that a ſhot of 30 pound of
3
Stoxe, is as big as the like hor of 80 l. of Iron and ii l. of Stone, is of the ſame
90
diameter 6 Inches, as a ſhot of 30 l. of Iron and 45 l. of Lead; the proportion between
X2
Lead and Scone, is as 4 to 1 ſo that one shot of Lead of 40 1. is of the beight as a gø (113
88
The proportion between Lead and Brals
, is as 24 to 19,
The proportion between Iron and Braſs, is as 16 to 1%.
1
1
I
1
the ſame diameter.
3
90
X
04771 212
90L45
3
Ву
!
}
+
1
1
.
Book V.
/
Inches. In
Quart.
1
I
I
2
I
I.
2
9
IO
6
O
310
SI
2
2
2
I2
2
3
that Line you ſhall have 30 pomalo
54
The Art of Gunnery:
By theſe Rules aforegoing you may Calculate with eaſe, if Iron flot be wanting, and
the other to be had, what height and weight either ſher of Lead, Braſs, or Stone, ought
to be of to fit any pieces of Ordnance ; and by the fame Rules here is a Table faithfully
Calculated; and doth ſhew the weight of any ſhot of Lead, Iron, and Store, from 2
Inches diam. to 8 Irches, and Quarters of Inches; the proper Stone for this purpoſe
is Marble, Pibble, Blew hand Stone ; (there may be a little difference of weight in:
ſome ſort of Stone : but theſe do never agree in weight; you muſt remember in load.
ing your Piece with a Shot of ſtone, you muſt not have ſo much Powder as you do with
Iron-shot, but abate according to proportion, as is between Stone and Iron.
Iron. Lead. Stone.
The uſe of the Table, to find
Poun. Ounc. Poun. Oun. Poun. Oun.
the weight of any shot of
7
Iron, Lead, or Stone frönk
9
2 to 8 Inches Diameter.
213
This Table is exactly Calcu-
lated, and the uſe thereof is very
3 3
caly; we will make it plain by
3
I 4
32 6
two Examples; I would know of
Shor of 6 inches, their weight in
3 3 7 7
Iron, Lead, and Stone : the firſt
4
S
IS
Column 'is' Inches, the ſecond
411 IO IO
16
Quarters of Inch. the third Poune
4 12
I 2
10
I 18
12 and Ounc. of Iron, fourth Pounds
4 13 14
14
55
9
and Ounces of Lead, fifth Porno
and Ounces of Stone.
5 17
26 216
5. I 20
I i 30 2
7
Enter the Table with 6 Inchis
52 23
2134
II8
II diam. in the firſt Column, and in
5 326
6139
919
14
6
of Iron, 45 pound of Leady II
04
6 II
| 0051 00 12
pound 4 ounces of Store, the weight
of 6 Inches diam. And likewiſe,
6 12
04
for 4 Inches diam. the weight of
6 13 42
0062
an Iron Shot is 14 pound, 140X16.
7 48
0072
of Lead 22 pound 5 ounces,
7I 53
0079
0820
Stone 5 pound ounces; and ſo of
7 258
0087
00 | 22 12 the reſt.
73.64
loo96
0024
8 71
106 8126
2
1414
1
1
I2S
I27
IOI
2 I
IS2
OQ 12
8
7
13
4
I 2
Sir
1
6
073
4
1514
22
1
8
8
0045
00 II
surumloons TANAN
30
34
3
12
00157
oo 14
00|IS
1
I 2
cle
00 18
of
00
oo
1
00
IO
SECT. VIII.
How by knowing the weight of one Piece of Ordnance, to find the weight
of another Picce being of that very shape of the same Metal, or any
other Metal.
Irſt, with a pair of Crallapers take the greateſt thickneſs of your Piece, as at the
Baſe-Ring; and alſo the Piece, whoſe weight you know not.
Example.
Admit a Braſs Saker of 1900 weight, hath his greateſt thickneſs 11 Inches; Now
I find the diam. of the other Braſs Piece, whoſe weight I know not, to be 8 : : then
always by theſe Rules:
1
If
+
1
1
1
܂
55
*
CI
:).
1
4
i
I
.
Shot, from 1 Inch and to every half in
Chap. XII. Tbe Art of Gunnery.
If the greateſt diam, and weight is given, to find leſs weight, or elſe the contrary.
As the Logarithm greateſt diameter, 11
306069
The Logarithm of the leaſt, 8,7
294200
The Differéſce increaſing.
11869
3
x 3 or the Triple of this Difference Subſtra£t
35607
From the Logarithm of the weight given 1900 3278359
Reſt the Logarithm of 837, the weight required, 292 228
Or extend the Compaſſes from 11 to 87 Inches diam, the ſame diſtance will reachi
from the weight given 1900 pound turned 3 times over ta 837 pound.
The Arithmetical way. .
C8 X 1900 ,
837 1. weight almoſt in Braſs.
But if the Piece had been Iron whoſe weight you fought, you muſt always do as
before with the Braſs, and find the difference of their Metals by the laſt Problem,
which is 16 to 18, then fay by the Tables,
As the Logarithm of Braſs, 18,
125.527
o?
is to the Logarithm of weight in Braſ1,8357 292272
So is the Logarithm of proportion of Iron 16 - en 120412
The Sum 412684
to the Logarithm of the weight in Iron 744-287757
Or extend the Compaſſes from 18 to 837, the ſame diſtance will reach from 16 to
744.1. weight in Irox.
Arithmetical way.
X 837 by 16
====744 l. of Iron almoſt.
Sect. IX.
How to make a Shot of Lead and Stone, the Stone being put in the Mould
in which the
Leaden Shot
ſhould afterwards be caſt, to be of the like Dia-
meter and Weight as an Iron Shot is of
Lead. Stone.
: It is found by experience, that if
Poun. Ou. Poan. Oun. Poun. Oun.
you take siparts Lead, and one part
Scone, ic will.come very near the mat-li
ilo
ter, wanting, not above 3 Ounces, i
60
which is nothing, reſpecting the diffe-
14.
rence you ſhall find in Pebble Stones....2
IZ
Here you have a Table how much
3
Lead, and how mach Stone muſt be
together, to make the equal of Iron 3
4
the firit and ſecond Column to 8 Inch.. 4_2
1417
Diameter; the third Columnis how 5
77
S
much Lead, the fourth how much
413
Stone, the fifth how much weight both
25 ols
30
6
6
o
together.
8
po
o 48
7 2 148
o 58
59
071
/
18
1
5
1
Both together.
Inches.
Quart.). 'n
2
8
2
o
2
a rol
2.
tals
4
3
2
12
2.15
OII
103
S
818
0
7
IO
7
812
IS
IO
IZ-
1
14
2 119
I223
6
22
38
olanco
OIO
!
QII2
CH
Hhh
SECI.
1
1
1
56
Ibe Art of Gunnery.
Book V.
Sect. X.
*
1
281291
1
1
-
kili
16
1
SO4
84
YOO
11.
Hors by knowing what quantity of Powder will load one Piece of Ordnance ;
to know how much will load any other Piece whatſoever.
Dmit you have a Saker of three luches three quarters at the bore dians. and it rea
quires 4 pound of Powder ; what will a Demi-Canox of 6 ; Inch. require? Work
by theſe Rules always.
As the Logarithm of 3 in diam.
257403
The Logarithm of 6 Inch. diam.
the difference increaſing,
23888
(3
The triple of the difference added
71664
The Logarithm of 41. of Powder, O OMNCES. 160206
to the Logarithm of 20 or 2016 L. of Pow. 2 31870
100--84-16 So that the Demi-Canon muſt have 20 pound 13 ounces for her Charge of Powder;
reduce the Fraction as before in the Margin into ounces. ·
By the Scale, extend the Compalles from 3.7 to 6 Inches diam. the ſame diſtance
turned three times over from 4. - will reach to 20 % pourd weight, as before.
1344
The Arithmetical way.
3 44L13 iyong
C6,5*4
201.13 Ounces of Powder for to load a Demi-Canon:
3
C
You are likewiſe to underſtand that the Demi.Canon ſhould be fortified ſo well as
the Saker by this Rule.
The diameter of the Saker is 3 Inches
- 257403
The cemi. Canox diam. is 6 Inches
the difference increaſing,
23888
(3
The triple of the difference by (3)
71664
added to the Logar. of 1600 weight of Saker 320412
gives the Logar. of 8332 the demi-Canon, 392076
Alſo by the Scale, and Arithmetick Rules, as in the foregoing Rules
you will find
the weight of the Demi-Canon 8332 pound, proportionable according to the saker;
but fuppoſe the Demi.Canon to be no more than 6000 weight, then you must uſe theſe
Rules.
The ſuppoſed weight of the Demi-canon 6000
377815
add the weight of the Powder well fortified, is 20
The ſum is
709704
Subſtract the weight of the Gur well fortified 8332 392074
leaves the weight of the Powder 15 pound,
317630
Fifteen pourd being a fufficient Charge for that Piece: or extend the Compaſſes from
6000 to 8332, the fame diſtance will reach from 20 to 15l. of Powder, as before.
.
281291
1
1
1
331889
The Arithmetical way.
6000 x 20 pound 13 oxnces.
15 pound almoſt, as before.
-
8332
Thus
1
1
į
'.
CHAP. XII.
I be. Art of Gunnery.
57
15
/
Rules you may
..:.:.61
find the Diao
meter of a sboc
.
CY1011
1
vii.
+
Thus you are always to take care of over- loading your Piece, which error many run
into, when they call a Piece a Demi-canon; they preſently load her with ſo much as is
allowed for ſuch a,Piece fo named, feldom examining whether the Piece have Metal
énough for falträ'Charge ; by which miſtake they endanger their own lives, and others
which ſtand near. Now, for eaſy plain Rules, I ſay you never had before laid down
· In'this manner to reſolve theſe things ; for: if you compare theſe Rules with Paih.
mitjeMaſter Guniher of the City of Worčifter, for any other Arc-of: Gunuery, you will
find a great
deal- of difficulty in Cubing anul. Extracting the Lube Röst, and with re-
ducing and Fractions (which here you may do five Queſtions for one that way, and
more true and near, therefore I compare them to his Rules.
.**.!!111) Misir
s How to make the true. difpert of any true bored Piece; of Ordnäicė.
Now we have found how to proportion Shótani Porder to any Piece of Ordnance By the lanie
true bored ; - Before we Load and Fire, let us find the true Diſpert. to directithe shot to
the aſſigned mark.
Girt the Piece about the Baſc Ring round at the Britch with a Thred, and alſo the with a string.
Muzele Ring at the Mouth, and divide them wợ meaſures into 22 equal parts, which
you may preſently do, by applying it to a Scale, that hath an inch divided into 10
parts and Divide the parts by. 7, and
Subſtract the greater out of the Jeffer, and.take
half the difference, is thie true Diſpërt.
As for Example.
Sappoſe when I have meaſured the length of each String, and Divided it into 22
equal parts, I find that 7 parts of the longer String is in inches and 7 parts of the
Thórrer is 9 inches
I ŞubItract 9 out of it and the remain is z; the half is i, which
is the true Diſpert
Another way to Diſpert any Piece.! :-
-:If you have a pair of Callipers; as in the general Figure ACB, as you take the
diameter of a Shot, and apply it to a Scale : Divided into 8 or 10 parts, to know the
Contents thereof; ſo with the Callipers take the greateſt thicknels or diam. of the
Bale Ring, and by your Scale ſee how much that is; as admit that the length of
the Linea:b;cid, where the diam. of the Bale Ring, then take the diam. of the
Muzzle Ring; as admit it be a, b, as you may try by the Figure of the Gun in the
general Eigure then Divide the difference bd into 2, equal parts, and one of them is
the Diſpert, put it upon the Müzzle of the Gunas CB, and ſtick it faſt on the top of
the Muzzle Ring with a little Pitch or Wax, and from the Baſe Ring at A in the Fi;
gure., to the top of the Diſpert at By take aim to the Mark. you would floor to,
that is the way to bit, but if Callipers be wanting, take a Stick that is ſtraight and flát
,
and 2 Strings with two Musket Bullers ár the end, and two Loops made at the other
end, the Stick being ſomething more than the diam." at the Bafe Ring, and put the Stick
upon the top of the Ring at the
Mazzls, as you ſee the Fig. HK on the Gun, and put the
Strings fo nearer:and farther, until they only touch the ſide of the matter of the muzzle
Ring, and mark the Loops on the Sțick, and put the Stick on the Baſe Ring, and do in
like manner, and mark the Sticks; and the Work will be the fame, as it were taken
by the Callipers, and the difference of the two Notches on the Stick will be ab the
Baſe Ring, and ab the Notches of the diam. of the Mozzle Ring, and half the dif-
ference bc or cd is the Diſpert, as before, if the Piece be true bored.
A fourth way to Diſpert a Piece of Ordnance.
If the Piece be noț Chamber-bored, take the Priming. Iron, and put it down in the
Touch-hole, until it reſt upon the Metal in the bottom of the bore, there make a mark
with the Baſe Ring; likewiſe apply the Priming Iron'to the bottom of the Metal at the
mouth, and ſo much higher as the mačk'is which you made at the Baſe Ring, than the
Muzzle Ring, the difference is the true Diſpert.
$
::o
31
2
1
and
+
L
>
í
Hhh 2
SECT.
7
!
5
58
The Art of Gunnery.
Book V.
SECI. XI.
1
;
laſt way;
How to know whether jour Piece be Chamber-bored.
Irft you may Diſpert your Piece the three firſt ways, and when they agree in one;
take that for the true Diſpert ; then with your Priming Iron take the Diſpert this
which done, compare it with the other Diſpert firſt found, and what it
wants, is the juſt difference of the Chamber from the Bore of the Piece,
Admit the Diſpert truly found by the two firſt ways be three Inches, as by this laſe
way is but two Inches, it ſhews that the Chamber differs from the true Bore on each
ſide one lächs ſo that if the Bore of the Prece be five" Inches high; the Chamber be-
ing one Inch on each ſide lower, is bur three Inches high : the like Obſervation we
would always have you to make, that you may not afterwards be deceived in making
Cartredges of Canvas or Paper to load the fame.
1
!
1
SE'E T. XII.
How to know what Diameter.every Shot muſt be of to fit any Piece of Ord
nance, or to chooſe Shot for Ordnance.
Ake the Diameter of the Bore of the Pieces, and Divide into 20 equal parts, and
one of thoſe parts is ſufficient vent for any piece, the reſt of the ing parts muſt be
the height of the Shot , but now adays moſt Gunners allow the shot
to be juſt one
quarter of an Inch lower than the Bore; which Rule makes the Shot too big for a Caron,
and too little for a Faulcon, but if the mouth of the Piece be grown wider, then the
reſt of the Cylinder within by often ſhooting; to fit Shot to ſuch a Piece, you muſt
trie with ſeveral Ransmers-heads, until you find the Diameter of the Bore in that
place where the Shot uſech to lie in the Piece ; and a Sbot of one twentieth part lower
than that Piece is fufficient, therefore let Gunners remember to triethe Piece, as di-
.
pected.
3
SECI. XIII.
1
+
How to find what Flaws, Cracks, and Honey.combs are in Pieces of
Ordnance.
Te
Here is one good way, as foon as you have diſcharged a piece of Ordnance, cover
the mouth of the Piece cloſe, and ſtop the Toxcb- hole at the inſtant time ; if there
be any unknown Cracks or Flaws which go through the Metal, a viſible Smoak will
come through thoſe Cracks and Flaws; if not, the Gun is norcracked.
There is a way to reflect the Sun-beams when he ſhineth, with a Looking-glaſs of
Steel in at the hallow Cylinder of the Piece ; for by this means a bright and clear light
will be within, and by that light you will ſee every Flav, Crack, or Honey-comb.
But this way you may fee at any time ; takea Stick ſomething.longer than the Piece
cleave the end of the ſaid Stick, for to hold an end of a Candle, light the Candle, and
pat it into the cleft end of the Stick, and put it into the Piece; by this light obſerve by
degrees whether from the one end to the other there be any of the foreſaid Flaws, Cracks
or Honey-combs in the Piece.
This is a uſual way likewiſe, if in ſtriking a Piece upon ſeveral places of the Metal
with a Hammer of fron, you ſhall at any Itroak hear a hoarte found, then without
doubt there is Honey-combs : but if in ſo ſtriking the Piece, you tlall'at every ſtroak
hear a clear ſound, then may you be ſure your Piece is clear of any Horey-combs,
Cracks, or Flaws.
SECT. XIV.
4
1
1,
1
i
i
The Art of Gunnery.
59
2
that ſhews the Centre of the bore to be at
CHAP. XII.
SECT. XIV.
How to find whether a Piece of Ordnance be true bored, or not.
firſt
, there muſt be provided a Staff
, and two Rammer heads upon the Staff, and
on the Rammers heads there muſt be two right Lines drawn upon them; that is,
Divide the two Rammer heads that are the juſt height, and fit the bore into two equal
parts oppoſite to each other, and draw Lines thereon ; the like do by the Staff, that
the Lines on the Rammer heads may ſtand alike, one at one end and at the other
end, as you ſee in the general Figure L M.
And let the Staff come through one of the Rammer heads about 9 Inches longer than
the Cylinder of the Gur; then lay a flat Stick on the Myzele-Ring, and hold the
ſide of the Quadrant on the Scale to the Stick, and it will by the String and Plummet
find the middle
, or upper and lower place of the Metal; or by hanging a Plumb-Line
and Quadrant before the concave, and the Stick on the top; then after you have found
the Point, and upper and lower place of the Metal, put the Rammer head L into the
Gun, and let one hold him hard, and right with the Line or Mark on the upper part of
the Gwn, and lower part with the Line on the Rammer head on the Staff above and
below, whilſt you put in a Priming Iron in at the Toach-hole, and ſtrike hard the
Rammer head, make a Mark; then pull him out, and apply the Line on the Ram-
mer head to the Mark on the upper and lower edge of the Muzzle of the Gun,
and you may preſently ſee how much the Mark is from the right Line of the Ram-
mer head, to the right hand, or to the left; that is, if the Mark is juſt on the
right Line, the bore is in the midſt : but if you find it a quarter of an Inch on the right
or left hand, ſo much lyeth the bore either to the right or left; and in Shooting, the
Piece muſt be ordred accordingly.
But now to know whether it is thicker upwards or downwards, or how the bore is
the way to know this, find the Diameter of the Piece at the Touch-hole, as is already
taught in 10 Chap. bend a Wire a little at the very end, that it may catch at the Metal
when it is drawn out, after the Wire is fitted thus, firſt put it into the Touch-hole til
it touch the bottom of the Metal in the Chamber ; then holding it in that place, make
a mark upon the Wire, juſt even with the ſaid Touch-hole ; afterwards draw
up
the
ſame Wire, untill it catch at the Metal at the top of the Chamber ; at that inſtant make
a mark upon the Wire juſt even with the Touch-hole : the difference betwixt the two
marks, is the juſt wideneſs of the Chamber, and the diſtance between the firtt mark,
and the end of the Wire, having half the Diameter of the Chamber of the Piece Sub-
ſtracted from it, will leave the half of the 'Diameter of the Piece, if the Piece be true
bored; but if this number be more then half the Diameter, then the bore lyeth too far
from the Touch-hole, and the upper part of the Meral is thickeſt, but if leſs, the under
part bath moft Metal.
One Example will make it very plain.
Suppoſe that the Metal at the Britch be repreſented by ABCD, and the Metal at the
Muzzle by eteh, and the bore of the Piece I, whoſe Centre is 1, or the bore K, whoſe
Centre is m: (and I find the Diam.of the Piece to be 21 Inc. at the Touch-hole, the halt
thereof is 10 inch. Then I find by a Wire the Diam.of the bore to be s inch. but the
bottom of the Metal is olsch. half the Diam. of the bore being 2 ; Inches to a 10'
makes
13 to the bottom of the Metal ; but if you add to 8; half the diam. of the bore 2
:
R, and the thinneſt of the Metal is undermoſt
, and there he is like to break firſt; beſides
,
it thews that you muſt add balf an Inich to your Diſpert of a true bored Piece
, te make
a Diſpert for the Piece to thoot well : but if you had found by the direction beforegiven,
that the an Inch had been lefs, as 1o only, and the greateſt part of the Metal had been
under, and therefore you muſt cut the Diſpert an Inch Shorter then a Diſpert made
for ſuch a true bored Piece; and likewiſe if you find by the Rammer head, and prick
with a Wire at the Touch-hole; an Inch difference to the right or left hand, as I or m.
that lide which is the thinneſt, yon muſt put the Diſpert cur an Inch Morter, the three
Figures makes all plain as it is written, as you may ſee by the direction of inches. .
SBCT. XV.
1
T
J
+
}
60
1
Book V.
The Arë of Gunnery.
ji
1
!
ASI
1
:,,
1
!
A
wi?
Lil
1)
;.11
CD
1
how's
! .
Blür:péu:VOIENTE:DUL:RET
: 1 pente
!
1
1
1
:
I
DATE: (51KW
Blo:11:31:1:1:1 GATIT:3:1:21TTI CRI:18:1:171):13:
1
1
C
1
+
1
Builo Uumului
1
1
!
1
SECT
1
!
1
!
+
3
1
f
1
Chap. XII.
The Art of Gunnery.
61
1
.
!
+
+
SECT. X V.
Of Iron Ordnance what quantity of Powder to allow for their Loading.
Ou muſt firſt Calculate a Charge of Powder for the faid Iron Piece, as if it had been
a Braſs Piece, and in caſe you want the weight of the ſaid Irox Piece, you muſt
find it as you were taught in Chap. ? ; and yýhen you have found it as is taught in Chap.
9, how much Powder will Load the fame if it were of Braſs, then juſt 3 quarters lo
much is ſufficient to Load an Iron Piece.
As for Eample
.
A Braſs Saker of 1500 weight requires 44. what will an Iron Demi-Culvering of
2800 weight require? Work as in the 9 Chạp. and you ſhall find 61. or 61.141
ounces, ſo well fortified as the saker, will ferye a Braſs Demi-Culvering for a Charge.
The which we will likewiſe examine by the Rule in the 7 Chapter. .
The Braſs Saker's
diam. is 3 til inthi Logarithm
257403
The diam. of the Demi-Culvering Braſs 4 inches 265321
The difference increaſing.
7919.03
The triple of the difference.
- 23754
The weight of the Saker added to it 1500 317609
gives the Logar. of the weight of of the Demi-C. Braſs 2 592 1. 341363
Or by the Scale, extend the Compasses from 3 7.00 4 av to the fame diſtance
turned 3 times from 15oo, will reach 2592 l. as before.
The Arithmetical way.
4C*15.00
is equal to the weight 2592 pound, which is
==== the weight ſuch a Demi-Culvering ſhould be of
310
that burneth 61.14 ounces of Powder.
To find what a Demicalvering of Braſs of 28 hundred will require, Work thus.
The Logarithmof 2592
- 341363
The Logarithm of 2 800
344715
The difference increaſing.
3352
The one third of the difference,
- 1117
The weight in Powder 6 l. added
283884
gives the weight 7 1. 8 ounces
285001
Or extend the Compaſſes from 2592 to 2800, the fame diſtance will reach from
6:41074 . as before.
The Arithmetical way.
2800 x 6
--=7 pound 8 ounces, as before.
1
1
?
2592
+
S
Of which number you muſt take 3 Quarters for a Charge for the
ſaid Demi-C#-
vering; thereof, being 5 pound 1 o ounces will be a fufficient Charge for ſuch a Piece;
and alſo whatſoever you find on the Scale, and in the Table in the third Chapter for
Braſs Pieces, take three quarters thereof for the Charge of your Iron Piece, it they be
near that weight.
1
Sacr.XVI.
62
1
Book V.
The Art of-Gunnery.
2
SE c f. XVI.
To know what quantity of Powder ſhould be allowed to a Piece of Ordnance
not truly bored.
1
thinneſt 4:3
C
thinneſt 4:3
of Met. 4:3
Dmit the diameter of the Metal of the Piece at the Touch-hole be 16 inches, and
the diameter at the bore is si inches, the weight of the Piece 48go, as you may
ſee By Chap. 7, ſuch a piece you may find in the ninth Chap. requires 1 1 i. for her
due Charge, being near two Diameters of her bore in Powder But by my Inftrum
msent in the general Figure, with the two Rammers heads at the two ends LM
(at the Rammer end that was in the Gun at the Touch-hole, I find by the
prick at S on the Rammer, the foule or bore to be i ined out of his place, or
i inch from the middle of the Metal; then I conclude, that the thinnest
part of the
Metal is 4 inches parts, and the thickeſt lide 6 and parts; by which it appears that
one ſide is juſt 2 inches thicker than the other ſide, as you may fee plainly by this Figure;
the Line AB divided is the diameter
or greateſt thickneſs at the Touch-hole,
every Diviſion ſignifies an inch from the
inward Circle to the outward Circle, is
the thickneſs of the Metal; the inward
B
...Circle ſignifies the bore of the Piece,
which you may ſee is juſt an inch from
Diani, 2:5
'
A
the true bore or Centre of the outmoſt
Circle; therefore you muſt work as if
Centre 6:8
the Piéce were fortified no more than
only ſo much as the thinneſt part of the
Metal is, which here doth appear to be
5:2
4 inches parts, the of the diameter of
Dem
14
the bore is 2 inches added, makes 7
from A to D, the Centre of the bore being the thinneſt part of the Metal, the whole
diameter being 14, which is the true diameter, by reaſon the thinneſt ſide of the Metal
is but 4 inches thick,
And by this you muſt proportion your Charge by the former being 16 inches, if
the bore had been placed at C in the true Centre, then evermore by theſe Rules.
10:4:16 The Logarithm of greateſt diameter 16 is --220412
The Logarithm of the leſs diameter 14:01. 214012
9(4L6
The difference decreaſing,
5800
x[o
3
The triple of the difference Subſtracted --
17400,
from the Logarithm of 11 l. Powder o oruces, 304139
Leaves the Logarishm of the Powder 7 i pound 1 86739
So that 7 peand'i or 6 ounces, is a ſufficient Charge for ſach a falſe bored Piece; or
extend the Compaſſes from 16 to 14, the fame diſtance 3 times repeated from 11, will
reach 7 its pound, as before.
The Arithmetical way.
C14 is 2744 * II
====7 pound 6 ounces, as before.
C 16 is 14096
4
1
1
>
1
.
SECT. XVII.
1
1
-
CHAP. XII.
The Art of Gunnery,
63
3
i
$467:
-YI.
How Moulds, Forms and Cartredges.gnare to be made for any ſort of
Ordnance.
Artredges ärtulually made of Canvas and Paper-Royal, firlé'take the height of thé
Ubore of the Piéce of your Scall a little leſs. part of an inch of the diameter for the
Vent, and three dzami. bf the Chamber of the Piece in breadtkj cut the Paper and the
Canvas; añid forth¢Canon in hảight to the whole Culdering, is allowed about 4 diam.
of the Pièce, from the Cultering to the Minicn; the Charge the length of two dianiet
and call from the Minion to the Baſe 3 diameters of Powder and make them, at firſt
about 4 diameters long, and according to the directions here given,' mark them of půc
a pound of Powder into each Cartredge, and meaſure how full it fils by your Scale for
each Gun in your ship, or Army and by that Rule you may know how to
make a Table, to make a Scale, to mark the Cartredge for the full loading, or diminiſh-
ing of your Powder., according to the goodneſs or: þadneſs of the Powder
and to the exraordinary over-heating of the Piece, having reſolved for what
ført of Ordnance care to ſerve you, and accordingly, to, have a form of Wood
turned to the height of the Cartredge, which
is the 7, 1 part of 22 the diameter of the
bore, and an inch longer than the Cartredge is to be, before, you paſte your Paper on the
form, firſt Tallow him, fo will ſhe Canvas and Paper Mip off without ſtarting or tearing:
if you will make for tapered bore Guns, your, Forms muſt be accordingly tapered, if
you make Cartredges of Canvas, allow one inch for the Seams ; but of Paper { of an
inch more than 3 diameters for the paſting. If once about the former, having a bottom
fitted upon the end of the former, and Cartredge you muſt paſte the bottom close,
and hard round about, then let them be well dryed and then mark every one with
Black or Red Lead; or Ink, how high they ought to be filled, which if you have no
Ladles, Scales, nor Weights, theſe diameters of the Bullets make a reaſonable Charge
for the Canon 2 - for a "ulvering.3; and for the saker 3. for the lefſer Pieces 3 of the
diameter of the Bullet, and let lome want of their weight againſt time they are over
hor, or elſe you may endanger your ſelf, and others.
;
1
Sec T. XVIII.
How to make. Ladles,. Rammers, or Spuoges for all ſorts of Ordnance.
Very Mr. Gunner doth, or ſhould know how to Trace, Cut out, and alſo make up
and finill all Ladles, Spunges, and Rammers, and direct others how to make, and
finiſh the ſame ready for uſe.
You have in the Table in the third Chapter the length and breadth of the Ladle,
anſwerable to eacli Gun in inches and parts, and you muſt allows a diameter more to
incloſe the head of the Staff within the Plate; the Button, or head of the Ladle-ſtaff
muſt be the height of the Shop almoſt; for Spunges, their bottoms and heads are to be
made of ſoft Wood, as Aſp, Birch, Willow, to be one diameter in length, and, or
a very little leſs of the height of the shot, and covered with Sheep-skins, Wool, and
nailed with Cooper's”. ails, that together they may fill the concave of the Piece,
Let the Bottom or Head of the Rammers be of good hard Wood, and the height, as
before one, and the length of the diameter of the Shot at one end next the Staff, it must
be ſo turned, that a Ferril of Braſs may be pur thereon, to ſave the Head from cleaying,
when you Ramme home the Shot, the Buttons muſt be bored for the Staff to be put in
and faſtned with a Pin through, and his length a Foot more than the concave of the Gun.
To make a Ladle for a Chamber-bored Piece, your Compasſes opened to juſt the
diam. of your Chamber within part of an inch, Divide the meaſure into two equal
parts, then ſet your Conspasſes to one of them, and by that diſtance draw a Circle'oni 2
Slat or Paper, the diam. of that Circle is - ſhorter than the diam. of the Chamber, and
take parts of the Circle for the breadth of the Plate of the Ladle; and for Cannons,
the length ought to be twice and parts to hold at two times, the juſt quantity of
Iii
Sec z
1
7
1
Powder.
/
64
1
The Art of Gunnery.
II Book V?
.
1
o
S2C%. XIX:
Honö the Carriage of a Pièce
Should be made.
M
Eaſure the length of the Cylinder of the bore, and once, and half that. length
Thould be the length of the Carriags, and in depth 4 diameters of the bore of the
Piece at the fore-end, in the middle 3 and in and ar the end next the ground 2, and
the thickneſs the diameter of the Shor; the Wheels ſhould be one half the length of
the Picca
in height; the Saker and Minion muſt, exceed the former by part, the Foula
con and Fanlcones by one fixth part, : Sca-Carriagos are made lęksas the Block-maker
that makes them hach. Rules for.
03:37::.
ravimo
Siu.c T. X X.
ܝܪܗ. ܕ ܀܂
1
L
f
"!
1
+
To Borrit
1
To know whether the Trunnions of any Piece of Ordnance are placed right!
МБ
Eafure the length of the Cylinder of the boſe from the Muzzle to the Britch,
Divide the length by 7, and Multiply the Qyotſent by 3, and the Product wití
ſhew you how many Inches the Trunnions muft ſtand from the loweſt part of the con-
cavity of the Piece, and you muſt know that the Trunnions ought to be placed, ſo that
of the Piece may be feen above, or ia that place where the Trunnions are ſet.
ig:02
SI C'T. XXI.
How much Rope will make Britchings and Tackles for any Piece.
N Ships that carry Guns, the moſt experienced Gunners take thiş
Rule.. Look how
many foot your Piece is in length, four times ſo much is the length of the Tackle, and
their Britchings twice the length; and if the Ropes, be ſuſpected not to be good, they
nail down Quoyners to the Fore-Trucks of heavy Guws, that he may not have any play;
and if britchings, and I ackles, and Q Hoywers Phould give way in foul weather, preſently
diſmount her, that is the fureſt way. The Rammers and Spunges are made of four Seranda
"Ropes beſt, and ſerved cloſe, and ſewed with Yarn, that they may be ſtiff to Ramme
home the Shot and Wadd.
Sacri XXII.
4 에
​ih ,
2
I
21
r
1
A
+
1
1
What Powder is allowed for Proof, and what for Action of each Piece
Or the biggeſt fort of Pieces and Canons; for Proof , and for Service her weight
of her Iron shot, for the Culvering the weight of the shot almoſt for Proof,
and for Action, for the saker and Faulcon, and for leſſer Pieces the whole weight of
the Stot, until they grow hot ; and for Proof, the leſſer Pieces give them I and of
the weight of the Bullet in Powder.
SICT. XXIII.
The difference between the common Legitimate Pieces, and the Baſtard.
Pieces, and Extraordinary Pieces.
Ging
pners call them Legitimate Pieces, as have due length of their Chaſe, accord-
ing to the height of their bores; Baſtard Pieces are ſuch as have ſhorter Chaſes,
than the Proportion of their bore doth requires and Extraordinary, are fuch Pieces as
have longer Chaſes, than che proportion of their bore alloweth; and theſe are called
Baſtard Conors, Culverings; and fo likewiſe of Saker and Fanlcon, which by your
Scale, and the Rule thereon, you may preſently find them.
Sect, XXIV.
.
1
1
1
1
1
*
Chap. XII.
Tbe Art of Gunnery
65
1
-
f
!
1
!
in:0...
SECT: XXIY:
How Powder is made, and ſeveral ways. to knop whether Powder be de-
eaged ob 10, by moi tirre or Age, in pårt, önin whole.
Owder was always made of Sait-peter, Brimſtons and Charícole ; but in theſe
latter times experiencë hàth Itill
mended the goodneſs or ſtrenģth of it more, that is
Wäs'in former times by much: but
briefly tilts , the beſt fore that is made at this pret
Pént time is made of ſix parts Sälc-peter, Brimſtone, Chár-cole, one part
.
si birthe Muſquetór Piſtol Powder is niow commonly made of Salt;peter five pafts,- ofie
part of Brimſtone, and one of Cole; Canon-Powder of Salt-peter four times as much
as of Brimſtone, and as of Cole. The reaſon why Piſtol-Powder being the ſtrongeſt
of 6 - 1 - 1 is not ſo good for the Canow as 4-1-1 the weakeſt, although you
take but ſo much of the Piſtol Powder as you find by an Engine to be of like ſtrength
with another quantity of Canon-Poroderi
The reaſon why Canon Powder is beſt for Ordnance, -is; becauſe it taketh üpa grea-
rer room in the Cylinder of the Piece, than Piſtol-Porder , for in taking
it hath the greater length or fortification of Métal about it in the
Pitch
111115,
Suppoſe a Saker require four l. of great Powder for her loading, and I would knoiv
how much Piſtol Powder is equal in ſtrength to fólir l. of Canon-Powder, trying by an
Engine made on purpoſe to try Powder, I find 3 l. of Piſtol-Powder ; therefore you
calily concieve, that 3 pounds have but 3 quarters of the Metal of the Piece to keep it
from breaking, when 4 pound had a quarter more Métal, than the other had.
Lipi
Nath. Nye'Mr. Gänner föund-by 'an'experiment made by him at Deriton tfie-37
of March Anno 1644, he loading a Saker-bore Piece of Iron, and the thickneſs of the 47 Pound of
Metal.
Ivietal about the Chamber was 2-inch. and load her with 4 pound of weakGanon- Pójväder
which filled the Cylinder of the bore 9 inch. juſt; which girch. in length, and two irich.
in thickneſs is 225 inches of Merál-about the Powder, which was 6 ounces more than
the Piece ſhould have had in proportion to Piſtol- Powder': He fired, and the Piece went
off fafe, and he faith, he loaded her again with one pound and of fine Powder almoſt,
which filled the bore but 2 inches and, and had to its Fortification but 6,8 inches,
which in weight is 15l. and when the Gun was diſcharged, it broke into divers pieces,
as there is witneſs enough in that Town,
The harder the Corns of Powder, are in feeling, by ſo much the better it is. Secondly, How to know
Gun-powder of a fair Azure or French, Ruffer colour is very good, and it may be judged gocd Powder:
to have all its Receipts well wrought, and the proportion of Peter well refined. Thirdly,
Lay 3 or 4 Corns of "Gun-powder upon a white piece of paper, the one three fingers
diſtant from the other, and fire one, if the Powder is good, they will all fire at once,
and leave nothing but a white chalky colour in the place where they were burned, neither
will the Paper be touched; but if there remains a groſneſs of Brimſtone and Salt-peter; it
is not good. Fourthly, If yon lay good Powder or the palm of your Hand; and ſet it
on Fire, it will not burn you. Fifthly, To know the beſt amongſt many forts of Pox-
der, make a little heap of every fort, and then ſetting thoſe heaps one from the other,
mark well when you put Fire into them, which of the heaps did rake Fire the ſooneft;
for that Powder that will ſooneſt be on Fire, fmoak leaſt, leave leaſt ſign behind it, is
the beſt fort of Gun-powder.
J: 11.15
/
}
11
}
1
1
:
liig
S5Ć T.
1
Foto
1
66
.
+
Boor V
The Art of Gunnery.
is ide!
W**
::2
1
1
*
ij
SË C T. XXV.
How to make an excellent good Match to give Fire to ang Ordnance.
Ake Cords made of Hemp that is not very fine, or of Tow which is better, Cal-
though it will conſurae ſooner and twitt irruntil you liave made the Srands as
in sa makis little Finger ; this done böyl the faid Cords in ſtrong Lye aſhes
, and a little
Sale-peter, until all the Lye be waſted, and then make it up, and take the feces or re-
mainer into your hand, and with the other draw the Match through twice or thrice,
then drie.it, and keep it for ſpecial uſes, for Vaults, Mines, and moiſt weather, and it
is very fit for your uſe any where.
Ś i c r., X.XVI.
How to make Powder it ſhall not waſt with time, and preſerve that as is good
to keep it from decaying,
Hat time ſhall not waſt it, take what quantity of Powder you will, and mix it with
1 Brandy, and
make it up in Balls, and drie
them well in the Sun, or in a warm place,
and keep them in Earthen Pocs well glazed until you have cauſe to uſe them; this Pow
der will not waſt with age, nor decay.
To preſerve, Powder that is good, all Gruners have, or ſhould have that reaſon to
keep their powder and Store in as good a drie place as is to be had in Porto
Ship con,
venient
, and every Fortnight or at moſt three weeks turn all the Barrels, and Carrjedges
Barrelled up for readineſs
, turn theni upſide down, ſo will the Peter be Diſperz into
every place and part alike; for if it thould ſtand long, the i'eter will deſcend-down-
wards always as it lies, and if it is not well ſhaked and moved, it will want of its ítrerigrh
at top very much, and one Pound at bottom with long Itanding, will be ſtronger than
three at the top; keep all Carcredges which are filled for the Piece againſt he is, lot in
Barrels by themſelves, that you may know them by a mark when need requires
1
1
1
1
1
::?
1
1
:
.
1
1
Sich. XXVII.
T
.
1
To renew and make good again any ſort of Gun-powder that hath loſt its
ſtrength by long lying, or moiſture, or any other means.
FREE
Irſt moiſten the ſaid Gun-porder with Vinegar or fair water, beat it well in a Mor-
tar, and then ſift it through a fine Sieve, or a Search;with every pound of Gun-powder
mingle one ounce of Salt-peter that hath been mealed; and when.you have ſo done, beat
and moiſten this mixture again, until you ſee by breaking, or cutting with a knife that
there is no ſign of Sali-peter or Brimſtone in it; moreover, corn this Powder when it is
incorporated with the Peter as it ought to be; then prepare a Sieve with a botrom of
thick Parchment made full of round holes, then moiſten the Powoder which ſhall be
corned with Water, put the ſame, and alſo a little Boul into the Sieve ; and when
you
have fo done, life the Powder fo-as che Boul rolling up and down in the sieve may break
the clods of Powder, and make it by running through the holes to corn; and if it will
not go through, you muſt beat it again until it will.
1
+
1
1
1
>
SECTIE
1
}
!
fr
.
1
Lt
GHAR. XII.
I bei Art of Gunnery.
67
I
-vin
1
is!!
T
sp
Sarcér.: XXVIII.
To make Powder of diven's Colours, and firſt to make White Powder.
Ake of Sult-peter 12 pares
; of Brimſtone two parts; and of Campllir öne part,
beat,and fift, and incorporate all theſe things together, and after you have' to done,
bcat tlieta again, aid ſo oft until you'are ſure they
are well
incorporated, then móilen
it'with Aqua Vitd and which you have thus doné, cořn the Ponder; as you are fäsight
before.
Tai make Red Powder :::910 1.2. also: ;
Take the ſame things
, and jork them as before directed for white Powder, and as
that was moiltred, with Aqua Vite, now you muſt moiſten this with Yineagaf, being
Red as Blood, which will make the Punyder likewiſe ſo in moiſtening of it, and then
corn it, as is before taught. 6.
To make any Coloured Powder.
!!!!!!!!!!
BoiltheViņeagarin ſuch tranſparent Colours as you would have the Powder to be of,
as if: Blew with Blew Bice, of Green with a little Verdigreace and the likers always
take care that the Colour berhot too thick, but very thin, or elſe it will weaken the
Powder that you do make. od VLT
!: Tit!,!
osar
Sect. XXIX.
son't
Fill's
1
veriline
|
1
!!
1
,*
sºr:-:fi:...
1..
1
OD 30 honi
si
of ſeveral ſorts of Salt-peter, and a way how to make a fort of Salt-peter
very excellent, with eaſe; and leſs-cost than any way.
W!! - oltinis
Rtificial Salt-peter is a mixture of many ſubſtances gotten with Fire and Watercout
A
the beſt of all is of Beaft-dung converted into Earth, in Stables, or Durghills of a long
time not uſed; and when it is to be made it is made with a great deal of Charge. Another
excellent fort of Salt peter is made on Flower that is called Plaſter that groweth on
Walls four parts, of Unlak'd Lime one part, and ſo boiled over the Fire with Water,
which is to no purpoſe to make relation how for to make full direction
will fill my live-
ſed ſheets too fåſt; but this one way, which is the moſt eaſie and leaſt coft, I will write the
Receipt thereof;" which is this
...
Take quick Lime, and pour warm Water upon it, and lex it.ſtand ſix days, ſtirring
it twice or thrice a day; and take the clear of this
. Water, ſet it in the Sun until it be
waſted, and the Salt-peter will remain in the bottom. To refine Salt-peter, and make
it fit for uſe, there is ſeveral ways, but this by:Fire Iſhall only write thereof. Do:thus":
take an Iron Potor Skellet, and fill it with Petery ſet it on the Fire, and cover ir cloſe
with an Iron Cover on the top, or with a Stone, when the Sale-peter is melted, cake
Brimſtone moſt finely
bearengand caſt ſome thereon, kindle it,ánd let it burn until all the
upper part be burned, which when eftected, will leave the Salt-peter clofe like to a
piece of Marble, for the Brimſtone will burn up the groſs victioufnels of the Salt/perer ;
It is to no purpoſe to give a further relation of this, by reaſon every Gunner may have his
Peter ready made refined and in Meal at the Powder-mens, or Chandlers; or it he is
conſtrained to make Peter'or Fowder, he may have ſeveral Books which give a full and
large deſcription of the making thereof, as Nath. Nye Tarta glid, or Nurton's, but
for what is uſeful for a Gunner in particular, is fufficiently ſpoken already ; therefore
let it fuffice now, having ſhewed fufficiently how to make Powder, and trie the ſtrength
of Powder ; to know what Shot and Powder is meet for every Piete, to find whether
the Piece. be true bored or not, to load a piece with difcretion, if not true bored to
make the Diſpert ; and alſo to know the difference betwixt Iron and Braſs Pieces. I all
come to touch how to make a good shor either of Point-blank, or at Random, with as
much eafe and plainneſs as ever was taught by any before.
SECT. XXX
1
3
1
1
!
T
68
IlBooKIN:
The Artufi Gunnėry.
t
܀
Be With
1
!
"
1
Secr. XX.X.
soviHow to Loağ.and Fire aiPiece of. Drdnarićetlikeinın Artiſta: ;?
B
Efore, je ſhootas , Mark, it is best to Load our Pjece, in whicl.fift.obfervethe
and.befareto lay your Budge Barrelui. Cartredgehof youporn to Wind-
ward e fryzur iLiestorand place your Liñſtock to life ward clear
the Tçach-holes and
Spungehe, well, ans: Atrike the Spunge on the Muzzleafbake off the foulnets tivo pa
three blows.
Then let him ſtand on the right ſide of the Grö, äna ibid the Barrel, ſo that his altiſtant
may thruſt in the Ladle being full, give it a ſhogy, then Itrike off the heaped Powder,
he being in the right lidt likevile, with His Body eleif of the Muzzles Put the Ladie
home rofilie Chambé,
Niidily holding yolix Thimb upol the upper part of the Ladie,
füll, then
carti che si dit until your Thumb be quite Ahderit, and
give'a ſhake or tivo to
clear the Pow.ler out of the Ladic; as you hale hini out, keep him upilike you may bring
no Powder out with the Ladle; then with the Rammer put the Powder home gently, and
after put in a good wad, and thruſt it home to the Powder, and give it two or three
ſtroakş, to gather the 199fe Powder, together, and it will. fire the better ; be ſure your
allikant have his Thumb on the Touch-bole all chis while; then pat in the Show with
the Rammer homegrand after him another Wad, cand then with the Rimmer give trvo
or three ſtrokes more to ſettle it home, that there may be no vacuity between the first
Wad Bullet, and laſt Wadde your Budge-Barrel and your ſelf ſtanding to Wind-
ward always, and your"Piece by the Diſpért,directed to the Mark, Prime her, and let
the Powder come from the Touch-hole to the Baſe-ring, your Leg, {tanding for-
wards and fire the Powder on the Baſe-ring, and draw-back your Hand; and you have
fired like the beſt of Gunners, but if you had given fire upon the Touch-hole; the
Porder there would have endangered to have blowed the Cole and Linſtock and all out
of your fiand; therefore, you muſt have a care:ofcagreat. Touch-hole.
!
to ai
SBCT. XXXI.
The difference of Shooting by the rižetal
, and by a Diſpertly a Right Range,
and at Random, by the Figures following.tsa: :.
Hooting by the Metal is the Figure A B, that is, admit you raiſed the Muzzle-ring,
Shooting by
and the Baſe-ring, and the Mark, and your Eye in a Right-line, if you put the
Scale inta the Muzzle with the Plummet:hanging to its you ſhall find it differ 4 or 5 or 6
degti according to the length or Mark of the pieces and in regard of the ſeveral diffe-
rences of the length and marks, or Diameter of her Baſevand Muzzle-ring, no certain
proportion can be generally alligned, yet før moft Rieces.it hath been well obſerved, that
the
Piece.directed by, her Metal, will ihvor about twice as far as when the Mark is level
and ſet by.a Diſpert &c; Quadrant and the Sight-line parallel to the Horizon, ſo that admit
à Piecë were laid-by the Metal of Baſe and. Muzzle-ring, and that it differed from a
right Level 6 degr. as you found by your Quadrant in the Scale, you fired at the Mark;
thie Gun fo layed, and meaſuring the diſtance you find.it.412 Paces, which is as much
beyond the Mark,as it is to it, which is the difference of Shooting by the Diſperc or Axis
of the bore in right bored Pieces following, this is chlled Point-blank; for if you ackrïow-
Shooting
by a Dilperto ledge the higher the Muzzle of a piece is elevated, the farther the shot is carryed in a
Right-line." There can be no Point-blank directly known, nor Rules to know them,
without you take a Piece and make 11 Shots out of her, and ſo by.it proportion a Table
,
as here is one following. You muſt have leave, if poſſible, to Shoot ſo many Shots in
a Piece.at fuch elevations with the like goodneſs and quantity of Powder, as will sake
you ſuch a Table of the Proportions of Right Ranges, called Point-blanks, before you
can have any gueſs certainly of Proportion for other Guns; but to make a good Shot:a6
a Mark, firſt be ſure your Guns Trunnions be placed right, the Carriage well made, the
Platforos
1
11.
il
10:
}
}
។
con
ce
::,!
the Metal.
1
1
?
5
>
.
The mixt or Crooked Motion.
1
F
I
2.48 to paces, right Rainge 10;..
D
E
A:
of Shooting Mira Comme ;' or by the mettall of the peice
412. Paces.
Wh
The way
B
مممممم
+
4
Shooting, punctually, Levill by adjpurt 206. Paces.
D
JOIHIN
LP
*yout
ܐ. ܘܟ
I
The violent Motion.
2123161
,
B
Motion
The Tlaturall
INTE
is
HE
1
th
900
Boo
700
600
500
400
300
200
Too Paces.
100
I
A
6
Book:V:page.68:69.
4
F
F
1
4
|
7
+
1
,
1
1
A!
I
1.
.
4
1
제
​1
1.
1
+
1
1
T
41264
$
2.7.8
and Quadrant on it and Plumb-line, "if it hang in the Mouth of the
Chap. XII. Fle:Art of Gunnery.
69
Platform clean ſwept, and that the ground be Level, and that the Carriage- A Table of
Wheels be one as high as the other, and whether the Axle-tree be placed Point-blanks.
jult acroſs the Carriage, or not; or whether the Piece be true bored or not, i deg.:tandoris
if it be a true bored Pieci, let your Diſpert on the Muzzle-ring gust over, O. 192.
then Centre of the Muzzle of the Piece, which may be done by holding a 17.299
Line right before the Muzzle, and two Sefcks or Notches máce iti Sticks put 2224
on the Muzzle-ring, and by the Line you may eaſily find the middle where
3,6244
to put the Diſpert; and likewilė if the Line touch the upper lide & lowér fide
of the Metal alike, you may be ſure the
Gost liés Level, or by the scale
piece, 6.1285
and no degrees of Altitude, but by the Long Side-lie thereof, be ſure the
71302
Pisce is Level, and will earry the Bullet Horizontally
in his violenc oyile, s. 310
therefore by your Crows and Standers by, or Matrolles ſet about the Piece 99337
to the Mark, a9 at D, that with your Eye two foos from the baté:ring 10 354
you may ſee the Mark on the upper part of the Balk: ring, the top of the 20 454
Difpečt, and the Mark or Turret you Shoot si in a Right-line, as CD, 30693
atid this you may call Point-blanik, and all Shooting in this Form, and no
49 855
other,
SO1060
On the upper part of the Baſe-Ring, the top of the Dilpëre; 'ati thë Mark or Turget To Shoot in 2
Right-Rarge,
you Shoot at in a Right-line, as CB; and this you may call Point-blank, and all
Shooting in this formu, and no other.
But for Shooting in a Right-line called the Right Range of a Buller out of
Right-kira:
for making of Batteries, or Shooting at Random at any advantage, you may make uſe an; Degree of
of this Table, until by your own experience, you have inade out of a Guw by Shopring Elevation.
firſt Level, and aftet ward from degree to degree to 10 degrees mouritüre, or more in a By the Rule of
Right-line.
Proportion.
The uſe of the Table of Right-Ranges, or Point-blanks before-going, it is foundi248 239519
by experience, that the Piece aſſigned at ſix degrees of mounture Shoot 200 Paces in a
a right or inſenſible crooked Line; 1.deſire to know how far the fame Piece will Shoot 10:354,25 4900
in a ſtraight Line, being mounted to 10 degrees? lay by the Table if 285 the Number
againſt o degrees giveth 200 paces, what will 354 the Number againſt so degrees give?
I anſwer, 2487% paces.
The Logarithm of 6 degr. is 28 5.
245484
bacl wards, is
The Logarithm of the Paces known 200
6 digr. againſt
230103
The Paces againſt 10 degr. is 354 is
2 54900
Table,
The Sum
485030
give the Logarithm of 248: Paces
239519
U ::
.
Siiri
that is, & far
as the Bullet
doth go in a
any Piece
pl.
laces,
200'230703
Sum 485003
285245484
Which done
285 in the last
ll
Orextend the Compaſes from 285 to 200, the fame diſtance will reach from 354
200*354
to 248 5 Paces as before
= 2:48: Paces, as beföré.
285
And the Figure is from E the top of the Caſtle to F the ſide of the Tower at
To degr. mounture, carries the Bullet violently 248 Paces.
1
.
SÉCT.
70
The Aidof Gunnery. II Book V:
Siva
1
+
+
5.
1
Jus:1; .
Papillit L
...
SECT. XXXII. T.
nigiri za
:511 Y DSi,....ini.
Horb to Ordego Dire& a Piece, and amend an ill shot that was mirdi,
either by the Keton Level, Right-line
, oy Advantage, or itlount.
Adonyos hath been before taught
again, and when
your Preçelies directly
against
Fter you have made one Shot, und find the piece carry juſt over the Mark, then
the Mark, obſerve how much the laſt ſtroke of the Shor is above die Marx, lo much
longer make your Diſpert, that the top of it may be juſt ſeen from the Britch of the
Piece in a direct Line with the ſtroke of the Shor; and being to fitted, Leyel yous
Piece with this new Diſpert to the aſſigned Mark, give Fire, and without doubt it will.
Atrike the ſame ; if the trſt Shot had ſtruck under the Mark, then bring the Piece in all
points
, as before ;, mark how much of the Diſpert'is over the Shot) and cut it juſt to
Thort, as being at the Britch, you may diſcern the top of it
, the Mark of the Bile-
Ring, and the ſtroke of the Shot in a Right-line, when you percieve it is of Inch:
length, Level thë' Pièce to the intended Mark as at the firſt, Prime, and give Fire.
If the firſt Shot had ſtruck on the right Hand of the Mark, to mend it, you muſt
I.evel the Piece'as furmerly, you ſtanding behind the Britch of the Piece, obſerve the
Itroke of the Shut over the Diſpert, that part of the Bafe-ring which you at that inſtant
looked over in a kight-line towards the Diſpert, and the ſtroke of the Shot, ſet up a Pin
with a little foft Wax on the Baſe-ring, ſo this Pin will be in a Right-line with the Dif-
perr and ſtroke, chen Level your Piece to the Mark intended by this Pin, and the firſt
Diſpert; and without queſtion you will make a fur: Shos; for when you Level by the
Meral of the Baſe-ring where the Pin is placed and the Mark; the Piece ſtanding at thär
direction, look over the top of the Diſpear from the Mark in the Bafe-ring, and you
.
thall find the Piece to lie juſt
ſo much to the left, as the former Shot ſtruck
, to the right,
from the intended Mark, which thould in all likelihood now ítrike the Mark.
?
)
But if the Shot be both wide abil too low, then you muſt uſe both directions, as
before taught to make the next Shot ;' firſt regulate the Diſpert by cutting it thorter,
according as the Ivark of the Shot is lower than the intended Mark, then by the laſt
Rule mend the Shooting wide ; theſe things done with care and diligence, you may
be ſure will mend a bad Shot.
1
Se c'T. XXXIII.
of shooting upon the Advantage or Random, at a Mark, beyond the Righi.
line of the Pieces reach, or Right-range of the shot, and of the Dead-
range for every Degree.
N the two laſt Sections we have Mewed for the Right-range ; now we come to Mhew
for the Dead-range; which confifteth of the right and crooked Range together jo
one, and then called the Dead range, which is the whole diſtance from the Platform
upon which the Piece intended. or aſſigned is diſcharged unto the firſt full of
graze of the Bullet on the Level-Line , or on the ground called the Horizontal
Plain, (by reaſon the different length of the Piece, and ſtrength of the Pow-
der increaſeth or diminisheth the courſe or fury of the Shot, and therefore
more difficult to be found out, but only by Experience, or Diagrams, Tables, or Scales
made by Experiment; now it vis ery diftiault, and a thing uncertain alſo to arrive hure-
in unto exactneſs, without ſome Experience at the Piece; and therefere every one that
will learn to Shoot at Random, muft draw his Piece in a Level Ground, where firſt
Shooting Level, he'muſt obſerve that diſtance in Feet and Paces; then Mount his Piece
to one degree, and mark where that shall Graze, and thus find the diſtance of every
degree from the Level to the 10 degree, and by theſe diſtances make a Table, to which
annes
71
CHAP. XII.
The Art of Gunnery.
annex the degrees againſt the diſtances, by which Table, and the Rule of Proportion;
or Logarithms, or Line of Numbers, find how far another Piece will convey her Shot
from degree to degree; but in caſe you cannot have the Liberty, nor Powder to do the
aforeſaid, you ſhall have a Table here that was made out of a Saker eight Foot long
(by Nath. Nye, as in the 40th. Chap. of his Book,) where he faith, he Loaded her
with three Pound of Porder:, the Shor at one degree of mounture was carryed 375
Yards, or 225 Paces; the next Shot was at five degrees Random, and at that mouriture
the Shot was conveyed 416 Paces; and the next tryal was at ſeven degrees mounture,
and the Random produced 5os Paces; the laſt tryal was at 10 degr. mounture, which
ſent the Shot 630 Paces of five Fort to a Pace.
Whilſt he made theſe Shots, he loaded the Piece himſelf with looſe Powder, exactly
weighed, and weighed the Wad alſo, and beat down the ſaid Wad with four ſtrokes
ſo near as he could by the fame ſtrength, as he did the time before; alſo he let the Piece
coot betwixt each Shot of it felf, ſtaying above half an hour betwixt each Shot, he pue
no Wad after the Bullet, becauſe thé Piece was mounted, he tryed the ſtrength of his
Powder, and noted it down, to compare with other Powder, to know its ſtrength by ;
and that is the way all Gunners muſt take, that intend to make good Shots at Random
All Mr Gunners ſhould be able to draw an exact Deſcription of the ſaid Garriſon, and
every object as lyeth near his works by the Rules of the 7th. Chap. of the Ars
of Surveying by the Sea-Compaſs in this sth Book; fo that he may know what is within
reach of his Gans, by which means he ſhall not be troubled to take Diſtances, but be
ready at all times to know his Diſtances by his Maps : then after he hath made one Shot,
he may make another Shot to any Diſtance he pleaſeth.
+
Example.
Suppoſe I know the Diſtance by my Map where the firſt Shot grazed to be 704
Paces, as you may ſee by the Figure out of the lowermoft Gun of the Caſtle from Ś
to the graze at A, the mounture of the Piece being 4 degr. how much muſt I mount
the Piece to convey her Shot 900 Paces, as you may ſee by the Figure B the Gun, to
C the Shots graze, or place required.
You muſt proportion theſe Diſtances of Random, to thoſe in the Table following :
ſaying, if 704 Paces require 370 Paces, as is in the Table at 4 degrees of Random,
what number will be found againſt the degree in the Table; I muſt Mount the Piece
unto 900, and work by theſe Rules.
Extend the
Compaffes from
370 x 900
the ſame will
473
701
704 to 370,
11
11
Il
reaci from goo
lo 473•
295424
The Logarithm of the Shot made 704 is
284757
The Logarithm in the Table againſt 4 degr. is 370) - 250820
The Logarithm of goo Paces Random is
The Sum is
552244
gives the Logar. of 473 Paces, which I look for in – 267487 the Table
of Randoms, but find no fuch number there, but the next leſs is 461, and the nex
greateſt is
$50 againſt 7 degrees, the difference between theſe two Numbers, is 44 and
461 is 12 leſs than 473, and 12 is almoſt the one fourth part of 44, and therefore it
thews that the Piece muſt be mounted at 6 degr. 15 min, or one Quarter to reach the
Diſtance of 200 Paces as from B or b to C.
$05
461
44
sos
473
32
Kkka
Here
72
The Art of Gunnery Book V.
2
}
of the Quadrant I have ſhewn you; and alſo obſerve how many degrees the Platform
A Table of
the proportion
Here is a Table of Randoms that was made out of the fore-named
of dead Ran Saker of 8 Foot long, and loaded with 3 Pound of Powder, and every
ges.
Gunner is adviſed, it poſſible, to get Powder, to make one by his own
Deg. Randoms. experience, and always to keep ſome of the fame Powder to try his
206 Proportions by ; the Rule in the 24 Section by any other Powder he
Thall have occaſion to uſe, for this is 1 of the material things belonging
I 225
274
to a Gunner, without which knowledg hec an never make a good Shor;
3 1 323
for at the time of a Leagure he muſt expect often to change his Poro-
der, as ſometimes you ſhall have 9 Pound of one fort as good as 15
of another fort, as by Inſtrument and Shooting you may have expe-
6
rience.
7 .sos
8 548
SECT. XXXIV.
9589
10 630
How to make an effettual Shot out of a piece of Ordnance
at Random.
Very one that hath Charge of a Gun, muſt at one time or other ger leave of his
ſure the Diſtance from the Platform to the firſt graze of the Shot, you muſt apply it to
this Table by the laſt Rules of Proportion in the laſt Section and find what deg. you ſhall
need Mount the Guito for any other shot at any other time, when you ſhall have oc-
caſion; when you have Loaden your Piece, as you are directed in Sect. 32, take the
Diſtance to the Mark in the XVI Chap. of the ſecond Book of the Deſcription and uſe
4 370
5 416
461
1
E
1
F
J
is higher or lower than your Mark by your Quadrant on the back-lide of your Scale;
after you have done that, then Calculate by the laſt Rules what degree the Gun muſt
be Mounted to, to reach the Mark, if the ſaid Work be under the Platform, subftract the
Difference found by your Quadrant, out of the degree of the Random; but if the ſaid
Mark be higher than the Platform, Add the degr. of that Altitude to the degree of the
Random, and at theſe corrected degrees Mount your Piece.
on a
How to Mount pour Piece by your Scalc and Quadrant thercon.
t
tha l'iece.
The uſe of the
To the ſide of the Scale or Quadrant is a piece of Braſs .fitted of the ſame breadth,
Quadrant on with two Screws, and holes fitted to ſcrew the Brafs Plate two Ixches of the former
back-ſide of length, without the Edge or Side of the Line of Numbers, for to take any Angle that
th: Scale in the is under the Line of Level, for if you put the Braſs into the Mouth of the Piece, the
third Chap. for
the Mounting of
Line of Numbers being next unto it, and put in the Tompkin into the Mouch likewiſe
to ſtop ir faſt in the middle of the Metal at the bottom, and then the ſtanders by raiſe
the Britch with Crows to what degree you pleaſe; and ſo likewiſe if the Mark or degr.
aſſigned be above the Line of Level, if the scale will not ſtand faſt by the degree
of
the Diameter that fits, the Bore, putting of it juft into the Mouth of the Piece, then
ſcrew the Braſs Plate to a hole made on purpoſe for the other ſide, and turn the degrees
of the Diameter to the Bore, and falten it with the Tompkin in the middle of the
Mouth, as before ; and ſo this Inſtrument will be moſt uſeful for all things as belong
to a Ganner, with leſs trouble and Charge, than any other that ever was made by any
other Men, and far more uſeful.
· Then the Inſtrument being in the Month of the Piece, as before directed, mark di-
ligently until the Plumb-Line, which proceeds from the Centre of the Quadrant, cut
theſe aſſigned degrees and Parts of degrees that you are to Mount the Genby, in the
Arch which is Divided into 90 degrecs in the outward Circle thereof ; your Gun fo
Loaded and fitted, as beforeſaid, make your Shot, for without queſtion, you
make a good Shot, and ſtrike or came near the Mark,
As
}
will
1
?
A
CHAP. XII.
Ibe Art of Gunnery.
73
As for Example.
you
Suppoſe you make tryal of your Gun as is ſpoken of in the laſt Section
32,
find that at 4 degrees of Random upon a Level. Ground the Shot is conveyed 704 Paces,
if
you
be called out in haft upon Service againſt a City, or other Fort, and being ap-
pointed to play your Guntowards it, you alſo find it to be beyond the reach of the
Right-range of your Shot, and
the Diſtance being 560 Paces; and alſo that the place
is lower than where you can Plant your Cun by one degree and or 10 min. then to
know the degree of Mounture, you may work as by the laſt Rule, if 704 gives 370
againſt four degrees, what will s60 give? the Diſtance to the Mark, it will give you
the Number 295 ; look for this Number in the laſt Table 295, or the neareſt Number
to it, and againſt that degree and Part of a degree, which muſt alſo be found by Sub-
ſtracting the neareſt leſs Number out of the neareſt great Number, the greateſt Num-
ber in the Table is 323, the neareſt leaſt is 274, the Difference is 49, the difference
betwixt 323, and 395, is 28, the half of 49 is 241, which thews that degree is 2
and a little above of Random, but becauſe the Mark is lower than the Platform, Sub-
Atract one degree or 10 min. out of 2 degr. 32 min. and the Remain is o degr. 22 min.
the true height the Piece muſt be elevated, to reach the Mark; but if the Shot graze to
the right cr left, you are to mend it by direction in Sect. 31, but ever by the Example
or Direction there.
Suppoſe the Shot graze over the Mark 20 Paces, Subftract this 20 out of 56o che
Diſtance, and Mount the next Shot according as if the Mark were but 5 40 Paces diſtant,
if 20 Paces too ſhort, make the next Shot as 5 80 Paces, that is the degree that is found
by that Proportion to reach ſo far.
!
Sac I. XXXV.
Irft, you
How to find the Right-Line, or Right-Range of any Shot diſcharged out of
any Piece, for every elevation by one Right or Dead-Range given for
the Piece aligned.
muſt have the help of this Table of Dead-Ranges, which was made by You may make
Experience of Mr. Norton, and the Table of Point-blanks, or Right-Ranges, was uſe of this Ta-
the fame Shots in a Right-line at every degree of Mounture, as this Table is the Dead- ble by the ſame
graze of the Bullet at the fame Shot with the ſame Gun, for he made 200 Shots for Rules in the 32
tryal.
Now although it be a thing very difficult, and likewiſe uncertain to arrive herein to
exactneſs, without ſome Experiments made with the aſſigned Piece and Powder ; yet to
come to a neceſſary nearneſs at firſt
, far furer than by accertain gaeſling by this Fable,
or by my Scale , and the Rules therein directed.
Se&. and Table
of dead-ranges
As for Example.
Admit you were to ſeek the Right-Range of a Bullet, that the Piece was fired at 30
degrees Mounture, and the Dead- Range of the Bullet was known to be 2200 Paces.
.
2200
X
693
210 Paces.
2150
3
11
.|
Kkk 2
Or
1
1
I
1
1
574
T-be Art of Gunnery
Book V
R
1
O
1
{
..
اور
ivici
1.610
Range of
}
::*
1
i
!
:
دارد به دل
ولو
1
КІ І
1849
A Table of Pro-
Or look in this Table of Dead-Ranges againſt 30 degrees is
portion of Dead-
Ranges.
42150)
-333243
Degr. Paces. And the Dead-Range given or knowit for 30, déğreti
192.
to be 2800 Paces.
334 42
08:298
The Number againſt 30 degrees in the firſt Table
of
21.09 - 404
Point-blanks 693
284073
?
-3.510
610
Y:
The Sum 41 618315
4;
Gives the Logarithm of 710 Paces, for the Right=2
518722
285072
6:.
6. 828
The Bullet carryed violently in a Right-Linejat 3:0 degr. Moun
934
8.;
1044
ture.
9:1-1129
ind
But admit the Level Right-Range is given, and the Right-Range
19.1214
of 30 degrees Mounture be fought.
IL 1296
1.2 I 394
Work by theſe Rulesa
13 1469
The Logarithm of Point-blank o degr. is 192
27833'0
14:1544
15 1622 The Num. againſt 30 d. is 693 in the Table Point-blank-284673
16 1686 The Level Right-Range of this piece is 197 Paces 229446
17. i 1744
18lib -1792
The Suña 5.113519
Gives the Log. of 711 Pačes for the Right-ranges required 285189
19
20 1917
And as the Numbers are in the Logarithm, fo you may do by
25 2013
the Line of Numbers.
30 2150
693
+ x +97
711 Paces the
35 2249
2289
Range in a
2296
192"
Right-Line.
45
2289
52
2283
SICT. XXXVI.
60
1792
To know how much of the Horizontal-Line is contained directly ander the
Right-Line of any Shot called the Right-Range made out of any Pięce
at any Elevation aſſigned.
Eit propounded to find what part of the Horizontal-Line lyeth directly under the
Right-Range of the Piese alligned at 30 degr. Elevation, the Right-Range for 30
deg. Mounture, by the laſt Rule is found to be 711 Paces, Work with the Complement
of the pieces Mountare 60 degr. thús always.
As the Radins 90 degr.
1000000
is to the number of Paces in the Right-Range 711 Paces 285186
So is the Compl. Sign of 30 dégr. Mountare, which is 60 degr...993969
to the Logarithm of the Horizontal-Baſe 619 Paces
Now
yon
find that 619 Paces lies under the Right-Range of the Shor; you may pre-
fently
find how much of the Horizon is contained under the Crooked-Range of the Shot,
you Subſtract 619 the Horizontal Diſtance out of 2200, the Randoms at the firſt
graze of the Bullet from the Piece, the Remainer is the Horizontal Diſtance 1581
Paces, which lies under the way of the Shot, as it goeth helically between the Right-
Range, and the natural or perpendicular motion, or before it make the firſt graze;
like in all other queſtions.
IH
.11
40
42 /
!
:
1
}
1
1
1
279155
11
is ...:
if
the
1
Secr, XXXVII.
1
1
1
1
1
75
: ازر . :: .:
1.
>
.
i
M
0
1
+
Chap. XII. Hibe Are of Gunnery.
e
silioria ,
& C3.XXXVII.
of the violent, érõokėd; and, natural Motion or course of a Shot diſcharged
out of any Piece of Ordnance affigned,
Y the thiqd and fourth Propoſition of the ſecond Book of Tartaglia, his Nova
Scientia; ſhews that every bodynequally titavy, as a Shot in the end of the violent
motion thereof, being Diletrarged out of a Prece of Ordnance, ſo it be not right up, or
right down, the Crooked-Range, hall join with the Right-Range, and to the natural
Courſe and Motion betwixt them bort.
+
B
X
1.11.
A
birinto
IT
CH
:
As for Example.
The Right-Range being all the Right-Line A B which is properly called his Violent
Motion, and BC will be the mixt or Crooked-Range, and CD the Natural Motion,
wherein from A to B is the furtheſt part of the Violent Motion, and from C to d the
end of the Natural Motion.
And in the ſeventh Propoſition of the fame Book, he proveth that every Shot equal-
y heavy, great or little, equally elevated above the Horizon, or equally Oblique or
Levelly directed, are among themſelves like and proportional in their Diſtances, as the
Figure following thewetla, as A:E:F is like and proportional in the Right and
Crooked-Ranges unto H : 1, and in their Diſtance or Dead-Ranges AF unto A: I.
And in his fourth and fixth Propoſitions of the fame Book, he proveth that every
Shot made upon the Level hath the mixt of Crooked-Range thereof, equal to the Arch
of a Onadrant 90 degr. and if it be made upon any Elevation above the Level, that
then it will make the Crooked-Range, to be more than the Quadrant.
And if that be made Imbaſed under the Level, that then the Crooked-Range thergof
will be an Arch leſs than a Quadrant,
And
4
76
Book V,
Tbe Art of Gunncry.
if
And laſtly, in his ninth Propoſition of the fame Book, he undertakes to prove,
one Piece be Shot off twice, the one Level, and the other at the beſt of the Random at
42 degr. Mounture, that the Right-Range of the length is but of the Right-Range
of the beſt of the Randoms, and that the Dead-Range of the Level is but as of the
Dead-Range of the beſt Random, whereto he that delires a further Demonſtration of
theſe Propoſitions in his ſaid fecond Book of Nova Scientia.
+
:
:
line
>
1
1
A
I
K
SECT. XXXVIII.
1
How to make a Gunner's Rule, being an Inſtrument which will ſerve to ele-
vate a Piece of Ordnance with more facility than the Gunner's Qua-
drant.
Ecauſe the Quadrant on the back-ſide of the
Scale cannot be conveniently uſed at all times,
eſpecially when the Wind blows hard, and being
near the Enemies Guns, the Plumb-Line is ſo long,
or too long before he ſtands ſtill; to remedy this
,
the Gunner's Rule was invented; the Figure thereof
you may here fee; this Ruler muſt be 1, or 1 2, or
14 Inch. long according as the Gun will require, it
muſt have a long flit down the middle thereof like an
Eye-vane of a Quadrant or Back-staff, the Head
thereof make Circular according to your Gun, as
you ſee in the Figure ; the Inſtrument is deſcribed
ſtanding upon his Britch of a piece of Ordnance;
3
in the middle of the ſmall narrow lit you muſt place
a Lute-ſtring, a well twiſted Thrid, or Silk-ſtring
may ſerve, this Bead muſt be ſet to ſuch an Ixch and
Parts, as you find is agreeable to ſuch a degr. as you
muſt Mount your Gun unto, and on one ſide the ſlic
you muſt place a Diviſion of Inches, and every Incb
into ro Parts Divided, and thus it will ſerve for all
ſorts of Guns, but if it be for a particular Gæn,
then on the other ſide you may place the degr, and
min. when you ſhall find by the length of your G:51,
incb. and Parts goes to make one degree;
but to uſe it with all ſorts of Ordnance, let it only be
Divided into Inches and Parts, the Bead ſtands if
Ir.cb. four in the Rule.
2
.
ZI
1
SECT.
77
the legii
The uſe of the Table
if your Gan be 8 Foot long one Inch on makes one degree; your Gun 12 Foot long 5 Inches into makes
2 degrees.
Or you may ſet your Bead to 5 i: Inch, to Mount him a degrees.
SECR
2
I
167
198
1412
7016
847
989
821
I 210
7811
2611
(1
40
I
222
LO7
443
388
&
12
8912
2 5113
The Art of Gunnery.
1
472
6518
8210
II
2414
I
715
8519
fitting is
the help of this
How to Divide the Gunner's Rule into degr. by help of a Table,
be Elevated to any degr. without the help of a
foot long to 14 foot long; and by
SECT. XXXIX.
other Geometrical Inſtrument phatſoever.
fit the Ruler for one Gun only, here is the Rule for the Deviation of
Note, this Table hath 11 Columns, the firſt shew's the length of the Piece in
Feet and half Feet, the other 10 in the Head is ro degr. and under is Inches and the
of an Inch, from i degr. to 10 degr. and ſo you may take them out of the
Piece from 5
Piece 1719 y
Quadrant, Ruler, or any
Table, and let them on the Rulers oppoſite ſide.
ing
Table,
8 Foot long.
I
683
81
18
) ܐ
The ler.. of the Picce 1 Degree.lz Degrees 3 Degreeift Degrees's Degreesls Degrees 7 Degrees 8 Degreesi? Degreestro Degr.
Feet and Feet. Tach.100 Inch.100 Inch. 100 Inchio d'Inch.soofiach.100 Inch.100 Inch, 10 olunch. 1ool Tach.1o.
Inch. 100
s Foot long I
312
613 817
1416
219 25110
281
S
Foot and half. I
283
4214
56S
201
6 Foot long
66
+
886
589
291
6 Foot and half. I
36 2
6
724
815 44
80ls 1719
5310
691
7 Foot long
944
4115
88,7
7713
7
7
Foor and half.
583
14.4
28,7
4210 9912
5514 14115 71
S
365 410 721
4010
7613 44:15 12116 32
S Foot and half.
585 3717 6
9511o
74.12
5314 3216
92
9 Foot long
6812
471 3713 2715
9
Foot and half. 2
0014
06
8
216
4120 4
10 Foot loug.
2016
4010
S4118
5021 8
1 Foor & half. 2 2014 4116
6918
4817
681
11 Foot long.
314
6216 939 2411
9120
14
u Foot & half. 2
42/4
269
68112
1014 5316 9519
37 21
12 Foot long.
67
5910
65IS
7120
2522
33
1 2 Foot & half.
6315 2017
8910 5213 15115 7818
6726 33
1 3 Foot long.
8210
9613
4419
40
13 Foot & half. 2 845 688
3614 20127 4'12
88;22
5628
14 Foor long. 9515
go 8
85111
75.17 7020 6523
6c126 5629 53
793
893
12117
8118
19:19
I
795
589
98
Io
Cli2
1014
gir8
2
30112
1014
3018
6114
28.15
73116
8113
8811
19
8 9122
10
88,16
5613
2
22:18
82123
80124
487
21
2
I 2!1 2
1817
5315
7825
2
4123
Chap. XII.
4127
48,21
2
488
745
7016
9224
68,27
for any
5211
7225
421
TN
100 part
2
8014
*
:
1
1
"}
78
The Art of Gunnery.
Воок V.
SECI. XL.
How to give Level to a Piece of Ordnance, with the Gunner's Rule at ang
:i
Degree of Random.
";
Y
Our Piece being Loaded in all points, as is before taught, and you have brought
the Piece in a Right-line with the Mark, the Diſpert being placed upon the Muzzle.
Ring; in like manner place your Ruler upon the Baſe-ring, and let one Itanding by hold
it, for the Foot of it fitted round to the Gux, you may be ſure to put it right, and you
may eſtimate on its perpendicular near enough, now having before the Diſtance to the
Mark you intend to Shoot at; and admit you have found it to be 461 Paces, and the
firſt Shot you made for Practice out of that Gun, conveyed her Shot at two degrees of
Mounture 2 74 Paces, then according to the Rules in the 32 Section, and the Tables
of Random, there I find 461 againſt 6 degr. which I muſt Mount the Gun to reach
461 Paces.
will require;
Then to find by this Table how many Inches, ard hundred parts of one Inch 6 degr.
look in the Table above, and find on the left Hand in the firſt Column the
length of the Piece 13 Foot juſt, under 6 degr. in the Common-Angle, you ſhall find
16. Inches, and to that height I fer the Bead on the Lute-ſtring, to 16.4 inch, or
16.0; for every Inch is Divided into 10 Parts, and every Part is ſuppoſed to be Di-
vided into to more ; then cauſe the Piece to be Mounted higher or lower, until you
bring the Bead, the top of the Diſpert, and the Mark all in one Line, ſtop the Piece in
that poſition with a Coyn, Prime, and give Fire.
If you will Shoot by the Meral of the Piece, Subſtract the height of the Diſpert
out of the Inches found by the Table, and the Remainer, Mount your Picce
unto it
the Diſpert be 3 Inches long, Subftracted from 16:44 found in the Table, leaves 1 2
, or 12 of an Inch, you muſt fer the height of the Bead to Shoot the ſame Die
ſtance, by the Muzzle-Ring without the Diſpert.
SECT. XLI.
1
How by the Table to give Level to a piece of Ordnance, without the
Gunner's Rule.
of
F you have not a Quadraxt, nor a Ruler, and would make a Shot at 4 degrees of
Elevation, look in the Table, and find the length of the Piece, which fuppofe to be
9 Foot and half, right againg 9 in the Angle under 4 degr. you ſhall have 7 Inches
to be the length of any Stick, cut, and ſet it upon the Bale-Ring, and bring the top
the ſaid Stick, the top of the Diſpert, and the Mark in a Right-line with your Eye, and
Prime, and give Fire, and you will make as good a Shot, as if you had the Ruler, and
Bead, or Quadrant; if you will have no Diſpert, take the Diſpert
, and Meaſure it upon
the foreſaid Stick at the
Bafe-Ring, and from it cut off its lenghth juſt, and the Remainer
you may uſe upon the Baſe-Ring, and it ſhall mount the Gun to 4 degrees, as before ;
aud bring the top of the Stick, the Muzzle-Ring, and the Mark in a Right-line, and
you may be ſure to make a good Shot , if the Diſpert were 3 Inches, that cut from 7
inshes, the Remain is 4 Inches, for the length of the Stick to be ſet on the Baſc-Ring,
for to Level the Piece without a Diſpert.
1
..
1
+
:
SECT,
1
1
CHAP. XII.
The Art of Gunnery.
79
SECT. XLII.
How to make a shot at the Enemies Lights in a dark Night.
U
Pon ſuch occaſion, to Shoot at a Light ſeen in the Night, Diſpert your Piece with a
lighted and flaming Wax-Candle, or with a lighted piece of March, that with
your Eye you may bring the Baſe-ring, the fired Match on the Muzzle-Ring, and the
Enemies Light in a Right-line, (or mark) then give Fire, and you will make a good
Shot.
SECI. XLIII,
How to make a perfect shot at a company of Horſe-men, or Foot-men paſſing
by the plare where Ordnance doth lie upon a Level-Ground; and aljo to
make a good shot at a Ship Sailing upon a River.
The
Ake a Piece that will reach the way or Mark in a Right-Line that the Horſe or
Foot are to paſs by, then your Gun Loaded ſo with Powder as it may preſently
take Fire, and Shot fit for that ufe'; and ſeeing a Tree, Buſh, or Hillock, or ſome turn-
ing croſs way for his Mark, and when the Enemies come near to that way in a Right-
Line with his Gun, give Fire : and at Shooting at a Ship in a River, he muſt put his
Piece to ſome evident Mark on the other ſide the River, and when the Head of the Ship
ſhall begin to be betwix the Piece and the Mark, and then give Fire.
SECT. XLIV.
.
How to cauſe the ſame quantity bosh of Powder and Shot, diſcharged out
of the ſame Piece, to carry cloſe, or more ſcattering.
N the Book of Mr. John Bate of Extravagants, he faith, take the quantity of a
Peaſe of Upiam, and charge it among the Cafe-Shot, and it will make the ſaid Shot
fie cloſer together, than otherwiſe it would; for Opium is of congealing and fixa-
this a Sea-man found by experience.
1
1
tive nature;
Sect. XLV.
How a shot which ſticketh faſt within the Concavity of a Picce, that it can-
not be driven home unto the Powder, may be shot out, without hurt to
the Gunner, or hurt to the Piece. :
VV
in as
Hen any Piece of Ordnance is Charged with ſuch a Shot as will not be
driven home unto the Powder, then the Gunner to ſave his Piece from
breaking, muſt ſo Imbaſe the Mouth thereof, or put hin under the Line
of Level, that fair Water for two or three days being put into the Touch-hole at ſeveral
times, may run into a Veſſel fet under the Mouth of the Gun, to ſave all the Salt-Peter
that was in the Powder : and then let the Gunner clear the Touch-hole, and
put
much Powder as poſſible he can, and Prime, and give Pire, and it will ſerve to drive
out the Shor.
But when a Shot hath lain long in a Piece, until he is grown ruſty, and ſo ſtick falt;
put ſtrong Vinegar in the Mouth of the Piece, and with the Rammer ſtrike the Shot
until it doth move, then put in Vinegar until it run clear through the Powder and Shot,
Prime, as before, and give Fire with good Powder ; and if it do not run through after
it hath ſtood 3 days, clear the Touch-hole, Prime, and give Fire.
LII
SEC
L
80
The Art of Gunnery.
Boox V.
!
!
1
S. CET. XLVI.
A Piece of Ordnance at the fame Elevation, and towards the ſelf-fame
place, with the like quantity:ef
. Powder and Shot, diſcharged ſeveral
times
what difference there is in their Ranges.
Here hath been a Piece diſcharged in the ſpace of an Hour feaven times, with the
T
like quantity of Powder, Shot, and Mounture, and their Ranges have been fonud
as followeth ; the firſt Shot was conveyed 418 Paces, the ſecond 438, the third 442,
the courth 4?4, the fifth 427, the ſixth 412, the feaventh 395; ſo that the greateſt
difference from the firſt Shot was '4 Paces; this every Graner muſt take notice of, if
he intends to Shoot well at Random ; the reaſon of theſe things is this, the firſt Shot of
Powder found, the Chamber of the Gun moiſt, and the Air quier, the ſecond Shot, the
Chamber was dryed, and the Gun in a good temper, and the Air moved towards the
Mark with the firſt Bullet, and having lefs aſſiſtance than the firſt, went beyond, and
made the beſt Shot ; and every Shot after, will come ſhorter and shorter, as the Gues
grows hotter and hotter ; the reafon is, by how much hotter the Piece is, by ſo much
the more attractive is the concavity of the Piece made ; and becauſe the Shot' is driven
forth or expelled with no other thing, than by the Air's exaltation, or Wind,
cauſed through the Sald-Peter; and therefore the oftneř the Piece is Fired, and the more
heat, the more attractive, which fuppleth and retaineth çonringally more of that
Wind, which ſhould ferve to expel the Bultes And therefore the vutúe expullive in
that Piece, doth more and more' decreaſe, and the Skol Ayeth no Wiili itat fwifineis, as
it did before in the 2 firſt Shots, which dryed and brought the Giw into his beſt temper;
but the third and fourth Shot is but little differeřice from the firſt, but the reſt will differ
f
every Shot.
.
Nicholas Tartaglia doth report, that many Shots being made at a Battery by a
Piece, it chanced by fome occafion, the Piece role up in fich fort that the piece
touched the ground with its Mouth; a little Dog running by, did ſmell into the Mouth
of the Piece; and aftera little time, was drawn almost to the further end of the conca-
vity : they pulled hing out almost dead this was done by the virpe Attractive.
,,;;;
:
1
<3
SECI. XLVII.
How to Weighs Ships funk, or Ordnance under Water
: or to know wka
empty Cask will carry any ſort of Ordnance over River.
No
Isholas Tartaglia hath well collected from the Learned Archimedes, and fath
calculated the Proportion of Stone, and other Metrals, what they will weigh in
the Water, and in the Air.
All Men know by, reaſon, that whatſoever is heavyer than ſo much Water, as the
body of the Metal thruſteth our of the place, or Vefteſ, will link; and being lighter
than ſo much Water, will ſwim.
Tartaglia faith, that ordinary Free ſtone weighing 93 Iin the Air, trill weigh büt
in the Water; which is near, asa isto f, between the Free-ſtore, alid'the Wateř. And
that Marble-ſtone that weigheth 7 l. in the Air, will weigh but 5 l. in the Water, whichi
is near as, 7 to 2, between the Marble
, and the Waçer.
ArdiIron and Trin; that in the Alf weighieth ribt. will weigh 16%. in the Winter
ſo it is as 19 to 3.12.11
Braſs weighing in the Air 65:1. will in the Water weigh so la sed to Braſs is to
Water, as 65 to 10.
And Lead weighing in the Air 30 l. will weigh in the Water but 27 1. 6o Lead is to
Water, as 30 to 3:
.!
bas &
And
--
>
Chap. XII.
I be Art of Gunnery.
81
; 'we
And laſtly, Gold in the Air being 171. in the Water, Briſtol, Septem. 1 2.1667
it will weigh but 16 l. fo Gold is to Water, 'as 17 to 1. by experience of a Spaniſh
He alſo layeth
down Rules to weigh Ships, or Guns, or any ship called ibe Viétory of
thing elſe in the Water, that hath not lain coo long, and Majork, Somke in Hung-
docked it ſelf in Oaze : for it the thing funk be upon Sands rode at the Pill; her Bur-
or Rockis, it will weigh the better. He deſcribes Vefſels then about 300 lins,
Loadeu to be brought to the place where the thing is funk, iveighed her with 4 empty
and a Globe of Glaſs put in a Frame of Wood, and a place Lighter; of 30 Tunsa
in the bottom of the Glaſs to put his Head into the concave, piece, by laſhing the Ligh-
he.may both ſee and breath, and by a Windleſs Rope, and cors at LowWater, Head
weight to link it, he may firſt let down the weight, and and Stern; and at High-
after have himſelf down in that Frame, that is in a form of water the Ship we heavil
an Hour-Glaſs
, to the bottom of the Sea, and do the work, A ſhore by a Grabbie,
and ſling the hip, and Guns, and when he will come up
erd did riſe as the Water
to the top again, to un-wind the Rope, and the Frame did flow; and the Low-wa-
will be guided upright, and he and it will come to the ter or two afterw.rd, che
top of the water very fafe, and faſten this Rope brought Ship was free, and ſwim-
from the Ships, and un-lead the Veſſels, then will they med; the ship and Water
Buoy up the Ship ſunk, or any thing from the ground. WAS Eſtimated to weigh
1000 Tuns, that the four
Or by a Float-Stage & Windleſs, Capſton, and Trunk of Lighers weighed; she bad
Leather made ſo thick, that no Water can come in, and with
ng Goods in, cut turned over
a pair of Glaſs Eyes faftned, that no Water can go through, as ſhe was
She was, having done
fitted to the Caſe of Leather within, and two Bladders blown to Carein.
at the brim or top of the Water, made faſt to the Caſe of
Leather to ſwim, the Mouth of the Caſe, while the Man goes down with Ropes fit to
The beſt weight
lling it, and makes them faſt at liberty, then hale him up after a time fit, and by is where the
your Veſſels, as before, and Float-Stage, Heave and Buoy up the thing funk.
Know this, that s Tun of Cask will ſwim a Canon of 8 or 9000 weight, 4 Tun
a Demi-Canon, 3 Tun a Culvering, and 2 Tun a Saker, with all things belonging there-
unto, as Planks and Ropes.
water cbs and
flows much,
SECT. XLVIII.
How many Horſes, Oxen, or Men will ſerve to draw a piece of Ordnance.
Or every hundred weight of Metal, one Man; fo a Piece of 8000 Pound weight
requires 80 Men, and as many more Men as the Carriage may be in hand weights;
for
every: oo weight of Metal, one Horſe, then 16 Horſes will draw a Cun of xooo
weight, but in the Winter 24; alſo 17 Yoke of Oxen is thought ſufficient to draw a
Piece of 8000 weight, but in the Winter, they had need be one third part more.
Lll 2,
SECT.
1
1
82
The Art of Gunnery.
Воок
V.
1
1
1
Sec T. XLIX.
Hom Gunners may take a Plott of their Garriſon, and every object near it.
HE
E may by the Compaſs and Ruler directed in the ſeventh Chapter of Surveying
of, Land, take the Plott of his Garriſon, and every noted place or way within Gun-
Mot, and draw it into a Map, and have it in a readineſs, and need not be troubled to
make. Diſtances every time he hath an occaſion to make a Shot , but by his Scale and
Map, may know if his Gun will reach any place thereabouts; and by the fourth Chap
of the General uſe of the Canox of Logarithms in Mr. Gunter's Works, learn the Pra-
Etice of Fortifications, there it is put down for him at large.
"!
41
1
1
..
나
​1
1
11
1
83
Rodi
baiset
ရင်
OF
A R T I F I C I AL
FIRE-WORKS,
FOR
Recreation,
9
A N D
SEA and LAN D-SERVICE.
CHAP. XIII.
SECT. I.
A Deſcription of the Mortar-Piece, and how to make one of
Wood, and Paft-Board (for a need,) Braſs and Irois one's
being wanting
He fame Metal that makes the beſt fort of Braſs.Ordnance, they
make Mortar-Pieces with, and by theſe Medfüres, if the Dium.
or Bore be 9 Inches, let the Mortar be one Foot and half' in
length, and let the Chamber in which you Load your Piece with
Powder be 3 Inches Diam, and 4 and a half deep; the thickeſt
of the Meral above the Tõuch-hole 3 Inches, and the upper
part thereof 1 Inch
To make the Mortar-Piece of ivood and Paft-Bourel
.
Provide a Wooden-Ruler of ſuch bigneſs as you deſire to make the Diam:ter of the
Moricr, then greaſe your Ruler well, thar the ſtuff may flip off that is put about hiin,
which is Paft-Boards and Canvas, and very well plyed with hot Glue; and after he idey
a little while on the Rowler, and another while off from the Rowler; and when this
kind of Trunk is very dry, piu it on the Ruler, and let it in a Lathe, and cut vif biors
ends of the Trunk with a Chizel very even, then turn a Foor thereto with a louder in
put the Trunk upon, and in the middle thereof make the Chamber for your intelis
if the Piece be s Ixches in the Mouth, let the thickneſs of the Part-Bourd Trunk.be
two inches thick, and 18 Inches long, the Britch or Foot 10, the Shoulder 2 inches
Jong, and 2 high, that when the Trunk is put on this Shoulder, and joyned wirn the
Food,
N
I
!
1
1
84
E
El
Artificial Fire-VVorks. Book V.
Wood, it may be juſt even with the fame; the Bore into which you put your Powder
muſt be two Inches high, and three deep, Plated with Copper, Lattin, (if it be poſſible)
as alſo all the reſt of the Wood that goeth into the Trunk; when you have
put the
Trunk into the Britch of Wood, nail it round about the Shoulder, by making holes
for the
Nails, and then driving in the Nails upon that Wood, that you made to receive
the Paſt-boards or Trunk; then cover both Wood and Trunk with good Belch-Cord
and Glue again, and let it be well dryed, it will laſt a long time, and with ſuch you
may Shoot Ballouns into the Air for Recreation.
1
1
*
SECT. II.
E
How to fit and prepare Granadoes for the Mortar-Piece.
He Shot of great Mortar-Pieces are moſt commonly one tenth part lower than the
Bore, becauſe of Cording thens, to ſling into the Mouth of the Piece; and for fear
of ſecret Cracks, which cannot be eaſily eſpyed, they are coated with Pitch, ſo that
being fitted and prepared, they do but juit fit the Bore.
How to make Fuſes.'
Every Ball hath an hole left to put in a Fuſe, or piece of Wood, juſt like a Faucet for
a Spiggot; this hole muſt be juſt one quarter of the Diameter of the wooden Fuſe,
which Fufe muſt be in length three quarters of the height of tbe Granadoe; make it
taper; and then filled with compoſition, and driven gently into the Powder that is in the
Ball, Icaving a little of it without: the Compoſition of this Fuſe is made thus ; take
one Pound of Powder, four Ounces of Salt-Peter, and one of Brimſtone, firſt beaten
to Powder, and lifted in a Sieve ſeverally, theſe Ingredients being mixt together, your
Compoſition is fit for uſe.
1
Se CT. III.
1
How to make Granadoes of Canvas for the Mortar.
T
He operation of theſe Granadoes made of Canvas is quite contrary to theſe already
ſet down : theſe are only Fit to Fire a Town, they are not of fo violent execu-
tion, as the precedent, yet altogether as coſtly in the making; for tħe making of then,
fit a piece of Canvas upon a round Bal of Wood or Form, fo big as you would have
your Granadoe, then take this Compoſition following ; four Pound of Salt-peter, two
Pound of Gun powder duſt, "and two Pound of Brimſtone ; all theſe incorporated,
and moiſtned with Oyl of Salt-peter ; fill your Caſe with this Compound, and cover it
with Cords, and pierce the Sack full of holes, and in every hole put an Iron Barrel,
Charged like a Piſtul; thefe muſt be driven into the Sack unto the head, then let there
be an hule about an Inch deep, which ſhall ſerve to Prime it with Powdet-duſt, moiſten
it with Oyl of Petrol; be ſure your Barrels have great Touch-holes that the rúlt
through time may not choak them, and they will be ready for ſervice many years.
។
1
SBCT. IV...
Hon Granadoes are to be charged in a Mortar, Aud Fired,
Ou muſt take great care in the Loading or Charging the Mortar, thus; firſt, weigh
Y the Powder to a brachn that you put into the Chamber, and after it pur'a gcod
cloſe Wadd of Hay, or a Tampion of Wood, then.cut a TupFo# the Ground that -
may juſt fill the bottom of the hole or bore of the Mortar next the Wadd ;- your Gra.
nadue being prépared with a coat oi Rich and Card, as before caught; fing it into che
Mouth of the Mortar'; obſerve to have the lule of the Granadse in the micdle of the
Bore, then go to the Britch, thruſt up a Wire, " in the Touch-hole to make.fürė, then
.
Prime
T
1
A
LITI
Chap. XIII. Artificial Fire-VVorks for VVar. 85
Prime with the beſt drie Powder you have ; for (believe me) this is a ticklish fort of
Shooting; without care, your Life, and Mortar-Piece is now at ſtake; but we will
give you very fure directions how to give Fire.
Provide ſmall Fuſes, ſuch as we taught you to make before for the Shells, but leför
about a quarter of an Inch bore, three quarters of an Inch thickneſs, and eight Inches
long, fill theſe with good Powder-duſt, moiſten it with Oyl of Salt-Peter but a little,
and
put
it in with an Iron Rammer, try whether you like the time, they continue burn-
ing; if tco flow, abate Oyl of Peter; if tvo fift, add more to it.
This being prepared, the uſe is, (viz.) thruſt the Pick of your Linſtock in at one
end of the Fule you mean to give Fire withal, bid one of your aſſiſtants come to one
ſide of the Mouth of the Piece, and give Fire to your Fuſe, wherewith Fire the Fuſe in
thu Mortar, and then with great ſpeed give Fire to the Touch-hole; theſe Fules are
very certain to give Fire, but Match doth ofttimes fail.
SECT. V.
T
Pitch it,
How to make Hand-Granadoes to be hove by Hand.
Here is good uſe made of Hand-Granadoes in affauits
, and Boarding of Ships,
and there be two ſorts of them made; the firſt is thewed already, the ſecond is made The Sdells die
by Sea-Gunners upon a Mould made with Twine, and covered over with Car. made of Glaſs,
tredge-Paper, and Muſquet-Bullets cut in two, put with Paſt and bits of Paper thick on
or nelld Clay,
or Paper
the out-lide; after you have doubled the Shells, Paſte on ſome at a time, and let it drie,
and then ſome more until he is quite full; then dip him in ſcalding Rozin, or Pitch,
and hang him up, and he is for your uſe, but you muſt have the innermoſt end of the
Twine. which muſt be left out at the ſmall hole for the Fuſe; and before
you
you are to wind it out, and ſtop the hole, and then Pitch it.
To Load them, fill theſe ſmall Shells with Gun-Powder, then make a Fuſe of one
Pound of Gun. Powder, lix Ounces of Salt-Perer, and one of Char-cole; or if you
will have thein of leſs durance, you may take the Compoſition made for the Fures be-
fore ſpoken of for great Granadoes, knock the Fuſe up to the head within 0.2 quarter
of an Inch, which is only to find it by in the night ; ſtop well the reſt of the holes, if
any Chinks are open, with ſoft Wax; then your firſt Shells muſt be coated with Pitch
and Hurds, left it ihould break with the fall, and be ſure when you have Fired the Fuſe,
ſuddenly to caſt it out of your hand, and it will do good execution.
SE c T. VI.
1
A
How to make Fiery-Arrows or Darts like Death Arrow-Heads.
MA
Ake your Head of Iron, Tharp and bearded, to ſtick fast; and to it have an Ar-
row or long thaft of Wood, and about the middle of that Head make faſt a Lin-
nen Bag in form of an Egg, leaving open at the end a hule, that it may be filled with
the Compoſicion following ; take one Pound of Peter, half a Pound of Gan-powder,
and as much Brimſtone in Powder ; all theſe Ingredients being mixt well, and mingled
with Oyl of Petriol; with this fill the Bag round about the Arrow-head, then let all
be bound about with Wire ; and for Priming of theſe , dip Cotton-week into Gun-
Powder wet with Water; and well dryed again before it is uſed, and let the Arrows or
Shafts be fo put into the Head, that when they be ſtuck in a Houſe or Ship-lide, or any
whcre elſe, the Man which endeavours to pull them out, may be deceived, and pull only
the Shaft, and leave the Heud to burn the Houſe, or Ship, or Mens Cloaths, or any
thing elſe ; if you throw or ſhoot it well, it will fire whatſoever combuſtible ſtuff or
matter Thall be near it, as Sails, Timber, Pitch, Tarrel places; and this will much aí
fault Eneries in ſtorming a Work, or Boarding a Ship
SECI
1
86
Artificial Fire-VVorks
Book V,
SIC T. VII.
1
How to make Fire-pots of Clay.
Ire-Pots, and Balls to throw out of Mens Hands, or with a Baſcula, may be made
of Potters-Clay, with Ears baked, and to it hang lighted Matches, and throwing
them, if it lighteth on a hard thing, it breaks, and the Matches Fire the Powder, and the
half Bullets of Muſquets contrived upon them, as before, diſperſes, and doth much
miſchief; their mixture is of Powder, Peter, Sulphur, and Sal-armoniack of each one
Pound, and 4 Ounces of Camphire pounded, and Searced, and mixed well together
with hot Pitch, Linſeed-Oy), or Oyl of Peter ; prove it firſt by burning it, if it be too
ſlow, add more Powder, and if it be too quick, more Oyl, or Rozin, and then it is for
your uſe.
1
SICI. VIII.
How to make Powder-Cheſts.
Ou
Y ,
and one longer and broader to the bottom thereof; between the three Boards
put
a Cartredge, then make it up like a Sea Cheſt, and fill it with Pibble-ſtones, Nails,
Stubs of old Iron, then nail the Cover on, and the end to the Decks, in ſuch a place as
you may Fire the Powder underneath throngha hole made to put a Piſtol in.
SECT. IX.
How to make Artificial Fire-Works for Recreation and Delight.
VV
E ſhall not deſcribe the Moulds in particular, being needleſs ; for ſuch Men
as are inquiſitive into theſe things, let them buy Mr. Babrington, or Bate,
or Malihus, or Nortor's Fire-Works, here we will lay down ſuch Rules, as ſhall be
as ſoon conceived without Figures, only a Rowler or Mould for to make the Paper
upon ;
and that may ſerve for all the reſt, they being made in the jame manner.
To mak good experienced Rockets our way, do thus ; get a Form or Rowler to be
turned in a Lathe, what thickneſs you pleaſe, and intend to make your Rockets, and
let his length be 8 times the Diameter; if it be of an Inch in thickneſs, the
length will be o Inches, put ſo many Rowls of Paper on this Form, until it is an
Inch thick, or make it aInch the whole then Paſte the upper ſide to the reſt , then you
muſt contract the Paper together an Inch from the Mouth, thus : dip an Inch of the
Caſe in Water, the Formor in him, and with Twine, about of an Inch from the end
gather it in; but let a Formor, or a thing near the bigneſs be put into his Mouth, while
you
draw it in with the Twine and choak it ; you muſt remember to leave a ſmall
hole to put in a Wire through the Compoſition half way the Rocket, as big as a Bod-
kin; then take out the Formor, and dry them, and they are for uſe at any time; the
Figure following makes all plain ; A is the Mouzh of the Rocker, B ſo far the Bódkin
muſt be thruſt up the middle; you muſt be provided with a ſmaller Bodkin, which when
your Rocket is filled with the Compoſition, and tyed to the Rod, you muſt thruſt this
Wire Bodkin in at the Mouth, ſtraight up to the midſt of the Rocket, having a care
not to thruſt it more upon one ſide than the other.
*
***
SECT,
}
Chap. XIII.
For Recreation.
87
S = c T. X.
To make the Compoſition for Rockets of any ſize,
He Reader may make uſe of theſe Rules, not upon truſt
out of Authors, but found by Practice and Experience;
and frit for Rockets of 1 Ounce, you muſt uſe only Canon-
Powder-duſt, being beaten in a Mortar, and finely Searſed;
this makes him rife very fwiit, making a great noik, but car-
ries no Tail. Theſe of more Operations are made by putting
one Ounce of Char-cole-dult to 8 Ounces of Powder ; this
Compoſition will hold for Rockets of one, two, and three
Ounces; but for theſe of four, take three Ounces of Char-
cole-dult, to one Pound of Canon-Powder-duſt, continuing
that Rule, until you come to Rockets of jo Ounces; and alſo
for Rockets of a Pound, take one Pound of Powder-duft, and
tour Ounces of Char-cole-duſt, and theſe are big enough for
any Recreation or Delight.
To fill the Rockets with this Compoſition,
Hold the Mouth downwards where it was Choaked, and with
a knife put in ſo much as you can of the Receipt provided for
that size at one time, then with a Rammer fitted to the Caſe,
and with a Mallet give three or four indifferent knocks, then
put
in
more Compoſition until it be full, every time knocking
the Rammer, as before, until the Compoſition come within
one Diameter of the bore of the top; then put down a piece
of Paſte-board, and knock it in hard, prick three or four
little holes therein; then put fine Piſtol-Powder in almoſt to
the top, and upon that another cap of Paper, upon which put
a Piece of Leather, that it may be tyed on the top of the
Rocket, and faſt Glued on; then get a ſtraight Twigg, and
bind it upon the Rocket with good Twine ; it muſt be no heavier, than being put upon
your Finger an Inch and a halt from the Mouth of the fame, that it may jult ballance
the Rocket, then it is prepared for ufe.
ܝܙܝܐ
a train of
To give Firc to one or two Rockets.
Set your Rockets Mouth upon the Edge of any piece of Timber, that ſtands fo high
from the ground, that the ſtick may ſtand perpendicular from it downwards or upon
a lide of a Wale or Carriage-wheel, or any dry place whatſoever; then lay
Powder that may come under the Mouth thereof; give Fire thereto, and you have done;
but to Fire more Rockets than one , that as one deſcendeth, the other may aſcend by
degrees : make this Compoſition following; of Roch-Peter eight Ounces, Quick-
Brimſtone four Ounces, and fine Powder-duſt two Ounces, which lay in a Line, from
one Rocket to another, they being placed ten Inches, or a Foot one from another ; give
Fire to this Compoſition, and you have your defire, if you did prick the Rocket with
the Wire, as directed; you ihall ſee how gallant one will mount the Air,when the other
is ſpent.
11 mm
SECT.
88
Book V:
Artificial Fire-Works.
SBCT. XI.
How to make flying Serpents and Rockets that will run upon a Line, and
return again.
Or this you muſt provide a ſmall Rowling-Pin about one quarter of an Inch in
thickneſs; upon which Roul ſeaven or eight thickneſs of Paper ; fill them four
Inches with Powder duft, ſometimes putting between the filling a little of the Compo-
ſition for Rockets of ten Ounces, and at the end of four Inches choak him ; fill two
Inches more with Piſtol-Powder, then choak the end up, and at the other end put in a
little of the mixture for Stars, which follows, and choak between them and the Com-
pofition, and it is fit for uſe ; but divers of thoſe with the Starry end downwards upon
the head of a Rocker and Powder-train to blow them out, when the Rocket is ſpent,
they will firſt appear like ſo many Stars ; when the Stars are ſpent, taking hold of the
Powder-duſt, they will run riggling to and fro like Serpents; and when that Compoli-
tion is ſpent, they will end with every one a Report, which will give great content to
the beholder,
I did omit to ſpeak of Runners in its proper place in the laſt Section, for that is the
Compoſition, which you muſt make them of, very carefully whether they be, double or
ſingle, or thoſe that carry Dragons, Men, or Ships, or other Shapes in motion, leaſt
they ſhame their Maſter; the Line muſt therefore be fine, even, and ſtrong, and being
rubbed over with ſoft Sope to make it ſlippery, and not eaſily to take Fire; Thoſe that
turn Wheels, may have a further addition of Roch-Peter in their Receipts to add plea-
fure and life to the beholders ; You muſt have a piece of Cane as long as the Rocket,
and bind to the Rockets, and ſo that ones Head may be to the others Vent, that when
one hath carryed the Cane on the Line to the end thereof, the other may Fire, and bring
him back again to the Tower or place where it was fired; theſe Figures are made wide
ſtrong Paper or Parchment, and with Lattin, and Wire, and Twine, until they be
brought into theſe Shapes, and then painted like Ships, or Dragons, or like the thing
it carries with it.
.
SE cT. XII.
How to make Fire-Wheels, or as ſome call them Girondles.
File
Irſt be provided with Spinning-Wheels or the like, made eaſy to run round
upon
its Axis, Horizontal, or Vertically ; and put Flags on the top of the middle,to
ſet out the 'Wheel; bind Rockets to the Wheel, and Crackers betwixt each Rocket,
with the Mouth of one towards the Tail of the other ; thus continued, until you have
firted the Wheel quite round; which done, cover them with Paper paſted over, and
coloured handſomly to ſet it out, that one taking Fire, they may not Fire all, and daub
Soap upon them quite round, leaving the Mouth of one of them open, to give Fire
thereco; the firſt Rocket being burned, will ſet Fire to the reſt one after another, keep-
ing the Wheel in a contioual motion, until they be all ſpent; you may bind Fire-
Lances to theſe Wheels, either upright, or near over athwart, which will make to
appear diverſity of Fire-Circles; you muſt take care to place your Wheels called Gi-
rondles at convenient diſtance from other Fire-Works, leaſt they ſhould make a confu,
fion, and ſpoil all the Work.
SECT, XIII.
1
Chap. XIII.
Artificial Fire-VVorks.
&q
SECT, XIII.
1
How to make divers Compoſitions for Starrs.
F:
Or Starrs of a Blew colour mixed with Red, the Compoſition is of Powder-mealed
eight Ounces, Salt-Perer fonr, Quick Brimſtone twelve Oances, Meal all theſe vec
fine, and mix them together with two Ounces of Aqua Viva, and half an Ounce of
Oyſ of Spike ; which let it be very dry before you uſe it.
Another Compoſition which will make White and beautiful Fire ; take Powder,
eight, Salt-Peter 24, quick Brimſtone two, Camphire one Ounce, Meal theſe Ingre-
dients, and Incorporate them; make the Camphire with dipping your Peſtil into a
little Oyl of Almonds, and it will Meal preſently, and keep it cloſe from the Air, or
elſe it will become of no uſe.
Another White-Fire which laſtech long, take Powder four, Salt-Peter 16, Brin-
fone eight, Camyhire one, Oyl of Peter tio Ounces, Meal theſe that are to be Mealed,
and mix them according to the former directions.
1
SECT. XIV.
T
How to make and uſe the Starrs.
Ake little ſquare pieces of Brown Paper, which fill with either of the foreſaid
Compoſitions which you like beſt, fold it down, rowling it til you make it round,
about the bigneſs of a ut or bigger, according to the ſize of your Rocket that you
intend them for, Prime them with drawing through them Cotton-Week, and they
are
prepared to make faſt to the Wheels : you may alſo make them thus; you muſt have a
Rowler which muſt be as big as an ordinary Arrow, which thall be to Rowl a lengtla of
Paper about, and Paſte it round, and dry it well, fill it with a Thimble, and thrult it
down with a Rowler, and then cut it in Thort Pieces about half an Inch long; then you
muſt have in readineſs either hot Glue, or Size mingled with red Lead, dip therein ons
end of your ſhort Pieces, leaſt they take Fire at both ends together ; bélides, it wil
Bot ſo eaſily blow out; theſe being thus done, ſet them to dry antil you uſe them, and
in the top of the Rocker, whereas in the 10 Section you were to fill ii with Piſtol-
Powder, now you must put rone, but a very little, and that is to blow one of the bits
of Starrs out, which muſt ſtand in the room of the Powder , and on the top of thui
another Tire, with ſtrewing a little Powder and duft ; and in like manner another, to
third or fourth, putting a little ſmall corned Powder between them, until you come
unto the top of the Rocket-caſe, there put a Paper over the Head of it
, and tie it clor:
about the top, that none of the Powder come from between the Stars; the Corton-
Week is ſuch as the Chaundlers uſe doubled 6 or 7 times, dipped in Salt-Peter Water,
or Aqua Vile, wherein ſome Camphire hath been diſſolved; or for want of either,
in fair Water, cut it in divers pieces, Rowled in Mealed Powder dryed in the Sur,
and it is done.
2
V
Sec T. XV.
How to repreſent divers forts of Figures in the Air with Rockets.
Hen you have divers Rockets to make for a great Fire-W'ork,let one be with
a Report, another with Starrs, another with Golden Hair or Rain, one with
Silver Hair or Rain, which it ſeems to be when you are right under; and
upon the Head of another Rocket place the Serpents, and they will make moſt delight-
VV
fil ſport.
Mmm 2
SE?,
90
Воок У.
Artificial Fire-Works.
SECT. X.V.I.
How to make silver and Golden Rain, and how to uſe them.
1
Y
Ou muſt provide ſtore of Gooſe-Quills, which being provided, you muſt cut then
off fo far as they are hollow ; the Compolition to fill them is, cwo Ounces of
Cole-duft, and one Pound of Powder well mixed; having filled many of theſe Quills,
you
Mall place them in the ſame place as I told you to put the Powder and Stars, put-
ting a little Piſtol-Powder to blow them out, as you did the Stars, and fill the top of
the Caſe as full of them as you can, with the open end downwards ; ſo ſoon as the
Rocket is ſpent, there will appear a Golden Shower, or Rain; or with the Compo-
fition for White-Stars filled in the Quills, will make a shower of Silver Rain.
1
.
.
4
Sec T. X VII.
How to make Fire-Lances.
M
Ake them thus ; firſt, you muſt make Cartredges, or Caſes juſt like the Caſes
for Rockets, only thoſe for a need may be made with Paft-Board, and Glued, as
they are formed round, but the former is better ; let them be filled with the drie Com-
polition for Stars in the 13th Section ; Prime them with wet Gun-Powder , the lower
end of the Cafe is ſtopped with a piece of Wood, to the end they may be nailed and
ſtirred when and where they dhall be uſed, the Wood being about three Fingers.breadtls
long out of the Caſe or Cartredge, or as long as you will.
:::
F
1
Soc. XVIII.
11
The manner how to make Balloons for the Mortar-Piece.
Ou muſt have a Formor or Ruler twice the length of the Diameter and of the
bigneſs, as you will have the inſide of your Balloon, and upon that Formor put
ſo many Paft-Boards, as you ſhall think ſufficient for ſtrength, then Paſte or Glue
them well together, and choak him at the end with a String, leaving a ſmall hole for a
Port-Fire, which muſt be made juſt like a Rocket, but no holes in it as the Rocker hath,
and of ſuch length as is fit : now to fill the Balloons, place all your Serpents within ic
together, with Stars, Rockets, and Crackers, leave very little room within the Cafe,
or Cartredge; and being filled, put in as much Powder-duſt as you can, that it may run
every where through the Chinks between the Serpents, Rockets, and Stars, that they
may all Fire, and that the ſaid Powder-duft may break the Balloon; theſe things thus
done,' choak up the other end cloſe, and Charge it in the Mortar, as we have taught
you to do the Canvas Granadoe in the fourth Section, and you may ſhoot it when you
pleaſe, and you will make moſt excellent delight to the Spectators, and credit to your
felf; for this is part of the way of Mr. Malibus's Fire-Works, which were the beſt that
ever I practiſed.
+
1
A
SECT.
1
F
了​一
​:
-
三
​生​。
A
*
section 17
Rocket single or doble
madetorin vpon
line from tower to lower
:
.
Section
sect: 4.
OTION
H
B
A
6 is E
Bee
The BallA is a Granado
coated armed ard bopdeg
th
Fire Bols of Clay wireares
and market iB
Secliore, y
Balls fofire and stick is D
made up same stuffe.com
7
section
Fire Arroyes ordarts sect:
FG is a sillinder Granada
of turned timber to
shoote fliers section
the zo
Vertual wheelew House
w you may vnder-
slandýform of horizon
dall wheeles or any other
ABCD are the monter
Peie e mould and Ballons
section &
AndM N is a wonder
Morterrorchambers as
m section 18
Piena die hele
o
:
6
F
.
DIRECT
G
100
M
G
NE
ដើម្បី
.
D
C
c
B
Booke V. ». 90.91.
.
:
:
:
:
'.'
主
​i
1
}
1
|
|
1
1
1
.
4
9
|
上
​f
i
4
1 引
​}
垂
​*
己
​4
}
1
r
.
-
1
|
作
​1
--
|
4
F
:
|
1"
:
1
1
JF
1
4
||
|
1
1
产
​青
​}
4
”
}
Chap. XIII. A Cure for burning with Gun-Powder. 91
SE cT. XIX.
A moši precious Unguent for any Burning.
D
Ivers Men in the Practice of Fire-Works one time or other chance to be burned,
or blown in the Face by Powder ; here you have Mr. Malthus's Salves, which
is known by often Experience to be very good; and will fully cure you.
The S ALVES.
Take freſh Hogs-Greaſe, or Lard, as much as you pleaſe, and boyl, and take off the
Scum, until there ariſe no more Scum; then ſet the Lard three or four nights abroad,
after which it muſt be walhed in running Water to take away the Saltiſh nature, and to
make it White ; then melt it, and keep it for your uſe.
Otherwiſe,
The White of an Egg, and freſh Butter being mingled together, and well beaten into
an Oyntment, is excellent goodi
SE cT. XX.
Another Salve moft Excellent.
TA
Ake a Stone of Quick Lime, and let it be diſſolved in clear Water, and when the
water is ſettled, pour it out gently from the Lime through a Linnen cloath, then
pui
as much Sallet-Oyl, as you have Water together, and beat it all to an Oyl, you may
keep it for ſuch uſes, and you have a molt Sovereign Care for all manner of Burning
whatſoever.
܀
;
1
The End of the Fifth Book.
TO
12
12
F
2
+
i,
လုံး ***
bobolo dai
proceso
9o
Search
ho
*** ၁၅
google
Τ Ο.
E
Cap. SAMUEL
S A MŰ EL S T U R MY
THE
1
A U T H O R,
For his Work and VVortb.
Slancia
Ince'mongſt all Nations Warit ſelf doth ſhew,
It Man behooves Wars Weapons for to know.
Who here may learn the Gunner's aiming Arts,
Which thy free Induftry to all imparts.
Therfitceſt Subject now it is by far,
At theſe times when ſuch Rumors are of War.
And fills the Ears, and couragës awake;
Go on then, and to thee this credit take, (write,
That he that Reads theſe things which thou doſt
May know a Gunner's part, though he never Fight.
And knowWars chiefeſt Engines uſe, and ſtrength,
In Bar, Cylinder, Axis, and in Length ;
In Touch-bole, Carriage, Wadd, in Shot and Charge;
Of Fire-Works; in brief, Thou giv'ſt in Charge
French, Spaniſh, Dutch, Italian, Vail your Caps
To the Author's skill in Mars his Thunder-Claps.
Cap. JOHN VINGEN T.
1
1
Τ Η Ε
1
70-90 90 90
OUXO
70
.
017OELOCUCI
1
T H E
MARINER'S MAGAZINE
OR,
STURMI'S Mathematical and Practical
ARTS
The Sixth Book,
Wherein is Contained,
The Definitions of the Circles of the sphere; With the manner how to Reſolve
all the moſt neceffary Propoſitions thereunto belonging, by a Line of
Chords and Signs; As alſo, by Chords and Tangents,
and half Tangents; and
likewiſe by
Calculation of the Tables of Artificial Signs and Tangents, and all uſual Aſtronomical
Propoſitions appertaining to the firſt Motions, being all of extraordinary life; whereof few of them
have yet been Treated fo plain in the Engliſh Tongue.
420
e
30
20
OTT
Circultis
Eguineti
AGRIUS
OI.2
Tunewsfeed
Stordal II
Tropicus
WR
Canoni
ols
Hori
zm
80.7coo4.30*20-10.940:20 130:4023000-170-810-6.90
440
Murni
U1111111! ITINIAI
60 SO. 470
Mimdi
Capricorni
alis
30 1:20.. IO
Circulus
Anfartinus
SPM
08 OL
09
os
WW
BUMNITRITONOWE
reu
Felices anime quibus hac cognofcere primam
Inque domos Superas Scandere cura fuit.
F
-
|
1
수
​,
1
.
1
성
​!
4
1
+
1
95
ราวการต่อ
stoch
ออกจอ
poor het
maibe sabe
de the
ooosha
4 dodnebor
Τ Η Ε
Denomination of Eight and Forty Conſtellations of
the Fixed S T A R S;
OR,
The Rudiments of ASTRONOMY,
Put into plain RHYTHM E s.
7
3
He Army of
Then comes the great Dog, at whoſe Tail,
Declares the Glory of God moſt high, The famous Argo ſpreads her Sail:
Seen and perceived of all Nations,
Above the little Dog doth Hame,
For whom the Latins had no Name;
In Eight and Forty Conſtellations.
Firſt, near unto the Northern Pole, Long Hydra on her Tail alow,
The Dragon and two Bears do role,
Carries the Pitcher and the Crow';
Whole hinder parts and Tails contain The Centaur holds the Wolf by'th bee!,
The luffer and the greater Wain. The Altar, and Ixion's Wheel,
The Hare, the Bear-ward, and the Crown; Are never ſeen of us; but here
And then comes Hercules kneeling down ;
The Southern Fish brings up the rear.
The Planets.
And next below a place doth take,
Under thoſe Fixed Stars above,
Great Serpentarius with his Snake.
Under the Harp of Orpheus,
Seven Planets in their Orbs do move ;
The Eagle, and Antinous,
The higheſt is Saturn, thirty year
The Silver Swan her wings doth ſpread He ſpends in compaſſing his Sphere;
Above the Dart and Dolphin's head; Twelve Jupiter, and Marsin twain,
Then fegafres comes on amain,
Sets forward, and comes round again.
Andromida follows in her Chain
Then in one year the Sun diſplays
Th: Triangle below her Itands,
Three hundred ſixty and five days;
And at her Feet in Perſeus's hands, And near a Quarter, which in Four
The Gorgon's head above are ſeen Encompaſſings, makes one Day more.
Her Parents Cepheus, with his Queen
Between the Sun and us, there Hy
Caſſiope, not far below,
Fair Venus, and ſwift Mercury;
Heniochus his Goat doth ſhow;
Theſe always near the Sun we find,
On his left Shoulder, in his Hand Not far betore, nor far behind.
Canft thou
He doth the ſtormy Kids command. The Moon is lowelt, who in ſeaven
bind the ſweet
Here in the zodiack begins (Twins (1) And twenty Days goes round the Heaven Influence of
The Ram (.) the Bull (o) the loving And about two Days more doth run,
looſe the
The Crab (,) the Lion (0) and Before the overtakes the Sun.
Bouds of Orz-
(Virgin (sup) tender ;
So twenty nine and half in all,
Canit thou
The Ballance) Scorpion (m) and Do make a Month Synonidal.
Bow-bender (7)
Theſe Planets make their courſe to thiEaft, bring forth the
Mazarroch in
Goat (Y,) Water man internet
) then Fiſhes. Though they be fafter hurled Weſt ;
his reaſon, or
twain (,)
And lix Degrees the reſt may ſtray,
cault thou
Shall bring you round to the Rami'y again. Belides the Suns Ecliptick way.
quite .-rétines
Ek's O1:S ?
In the Southern Herniſphere,
The Circles of the Sphere.
The monſtrous Whale above the rest,
Six greater Circles nark you thall,
Eridanus ſcarce wets his Breaft,
Which equally divides this Ball;
Over the Hare Orion bright,
Juſt in the midſt in between the Poles,
Sparkles in a cold Winters Night;
From Eaſt to West the Equ?!95 Roles.
Nnn
The
Job 7. v. 31.
the Pleiadies,
Ol
on ?
ver. 32
Fifteen
Images appear
Ae
96
+
!
The Ecliptick cuts him, and doth ſlide Through both the Poles o'th'World do paſs
Twerity three and half Degrees alide. And the Equinoctial down-right croſs;
Horizox even with the ground,
And Nineſcore Parallels have that Line,
From Stars below our light doth bound. By which Stars North and South decline.
Meridian ſtraight upright doth riſe, The Ecliptisk hath his Longitudes,
Parting the Eaſt and Weſtern Skies. And Parallels of Latitudes
Two Colares through the Poles do run, For Stars ; But in Geography,
Quartring the Circle of the Sun; The Towns beſides the
Equator lie.
One where the Spring and Fall begin; Over our Head, and under, Feet,
Th’other where longeſt Days come in :
The Nineſcore Azimuths do meet;
Four leſler Circles mark them well, And here as many Parallels,
Are to th’Equator Parallel.
Of Altitude Horizontels.
Two Tropicks bound the Sun's high way, Longitudes and Meridians all,
Shewing the long'ſt and shorteſt Day.
And Azimuhs great, Circles call ,
The Arttick Circle cuts the Bears,
But all the Parallels in Heaven,
Th’ Antarktick oppoſite appears.
Being leſſer cut, the Globe uneven,
Meridians half twenty four,
Degrees three hundred and threeſcore
For Hours, and for Degrees Nine ſcore; Hath every Circle, and no more.
!
The Heavens declare the Glory of God, and the Firmament ſhew-
eth his handy-work, Pfal. 19
When I conſider the Heavens the work of thy Fingers, the Moon,
and the Stars which thou haſt ordained, what is Man, that
thou art mindful of bim? orihe Son of Man, that thou viſiteſt
him? Pfal. 8.
t
1
1
1
$
1
STORM T 'S
1
t'
97
964
Photo
၌ အ
do 10 debe other boobs bobobet
autofocus on todella
obok ဝသ
ܘܗܽ
bahor
( 2006
ogbogbo
S T V R MY 'S
MATHEMATICAL and PRACTICAL
A R T S
The Sixth Book. .
Wherein is contained a Definition of the Circles of
the Sphere, with the manner bom to Reſolve all the
most neceſſary Propoſitions thereunto belonging, by
a Line of Chords and Signs, or by Chords and
Tangents; as alſo by Calculation by Tables ,
CHAP. I.
1
N the former Books are contained fuch Pro-
blems Geometrically, as are moſt neceſſary
for every profeſled ingenious Artiſt to un-
derſtand and Practice
; Now to the end
the Practitioner may Elevate his Thoughts
to the contemplation of thoſe Glorious Bo-
dies, the Sun, Moon, and Stars ; I Mall here
in this place give a brief Survey of the firſt
Rudiments of Aſtronomy, for the help
of
young Practitioners and Mariners, for
whom chiefly I take theſe pains; 1 thall give
a brief and ſuccin&t Explanation of the ſe-
veral Circles of the Sphere, better than we
could in the foregoing RHYTHMES
to be underſtood, and then thew how to re-
folve the moſt uſual and common Problems
thereof; and after the Art of Dialling Geométrically and Inſtrumentally, and by Cal-
culation, as promiſed.
1. This is to be underſtood, that Aſtronomers do imagine 1o principal Points, and
10 Circles to be in the hollow inſide of the firſt Moveable Sphere, which are commonly
drawn upon any Globe or Sphere, belides divers other Circles which are not delineated,
but only apprehended in the Fancy.
OC
Non 2
The
.'
1
98
Spherical Definitions in Aſtronomy. Book VI.
The Points are the two Poles of the World, the two Poles of the Zodiack, the two
Equinoctial Points, the two Solftitial Points, and the Zenith and Nadir.
2. The Poles of the World are two Points, which are directly oppoſite to one
another, about which the whole frame of Heaven noveth from the Eaſt into the weſt,
whereof that which is ſeen on the North-ſide the Equinoctial, is called the Arctick-
Pole.
3. And the other directly oppoſite to the former is called the Antrittick, or Souch-
Pole, and can be ſeen only on the South-lide of the Equator ; a right Line imagined
to be drawn from the one Pole to the other, is called the Axis or Axle tree of the
World.
1
į
1
4. All other Lines drawn through the Centre of the Sphere, and limited on each ſide
of the ſurface of the Sphere, is a Diameter, but not an Axis, unleſs the Sphere move
round about it.
5. The Poles of the Zodiack are two Points directly oppoſite to each other, diſtant
from the North and South Pole 23 degr. 31 min. and are Diametrically oppolite, on
which the Heavens move, from the Weſt into the Eaſt.
6. The Equinoctial Points are in the beginning of Aries and Libra, to which Points
the Sun cometh the 11 of March, and 13 of September, and makes the Days and Nights
of equal Length in all places in the World.
7. The Point of the Summer Solſtice, is the beginning of Cancer, which the Sun cometh
unto about the 12 of June, and longeſt day, to the Inhabitants on the North pårt of
the World, and the ſhorteſt day to the Inhabitants of the South.
8. And the Point of the Winter Solſtice , is the beginning of Capricorn, to which
the Sun cometh the II of December, and maketh the Thortert Days of the Year to the
Inhabitants of the North Hemiſphere, and the longeſt to the Inhabitants on the South-
ſide the Equinoctial.
9. The Vertical Point, or Zenith, is the Point directly over-head, and is the Centre
or Pole of the Horizon, 90 degr. every way from the Horizon.
IC. The Ņadir is the oppoſite Point under our Feet.
Circles of the Sphere.
The Ten Circles are as followeth ; The Equinoctial, or likewiſe called the Equator,
which is the chief Circle in the Sphere, dividing the Heavens in the middle between the
two Poles; the two Points of Aries and Libra, cut this Circle in oppoſite Points, and
make the Days and Nights of equal length over all the World.
2. The Meridian is a great Circle paſſing through the Poles of the World, and the
Zenith of the place, the Sun when he comes to this Meridian, it is Noon; the number
of Meridians is as many as can be imagined Vertical Points from the Weſt to the Eaſt,
whereof the Coſmographers have deſcribed, 180.
3. The Horizon is diſtinguiſhed by the names of Rational, or Senſible; the firſt is a
great Circle every where Equidiſtant from the Zenith, and divides the ſuperior or upper
Hemiſphere, from the lower, and by chance are diſtinguiſhed by the names of Right,
Oblique, and Parallel-Horizon.
A Right-Horizon have the Inhabitants under the Equator, who have the Horizon
paſſing through the Poles of the World, and cuts the Equator at Right-Angles.
An Oblique-Horizon is ſuch an one as cuts the Equinoctial Oblique, or aflope, or hath
any degr. of Latitude from the Equator.
-
į
1
1
1
A
1
Chap. I. Spherical Definitions in Aſtronomy.
99
KE
A Parallel-Horizon is one that hath the Pules for the Zenith and Nadir, and the
Equinoctial for the Horizon.
The Senſible-Horizon is a Circle that divideth the part of the Heavens, which weise,
from the part we ſee not; called a Finitor.
4. The Zodiack is a great Circle, that divides the Equator into two equal parts ; the
Points of Interſection are Aries and Libra, the one half declining toward the North,
the other to the South 23 degr. 35 min. his ordinary breadth is 12 degrees; but later
Writers make it 14 or 16 degrees by reaſon of the wandring of Mars and Venus.
In the middle thereof is a Line called the Ecliprick, from which the Latitude of the
Planets are numbred both Northward and Southward; the Circumference of this
Circle containeth 360 degr. which is divided into 12 equal Purts called Signs, every one
repreſenting ſome living Creature, either in Shape or Property, as you read in the De-
nominations, and alſo every Sign containeth 30 degr. and every degree containeth 60
min, and every min. 60 ſeconds, and every ſecond 60 thirds.
:
The Names and Characters of the 12 Signs.
8
NE
V
Aries.
IL
Gemini.
12
L.O.
Taurus.
Cancer.
Virgem
m
1
ve
*
Libra, Scorpio. Sagitarises. Capricornw. Aquarius. Pifccs.
s. The Six uppermoſt are the Northern, and Six undermoſt the Southern Signs.
Of the Colures,
6. Theſe are two great Circies, or two Meridians paſſing through the Poles of the
World, croſſing one the other at right Angles, and dividing the Equinoctial and the
Zodiack into four equal parts, making thereby the four Seaſons of the Year,
7. The Solſtitial Colure is as before, a great Circle drawn through the Poles of the
World, the Poles of the Zodiack, and the Solſtitial Points of Cancer and Capricorn,
ſnewing the beginning of Summer and Winter.
8. The Equinoctial Colure, is a Circle palling by the Poles of the World
through both the Equinoctial Points of Aries and Libra, Thewing the beginning of the
Spring and Autumn, when Days and Nights are equal.
9. The T. opick of Cancer is a leffer Circle of the Sphere, equally diſtant from the
Equinoctial to the Northward 2 3 degr. 31 min. 30 Secom.ls, wherein when the Sun is,
he is entring Cancer, and making his greateſt Northern Declination.
10. The Tropick of Capricira is alſo a leſler Circle, equally diſtant from the Equi-
noctial Southward 23 deg. 31 min. 30 ſeconds, to which when the Sun cometh,
which
is the icth of December, maketh his greateſt Southern Declination.
11. Of the two Pole Circles.
Theſe are two leſler Circles, diſtant ſo much from the Poles of the World, as the
Tropick of Cancer and Capricornos is from the Equinoctial 23 degr: 30 min. which are
the Pole Points of the Zodiack, which moving round the Poles of the World, deſcribe
by their motion the faid two Circles; that about the North-Pole is the Arctick Circle,
and that about the South the Antarctick Circle.
12. The firſt Six are called great Circles, and the other Four leſler Circles; by the
Centre of a Circle is meant a Point or Prick in the middle of a Circle, from whence all
Lines drawn to the Circumference are equals and are known by the names of Radius.
1:.
That
1
.
100 Spherical Definitions in Aſtronomy. Book VI:
13. That is ſaid to be a great Circle, which hath the fame Centre as the Sphere,
and Divides it into two equal halfs, called Hemiſpheres ; and that is a leſſer Circle which
hath a different Centre from the Centre of the Sphere, and Divides the Sphere into tiró
unequal Portions or Segments.
14. Of other Circles imagined but-nor deſcribed in a material Sphere or Globe.
Such are the Azimaths, Almicanters, Parallels of Latitude and Declination.
Azimuth or Vertical Circles paſs through the Zenith, and Interſect the Horizon
with right Angles; wherein the diſtance of the Sun or Stars from any part of the Meri-
dian are accounted, which are called Azimuth, and the Enft and Weſt is called the
Primc Vertical Azimuth.
15. The Sun or any Star having Elevation or Depreſſion above or below the Horizon,
are then properly ſaid to have Azimuths; but if they be in the Horizon, either riſing
or ſetting, the Arch of the Horizon between the Centre of the Sun or Star, and the true
Points of Eaſt and Weſt, is called Amplitude.
16. Circles of Altitude called Almisanters, are Circles Parallel to the Horizon,
and Interſect the Vertical Circles with right Angles, which are greateſt in the Horizon,
and meet together in the zenith of the place, in which Circles the Altitude of the Sun,
Moon, or Stars above the Horizon are accounted, which is the Arch of an dzinsuih,
contained betwixt the Almicanters, which paſieth through the Centre of che Sun,Moon,
or Stars, and the Horizon.
17. Parallels of Declinations are leſſer Circles, all Parallel to the EquincEtial, and
may be imagined to paſs through every degree and part of the Meridian, and are de
fcribed upon the Poles of the World ; in like manner, the Declination of the Sun or-
Star is meaſured by the Arch of the Meridiar between the Sun and Star, and the Equia
noctial.
18. Parallels of Latitude in the Heavens, are leſſer Circles deſcribed upon the Poles.
of the zodiack, or Eclipsack, and ferve to define the Latitude of a Star, which is the
Arch of a Circle contained betwixt the Centre of any Star ur Planet, and the Ecliptick
Line, making right Angles therewith and counted toward the North or Soussh Poles
of the Ecliprick, the Sun never paſſeth from under the Ecliptick-Line, is ſaid there-
fore to have no Latitude.
19. Longitude of the Sun or Stars is meaſured by the Arch of the Ecliprick, com-
prehended between the Point of Aries, and a ſuppoſed great Circle paſſing from the
Poles of the Ecliptick, and the Sun or Stars Centre, and accounted according to the order
and ſucceſſion of the Signs.
20. Longitude on the Earth, is an Arch of the Equinoctial contained between
any
aſſigned Meridian where it begins, and the Meridian of any other place, but accounted
Eaſtward from the firſt place, as the Right-Aſcention ; but in my Tables it is counted
Eaſt and weſt from the Meridian of the Lands-end terminating at 180 degrees.
21. Right-Aſcention is an Arch of the Equinoctial accounted from the beginning of
Aries, which cometh to the Meridian with the Sun, Moon, or Stars, or any portion
of the Ecliprick ; and by it there are Tables made for the Sun, Moon, and Stars to know
the time of their coming to the Meridiár, as by the help of the hour of the Star, the true
time of the Night.
22. Oblique-Aſcention, is an Arch of the Equinoctial between the beginning of Aries
and that
part
of the Equinoctial, that riſeth with the Centre of the Sun or Star, and any
portion of the Ecliptick in any Oblique-Sphere.
23. Aſcentional-difference is the Arch of Difference between the Right- Aſcentior,
and the biblique-Aſcention, and thereby is meaſured the time of the Sun or Stars before,
and after Six.
24. Elevation of the Pole is the Height thereof above the Horizon, which is equal to
the diſtance between the zenith, and the Equinoctial, whoſe Complement is equal to the
diſtance of the zenith from the North or South Pole, or the Elevation of the Egnator
above the Horizox; theſe Circles I have inſerted, to the end they may be perfectly known;
for without them, the Reader cannot well underſtand the following Problems of the
Sphere that are depending thereon.
1
!
*
1
1
1
CHA P. II.
...
.
IOI
T
3
the Arctick.Circle, about the Pole North, Xħ the
darbas hoopisto
poboador 2004 အရ၍
అందం అంతం လို့ ခံလို့ရှိဝသို့ဂလို့ရှိရာသို့လို့
CHAP. II.
The Projection of the Sphere in Plano, repreſented
by the Analemma, and the Points and Circles
before deſcribed.
He Sphere may be Projected in Plano in ſtraight Lines, as in the Ania
lemma, if the Semi-diameters of the Circles given, be Divided in ſuch
fort as the Line of Signs in the Fundamental Diagram of the Scale.
This Scheme is fitted for the Latitude of Briſtol si degr. 28 min.
and repreſents the points and Circles of the Sphere before deſcribed.
Take with your compaſſes the Chord of 60 degr. and upon the Centre C deſcribe
the Circle HZON (2.) Draw the Diameter HCO which repreſents the Horizon
and at Right-Angles thereunto, croſs it with another Diameter 2 CN.
Then with the Latitude of the place, prickoff
sa degr.28 min. from Oto N, and
from Htos, and of the ſame Line of Chords, take the Complement of the Latitude
38 degr. 32 min. and prick off from HÆ, and from 0 to Q, and draw NSC and
ECO
Then take the Suns Declination 23 degr; 31 min. and prick off from £ to G and T,
and with the like Chord do the fame from N to Y and g, for the Polar-Circle, and the
like do from Q_to D and P, and from S to X and h; and through theſe Points draw
Parallels to the Equator Yg, and TSD, and Ghip, and X h.
And through the Centre draw the Ecliptick-Line TGP, and draw RS Parallel
to the Horizon HCO, which is the Parallel of Altitude of the Hour of Sis; and at
any other diſtance, draw Parallels of Altitude Eif.
(1.) Thiis are the Points before defined, repreſented in this Diagram, N is the
North-Pole- Aretick, S the South-Pole-Antarctick; & the North, x the South-Pole
of the Ecliprick; C both the Equinoctial-Points of Aries and Libra.
T The Point of the Summer-Solſtice --- P the Point of the Winter-Solffice, Z the
Zenith over our Heads, N the Nadir-points under our Feet.
(2.) The greater Circles are HCO the Horizon, ZCN the Axis thereof, or
Prime Vertical Azimuth of Eaſt and Weſt; HZON the great Meridiar, and alſo
the Colure of the Summer and Winter-Solſtice, - ECQ the Equinoctial; TCP the
Ecliptick; SCN the Axes of the World, the Hour-Circle of 6; an dlſo it repre-
fents the Celare of the Equinox es.
(3.) The leffer Circles are there reprefented, TD the Tropick of Cancer; GP
Antarctick-Circle, or South-Pole.
(4.) Other Circles not defcribed upon Globes, are there repreſented ; Ef repre-
ſents a Parallel of Altitude called an Almsicanter : phe Prickt Arches zó, and si be-
ing Ellipſes repreſent the Azimuths, or Vertical-Circles.
The
102 The Analemma repreſenting the Sphere. Book VI.
The Projection of the sphere in Plano, by straight Lines or Signs.
!
780
Zame
उगा
습
​Æ
1
1
T
40
M
8
1
8
R
Fit
H Η
907060 50 40 30 20
IO
TO 420 30 50
50 07090
T
h
X х
for...fenocooler
n
.
1
P
!
N
1
he
1
The Prickt Arches from the Poles, repreſent the Meridian or Hour-Circles, which
are alſo Ellipſes; the Drawing thereof will be troubleſom, and for that reaſon
is not mentioned; and how to 'ſhun them in the reſolution of any Propoſition of
the Sphere, by Chords ſhall be ſhewn in the ſeveral Queſtions following; But the Sphero
may be Projected in Plano by Cireular-Lines, as in the general Aftrolobe of Gemma
Trifius, by help of the Tangent, and į Tangents in the Fundamental Diagram of the
Scale, and by the Directions in the 4 Book 12 Chapter beforegoing, and will reſolve
the ſame things; the directions Thall be one and the fame, in both, in Letters, and
repreſents the fame.
1
1
1
1
3
.
Th
1
។
..
Chap. II. The uſe of the Line of Signs & Tunçuris. 107
The Convex Sphere, by Circular Lincs.
Z
y
T
no
N
ÆC
E
9
G
ma
H
n
D
SE
X
2
2
!
din
P
,
h
>
Any Line drawn Parallel to the Equinoctial E Q, as p9, TD. Yg: doth repre-
fent Parallels of Declination.
And
any
Line drawn Parallel io the Ecliptick TP, repreſents a Parallel of Laticidi
of the Stars and Planets in the Heavens.
(5.) Divers Arches relating to the motion of the Sun, and ſeen upon the Globos,
and found by Calculation, are in the Cont:ex-Sphere, repreſented in Right-Lines, and
in the Concave-Sphere by Circular-Lines.
(1.) The Suns Amplitude, or Coaſt of Riſing and Setting, from the Eaſt and Weſt
in the Anaiemma, is repreſented by CL in North Signs, and by CF in South
Signs.
:: (2.) His Aſcentional-difference, or time of Riling and Setting from Six in Summer
by SÍ, in Winter by FH.
(3.) His Altitude at Six in Summer by RC, and his Depreſſion at Six in Winter,
by @%.
(4.) His Azimuth at the hour of Six in Summer, by R S, or C1, equal to h} in.
Winter,
(:) His Vertical-Altitude, or Altitude of Eaſt and Weſt, by MC in Sumcier,
and his Depreſſion therein in Winter by CN
(6.) "His hour from Six being Eaſt and Weſt, in Summer by MS, in Winter by
N.
(7.) His Azimuth from the Eaſt and Weſt upon any Altitude, is repreſented in the
Parallel of Altitude by the Convex-Sphere, where it Interſects the Parallel of D. cinz-
cion by lo; but in the Concave-Sphere may be meaſured on the Horizon Ho, as CV,
or C1, meaſured on the Line of half Tangents.
(S.) The
Ooo
1
104
Book VI.
The uſe of the Line of Signs, &c.
1
(3.) The Hour of the Day from Six, to any Altitude, is always repreſented in
the laid Point of Interſection, in the Parallel of Declination, hereby 90, or in the
Concave-Sphere by So; and all theſe Arches thus repreſented in Right-Lines, are the
Signs of thole Arches to the Radius of that Parallel in which they happen, being ac-
counted in the midft of that Parallel.
How to meaſure the Quantities of thoſe reſpective Arches by a Line of Chords, and
Signs, and by Half. Tangents ; and conſegnently thereby to reſolve the moſt uſeful
Caſes of Spherical Triangles; as alſo by Calculation, is what I inteird ſhall be the fub-
jest of the Pages, viz. and the Art of Dialling bý a Gnomical Scale.
The former Sphere or Scheme doth repreſent the Trianglés commonly uſed in Cal.
culation.
Thus the Right-Angled Triangle CK 0, Right-Angled at K; ſuppoſing the Sun at
o, is made of CK, his Right Aſcention, ok his Declination ; - K Co the Angle
of the Ecliptick, and the Eqdinoctial being the Sùns greateſt Declination 23 deg. 31,
Cok, the Angle of the Súns Meridian and Ecliptické.
In the Right-Anglēí-Triangle LON, Right-Angled at 0; fuppofing the Sun at
LON is the Elevation of the Pote, NL the Complement of the Suns Declinacion,
LO the Suns Azimuth from the North.
I NO the Hour from Midnight, or Complement of the Aſcentional: Difference,
NLO the Angle.of Poſition, that is, of the Sans Meçidian with the Horizon, and of
the like parts, of their Complements, iş made the Triangle CML,
In the Right-Angled-Triangle CIS, Right-Angled at 1; ſuppoſing the Sun at S,
there is given CS his Declination, i3 his Altitude at the hour of Six, CI the Suns
Azimuth from the Eaſt and Welt at the hour of Şix, 1CS the Angle of the Poles
Elevation, CSI the Angle of the Süßis Poſition.
In the Right-Angled-Triangle CCM, fuppoſe the Sun at M; d M the Suns De-
clination, Cå his hour from Six, in the Altitude being Eaſt or Weſt, HCM the
Latitude, dMC the Angle of the Suns Pofition.
In the Oblique-Angled Triangle ZON, if the Sun be at O. ZN is the Com-
plement of the Latitude, and Nw the complement of the Suns Declination, or di,
Itance from the Pole. Ź the Comple-nt of the Suns Altitude, or height; ZNG
the Angle of the Hour from Noon; NZO the Sans Azimuth from the North-part of
the Meridian ; ZON the Angle of the Suns Potition.
And thus we have ſhewed how the fo.mer Diagram or Analemma repreſents the
spherical i riangles uſed in Calculatio: whey, or the Six parts in each Triangle,
it
any
three are given, the reſt may be fome oy walculation rom the Proportions, and
that either by Addition and Sut ftracis. wy che Aricciai Signs and Tangents, and
what is resolved by either of theſe ior, o tables, we will reſolve with the firſt Tables,
and with Scale and Companies, that you may see the near agreement betwixt shem.
:
W*
CHAP. III:
105
99006
sono
pohoto
loco
boot
ാറുമാറ്റം
sebab hambo
st
CHAP. III.
1
T
1
How to Calculate the Sun's true place.
His Propoſition is propounded, in the firſt place, becauſe many others
depend upon it ; According to the Hypotheſis of Ticho, it is to be lug-
geſted, and there is afcribed to the Sun a Triple Motion ; firſt, a Mo-
tian upon his own Centre, whereby he finiſheth one Revolution in 26
Days.
2. A daily motion from the Eaſt into the Weſt, whereby he cauſeth the Day
and Night.
3. An oppoſite Motion from the West into the Eaſt, called the Annual Motion,
whereby he runs once round in a Year through the whole Ecliptick, moving near a
degree in a Day, and thereby caùſeth the ſeveral Seaſons of the Year.
The two later Motions are fancied out unto us, by a Man turning a Crane-Wheel,
or Grindəftone 365 times round, while a Worm ſtruggling againſt, and contrary to
that Motion, creeps once round the contrary way, but Obliquely and a flope; that is,
from the further ſide of the Wheel, towards the hithermoſt, and by this Motion the
Sun is ſuppoſed to deſcribe the Ecliptick-Line, and continually to inſiſt in this Courſe;
the other Planets, except the Moon, moving round him, or following after him, like
Birds flying in the Air, being ſubject to his Motions
, and many of their own belides
many of which Motions are removed by the Copernican ſuppoſition of the Earths Mo-
tion, which is a ſubject of much controverſy among the Learned; However, be icei-
ther the one, or the other, the Propoſitions hereafter Reſolved, vary not, by reaſon
thereof : And ſo the Sun being ſuppoſed not to vary from under the Ecliptick in re-
ſpect of Latitude; the Propolition and Quære, in effect is what Longitude he hath
therein, from the neareſt Equino&tial-Point, which may be found within a Degree for
his Courſe in each day, from his Entrance into each Sign, from that day of the Month.
3
1
1
1
po Aries, March 10.
8 Taurus, April 9.
I Gemini, May 11.
Cancer, June 12.
I Leo, Fuly 13.
The Day of the Month
m Virgo, Auguſt 13.
the Sun enters into each i Libra, September 13.
Sign.
m Scorpio, Ołtober 13.
7 Sagitarius, November 11.
Yo Capricornus, Decem. Il.
mons Aquarius, Fanuary 9.
# Pisces, February 8.
1:. If the number of the Day of the Month given, exceed the number of that Day,
in which
the Sun enters into any Sign; Subſtract the leſſer Number from the greater
and the Remainer is the Degree of the Sign, that the Sun poſſederh.
000 2
PROBL
106 To Calculate the Suns place in Aſtronomy. Book.VI.
PROBL.I.
ܕܐ
•
6 45.
Example.
N the 1 2 of May I would find the Suns true place by the former Rules ; The Sun
enters Gemini May 11, which Subſtract from 12, the Remainer is
,
which
Thews the Sun to be in i deg. of Gemini the third Sign; that is, 61 degr, from the next
Equinoctial-Point.
2. Examples
Let it be required to know the Suns Place the 4th of November ; on the 11 day of
that Month the Sun enters Sagitarius, and the 13th day of September he enters Libra:
betwixt the'1 3 of September, and the 4 of November is 52 days, and conſequently 52
degr. from the Equinoctial-Point Libra; then jo taken from 52, there remains 22
degr. the Suns place in Scorpio, which is the thing required.
But here is a ņearer Rule yet than this, to find the Suns place exactly, and that is by
Mr. Vincent Wing's Hypotheſis, and Tables in Aſtronomia Britanica, how to Calculate
his true place from Earth, the Rule is,
Firſt
, enter the Table of the Middle-Motion of the Sun, and write out the Epocha
nexi going before the time given, under which fet the Motion diſtinętly belonging to
the Years, Months, and Days, and Hours, and Minutes, if any be; (only in the
Biffexstile or Leap-Years,) after February a Day is to be added to the number of Days
given ; then adding them all together, the ſum will be the Middle-Motion of the Sun
for the time given.
As for Example.
Suppoſe the time given be the 12 of May at Noon 1667, at which time the Suns
place is required.
Time given. Longitude o Apog. o
SD MSS
SS DMS
The Epocha 9 20 24 493
1661 years In- 11 29 33 070
6'io
cluding 6 Mar. 3 28 16 39
Days 12. o II 49 40
The Suns Mean-motion, or Longitude. 2 00 04 153
2 00 04 153 6 si 37
2. Subſtract the Apog. of the Sun from his Mean-Longitude, and the Remain will
be his Mean Anomaly.
Example
SD M S
The Suns Mean-Longitude is
04 IS
The Apogeum Subſtracted
The Suns Anomaly.
38
With the Suns Anomaly enter the Table of his Equation with the Sign on the Head
and the deg. deſcending on the left hand, if the Number thereof be under 6 Signs ; but if
it be more than 6 Signs, enter with the Signin the bottom, and the degr.aſcending on
the right hand , and in the common-Angle you have the Equation anſwering thereunto ;
only you muſt, if need require, remember to take the Proportional part.
Example.
In the Table of Equation anſwering to 23 degr. is
51
The Suns Mean-Longitude, add
04 15
The Suns true Longitude.
Therefore the true place of the Sun is in i degr. 16 min. 6 ſeconds of Gemini.
Another Example.
In the Year 1583 March 14 at Noon, in the Meridian of Vraneburg in Denmark,
thrice Noblç Ticho-Brahe, moſt excellently obſerved the Suns true place in 3 deg. 17
min. 42 ſeconds of r. The time at Londow was 1583 March 13 day 3 h. & 2.
O
20
1
2
2
OO
1
3 06
SI
37
IO 23
12
|
A
S
SD
OT
M
II
2
OO
2
01 16
16 06
+
.
A
1
1
1
A Table of the Suns Mean Motion.
Longit
. Apog. Long.
SM
i
H
M
SDM SIM S D M S
Lon
M S
1
0
O
O
1
I 19
1
I
0.
O
2
O
I
I 2.4
O
1
O
1
5 OS
I 29
I 31
1 34
IO
I
0
оос
2
10
I 39
II
IO
0
0
ܕܐ
I 43
00 30
A Table of the Suns Mean Motion, .
The Apoche, or Radius.
Mean Motion in years under 20
Longit. ©
Apog. O
Longit. O Apog.
Years.
Year.
SDM SSD M S
S D M S M S
Ch.119 07 59.5112 08 20 03
II 29 45 40
15819 19 48 55 3 05 22 55
IT 29 31 20 04
160119 19 52 5413 50 43 28 3
!1 20 16 59
3 OS
162119 20 06 52
20 06 5213 06 04 00 L 400 00 01 48
4 07
164119 20 15 513 06 24 33 511 29 47 28
16613
20 24 4913 06 45 05
6II 29 33 7 6
168119 20 33 4813 07 OS 38 211 29 10 47 7 I 2
L 8100 00 3 35 8
13
9 II 29 49 IS 9 54
Mean Motion in Years above 20.
10 11 29 34 55 16
II 29 20 35! II 18
00 00 05 23
12
ooo 08 5910 20 33
000 17 57000 41 05
40
1311 29 51 03 13 21
60 000 26 5610II OD 38
14 II 29 35 43 14 23
80 000 35 5410 II 22 10
ISII 29 22 23 IS 25
L 16100 00. 07 II 16 26
100 10 00 44 530 II 42 43
17 11 23 52 51 17 28
O 02 29 45 10 03 25 26
18 11 29 38 31 18 29
30010 02 14 38005 08 08
400 10 02 59 3110 50 51
19112924. TO 19 31
Oo oo 08:59 | 20 33
5000 03 44 23 10 08 33 34
600 o 04 29
16 O 10 16 17 Mean Motion in Months.
700 005 14 09 10 11 59 00
800 0 05 59 01 0 13 41 42 Febr. for oa 33 18 00
Jamu. loo oo oo oo
yooo 06 43 5410 15 2.4 26 Marc. or 28 09 1!
oo os
OO IO
10000 07 23 47 10 17 07 08
20000 14 57 341, 041415 May 03 28 io 390
April 02 28 42 306 is
3000 O 22 26 21
June 104 28 49 58 oo
0025
40000 29 55 08 2 08 28 30
July 105 28 24 07 00 31
552
6000 1.14 52-4213 12 42 44 Sept. oj 29 30 4400
57 25
42
Oto.8 29 04 54 00 47
Nov. 09 29 38 12 53
Dec. 10 29 12 22
59
1310
L 12
I
46
20
20
I
O
O 59 OS
2 24131 17
001 SS 17! Do 20 4 5632
30 0257 25
3 7 24133
1 2 1
410 03 56 33
4
9 51 31
O 04 SS 42
5 I 2 1935 I 26
6
Oo5 54 50
6 0 14 47 36
2006 53 581 DI 7 17 1537
8
@ 07 53 0710
8
0 19 43 38
9 O 08 52 IS
0 1 9 22 II 39
I 36
IO ooy SI 23
O 24 38 40
10 so 31
027
6141
I 41
11
49 4010
1310 29 34 42
12 48 491
O2 13
o 32
243
14'0 13 47 57
141 0 34 30 44 I 48
ISO 14 47 OSO 2 IS o 36 5845
SI
160 15 46 1310 316 0 39 25 140
1716 45 22 0317
O 41 53147
1810 17 44 300
18 0 44 21 48 I 58
1910 18 43 38 10 3119 O 46 4049 2 01
20 0 19 42 47! O 3 20 O 49 1750
21 10 20 41 55
O SI 45 Si 2 06
o 21 41 03
4 | 22
o 54 1352 2 08
2.3 10 22 41 12
O 4 23
56 4053
240 23 40 20
0424
o 59
250 24 39 2804
O4 125
IOI 34155 18
26 0 25 38 37
0426 I 04 0456 2 18
270 26 37 45 427
I 06 3257 2 20
28 10 27 36 53
I 09 00 58
28 35 02 5 29
I II 2759
30 10 29 35 io lo s 130 I 13 55 160
2 28
31 OOO
00 34 18 os (Mm se. th. I See See
m . I th.
تم ا ر د بیا بیا بیا با ما با
I 53
I56
200
2 03
Bl
5510
L 30
321
22
2
IT
854
2 13
OO
00
2
O O
S 2-8
I 21 21 22
20
2 23
2 25
29
A Table of the Suns Equation.
00
oo
The Calculationi
Apog. o
1
SD
SDM S
Long.
Mi Š
w
3 22
22 55
The Apocha.
T 1581.
Years added 2
March.
Days 13
Hours 23
Minutes 8
IO
9 19 48 55
IT 29 31 20
I 28
9. 1.1
I2 48, 48
56 40
20
2
+
.
OO
2
2
2
3 5 25 II
I IS 14
3 S-125 II
The Suns Mean Motion,
Apogeum Subſtract.
The Anomaly of
The Equator added to i 15' 14
The Suns true place,
with Obſervation.
So 03
SI
p. 3 18 5
Sig. o Sig.
o Sig. 1 | Sig. 2 Sig.
31 Sig. 4 Sig. 5
E Sub, IE Sub. Sub. E Sub. Sub. E Sub.
IDM SID MSID MSID MSID SMDM SI
oyooo105932 11 44 2.812 25411 482311 3 26130
10.25 1 121 145 34 22 501 47 201
I I 32/29
210 940 3.10 146 382 2561 1 4675 0 59 37 28
310 6121 457 147412
255
I 45 90574027
4. 816 6 42 1 48 402
6 42 1 48402 252 11 44 105542 26
51 010191 8261 49 382
8 261 49 38/2 2 471 42 50 0 534325
6101222 1 10 9115o 342 2 39 1 41 37 051 43 24
1101425 1.115111511912 2291140 22
229 140 22 0 49 42 23
801628113 32 152 222 2 17! I 39 410 45 391 22
9101830 IIS III 53132
I 37 450 45 35 1 2 1
100?032116491154 I 1 46 1 36 24 0 43 31 | 20
11 22:34 1 1:8 26 11 5443 i 2 I 29 I 3.5 3 041 2619
120.24377 20 0211'55312 I 7 7!3339 039 2018
13.0 26-39 1 21 36 14 56142 0441 1 32 30371417
1402841 123 91356.552 0181304435 6/16
15 10 30 42 | 1 24 41 1573415949 129133258115
161 32431 261115810159 19 1 2741030 50 14
17 0 34 44 11 27 40158.44 1 58 47 1 26 70 28
41
13
1810 36 431 29 1159-16115812 1 24 37|0 2531 12
1910 384" 1 30 34 1594515735 11 22 560 24 21 11
2010 4038 1315812 01411565611 21181022101.10
21 03335 1 33 202 0401 5614111938 1959 9
220 34 811 3741 10415530100756101748
23 0 4627 11 360 I 261 54 44 116 12 0 15 36 7
241 048 22
1 461 153.561 1426101333
6
351OS0161 3833 2 2 3 153 si 1240 III
26 10 52 091 39.482
152 12
I
10 520
4
271 054 I I4I 12 2 301 IS118119
3
281055 52 I 42 I 2 2 2 40|ISO 21
50211711
7 Illo 429
2910 57421 43 21 2 2 48 149 231
1
0594211 44 2812 25411 48 23 1 33610 Oo
Add Add Add Add
Add
Sig. 11 Sig. 10 i Sig. 9 | Sig. 8 | Sig. 7
7 | Sig. 6
Agreeing
Z
(3) Example the time given the 10 of April 166 at Noon, and
admit by the former Rules we have found the Suns Mean Motion 29 degr.
o min. 30' his Apogeum 3 s. 6 d. 49 m. 295. his Anomally 9 s. 22 d. 11'
I;
firſt find a Proportional part, the Equat. anſwering to 22 s. I 8. 52' 22"
The Equator anſwering to 23 d. 1 51 29
their difference
Then I ſay, if i deg. or 60 min. give 53 ſeconds, what ſhall 11 of the
Anomaly give? by the Rule of proportion,
. 60: 53 11
(4)
53
58(3)9-
I
37172
59
S
2 18
8 57
92
643
43
3
578
2 IS
I
(
0
30
te siendo
.............
+
I
The Convex. Sphere, which reſolves all the moſt uſeful Problems in Aſtronomsy,
by the Direction of 13 Problems following.
N
팀
​I
и
8
UT..
M
K
8
ma
At
-
h
H
c
Р
X
l.com
m
-foodporn
X
$
1
W
BOR
n
- *
The Goncave-Sphere, which reſolves 13 Problems, vizi and by then may be
reſolved, moſt of the uſeful Problems in Aftronomy.
2
!
X
e
* minus
7
E
d
B
B
MR
S
H
00
$
n
L
म
BR
o
CHAP. III.
The uſe of the Line, &c.
107
Multiply 53 by 11, the Product is 583, which Divide by 60, the Quotient will
be 9", and becauſe the Equation decreaſes ! Subſtract it from the Equa. anſwering
22 d.gr. which is i d. 52 - 12" for the true Equation deſired, which according to the
Title, being added to the Suns Mean-Longitude, giveth the true place of the Sun re-
quired.
Example.
S
D
29
00
30
The Suns Mean-Longitude.
The Equation added.
The Suns true Longitude.
52
12
I
00
52
42
D.
Therefore the Suns true place is in 0 52:42 of Taurus ; theſe Examples are ſuffi-
cient for Direction, to find the Suns true place at any time,
PROBL. II.
i
The Suns Diſtance from the next Equinoctial-point; and his greateſt De-
clination being given, to find the Declination of any point required.
VV
Ith your Compaffes take the Chord of 60 degr. upon the Centre C, deſcribe
the Circle HZON, and draw the Diameter HCO, which repreſents the
Horizon, and at Right-Angles, or perpendicular thereunto, draw Z CN, the Vertical
Azimuth of Eaſt and Weſt, and take the Latitude of the Place, as in this Example, is
Ś I d. 28 m. and prick it from Oto N, and from H to S, and draw the Axis or
Meridian of the Hour of Six NCS, then prick from Z to E, and from Nto est.
degr. 28 min, and draw the Equinoctial Line & CQ, then the Suns place being given,
take 23 deg. 31 m. and prick from E to , and from Q_to
P, and draw the prickt Line
SCP, then take the Suns Diſtance from the next Equin.-Point, which in this Example
Mall be 61 deg. 18 m. out of the Line of Signs, and prick it from C to C, and through
0 draw a Parallel-Line to the Equinoctial
, as TD,
and it hall be a Parallel of Decii.
nation, and where it cuts the outward Meridian, as at T; apply the Diſtance E T to the
Line of Chords, and you have the Declination 20 degr. 30 min. which was required
Or you may take the neareſt Diſtance from o to the Equator, and apply it to the Line
of Signs, and that will give you the Declination 20 degr. 30 min. as before ; and if
through you draw a Line Parallel to the Horizon HO, as ef, it is a Parallel of
the Suns Altitude, and ſo have you the Sphere Orthographically in Right-Lines in the
Convex-Sphere; and if you follow the directions of the uſe of Tangents, and half Tan-
gents in the 12 Chap. of the fourth Book of the Deſcription of the Globe in Plano, you
have the Sphere projected in Plain and Circular Lines, and fitted for the uſe of divers
Queſtions; the Direction in both Spheres by the Letters ſignify the ſame thing ; but
obſerve what you are directed by Signs in the Convex-Sphere, is likewiſe to be done by
Tangents in the Concave-Sphere.
By the Tables in the Right Angled Triangle CKO; we have given, firſt the Hy-
Co 61 degrees 18'; fecondly, the Angle K CO 23 degr:31', hence to
find ko, the Rule is, as the Radius is in proportion
to the Sign of the Suns greateſt Decl. 2 3 d. 31'K CO
So is the Sign of the Suns diſtance from the next Equinartial.Poiix
994307
61 degrees i 8 min.COM
to the Sign of his Declination required 20 degrees, 30%
954406
Or extend the Compaſes in the line of Artificial Signs from 90 degr. to 23 degr. 30
min, the fume extent will give the diſtance from the Suns Place, to his Declination.
The
porhenafc
IO
960099
Ке
I
1
44
$
MWL
1
108
The ule of the Line of Natural and
Book VI
1.
Z
GO
'n
N
Æ
dt.
M
e
700 focsacfo.com
S
B
699
H
X
+
2
foculA
S
I
1
The Sun being either in i deg: 18
min. of Gemini, or 29 deg. 42 min. of Capricorn,
or i degr. 18 of Sagitarius, or 28 degr: 42 min. of Cancer, that is, 61 degri is
mim from the next Equinoctial-Point, the Declination will be found to be 20 degrees
30 minutes.
PROBL. III.
I
1
J
4
A
6
Having the Suns greateſt Declination, and his Diſtance from the next Equi-
noctial-Point; to find his Right-Aſcention.
I
N the Foregoing Scheme, having drawn the Parallel of the Suns Declination TD,
paſſing through the place at * the extent se, is the Sign of the Suns Right-Aſcen-
tion from the next neareſt Equinoctial-Point, to the Radius of the Parallel TD, and
therefore place the extent ST from C to X, and upon X as a Centre with the extend
se, deſcribe the Arch at k; a Ruler laid from the Centre juft touching the extremity
of that Arch, finds the Point N in the Limb of the Meridian or Quadrant, and the
Arch ON, applyed to the Line of Chords, is so degr. og min. and ſo much is the
Suns Right-Aſcenlion in the firſt quarter of the Ecliptick.
In the Triangle CK we have given as before, (1.) the Angle of the Suns grea-
teſt Declination KC® 23 degr. 31 min. (2.) the Longitude of the Sun from the
next Equinoctial-Point Aries cô 61 degr. 1 8 min. hence to find the Suns Rght-Ar-
fcention, the Rule is;
As the Radius
to the Tangent of the diſtance 65 degr. 18 min.Co - 1026162
So is the Co-Sign of the Suns greateſt Decl. 23 deg. 31' KC0--996234
to the Tangent of the Right-Aſcention CK 59 deg. 9m.- 1022 396
Or in the Concave-Sphere ; it you draw the Meridian from N through to S,
whole Centre will be found upon the Equator, it will cut the Equinoctial in K; mea-
ſure the diſtance GK on the Line of half-Tangents, and you have s9 d. 09', as before.
Or extend the Compaſſes from 90 d. to 66 d. 29, the fame diſtance will reach from
deg.18 15.to 59 deg. 9 min, which is the Suns Right Aſcention in 61 deg. 18 I.
I
- IO
6
Buc
1
I
1
.
1.
7
Chap. III.
Artificial Signs and Tangents
.
109
;
But this you are to obſerve, that if the Right-Aſcention fought, be in the ſecond Qua-
drant on 1, then you are to take the Complément of the Arch found to 180 deg. if in
the third Quadrant a mi, adde 180 deg. to the Arch found; but in the laſt Quadrant,
Subſtract the Arch'found from the whole Circle 360 degr. and you shall have the Right-
Aſcention deſired.
Example 2.
The Sun in 28 degr. 42 min. of so, that is
, 61 degr. 18 min. from the Equinoctial
Pointed the Rule'is as before. As the Radiss is to the Co-Sign of the greateſt Decli-
nation, ſo is the Tangent of the Suns diſtance from the next Equinoctial-Point 61 degr.
18 min. co the Tangent of 59 degr. og as before, which taken from 180, is 1 30 deg.
simin. which is the Suns Right-Aſcention in 28 degr. 42 min. of Cancer.
Example 3.
The Sun in i degr. 16 min. off, thac is, 61 degr. 18 mix. from the next Equinoctial-
Point is, the work is the fame as before; therefore to the Arch found, I add 180 degr. a
Semi-Circle, fo 58 deg. og mix and 180, makes 239deg. 09 min. the Right-Aſcention
of the Sun fought in i deg. 18 min. of Sagitarius F.
Example 4.
The Sun in 28 degr. 42 min. of Capricorn, 61 deg. 18 min. from the next Equi-
noctial Point r, the operation is the ſame with the former Example; wherefore Sub-
ſtract the Arch found 59 deg. 09 min. from the whole Circle 360 deg. and there will
remain 200 deg. si min, which is the Suns Right-Aſcention in 28 deg. 42 min. of y
Capricorn.
PROBL. IV.
#
The Elevation of the Pole, and Declination of the Sun being given; to find
tbe Aſcentional-Difference.
THE
His is repreſented in the Figure by SL in the Parallel of Declination, and it is
therefore to be reduced into the cominon Radins, therefore take the Radius of the
Parallel ST, and prick it from C to X, as before ; then take the extent SL, and
ſetting one Foot upon X, with the other draw the part of an Arch at a, lay a Ruler
from C; that it may juſt touch the outſide thereof, and it cuts the Circle in d, and
tåke thé Chord or Extent Hd; and you will find it 2 8 deg. o min. which being con-
veried into Time, is an Hour 52 min. and ſo much doth the Sun Riſe before, and Ser
after Six in Summer, but ſo much doth he Riſe after, and Set before Six in Winter,
when he hath the fame Declination South.
In the Right-Angled Spherical Triangle SLC are known.
I. SCL the Com.
plement of the Poles Elevation 38 dega 32 min. 2. The Suns Declination 20 deg. 30
min. hence to find the Aſcentional Difference SĖ.
As the Radius 90 deg. is in Proportion
to the Tangent of the Latit. 5 i deg. 28 min. SCL-1009887
So is the Tangent of the Suns Declination 20 deg. 30'SC-957273
to the Sign of SL the Aſcentional-difference 28 d. oo m.-967160
Extend the Compaſſes on the Artificial-Lines of Signs and Tangents
, and you will
find it, as before; or if you take the diſtance NS, and prick it from Stok, and lay
the Ruler from Cover L, it will cut the Arch of the Meridian in a; then Meaſure the
diſtance Nd on the Line of Chords, and it will be 28 deg. oo min. as before found,
that is one Hour 52 min.
10
t
PROBL:
1
1
110
Book VI.
The uſe of the Line of Natural and
PROBL. V.
The Suns Right-Afcention, and his Aſcentional-difference being given;
to find bis Oblique-Aſcention, and Deſcention.
TO
perform , theſe
1. If the Suns Decli-
nation be North, you muſt Subſtract the Aſcentional-difference from the Right-
Aſcention, and the Remain will be the Oblique-Aſcention; but if you add them toge.
ther, the ſum will be the Oblique Deſcention. 2. If his Declination be South, add
the Aſcentional-difference, and the Right Aſcention together, the ſum will be the
Oblique-Aſcention ; but if you make Subſtraction, the Remainer will be the Oblique-
Deſcention.
Admit the Sun is in the i deg. 18 min. of Gemini by the ſecond Problem, his Right-
Aſcention is 59 deg. o9 min, and his Aſcentional-difference by the 4 Problem, is 28
deg.: o min. therefore according to the firſt Rule, becauſe his Declination is North, the
difference thereof 31 deg. 09 min. is the Suns Oblique-Aſcention, and the ſum of them
87 deg. og min. his Oblique-Deſcention.
e
PROBL, VI.
V
I
Y
↑
To find the time of Sun-Riſing, and Setting, with the length of the Day
and Night.
Ou muſt find the Afcentoinal-difference by the 4 Problem, which conveited into
Time, allowing 4 min. of an hour for every degr. and
feconds for every min.
and the ſum of Hours and Minutes, is his difference of Riling and Setting before
or after the hour of Six, which was found before to be 28 deg.or.i hour 52 min.
Therefore when the Sun is in Northern Signs, add the ſum to Six, and the Total is
the Semi-diurnal Arch, as in this Example, is 7 hours 5'2, or time of Sun-ſetting,
and Subſtract it from Sis, and the Remain is 4. b. 8.74 the time of Sun-Riſing ; double
7 ho. s'2 m, it is I's ho.44, the length of the Day; Subſtract it from 24 bo. oom.
and the Remain is 8 ho. 16 n. the length of the Night the 12 of May, in Latitude
deg. 28 min. at Briſtol.
Bur if the Sun is in Southern Signs,' make Subftraction, as in this Example, the
Sun having 20 deg. 2c min. South Declination, or in s deg. 18-tix: 1; Subſtract i ho.
S'2 from 6, the Remain is 4 ho. o8 m. for the time of Sun-Setting, double it, and it
is 8 ha. 16 m. the length of the day; add 1 ho. 52 m. to 6, the fum is 7 ho. 52 m.
is the time of Sun-Riling ; double it, it is is bo. 44 m. the length of the Night, in
Latitude si deg. 28 m. in i deg. 18 m. of Sagitarius.
SI:
+
.
PROBL. VII.
1
The Elevation of the Pole, and Declination of the Sun being given ; to find
his Amplitude.
Eaſure the extent CL with the Compaſſes in the Line of Signs, and it will reach
to 34 degr. 40 min. and ſo much doth the Sun Riſe and Set to the Northward of
the Eaſt and Weſt' in the Latitude of Bruviai, when his Declination is 20 deg. 30
þur he Riſeth and Setteth ſo much to the Southward of the Eaſt and West,
when his Declination is ſo much South.
Now on the Concave-Sphere, the extent CL on the Horizon, applyed to the Line
of half-Targents, is 34 deg. 40 min. the Amplitude, as before.
If
win.
North;
Cap. III. 21:Artificial Signs and Tahgents:
III
thús..
.16
c't
..
pino
L
them and then
iii
If the Suns Parallel of Declination doth not meet with the Horizontal-Line HO,
as in Regions far North, the Sửn doth riot Riſe not Ser.
In the Right. Arigled Spherical-Trainġle LOC of the 4 Problemi, having the Argil
ICO, the Complement of the Latitude 38 degt: 32 min. and L'O the Suns Decli-
nation 20 degr. 30 min. in 1 d. 18 m. II his Amplitude by Calculation may be found
The Artificial - As the Co-Sign of the Latitude's 1 deg: 2.8 poin, l'co - 979445
Lines by this
is to the Kadius go degr.
Rule anſwers So isthe Sign of the Declination 20 degr. 36:10
954432
the fame. to the Sign of the Amplitude CL 34, deg. 40 min.
974986
? This Rule is the comnion Rule Mariners make uſe of for the finding of the Variation
of the Compaſs at Sea, by comparing the Coaſt, or bearing of the Sun, obſerved bý áin
Amplitude or Azimuth-Compaſs at the Suns Riſing or Setting, and by his bearing,
found by thèſe Rules beforegoing, the difference Theweth the Variation.
2. :
As for Examples
Admit you obſerved the Suns Amplitude of Riſing or Setting by your Compaſs in the
firſt Chap. and fifth Book of the Art of Surveying deſcribed:? i?
And by the Compaſs found, the Magnetical Amplitude: 4.5 1.55 compl. 44 d: 05N.
By the Rules beforegoing find the true Amplitude, --- 34 d. 40 compl. 55. d. 20' N.
Subſtract the leſs out of the greater, the difference is dois m., Variation,
And by reaſon the Magnetical. Amplitude is more than the true. Amplitude " there-
fore the Variation is 11 degr: 15 min. which is one Point Weſterly, and ifłyouare:
bound to a place that bear Northof. you, you muſt Sail upon the North by West Point
or if you bare Weſt, you muſt Sail W and by S, and if South, the Courie muſt be south
and by Eaſt; or if you bear Eaſt, then the Courſe muſt be Eaſt and by North, to make
good á North, or Weſt, or South, of Eaſt'Courrë'; and ſo of all the reſt of the Points
you muſt allow in like manner.
2. But admit the Magnetical Amplitude obſerved by the Carpaſs, were buti 2:3:das.
25 min, and the true Amplitude by the former Rules found to be 34 degr. 40 min. the
upper-Subſtract from the lowerj
. the difference is a degr. 15) mini and by reate the
Magnetical Amplitude :is leſs than the true Amplitude, and the difference i idegruz.
which is one point Variations Eaſterly; and forthe North Point is the Nby.E poist;
and NE is the N E:by. E; and Eis Ebys, and South is S:by W., and W is Wilý N
Point of your Sailing Compaſs, when you have ſuch a Variation, and the Complenten:
of the Amplitude isihe
. Suns Azimuth from the Northvor South parted the Meridian;
according as your Declination is.;. And this is ſufficient for:ani Example to find tbe Vari-
ation of the Compaſs in any place or time.
......,
As likewiſe by his Oblique-Aſcenſion, and Aſcenſional differences or by the syns
bţing Eaſt and Welt by the following Rules
, or by, the Suns Azimuth at the hour of six ;
as likewiſe his Azimuth at any other time or place obſerved, as hall be hewn for the
telpand benefit of young lariier's
,!;!:,: Arish; Mesin
!
.
1
.
2
silo:
:
+
1
.':
!!! I i
.:!
!
e nii
.
H
I
• A3791.1
PPP
PROBL
I
9
112 The uſe of the Line of Signs & Tangents. Book VI.
1
t
PROBL. VIII.
1
I
Having the Latitude of the Place, and the Suns Declination, to find the
time when the Sun cometh to be dve Eaſt and Weſt.
N the Parallel of Declination, the hour from Six is sepreſented by SM; with that
citent upon
the Point X draw the Arch b the Ruler laid from C to the outward edge
of the ſaid Arch, cuts the Circle at(e) tbc diſtance Oc applyed to the Line of Chords
Theweth 17 deg. 30 min. it converted into Time is I h.9 m. 20 ſec. and ſo much after
Six in the Morning, and before Six in the Afternoon, will the Sun be due Eaſt and Weſt,
by the Concave-Sphere, if you lay a Ruler from A over M, it cuts the Limb in ()
mealure Ne on the Linc of Chords, and it is the fame 17 deg. 20 min. the Rule by
Artificial Signs and Tangents holds as by Calculation, 'viz.
r
Suppoſe the Latitude se degrees 28 min. and Declination North 20 degrees 30 min.
thereforc in the Right-Angled Spherical Triangle ZN M are given (1) ZN the Com-
plement of the Latitude 38 degr. 30 min. (2)
NM the Complement of the Suns Decli.
nation 69 degr. 30 min. Then I ſay.
As the Radiu go is in proportion
IG
To the Co-Tangent of the Declination 69 degr. 30 min. NM -957273
fo is the Tangent of Z N compl. of Latitude 38 degr. 32 min. 990112
is to the Co-Sign of ZNM 17 deg. 20 min. which Reduced
is 1 b.9 m. of time, as before.
} 9.47385
\Vhich sb.9 m.added to 6 b. isgh.gm. the moment in the Morning, the sun will
be due East; and if you Subſtract 1 b.9 m. 20 jec, from 6 b.oom, and the Remaia
will be ak.gom. 40 fer. the moment in the Afternoon the San will be due Wet.
Puos 1.IX.
i
The Elevation of the Pole, and the Declination of the Sun being given; to
find the suns Altitude when he is dne Eaſt and Weſt.
Eaſure the extent CM on the Vertical-Circle, and apply it to the Line of Signs,
Spherc, and applyed to the Line
of Tangents, ſhews the ſame number, and fo much is
.
his Altitude fought in Summer ; but when he hath the like Declination South, then fo
much is his Depreſſion under the Horizon in Winter, when he is Eaſt and Weſt, if the
Suns Parallel of:Declination TM doch not meet with the prime Vertical Circle CZ,
the Sun cometh not to the Eaſt and Weſt, as it happeneth many times in ſmall Latitudes,
or Countreys betwixt the Tropicks.
1
1
In the former Diagram, the Suns Altitude when he is duc Eaſt and Welt, istheved
by the Arch CM, wherefore in the Triangle CVMwc have given, (1) the Suns De
clination VM 20 degr. 30 min. (2) thc Angle of the Poles Elevation MCV si deg.
28 min. to find his Altitude CM; I ſay,
This Rule will hold As the Sign of the Angle of Latit. şı d. 28 m.UCM 989334
is to the Sign of the Declin. 20 degr. 30 min. UM 954432
by the Artificial
Lincs.of Signs and
So is the Radius 90 degr.
. 10
Tangents.
to the Sign of the Altitude 26 deg. 37 min. CM-965098
PAOL. X.
11
1
ن : /3
41
Chap. III. The uſe of tbe Line of Signs & Tangents. 113
1
PROB Li X.
The Elevation of the Pole, and Declination of the Sun being given; to find
the Suns Altitude at the Hour of Six.
THE
Ake the neareſt diſtance from S to the Horizon CL, and apply it to the Line of
Signs, ſheweth the Altitude to be 15 degr. 54 min, or the fame taken of the Con-
Gave-Sphere, and meaſured on the Line of Tangents, theweth the ſame, and ſo much
is his Depreſſion under the Horizon at Six, when he hath South-Declination 20 degr.
30 min.
10
In the Concave-Sphere, you may ſee all the Triangles plain, and we have known in
this Triangle ZSN, (1.) The Complement of the Latitude ZN 38 degr: 32 m.
(2.) the Complement of the Suns Declination NS 69 degr. 30 min. to find the
Hypotenafe 2$; therefore I ſay,
As the Radius go degr. is in Proportion
To the Co-Sign of 69 degr. 30 min. NS
954432
ſo is the Co-Sign of 38 degr. 32 min. ZN
989334
To the Sign of the Altitude 74 degr. 6 min. Soo
943766
Whoſe Sign is 15 degr. 54 min. is the Suns Altitude at the hour of Six, when he is
I degr. 18 m. off in Latitude si deg, 28 m. Extend the Compaſſes from 90 degr. to
20 d. 30', the fame extent will reach from the Latitude 52 deg. 28 min. tà is deg. 54
m, as beforc.
)
PROBL. XI.
Having the Latitude of the place, and the Declination of the Sun given ; to
find the Surs Azimuth at the Hour of Six.
2
His is repreſented in the Convex-Sphere by VZ in the Parallel of Altitude of the
Sun VSB; Prick VB from Cto W, and with the diſtance VS draw the Arch
upon W at h, and lay the Ruler juſt touching the faid Archi
, cuts the Circle in Y ; the
distance HY meaſured on the Chords, theweth the Azimuth, or the diſtance Goo,
on the Concave-Sphere, applyed only to the Line of { Tangents, thews the Azimuth
to be 13 deg. 07 min. and ſo much is the Sun to the Northwards of the Eaſt and Weſt
of the hour of Six.
In the Right-Angled Spherical-Triangle ZNS of the general Diagram, we have
known firſt, ZN, the Complement of the Latitude 38 degr. 32 min. (2.) NS the
Complement of the Suns Declination 59 degr. 30 min. to find the Azimuth of the hour
of Six, repreſented by the Angle NZS.
I ſay,
As the Radisi go is in proportion
to the Compl
. Sign of the Latitude 38 deg: 32'ZN 979446
So is the Co-Tangent of NS 69 deg. 30 min.
957273
to the Co-Tangent of the Azimuth NZS 76 d.53-936719
Or extend the Compaſſes from the Co-Sign of the Latitude to the Radius ; the fame
extent will reach from the Tangent of the
Declination, to the Azimub 76 deg. 53 min.
as before; the Suns Azimuth from the North part of the Meridian in the Latitude of
Si degr. 28 min. and Declination 20 degr. 30 North, (13 degr. 07 min is from the
Weſt.)
Ppp 2
PROEL XII,
IO
A
1
114
Book VI.
The uſe of the Natural and
PROBL. XII.
Having the Latitude of the place of the Suns Declination, and his distance
from the Meridian being given, to find the Suns Altitude at any Time
aſſigned.
Y this Caſe may be found the Suns Altitude on all hours, and the diſtance of Places,
B
in the Arch of a great Gircle ; for the Suns Altitude on all hours thereby is meant,
that if the hour of the Day, the Declination and Latitude be given, the Suns Altitude
proper to the hour, er his Depreſſion may be found.
Take the Chord of 60 degr. and deſcribe the Arch HT POD, draw the Horie
zontal-Line HCO, and from 0 to P prick of the Chord of the Latitude sr degr. 28
min. and from P to T and D fer
of the Complement of the Suns greateſt Declination,
66 degr. 29 min. and draw the Parallel of Declination TD, and the Axis CSP, or
the Meridian of the hour of Six; then draw the Radinis TC, which is the Ecliptick-
Line, and take off the Line of Signs, and prick
30
45<for the
Godegre.com 2
down,
3
from 6 before it, and after it.
60 Hours of --40
95
Then take the neareſt diſtance from 15 degr. to CS, the Meridian of the hour of
Six, or Axis, and prick it from Stos and 1 ; and lik-wie take the neareſt diſtance
fro 30 to CS, and lay it from Sto 8 and 4; and in like manner do'with the reſt, then
will the neareſt diſtance from 45678910 II to the Horizontal-Line HCO be
the Signs of their reſpected Altitude ;
fo the Altitude
AĆ
Si deg. 34 min.
5
In Summer will be
30
Lis
for the hours 65
12
62 .
5
P
3
45
IL
30
+
1
S
15
'HI
C
L.
7
LO
at म
45
1
S
75
Т»
And
Chap. III. Artificial Line of Signs and Tangerts
. 115
7
And ſo much is the Suns Depreſſion under the Horizon at the hours of 3, 7, and 6 in
Winter; as you may foon trie by the fame DiviGon on the Parallel of the Suns greated
Declination Southward DST 27:23
}
The Summer Alt. for the h. 9 45:42
of
53:45
59:42
36:42
are
IO
II
,
are
10:28
II
3,2,1,
And the Winter Altitude for
9 2
5 d.13
the Hours of
Io
213:48
And ſo much is the Sun depreſt under the Horizon in Summer at the hours of
from Mid-night, as you may ſoon find it by the neareſt diſtance from 9, 10, 11, in the
Line LD to the Line LO; if you apply that diſtance to the Line of Signs, you may
draw the Parallel of Altitude through each hour, as the Example is through 9 hoef
and meaſure He; or Of on the-Line of Chords, and it is 45 degr. 42 min. the Suns
Altitude at 9 a clock the i 1 of June.
2. But admit the Latitude were 51 deg. 28 min. and his Declination oo deg. oom.
and ſuppoſe, for example, ſeek his diſtance from the Meridian is 2 ho. 44 min. or 41
deg. o'o and the Sun upon the Equinoctial the i 1 of March, and 13 of September.
Upon the Equinoctial at K is the Sun repreſented the 11 of March, or 13 of Septem-
ber, and diſtance from the Meridian 2 bo. 44, or 41 degr. oo min. If in the Convex-
Sphere you lay Se, from C to K the Right-Aſcenſion 59 deg. 9 min. and through
K
you
draw a Parallel to the Horizon, as d M, it will cut the Meridian in d, and mean
ſure the diſtance Hd on the Line of Chords, and it is 28 degr. 03 min, the Suns Alti-
rude required, or the neareſt diſtance from K to the Horizon-Line HCO applyed to
the Line of Signs, would have thewed the fame. In the Concave-Sphere, if you
rake
the diſtance CK, and draw a ſmall Arch atr; and take the neareſt diſtance to the Azi-
muth-Line of Eaſt and Weſt ZC from K, and with that diſtance turn the other Foot
over, and croſs the Arch at (",) and through K and r draw a Circle of Altitude, it will
cut the Limb in E and F meaſure HE, or O F on the Line of Chords, ſheweth the
Altitude 28 degr. 03 mix. as before.
!
!
In the Concave-Sphere, you may ſee the Triangle ZÆK; and we have given firſt
Æ Z the diſtance from the Zenith to the Equator, equal to the Latitude of the place,
si degr. 28 min. Secondly, ÆK the Suns diſtance from the Meridian 41 deg. to find
the Suns Altitude from K to the Horizon.
IO
I ſay, as the Radias 90 deg:
is to the Co-Sign of the Suns diſtance from the Merid. 42 d. o' ÆK - 987777
So is the Co-Sign of the diſtance from the Equ. to the Zen. 51 d. 28 m. £2979446
to the Co Sign of Z K 61 degr. 53 min.
967223
Whoſe Complement is the Altitude Kp 28 deg. 03 min. which was required ; you
may in your practice draw a particular Figure for the Queſtion, and ſhew
Tri-
angle by it felf, by the Line of Chords, and half Tangents.
Secondly, when the Sun is in North Signs, r8 IS1 M.
Let it be required to find the Sons Altitude at 10 a clock and 2 m. paſt before Noon,
when the Sun is in entrance of Gemiui, in Latitude 5 degr. 28 min.
every
1
d
First,
!
1
116
The uſe of the Line of Natural and
Book VI.
Firſt, By the Convex-Sphere, the neareſt diſtance taken from the Suns place, to the
Horizon H'C, applyed to the Line of Signs, theweth's i degr. 12 min. the Suns Alci-
tude required; Orto find his diſtance from the Meridian, take ST, and prick it from
Cto X; then with the diſtance Son X as a Centre, draw the touch of an Arch at
K, a Rúler laid over the Centre, over the outward edge of the Arch, cuts the Arch of
the outward Meridian in (*;) then meaſure On on the Line of Chords, and it is 60
deg. 2 min. So, whoſe Complement is 29 deg. si min. the Suns diſtance from the Me-
ridian O T; Wherefore in the Triangle NZO, we have known, (1,) ZN the
Complement of the Latitude 3 8 deg. 32 min. (2,) Ne, the Complement of the Suns
Declination 69 degr. io min. (3.) the comprehended Angle ZNO, the diſtance of
the Sun from the Meridian 29 deg. 58 min. to find Z, and hereby the Suns Altitude
920, I ſay,
As the Radins 90
10
is to the Co Tangent of the Latitude si deg. 28 min. NZ
990112
So is the Co-Sign of the Angle from the Meridian 29 deg. 58 m.ZNO
993767
to the Tangent of the fourth Ark 34 deg. 36 miv. Na
983879
From the Complement of the Suns Declination N*69 deg. 30 min. Subſtrat Ng
34 deg. 36 mix, there remains 34 deg. 54 min.
As the Co-Sign of the fourth Ark. 34 deg. 36 min. Ng
991547
is to the Compl. Sign of the Latitude 38 deg. 32 mix. ZN
So is the Co-Sign of the fifth Ark, 34 deg. 54 min. 9.
991 3.89
1980723
to the Co-Sign of Z 38 deg. 48 min.
989176
· Whoſe Sign issi deg. 12 min. po the Suns Altitude above the Horizon at 10 a
clock, and o2 m. paſt, and 1 a clock 58 min paſt in the Afternoon, when he is in i
deg. 18 m. of I in Latitude of si deg. 28 min. North,
Now if you follow the Rules before-going, you may find the Suns Altitude by the
Line of Chords, and half. Tangents by this Figure to be the ſame.
989334
Z
TER
€
Æ
1
B
Joondoo.f.....
s
H!
hiir
18
!
24 와
​1
&
S
.
12
>
Suppose
1
CHAP. III. And born to Calculate in Aſtronomy.
117
10.0;
Suppoſe the Sun in the Southern Signs am 7. ve H in the oppoſite Point to the
former, having South-Declination 20 degr. 3 oʻmir: and be alſo diſtant from the Meri-
dian 29 d. 58 m. take the Declination 20 deg: 30 min. and prick it from £ to B, and
Q.to R in both the Spheres, and draw the ſtraight Line in the Convex Sphere B MR,
and cake from the Suns place in the oppoſite Sign in the Parallel of Declination, his
diſtance from the Meridian YR, and prick it on the other ſide of the Parallel from B
and the neareſt diſtance to the Horizon-Line HC, applyed to the Line of Signs
thew's the Altitude to be 13 d. 23 m. and in the Concave-Sphere take of the I ine of half
Tangents, the Declination 20 degr: 30 mix, and lay it from the Centre Cto M, and
the Atis CS continued, take the Complement of the Declination of the Line of Se.
Catics, and place it from Con the continued Line or Axis, and that will be the Centre
of the Parallel of Declination; or if you take the like Complement 69 degr: 30 min.
of the Line of Tangents, and put it frorm M on the Axis; it will be the Centre of the
Parallel of Declination, therefore drawit BMR, and it will cut the Meridian in
the place wherc the Sun is.
Now to find the Suns Altitude or Ark (10) or Ze; therefore to find how much
it is you muſt find the Pole of the Circle No2, which is done aftef this manner.
Lay' a Ruler from Z to h, and it will cut the Circle in ; then take go deg: and prick
it from 4 to 7, then lay a Ruler over from 2 to $, and it ſhall cut the Hòrizon in 8;
which Point is the Pole of the Circle z hn,
To meaſure the Ark Zº, you muſt lay a Ruler upon and ®, which will cut the out-
ward Circle in the Point X, ſo ſhall X Z meaſured on the Line of Chords, give you the
quantity of d.contained in the Arch XZ, which will be 26 d. 39° equal to the Comple:
ment of the Suns Altitude/ 'I have been the larger in this precepr, that it may
be a Rule
of Dircation, to thew.howthe Ark of any great Circle of the Sphere ; the ſides of all
Spherical Triangles being luch, may be macaſured whatſoever, by his operation in the
Concave-Sphere.
Obſerve the Fígare we have given in the Oblique-Angled Triangle ZN8. 1.NZ
as before, the complemenr of the Latitude 38 dég. 32 min. 2. No 110 deg. 30 min.
the fame diſtance from the North-Pole. 3. the Angle ZN 29 deg. 58' to find the
Altitude he.ch
As the Radiwu 90
is to the Tangent of NZ 38 deg. 32 min.
So is the Co-Sign of the Angle, ZNO 29 deg.ss'
993767
to the Tangent of N9.34 deg. 36 min. 4 Ark, as before, -983879
From the Ark No iro deģ. 30 min. Subſtract the Arch Ng 34 deg.36 min.
and there remains -75 deg. 54 min.
As the Co-Sign of N, 34 dag. 36 min.
is to the Co Sign of ZN 38 degi 32 min.
989334
So is the Co-Sign of tle Sfth Ark q. 75 deg. 54 min 938670
1928004
to the Co-Sign of Z8 76 deg: 37 mir.
Now the Complement of Ze, is a ť deg. 23 min. Which is the Slips Altitude
required.
1
..:
IO
990112
1
1
991547
936457
1
!
1
PROBLE
1
.
1
118
1
The uſe of the Line of Natural and Book VI.
PROJ L. XIII.
5
with
ins
4
1
The Suns Altitude, and his Distance from the Meridian, and Declination
being given; to find his Azimuth.
Emonſtrate the Queſtion by the Line of
. Signs and Chords, with 60 degr. draw
, , ;
and of B Parallel to the Horizon; then draw the Axis CN, and the Equinoctial
CE at Right-Angles or 90 degrees from N, as. NJE; then draw the Parallel of De:
clination TL and (BR) by the Line of Chords, and Signs, then take the exteņtil: Ex
and ſet from Cto P, and upon P with the extent 1 , draw the Arch (oo) a Ruler laid
from C, juſt touching that Arch, cuts the Limb (0.) the Arch Ho meafured on the
Line of Chords, is 41: degr. 42 min, and ſo much is the Sun to the Southward of the
Eaſt and Weſt, the Complement is 43 deg. 18 min. and ſo much is the. Suns Azimuth
from the South part of the Meridian Weltward. Now by the Concave-Sphere, if you
find the Centre of the Aximuth Circle on the Horizontal-Line at 0, and draw. the
Circle. from 2 through N; meaſure the diſtance Co on the Line of half-
Tangents, and it will be the Suns Azimuth from the Eaſt and Welt, as before, 45 deg.
42 min. (Complement 48 deg. 18 min. from the South)
,:!
..
1
1
1
Gril: :smi
1
viviis:::1 30ogii'isins
17:1).
!!
b
T
: زر .
200911033090), 1/4: :..
Zy lisiloliniuti il !::: 7
i
1r lirinin
II
:
TV?
j
។
A
Home
+
*1
.
rin
:: both
1
::
+
SEKS
1
DL
BY
1
1
: ; ܪ
it.
I
que
R
t
;
9
L.
*
-
In the Oblique Spherical-Triangle ZN, we have known. 1. Ze the Com-
plement of the Suns Altitude 38 degi 148 min. 2. The Angle ZN 0.29 degy..58 m.
the Suns diſtance from the Meridian." 3. The Complement of the Suns. Declination
NO 69 deg. 30 m.
Now I work thus
As the Complement-ſign of the Altitude 38-degr: 48 min. Ze 979699
is to the Sign of the Angle from the Merid. 29 deg: 58m. ZNO 969859
So is the Compl. Sign of the Declination 69 deg: 30 NO
1967011
to the Sign of the Suns Aximath NZ 48 deg. 18 min, 9,879 12
Now admit the Altitude were 13 degr. 23 min. his diſtance from the Meridian 29
Set Vy from deg. 58, and his Declination South 20 deg. 30'.
o 3," and "By the foregoing Rules, you will find the Arch Hwy, meaſured on the Line of
with vo draw Chords 6 i deg. 15
min. the Suns Azimuth from
the Eaſt and Weſt, whoſe Comple-
t'e Arch at d. ment is 28 deg. 45 min, the Azimuth from the South part of the Meridian,
Os
997158
71
1
1
}
Chap. III. And bow to Calculate in Aſtronomy.
119
Or on the Concave-Sphere draw the Azimuth Circle ZON, and meaſure on the
Horizontal-Line Chi, applyed to the Line of half-Tangents, is 61 deg. 15' as before,
and the Azimuth 28 deg: 45 min. H h from the South part of the Meridian ; and asi
deg. 15 min. his Azimuth from the North part of the Meridian Oh.
Obſerve the general Diagram of the Concave-Sphere, you have in the Oblique-An-
gled Spherical Triangle RNO. 1. Zo the Complement of the Suns Altitude 26
76 deg: 37'. 2. The Angle Z NƏ 29 deg; 58 min, his diſtance from the Meridian.
3. N110 deg. 30', the Complement of the Suns Declination. The Rule is to find
the Azimuth,
As the Complement-Sign of the Altitude Zo 76 deg. 37 998804
is to the Sign of the Angle from the Merid. ZNO 29 d. 58 m. 969853
So is the Complement-Sign of the Declination No 69 d. 30 m.- 997158
1967011
to the Sign of the Suns Azimuth NZ 28 degr. 45 min. - 968207
PROBL, XIV.
The Poles Elevation, with the Suns Altitude and Declination given ; to find
the Suns Azimuth,
1
2
draw a Diagram by the Rules beforegoing in the Convex-Sphere, that is, the
Elevation of the Pole ON $r deg. 28 m. the Suns parallel of Altitude vo V 13 d.
23 m. and the parallel of the Suns Declination 20*digr; 30 min. BR; then take v V,
the parallel of Altitude, and prick it from Cto oo then take the diſtance ye, and
on & as a Centre, draw the Archat (2), a Ruler laid
over C, and the outſide of )
Thall cut the Arch in (^,) then meaſure the diſtance (HW) and the Line of Chords
gi deg. 15 min. as before.
Or dsaw the Concave-Sphere, and the Axis or Elevation of the Pole ON; and
likewiſe prick from Hand on both ſides the Parallel of the Suns Altitude vo V 13 deg.
23 min. and take the ſame number of the Line of half-Tangents, and prick it on C2,
the Prime Vertical Line of Eaſt and Weſt from Cupwards io (v.) and draw the parallel
of Altitude, you, whoſe Centre will be upon the Vertical CZ continued or found,
by taking the Complement of the Altitude 76 deg: 37 min. of the Line of Secants, and
prick it from Con the Vertical-Line continued, and that is the Centre; draw, as (you)
thence draw the Parallel of Declination 2 o deg. 30 min. by the former direction from
Æto B, and from Oto R, and take zo deg: 30 min. of the Line of half-Tangenrs, and
prick it from Con the Axis to (m.) and draw BOM R the Parallel, whoſe Centre
is found upon the Axis continued, as before, by the Complement Secant of the Decli-
nation 69 deg: 30 min.
Then where the Parallel of Altitude and Declination croſs each other, which is at e,
there is the Sun aç that time, therefore draw the Azimuth-Circle, as before (12 Probl.
directed,) from Z through to N, and it will cut the Horizon in Hi then meaſure Ch,
and it is the Suns Azimuth from the Eaſt and Weſt, which applyed to the Line of
half-Tangents, thews.bi deg. 15 min. as before, whoſe Complement is 28 deg: 45 min.
the Azimuth from the South ; in like
männer meaſure all Azimuths from the Prime
Vertical on the Horizon,
By Calculation ; Firſt, conſider the Declination of the Sun, whether it be towards thie
North or South, fo haye you his diſtance from the Poles; then add this diſtance, the
Complement
of his Altitside, and
the Complement of your Latitude all three together,
and from hialf the ſum Subſtract the diſtance from the Pole or Complement of his De
clination, and note the difference.
Q.99
Look
i
70
120 The uſe of the Line of Signs & Tangents. Book VI.
2. The
5
Look well and obſerve the general Diagram in the Oblique-angled Triangle.
ZON the Complement of the Suns Altitude is 20 76 deg 37 min.
Complement of the Suns Declination is Nº 69 deg. 30 min. and the Complement of
the Latitude ZN 38 deg. 32 min. which known, you may frame your operation, thus,
Decl. South 20 d. 30m. The diſtance from the Pole 110 deg. 30 min.
Suns Altit. 13 d. 33 mil Complement, or Sign is 76 : 37,20 998804
Latit, North
1.28 m. Complement, or Sign of 38: 32, ZN 979446
( 37 deg. 18 m.
Sign, or fum is
225 : 39,
1978250
all
The Sign of 67 deg. li', or half fum of (3) 112 : 49
996461
So is the Sign of 2 deg. 19 min. or the difference
Add the Radius (2)
From this Sum Subſtract 1978250 the fourth Sign
2857123
Take the half with the Radius (it is the 7th. Sign)
1878873
The half doth give the Mean-proportional Sign 14 degr. 22 min. 939439
SI
2 : 19
860662
2000006
And the double of 14 degr. 22 min, is 28 degr. 44 mix, the Azimuth of the Sun
from the South part of the Meridian; and it taken from 180 degr. oo min. leaves 15 I
degr. 15 min. the Suns Azimuth from the North part of the Meridian, as before.
An Example for North Declination anſwerable to the first in (13 Probl.)
By the former Rules, you may find the Azimuth in each Sphere by Inſtrument. The
Suns Declination North 20 degr. 30 min. his Altitude si deg. 12 min. the Latitude SI
degr. 28 min..
Obferve the Diagram in the Oblique-Triangle ZoN the Complement of the Suns
Altitude 2 o 38 deg. 48 min. the Complement of the Suns Declination, NO 69 deg.
30 min. the Complement of the Poles Elevation ZN 38 deg. 32 min. which known,
the operation may be thus framed.
.
1
1
Suns Decl. North 20 d. 30 m. Compl. 69 d. 30°
As the Radius
Altitude
SI : I2 Compl. 38 : 48 : 979699 is to the Co-Sign of Alt.
Latitude
SI : 28 Compl. 38 : 32 : 979446 fo is the Co-Sign of Lat.
The Sum 146: 50:1959145 to a fourth Sign 2 2 4.59
The Sum.
73:25:998154 as 4 S. is to the S. offum
The difference 3:55: 883445 ſo is the Sign of the differ.
Add the Baſe 69 deg. 30 NO (2)
2000000
3881599
The half of it 1922454 to'a 7th. Sign 9 d. 39m
So is the Co.Sign 24 deg.09 min.
. 961227
The double of it is 48 deg. 18 min. the Suns Azimuth from the South part of the
Meridian, as before found, or 131 deg. 42 mins from the North part of the Meridian.
I have ſet down theſe two Examples thus particularly, to ſew the agreement with the
this note, that generally in all Spherical-Triangles where three ſides are
known, and an Angle required, make that fide which is oppoſite to the Angle required,
to be the Baſe, and gather the Šum, the half Sam, and the difference, as before.
Having theſe means to find the Suns Azimuth, we may compare it with the Magne-
tical Azimuth, and ſo find the variation.
former two;
}
1
.
1
Għap. III. And bom.to Calcultate in Aſtronomy.
121
As for Example.
Admit the Magnetical Azimuth by the Needle, is
59 drg. 33
And the Suns Azimuth found; as before, is
48--18
The Variation is the difference Weſt
- Il deg. 15
So the Magnetical Azimuth being more than the true Azimuth by 11 deg. 15 mix.
which is one Point of the Compaſs; therefore it ſhews the Variation to be one Point,
or 11 deg. 15 min. Weſterly.
And ſuppoſe the Courſe by the Compaſs be Eaſt 8 Points from the North or South,
or 90 degrees; and let the Variation be ir deg. 15 min. to the Weſtward.
I demand the true Rumb;
Mr. Borough obſerved in 1580 d. 7 The Magnetical Rumb
!1 deg. 15 min. Variation Eaſterly Subſtract the Variátion Weſterly - 11 d. 1s
in Line, Houſe, Fields.
there remains the true Rumb NE 78 1.45
Mr. Gunter 1662 found 6 deg. 15
So that if the Variation be Weſterly, you may concieve by looking upon the North-
Point; by the Variation one Point, that it being Weſterly, it is always accounted to
the left hand, fo the North-Point, is one point to the right hand of his true place; and
you muſt Sail N by W, to make good a North way, and w by S to be a good Weſt,
and Sby E to be a direct South, and E by N to make good an Eaſt Courſe, which will
make an Angle with the Meridian of 78 deg. 45 min.
.
god. oon.
1
7
.!
2 Examples
Bat ſuppoſe the Magnetical Azimuth by the Needle had been
And the Suns Azimuth found, as before, to be
371.03
48-IS
Subſtract the lefſer out of the greater, the difference NE
IId.is
And in regard the Magnetical Azimuth is leſs than the true Azimuth by ir deg. 15
min, therefore the difference and variation is Eaſterly one Point, which is ur deg. is
min. and conſequently all the Points , ſtand 11 deg. is min. or one point to the left
hand out of their true Places; and therefore, to make good a Northi Courſe, you muſt
Sail by your Compaſs N by E ; and an East Courſe, Sail E by South, and South Sail
Sby W; and to make good a Weſt Courſe, Sail W by N; and ſo it is to be underſtood
of all other Courſes or Points; for in this Example, the true Courſe makes an Angle
with the Meridian of 48 deg. 18 min.
The Year 1666 at Briſtol, in Rownam Meadows, My felf and Mr. Phillip Stainard,
and ſome other friends Maſters of Ships, took with us a Quadrant deſcribed in the 16
Chapter of the Second Book of 20 Inches Semi-diameter, and one Needle, and one A-
zimuth Compaſs, deſcribed in the Firſt of the Fifth Book, the Needle about 9 Inches
long, the Chard 8 Inches ; and in the Afternoon we made theſe Obſervations follow-
ing
Declin. 23 d. 30'dift. 66 d. 30
Altitude 44: 20 Com. 45 : 40
As the Radius is in proportion
Latit. 51 d. 28 m. Com. 38: 32 To the Co-Sign of the Latit. 38 d. 32' 970446
The Sum -
So is the Co-Sign of the Alt. 45 d. 40 985447
ISO 42
To the fourth Sign 26 d.27 m.
The half Sum
964287
75 21
The differencce -- 8:51
Q992
Add
IO
L
11
122
Book VI.
The uſe of the Line of Natural and
1
As the fourth Sign 26 degr. 27 min. 954893
Add the Radius to the Sum
is to the Sign of the half Sum 75 d. 21'
Take the half is the Sign of So is the Sign of the difference 8 deg. 51'-918709 add
998564
half the Ark.
1917273 ſum
to the ſeventh Sign 19 deg. 31 min. (add Rad.) 1952 380
the half is the Mean proportional Sign of 35: 19976190
Obſer. Magnetical | Suns true Variation Which doubled, is 70 deg. 38 min.
Altitude. Azimuth, Azimuth.
VVefterly. the Suns Azimuth from the South part
Gr. M. Gr. M. Gr. M. Gr. M. of the Meridian, or 54 degr. 41 min.
the Complement of 35 deg. 19 min,
44
oo 170
380I
doubled, is 109 deg. 22 min. the Suns
39
0078 2401 36
Azimuth from the North part of the
50190
88 26
34
Meridian; and fo of the reſt, as they
27
42195
00193 36
are ſet down in this Table, viz. from
the South part of the Meridian,
23
20103
23
23
1
20172
22
308
31
OC
I
00
I
24
00
IOI
I
PROBI. XV, .
1
زورونا
VV
To find the Altitude of the Sun by the shadow of 4 Gnomon fet perpendicular
to the Horizon by Scale and Compaſſes ; as alſo by Calculation.
Mon
Ith your Compaſſes on a piece of Board, deſcribe the Circle ABCD, place it
Horizontal with a Gnomen in the Centre O, croſs it with 2 Diameters; then
turn the Board, until the ſhadow be upon one of the Diameters, at the end of the ſhadow
make a Mark, as here at E ; lay down alſo the length of the Gnomon-Pin or Wire from
the Centre on the other Diameter from O to F, draw a right-I ine from Eto F, as EFH;
then with the Chord of 6o deg. ſweep the Arch G H upon E as a Centre ; apply the
diſtonce GH the Arch to your Line of Chords, and that will give you the Altitude of
the Sun required, as in this Example will be 52 deg. 53 min.
B
>
TH
i,
37
t
1
1
G
E
I
.
K
2
D
As
1
[
I
Chap. III.
Artificial Signs and Tangents. qu :
.: 123
1
1
4
As the parts of the ſhadow 2 8
144715
are to the parts of the Gnomon 37 156820
So is thc Radius 9o deg.
1000000
to the Tangent of s2 deg. 53 min. IOI ZIOS
So the Pin or Gromen OF being 37 parts, and the ſhadow OE 28, fuch equal
parts, the Altitude will be found to be 52 degr. 53; or the Gnomon being 28, and the
Shadow 37 parts, the Altitude will be 1 K 37 d. 07 m. or the ſhadow being 83, the
Gnomon or Staff 100, the Tangent of the Angle will be 50 deg. 18 min. 20" the Alti-
tude of the upper edge of the Sun or Angle HEG; from which, taking the Semi-dia-
meter of the Sun 16 m. 27', there remains sod, 1'59" the true Altitude of the Centre
of the Sun.
After this manner, if you obſerve the greateſt Meridian-Altitude of the Sun the ir
of June, and 10 of December, you ſhall by the difference of them find the diſtancco fthe
Tropicks; the greateſt Declination of the Sun, and Elevation of the Equator, and La-
titude of the Place.
As for Example.
At Londos the greateſt Meridian-Altitude of the Sun is 61 deg. 59" 30", and the
leaſt 14 deg. 56' 30".
The Suns greateſt Meridian-Altitude taken Fosne II, is 61 d.59 : 30"
The Suns leaft Meridian-Altitude taken December 10
14:56:30
The diſtance of the Tropicks, take the half of
47:03:00
And it is the Suns greateſt Declination, Subſtracted from the Alt. 23 : 31 : 39
Leaves the Elevation of the Equator,
38 : 28 :
Whoſe Complement is the Latitude of the Place, SI : 32 : 0
t
O
PROS L. XVI.
Having the Latitude of the place, the Suws Declination, and the Suns Altia
tude; to find the Hour of the Day.
1
}
1
Y the Line of Chords and Signs, by the Convex-Sphere, fer the extent ST, from
B
C to X,
and
upon
X as a Centre, with the extent So, draw the Arch K, a Ruler
laid from Ć juft touching the Arch, finds the Point (",) the Arch (on) meaſured on
the Chords, theweth 61 deg. 10 min, the Suns diſtance from the hour of Six, viz. 4
ho. and almoſt one min.
The Rule is by the Tables ; Add the Complement of the Suns Altitude, and the
Complement of the Suns Declination, or Diſtance of the Sun from the Pole, and the
Complement of your Latitude, all three cogether, and from half the lum; Subſtract
the Complement of the Altitude, and note the difference.
Thus in our Latitude of Briſtol si deg. 28 min. the Declination of the Sun, being
20 deg. 30 min. Northward, and the Altitude si deg. 12'. I find the Sun to be 29
deg: 50 min. from the Meridian, as by this Example.
Altitude of the Sun si d. 12, the Compl. 38 d. 48, As the Radius, 90
Declination North 20 : 30 the diſtance from the Pole 63 30. to the Sign of the Suns
diſtance from the Pole.
097158
Latitude North 51:38 the Compl. is 38: 32 fo is the Sign Compl. of Lat.979414
The ſum of all three 146:50 to a fourth Sign
The half Sum 73: 25 as the 4 Sign is to the Sign of half ſum 998154
The difference
34:37 ſo is the Sign of the difference 975441
197359)
To a feaventh Sign add the Radius
1997023
Take the half
, is the Sign of 14 deg. 55 min. van
998511
IO
976572
I
The
1
.
1
124
+
The uſe of the Natural Lines of Signs, Book VI.
The Mean Proportional between this ſeventh Sign, and the Sign of 90 ; that is,
add the Radius to the ſeventh Sign, and take the half, and it will be the sign of the
Complement of half the Hour from the Meridian, which in this Example is found to
be 14 deg. 55 min. the double of that is 29 deg. so min. which converted into Hours,
doch give almoſt cwo Hours, it wants but 40 Seconds.
PROBL, XVII.
Having the Azimuth of the Sun, the Altitude of the Sun, and the Declina.
tion
to find the Hour of the Day.
1
Hus the Declination being 20 deg. 30', the Altitude si deg. 12 min. the Azi-
T be 18 .
find the time to be 29 deg. 58', that is, almoſt 2 Hours wanting & Seconds ; fo the
difference is 32 Seconds; that is, by reaſon of the ſeveral Operations, which is near
enough for the Mariners óſe.
As the Co-Sign of the Declination 69 deg: 30min.
997158
is to the Sign of the Azimuth 48 deg. 18 min.
987311
So is the Co-sign of the Altitude 38 deg. 48 min.
979699
to the Sign of the Hour 29 deg. 58 min.
1967010
969852
A
PRO3 L. XVIII.
How to find the Right-Aſcenſion of a Star, and the Declination of a sinta
having the Longitude and Latitude of that Star given.
P
Roject the Sphere Geometrically, that is, draw she great Meridian with a Che..
of 60 deg. you may draw the Horizontal Line Ho, and Vertical ZN, then i
the Latitude si deg. 28' from 0 to N, and draw 2 E, and from H to S, and from
Nto Q: and draw the Equinoctial-Line ECQ, then take the diſtance of the Pole
of the Ecliptick, from the Pole of the World 23 deg. 31 min. and prick it from N to
P, from & to V, from Sto A, and from Qtos; then draw the
Ecliptick - Line ve,
C, S, on which you muſt put the Longitude or Diſtance of the Star, 'from the rext
Equinoctial-Point, as in this Example; The Star in the Mouth of the great Dog Siricos.
his Longitude is found by the following Rules to be 9 deg 32 min. of Cancer and his
Latitude is 39 deg. 30 min. South, a Star of the firſt Magnitude ; take 9 deg: 31 min.
out of 90 deg. the Remain is 80 deg. 28 min. the diſtance of the Star from the next
Equinoctial Point C, prick that from Cto K on the Ecliptick, and draw the Circle of
Longitude P, KS, then prick the Latitude 39 deg. 30' from we to D, and from 5 to F
by the Chords; then take the ſame number of the half-Tangents, and prick it from
to M, and draw the Circle of Latitude of the Star parallel to the Ecliptick, as DMF
and where this Parallel cuts the Circle of Longitude, as at * that is
, the place of the
Star; then draw the Meridian-Circle from the North Pole through * the Interſection
to the South-Pole, and it cuts the Equinoctial in R; meaſure CR on the Line of half-
Tangents, and it gives the Right-Aſcenſion Complement to 180 deg. which is 82 deg.
21 mix, the Complement is 97 deg. 39 min, the Right-Aſcenſion deſired.
Now to find the Declination of the Star; lay a Ruler over Pand the Ecliptick, at K,
and it will cut the Arch in L, take a Quadrant 90 deg. and prick it from L' toľ ; lay
a Ruler over p and P, and it will cut the Ecliptick in 0; lay a Ruler over, and *,
and it cuts the Lamb ine ; meaſure Qe on the Line of Chords, and it is 16 deg. 14.03.
the Stars Declination required, By the Concave-Sphere; the Convex-Sphere, will not
ſo conveniently ſhew the true Scituation and Place of the Stars as this, and there-
fore it is omitted
Ву
1
}
1
Chap. III. And bow to Calculate in Aftronomy.
125.
។
B
ng
H
1
y
K 12
69
Di
M
S 33
1
A.
11
I
By Calculation
The Stars have little or no alceration in their Latitude; bnt in their Longitude they
move forward about 1. degr. 25 min, in a hundred years, which is 85'.
By Noble Ticho, his Tables of Longitude and Latitude of the Stars, rectified by
himſelf, to the beginning of the year 1601.
The Latitude of the moſt bright Star Sirius in the Mouth of the great Dog, is 39
deg. 30 min. and his Longitude is 8 degr. 35 mix. 30" of ; I delire the Stars true
Longitude, or to be rectified for this preſent year 1667.
You muſt work by the Rule of Proportion, thus, if 100 give 85 m. what ſhall 66
the difference in years betwixt 1601, and 1667 give ? Multiply, and Divide, and the
Quotient will be 56m. 6'added to the Longitude found in the Tables of the Stars in
the ſecond Book on the back-ſide the Nocturnal, which 8 deg.35 min. 30" makes 9 d.
3136' of Cancer, the Longitude of the Star Sirius this year 1667; and fo work to
find the Longitude of any other Star in any other ycar, paft, or to come.
Take 9 deg. ğz' the Longitude of the Star, out of 90 deg. there remains so degr.
28', his diſtance from the next Equinoctial Point ; which being knowo, the Firſt
Rule is,
As the Radius 90
is to the Sign of the Stars Longitude from the next Equ. Point 80 d. 28 CK 999396
So is the Co-Tangent of the Stars Latitude 39 deg: 30' A*
1008389
to the Tangent of the fourth Ark so deg. 06 min.
1007785
Compare this fourth Ark with the Arch of Diſtance betwixt the Poles of the Eclip-
gick, and the Poles
of the World 9.3 dég. 31 min. ifikie Longitude
and Latitude of the
Scar be alike, as in North Signs V8 IS Szuk, and the Latitude is on the North-ſide
the Ecliptick Or if the Longitlide be among the Southern Signs, as 4m 7 19 23 *, and
the Latitude Southward, then ihäll the difference between the fourth Ask found, and
the diſtance of the Poles' 2 3 deg. 31' be your fifth Ark.
:
I
IO
1.
But
71
126 Tikeruleiofibe. Natural Lines of:Signs, Book VI.
But if the Longitude and Latitude ſhall be unlike, as it is in this Example; as the Lon-
gitude in a Northern Sign, and the Latitude South, or the Longitude in a Southern
Sign, and the Latitude North , then Add this fourth Ark found, to the diſtance of
both Poles 23 deg. 3 r min. the ſum of bath ſhall be the fifth Ark.
t
Then the Rule is,
As the Sign of the fourth Ark so deg.06
988488
is to the Sign of the Fifth Ark 73 deg. 37
998199
So is the Tan. of the Stars Long. from the next Equin. Point 80 deg. 28 m. 4077484
to the Stars Right-Aſcenſion from the next Equin. Point 82 deg. 21 m. 2075683
82 d. 2 1 m. Subſtracted from 90 d. or 1 8o leaves 7 d. 39' which added
to 90 de the ſum is 79 d. 39'
the Right-Aſcen. of Siriu required
. --} 1087195
}
Then the Rule to find the Declination, is,
As the Co-Sign of the fourth Ark 50 deg: 06-980716
is to the Co-Sign of the fifth Ark 73 deg. 37' — 945034
So is the sign of the Stars Latitude 39 deg. 30%, 980351
1925385
to the Sign of the Stars Decl. required 16 d. 14' 944669
988740
You have the proof of the Work by the foregoing Geometrical Rules; or you may
take this by Calculation, if there be no former
errour, the Proportion will hold.
"As the Co-Sign of the Latitude 39 deg: 30 min.
is to the Co-Sign of the Right Aſcen. from the next Equ. Point S2 d. 21'912424
So is the Co-Sign of the Declination 16 deg: 14 min.
998233
1910657
to the Compl. Sign of Longit: ftom'the next Equinoctial 80 d. 2,8 m. 2
or Sign of the Longitude 9 deg: 32 min. as was at firſt given. - $ 921917
by theſe Rules and directions work, to find the Right-Aſcenſion, and Declination of
any Star, which you deſire to know,
wid
PROBI. XIX.
i.
I
:
.
+
9
5
Having the Declination, and Right- Aſcenſion of a Star; to find the Longi-
tude and Latitude thereof.
I
N the former Diagram of the 18th Problem, you have the Right- Aſcenſion of the
Gliſtering-Star in the great Dog's Mouth called Sirius ;. CR 82 4.21 m. take it off
the Line of half-Tangents, and prick it from Cro'R ; and you have alſo the Decli-
nation drawn Bye; then draw the Meridian-Circle'from N, cutting the Point R, the
Stars Right-Aſcenſion from the next Equinoctial. Point, and Parallel, of Declination in
*to S the South-Pole; then take the diſtance of the Poles of the World, and the Poles
of the Ecliptick 23 d., 31, and prick from N to P, and from Ætov, and from Sto
A, and from Q to and draw wy'C's the Ecliptick-Line; then draw the Circle
of Longitude through P; through the interſection of the Parallel of Declination, and
Meridiãn, which is the body of the Star to A, and it will cut the Écliptick in K; mea-
ſure Č K on the Line of half-Tangents, and you have the Longitude of the Star from
the next Equinoctial-Point 80 deg. 28 min.
And
+
IF
1
Cap. VI. And how to Calculate in Aſtronomy: .
127
ܪ
!
And to firled thie, Iatitude if you lay a Ruler over P ånd K, it will cut the Limb in
I, prick 90 deg, from 5 to , and lay a Rulėr över P and 4, and it will cut the Eclip-
tick in the Point 3 ; lay a Ruler over 8 and * and it will cut the Limb in F; apply
the diſtance or to the Line of Chords, and it will be 39 deg. 30, the Latitude re-
quired. Li agrilis
Thathe Triangle ZRC by Calculation, we have 1. the Angle ZC R; tkie diſtance
Poles zi minpilz, the lide CR 8z degr. 21 " the Right-Aſcenſion
from the next Esquinołtial-Poitit, then seaſon muſt guide yols as by theſe Rules, to
firia the Longiture and Latitude of #Star.
IO
As the Radius 90 deg.
is to the Tang. of the Angle ZCR:23 d. 31 m.the Poles diſtance - 963864
So is the Sign of Right-Aſcenſion CR 82 d. 2 1' from the next Point -999611
to the Tangerit of ZR 23 deg. 20 mini
963475
ti
Which 2 3 degy:20 add to the South-Declination 16 degr. 14 min. makes 39 deg.
34, ZR; bur it the Decliparion had been Northerly, you muſt have Subſtracted the
Arch, found out of it, and the Remain had been ZR; but if it had been more than
tle Declination; Subſtract itour of the Arch found; "and the difference had been ZŘ;
ſo.with reaſon Wave thie Rule; as occaſion requireth:
Therf to find the Aggle CZR, and the ſide CZ, the Rule is,
Ästhe Sign of ilte fourth Ark ZR 23:deg: 2009
959778
is to the Sign of the Angle of diſtance of the Poles ZCR 23 d. 3,17T960099
So is the Sign of Right-Afcen. from the next Equ. Point CR. 82 d. 21 m.999611
7)
1
3
M
1
13
to the Sign of CZR 86 deg. 49 min.,the fifth Ark
1959710
999932
L
1
IO
1.
Then
As the Sign of ZCR 23 deg. 3 1 min.
960099
is to the Co-Sign of RC 23 deg.:20 min. 959771
So is the Sign of CŘ Z 9 degr. Radius
to the Sign of CZ 83 degr. oz min.
999679
Which Anglé CZR 86 deg. 49' is equal to the Angle *Z K.
Then to find the Latitude of the Star,
As the Sign of * KZ Radius 90 degr.
is to the Sign of 39 deg. 34' the 4 Ark and Declination R* 982412
So is the Sign of *ZK 86 deg. 49 min.
999932
to the Sign of the Latitude of the Star deſired *K 39 deg. 3'0m, 980344
And laſtly; to find the Arch Z K, and by it conſequently the Longitude C K.
As the Tangent of the Angle ZK* 86 deg. 49 min.
1125479
is to the Radius 90 deg:
IO
So is the Tangent of 2* 39 degr. 34
991713
to the Sign of Z K 2 deg, 35'
866234
Rrr
Therefore
"M
1
1
1
1
1
4
128 The uſe of the Line of Signs & Tangents. Book VI.
1
Therefore, if you Subſtract 2 degr. 35 min. out of the Arch CZ,83 deg.03 min.
the Remain is 80 degr. 28 min. the true Longitude of the Gliſtering Star Sirin in the
Mouth of the great Dog, from the next Equinoctial
. Point at the time given,
But if the Meridian-Circle had cut the Ecliptick at K, and the Circle of Longitude
at Z; then in ſuch caſes, add the Arch found 2 K to CZ, and the ſum had been the
true Longitude from the next Equinoctial-Point C;; and ſo work with reaſon, to find
the Longitude and Latitude of all other Stars, as by reaſon you did Subſtract Z K from
CK; therefore the Sign was above Quadrant, if you Şubſtract his diſtance from the
Vernal-Equinox 80 deg. 28 min. from go.d. the Remajn is 9 deg: 32 of Cancer, the
Sign and deg. the Star is in, as before.
1
i
i
PRO. I. XX:
;?.
.
Having the Meridian-Altitude of an unknown Star, and the diſtance there-
of from a known Star; to find the Longitude and Latitude of the unknown
Star.
impia
1.N the 16th Chapter of the ſecond Book of, Harmonicon Cælefte, Mr. Vincent Wing
hath this Example, and Obſervation , made by the phenix of Aſtronomy Tiche-
Braghe in the year 1977, which we will borrow for an Example ; it being a uſeful
Rule for all Ingenious Navigators, for by it they may find the Longitude and Latitude,
and conſequently, by, the gregoing Rulesz the Right-Afcenfion, and Declination of
thoſe good Stars for their uſe, that are in the South Hemiſphere, viz. as they have been
named by the Portugals; the South-Triangle, which Conſtellation hath 5 Stars, one of
che Eaſtermoft corner, which comes laſt to the Meridian of the ſecond Magnitude. The
Crasie, in which there is 13 Srars on the left Wing, and another on the right ſide the
back of the ſecond Magnitude. The Bhanix, 15 Stars, the Water.Serpent hath 15
Stars. The Dorado, or Gilt-head-Fiſh, fituate in the very Pole of the Ecliptick; and
in that Conſtellation is 4 Stars. , The Chamelion, with the Flie, in which is 13 Stars ;
The'Bird of Paradiſe, in which is' 12 Stars, the Peacock, in which is 15 Stars ; one in the
head of the ſecond Magnitude ; the Naked Indiar, in which is 12 Stars; and alſo the
Bird I aican, or Braſilian Pye, in which Conſtellation is 7 Stars, two of them of the
third Magnitude: Alſo two uſeful Stars for Navigators; ' in one Conſtellation, which
are Noah's Dove, which containeth 1 1 Stars, of which there are 2 in the back of it, of
the ſecond Magnitude, which they call-thie Good Mesſengers, or Bringers of good News,
and thole in the right-Wing are conſecrated to the appeaſed Deity; and thoſe in the left
to the retiring of the Waters, in the time of the Deluge; and they come to the Meri-
dian about half an hour before the great Dog; and by the Globes are about 21 degr.
zo' diftant, from the neareſt in the back; but I would have the Sea-men take him exact,
as likewiſe a good Conſtellation called the Crane Grus, or the Flamengo, as the Spa-
niards call its this Aftenſine conſiſteth of 13 Stars, and hath 3 Stars of the ſecond
Magnitude, that in the head is called the Phanicopter Eve, and the other are on his
Back, atid the other in ſiis left Wing;. Theſe Stars I would deſire thole Mariners that
Sail to the Eaſt or Weſt Indies; to take the Meridian-Altitude thereof, and their
diſtants from any known Stars, and by it you ſhall have all the reft; for many times
I havg beenSailing between the Tropicks, and for 1-2 days together have had no Meri-
din-Altitude of the Sun, by reaſon of cloſe and cloudy weather, which is bad for thoſe
that are bound to ſmall
. Ilands, and Cape-Lands; therefore to the Southward, as well
as to the Northward, theſe Stars will ſtand them in great ſtead, and ſerve their turn, as
well
As the Sun, to find the Latitude thereof.
The Rule is thus - About the end of the Year 1577, 'Ticho obſerved the diſtance of
il:e little Star in the breait of Pegaſus from the bright Star of the Vulture, to be exactly
45 degrees zi', and by the Meridian-Altitude thereof, he found the Declination thus.
Ticho
{
i
3.
ܪ
}
Chap. III. And how to Calculate in Aſtronomy.
190
tude of the Equa-
tor.
Ticho obſerved at Uranibürge the Star in the breaſt of Pegaisis,
and found his Meridian-Altitude
56 d. 32'rs.
The Latitude of Ilraniburge is 55 d. 54' the Compl. 34 06 Subſtract the Alci-
The Declination is found to be
22 : 26
To it add the Complement of 56 deg. 32', which is 33 : 28
And the ſum is the Latitude of Hraniburge 55: 54m.
So the Declination is found to be 22 deg: 26 min. North, which being given, the Lon.
gitude of the ſaid Star is to be inquired.
Therefore in the Oblique-Angled Triangle (of this Diagram) FOL is known.
Ji!.
1
1
I
!
1
H-
ilj! :)... GURU
:22
E
O
។
Oi!!
M
i
:
3
1
+
.;
..
;;
}
'n
1
1
S
4
i.
*T
1
1
L
}
+
:
Firſts...
The Complement of the Declination of the bright Star of the Vallare 8ż degrees
8 min. FL.
Secondly,
FO, the Complement of the Declination of the Scar in the Breaſt of Pegafon 67
degr. 34.
Thirdly,
OL the diſtance of them 45 degr. 31'.
By this Rule the Angle at F, which is the difference of their Right-Afcenfion, will
be found to be 44 degr: 54", as will be here demonſtrated.
Rer
The
a
1
1
1
1
.!
}
130
The uſe of the Natural Lines of Signs, Book VI.
+
سنة
***
The CoSign of Declination FL, is 82 deg 08 mix. 999589
The Co-Sign of Declinat. FO, is 67
34:
996582
The difference is
14-34
(1.) The ſum is
1996121
(2.) The Quadrat. or (2) the Rad.
2000000
:.
The Baſe or diſtance of the Stars LO
45 d., 31
41 Thic difference of their Declin, F L and FO 14:34
The Sum
60:05
The difference
30': 57
The half of the Sum 30 d. 02 30" Sign 96995 1
The half of the difference como 15: 28 30 Sign --- 942621
The (3) Sum is
1912572
1
.
Then ſay,
As the firſt Sum
1996171
.
is to the double or Quadrăt of the Radius - 2000000
So is the third Sum
1912572
'to the double or Quad. of the Sign of half &T 1916401
The Angle fought, which Bi-fected, gives the Sign of 22 d. 27' 17" 958200
}
1
1
1
Which doubled, is 44 degr. 54' 34", is the Angle LFO, which is equal to the
Arch DE, the difference of their Right-Afcenfion, which I add to the Right-Afcen-
fion of the bright Star of the Vulture, 292 degr. 35', and the Sum is 337 degr. 291
34", is the Right-Aſcention of the little Star in the breaſt of Pegaſu.
Then having the Declination of this Star 22 degr. 20, and the Right-Aſcenſion
337 degr. 29 34", the Longitude of the ſaid Star by the laſt Problem (19) will be
found to be 18 degr. 36. 4, and the Latitude thereof 29 degr. 24 min. North.
Now to draw the Diagram by Chords, and half-Tangents Geometrically, with the
Chord of 60 degr. draw the Circle ; then draw , , the Ecliptick 23 degr. 31' ,
S, and by C draw the Equator; then by the Parallel of Declination, and Right Afcen-
fion of the Vulture, will find LO, therefore if you put the difference of Aſcenſion
from v to E 22 degr. 30' 20", from the neareſt Equinoctial-Point, and draw the
Meridian-Circle FÉS, and it will cut the Parallel of Declination ato draw through
O, as PON the Circle of Longitude, and meaſure r X on the Line of half-Tan-
gents, and it is 11 degr. 24 min. from the neareſt Vernal Equinox, Subſtracted from
30 degr..leayes 18 degr; 36.0f * for the Stars Longitude"; and as before directed
you may find the Latitude je na to bc 29 degr. 24.
1
|
f
j
i
1
?
PROBL: XX.I.
1
F
1
Chap. III. And bów to Calculate in Aftronomý.
}
131
::.orT
1
PROBL. XXI.
To find the Parallax of Altitude of tlc Sun, 110011, or Stars.
:.* DR.
He true Altitade of the Sun, Moon, or Stars,..ought to be obſerved in the Centre
of the Earth, (if poſſible) wherero the Tablestare conformed ; but becauſe we
dwell upon the Superficies of the Earth 4000 Miles deareſt, origáz: Engliſh Milęs
from the Centre of the Earth ; therefore, the Planets ſeemi. Iower to uss than indeed
they be ; and therefore to find the true place of the .o *you mult. draw a Righis-
Line from the Centre of the Earth, through the Centre of the Sun, Moon, or Stars';
but the apparent viſible place is determined by a Line drawn from tho Eye, through the
Centre of the Star ; therefore the Parallax of a Star is an Arch of a great Circle, (paf-
fing by the Zenith, and the true place of the Star, the :Arch of the time Circle interi
cepted between the true and apparent place.
A Figure or Schemse ſhewing what the Parallax, or diverſity of Aſpects is.
1
N
I
.
1
M
1
1
T
C
E
P
8
H
B
1
In this Figure, C denotes the Centre of the Earth.
D the Place or Superficies of the Earth, from whence the cor* is ſeen,
cand 7, their Place in their Orbs.
C&I, COM, CON. she Lines of their true Place.
DIG, DOK, DOM, the Lines of their viſible or apparent Places.
Hence the Angle made by the Interfection of the ſaid two Lines through the Body
of the Planet, is the Angle of Parallax, that is to fay, in the Angle C&D, which
is equal to the Angle ISG, in the othe Angle of Parallax, is the Angle COD;
and laſtly in the Moon it is the Angle COD, or NOM, which is all one.
By this it is manifeſt, the nearer a Star is to the Horizon and Centre of the Earth,
the greater is the Parallax ; and hence it is, that the Orbit of the Moon being neare
to the Earth, her Parallax is greateft, and moſt perceptible, becauſe the Semi-Diameter
of the Earth bears a fenfible proportion to the Semi-Diameter of the Moons Orbit,
though it be very little, or nothing at all in compariſon of the Orbs of ħ 4, and the
fixed stars, which is cauſed by the Interval and
vaſt diſtance which is between them
but this laſt Problem hath no relation to the common uſe of Mariners, therefore I ſhalí
not infiſt any further, but refer the Reader to Harmonicon Corlefte, where there is a full
diſcourſe, and Rules relating thereunto.
But
1
11
1
1
1
11
!
1
132 The uſe of the Line of. Signs & Tangepts. Book VI,
+
1
i
But theſe Aſtronomical Propoſitions as I know to be uſeful for Sea-men, I have
here inſerted ; for the firſt and ſecond will find the Suns Place and Declination, together
with the 15 Probl. will find the Latitude.
The third, fourth fifth, and fixth will find the Suns Rifing and Setting, as the 7, 8,
9;15, 17:13; 14; to find the Variation of the Compáſs, and the i 2 to find the Sans Alti-
Lude at any time alligned, and the deſt being very uſeful Rules of thie Stars, by which
you may have the Hour of the Day, and Night) for having the Latitude of the Place,
with the Declination and Altitude of the Sun; orany:Star, they may find the Hour of
thic Sun.or-Star from the Meridian:by.the 16 and 1 Problem, then comparing the Right-
Afcenfion of the Sun, with the Righr-Afcenfion of the Star, they may have the Hour
of the Night.in all theſe Propofitions, I have been as plain and aš briet, as the ſeveral
Refolutions thereof would permitme, and I do with the Practitioner as much delight
in the Practice, as I have hadiin tfie Çompoſing of it.
+7
3
t
1
1
1
The End of the Sixth Book.
1
+
rice.
1.
!
.
1
.
1
4
L
!
1
{
1
t
1
1
1
广
​1
有
​年
​上
​}
十
​}
{
1
1
.
|
{
}
1
'.
1
{
1
|
自
​f
{
|
[
4
1
{
1
|
1
{
-
}
中
​1
1
1
1
|
1
1
1
1
1
}
1
The Seventh Book:
THE
Mariners Magazine,
OR,
STURMY's Mathematical and Practical ARTS.
THE
1
OF
A RT
DIALLING
Gnomical Scale
:?
BY THE
J J
l
1
AS ALSO BY
CALCULATION.
SHE WING
The Making of all ſorts of DIALS, both
within Doors and without, upon any Walls;: Cielings, or
Floors, be they never ſo'ùregulār, whércſoever the Director
Refle&ed Bcams of the Suħ may come for any Latitude.
:1
+
.
A N D
How to find the true Hour of the Night, by the Moon and
Stars : And how to Colour, Guild, and Paint Dials;
And how to faften the Gnomon in Stone or Wood
Never before made ile-plain to the meaniëft Capacity.
By Capt. SAMUEL STUR M. Y.
LONDON-Printed Anno Domini MDCLXIX.
1
14
Dj
r
t
1
Chery
IV
)
1
1
3.
44
1
1
☆
*
25
Z
8
F
uk
12
.
r
11
1
Transit Hora Siriemora
.
8 9 10 11 12 1 2 3 4
5
ti
22. z 3
1
}
8\.Sic tranſit gloria MunditAr:1667/4
VIL
JVI5
8
3
7/
be
F
VI
s
V
10
1
VIL
III 81
??
10.
1
11.12
17
IX
1.1.72
riX XI I. I
Zo 11221232
+
8
1
8
GA
1;! Minden
4
2. 10. 11. 112 111 211030
1
won bric:
1
Recict 03
NOV 111 TITIVI
para
.
1
.
UA
Horologium vitæ.
Latus ad occafum nunquam rediturus ad ortum
Wivo hodie, moriar cras, keri natus eram
.
}
1
!
To my much Honoured Friend
1
Ifaac Morgan Eſq;
Collector of his Majeſties Cuſtoms in the
Port of Briſtol.
►
N
2
.
:
1
Si R.
Ot to inform your fudgment in any
thing concerning the Subjeet Mat
ter of theſe my poor Labours (jour
Wiſdom and approved Knowledge
in all Learning being ſo general,
that I can add nothing unto it) but to inform the
World how much I honour you and your Vertues,
and by how many Obligations I ſtand engaged to
you for the many ſignal Favours you have vouch-
Safed me ſince the time I came firſt into this Port,
I dedicate this part of my Book, as the proper
Part of the fruits of my ſpare time, in twenty ſe-
ven Dials; unto- jou preſented as an unworthy
New-years-gift; and that Dial-piece being the
Subject of the mhole Art of Dialling, I will naine
the Dials, that I may charge you to Patronage no
more than you had; viz, Eight Verticals and De-
cliners, Eight Recliners and Incliners, and Eight
Décliners, Recliners, and Incliners, a Globe
with two Pole-Dials and one Shadow-Dial, made
on a Piece of Freez-Itone, as is ſeen in the Fron-
tiſpiece the Gromon or Stile fafined by me,
and likewiſe Painted and Guilded, which is well
Аааа 2
known
{
H
3
I
The Fpiſtle Dedicatory.
known by jou and many others: And being de-
fired by forne Friends that ſaw my way, and this
Piece of Dialling, to Print it, by their impor-
tunity, according to the beſt of my judgment, I
have ſo done ; and if any way profitably, then ac-
cording to minė own deſire. As it is, I have made
bold to make choice of you for the Patronage
thereof, that it may gain the more Credit by your
Protection ; And if any Mall be offended at this
Work,my Device ſhall be a Dial,with this Motto,
ASPICIO UT ASPICIAR; only to all Fa-
pourers of ART I am direct, erect, plain, as I
am, Sir, to you, and deſire to be,
it
N
SIR
ii.
?
3
Your humble and affectionate
Servant to be commanded, ,
St. Georges the pill,
¥ 29 Aug. 1667
!
1
SAMUEL STURM Y.
1
1
1
Book VII.
I
2
THE
1
Art of Dialling.
The FUNDAMENTAL DIAGRAM of the
DIALLING SCALE,
3
90
810
1
710
이
​To
60
60
50
50
D
Sinės
80,9
70
70
30
60
8
oz
60 slo
OL
410
270
20
30
por
5
Chordis
30 40 50
Ghoren Line
310
10
10
.
Inclination
of Meridians
3.
ماه های
7
90
А
4:1
5
3R
Line
VOB
Hour,
5.
fol:55 and fol: 1:6:9
MA
His Diagram of the Giaponical Scale I hayedefcribed Book 2. Chip. 3. to which
I ſhall only deſcribe the Lines on the Scalc, viz.
There are ſix Scales or Lines on this Inſtrument,
1. A Scale or Line of Chords, A D, in Degrees.
2. A
T
I refer you.
1
The Art of DIALLING. Book VII.
R
4
3. A Gnomon Scalc or Line, as BD, the uſe thereof you may ſee in Chap. 7. and ſo
forward in Degrees.
3. A Scale or Line of fix Hours, for drawing the Hour-lines in any Dial, divided
into every ro Minates; thcuſe you may read from Chap. 4. forward, as B E A.
4. A Scale of Inclination of Meridians, divided in Degrees as the Diam. BA;
the uſe thereof is in Chap. 12. and ſo forward.
5. U pon the clier Tide are tyvo Lihes or Scalesz fór the inlarging the Hour-Lines on
any Plain? Thic greater Pole is marked with the lefſer is marked with, called
the leſier Pole for diſtinction fake; and theſe Lines are divided into every 10 Minutes,
but may be by the Table into every 5 Minutes.
6. Theſc Scales or Lines are on the Mathematical Scale, with the reſt that are deſcri
bed Blok 2. called The Scale of Scales,
NAME IT!
1
Τ Η Ε
I
1
ARGUMENT
R
Eeader, read this ;-for I dare this defend,
Thy poſting Life on Dials doth depend.
Conſider thou, kon quick the Hour's gone;
Alive to day, to morrow Life is done.
Then uſe thy time, and always bear in mind,
Time's Forchead hairy is, but bald behind.
Here's that which will decline to thee, and now
How quick Time rans, how faſt thy Life dotb go.
Yet be ingenious, learn the Pra&tick Part,
And so attain to Practice of this Art :
Whereby you ſhall be able for to trace
Ont such a path, where Sol shall run his Race;
And make the greater Coſmus to appear,
According to each Seaſon of the rear.
:
1
C H - P. I.
CH
The Preface of the kinds of Dials,
1
A
1
Lthough Gnomoniques pertain to Aſtronomy; yet I think it not amiſs for the
caſeofche Reader, to place theſe in a diſtinět Book by themſelves.
Sun-Dials may be reduced to two forts. Some hew the Hour by the
Altitude of the Sun, as Quadrants, Rings Cylinders; and for the making thereof,
you muſt know the Suns Altitude for every day, or at leaſt every tenth day of che
year, and for every hour of thoſe days.
The other ſort ſhew che hour by the ſhadow of a Gnomon or Stile parallel to the
Axis of the World ; and of that I treat chiefly in this Book. Theſe be all Projecti-
ons of the Sphere, upon a Plane which lies parallel to ſome Horizon or other in the
World. And if upon ſuch a plane the Meridians only be projected, they ſhall ſuf-
ficc to ſhew che Hour, without projecting the other Circles, as the Ecliptique,
the Æquator with his Parallels of Declination, the Horizon with his Almicanters
andrazimaths, which are ſometimes drawil izpon's
Dials more för Ornament chân
for Neceſſity.
:: ? slizam, b) ilus ir
Pils : iO;I:2 x1
CH, P.
1
777
03
1:1
1
.:
!
1
CHAP.II. The Art of DIALLING.
.
3
CHA P. II.
1
1
Theorems premiſed.
F
Or the better underſtanding of the Reaſons of Dials, theſe Theorems would
be known.
I. That every Plane whercupon any. Dial is drawn, is part of the Plane of a Great
Circle of the Heaven, which Circle is an Horizon to ſome Country or other ; That
the Center of the Dial, repreſentech the Center of the Earth and World; and the
Gromon wliich caſtech the Shade, repreſentech the Axis, and ought to point directly
to the two Poles.
i
/
II. That theſe Dial Planes are not Mathematically in the very Planes of Great Cir-
cles; for chen they ſhould have their Centers in the Center of the Earth, from
which they are removed almoſt 4000 miles'; and yet we may ſay they lye in
the Planes of Circles parallel to the firſt Horizen, becauſe the Seinidiameter of the
Earth bearech ſo ſmall proporcion to the Suns Diſtance, that the whole Earth may
be taken for one Poiîtor Center; without any perceivable Error.
III. That as all Great Circles of the Sphere, ſo every Dial Plane hath his Axis, which
is a ſtraight Line paſſing through the Center of the Plane, and making Right An-
gles with it; and at the end of the Axis be the two Poles of the Plane, whereof
that above our Horizon is called the Pole Zenüb, and the other the Pole Nadir of
the Dial.
FL
..
I
IV. That every Plane hath cwo Faces or Sides : and look what reſpect or ſituation
the North Pole of the World hath to the one ſide, the ſame hạth the South Pole to
the other; and theſe two Sides will receive 24 Hours always: ſo that what one side
wantech, the other side ſhall have; and the one is deſcribed in all things as the
other.
V. That as Horizons, lo Dial Planes are with reſpect to the Ægvator divided into
firſt, Paralel or Äquino£tiál; ſecondly, Right ; thirdly, Oblique Planes.
VI. A Parallel or Polar Plane makech no Angles
with the Aquator, but lies in the
Plane of it, or parallel to it ; that is, hath the Gnomon erected on the Plane ac
Right Angles, as the Axis of the World is upon the Plane of the Aquator: be-
cauſe the Axis and Poles of the Dial are here all one with the Axis and Poles of the
World, and the Hour-lines here meet all at the Center, making equal Angles, and
dividing the Dial Circle inco 24 equal parts, as the Meridians do the equator, in
whole Planc thc Dial lies.
1.
1
1
VII. A Right Horizon or Dial Plane curtcth the e£qnator at Right Angles, and ſo
cutseth through the Poles of the World, that it hath che Gremon parallel to the
Planc, and ſo the Hour-lines parallel one to another; becauſe their Planes
, though
infinitely extended, will never cut the Axis of the World :: yec have choſe Dials a
Center, though not for the meeting of the Hour-linės, viz. through which the
Axis of the Dial Circle pafíechi, cutting the Plane at. Right Angles, and cutting
allo (necr enough for the projecting of a Dial) the Circle of the World.
VIII. An Oblique Horizon or Dial Plane cutteth the Æquator at Oblique Angles ;
that is, hath for their Gnomon the fide of a:Triangle, whoſe Angles vary according
to the more or leſs Obliquity of the ſaid Horizon : and the Gnomon shall always
make an Angle with the Plane, of ſo many Degrees as the Axis of the World
maketk with the Plane, or as either of the Poles of the World is elevated above the
plane.
IX. Every
'
1
1
1
4
The Art of DIALLING. Book VII
IX. Every Oblique Horičom is divided by the Meridians or Hour-circles of the Sphere
into 24 unequal parts; which parts are always leſſer, as they are ncerer to the
Meridian of that Horizon or Plane; and greater, as they are farther off: and on
both ſides of the Meridian of the Plane, the Hour-circles which are equally diſtant
in time, are alſo equally diſtant in ſpace. Whence it is, that che diviſions of one
Quadránt of your Dial Plane being known, the diviſions of the whole Circle are
likewiſe known.
X.. The Hour-lines in an Oblique Dial, are the Seations of the Planes of the Hour-
circles of the Sphere, with the Dial Plane: and becauſe the Planes of Great Circles
do always cut one another in Halves by Diameters, which are ſtraight Lincs paf-
12 I
II
2
Іо
NT
3
ų
6
8
1
KA
1
ſing through the common Center; therefore Lines drawn from the Center of the
Dial, to the Interſections of the Hour-circles with the Great Circles of the Plane,
ſhall be thoſe very Sections, and the very Hour-lines of the Dial.
XI. Every Dial Plane being an Horizon to ſome place in the Earth (as was ſaid Theo-
rem I.) hath his proper Meridian, which is the Meridian cutting through the Polcs
of the Plane, and making Right Angles with the Plane. If the Poles of the
Dial Plane lie in the Meridian of the place, then is the Meridian of the Plane all
one with the Meridian of the Place, and the Gnomon or Style ſhall ſtand erected
upon the Noon-line, or Line of 12 a Clock, as in all direct Dials Buc if the
Planc decline, then ſhall the Subſtyle Linc, or Line which the Gnomon ſtandech
upon, which is the Meridian of the Plane, vary from the Line which is the Meri-
dian of the Place; and this Variacion ſhall be Eaſt, if the Declinacion be Welt of
the Plane: And contrarily, becauſe rhe Viſual Lines, by which the Spliere is pro-
jected on Dial Planes, do, like the Beams of a Burning-glaſs, interſect or croſs
one another in a certain Point of the Gnomon (to be alligned at pleaſure, and cal-
led Nodes) and ſo do all place and depaint themſelves on the Dial Planc, beyondthe
Audus, the contrary way:
XII. Dials
1
!
.
1
1
1
A
1
CHAP.III, The Art of DIAL L'ING.
1
XII. Dials are moſt aptly denominated from that part of the Sphere where their Poles
lie, though ſomc Authors have choſen to denominate them from the Circles in which
their Planes lie; as che Dial Plane which lieth in the Aguinoctial, or Parallel to it,
is called by many an £quinoctial Plane; but I concur with thoſe who would ra-
ther call it a Polar Plane, becauſe the Poles thereof are in the Poles of the World.
T
.
5
CHAP..JII.
How to make the Polar-Dial, and how to place it.
He Plane of the Polar Dial lieth in the Æguinoctial, where the 12 chief
Meridians or Hour-circles divide both che Æquinoctial and this Plane into
24 Hours of equal parts; the Gnomon ſtands upon the lencer ac Right An-
gles with the Plane."
Firſt draw che' Horizontal Line A B, and croſs the ſame af Right Angles with the
Line, CD; now on the Center at G, with the Chord of 60 Derices, or with che
Tangent of three hours, you may deſcribe the Circle A C-BD, jana about it make
the Square
EFHI, then take out of the Hour-line One Hour, and lay it from
each
corner, as. E F H I both
ways: alſo do the like with two hours as you fee_done, and
from the Cenyer at G draw Lines to thoſe Hour-points:-10 ſhall/yqu have the Hour-
fines in che Æquinoctial Dial; C D being the Meridian or 12-a dock Line, and AB
the Eaſt and Weſt Line, ſerving for 6 in the morning at B, and 6 in'che afternoon ar
A, and ſo number the reſt of the Hours in order . You need draw no more hours chan
from 4 in the morning unco 8 açnight, for chis Latitude of Briſtol, being neer: 51 d.
30 min. if
For the Gnomon or Stile, you muſt have a {traight Pin or Wyre ſet upright in the
Center, of ſach length as you ſeeconvenient ; but if you will have it of ſuch a length
as may neither be too ſhort nor too long, then take this Rule.
Horo by Calculation to find the length of the Stile, and Seinidiameters of
the Parallels of Declination.
I
F it were required to proportion the Stile to the Plane, ſuppoſe che Semidiameter
of the greateſt Parallel upon the Plane were buc 6 Inches, and the Parallels ſhould
be the sd. of Declination, the Rule is general.
1
.
TOD0000
*
As the Tangent of 45 deg.
Is to the Tangeut-parallel of Declination 5 deg.--. 894195
So is the Semidiameter of the Plane 6 inches O A
277813
To the length of the Stile 53 partsa
172010
which ſhews that the length of the Stile muſt be as parts of an Inch divided into
Ico parts.
1
How to find by the length of the Stile, the Semidiameter of the Par
rallel Circles of Declination.
Suomi
Uppoſe the length of the Stile above the plane co be ro inches
, and you were to
find the Semidiameter of the Tropick, whoſe Declination is known to be 23 deg.
30 min. the Rule is for this and any other Declination,
As the Tangent of 45 deg.
1000000
Is to the length of the Stile 10 inches-
Iooooo
So is the Co-tangent of Declination 23 deg. 30 min.--1036169
To the Semidiameter of his Circle 23 inches 136169
Вь
1
f
1
which
1
}
*
1
4
9
11
2
ܟܫܝ ܀
1
1
I ILX IX X XI IA
6
The Art of DI ALLING, BOOK VII :
which ſhews the Semidiameter of the Tropick to be a3 inchies : So if the Declination
be 20 d. the Semidiameter will be 27 inches; if 15 d. then 37 *..; if 10 d. theu
56.776.; if s d. then 1145; and ſo of any other height of the Stile: as admit it
were parts of an inch high, then the Semidiameter of 23 deg. 30 min. would be
and for 20 deg. it will be in and for 15 deg. 17.; if 10, then 22; if
5 deg.then 6 inches; if Stile be, '4 and 7, the Semidiameter 23 deg. 30 min. is..
parts, as you may ſee the Figure inakes all plain ; and ſo of any other.
F
B
15
T
Inom firiem H.
D
INC
1
1
1.
TO
LL
.
IL
1
1
1
Of all Dials this is the plaineſt; for it is no more but divide a whole Circle into
24 cqual parts : and chis is the very ground to all the reſt.
With this Dial, the Hour-lines being equally divided into 24 equal parts, on the
inner circle you may make a Mariners Compaſs, with the 3 2 Points drawn upon it, to
know in all Latitudes whether the Moon being upon ſuch a Point makech High-wa-
ter; or upon what Point the Moon muſt be, when at thoſe Places ſer cogecher it ma-
keth High-water or Full-Sea.
For to know upon what Point the Moon is, may be done two manner of ways;
by ſetting it by the Compaſs, or by reckoning according to the age of the Moon; and
thc Hour of the day. The feețing according to any Point, may not be done with a
common flat Compaſs as the Mariners ſteer by (as many, wanting better reaſon,chink
they snay, to their great miſtake) by reaſon is dóch only divide the Horizon into
equal Points, and thewech in what Vertical Circle or Azimuth the Sun or Moon
ſtands : But this muſt be done with a Compaſs, which being elevated according to
the Superficies of the Æquinoctial, dividech the Æquinoctial fo likewiſe inco equal
parts, as the common flat Mariners Compaſs doth divide the Horizon. Such alı
Aquinoctial Compaſs, with a Dial in, as aboveſaid, is of faſhion 'as hereafter fol-
lowcth pourtrayed. Whereof the Wheel A B C ſheweth the Superficies of the
Aquinoctial, the Wyre ED the Axle-trec of the
World. The foreſaid Wheel muſt
be all alike marked on both ſides, as well under as above, with the 32 Points of the
Compaſs, and with twice iz Hours: and right againſt the Eaſt and Weſt ac
L and M, muſt ſo hang upon cwo Pins, as upon an Axletree, that it may be tarned
r
oder so
.
LIP
1
1
:
I
1
CHAP Vi The Art of D IA LLING
7
E
M
23
127
B
G
8.9
TOOB
D
2012
ETI
F
A
Ad
K
07
O
up and down, and thc Wyre at the under end at D, by the Quadrant FDG, niay
be ſet unto any height of the Pole. If then you ſec. ſuch a Compaſs with the under
bottom level, the Line HK Norch and South, viz. H to the North, and K to the South,
and the under end of the Wyre right againſt ſuch a Degree of the Quadrant F G, as
the height of the Pole that you find your ſelf in, then ſhall the Wheel ABC ſtand
cqual with the Superficies of the true Equinoctial, and the Wyre E D with the Axle-
cree of the World; and the ſecting by ſuch a one, and a common Compaſs, giveth
great Difference. And the nicerer the Æquinoctial, the greater; as may be underſtood
by the Examples following:
Ε Χ Α Μ Ρ Ι Ε Ι.
IN
N the height of so deg. or thereabouts, the Sun being in the beginning of Cancer,
ac his greateſt Declinacion to the North, by a common Compaſs cometh not before
· half an hour after ſeven of the Clock to the Eaſt, and at half an hour after four to the
W«ft; that is, he gocth from the Eaſt to the South and round to the Weſt in nine
hours; but from the Weſt through the North, until again in the Eaſt, in 15 hours.
Ε Χ Α Μ Ρ Ι Ε ΙΙ.
N N thc hicight of 30 deg. he cometh a little before half an hour paſt nine of the
clock to the Eaſt, and a little after half an hour paſt two of the Clock in the
Weſt, and forgoech in leſs than five hours and a balf.fram the Eaſt chrough the South
to the Wet; buc from the Weſt through clic North, until again in the Eaſt
, hego-
eth more than 18 hours. Thriedly, Being under the Line, and the Sun having no
Declination, he ariſeth in the morning right in the Eaſt, and ſo riſing higher and
higher, continuech"Eaſt until that he gotth over qurthcads through the Zenith into
the West; and fo continueth went ſtill going down Welt
until he cometh again
to the
Horizon: and ſo according to a flat Compaſs he is the one half of the day Eaſt, and
the other Welt, without coming upon any other Point. It is not ſo wich cluis Æqui-
noctial Compaſs
. The Sun and Moon go always a like time on cyery one Point of the
Compaſs, to wit
, from the Eaſt to the South 6 hours, from the South to the Weſt
6 hours, from the Weft through the North to the Eaſt in twice 6 hours
.
This Dial will Terve for all Latitudes, if you put the end of the
Wyre at Dg to the
height of the Pole,or. Latitude of the place as befofefaid, lo che ſhadow of the other
end at E will fall upon the Holurs and rue Roines of the Compaſs, all the time the
Sàn is to the North of the Aquinoctiál: but when the Syn is to South of the Æqui-
noctial, you muſt look for the Hours and Points of the Compaſs upon the under lide
of the Dial.
Bbbb 2
CHAP .
i
>
f
1
1
។
:
9 JO1112 1 2 3
1
L
8
The Art of DIA LLING, Book VII
+
CHA IV.
How to make the South Æquinoctial Dial, or Polar Plane.
T
!
He Æquinoctial Dial we call that which hath his Poles in the Æquino&tial
Circle, of which there be three kinds.
1. The Direct or Sotich Æquinoctial Dial, which faceth the Meridian
directly, not looking from him to the one ſide more than to the other, having his
Poles in the
Interſectians of the Æquinoctial and Meridian.
2. The Eaſt or Welt Æquinoctial Dials, which may alſo be called Æquinoctial
Horizontal D als; for alt Horizontal Dial declining juſt 90 Degrees
from the South
or
North, becomes an Æquinoctial Dial, as well as Horizontal, becauſe there is his Polar
height, upon the Interſection of the Horizon with the Aquinoctial: and though
this Dial be of kin to botli, yet-his-Gnomon shows that he ſhould be ſorted rather
with the Equinoctial Dials,than with the Horizontal. Theſe two ſorts are regular,
having the Poles in the four notableft Points of the Æquator. The third is ſomewhat
irregalar, but may be brought to Rule.
How to make the firſt of theſe, draw the Horizontal Line AB, and about the
midſt at C let fall the perpendicular CD, which is che Meridian or 12 a clock Line.
Let CD be equal to a Chord of 60 Degrees, or the Tangent of three hours, and
through D draw the Line F.E, parallel to A B : make alfo D E and C B cqual to D,
ſo have you a truc: Square CDEB." Now.cake one hour with your Compaſſcs off
your Scálc, and lay the ſame both ways from E towards B and D, as E 1.Do che like
with rwo hours, and draw the pricked Tangent-lines from C to theſe Marks.
Next, Let the length or height of the Gnomon or Stile be GH, equal to CH, or
3 hours; ſo drawing a Line chrough G H, parallel to the Horizon, you ſhall find it
cut the former Lines drawn to the Center C, in the Points l, m, n, o, p: through
which Points, if you draw Parallel-linės to the ri a clock Line CD, you ſhall have
all the afternoon hours as far as V: and the inorning hours muſt be drawn in like
manner and diſtance, to the left hand or Weſt fide, beginning from 7 in the morn-
ing unto 1:2, as in the Figure following:
Note
, that the height or length of the Stile is always 3 hours from the Meridian,
as you ſee HG, which you may make with Copper or Braſs Plate, or Iron, in form
as you ſee ſhadowed, whoſe breadth on the cop is here HR, which
may
be made
more or leſs as you pleaſe.
This Dial will ſerve in any Latitude, if the Plane be placed parallel co the hour of
6, ſo that the Planc be even with the Pole of the World.
1
1
C
B
I
H
Za
ER
P
주
​18
SƏM
KEast
1
R
E
и
2!
8
1
4
t
1
4
Howy
41
H
1
I
1
+
1
TV
L
1
1
i
CHAP.V. The Art of DIALLING.
1
9
+
How to calculate the Height of the Stile, and the Points of Hour-diftance
from the Meridian.
were required to put on all the Hours from 7 in the morning to s in the even-
ing; here we have 5 hours and 6 inches on either ſide of the Meridian, wherefore I
allow 15 Degrees for an hour. The Rule to find the height of the Stile is,
As the Tangent-compl. of the given Hour 15 deg.ca
TO57194
Isto half the Horizon or Diſtance from the Meridian 6 inches 277815
So is the Tangent of 45 Degreesome
1000000
To the height of the Stile 17 inches and parts
220621
And likewiſe the diſtance of the Hour-points of 9 and 3 from the Meridian will be
1, or Ilinch and 61 parts of 100,
How to finid the length of the Tangent between the Subſtile and the
Hour-Points,
.
1
}
1
.
of 100,
Aving found the length of the Stile in our Example to be inch 61 pares
chen in this Example, as we find the firſt Hour, fo find the reſt.
1000000
As the Tangent of 45 degim
Is to the Tangent of the Hour from the Meridian 15. degia
942805
So is ie height of the Stile, 1.. inchesama
220621
Tophe length of the Tangent-lixe between the Meridian òr Sub-?
163426
ſtiler 4 inch
I
1
An. Po
Hours
Tang:
In. par.
deg. mi.
1 2
9
O
o
O
II
I
O
IO
and the Hour-point of 1 and of 11
--a-clock, And fo of the reſt,take them
off a Scale of an Inch divided into
100 parts, and prick them from C
and D both ways to B A and EF,
and draw the Hour-lines parallel to
the Meridian.;, and ſo do with the
reſt, until it be finiſhed, as you may
fee by the Table.
15
30
45
60
ana w N
43
93
.I
- 61
2 79
6 O
Infinit.
ܟܗ ܙܘ
O
75
90
O
Si
1
!
1
CHA P. V.
How to make the Eaſt Equinoctial Dial, or the welft Lat. 51 d. 30-m.
His Plane is a right Horizon of thoſe people who dwell under the Æquacor,
diſtant from us go deg. of Longitude; as the South Æquinoctial Plane of
ti the laſt Chapter was the Horizon of thoſe who dwell under the Æquatoriin
the fame Longitude with us: Therefore theſe Dials are in all Poincs alike, only the
Subſíler Line, which in the South Æquinoctial Dial is at 12, is but 6 in the morning
for our Country, becauſe of the difference of Longitude.
To pourtraiót this on a Wall or Plane, firſt draw the Horizontal Line A B; then
upon the Center C deſcribe the Semicircle ADB, whereon lay the Latitude of the
place si d. 30 m: from A unco D; ſo drawing G D continued, you ſhall have the
Hour of 6: chen with your Compaſſes take off your Scale 15 deg. of the Line of
Chords, and turning them off 6 times, divide the Arch D F into 6 equal parcs, and
draw
1
ܙ
1
i
1
.
10,
The Art of DKALLING. Book VII.
you had
1
draw prick'd or blind Lines to choſe Diviſions, which would be all one as if
donc it thus, CD bemg equal to the Chord of 60, or Tangent of 3 hours, you
Thall make the Quadrant or true Square equal to the fide chereof C DEF, and from
the corner at E, you thall lay down both ways towards D and F the hours of 1 and
2, from whence draw Lines to the Center C : Next make choice of the length or
lieight of your Pin or Stile, which you muſt lay down from C to G on the 6 hour
Line, drawing from the Point G a Linc perpendicular to the Line of 6, or parallel
to the ſide CF, as GH, which curs che former Lines in the Points I KLMH;
through which Points drawing Lines parallel to the hour of 6, you ſhall have the
morning hours from 6 to 11, and the hours before 6, from 4 in the morning, are
equal as from 6 to 7 and 8.
A Horisontall line C Anº Domº1669 *B
7
1
Gigi
1
w
Soitih
F
B
2
Nortti
*
I
VI
VI
VA
VIU
IX X
E
1
XI
X
1
3
1
1.
1
11
1
.
+
Ang. Po.
Hours,
How to make the weſt Aquinoctial Dial.
THA
*He Weſt Æquinoctial Dial erect, ſerving for the afternoon, is drawn by the
fame Rules contrariwiſe like the Eaſt in all points, only it ſhews but the after-
noon hours, as the Eaſt thews the forenoan hours: When you have drawn on paper
the Eaſt Dial, and ſet it by gueſs in its fcicuacion, go on the Weſt ſide of it, and you
may ſee through the paper the picture by reflection of the Weſt Dial; and ſo will
the picture of the backſide of the Weſt fhèw you the true picture of the Eaſt Dial.
The way to calculate
the height of the
Stile, and the diſtance of the Hour-lines
Tang
from the hour of 6, is the ſame as in the
deg. min.
In.
par.
lalt Chapter of the Polar Plane : For
57
68:
Suppoſe the length of che Stile to be 10
4 8
inches, chen che length of the Tangent-
45
line belonging to the firſt hour will be
3 inches and 68 parts of 1co, as you ſee
17 32
in this Table for the reſt of the hours,
75
37 32
which taken off 2 Scale of equal paxts,
Infinit.
and prick'd from the Aquinoctial from
C cowards F, and likeiviſe upon the parallel DE: fo you will make a Dial ail, one
as by the former way, which is good proof, if you draw the Hour-lines through theſe
two Points ; and lo of the reſt.
СНАР.
1
2
15
30" 0
5 77
IO
0
60
3 9
I 2
II I
12
O
go
O
11
1
1
!
1
H
1
1
L
1
i
<
CHAP.V. The Art of DIALLING,
Cé. VI.
of ibékinds of Oblique Dials.
..
5
W
Hat an Oblique Dial is, and why it. hath been ſo called, hath been
thewed Chap. 2:
SRegulár.
Theyl be
Irregular.
The Regular lie in ſome notable Circle of the Sphere; as firſt!che Vertical Dial,
whoſe Plane liech in the Horizon, for which cauſe miany call it che: Horizontal Dial.
Secondly, the South and North Horizontal Dials,, ivhole Plane liech in the Eaſt Azi-
muth, and is commonly called the South or North erect direct Dial. As for the Eaſt
and Weſt Dials, they belong to another place, as wás ſaid.Chap. s.
The Irregular are ſuch as lic Oblique to the Horizon, as Reclining'or Inclining, Di-
als; or elíc lie
Oblique to the Meridians, as Decliners.;.of elle Oblique to boich
Recliners or Incliners declining, which are eſteemed che hardeſt of all, becauſe of
their double irregularity.
The Declination of a Plane is the Azimuthal Diſtance of his Poles from the Meri-
dian of the place Eaſt or Weſt.
The Reclination is the diſtance of his Poles from the Zenith and Nadir of
your
place.
Inclination is the neereſt diſtance of the Poles of the Plane from your Horizon ;
and whatſoever the reclination of the upper face of a Plane is, the inclinacion of the
lower face is the Complement thereof.
1
1
1
:
CHA P. VII.
1
How to make the Vertical Horizontal Dial.
LT
HII
D
Raw firſt the Horizontal-line A B, which is the Hour-line of 6; then take Take off the
the Latitude of sid. 30 m. from the Gnomon-line, and lay it down both scale of Hours
ways from the Center C, as to A and B : Next take the whole Hour-line of the balfs and
6, fixing one leg of your Compaſſes on A, deſcribe a little Arch towards D ; do che quarters, axd
like from the Point B, croſſing the Arch ac D; ſo draw the Lige A Dand B.D. Now prick from D
upon
theſc Lines you muſt tranſport the 6 hours from D unto A, and alſo from D Hours, and
unco B, as you ſee by the Figures 1, 2, 3, 4, 5, from whence drawing, Lines from dramo the quar-
the Center C, you ſhall have
the Hours as you ſee numbred from 4 in the morning ters in the Dial
uncil 8 in the afternoon, which ſufficeth for this Lacicude si d. 30 m.
As you were dic
For the height of the Stile, cake off your Line of Chords with your Compaſſes Hoteis
.
the Latitude of the Place 51 d. 30 m. and lay from Kto E, from 12 to neer 4, and
{o drawing CE, you have the height of the Scile, which may be made in Braſs or
Copper Plate, as you ſee ſhadowed in the Dial following.
Thus by Calculation,
As the Sine of god. Is to the Sine of the Latitude 51 d.30 m.
989354
So is the Tangent of the Ho. 15 do-
942805
To the Tangent of the Hour. line from the Meridian 11 d. 50 .-932159
retted for the
- -
:
+
as
1
1
•
11
1
1
12
The Art of DIA L LING. BOOK VII
ID
I
I
***
j
I
K
7
E
u
1
Alcun
AL
HB
;; ; ܫܝ ܙܪ
Z u
.
I
S
हो
+
Hours.
Gr.
mi.
Gr. mi.
Tangents.
1
I2
20
00
I
As in this Table, which rake from the
Line of Chords, and prick from the
Meridian 12 from K on cach fide, the
Degree or Tangent of each hour; And
by the fame Rule you may find the
Quarters, and that you may prick off in a
like manner
which is a way how to
make an Horizontal Dial, as before, Laci-
cude si deg. 30 min.
2
00
TS OB
30 00
45 00
60 00
75
100
90
II 50
24 20
28 3
53 35
3
4
5
.6
1
71 og
90
ܕܽ
051
Arn
1
CHA P. VIII.
A South and North Erect DireEt or Horizontal Dial, and how to make it.
T
His belongs co an upright Wall looking full North or South, and the Plane
of it lies in the Eaft Azimuth.
Firſt draw the Horizontal-line A B which ſervech for 6 in the morning
at A, and 6 in the afternoon at B; then from the Center lay down from the Gno-
mon of the Latitudes Complement 33. 30 both ways, as to A and B: Now with the
whole Line of 6 hours from A deſcribe an Arch cowards D, and with the ſame di-
ſtance from B croſs the ſame Arch, and draw the cwo Lines A D and BD, whereon
from D you muſt tranſport the hours, as you ſee by the Figures 1, 2, 3, 4; 5;
drawing Lines through thoſe parts from the Center C, you ſhall have the hours from
6 a clock in the morning to fix a clock in the afternoon.
With the Chord of 6o on the Center C deſcribe the Semicircle A D B from which
Line of Ghords take the Complement of the Latitude 38. 30, and lay down froin
Compl. 38 3o the Meridian at E unro F; ſo drawing CF, you ſhall have the height of the Stile
above
So.
90
Lat.
SI 30
i
Limony
VTIT
.
CHAP.X. The Art of DIA L·LING.
13
above the Planc. This, if it be for a large Dial, as againft a Wall, is beſt to be made
of a Rod of Iron; for ſmall Dials a Braſs Plate is beſt, and your Dial is done.
N
А.
V
B
VI
*
4.
*F
DS
XXXII. 2112
*I
D
1
*
}
2
.
This Dial ſhows the Hours from 6 in the morning to 6 at night: The other hours
before and after 6, as far as four and eight; belong to the North face of this Dial.
Becauſe the Almicantars, may oft obſcure the Interſections of the Hour-circles, you
may avoid thac if you reduce this Dial co a Vertical Dial, for the South Horizontal
Diál, being the Vertical Dial of thoſe people who live go degrees Southward from
us, that is, in 38 d. 30 m. of South Latitude.
Secondly, For the North face, imagine you had for the Gnomon a Wire thruſt
allope through the center of the Plane from this, Southſide Northward, and you will
preſently conceive, that in the North Dial the Horizontal r6 a clock. Line žvill be
loweſt, and that the Stile or Gnomon will cațn upwards towards the North Pole, as
much as it turned downwards on the other ſide, and that all the Hours. Save 6 in the
morning, and 627, 8 at night, may be left out in our Latitude, becauſe the Sun ſhi-
neth no longer upon it; and thoſe Hour-diftances you
may find and ſet off from 6
a clock Line, as you did the Hours of like diſtance in the South face. Note in a South
erect direct, or a South creat declining Dial, the Stile always points downwards; buc
if ic be a North erect declining Dial, the Scile points upwards.
(
cccc
CHAR
14
The Art of DIALLING. Book VII.
CHA P. IX.
How to make a South inclining 23 deg, in the Latitude of so deg.30 min.
!
S
+
Incl. 23
Uppoſe that the Plane be ſo inclining; tħat the face thereof be tovards the South,
and che North part be elevated 23 deg. above the Horizon, and that the South
part be dipped as much under the Horizon; then to find the height of the Stile
d. m. above thc Planc, you muſt ſubftract the Inclinación 23 deg. from the Latitude of the
Lat. 51 30 Place, which is here's i deg. 30 min. ſo the Remainer being 28 deg. 30 min. ſhall be the
height of the Stile. Now for drawing the Hour-lįnes, you ſhall do no otherwiſe
W. Srilc.28 30th
30 than you have done before in making the Horizontal Dial according to the Stiles
height 28. 30, as you may perceive in the Dial following.
Note, That if the Inclination of the Plane be more than the Latitude, then
you
muſt ſubſtract the Latitude from it, ſo there ſhall remain the height of the Stile above
rhe Planc.
Barif the inclination be South, that ſo the upward face of the Plane looks North-
ward, then you are to add the Inclination to the Latitude of the Place; and if ic
exceed 90 degrecs, you muſt then ſubſtract it from 180 degrees, ſo fhall you have
the Poles height above the Planc towards the South part of the Dial. The Figure of
this Dial followeth.
1
(
I
X XXXI I I
M
R
3
4
Ά
B
South
1
ve
correlacions trentai...
CH AP.
--
CHAP.X: The Art of DIALLING.
1
15
CHAZ: X.
How to obſerve the Declination of any Declining Plane:
A
LL perpendicular Plancs, as Walls
, lie in the Planes of one of the Azimnůchs;
which Planes cut always bh Zenich' and Nadir, and the Center of the
Earth, as in the Figure 24 cnith and Nadir E ŚWN. Horizon EW is
the Baſe or Ground-line, or any Horizontal Line, drawn upon a Wall or Plane, look-
ing full South or North: his Poles are at S and N in the Meridian ; wherefore he
declineth not, but lieth in the Eaſt Azimuth E W. '
.
piz
1
.
ܨ ܫܪ ܀
I
B
1
N
E
A
SP
A B is a Wall or Plane declining Eaſt by the Arch SP, to which A B or WE are
equal: for ſo much as the Wall bendech from the Eaſt Azimuth, ſo much doth his
Pole at P decline or bend from the Meridian.
Now to find how much any Plane declineth, and ſo in what Azimuth he lies, one
good way is this. When the Sun begins to enlighten the Wall, or when he leaves it,
then is tlic Sun in che ſame Azimuth with the Wall : therefore take at that inſtant his
Altitude, and thereby get his Azimuth, according to Clap. 14.of the Sixth Book, ſo
you ſhall have tlie Declination of the Wall.
Another way, if you have not time until the Sun cometh unto the Azimuch of the
Wall, or the Vertical of it, which cutteth the Pole thereof, then get thic Suns Azi-
muth as before when you can, and at the ſaine time obſerve by the light of your Cir-
cumferenter the Suns Horizontal Diſtance from the Pole of the Plage ; and by compa-
ring of thoſe together, you may caſily gather the Declination of the Wall: As in
Example.
I obſerved the Sun to be gone Weſt from the Pole of the Plane 72 deg. and by tlie
Alcitude of the Sun then taken, I found his Azimuth 62 deg. Here I reaſon thus:
The Sun is gone from the Pole Vertical of the Wall 72 deg. and from the Meridian
62 deg. therefore the Meridian lies between the Pole of the Plane, and the Sun: And
becaule 0 Pis 72, and OS 62, cherefore SP the Declinacion of the Plane is to deg.
the Difference of 12 and 62; and the Declination is Eaſt. for the Sun is nccrer to the
Meridian, clan to the Vertical of the Plane.
Cecc 2
And
1
L
1
4
4
A
The Art of DIA DEING BOOK VII.
And thus if you draw a rude Scheme of your Caſe, you may ſoon reaſon out the
Declination, better than do it blindfold, by the Rules commonly given.
And by thoſe two laſt ways you may take the Declination not only of upright
Planes, but of. Recliners allo.
:
1
1
How to take the Declination of any wall or Plane, without the help of
# Needle or Lor 'stone.
}
1
2
B
Or Example. SuppoſeSND E repreſent a Wall, or the face of thc Plane where-
on I am to make a Dial, and I deſire to know the Declination thereof from the
Meridian Eaſtward or Weſtward.
If you have no Inſtrument, take a plain Board, having one planed or ſtraight
fide or edge, which Board let be repre-
ſented by DEVQ: apply the ſtraight
edge of the Board ED to the ſide of the S
N
Wall or Plane, as in this Figure, and in
the middle of the Board at C, I ſet one
D
foot of the Compaſſes, and the other
opened to 60 deg. of my Line of Chords,
I deſcribe the Circle Ż BHA. In the
Center C, I erect or place a Srile or Wire,
as CO perpendicular to the Horizon, pla-
cing the Board as neer Horizontal as I can.
I find by obſervation, that the ſhadow of
the top of the Pin or Wire toucheth the
V
Circle in the forcpoon at the Point. B,
where I make a little mark; and like-
wiſe I obſerve in the afcernoon that it
touchetk che laid Circle in the Point A: Then I meaſure the half thereof from B or
A to X, and drawing a Line through the Center to X, as KCX, you ſhall have
the Meridian Line exactly deſcribed K Cx. Laſtly, I take the Diſtance Z X, which
I apply to my Scale of Chords, and; find the Arch thereof 18 deg. 10 min. and ſo
much is the Declination of th Plane E DNS, which you may ſee by clie Meridian
Line X K to be towards the Eaſt; therefore it is a South Plane declining Weft 18 d.
10 m. This way is the moſt caſie way, and requires time for the making the cwo Ob-
ſervations; therefore I will lay down ſome other ways, that may reſolve at one mo-
ment, or at onc obſervation.
G
K
$
9
+
[
How ta find the Declination by the Needle, whether the fir be
clear or not.
1
Apply the North ſide of the Inſtrument wherein the Needle is placed unto the
Wall, and hold it Horizontally as neer as you can, chac che Needle may have li-
berty to play to and fro; and when it ſtands, obſerve upon the Limb of the Chard
over which it moves, upon what Degree the Needle ſtands; for that is the Declinati-
on of the Plane, reckoned from the South Point of the Needle: And if you would
know the Coaſt, obſerve, That if the Needle ſtand upon the Eaſt ſide of the Meri-
dian Line, then is the Declination Weft; but if it ſtand on the Weſt ſide of the
Meridian Line, the Declination is Eaſt. By the Sea-Compaſs deſcribed Book V. as
it hangs in the Box, you may alſo find the Declination. Set the flit of the Brafs Die
amccer North and South, as before directed; then ſci the ſquare ſide of the Compaſs-
box next the Plane, reckon outwards 180 deg. and ſet the Index to it; fo reckon the
number of Degrees betwixt 180 che Index, and che Meridian, and chat number of
Degrees is the Declination of the plane required ; and by the Chard you may
ſee
what Coaſt it is, that is, whechier he declines from the North or South Eaſtward or
Weſtward. And note, That all Lines parallel to any Horizontal Line be Horizontal,
and all Lines parallel to Vertical Lines be allo Vertical.
С НАР.
1
CHAP.XI. The Art of DIA L»E ING.
17
1
H Н
4
C H 4 g. XI.
i en
How to inake a Declining. Horizontal Dialzori South erect declining from the
South Eaſtwards 32 deg. 30 min. in the Latitude of 51 deg. 30 min.
Ere chrec things-are required; for beſides-che-Diſtance-ofthe- ſeveral hours
froin 12, and the Elevation of che Gnomon, which are requiſite to the ma-
king of all direct and regulär Dials, we muſt here alſo know the Declinati-
on of che Gnomon, which ſome call the Diſtance of the Subſtile from the Meridian,
or the diſtance of the Meridian of the Plane from the-Meridian of the Place. For
in all Dials thic Noon-line in the Méridian of the Place, projected on the Dial, and
in all Horizontal or Mural Dials norrectiningor inclįning, the Noou-linc is a Perpen-
.dicular cutting the Center of the Dial, how much foever they declínie.
Bac declining Dials which look awry from our Meridian, havca Meridian of their
Ow11, which is called the Meridian of the Plane anche Subſtile becauſe the Srile or
Gnomon ſtands upon ic) avid is indeed the Meridian of that Place where this Decli-
ning Dial would be a Vértical Dial, and wheretlić Subſtile would be Noon-line; and
to this Subſtile the Hours of the Plane are always ſo conformed, that the necrer they
be to the Subſtile, the narrower are the Hour-ſpaces ; and contrarily, becauſe the
Meridians do cut every Oblique Horizon, that is thickeſt ncer the Meridian of the
place; and this Declining Dial being a Stranger with us, followeth the faſhion of his
own Country, and ſo hach his narroweſt Hour ſpaces neer his own Meridian, rather
than ours: And now, as that is the Meridian of our place, which cutteth our Horizon
at Right Angles, paſſing through his Poles, Zenith, and Nadir; ſo the Meridian of
any Plane is that which cuttch the Plane at Right Angles, and paffech through his
Poles.
Before we draw the Hour-lines in theſe ſort of Dials; it will be very convenient to
thew a general way for all Latitudes in a Diagram by it ſelf, and how to find the
Subſtiter Diſtance from the Meridian or 12 a clock Line, and the height of the Gno-
inon of Scile above the Planc. Firſt, Draw the Horizontal-line AB, and upon the
Center at C, take off your Scale with your Compaſſes a Chord of 60 Degrees, de-
fcribe che Semicircle ADB, and with a Chord" of 90 you may lay from A to D,
and froin B co D, ſo thall you draw CD from the Meridian-line of 12 a Clock
Then take the Coinplement of the Latitude 38 deg. 30 min. and lay from D to E, and
lo draw E F parallel to the Horizon A C; next cake the Declination of the Dial 32 d.
30 94, and lay from D to G, drawing the Radir O G thercon, you muſt lay che Di-
ſtance E F froin the Center at C, as CH. Now with the neereſt diſtance from H to
the Meridian CD, as HI, inake FL; and drawing a Line from C through L, it
will cut the Limb in the Point M; ſo meaſuring D M on the Line of Chords, you
Thall have the Subſtiler Diſtance 23 dog. 8 min. all which you may ſee in this Scheme
following
By Calculation,
As the Radiu's go deg.
Is to the Sine of the Declination SE 32 deg. 30 m.
So is the Co-tangent of the Latitude 5ı d. 30 m.
To the Tang of the Sulſtiler Dift from the Meridian 23 d.8 m.- 963081
For the height of the Stile, take the necroft Diſtance from H to the Horizon K,
and lay the ſame from L to cut the Arch in N:So meaſure MN, you ſhall have on the
Line of Chords the Height of the Stile ricereft 3 deg. 40 min.
By Calculation, viz,
As th: Radims 90 deg.
Is to the Go-fine of the Latitude gi deg. 30 min.--
979414
So is the Co-fine of the Declination 32 deg. 30 min.
992602
To the Height of the Stile 31 deg. 40 min. amma
Те
j
1
1
IO
973021
990060
1
-10
".
.
97 2016
1
4
N
Jl.
18
The Art of DIA LLING. Book VII.
1
To find what Hour or how much Time the Subſtiler is diſtant from the
Meridian or Inclination of Meridian.
2
A
B
1
stile
म
1
programmieru
Projets gras
E
F
0
G
M
1.
D
r
TA
1
Ake the neereſt Diſtance from Mto FE, and lay it on the Meridian from Fro
0: Then take the Diſtance from O unto G, and lay it from O unto P on the
Meridian ; ſo the Diſtance from Pro M, meaſured on a Line of Chords, will be found
to be 39.drg. 9 min. or cliercabours; which in time, allowing 15 deg. for an hour,
and four. Minuces to a Degree, you ſhall have 2 ho. 36 min. 36 ſec. which is the di-
ſtance of the Subſtile Line from the 12 a Clock Line, which in this Dial is, between
Dand 10 of the Clock in the morning. And by Calculation,
10
As the Radius go deg.
Is to the Şine of the Latitude 50 deg. 30 min.
So is the Co-tangent of the Declination 3.3. deg. 30 min.
To the Co-tangent of the Inclination 39 deg:9 min.
989354
1019581
1098935
Thus is ſhadowed a Geometrical way, and by Calculation, for any Latitude: But
for one particular Latitude, Mr. Philip Staynred, which firſt compoſed the Scale and
Gnomon Line, and Inclination of Meridians, and the greater and leſſer Pole on the
Dialling Scale, for 37 ycars ſince, as I have ſeen by him calculated, and the Projecti-
on Geometrical in his Study: he hach for the more cafe fer two Lines upon che Dial-
ling Scale, as he uſually makes, to find the Subſtile for the Latitude of 51 deg.30 m.
againſt the Lines ſtands the Letters Sub or Stile joyved with it ; ſo if you cake from
off the Subſtile-line che Declinacion of the Dial, and lay it from D unto M, which
in the laſt Example was 32 deg. 30 min. you ſhall find it to reach in the Diagram
from D unto M, as in the Line of Chords 23 deg for the Subſtilc, as before. Alſo,
the other Line nored with the word Srile, you thall likewiſe take from thence the
Declioration 32 deg. 30 min. which you ſhall find to reach in the Diagram from M
unto N, or in the Line of Chords 31 deg. 40 min.
CHA P. XII.
1
1
How to draw the Hour-Lines in a Declining Horizontal-Dial, or South erect,
declining 32 deg. 30 min. from the South Eaſtward, the Latitude being
51 deg. 30 min.
Inft draw tlič Horizontal Line A B, and on the Center at C deſcribe the Semi-
circle A E B, with thc Chord of 6o deg. and from A and B lay down go deg.
unto E; o Thall you draw ĆE the Meridian Line or Hour of 12 ; then in
the
4
F
}
1
1
I.
6,
CHAP.XII The Art of DIA I LING.
19
1
the former Diagram take the Subſtile diſtance DM 23 deg. and lay the ſame in the
Dial following from Eunto F, and from the Center C through you ſhall draw
CF K the Subſtiler Line. Next take the Chord of go deg. and lay ic from F both
ways upon the Arch, lo ſhall you draw the Gnomon Line GH, whereon from C,
with the Sciles Altitude before fount, 31 deg. 40 min. taken from the Gnomon Linc,
you ſhall make CG and CH. Then take the whole Linc of 6 hours, and with
the fame diſtance from G deſcribe an Arch at K, and with the like from B croſs the
fame Arch, and draw the Lincs G K and HK, which laſt Line cuts the Meridian at
N. Now if you meaſure K N on the Hour-lines, you ſhall find it ncer 2 ho. 36 m.
as you found in the laſt Diagram. Then take one Hour more, which is 3 bo. 36 min.
and lay the ſame from K unto M; and ſo increaſing one Hour more, you ſhall have
the Hour points 1 and i; allo diminifhing one Hour leſs than a ho. 36 min. which is
i ho. 36 min. the ſame will reach from K to O, and ſo 38 min. from K to P. Now as
you have divided K H, the very famc diſtance as is from K towards H, muſt be from
Ġ towards K; ſo drawing the Lines from the Center C through thoſc Points, you
ſhall have the Hour-lines, as you ſee in the Dial following.
IT
I IXXX XVII
UT
1
VI
ul
IL
lis
A
ir
1
fin
1
I
7
VI XXXX I
w
AR
:.
K
-10
By Calculation,
As the Radius go deg.
Is to the Sine of the Stiles or Gnomors Height 310.40 m.
972013
So is the Tangent of the Dift. of an Hour from the Suft.9d.9 m.- 920701
To the Tangent of the Hour-Arch from the Subſtile 4 d. 50m. 2
892714
betwixt the 10 a clock Lipe, and the Sult. Line on the Arch
By
1
*
6
SI
8
20
II
10
II
9
I2
23
do o
I
2
2
You may apply this Diſtance to the Line of Inctuon of Meridians, and it will
20
The Art of DI ALLING. BOOK VII.
By this Rule was this Table made;
Egual Dift. Hour Arches.
and by the fame you may make one for Hours.
any Latitude, and for any Declining
D. M.
D. M.
Dial; and you may by it prove your
4
80 SI 72 57
former Work: for if you prick from the
5 65 51 49 30
Subſtiler Line Fiche Chord of 4 deg.50m.
50 SI
32749
and draw a Line from che Center, it will
7 35
2046
be the Hour-line of 10 ; and prick the
sr
18
Chord of 3 deg. 5 min. from the Subſtile,
9 S SI 3 5
and draw a Line through that Point to
Meridian. Subftile.
the Center, and it will be the Hour-line
9 9 04 50
of 9 a clock; and ſo of the reſt , as you
.24
13 IS
find them in the laſt Column.
39 9
Note, Thar the Height of che Stile FS
54 9
36
being equal unto MN in the former Dia-
69.9
54
gram, which is the Chord of zi d. 40 75. 3 84.9.
78. 57
now becauſe the Plane declines-Eaſt,
there 11 I
fore the Gnomon ſhall decline Weft: for
the Dial being ſuch a Projection of the Sphere, wherein all chęúſual Lines croſs in the
Nodus of the Gnomon, and thence diſperſe themſelves again cowards the Plane;
therefore that which is Eaſt in the Sphère, will be expreſſed Weft on the plane, and
contrarily, as was ſaid Chap. 2: Theorem 3. Alſo I conſider, that howſoever the
Plane be surned Eaſt or Weſt, the Gnomon place is fixed, becauſe it is a part of he
Axis of the World, or a Lind Parallel to ſite Now"therefore I turn a South Dal,
and make him decline Eaſt, and hold the Gnomonlunmovable, clië Weft ſide of the
Dial will approach ncerer to the Gnornan, as reaſon and ſenſe will require. Likewiſe
the Hours which are found on the ſame ſide of the Meridian or Noon-line with the
Sabſtile, muſt be for the ſame way with it from the Noon-line in the Dial.
And if you would draw the North Dial of this Plane, do but prolong thoſe
Hour lines, and the Subſtile upwards beyond the Center, and you have the North
Dial beyond C, or abovethe Horizontal Lincta Bs as the South Dial below it. And
note, Becauſe the Sun ſets after 8-á. Clåck in Summer, cherefore the three hours next
before and after midnight, may be lefy out in this Dial, and all others which muſt
ſerve in our Latitude.
This is the meſt
ready way to delincate thè oppofítc
. face' of any'Dial. Nire, That
if a Wall decline from the South Eaſtwards 31 deg. 30 min. therefore the Plane which
lieth 90 deg. from his Pole; is in the 32. Ažimuch from the Eaſt Northward.
Note this well: Extend the Compaſſes as before from K to N, che Interſection
of the Meridian
with the Line K Hac N; before found to be 2 ha. 36 min which
converted into Degrees, by allowing 5 Degrees to an Hout, and 4 Minutes
to a Degree, it makes 39 deg. 9 min. which 39. deg: 9 min. Thew me the Difference
of Longitude berween our Country and the Country of this Dial...
1
am
5
give you
che Diſtance before-39 deg: 9 min.
Nore, I allow this Countries Longitude to be 27 deg. 44 min. at Briſtol, to the
Eaſtward of the Grand Meridian Fowers and Calfs one of the Iles of Azores, which
added to 39 deg. 9 min. fhcws the Longitude of the Councry of the Dial to be 66 4.
53 min. Eaſtward, and Latitude 31 deg. 40 min. which I find by my Globe is in the
Delarts of Arabia at Afickia necr Soar.
1
.
1
CHA.
"
1
!
4
1
CHAP.XIV, The Art of DFALLING,
21
CHA P. XIII.
How to obſerve the Reclination or Inclination of any Plane.
W
Hit Reclination and Inclination are, hath been ſhewed Chap. 8. you
will have it following in a Diagram by it felf.
All Reclining aud Inclining Planes have their Baſes or Horizontal
Diameters lying in the Horizontal Diameter of ſome Azimuth; but the top of the
Plane leaneth back from the Zenith of your place in the Vertical of the Plane" (which
is the Azimuth cutting the Plane at Right Angles) ſo much as the Reclination hap-
nech to be: and the Polc of the Plane,on that fide the Planè inclines to is ſunk as much
below the Horizon, as the top of the Plane is funk below the Zénith; and the oppoſite
Pole is mounted as much.
Ler ESW N be Horizon, Z the Zenith, E W the Horizontal Diameter of the
Plane and of che Eaſt Azimuth, EOW a Plane not declining bue reclining Souch-
wards from the Zenith by the Arch Z O 45 deg. and his oppoſite Face inclining to the
Horizon according to the Arch OS 45 deg. the Pole of the reclining Face is at Pin
the Meridian CP, which here is alſo Vertical of the Plane, and is clevared 45 deg.
in the Arch NP, equal to the Arch of Reclination 2 O, the Pole of the inclining
Face is depreſied as much on the other ſide under the Horizon.
To find the Quantity of the Reclination, you ſhall draw a Vertical Line on the
Plane by Chap. 3.
and thereto apply a long Ruler, which may overſhoot the Plane
either above or below : to that Ruler apply any Semidiameter of a Quadrant, and
the Degrees, between char Semidiamererand che Plumb-line; ſhall be the Degrees of
Reclination. Or ſtick up in the Vertical Line two Pins of cqual height, and perpen-
dicular, and placing your ſelf cither above or below the Plane, as you find moſt cafie,
direct the sights of your Quadrant to the Heads of the two Pins, being in a righe
Line with your eye; and the Plummer ſhall ſhew che Reclination on the side of the
Quadrant, and the Inclination, which is always the Complement thereof, on the
other.
5
CHAP. XIV,
To draw the Hour-Lines in all Declining, Reclining, Inclining Planes.
I
1
F a Plane ſhall decline from the Prime Vertical, and incline to the Horizon, and
yet not lie even with the Poles of the World, it is then called a Declining, In-
clining Plane. Of theſe there be ſeveral ſorts, you may ſee 19 Planes in the
following Diagram, and directed how to know them in the 25th Chaptır: but to
fhew the Recliners in order as they come, viz. the North Recliner 45 deg. and South
Incliner falls between the Æquator and the Tropick of %, as the Circle ÉQW;
che
Souch Recliner 45 deg. and North Incliner, falls between the Horizon and the Pole,
and is repreſented by the Circle E A W. The Eaſt and Weſt Recliner and Iucliner
45 deg. may be ſeen by the Circle NVS, the Inclination may be Northward 45 deg.
and declining 45, as the Circle FKC, or the Plane may decline S 45 Weſtward, and
recline 45 deg, from the Zenich, asche Circle CLF. If your Inclination or Reclina-
tion fall more or l-fs
, you may ſee the way. Each of theſe Plancs have two Faces, the
upper toward the Zenith, the lower towards thé Nadir, wherein having the Latitude
of the Place, and the Declination, with
the Inclination of the Plàne, you are farther
to conſider what muſt be found before you can draw the Dial, which will follow in
order, and is repreſented in this Fundamental Diagram; only I will mention the
Arches and Angles in the hardelt, which is a South declining Weſt
45 deg. and recli-
ning from the Zenich 45 deg. In ſuch you muſt conſider,
Dddd
1: The
1
1
www
1
!
22
The Art of DI A L'LING, Book VII.
1. The Arch of the Plane berween the Horizon and Meridian CO or FO.
2. The Arch of the Meridian berween the Horizon and the Plane ON or Os.
3. The Angle of Inclination between the Meridian and the Plane CON or FOS.
4. The Subſtile-diſtance from the Meridian OR or OR.
5. The Height of the Pole above the Plane or Stiles Height P R.
6. The Inclination of both Meridians or Angles at P.
7. The Difference of the Hour from the Subſtile. But firſt we will deſcribe the
Diagram.
1
hie
1
S
t
80
go
018
112
09
2
mo
40 od slo
of 액
​&
.
K
Ho
1
to
E
+
W
Hi
R
01
20
O
th
...
of a of
05
112
70
80
ok
9)
018
°
N
To Deſcribe the DIAGRAM.
!
.
+
He Deſcription of this Diagram is ſet down at large by that worthy Mathematici-
au Mr. Edmond Gunter, in the use of his Sektor, Chap.3. But for this
fuffice, if it have the Vertical Circle, the Hour Circles, the quacor, and the Tro-
picks firſt drawn in it; other Circles may be ſupplied afterward, as we shall have ulc
of them ; and thoſe may be readily drawn, as I have borrowed of liin, in chis
.
manner.
Lec the outward Circle, repreſenting the Horizon, be drawn and divided into
four equal parts, wich S N the Meridian, E W the Vertical, and cach fourth part
into 90 deg. That donc, lay a Ruler to the Point ș, and each Degree in the Qua-
drant EN, and not the Interſc&tions where che Ruler croſſeth the Vertical ; ſo thalli
the Semidiameter EZ be divided into other 90 deg. and from thence the other Semi-
diameters may be divided in the ſame fort; thoſe may be numbred with 10, 20, 30,
from E towards C; and for variecy, with 10, 20, 30, from C towards W! But for
the Meridian, the South part would be beſt numbered according to the Declination
from the Equator, and the North part according to the Diſtance from the Polc.
Then with reſpect unto the Latitude, which here we ſuppoſe to kc si deg. 30 min.
open
1
a
X1V
CHAP.X.V. The Art of DIA LLING:
23
+
!
open the Cornpaſſes unto 38 deg. 30 min. from C toward W, and prick them down
in the Meridian from Cunto P ; ſo this Point Pfhall repreſent the Pole of the World,
and through it muſt be drawn all the Hour-circles,
Having three Points E PW, find their Centers, which will fall in the Meridian,
a little without the Point S, and draw them in a Circle EPW, which will be the
Circle of the Hour of 6.
Through chis Center of the Hour of 6 draw an occult Line at length parallel to F
w, to this Line ſhall continue the Centers of all the other Hour-circles: where the
Circles of the Hour of 6 crofſeth chis occult Line, there will be clic Centers of , and
3 their Hour-circles.
The diſtance between theſe Centers of 9 and 3, will be equal to the Semidiameters
of the Hour.ircles of soand 2: where theſe two Circles of 10 and 2 ſhall croſs this
occult Linc, there will be the Center of 7 and s. And again, take with
your
Com-
paſſes off che Diameter E W 75 deg. under w, and turn the Compaſſes three times
over on the occult Line, from the Center of the Hour of 6, and you have the Center
of the Hours of 1 and 11. Again, take 78 deg.from E cowards C, and lay ic boch
ways from the Center of the firſt Hour-circle of 6, on the occult Live, and you have
the Center of the Hour-circle of 4. So practice for any other Latitude,
The Hour-circles being thus drawn, take şi deg. 30 min. from C toward w, and
prick chein down in the South part of the Meridian froul Cunto A, and bring the
third Point E A W into a Circle; this Circle ſo drawn ſhall repreſent the Æquator.
The Tropick of Cancer is 23 deg. 30 min. above the Equator, and 66 deg. 30 min.
diſtant from the Pole; and ſo in chiš. Lacicude it will cross the Souch part of the Me-
ridian at 28 dég. from the Zenith, and the North part of the Meridian 15 deg. be-
low che Horizɔn. Take therefore 28 deg. from C towards W, and prick them down
in the Meridia’ı from C unto D, ſo have you the South Interſc&tion: Then lay the
Ruler to the Poin: W and 15 deg., in the Quadrant NW, numbréd from N toward
W, and note where it croffech the Meridian
; lo ſhåll you have the
North Inirerfect:
on. The half way between theſe two Interſections in the Meridian Line is thic Center
of the Tropick of Cancer: Which being truly drawn, will croſs the Horizon, before
4 in the Morning, and after S in the Evening, about 40 deg. Northward from E and
W, according to the riſing and ſetting of the Sun at his entrance into Cancer.
The Tropick of Capricorn is 23 deg. 30 min. below the Æquator, and 113 deg.
30 min. diſtant from the North Pole; ſo that in this Latitude ir croſſech the South
part of the Meridian at 75 deg. from the Zenith, and the North part of the Meridian
at 62 deg. below the Horizon. Take therefore 75 deg. towards w, and prick them
down in the Meridian from Z unto 19; ſo have you the South Interſection : Then lay
the Ruler to the point w, and 62 deg. in the Quadrant NW, numbred from N to-
wards w, and note where it crofſeth the Meridian, ſo ſhall you have the North In-
terſection: the half way thall be the Center, whereon you may deſcribe the Tropick
of Capricorn vg. This Tropick will croſs the Horizon after 8 in the morning, and be-
fore 4 in che Evening, about 40 deg. Southward from E and w, according to the
ring and ſetting of the Sun at his entrance into Capricorn. Now we will proceed co
draw the Hour-lines in a North Recliner and a South Incliner, and ſhew the Height
of the Stile above the Plane on the Meridian, and a Souch Recliner, and a North In-
cliner; and ſo in order to the reſt.
i
CHA P. XV.
How to make a North and South Reclining Dial.
1
T
T
Hc Baſc or Horizontal-line of ſuch a Dial licch in the Eaſt Azimuch, and
his Pole in the Meridian, as you may ſee Chap: 14.
The Plane of the 14 Chapter was a North Plane reclining Southward
45 deg. the Zenith is diſtant from the North Pole 38 deg. 30 min. the Complement
Dddd 2
of
1
+
!
24
The Art of DIALLING, Book VII.
+
of the Latitude si deg. 30 min. toward the South, and I ſee che Reclination is 45 deg.
more Souchward, becauſe I ſee my Plane reclines ſo much that way. I add the com-
plement of the Latitude 38 deg. 30 min. and the Reclination 45 deg, together, and I
ſee then by the ſame the North Pole is elevated 83 deg. 30 min. which is the Height
of the Stile above che Plane on the Meridian; which 83 deg. 30 min. taken off the
Gnomon Line of the Scale, you may proceed and draw the Dial, and lay that on the
Line W E, and work as you did in the other Dials.
The oppoſite face to this is the South Incliner ; and if you would draw it, do but
prolong thoſe Hour-Lines, as was ſaid in Chap. 13. and you have the South Recliner
below the Horizontal Line WE.
Note, Had this Reclination been gi deg; 30 min. and che Complement of the La-
titudc added to it would have made go deg.chen it would have fallen into the Plane of
the Aquinoctial, and ſo the Dial would have been a Polar Dial, and all the Hours
would have had equal ſpaces, and the Gnomon would have ſtood perpendicular,
which are the properties of a Polar Dial, as hath been ſhowed Chap. s.
-
ܙܝܙܗ
1
F
ALE
6
E B
s
VIZ
g
IX
X
XI XI
I
T.
d.
As for the South reclination, which is 45 deg: which ſubftract from the Latitude
5l 30
si deg. 30 min. and you have the height of the Gnomon or Scile above the Plane,
45
oo which by reaſon the Hour-lines will be lo neer together, continue them, and cut them
off as they may fit your Planc, by leaving out
one of the Hours, or more, as you
6 30
will; ſo will you have the Stilc a handſom height above the Plane, and the Hour-
circles a good diſtance alunder: But for the North Incliner, his Lines and Stile mult
be drawn from the Center of the Plane, although they do come neer. If the Recli-
51 30 nation had been 31 deg. then you ſhould have ſubſtracted the Reclination from the
Latitude s ! deg: 30 min. and the Remainder would have been 20 deg. 30 min. the
30 Height of the Stilc or Gnomon ,
d.
31
CHAP.XVI The Art of DIALLING.
25
C H A P. XVI.
How to make an Eaft or Weſt Reclining or Inclining Dial.
A
S it hath been ſhewn Chap. 15. That the Baſc or Horizontal-line of a South
Recliner lieth always in the Eaſt Azimuth ; ſo the Baſe of an Eaſt Recliner
lieth always in the Meridian of the Place: And as all Declining Planes lic
in ſome Azimuth, and croſs one another in the Zenith and Nadir, by Chap. 13. So
theſe Reclining Planes lic in ſome Circle of Poſition, and croſs one another in the
North and South Points of the Horizon ; which being conſidered; theſe Eaſt Recli-
ners, Weſt Incliners, and Weſt Recliners, and Eaſt Incliners, ſhall be made as eaſily
as the former.
For theſe Eaſt Recliners be in very deed South Decliners to thoſe that live ga deg.
from us Nortliward or Southward, and liave one of thoſe Poles elevated as much as
the Complement of our Latitude; for the perpendicular or Plumb-line of thoſe peo-
plc is parallel to the Horizontal Diameter of our Meridian.
1
EXAMPLE.
I
Have an Eaſt Plànc reclining 45 deg. which I would make a Dial.
In the former Diagram I number 45 deg. from E to F, and then lay a Ruler
from N to F, and it will cut the Semidiameter Z W in 45 deg. in V. And then draw
the Arch SVN, which Circle ſhall repreſent the Plane propoſed.
Then thic Arch of the Planc bcoween the Horizon and the Subſtiler Diſtance is re-
preſented in the Diagram by NQ and may be found by reſolving the Triangle ON
P, wherein che Angleat Q is known to be Radius, and the Angle at N to be Recli-
nation, and the Angle ac P the Latitude: Then work thus,
As the Radims or Sine of 90 deg. l-
-1000000
Is to the Sine of Reclination 45 deg. N-
984948
So is the Tangent of the Latitude si deg. 30 min. PN-
1009939
To the Tangent of the Subftile QN 41 deg. 38 min. 994887
.
Or upon Gunter's Ruler, Extend the Compaſſes from the Sine of go deg. to che
Sine of 45 deg. the ſame will reach from the Tangent of the Latitude 51 deg.
30 min. to ncer41 deg. 38 min. as before, in the Line of Sines; and ſuch is the Subſtiler
diſtance.
Secondly, The Height of the Pole above the Plane may be repreſented by the Arch
P Q and may be found, by which we have given in the Triangle QNP: For,
As the Sine of 90 Q-
I000000
To the Sine of si.deg. 30 min. PN-
989354
So is the Sine of Reclination 45 deg. N.
984948
:
the Plane P
1
+
Extend the Compaſſes from the Sinc of 90 deg. coche Sine of șr deg, 30 min. the
ſame Extent will reach from the Sinc of Reclination 45 deg. to 33 deg. 36 min. as bo-
fore, which is the Height of the Stile..
Thirdly, The Inclination of Meridians (or indeed you may call ic Longitude) is
here repreſented by the Angle P QN; for having drawn the Arch of the Meridian
of the Plane SON, or let fall a Perpendicular PR, and that from the Pole unto
the Plane, this perpendicular ſhall be the Meridian of the Plane; ſo that from Q to
N is the Diſtance of Inclination of both Meridians, which will be found as before :
For,
As
!
t
26
The Art of DIA LLING. Book VII.
989354
1000000
As the Sine of si deg. 30 min. P N-.
To the Sine of 90 deg;
So is the Sine of the Subſtiler Diſtance 41 deg. 38 min.
To the Sine of Inclination of both Meridiassa
971594
which will be found to be s8 deg. 40 min. N PQ.
Extend the Compaſſes from chic Sine şi deg. 30 min. to the Sine of 90; the ſame
Extent will reach from 41 deg. 38 min. the Subſtiler Diſtance, to 58 deg: 40 mix and
furch is the Angle PQN of the Inclination, berween the Meridian of the place and
thc
proper Meridian of the Planc: which reſolved into time, doch make about 3 ho.
54 min. and ſo the Subſtiler muſt be placed neer the Hour of 8 in the morning.
to
III V
V
VEL
1
K к
1.
1
UUED
I HR
S
3
포
​X
1
X
1
1
1
VI V V MIUI
TU
..
1
:
For to draw the Hour-lines on the Plane, firſt draw the Horizontal-line SN: Then
take off your Line of Chords with your Compaſſes the Chord of 60 deg. from your
Scale, and (wcep the Semicircle: Then cake off your Linc of Chords with your
Coin-
paſſes the Subſtiler Diſtance, and lay it from N on the Arch to A; Then draw through
the Center aud A in the Arch, the Subſtiler Line, croſſing it in the Center at Right
Angles with the Line KF: Then cake off with your Compalies the Height of the
Pole above the Plane, or the Sciles Height 33 deg: 36 min. from the Gnomon Line of
the Scale, and lay it from the Center of the Dial both ways from K to F. Then take
the whole Linc of 6 Hours, and ſweep the two ſmall Archies from K towards G, and
che like from F toward G? Then draw the Lines KG and FG: Then extend the
Compaſſes from G to N, and apply it to the Hour-line, and you ſhall find the Indi-
nation of Meridians to be as beforc'; ho. 54 min, Then take 4 ho. 54 min. and lay it to
O, and the ſame Diſtance from K unto o toward G; and the like do with 2 bo.
54 min. and make ſuch marks on the Line G F and KG as you ſcc in chc Figure to
1
draw
)
A
CHAP.XVI), The Art of DIALLING.
27
draw the Hour-lines by; and then take off the Line of Chords 33 deg. 36 mir. the
Stiles Height, and lay from A to B; ſo drawing the Hour-lines, and you have done.
And then you may ſee as in a Glaſs the Weſt Recliner, the oppoſite Face, as you were
Thew'd before Chap. 13. that is, ſtrike the Subſtiler Line, and all the Hour Lines
through the Center, and the ſame Figures to every Hour beyond the Center, which you
had on the firſt fide, and ſet the Gnomon upon the Subſtile downwards, to behold
the South Pole, and you have done both : So have you on the back ſide, looking
through the Paper, the Weſt Recliner and Eaſt Incliner, if you draw in the like
manner, or prick on the back ſide, for 11 in the Eaſt I in the Weſt Recliner, and ſo
contrarily of the reſt.
C H 4 P. XVII.
How to find the Arches and Angles that are reqaiſite for the making
of the Reclining Declining Dial.
Efore you can intelligently make a Reclining Declining Dial, which is the moſt
irregular of al, having two Anomalies, viz. Declination and Reclination,
you
muſt be acquainced with thoſe chree Triangles in the Sphere, wherein cer-
tain Arches and Angles lie, which are needful to be known. I adviſe you firſt to draw,.
though it be buut by aim, an Horizontal Projection of the Sphere, ſuch as here I have
drawn for a South declining Weſt 45 deg. and reclining from the Zenith 45 deg. in
the Latitude of si deg. 30 min. which thall be our Example. The ſame alſo is thewed
in the Fundamental Diagram; only I ſhow you this, to let you know, therc is ſeve-
ral ways to the Wood bclides one.
1
N M
1
5
Æ
1
PIA
P
E
H
.
1
mis
+
Q
Lu
1
ܪܙ
1
28
The Art of DIALLING. Book VII.
2
95; the Jame Extent will reach from the Co-
In this Figure the Arch FL Cisthe Plane, Z L the Reclination thereof, F E the Baſe
or Horizontal Line of the Plane,and Æ n the Vertical of the Plane,cutting it right at L,
and cutting the Pole thereof at H: for n is the Pole of a Plane erceted upon F E; bue
the Pole of the Reclined Plane FLE is Hng; or Sn is the Declination of the Plane,
PHm the Meridian of the Planc, cutting the North Pole at P, the Plane at Right
Angles at R, and the Pole thereof ac H.
In the firſt Triangle FN () you have given F N 45 deg. the Complement of the
Planes Reclination N, the Right Angle of our Meridian with our Horizon F, che
Complement of Reclination 45 deg. whereby you nday find FO the Oblique Aſcen-
fion, or the Arch of the Plane becween the Horizon and our Meridian, that is, how
many Degrees chc Noon-line ſhall lie above the Horizontal-line. Alſo you may find
NO the Perpendicular Altitude of the Noon-line, or the Inclination of the Noon-
line of the Dial to the Horizon, which caken out of si deg. 30 min. remains the
Arch of the Meridian betweep the Pole and the Plane. But note, That when this
Alcitude of the Noon-line NO is equal to NP the Elevation of che Pole, then is the
fecond Triangle PR O quite loft in the Point P, and the Plane becomech then a De-
clining Æquinoctial Plane: Alſo you may find the Angle at O, called the Angle of
Inclination between the Meridian and the Plane. In the ſecond Triangle ORP you
have given O as beforc, R the Right Angle of clie Plane with his Meridian, O P'the
Poſition Laticude, that is, the Latitude of the Place wherein the Reclining Plane
ORLEQ_ ſhall be a Circle of Poſition; this is given if you ſubſtract NO, the Al-
titude of the Noon-line, as before, out of the Latitude ; and hence may be found
OR the Declination of the Gnomon or Subſtiler Diſtance, or Diſtance of the Meri-
dian of the Plane from the Meridian of che Place ; Rathc Elevation of the Pole
above the Plane, in the Plancs own Meridian or Sriles Height 3 P the Angle be-
tween the Meridian of the Plane, and the Meridian of the place. This Angle is
called the Difference of Longitude, becauſe it ſhews how far the Places are diſtant
from us in Longitude; wherein this Dial ſhall be a Direct Dial, without Declina-
tion, having his Gnomon in the Noon-line of the Place, and thews alſo how many
Degrees of thc Planc comes between the ſaid Meridians. Let this be well obſerved by
Learners.
Hence may be found, if you will, the phird Triangle PZH: You have given P Z
the Compleinent of our Latitude, ZH the Complement of the Planes Reclination,
Z the Supplement of the Planes Declination.
Alſo hence may be found H Pixwhoſe Complement is P Riche Elevation of the
Pole above the Plane, the Difference of Longitud, H, whoſe meaſure is R L, the
Arch of the Plane between the Méridian of the Plane or Subſtile, and the Vertical
Line of the Plane; the Complemerit thereof is R F, the Subſtiles Diſtance from the
Horizontal Line of the Plane,
Every Arch and Angle is given and may be found by the Problems of Spherical
Triangles, as before; but we will make ſhort our buſineſs.
Firſt find the Arch of the Plane between the Horizon and Meridian Fo.
As the Sine of 90 N-
1000000
To the Sine of Reclination for the Zenith F 45
984948
So is the Co-tangent of Declination 45 NE
-1000000
To the Co-tangent of F0°36. 16
984948
taken out of 99, there remains for FO 54.44. which is the Arch of the Plane be-
tween thc Horizon and Meridian.
tangent 45, to 35. 16, as before, by the Ruler.
Secondly, The next to find is the Arch of the Meridian berween che Pole and the
Plane; but we vvill by oppoſite Work find it thus,
As the Sine of the Angle at N 90-
1000000
To the Sime of his oppoſite Side FO 54.44--
991194
So is the Sine of the Angle at F 45
984948
To the Sine of the Side NO-35 deg. 16 min.
976142
Or,
:
!
1
*
CHAP.XVII. The Art of DIAL LIN G.
29
Or, Extend the Compaſſes from the Sine of 90 deg. to che Sine of 54 and 44 ;-the
ſame Extent will reach from the Sine of 45 deg. to 35.16 NO, which Subſtrated
from the Latitude si deg. 30 min. remains the Arch of the Meridian between the
Pole and the Plane O Pio deg. 14 min.
Thirdly, Now you muſt find the Angle N OF, which will be ROP, which is
called the Angle of Inclination betwecn the Meridian and the Plane, thus :
1
As the Sine of FO 54. 44
008805
To the Radius or Sine of go
1000000
So is the Sine of ihe Side FN 45-
984948
To the Sive of the Angle ROP 60. O
993753
Extend the Compafles from the side of 90, coʻche:Sine of 54:44; the ſame Ex-
tent will reach from the Sine of 45, unto 60, as before-found, the Angle of Inclina-
cion berwveen the Meridians and the Plane.
Fourthly, The next to be found is the Subſtile Diſtance, or the Meridian of the
Plane from the Meridian of the Place.
As the Sine of the Angle at R 90
1000000
To the Co-fine of the Angle at O 30.0-
269897 !!
So is the side of the Tangent PO 16.14
946412
To the Tungerit of the Sulfiile OR 8 deg. 17 min.
916309
And thus extend the Compaſſes from the Sine of 90 deg. to the Co-line of 30 deg
the ſame Diſtance will reach from the Tangeas of 26714, to the Tangent of the so
stile from che Meridian 8 deg. 12 min.
Fifthly, Next find the Elevation of the Polda bo
above the phone of Scilés Height : Por
As the Sine of so deg. at R
To the Sine of his Oppoſite PO 16. 14...
944645
So is the Sine of the Angle at O 60
993752
To the Sine of the Stiles Height above tht Planë 14. I 438398
Or, Extend the Compaſſes from the Sine of godeg, to the Sine of 16. dég. 14 min
the ſame Diſtance will reach froin the Sine of 60 сhe Angle at 0, to the Height of
the Pole above the Planc 14 deg. I min. or Stiles Height, as before.
Sixthly, Now for the Difference of Longitude or Angle ac-P, or Inclination of
thc Meridian of the Plane to the Meridian of the Place, it is thus.
As the Sine of the Arch PO 16. 14
944645
Is to the Sine of 90 deg. R-
1000000
So is the Sine of the Subſtile Diſtance 8 deg. 19-
915856
To the Angle as Por Difference of Longitude 3 i deg. 1 min. +971211
Excend the Compaſſes from the Sine of 16 deg. 14 min. to the Sind of 90 deg. the
famc Excent will reach from the Subſtile Diſtance 8 deg. 17 min. fo the Inclination of
boch Meridians 2 ho. 4 min, as before.
Seventhly, The Diſtance of the Hours from the Subſtiler are here alſo repreſenced
by thoſe Arches of the Plane which are interſected between the propěr Meridian and
the Hour-circles. The Angle ac R, becween the Pole and the proper Meridian, is a
Right Angle;
che Side R P is the Height of the Pole above the plants and then the
Angles at the Pole between the proper Meridian and the Hour-cistles may be gathered
into a Table: For,
As the Sine of go R,
To the Sine of the Height of the Pole above the Plane 14. 1.
So is the Sine of the Angle at i be Pole 16. 14.
To the Sine of the Hoxr-line from the Sabftiler 3 deg. 50 min.
Excend the Compaſſes from the Sine of go, to the Sine of 14. 1; the ſame Excent
will reach from 16.34, o 3 deg. so min.
The
Ееее
30
The Art of DIAD ZING. Book VN.
ZIZ I OJ
The Arch of the Plane between the Horizon and the Meridian FO 54 deg. 44.
The Arch of the Meridian between the Horizon and the Planc ON 35. 10.
The Angle.of Inclination becween the Meridian and the Plane FON 60.
The Subſtile Diſtance from the Meridian OR 8 deg. 17 min.
2: The Heighoof the Stile above chiePlane PR 14 deg. 1 mio.
The Inclination of both Meridians or AnglearP 31 deg. I min.
The Difference of the Hour from the Subſtile 3 deg. 50 min,
I
C-H A P.; XVIII.
1
11
is
i'!
فت:ر .
N
i Ham to dramaesthe Reclining Declining Dial.
Ore, You have got the chree principal Arthes for che drawing the Hours.
1, The Arch of the Plane between the Horizon and the Meridian.
54 deg. 44 min.
2. The Subſtilc Diſtance from che Meridian of the Place OR 8 d.17 m.
3! The Heighc of the Pole above the Plane or Stiles Heighị PR 140.10.
How to draw the South declining Weſt 45 deg, and reclining from the Zenith 45 d.
is thus: : Firſt draw the Horizontal Line Fe, with your Compafies cake off the Scale
and Line of Chords the Radius oo deg. and ſweep the Circle FQ;RP c; clien take
I
4
on VG
3:
.
6
7
1
1
IR
+
LE
AP
00
+ 1
II
F
Horizontal
e
Line
K
6
1
!
9
off the ſame Line of Chords 54 drg. 44 min. the Arch of.che Plarie between the Horia
200-and chc Meridian, and lay it from Fon the Arch to 0: Then take off
your
OF
Line
LI
--
3
CHAP.XIX, The Art of DIALLING,
31
1
of Chords 8 deg. 17 min. the Subſtile Diſtance, and lay it from 0 to R; then from
the Center draw the Subſtile Line CG; then take of the Quadrant go deg. of clie
Line of Chords, and lay it both ways from R to M and N, and draw that obſcure
Line; then take off che Poles Height above the Plane or Stiles Heighe 14 deg. I min.
from the Gnomon Line of the Scale, and lay it both ways from C coK; and then
with your Compaſſes cake off your Scale the whole Line of 6 Hours, and from K co-
ward'G, and from I toward G, ſtrike the owo ſmall Arches; then draw the Lines
KG and I G; then extend the Compaſſes from G to the Meridian Line ac L, and;
apply that Diſtance to your Scale and Hour-line, and you will find it to be 2 hv. 4 mi
which is the Inclination of Meridians or Angle at P, as before: Then lay that Diſtance
from L toward G, and take (as before directed in Chap. 17.) one Houir loſs, and lay
it off in the ſame manner from G toward K, and from I coward G; and the likedo
with 3 ho.4 mir. until you have put all the 6 Hours on the Line from Gco K, and
from I to G: Then draw the Hour Lincs according to your Plane, whether it be a
Triangle, or Circle, or Square, or what ſhape focver , then lay off the Height of
Stile 14 deg. I min. from R to P, taken off the Line of Chords, and draw it from
the Center, and cut it fit to your Plane in what thape you will, as you may fee in
the Figure, and your Dial is done. The oppoſite Face or Incliher to the Horizon is
but to continue the Stile and Subſtile and Hour lines tlırough the Center, as you may
fee, and you have it. In like manner you may draw the South Eaſt Decliner 45 deg.
and Reclining from the Zenith 45 deg. if you once draw on Paper this before, and the
Hour-lincs over the faine, and che like the Subſtile, and the Stilcro ſtand
uponi
the
Subſtile upright, as of the reſt: And let the higheſt. part of the Stile bë çowards the
North Pole, pointing upwards; and where chc Hour of 10 is in the South Weſt Re-
cliner, on the back fidc put z'a clock; and for at puc , and for 1 put 11, and
for 2 put 10, and ſo contrary all the reſt. And if you obſerve your Work, you
have the South Weſt and the South Eaſt Recliners, àud the North Weſt and Nortli
Eaſt Incliners; or you may draw them by what was given in the firſt, in the ſame
manner.
1
your
}
.
CHAP. XIX.
How to find the Horary Diſtance of a Reclining Declining Dial.
Y
Ou have ſeen Chup. 17. how caſily Eaſt and Weſt Reclining Dials are to
be made; and by the Figure in Chap. 18. how shey fall out to be Circles
of Poſición, os you may ſee by FOR CO.
I will ſhow you how all reclining Dials may be reduced to Eaſt or Welt Recliners,
for ſome Latitude or other ; and ſo the Hour-diſtance found by the Method of
Chat. I7.
The Circles of Polition, as have been ſhowed, do all croſs one another in the North
and South Points of the Meridian: Now therefore by the Poinc 0, where the Plane.
cuts our Meridian, draw a new Horizon, as O BQC, and then ſhall you
fee
your
Planc in chat Horizon to be a very Circle of Poſition.
But now we are gotten into a new Latitude OP, called before Chap. 18. the Pori-
tion Latitude; and we liave here a new Reclination: for wliereas this Plane reclineth
in our Latitude ZF L 45 deg. his Poſition Reclination is O, viz. ZOL or POR
68 deg. In the making of this Dial therefore you ſhall forget your own Laticus,
and the Planes Reclinacion in your Horizon ; and with this new Latitude and Reclina
tion inake the Dial after the manner of the Eaſt Recliner, Chap. 17. not regarding the
Declinacion at all: for the Baſe of chis Plane is now fallen into the Horizontal Lime of
the Meridian ; and his Declination being a Quadrant, he is become a Regular Plane,
and neither his Declination nor Reclination ſhall much trouble
you.
How to place your Noon-line from the Horizontal or Vertical Line of tlie Planc, you
have found alrcady.
Еece 2
Note,
1
1
+
t
1
32
The Art of DIALLING. Book VII.
Note, Your new Latitude is PO 16 d. 14 m. then you know your Plane is the 60 d.
Azimuch from the Axis, becauſe POR is 6o deg. as before 90 deg. farther from the
faid Azimuth you have chc Pole of the Plans, and therefore is the Meridian of my
Plane, and ſhall make che Subſtile of my Dial OPR 3 deg. 1 min. his Diſtance from
the Meridian of the Place in that Æquinoctial, and is therefore the Difference of Lon-
gitude, as before.
Then have you the Side P R the Height of the Stilc 14 deg. I min. or. Elevation of
the Gnomon, as before: Likewiſe have you OR the Declination or Subſtile Diſtance
from the Meridian 8 deg. 17 min. as before, and you may proceed to draw the Dial
in like manner as you have been directed in Chap. 16.
1
N M
PA DO
W
H 五
​1
1
0i
B
mas
E
CH Äri XX.
To draw the Proper Hours of any Declining Dial.
Very declining Plane, whether it recline or not, hach two great Meridians
,
much ſpoken of.
I The Meridian of the Plane, which is the proper Meridian of that Country
to whoſe Horizon the Plane lieth Parallel.
2 The Meridian of the Place, which is the Meridian of our Country in which your
ſet
up this Declining Planc, tó ſhew the Hours; and fo either of theſe Meridian
Dials
may be conformed.
How to draw the Hours of our Country on ſuch a Plane, is the harder work, be-
cauſe
money
1.
1
CHAP.XXI. The Art of DIALLING,
33
cauſe the Plane is irregular co our Horizon : yet I ſuppoſe I have made the way very
eaſie in the former Chapters
. But to draw the Hours of the Country to which the
Plane belongs, is moſt caſie; for if you take the Subſtiler for the Noon-line, and the
Elevation of the Pole above the Plane for the Latitude, you may make this Dial in all
points like the Vertical Dial, after the Precept of chap. 8.
CH A P2 XXI.
ܚܪ
To know in what
countrý awy Declining Dial fhall.ferve for å
Vertical
I
F the Dialdcoline Ealt, add the Difference of Longitude foundin. Chap. 13.0
18. to the Longitude of the Place, and the ſum of the exceſs above 360 is the
number of the Longitude fought. If the Dial décline Weſt, fubftract the ſaid
Difference of Longitude out of the Longiruịc of your Place, and the Difference is
the Longitude inquired: but whçói che Longijude of your placc happens to be leſs than
the Difference of Longitude, you muſt add có it 360 degan before you fubftract the
Difference of Longitude. Note, The Elevation of the Pole above clië Plane, or
Stiles Height, if the Latitude of the Place inquired.
Example. The Declining. Plane of Chap. 13. will be a Vertical Plane in the Lon-
gitude of 66 deg. 46 min. in the Døfartsok. Arabir nēer Zoar: and the declining rc-
clining Plane of Ghap. 18.& 2 parallel to the Horizon of thoſe that fail in Lon-
gitude 357 deg 41 min. and North Latitude 14 deg. that is, as the Terreſtrial Globes
and Maps Thew me, between Bonaviſta, one of the Cape Verd INànds, and Barbadoes.
+
CHA P. XXII.
.
How to find the Arckes and Angles which are requiſite in a North
Decliner Recliner, and a South Decliner Incliner.
1
I
14
Could not paſs by this Example of the North Recliner Decliner, and South In-
cliner Decliner, alchough it is ſhewed in the Fundamental Diagram; but it
may be too obſcure, and harder to be apprehended by the Induſtrious Practitio-
ner there;' therefore I would adviſe him to draw a Scheme of the Dial, as was ſhewed
Chap: 15.6 18. or draw the Fundamental Diagram in Paper, and with a ſmall
Needle prick the Hour-lines, Horizon, and Meridians, Æquinoctial
, and the Tro-
picks; and then you have a Figure ready to be ſtamped with a little Charcoal-duft
as often as you have occaſion: Or if you apprehend your Work in any manner, the
Figure following, or the like, may ſerve your curn, to ſhew you the Angles you are
to find, and Arches for the making of your Dial. I ſhall be short in this, and refer
you to chap. 15. a 18.
The Circle ESWN is our Horizon, as before ; NS our Meridian, FLC the
Plane, 2 L the Reclination thereof, F C the Baſé or Horizontal Line of the Plane,
ÆN the Vertical of the Plane, cutting it right at L, and cutting the Pole thereof ac
H: for N is the Pole of a Planc crect upon FC; but the Pole of the Reclining Plane
FL Ç is H; SE or n N the Declination of the Planc.
Now you ſee your three Triangles all adjoyning in this Scheme, viz. FSO and
OR P rectangled at Sand R, and PZ H obtuſe angled at Z.
It is truc, That the ewo first may do the Work, and ſo we will be brief, Obſerve,
you are to find as followech.
1. The Arch between the Horizon and the Plane FO
2. The Arch of the Meridian between the Horizon and the Plane SO.
co
3. The
1
국
​f
34
The Art of DIALLING, BOOK VII.
3. The Arch of the Meridian between the Pole and the Planc PO.
4. The Angle of Inclination between the Meridian an 1, the Plane FOS.
5. The Angle of Inclination becween both Meridians OP R.
6. The Subſtile Diſtance from the Meridian OR.
7. The Height of the Scile above the Plane PR.
1
$
I
F
Æ
8
A
W
R
***
8
11 M
7
Zr: a
I OF
N
1
1
?2
1. To find the Arch of the Meridian between the Horizon and the Plane, it is thus.
1
1000000
1
As the Sine of 90 at S-
1000000
To the Sine of Reclinatien at F 45
984948
So is the Co-tangent of Declination F S 45
To the Complement Tangent of FO 35 deg. 16 min.
984948
Or chus: Extend the Compaſſes from the Sinc of 90, to the Sine of 45; the ſame
Extent will reach from the Tangent of 45 deg.co che Sine of 35 deg. 16 min. as be-
fore'; which ſubſtracted from 90 deg. there remains 54 deg. 44 min. the Arch of the
Planc between thc Horizon and Meridian F O.
2. To find the Arch of the Meridian between the Polc and the Plane, firſt find
the Arch of che Meridian between the Horizon and the Plane So chus.
ind}
As the Sine of go at S
1000000
To the Sine of the Arch of the Plane between the Horizon and
991194
Plane FO 54.44
So is the Tangent of Reclination at F 45-
1000000
To the Tangent of the Arch of the Meridian between the Horizon?
991194
And Plabe SO 39 deg. 14 min.
Or thus: Extend the Compaſſes from the Sinc of 90, to the Sine of 54 deg. 44 ms.
the ſame Extent will reach. from the Tangent of 45 deg. to the Tangent of SO 39 d.
14 min. by the Tables found before ; which 39 deg. 14 min. taken out of 90 deg.
there
1
1
4
1
.
1
1
CHAP.XXII. The Art of Dia Z'LANG.
35
છે
A
...19
. I
!!!
T
: и
1
.
+
there remains so deg. 46 min. the Arch of che Meridian berween the Zenich and the
Plane OZ ; which being added to the Complement of che Latitude Z P, there will
be 89 deg. 16 min. for the Arch of the Meridian between che Pole and the Plane Pom
3. For the Anglobecween chc Meridian and skis Rļaue: F60SQËZOL, it is
As the Sine of the Arch FO 54 deg. 44 min.
T. 991 194
To the Radius or Sine of godeg. Sa
1000000
So is the Şine of the Declination FS 45 degos-
98494
To the Sine of the Angle at 0 60.0.
993754
Ór,
Or, by Gunter's Rule, Extend che Compaſſes from the Sinc of s4 deg: 44 min.
to the Sine of 90 deg. the ſame Extent will reach from the Slue'ofi
45 deg: 0;' co-ahe
Sic of 60 deg. o, the Angle between the Meridiaá' and the Plane. V * :.
4. Before we find the Inclinations of bochi Meridians, or the Difference of Longi-
cude or Anglc at P, we will find the Subſtiler Diſtánco O Rim...:1,"
*!1961C5
As the Sine of the Angle at Rgodega 1000000 mm
To the Co-fine of the Angie.at 0 60 deg. ?969897
So is the Tangent of the Side PO 89 deg. 1.6:min.
-118917013
To the Tangent of the Subfifile OR 88'deg: 32 min dahiij9r7652!.
Extend the Compaſſes from the Sine of 90 deg. to the Sing of 30 deg. the ſame
Excent will reach from the Tangent of 89 deg. 16 min. to che Tangent of the Subſtilo
88 deg. 32 min.
5. Now for the, Inclination of both Meridians, or Difference of Longitude, or
Angle at P, it is thuis. For,
As the Sine of the Arch PO 89 deg. 16 mi
999996
To the Rádius or Sine of 90 aiR
So is th: Subftiler Diſtance OR.88 deg: 32 min.
999985
To the Sine of the Angle of Inclinations bet pieen both Meridians
88 min. of Longitude
Which converted into Hours, by allowing 15 deg, to onc Hour, 4 Minutes to a De-
gree will
be s ho, 55 min. as you may find by the Line
of Inclinacion and Hours on
6. For the Height of the Pole aboyc thc Planes, or Stiles Height;
As the Radius, or Sine of godog,
1060000
To the Sine of the Arch PO 89 deg. 16'min.
999996
So is the Sine of the Angle at o 6odegoms
993753
To the Sine of the Stiles Height PR 60
993749
the Height of the Pole above the Plane or Stiles Height. And thus if you obſcrvo
what was ſaid before in the South-Decliners Recliners, and now for the North Decli-
ners Recliners Incliners, you have the Propoſitions for any ſorts of Reclining Decli-
ning Inclining Dials
, and how to find the Arches and Angles
, as before, fitting for the
making of them. Now we will proceed and draw the Dial.
I 000000
. 3. 999989
your Scale.
terpen
Goldeg.
1
}
CHAT
+
1
1
.
?.
なれい
​1
26
3
The Art of D. I\A EL TNG. BOÓK VÍT.
1
.
XXIII..
(
is no
I
A
Orain
CHAR.
Howo (to-drappi the Declining. Inclining Dial:
is wað faid-Tháp. 18. the itiree prišcipal Atches for draiving the Hour-lines
o'in a South declining Weſt and inclining to the Horizon, and a Norch Recli-
ncr Decliner, as before, or South Eaſt Incliner, or Nòrch Weſt Recliner, are
theſe
three.
1. The Arch of the Plane becween the Horizon and Meridian 54 deg. 44 FO.
2. The Subſtile Diſtance from the Meridian of the Place 88 deg: 32 O R.
3. The Height of the Polc above.che Planc, or Stiles Height 60 Heg. PR.
And ſo follow the Directions of Chap. 19. and di'aw che Dialas you ſee. In whic
fkapelocver the Plančiss proportion thé Hours and Subſtile to your Plane, and let the
Gromon or Stile ſtand upright on the Subſtile Line; and ſo have you the lower
Face
the South Welt Incliner to the Horizon 45 deg. Recliner the like North Face, and de-
clining:45 degras before, and your Dial is done.
Nocc, This
. South-Weſt Incliner-Recliner is parallel to the Horizon of thoſe that
live on the South Land in the South:Sea ar Térfà viſta Decleros, ivi Longitude 297,
and South Latitude 60.necr.che Scraights of Magellanicum.
1
1
.
I'.
uli
'urrains
31':
1
17 oci8 Trimine 2014
31 21
I
1
4
12!!
1
1
DES
P
4
1
91
1
1
F
0
0,7
12:01 1
1
e
f
Bi
-- I
Soi!?
t
61
ܘܐ
iix
R
B.
C
1
.
And the North declining Eaſt and Recliner will be parallel to the Colducks; nece
Tartaria, in Longitude 114 deg.and Latitude Norch. Go'deg. as the Globe thews me,
and I have ſhewed you how to know the like in Chap. 21.
And obſerys, The's ho. 55 min, or 88 deg. 45 Difference of Longitude, as before,
fhews that the Sun riſes and comes to the Meridian, or 12 2 Clock with us ac Briſtol,
Ś ho. 55 min. before it doch with thoſe Inhabitants in the South Sca, as before : And
on the contrary, the Sun is riſen or on the Meridian with the Coffacks to the Eaſtward
of us, the like time as the Sun riſes with us before ir doch with thoſe of TerroViſta
Decleron in the South Sca: So that you ſee of what uſe the Line of Inclination of
Meridians on your Scalc is, as likewiſe of all Declining Dials.
САР.
A
1,
1
:
1
I
*
7!!!
018
1
1
1
7
CHAP.XXIV, The Art of DIA LLIN G.
1
.
37
CHA P. XXIV,
:
How to know the ſeveral ſorts of Dials in the Fundamental Diagram.
T
Heſe ſeveral ſorts of Planes take their denomination from thoſe. Great Circles
to which they are Parallels, and may be known by their Horizontal and
Perpendicular Lines, of ſuch as know the Latitude of the Place, and the
Circles of the Sphere.
1. An Æquinoctial.Plane, parallel to the Æquinoctial, which paſſeth through the
Pojnes çf Eaſt and Weſt, being right to the Meridian, but inclining to the Horizon,
with an Angle equal to chc Complement of the Laticude; this here is repreſented by
EOW.
2.. A Polar Plane, parallel to the Huur of 6, which paſſech through the Polc and
Points of Eaſt and Weſt, being right to the Aquinoctial and Meridian, buc incli-
niig co thc Horizon, with an Angle equal to the Latitude; this is here repreſented by
EPW.
3. A Meridian Plane, parallel to the Meridian the Circle of the Hour of 12,
which paſlech through the Zenith, the Pole, and the Points of Souch and North,
being right to the Horizon, and the Prime Vertical; this is here repreſented by
SZN.
4. An Horizontal Plane, parallel to the Horizo11,, liere repreſented by the outward
Ciclo E SIN.
+
1
1
i
1
S
1
8
OZ
11
9
2
70
re
9
3
oft
4o 5o bolo
18
30
+
30
OK
po
2
1
:
a
th
1
lot
W Ꮃ
4
510
oft
R;
O
40
А.
som
4/0 *
20
of a
!
Hola
og
7
기이
​80.
08
1
9
N
15. A South and North erect dircet Dial,parallel to the Prime Vertical Circle, which
palleth through the Zenith, and the Points of Eaſt and Weſt in the Horizon, and
ffff
is
។
1
+
1
i
1
18
The Art of DIALLING. BOOK VIL .
1
is right to the Horizon and Meridian; tharis, makes Right Angles with them both:
this is repreſented by EZW.
6: A South Declining Plane Exftward is repreſented by B D.
7. A South Incliner and North Recliner is repreſented by E QW.
&i*Th¢Soutli Recliner and North Incliner is repreſented by EAW.
9. A Meridian Plane, which is the Eaſt and Weſt Incliners and Recliners, and
from the Zenith parallel to any Great Circle which paſſech through the Points of
South and North, being right to the Prime Vertical, but inclining to the Horizon;
this is repreſented by SÝN.
10. A declining, reclining, inclíning Plane, which is parallel to any Great Circle
which is right to none of the former Circles, bue declining from the Prime Vertical,
reclining from the Zenich, inclining to the Horizon and Meridian, and all the Hour
Circles; this may here be repreſented either by FLC or FK C, or any ſuch Great
Circles which país neither through the South and North, nor Eaſt and Weſt Points,
nor through che Zenith nor the Pole.
Each of thoſe Planes, except the Horizontal, and South ioclining 23 deg. hath two
Faces whereon Hour-lines may be drawn; and ſo there are 19 Planes in all. The
Meridian Plane you ſee hath one Face to the Eaſt, and the other to the Weſt: Re-
member, that it is an Eaſt and Weſt Dial. The other Vertical Plancs have one to
the South, another to the North; and the reſt, one to the Zenith, and another to the
Nadir. What is ſaid of the onc, may be underſtood of the other.
|
1
CH A P. XXV.
How other Circles of the Sphere beſides the Meridians may be projected
upon Dials.
T
!
4
1
He Projection of ſome other Circles of the Sphere beſides the Meridians
(chough it be not neceſſary for finding the Hours, yer) may be both an Or-
nament to Dials, and uſeful alſo for finding the Meridian, and
placing the
Dial in its due fituation, if ic be made upon a movable Body, as ſhall hereafter be
ſhewed,
The Circles ficreſt to be projected in all Dials for thoſe purpoſes, are, the Equator
with bris Tropicks, and other his. Parallels, which may be accoanted Parallels of
Declination, as they paſs through equal Degrees
, as every 5 or 10 of Declination :
Or Parallels of che signs, as they pals through ſuch Degrees of Declination as the Suu
declinch when he entrech into aoy Sign, or any norable Degree chcreof; or Paral-
lels of the Length of the Day, as they paſs through ſuch Degrees of Declination
wherсin the Sun increaſeth or decreaſeth the Length of the Day by Hours or half
Hours.
Alſo the Horizon, with his Azimuths and Almicancars, are as an Ornament to
Horizontal and Vertical Dials, and are likewiſe uſeful for projecting the Æquator
and his Parallels in all-Dials. My intent is to be brief in this Trcatiſe of the Furni-
ture here following, becauſe I will have a Preſident of ſome other Country Dials :
I ſhall therefore think it ſufficient if I ſhew you one way to furniſh any Dial with
the Circles of the Sphcre, leaving you to deviſc others which I could have thewn.
1
7
1
CHAP.
5
i
TI
!
1
CHAP XXVII. The Art of DIAL UNG.
99
1
CH A P. XXVI.
How to deferibe on any Dial the proper Azimuths and simicantATS.
of the plane.
11
F
។
1
Rom any point of the Gnomon taken at pleaſure les fall
' af Perpendicular upon
the Subſtile ; that Perpendicular thall be påre of chië Axis of the plane, and
ſhall be repured Radius to the Horizon of
che Horizon of your Planc. The top of this
Radius in the Gnomon is called Nodms, becauſe there you muſt ſet a Knot, Bead, or
Butron, or elſe cut there a Norch in the Gnomon, co give ſhade; or cut off che Gno-
mon in the place of the Noduso-that the end máy give the ſhadow for choſc Linca-
ments. Let ilot your Nodus tand too liigh above the Plane, for too great a parc of the
Planes day ; nor Icc ic ſtand too low, for then che Linçaments will run too cloſe
together : a mean muſt be choſen.
Ac che foot of this Radius cake your Center and deſcribc;á Circle of the Plane, Tropicks and
and divide it into equal Degrecs,., and from the Center draiv Lines through thoſe other kircles of
Degrees infinitely, that is
, ſo far as your Dial Plane will bcar; cheſc Lines ſhall be Declization is
the Azimuths of the Horizon of the Planc, and ſhall be numbred froņ'hiş Meridian Shewed in chap-
or Subſtilc.
3. .
Divide any of theſe Azimuth Lincs inco Degrees, by Tangents agreeable to the wbere they be
ſaid Radius ; and having made a prick ac every Degree, through every of theſe pricks calculated and
you ſhall draw parallel Circles
, which ſhall be Almicàntars or Parallels of Alcitude, drama by belo
to be numbred inwards : ſo that at the Cencer bé go for the Zenith, and from che of 4 Scale of
Center outwards you ſhall number 80, 70, 60, until you come wichin 10 or 5 deg. Equal Parts.
of thic Horizon for the Plane is too narrow to receive its own Horizon, or chic pa-
rallel neer, if the Nodus have any compcrent Alcicude.
1
noctial Diil,
1
0
+
CHA P. XXVII.
1
Horo to deal with thoſe Planes where the Pole is but of ſmall Elevation,
and how to enlarge the stile thereof.
1
S
1
L
1ml
Uch Planes whoſe Sciles or Gnomons lic low, cannot have their Hour-lines di-
ſtinctly ſevered, unleſs the Cencer of the Dial-beplaced-oue:of che-Plage, as
you may,ſce in Chapter 18. Now to, inlarge the Stile in ſuch Dials, there
are two Lincs or Scales in the large-Mathematical Scalej which I call Polar.or
Iangent
Lincs.of cwo ſeveral Radius's, the larger of them marked thus *, and the leſſer of
them marked ng as you may ſec in die Second. Book, :Chapi 3: -The-uſe whercof I
thall here ſhew you.
In Chapter 18. of chis Book there is deſcibed a Planc char declines from the Soul,
Weſtward 45 deg, and'reclitics. from.che Zeriich :45' deg. Nộto fùppoſe is
Were re-
quired to inlarge chat Stilc and Dial.
:.:
Firſt you thall find
that there is given the Arch of chie Plásič bietween the.Horizen
and the Meridian, 54, deg. 44 min: Secondly, The Slıbftiler
Diſtance from the Mezi-
dian!" ro'chie Subftite is
. & deg. 1.7min. Thirdly: The Height of the Pole above clie
Plane is sadegh i mie Fourchlys
The Difference ofí Longirude or Inclination of Me
ridiánfis 31 degimin, as you may fee in Chap.1; Thele Being found, yog inuſt
chus proceed in the delineation of your Dial...
Firſt with your Compafles cake from off your Scale a Chotd of 60 deg: and, on
the Center Cfwcapuche Arch H L R:P; and draw CH che Horizontal Line blindly:
Then from H fec; off the ſeveral Diſtasices before found, upon this Arcli of a Circle
viz: Firſt, H L 54 deg: 44 min. for the Arch of the Plané berwésii clic Horizon and
the Meridian. Then (čc off L R 8 deg. 17 min. for the Diſtancdf- the Subſtile from
Efff 2
clic
1
.
I
4
7
si.
1
1
1
The Art of DIAL EST-N G. Book:VII.
1
1
1
women
I
the Meridian, and there draw chc Line of the Subſtile CRB. Thirdly, From this
Point R ſec off RP, which is 14 deg: I min. for che, Height of the Srile above the
Subſtile, and draw che prick'd Line CPK for the Stile or Axis.
Now this Stile being but ſomewhat low, for the enlarging hereof, firſt chuſe
, ſome
convenient place in your Subſtiler Line, as in chis Example at B, and there draw the
Line FB A ſquire wiſe to the Subſtiler Line. Then take with your Compaſes from
off your larger Polar Line marked the Diſtance of three Hours, and prick it down
in this Line from B to I; then from this point I take with your Compaſſes the
ncerelt diſtance to the Line of the Scile, and with that diſtance draw the Line It pa-
rallel to the Stile, and this will be the Line of the Stile inlarged.
1
4
+
1
3
;
1
I
I
A А
12
3
1
G
+
Substile
Stile
Stileinlarged
I2
F
T
oly
1
10
110
ER
Il
19
JY?
.
TH
1
ma
K.LT
der's
101
mi
venit
22.
bang
IVIA
+
1
L
Now co ſet on the leſſer Polar Line; char ſo you may draw the Holir-libes, firſt take
from off your lefſer Polar Line marked mi, che diſtance of 3. Hours, and with chac
diſtance draw a little white Line parallel to the Line of the Stile, and note where it
cuts the Line of the Stile inlarged, which is in the Point R. Mark this point R well,
and through this. Point R draw the Line O G parallel to the former Polar, Line
FBĄ. And now if you raķe th; diſtance ER in this Line, and meaſure it on your
Scale, if you have done your work right, you shall find it juft equal to 3 Hours
upon the leſſer Polar Linc. And now by theſe two Polar Lines you muſt draw the
Hour Lincs after this manner.
Firſt you muſt conGider the Inclination of the Meridians, which in this exampsaisi
31 deg. 1 min. which reduced into time, is 2 ho. 4 min: or 2 of the Clock and 4 mines
in the afternoon, from whence you may frame a Table for your direçtiğii in plăcing
the Hours, after this manner.
Now
1
1
*10.2) :
.
.
1
1
1
GHAP, XXVIII. The Art of DPÅ LITIN G.
141
1
XII
2
I
II
Now according to this Table, take the diſtance of theſe
Hour-lincs firſt out of your larger Polar Ling %, and The Homr-(Tb:di
prick them down in the Line FB A, from the Point B
lines of the from the
cowards F and AB Thus the Line of II muſt be ſec
Plane.
Subpild
4 min, from the Subſtile B towards F; the Line of I
Ho.
muſt be ſer i ho. 4 min. from the Subſtile B toward F;
Mi.
the Line of XII muſt be ſer 2 ho. 4 min. from B cowards X 4 F 4
F. And ſo on the other ſide of the Subſtil, thc Line of
XI
3
4
III muſt be ſer 56 min. from the Poinç B toward A; the
41
Line of IIII muſt be ſet 1 ho.56 min. from B toward A.
I:
4
And ſo you muſt do till you have let down all the Hour-
the Linc FBÁ, taking them out of the larger
Subſtile
B:
Polar Line upon your Scale.
..:50
Having chus ſet down the. Hour-lines in the Line
IIII
$ 6
V
FB A, you muſt let them allo in thc Linc O EG, taking
56
their Diſtances off from your lefſer Polar Line , as be-
VI 3 A
56
fore you did from the larger Polar Line, making uſe of
the Table to direct you, as before.
sil
And thus having made marks for the Hour-lines in both cheſc Tangent-lines; yok
muſt draw the Hour-lines through theſe Marks; and ſo ſhaping your Dial into a
Triangls, Square, or Circle, as your Planc will beſt allow, number the Hour-livies
with their proper Figures, and to finiſh your Dial, as the Scheme will direct you
better than many words.
:D
ork's
lines upon
I
1
Į
i'
1
...1: vich
!!11 goly
CH-Ä Pi XXVIII.
Ariother Example, How to Inlarge the Stile in a South Dial, reclining
45 deg. from the Zenith Northard.
1
8
9
10 II I2 I 2, 3
i
ia
Ele dati 01:1
tror
.:
F
q
A
016
ir
.
skile
A Silbs file
* รร'
Llity."
!
زر
enn
1
sfilemnlarged
6:31
../i16.
??
111:
i
F
öl
Linieves
.
E
G
1
min
7
4
O
29
ܕ.
1
+
Qi:
3
HE Stile or Gnomon in this Example is very low, lying véfy sheer toth Pole
of the World, as you may fee before in the fundamentai Diagram. This
Dialionly rášlíñics from the Zénith ; and thercfore to know the Sciles Height,
you
17
w...
1
1
}
1
។
42
The Art of DIA LLING BOOK VII.
you need only ſubftract the Reclination from the Lacicude of the Pole; which being
58 deg. 30 min. she Reclination 45 deg. ſubſtracted out of it, chere remains 6 deg.
30 min. for the Height of the Stile.
This Dial hath no Declinacion, and therefore the Subſtile muſt be the Meridian-
lino; draw thar-Line cherefore firſt about the middle of the Plane, and then with a
Chord of 60_deg, deſcribe a ſhort Arch RP and from R prick off 6 deg. 30 min.
according to the Height of the Stile before-found, and thereby draw the pricke Line
CP from the Center C, repreſenting the leſſer Scile of the Plane.
Now make choice of ſoine fit place upon the Subſtiler Line;" as B, and there croſs
the Meridian at Righ¢ Angles with the Line FBA: Then cake from off your larger
Polar Scale * the diſtanc: of 3. Hours, and prick it from B toq. Then take the
neercft diſtance from this point q, co che Line of the Srile or Axis, and therewiclı
draw a Line parallel to the Scile, and this ſhall be the Line of che Stile inlarged.
Then take the diſtance of 3 Hours off from your leffer Scale marked hood, and with
that diſtance draw a blind Lioc parallel co the Meridian or Subſtile, and mark where
it croffech chc Line of the Stile Inlarged, which is in the Point r: Then cake the
acerct diſtance from this Poinc , to che former Polar Line FB A, and ſo draw
the Linc O E G parallel chercunto, chrough the Point q : This ſhall be your leſſer
Polar-Line.
Now to ſet off the Hour-lines upon theſe two Lines, fint cake with yoar Compaſſes
he diſtance of one Hour off froin your Polar Scale, and prick ic boch ways
from
the Poinc:B coward A dod' f:: Likewiſe do the ſame for 2, 3, and 4 Hours, and prick
chem down in the Line FBA.
Then take the diſtances of each Hour alſo off from the leſſer Polar Scale H, and
prick them down from E on both ſides the Meridian Line, in che Line OEG. Then
draw through theſe Poincs the Hour-lines by their ſeveral marks, as you may fec in
che Figarc, and put the numbers of the Hours thercanto; ſo your Dial will be
finiſhed, as in the Figure. And if you well underſtand this, you may do the like
in any other Dial which ſhall aced inlarging... vi
→
ini
!
1
сн.р. ХХIХ.
Home to make a Vertical Didd upon the Cieling of 4 Floor within Doors,
where the Direct, Beams of the Sun never come.
1
1
T
1
I
He greateſt part, and as much as you (hall uſe of the Vercical or Horizontal
Dialy deſcribed chap. 8. may by reflection be curned upſide down, atid
placed upon a Cicling; btcche Center will be in the Air wichouc dobrs.
The fifft thing to do is to falten a piece of Looking-glaſs
; as brood as a Groat or
Sixpence, ſec level; or a Gally-poc of Fair-water, which will ſee it ſelf level; being
placed upon the Sple of the Window, ſhall ſupply the uíc of the Now in the Gno-
mon ; and the Beams of the Sun being reflected by the Glaſs or Water, ſhall ſhew
the Hours upon cheCieling
Before you can daw a Figure for chis Dial well, I would adviſe you firſt to find
che Meridian of the Room, which may be done thus.
Hanga Plumb linç in the Windbu, directly over the Nodus or place of the Glaſs;
for the shadox_whichche Plumb-line gives upon-the-Floor'a Noon," is the Meridiah-
linc lought; and by Ruler, or a Liệe fttecched upon it, you may prolong is as far
as you ſhall nccd.
Then take che perpendicular Height chereof from the Glaſs co'che Cieling of the
Room, which ſuppote it be 40 Inches, as D-B, the Glaſs being fixed at D7 Now
from B draw the Meridian-line upon the Cieling, which ſhall be repreſenred boch
ways continued, as ABCK, and from, Derecta
Perpendicular co D.C.::Or-le-a.
ftander bý topone end of a Thred on che
Glaſs, acD; extead, the
fame to the Meri-
dráh-liac, moving thic end of theiltring Qories and longer upon the Meridian, till
anothe
។
113
1
#
2137
.!!Y
1
+
1
1
11
43
CHAP.XXIX, The Art of DIALLING.
another holding the side of a Quadrant, ſhall find the Thred and-Plummet to fall
directly upon the Complement of the Latitude, which in this example is 38 deg.
30 min, and that is the Interſection of the æquinoctial. Then raiſe the perpendicu-
lar, as D A ; take a Chord of 60 deg. and from A ſweep che Arch at P, and, from
Play down che Larisude si deg. 30 min.co q, and draw the Line A D.
3
K
WI
. Meridianline B
А
See
اند
Au
E
3381.30
TO
1
L
51-30
M
20
F
Gnomonline-fiaken of
-309
1
G
M
کہ
460
02-ISA
tayo
80
TITI
190
A
NEX
-100
LITO
IZO
130
1
I
V21
0
VA
PINO
FIN
LIGO
Iyo
P
Q :180
Then let fall a Perpendicular from-C, as CEFGHI, which is the Æquinoctial
Line ; and ſo likciviſc draw a Parallel Line to the Meridian of 40 Inches at D. Now
note, That the Hours muſt be drawn all one as the Horizontal Dials are.
Then draw a Line Parallel to the Equinoctial, as K O, at what diſtance you think
convenient, on che Cieling of the Room, which let be here so Inches, as C K, as you
may meaſure by the Scale. Now for the placing of the Hours on the Cieling of the
Room, you muſt meaſure how much by the scale of Inches: cach Diſtance on the
Æquino&tial Line is from C to E, and from Cro F, and from Cto G, H, and I, and
likewiſe on the Parallel from K, collecting them into a Tablc ;. Co will it be ready to
tranſport on the Cieling.
Here I have niade the Table.
I
1
Inches. par.
Inches. par.
17
38
7
7
O
CE
CF
C G
CH
СІ
27
59
101
173
63
K L
K M
K N
KO
KP
$
3
4
5
III
222
385
8
So
2
1
I
The Art of DIA LE'İN G. Book VII.
44
+
1
+
So by chis Table you ſtiall find the Diſtance CE, which is from 12 to 1 in the
afternoon, or it in the morning, to be 17 Inchies; which you may prick upon the
Cieling. Likewiſe K L on the Parallel, between 12, 11, and 1, will be found to be
27 Inclics parts of an Inch in 10 pares ; which Hour-line mark out upon the Ciel-
ing. Then draw a ſtraight Line through thoſe two Points L and E; this Line conti-
nued ſhall be the firſt Hour from the Meridian, which is II in che forning, or I in
the afternoon: SQ do for all the reſt of. che Hours:
Now by this you may know.baixfär_che Center is without the Window; meaſure
it, and you will find ie 31 Inches from A to B, and from B to C 52 Inches, and
from C to K 50 Inches, as before. I hope now I have given the Practicioner content,
in making this ſo çaſic to be underſtood, although I may be condemned by others.
I will give you 'one Example more, to find the Meridian Line on the Cicling,
which is this. Fit a plain ſmooth Board, about a Foot ſquare, to lie level from the
Solc of the Window inwards; then-neer the ouċward edge thereof make a Center in
the Board, in the very placc of Nodus, or a little under it: Then by Chap. 3. get the
Meridian Line from the Glaſs on the Board; after you have drawn che Line on the
Board
upon the Center, deſcribe as much of a Circle as you may with the Scmidia-
meter of your Quadrant, which Circle ſhall be Horizon ; then from the Meridian you
may with your Degrees,on the Quadrant graduate your Horizon inco Degrees of Azi-
muchs both ways as far as you can.
Next you may deviſe to make your Quadrant ſtand firm and upright upon one of
his ſtraight Sides,' ivhich I will call bis Foot for this time ; and that you may chus do,
take a ſhort ſpace of a Ruler or Tranſom, and ſaw in one ſide of it a Notch perpendi-
cularly, in which Norch you may ſtick faſt or wedge chc heel of che coe of your Qua-
drant, in ſuch ſort as his Foor may come cloſe to the Board, and the other Triangular
Side or Leg may ſtand perpendicular upon it. ' Let the Foot be round, and with
your Compaſſes ſtrike a Circle round it : when you have ficted the Diameter of the
Foot on thc Meridian Line on the Board, draw a Circle round the Glaſs, char ſo you
may ſet the edge of the Circle according as you may have need, for to lay off the Suns
Altitude at every Hour. Now to find the Meridian on the Cieling, you may make a
Table for the Suns Altitude every Hour of the Day, in this manner as here is for the
Latitude si deg. 30 min. and place the Foots Diameter directly on the Meridian of
the Board, and clevate the Quadrant to the Tropịck of Capricorn, which in this Laci-
cude is 15 deg.
1
1
1
1
1
}
4
1
OL
A Table for the Altitude of the Sun in the beginning of each Sine, for all the Hours
of the Day, for the Latitude of si deg. 30 min.
+
1
O
1
Hours.
Gemini. {TANTHS,
Cancer.
Aries. | Piſces.
Piſces. | Agwari.
Capric.
Leo. Virgo.
Libra. Scorpio. Sagitta.
I 2
62
oj 58 45 150
038 3027
Oj 18
18
II.
159
59 43 56 34 48 12 36 5825 4017 613 53
IO 2! 53 45 50
5543 12 32 37 21 511 13 38 10 30
3 45 42 43
0 26 3115 58 3 12's 15
8
436 41 | 34 13 | 27. 31.188 8 33 IS
7 5 27 17 24 56 18
56 18 18
9 37
6
6. 6 18 11 15
15 40 9
S
7
933
II 37
4 8
I 32
2 I 40
50 55
61 36
1
6 50
1
on to
110.
Let a ftander by ſtop on the Glaſs a Thred, and extend the other part ſtraighe
the Cieling, che Thred touching only the Plane of the Quadrant, and making
Angle with it, but held parallel; and where the Thred thus extended touches the
Cicling, make a Point; then the Quadrant unmord, elevated co 62 deg. of Alcicude,
and cxtend the Line, and make another Point as before ; and between capſe two
Points draw a ſtraight Line, and that'ſhall be your Meridian, and ſhall be long
r2
enough
1
**......
.
45
A
2
:
orang
CHAP.XXX, The Art of DIA ELING,
enough for your uſe. Then elevate che Quadrant to 38 deg. 30 min. and hold the
Thred co ché Meridian on the Cieling, and where he touches mark; and croſs the
Meridian at Right Angles with an Infinite Line, which ſhall be the Æquator: So
you may do as you did before. but if the Plane of the Cieling of the Wall is inter-
rupred, and made irregular by Beams, Wall-plates
, Corniſhes
, Wainſcot; or Chim-
ney-piece, and ſuch like Bodies, I will ſhow you the Atomedy to carry on your Hour-
lines over all.
Extend the Thred from any Hour-line to the Tropick of Cancer in the Cicling, as
you were caught before, and fix ic there; and extend another Thred in like manner to
che Tropick of Capricorn, wherc-ever it ſhall happen
beyond the middle Beam, or
quite beyond the Cieling upon the Wall
, and fix the Threds alſo. Then place your
eye
ſo behind theſe Threds, chat one of them may cover another; and ac che fainė
inſtant where the upper Line to your ſight or Imagination cuts the Cieling, Beam,
Wall, or any irregular Body, about the end of the lower Line, there thall the Hours
line paſs from Tropick to Tropick: Direct any By.ſtander to make Marks, as many
as you ſhall need, and by thoſe Marks draw ché Hour-lines according to your deſire.
This is in Mr. Palmer, pag. 102.
If thc Arch of the Horizon, between the Tropicks, be within view of
your
Wino
dow, you ſhall draw the fame on the Wall co bound the Parallels. The Horizon
Aicitude is nothing, and therefore it will be a level Linc: and the Suns Azimuth
when he riſeth, commonly called Ortine Latitude, is in Cancer 40 deg. Eaſt North-
ward, and in Capricorn as much Souchward; and cheſe will be reiected to the contra-
ry Coaſts on the Dial.
.
as
1
1
CHÄ P. XXX.
+
How to make an Univerſal Dial on a Globe; and to cover it, if
it be required.
}
1
A
1
.
.
Globe, faith Euclid, is made by the turning about of , Semicircle, keeping the
Diameter fixed. This Dial, if Univerſal, will want the aid of a Magneti-
cal Needle to ſee it, and it muſt move on an Axis in an Horizon, as the
uſual Globes do; whole Æquator let be divided into 24 Hours, clie proporcion of the
Day Natural.
You may ſee the Figare on the top of the Dial in the Title, but that you cannor
ſee the two Poles, and the Scmicircle, and the Horizontal Circle.
You may imagine this Globe ſet to the Elevation of the Pole, as that is,
with two Gnomons of the length of the Suns greateſt Declination, proporcion-
ed to chc Poles Circle, with the 24 Hours, according to the 24 Meridians, and
ferves for a North and South Polar Dial.
But in the Meridian let be placed the 12 a Clock Line; chen turn the Semi-
circle till it caſt no ſhadow: xhen doch it croſs the Hours, which Hours are
drawn from the Pole to each of the 24 Diviſions, as before.
IE
you deſire to cover the Globe ; and make iocher Inycations theicon pt fifti
learn here to cover it exactly With 2. Pair of Compaſſes bowed towards the
Points (like a Pair of Calapers the Gunners uſe meaſure the Diameter of the
Globe you intend to cover ; which being known, find the Circumference thas.
Mulciply thé: Diameter by 22, and divide the Product by 7, and
you
have
Let the Circumference found be the Line E F, which divide into 12 Equal
parts ; draw the Parallel A B ärid CD, ac the diſtance of three of thoſe, Parts
from E to A and from F to C'; then by the outward Bulks of chofe Arches
draw the Line A B and CD.
Gggg
And
your delire.
13
1
1
4
A
46
The Art of DIALLING. Book VII,
1
"G
-
1
4
3
(T
I
TO
D
And to divide che Circumference in-
to 12 parts, as our Example is, work
chus,
Ser your Compaſſes in E, and make
the Arch FC: The Compaſſes fo open-
ed ſec again in F, and make Arch E
A ; then draw the Linc from A.co F,
and from Eco C. Then
Then your com-
paffes opened at any diſtance, prick
down one part leſs on both thoſe flanc-
ing Lines, than you intend to divide
thereon ; which is here ir, becauſe we
would divide the Line E F into 12:
Theo draw Lincs from each Diviſion
to his oppofite, that cuts the Line EF
in the parts of Diviſion. -
But to
proceed, It is Mr. Morgan's Conceit,
Page 116. Continue the Circumference
at length to G and H, numbring from
E towards G 12 of choſe Equal Parts,
and from F towards H as many, which
Thall be the Center for each Arch; ſo
thoſe Quarters fo cut out, ſhall exact
ly cover the Globo, whoſe Circumfe.
rence is equal to EF.
Thus have you a glance of the Mä-
thematicks, ſtriking at one cluing through
the ſide of another :. For here ons Fi-
gure is made for ſeveral Operations,
to ſave the Preſs chc charge of Fi-
gurcsac
1
IO
1
5
3
2
1
B
F
도고
​3
;
$
6
1
+
ta
a les
i
H
!
II
1
C'H A P. XXXI,
How to make a Dire& North Dial for the Cape of Good Hope, in South Lati-
tude 35 de and Longitude 57 d. to the Eaſtward of Flores and Corvo.
His Dial is made all one as the South Dial you may fee Chap. 9. Only obſerve
this, That you are 35 deg. to the Souchwárd of the Æquinoctial
, and that
the Sun, when he is on the Tropick of Capricorn, wants 11 Degrees of the
Zenith of chat Place Northward : As the sun goes always to the Southward of us in
England; ſo it goes to the Northward of, them therefore muſt che Scile or Gnomon,
point-downward in the North Face, and upward in the South Face. So likewiſe as
in our Souch Dial the afternoon Hours are out on the Eaſt Gde the Dial Plane and
the
T
1
!
1
,,h
NE VIVIX,
.
47
I 12. I
JILL IT I. I TIY X
CHAP.XXXI. The Art of DIA LLING.
this morning Hours on the Weſt fide; ſo in their North Dials chey will ſtand con-
crarily, by reaſon the Sun caſts 2. Shadow (as the Plane muſt ſtand there) in the
morning to the Weſt fide, and in thc afternoon to the Eaſt: So you ſee the Plane is
only curned to face the Sun. If you do but coacciye in your miod how the Sun
s
1
amb
m
1
1.
+
9 IO
00
AA
B
A
1
1
11
1
'
+
XX
t
E
4
1
caſts his ſhadow, you may as eafily make all ſorts of Dials on the South fidc of the
Æquinoctial, as on the North ſide: but that the People there have neither Order, Po-
licy, Religion, nor Underſtanding in Mathematical Arts or Sciences. The Africans at
the Cape of Good Hope are of a ſwarthy dark colour, and made black by daubing them-
felves with Grcaſe and Charcoal; they are fo wedded to ſuperſtition, that ſome adore
the Devil in the form of a bloody Dragon, others a Ram, a Goat, a Leopard, a Bat,
an Owl, a Snake, or Dog, to whom they ceremoniouſly kncel and bow. So much
for Æthiopia, and for Dials for them ; only you fee che manner, and that the former
Rules ſerve for any Latitude.
1
Ġggg
i.
CIK A e.
1
1
48
+
21
+
4
H
MOON- 9 da.'old.
O Di. 3o
10 30
for it will ſtrike what Hour of the Day or Night it is, and then leave off ſtriking, and
The Art of DIA È LING, BOOK VII.
CHA P. XXXII.
How to find the Time of the Night by the Moon ſhining upon a Sun Dial.
Aving the Age of the Moon by the Epact; as for Example, the 9t11
. day of
Auguſt 1665. the tipare to which add 9 che day of the Month, and 6
the Months from Marbh makes 38; from it ſubſtract 29 a whole Moon,
wich 12 ho.44 min. which mulţi ply-chiç remaining 9 by 4, makes 36; that divide by
5, and you have 7 ho. and 12 mint the odd Llnite, and 24 min. for the half day, or
12 hours added togecher, makes 7 do 36 mine for the Moon being Souchi.
Now having the Moons coming to shie Sauch by the former way, add this Souching
of the Moon and the Shadow of the Moon upang Dial cogccher, and cliat is the
cime of the Night. If the Sum seed 12 Hoix tal couly the overplus.
Or chus you maydo: If the Moon be p days old at Moon, die will be 9«lays and an
So. 178.3om. half at night; therefore you may add abolre a quarter or half an hour clicrcanto, as
it is more early or late in the nigbac, and add the Sodthing af bad Moon, which makes
at Nigbe.
7 ko
. zemin added to the shadow of the Moon 3 ho
. upon the şuın-dial, it inakes
Io ho 30 min. for the time of cha night: $o you ſee there is Opin. difference betwixt
theſçtwo kvays, which cangor well be ekonated, but-êther way will give neer enouglı
fatisfaction for the citne of the Ni
CHA P." XXXIII.
Hopia tę find the Hour of the Day or night by x-Geld Ring, and a Silver
Drinking Borol, or Chels or Braſs, ox Iron, or Tirall.
Aving a Gold Ring and a Siljer Drinking Bowl, také á pall Thred or Silk
and meaſure the compass of the top
of the'şilver Bom Glaſs, or other Vel-
rel, which will be a convenent lcrigių for your yle: Then Pile
this Thred
through the Ring, and tic the ends the cof togelitr, taking up as little as you can with
the knots. Put this. Iured over your Thumb, where yod feel the Palle b:ac, upon
the lower Joyne ir maybe; slicu. Otrdkçlı oy your hand, and hold it ſo thac che in-
ſide of your Thumb may be upwarty and hole gour Hand fo over the Bowl, that
the Ring may liang as neer the middle of the Bowl as you can gueſs: and you shall ſec
that the beating of your Pulſe.(holding your Hand a while as fill as you can) will
give a 'motion to the Ring, cauling it to Twing croſs the Bowl by Degrees more and
more, till at laſt it will beat again..che Sides chercof.
Now mark when it begins to ſtike, and cell the ſtrokes as you would do a Clock;
ſwinging alſo by degrees: Which hacíı been approved of by the experience and judg.
ment of many.
H
A Good Obſervation.
E
inay
take notice, Thac chere is no Dial can thew the crack time, without
the allowance of the Suns Semidiamccer, which in a ſtrict acceptacion is triie.
But hereto Mr. Wells liath anſwered in page 85. of his Art of Sh.:dows, where faith
he, Becauſe the Shadow of the Center is hindered by the Srile, the Shadows of the Hinr-
line proceeds from the Limb, which always precedech the Crnter one minute of tim", an-
Swer able to rs minutes the Semidiameter of the Sun : which to alloy in the Height of the
Stile mere erroneous ; bes: there may be allowance in the Flower-line, detrálling from the
true efquinoctial Diſtance of every Hour or 15 degrees, 15 minutes. But I will
farther with this Subject, to put the Learners in doubt of the true Hour ; for this is
as ncer a way which I have thewed you, as any projected upon Dial Planes. Your
may feca Geometrical Figure of it in iny Problems of the Spherc,
CHAP
Sono
!
1
GWAP XXXIV, The Art of DHA DHIN G.
49
.
F
A
T
is.
I
Tini
Court
O UNA P 10XXXNX
nils : - How to Paint the Dials whichlsyou-make, :)
012
Hi,
:..!?"."; dierba: blod
Lthough:I' never ſaw-ary inár Make aliat not paincore, but wliat
wiat made
painted, and guilded my ſelf; as you may ſee che Pied ifke Title on which
I made 16 Dials; and put in die Braſsendtons or Stirred into the free
with Lead, and guilded them with all the Figüres, and on tlft Ulfbedreiv the £quia
noctial Circle, the Ecliptick, the Tropicks, and Polar Circles, and the 24 Meridians,
tvith ſuch Conſtellations in the dark and Souch Memiſphere, and Stars, in ſuch co-
lours as was fic to ſet out the Dial, with Pole Dials, and Globe Diales Chap: 33
To Pdisit And Finiſh the Didls, ready to be set up in the place stimoline
on lyrhive it, whi?
FO
Or to faſten the Gnonion to the plane, he is of wood or Freeſtone, you muſt
have a finall thin Cliiſel, op Googc, or Gittbret, as iš fit for the Stile, be it round
as a Rod of Ironor 2 piece of Brnis,les in wich a Foge am inchu and half or magontato
as you will į avid'iniche Wopet makeuchi ligele Morriles ar mußt the bregdely and in the
of Hic Foor of Elie Stile: and ifjiç çames cloroy, coflich jephthe pher, fids, shen
ic is faſt:
bna
If it is in Freiſtònič, your Dial drawn firſt in Paper, lay it upon the Planc as it
ſhould be ; then cut out the Subſtile-line usiness its breadch as you can, and only
leave ſo much as will juſt hold is together. The paper láid as before on the Plane, withi
Blac's lead Penſili or ſuch like, draw che Şubſtilcising whers. ig kaadiqche Papety
and with a ſmall Chiſel make ſuçli Mortiſes in chat Ling as a snwcable to he
Fööt of the Stile; and crook liis Foot, and put it into ics place with almall Lagisand
ſome Lead melted, put the Stile perpendicular with the Plane, and pour in the Lead
into che Mortiſe until it is full, and when it is cold, then with a blunt Chiſel har-
den the Lead in one Inch ſide of the Stilc or Gnomon: And if the Morriſc ſhould
be too wide, er broken, and not even snough with the Rlang, then avez fome Flower
of Alabaſter, as joy inay have ie fit for that purpore at any Melonis,
wild as 488H as 'cis
we make a Plaſter, and ſo ſmoosh it and Ipread imeven and plain with the Plane;
it is preſenly dry. Now have you the Stilc or Gnomon as faſt as if it grew there.
: To Painc chem, you muſt firſt Prime them: The Prime is made chus. Take an
equal quantity of Bole Armoniack and Red Lead, well ground together with in-
feed Oyi, and sell rubb'd in with a Bruſh or Penſil into the Plans; that being
diy, for thic outſide Colour, ic is. White Lead, or Ceruſe well.ground fogęchei wide
Linseed Oyl. How to know the beſt
. Buy the White Lezd, and grind i saj?
Powder
, and put it into Water until it become as thick as Pap, and let it dry chrá
it is for your uſe.
For the Hour Lines a Vermillion, and a part "Red Lead, well grand together
with Linſeed Oy!, with a ſmall quantity of Oyl of Spike, or Turpenting that will
dure, and make tlic Lines thine,
Fora Gold Border, Rub the Þorder well wiçli çhiç whitę Ceräte Paint;, be ſursis
thick in the Border : Then with Blew Smalcs ftrew ycry chick the Border
wliile it is wet; and when it is dry, wing that which is looſe off, and ſave it in a par
per and for the reſt that clings, it is faſt cnouli, os
Take Red Lead and White Lead, and as much Red Lead, agzin as Whire, or Ycha
low Oker, well ground with Oyle of Spike of Turpencine; this is the Siſe : Then,
draw with chac tlic Figure you would
have isi Gold, and when it is ſo dry that it
will not come off on your fingers by a ſlight couch, lay on the Gold; and when it
is thorowly dry, wing ic off.
How to make a good Black, to ſhadow or makc Figures. Grind well with Linſeed
Oyi Lam-black, with ſome Verdigreaſe, and thac iş a, firm Black. Thc like you may
do with all other Colours, as you fancy for ſuch Work.
1:01:
. ..
ü
1
+
+
i
be very
III
។
50
The Art of ĐIẢ " IN G.BooKVII.
F
mi
7
1
f
.
4
1
V Receipt for Red trik.
Irſt, Sceep onc penny-worth of Brazcel Wood call night ih? Piſs or lirin; chen
boil it well and ſtrain it; then bruiſe two penny-worth of Cochincel, and boil
iſ, and put in it the bigneſs of a Hens Egg of Rocli Allom, chat brings it to a colour,
and then it is for your turn:
To Paint Freeltous, waſh the Stone with Oil, and it will laſt; and then, all the
Coldursbefore may be uſed, as directed.
1:7
How to cleanſe a picture.
it sic
Tako blew Smalis
, cemper is in water, and rub the Pitture with ic, and after
wipe it with a Lionen Cloth, which Clach hould be dipp din Beer, or other-
wiſe widi a dry cloth,' and it is clean.
To cleanſe a Gold Boräer
W
Alh it with Beer, and dry it, and then cleanſe is with Linſeed Oyl.
Maſticóus is a fine Yellow, ground with loine Oyl of Spike or Turpeuşine,
*** Bice is a good-Blew Colouring, to be ground with Linſeed Oyland Red Lcad.
And Spaniſh Brown will makca laſting Colour for, Courſe Work.
To grind Gold to write and Paint. ::-
Akę as many Leaves of Gold as you pleaſe, Honey three or four drops; mix and
grind chcſcand keep it ioCome Bone Veliels
. If you will ivrice ivich ic, add
fóme Gum-water, and it will be Excellent.
be
F
j
:1
-
1
!
+
ji
!
37
in
L
Some. Uſes of the following Tables of Logarithmes,
Sines, and
Tangents.
Mongſt the many admirable ways that have been from time to timc invciited
for propagating the Arts Mathemacical, and eſpecially chat of Trigonometry,
Logarithmes, invented by the Right Honourable the Lord Napier, Baron of
Marcheſton, may challenge the priority, and chic Tables of Artificial s'ines and Tan-
gents, “compoſed by Mr. Edmund Gunter Profeſſor of Aſtronomy, in Greſham-College
London ; for that they expedite the Arithmetical Work in moſt Queſtions ; Multipli-
cacion being performed by Addition, and Diviſion by Subſtraction, the Square Rpoc
extracted by Bipartition, and the Cubique Roor by Tripartition : Só that by help of
chele Numbers, and the aforeſaid Sines and Tangenes, more may bic performed in the
ſpace of an Hour, than by Natural Nambers or by Vulgar Aciilimerick can be in ſix.
Now of what frcquent uſe the Doctrine of Triangles, both Plain and Spherical, is in
Aſtronomy (for the Reſolution of which the Tables Following chiefly ferve) let the
precedent Work teſtifie. And as Mr. Newton in his Mathematical Inſtitutinns, or in
the ſame form as Mr. Vincent Wing in his Harmonicon Cæ!efte, are theſe Tables follow-
ing: And therefore I think it noc amiſs here in this place to infert ſome few Propoſiti,
ons, to ſhew the uſe of the Canón and Tables of Sines and Tangents following.
PROBL. I.
How to find the Logarithmes of any Number under 1000.
1
+
!
1
ľ,
3
which Columns, having the Lċccer N at the Head chereof, are all Numbers luc-
cettively continued from 1 to 1000: So chat to find the Logarithmes of any Num.
:ber,
1
1
+
V
4
1
.
!
+
CHAP.XXXIV. The Art of DIALLING,
51
:
ber, is no more but to find the Number in the firſt Column, and in the ſecond Co.
lumn
you Thall have the Logerithe anſwering thereunto.
Example. Let the Number given be 415, and if it is required to find the Loga-
rithme chcredf, in the Table of Logorithmes
, in the firſt Column thercof, under the
Letter N, I find the Number. 415, and right againſt it in the next Column Ipfind
618048, which is the Logarithme of 415. In che ſame manner you may find the
Logarithme under 1000; as che Logarithme of 506 is 704151, and the Logarithime of
goo is 954243, etc.
...:1
Buc here is to be noted, That before every Logarithme muſt be placed his proper
Characteriſtick; viz. If the Number consiſt but of one Figure, as 'ail Numbers uin-
der 10, then the Characteriſtick is o; if the Number conſiſt of twy Figures, as all
Numbers between 10 and 100, then the Characteriſtick is I; if the Number confift
of three Figures, as all Numbers berwixt 100 and 1000, then the Characteriſtick is. ??
2 ; and if the Number conſiſt of four Figures, as all between 1000 and 10000, the
Chara Steriſtick muft be 3. In brief, the Characteriſtick of any Logarithme mult con-
fiſt of an Unit leſs than the given Number conſiſteth of Digits or Places : And by ob-
ſerving this Rule, the Logarithme of 415 will be 2.618048, and the Logarithme of
506 is 2.704151, and the Logarithme of goo is 2.954243, c.
PROBL. II.
.
A Logarithme being given, to find the Abfolute Number thereurto belonging, by the
former Obſervation ; the Characteriſtick will declare of what Number of Places
the Abfolute Number confifteth.
Example. Ler the Logaritbme given be 2.164353; now becauſe the Characteria
ſtick is 2, I know by it the Abſolute Number confiftech of three places, and therefore
may be found in the ſecond Column of the Logarithmsc Tables, having o at the top
thereof, againſt which I find 146, which is the Abſolute Number anſwering to the
Logarithme of 2.164353.
0
3
i
1
4
ܨܝ ; z
PROBL. III.
How to find the Logarithme of a Number that confifteth of four Places.
You muſt find the three firſt Figures of the given Number in the firſt Column, as
before, and ſeek the laſt Figure thereof amongſt the great Figures in the head of the
Page; and in the common Arca or meeting of theſe two Lines is the Logarithmse you
delire, if before it you add or prefix its proper Characteriſtick.
Example. Let it be required to find the Logarithme of 5745; I find 574, the
three firlt Figures, in the firſt Column, and 5, the laft Figure, in the head of the
Table; then going down from ş in the head of the Table, uncil I come agaimt 574
in the firſt Column, there I find 759290, before which I place 3 for che Characte-
riſtick, whidi is 3.759290, and that is the Logarithme ſought for.
1
1
1
I
PROBL. IV.
Ang Number of Degrees and Minutes being given; to find the artficial Sine and
I angent thereof.
Admic ic were required to find the Sine of 21 deg, 34 min. I turn to the Sines in
the Table, and in the head thereof I find Degrees 21; then in che firſt Column (under
M) I find 24, and right againſt it is 9.562146 for the Sinc, and 9.593170 for the
Tangede of 21 deg. 24 min. But ſuppoſe it were required to find the Sine or Tan-
genc of 56 deg. 35 min, look for ålrelle under 45 deg. are found in the head, and the
odd miss. in the left hand; and all above 45 deg. are found in the foot of the Table,
and the min. in the laſt Column toward the right hand; as in this Example the Sirof
56 deg. 35 min.'is 9.921524, and che Tangent is 16.180590.
1
PROBL.
ļ
}
1
42
i
-
I
1
1
52
}
The Art of DIALLING, Book VII.
PR
1
1
5
PROBL. V.
If any Sine or Tangent be given, to find what Degrees and Minutes anſwer thereunto.
Suppoſe 9.584663. were a Sine given, I look for the Number in the Table of Sines,
and I find it Itand againſt 22 d. 36 m. and therefore is the Sine thereof. As admit
9.624330 were a Tangent given, look for the Number in the Column of Tangents, and
1 find ſtand againſt it 22 d. 50 m. The ſame muſt be donc for Sines and Tangents in
the foot of the Tables.
I
1
Ovid. Lib. Mere
Cancta fluunt omniſque vagans formatur Imago.
Ipfa quoque affidno labantur tempora motu.
All things paſs on : 'Thoſe Creatures which are made
Fail, and by Time's aſſiduate motion fade.
Much like the Running Strcam, which cannot ſtay,
No more can the light Hours that poſte away.
But as one Billow, haft'ning to the Shore,
Impels another, and ſtill that before
Is by the following driv'n ; ſo we conclude
Of Time, it ſo flics, and is fo. purſu'd.
Thc Hours are always new; and what hath been
Is never more to be perceiv'd or ſeen :
Thar daily grows, which had before no ground;
And Minutes, paſt once, never more are found.
1
1
}
1
Lib. Eles. I,
Labitur occulte fallitque volubilis ætas.
The frecting Age deceives, and ſtealing glides ;
And the ſwift Year on looſe-rein'd Horſcs rides.
Quid non longa dies ? quid non confumitis Anni ?
外
​;
1
5
r
I
The End of the Seventh Book.
(
1
1
+
1
1
1
1
1
1
C A NON
1
TRIANGULORUM
LOGARITHMICUS.
OR,
AT ABLE of ARTIFICIAL SINES and
TANGENTS to every Degree and Minute
of the QVADRANT,
The Common Radius being 10,000000 .
By Capt. SAMUEL STURM Y.
M
28.22
4gost 216
64.19
38.30
29 28
65-44
0 19.1
40 0'
31.34
473
47.40
yo
R
34.77
P
1
30.28
38.53
$
The Deſcription of the TRIANGLE.
Let ZPS repreſent the Zenith, Pole, and Sun, ZP being 38 deg. 30 min.
Complement of the Latitude, PS the Complement of the Suns Declination
70 deg. and the Complement of the Suns Altitude ZS 40 deg. oo min.
the Angle at Z ſhall ſhew the Azimuth, and the Angle at P the Hour of
the Day from the Meridian: Then if from Z to PŚ we let down a Per-
pendicular,as ZR, weſball reduce the Oblique Triangle into two Rectangled
Triangles ZR P and ZRS: If from S to ZP we let down a Perpendicular
SM, we ſhall reduce the fame ZPS into two other Triangles, as SMZ
and SMÝ Rectangled at M. wbatſoever is faid of any of theſe Triangles
,
the ſame holdeth for all other Triangles in the like Caſes.
g
1
London, Printed by E.Cotes, Anno Dóm, 1669.
Canon Triangulorun Logarithmicus.
4518.1.169261 9.999963 18.116963 11.3830371 18.48484819.99979718:48 5050f11.51495alis
Degree o.
Degree 1.
MI Sine 1 Co-line |Tangent Co-tång. Sinc 1. Co-fine |Tangentl:Coting. M
ol0.000000 10.000000 0.000000 Infinita, 3.24185519.99993418.141921111.75007167
116.4637261. 9.999999 6.463726 13.5 36274 8.24903319.99993215.24910211.750898 59
26.764756 9.999999 6.764756 13.235 244 13:256094 9999929 3.256165|11.743835158
36.940847 9.9999996.940847 13.0591533.2630429.9999278.263115|11.736885152
417.065786 9.99999917.06578612.934214 3.26988119.999925 8.25995611.73004466
$17.1626961 9.99999917.162696112.837304 8.27661419.9999228.276691111.723309155
617.2418771 9.99999917.241878|12.758122 3.28324319.99992018.283323111.716677154
7 7.3088241 9.999999 7:308825|12,691175
.6911751 3.28977319.99991818.289856 11.716144.53
877.3668169.9999997.366817 12.633183 3.296207 9.99991518.296292 11.703708151
917.417968 9.9999997.417970 12.582030 8.30254619.999912 8.302634 11.6973665-1
107:463726 9.99999847.463727 12:536273) 8.30879419.99991c|3.308884111.
.691116550
17.505118 9.99999817.505120112.49488. 9.3149549.999907/8.315046111.634954149
127.542906 9.999997 7.542909126497091 8.321027 19:999909 8.3.211,2211.67887348
1317.577668 9.999997 7.57727262.422328 8.327016 9.9999028.327114111.67288647
1417.609853 9.999996 7.609857|12.390143 8.3329249.9998998.33302511.666975146
1517.639816.9.9999967.639826.12.360188.33875319.999897 19:338856/15.661144145
167.667844 9.99999517.667849122332151 8.344504|9.99989418:344610418.65539944
17 7.694173 9.9999957.694179112.305821 3.350180 9.999891 8.35028911.649711 43
187.718977 9.9999947.71900312:281997 8.355783/9.9998888.35589511.64410542
197.7424771 9.9999937.742484 12.257516 8.361315 9.9998858.361430 11.638570 41
2017.7647541 9.999993177764761.12.235239 8.366777 9.999882 8:366355111.633105149
2117.785943 9.99999217.785951|12.214045 8.37217119.999899/8.372293211.627708139
227.806146_9.999991.17.806145.412,193845 8.37749919.9998769.377622 11.622378 38
2317.8294511 9.999990 7.825460 12.174540 8.3827629.9998738.382889 11.617111137
24 7.843934 9.99998212.843944 12.156050 8.3879629.99987013.38809211.611908 36
2517.861662) 9.99998915.861674 12.138326 8.39310119.99986718.393234|11.606766135
2617.5786951 9.99998917.878700|12.52129: 8.39817919.99986418.398315|11.601685134
27 7.895085 9.9999877.895099 i 2.104901 8.4031999.999861 8.403338 11.596662 33
28 7.910879 9.999986 7.910894 12.089106 8.40816119.99985818.40839411.59169032
297.9:6119 9.9999857.926194 1207-3866 8.473068 9.99985418.413213|11.586787131
3017.940842 9.99998317.940858112.059142 8.41791919.99985118.418068|11.585937130
3117.955082, 9.9999821.7. 98510012.04A900 8.42271719.9998488.422869|11.577131124
327.968870. 9.9999817-968889 12.031111 1.427462 9.999844 8.42761811.572382 28
3317.982233 9.999980 7.98*2253 12.017747 3.4321569.999841 3.432315 11.56;695 29
347.995198 9.999978 7.995215 12.004781 13.436800 9.999838 5.436962|11.56303826
35/8.007787 9.99997817.007810 11.9921913.44139419.9998343-441560 11.558440125
3618.020011 4.9999768.020044111.979956 2.44594119.99983113:440110101.553990124
37 8.031919 9.99997518.03194511.9680551 14.450444 9.9998278.450613 1.549387 23
388.043501 9.9999738.04352711.9564731 3.4548939.999824 8.455070111,544930 22
398.054781 9.999972 8.05 487911.9451813.459301 9.99982€ 8.459481 1.54051921
4018.065776 9.99997110.005800|11.934194) 18.46366519.9998168.463949|11.536151 20
4118.0765001 9.99996918.076531111.923469 8.4679859.99981248.46817211.531828119
42 8.086965 9.9999688,086997 17.9130031 8.4722639.9998998.47245411.52754618
43 8.097183 9.9999668.097217|11.9027831 8.476498 9.999805 8.476693|11.523307 17
44 8.107167 9.99996413.197203 11.892797 8.4806939.999801 8.480892 11.51910816
4618.126471; 9.99996118.126510 11.8734908.48896319.99979418.489170111.510839/14
47861358109-9999598.135851111.864149 8.49304019.999790 8.493250 11.50675013
488.144953) 9.9999588.14499611.855004118.497078 9.9997868.497293|11.90270712
4918.153907 9.999956|8:153952111.8460481 8.501.98019.999782;8.501298|11.4987021
58.3.16168119.99995418.162737111.837273 18.50504$10.99977818.505267111.494733)
5118.1.712801 9.99995218:171328|11.828692 8.50897419.999774/8.509200|11.499800 9
5.28.179713 9.9999508.17976311.820237 8.512867 9.9997698.5 1309811.486902
5318.187985) 9.9999488.188036 11.811964 18.5167269.999765 8.51696111.4830397
548.196102 9.9999468.19615611,803844 8.520551 9.99976118.520790 11.4793106
$518:2040701:9.999944/8.20412611.795374 8.52434319.9997568.524586)11.47541415
5618.2118951 9.99994218.21195 3111.788047 8.52810:19.99975318.528349111.4716514
57 8.219581 9.999940 8.219641 11.7803598.531828 9.999748 8.532080 11.467910 3
15818.227134 9.999938 8.227195 11.772805 18.5355239.99974418.535779|11.464221
59 8.234557 9.999936 8.234621|11.765379 18.5391869.9997408.53944711.460553
6018:2418551 9.999934 8.241921111.7580791 8.54281919.999735 8.54308411.45691610
Mi Co-fire Sine Co-tang | Tangene Co-lineSine 1 Co-tang. I Tingent.M
Degree 88.
1.
.
1
1
I
1
Degrec 89.
*
Canon Triangulorum Logarithmicus,
4
15718.7115079.99942418.712083/11.287917) 18.8381309.998967/8.93916311.1608377
Degree 2
Degree 3.
M| Sine 1 Cô-rine [Tangent Ca-tang. Sine | Co-fine (Tangent. Co tang. IM
13.5428 1919.99973518.543034 11.456916 8.71880-19.99940418.71939611.-80604160
118.5464229.999731 8.546697 11:453309: 18.72120419.99939818.72180611.278194159
218.549995 9.999726 8.55026811.449732 8.7235959.999391 8.724254 11.27579615
318.553558 9.999722 8.55381711.446183 18.7259729.999384 8.72658441.273412157
418.5570549.99971718.55733611.442664 8.728336 9.99937818.723959-11.27104:56
$18.56054019.99971313.560827111.439172 8.730688/9.9993718.73 317|11.268683/15
68.56399919.99970818.564291(11.435.709 18.733027/9.99936418.73366311.26633715
:78.56743119.9997033.567727 11.4322721 18.7353549.99935718.73599611,2640045
88.5708369.99969918.571137 11.428863 8.737667 9.99935018.736317 11.2616835
1918.574214 9.999694 3.57452011.4254808.7399699.99934318.7406:611,259374151
10 8.577566 9.99968913.577877111.422123 8.74225919.9993318.742922.11.257078150
1118.58089-19.99960513.58120811.418792 8.74453619.999339 3.745107111.254793149
128.58419319.999650 3.58451411.4154868.746801 9.99933a 8.747479 11.252521 48
138.5874699.9996758.587795 11.412205 8.74595519.999315 18.749740 11.25024047
145.5907219.99967018.591051 11.408949 8.75129: 9.999308 13.7598911.24801146
1518.5939489.99966518.594283|11.405717 8.753528/9.99930113.754227|11.24577345
1618.59715219.9996608.597492/11.402508 8.7567479.99929418.75645311.243547144
1718.6003329.999655 8.600977 11.399323 8.7579559.99928613.758668|11,241332 43
188.603488 9.99965018.60383811.396161 8.7601819.99921938.76087211.23912841
1918.6066229.999645 8.60697811.393022 8.762337 9.99927284763065111.-36935 41
2018.60973419.9996408.610094 11.389906 8.76451119.999265185765246|15.234754140
2118.61282319.99963518.613189111.386011 8.76667519.99925718.767417411.232583139
228.61539119.9996298.616267 11.383738 8.7688289.99925078.769578|11.230422 38
2318.618937 9.999624 8.61931311.380687 8.7709709.9992428.771727 11.228273137
24 8.6219679.9996198.62234311.377657 8.773101 9.999235/5.773866 11.229134 36
$ 8.62496519.99961418.625352111.374648 8.77522319.999-2718.775995111,224005135
1618.62794819:99960818.628 340 11.371660 8.777.33319.9992108.778114111.321886
278.630911 9.9996938.63130811.368692 8.779434 9.999.212 3.78322211,219773 33
283.6338549.999597 8.634455 11.365744 8.78415242-9992048.782320 11:21768032
29 3.6367769.999592 8.6371845.362816 8.783605 9.999197 8.784404 11.215592131
308.63967919.99942618.640093|11.359907 8.78567519.9991893.78648611.213514130
3118.64256319.9995 010.64293211.357917 8.78773619.99918115.788554411.21 1446129
3218.645423 9.9995758.64585311.354147 18.789787 9.99917418.790613|11.20938728
33 3.648274 9.9995708.648704 11.351296 8.7918289.9991669.792662 11.207338 27
348.651102 9.9995648.651538 11.348463 8.793859 9.99915818.794701 11.205299 26
3518.65391119.999558 3.65433(11.345648 8-79588119.9991508.796731|11.203269125
3613.65670219.99955318.657149711.3428511 8.79789419.99914218.798752(11.201248124
37 3.659475 9.9995 47 8.65992811.340072 8.799897 9.999134 8.800763|11.199237
3818.662230 9.9995418.66268911.337311 8.80189119.9991268.80276511.197235 22
3918.6649689.9995358.665433 11.334567 8.8038769.9991188,807458 11.195242/21
40 8.667639 9.99952918.668160111.331840 8.80585-19.999110/8.806742111,193258/20
4118.670393 9.99952313.670869111.329130 8.80781919.99910218.808717/11.191283119
428.673080 9.999518 8.67356311.326437 8.809777/9.9995948.81268311,18931718
438.675751 9.9995128.676239111.323761 8.811726 9.9990868.812641 11.187359117
448.678405 9.999506 8.678899|11.321100 8.813667 9.9990778.814589 11.189411 16
4518.681043 9.999499 8.64154411.3184568.81559819.99906918.816529111.183471 15
4613.68366519.99949318.68417211.31582811,8.81752219.9990618.81846111.181539
47 8.6862729.999487 8.68678411.313216 18.819436 9.99905 28.82038 411.17961613
4818.688892 9.999481 8.689381|11.310619 8.821342 9.999044 3.822298111,177702 12
8.6914389.9994758.691963|11.308037 18.8:3240 9.9990368.824205 11.17579511
50 8.693998|9.99946918.694529|11.3054711 8.8251309.999927/8.826103|51.17389710
5118.696543 9.99946248.697081/11.302919 8.82701119.99901918.877992|11.1720089
$218.69907319.9994568.699617 11.300383 8.8288849.999010 3.82987411.170126 8
5318.70158919.9994508.702139|11.29786118.83074919.9990028.831748!1,1682547
54 8.704090 9.9994438.704646 11.295354 8.8321069.9989938.83361311.1663876
5518.7065769.999437/8.707139|11.292560 8.83445619.998984/8.83547111,16452915
5618.709049|9.99943113.709618)11.290381 8.836297|9.99897618.837321|17.1626791
588.7139529.999418 8.754534 11.285466 8.8399569.9989588.540998 11.1590022
598.716383 9.9994118.71697211.28 3018 8.841774 9.998940 8.842825 0.137175
608.718800 9.999404/8.719398|11.280604 8.84358519.99894118.84464411.15535610
Co-fine I Sine I co tang (Tangene Co. fine | Sie | Co-tang. 1 Tangent M
Degree 87.
Degree 86.
(a 2)
49
1
Canon Triangulorum Logarithmicus.
618.89429119.99888718.855403111.14459? 18.950814:9.9982778.950597411.049403541
Degree 4:
Derlee 5.
M Sine
Sine I Co-line |Tangent! Coting! Sine | Cr-fine (Tangent! (o-tang.
018.84358419.99894118.844644|11.155356 8.94029619.99834418.941952/11.058048160
118.84538719.99893118.846455111.1535451 8.941738 19.99833318.943404111.056595159
28.8471837.9989238.848240 11.151740 18.9431744.9.)836218.944852 11.05514858
3 8.848971 9.998914 8.85005711149943 3.94460019.993311 8.946295 11.05 3705 57
418.85075119.998905 3.851846 11.148154 2.94603419.9933038.947734 11.05 2266 56
518.85252519.99889613.853628111.146372 3.95745*19.99828918.94916811.050832155
14
1
78.8560499.998878/8.857171 11.142829 1.9502875.9932658.952921 11.04797953
88.85780119.9988698.85893211.141368 18.9516969.99*2558.953441 11.046559152
918.859546 9.9988608.86068611.139314 3.953099 7.99324318.954856 11.045144 51
103.36128319.99885119.862433111.137567 8.95449919.99823213.956267111.043733156
1118.8630149.99884118.864173|11.135827 8.955894'9.9982208.957674/11.042326
128.86473819.9988328.865906 11.134094 8.957284,9.9982098.95907511.04092548
13 8.8664549.998823 8.86763211.132368 8.9586709:9981978.960473|11.039527 47
8.8681659.998813 3.869351|11.130649 8.9600$29.998186 8.96186611.038134 46
1518.8698689.998804 3.87106411.128936 8.96140919.99317415.963254191.036740 45
1618.87156519.99879513.872750111.127230 8.96280119.99316318.964639111.635261/44
178.873255 2.9987858.87446911,125531 18.9641709.998551 8.966019 11.033981 43
188.87493819.998776 8.876162 11.123838 8.9655349.999139 5.967394 11.032606 42
19 8.876615 2.9987668.877849 11.122151 8.96689319.9981288.968766 11.031234 41
2018.87828519.998757 8.879529111.120471 8.96824919.998 r06 9.970133111.029867 40
2118.879949|9.99874718.881202/11.118798 8.96960019.99010414.97149511.028505139
2218.881607 9.99873818..882869 11.1171311 8.9709479.998092 8.972855 0.027145 38
238.8832589.998728 8.884530111,1154748.972289 9.9980808.974209 11.02579137
248.88490319.998718 8.88618511.113815 8.9736269.9980688.975560 11.024440 36
25 8.88654219.99870818.887833111.112167 8.97496219.99805 68.976906111.023094135.
2618.88817419.998699|8.889476111,110524 13.97629319.99804415.978248111.021752134)
278.889801 9.998689 8.89111211.108888 18.977619 9.9980328.97958611.020414 33
28 8.8914219.9986798.892742 11.1072588.978941 9.9980208.980921|11.01907932
298.893-359.9986698.894366 11.105634 8.980259 9.9980988.982251 11.01774931
30/8.89464319.99865918.895984111.104016 18.98157319.99799618.983577 11.016423130
3119.89624619.99864918.897596|11.102404 13.98288319.997984/8.984899|11.015101129
3213.897842 9.9986398.899203 11.100797 2.9341899.99797118.986217 11.01378328
33 8.8994329.998629 3.900803 11.09919; 23.985491 9.9979598.987532 11.01246327
348.901017 9.99861918.902398 11.097602 3.93678919.997947.3.988842 11.011158126
3518.9025969.99860913.903987111.096013 9.98808319.99793518.990149|11.909851125
3618.90416919.99359918.995570111.094430 3.98937414.99792213.991451|11.008549124
3718.9057369.9985898.907147 11.092853 13.990660 9.99791018.9927501.00725023
3818.907297 9.99357718.908719 11.091281 3.9919439.997897 3.994045|11.005955122
398.9088539.998;683.910235 11.089715 3.9932289.9973853.995337|11.004663 21
4018.91040419.94855818.991846111.0891541 2.99449719.99787318.996624111.003376120
4118.9119499.99854818.913401|11.086599 3.99576819.99786018.997908|11.902092119
428.913488 9.9985378.91495111.085049 13.9970369.9978478.999188 11.090812 18
438.9150229.99352718.916495 11.0835051 3.998 299 9.9978359.03046510.999535119
8.916550 9.9985168.918034 11.081966 8.99956019.9978229.001738 10.998 262 16
8.91807319.9935068.919568111.080432 3.00081619.99780919.003507 10.996993|15.
4618.91959119.99849518.921096|11.078994 19.002969;9.99779719.004272/10.995728114
478.9211039.9984858.922619 11.077381 19.003318 9.9977849.005534 10.994466 13
48 8.922610 9.998474/8.924136 11.075864 9.0045639.99777119.006792 10.99320812
4918.924112 9.998464/8.925649 11.074351 19.005 805 9.997758 9.008047 10.991953/11
5018.92560919.99845318.927156111,072844! 19.00704419.9977329.009298 10.990702110
5118.92710019.9984428.928658|11.071342 19.09827819.99773219.010546|10.98945419
5218.92858719.998431 8.93015511.069845 19.00951019.99771919.011790 10.988210 8
$38.930068 9.998421 8.931647 11.068 3539.010737 9.997706 9.013031 10.986969 7.
5418.931544 9.998410 8.933134 11.066866 19.0119629.99769319.014268 10.9857326
301519.99839918.93461611.065384 9.0131829.9976809.01550210.9844985
5618.93448119.99838818.936093|11.063907 19.01439919.99766719.01673210.9832681 4
578.935942 9.9983778.937565|11.062435| 19.015613 9.9976549.017959 10:982041 3
5818.937398/9.9983668.939032 11.060968 19.0168249.9976419.01918310.980817
5918.938850 9.993355 8.940494 11.059506 19.018031 9.9976289.020403 10.979597 I
6018.94029619.9983448.941951/11.0580481 19.01923519.99761419.02162010.978380
1 Co-line Sinc 1Co-tang. | Tangent Co-finel Sine Co-tang (Tangent M
Degrec 85.
Degree 84.
1
44
8
2
I
Canon Triangulorum Logarithmicus.
Degree 7.
2
Degree 6.
M! Sinc | Co-ipe {Tangentl Cơ-tang. Sine 1 Co-ſine (Tangent Co-tang. I
09.01923519.99761419.021620/10.9783809.08589419.99675119.089144110.91085616
19.02043519.9976019.022834 10.97716619.08692219.99673519.090187110.909813159
29.021632 9.9975889.024044 10.9759569.087947 9.9967209.091228 10.908772158
39.022825 9.9975749.025251 10.9747499.0889709.996704/9.092266 10.907734157
4.9.0240169.9975619.026455 10.973545 9.089990 9.99668919.09330210.906698156
519,052037.99754819.027655119.972345 2.09108819.99667319.094336 10.90566455
619.02638619.9975349.028352|10.9711489.09202419.99665719.095367|10.904633154
7 9.027567 9.997520 9.030046 10.969954) 9.093037 9.9966419.096395 10.90360453
$9.0287449.9975079.031237 10.968763 9.094047 9.9966259.097422 10.90257852
99.029918 9.9974939.732425 19.9675751 9.0950569.9966102.098446 10.90155451
103.03101919.99748919.033609|10.9663919.09606249.99659419.099468 10.990533150
119.03225719.9974669.03479110.9652099.0970659.9965789.100487110.899513149
129.0334217.997452 9.03596910.964039 9.098066 9.99656239.10150410.898496 48
1319.0345829.997439 9.037144 10.9628569.0990659.996546 9.10251910.897481 47
1419.03574119.997425 19.038316 10.961684 9.100062 9.996530 9.10353210.896468146
1519.036896 9.99741119.039485|10.967513 9.10105619.99651419.104542110.895458145
169.03804319.997397|9.04005110.959349 2.10204819.9964989.105550 10.894450 44
17 9.03919719.99738319.041813 10.958187 9.103037 9.996432 9.106556 10.893444 43
18 9.040342 9.9973699.04297310.957027 9.10402519.9964659.107559 10.892441 42
199.0414859.99735519.044130 10.955870 9.105010 2.9964499.108567 10.891440 41
2019.04262519.9973419.04528410.954716 9.105 99219.99643319.109559110.890441 40
219.04376219.9973279.046434 10.953566 9.10697319.99641719.110556 10.889444139
2219.044895 9.9973139.04758210.9524181 9.107951 9.9964009111551 10.888449 38
239.0460269.997299 9.048727 10.951273 9.108927 9.996384 9.112543 10.887457 37
24 9.04715419.9972859.049869110.95013119.1099019.99636819.113533110.886467 36
2519.04827919.99727119.05100810.94899: 9.11087319.99635119.114521110.885478135
2619.049400 9.99725619.052144|10.947856 2:11194219.99633519.115507110.884493134
27/4.050$199.997242 9.053277 10.946723 :113302-996318 9.116491 10.883509133
28 9.0516359.997228 9.05440810.9455921 9. 1137742.9963029.11747210.882528131
2919,052749 2.997214 9.055535 10.944465 9.114737 9.996285 9.11845210.881548 31
3019.05385919.99719919.056640|10.943340 9.61569819.9962699.119429110.88057130
3119.05496619.9971859.057781|10:942219 9.11665619.99625219,120404|10.87959629
329.056071 9.99717019.058900 10.9411009.1166129.996235 9.121377 10.878623 28
339.0571729.9971569.060016 10.9399841 9.1185679.99621819.122348 10.877652/27
3419.05827119.997141 9.061130 10.938870 19.11951919.996202 9.123317 10.876683/26
3519.05936719.99712719.062240|10.937760 9.12046919.99618519.124284119.87571625
3619.06046019.99711219.063348 10.936652 19.12141719.99616819.125248|10.874751124
37 9.061551 9.9970989.06445310.935547 19.122362,9.996151 9.126211 10.873789123
38 9.06263819.997083 9.065556 10.934444 19:12330619.9961349.12717210.872828 22
3919.06372319.99706819.066655 10.933345 9.12424819.996117 9.128130 10.87187021
4019.06480619.997053|9.06775210.932248 2:12518719.99610019.12908710.870913/20
41 9.065885 9.99703919.068047 10.931153) 19.12612519.99608319.130041|10.869959119
429.066962 9.997024 9.069938 10.930062 19.127060 9.9960669.130994 10.86900618
4319.0680369.9970099.071027110.928973 19.12799319.9960499.131944 10.86805617
4419.069107 9.99699419.07211310.92788719.128925 9.996032 9.132893 10.867107 16
4519.07017619.99697919.073197110.926303 9.12985419.99601519.133839.10.866161115
4619.07124219.99696419.074278110.925722 19.13778119.9959989.134784[10.86521614
479.072306 9.9969499.075356 10.924644 9.131706 9.995980 9.135726 10.86427413
4819.073366 9.99693419.076432 10.923568 19.132630 4.995963 9.136666 10.863334 12
49 9.074424 9.99691919.077505 10.922495 9.13355119.995940 9.137605 10.862395 11
509.07548019.99690419.07857610.921424 9.13447019.99592819.138542 10.86145810
$119.
.07653319.99688919.079644|10.920356 9.13538719.99591119.13947610.86052419
529.0775839.9968749.080710110.9192909.136303 9.995894 9.140409 10.8595918
53.19.07863119.99685819.08177310.918227 9.137216 9.995376 9.141340 10.8586697
5419.079676 9.9968439.082833 10.917167 9.13812719.995859 9.14226910.8577316
5519.08071919.9968289.933891110.916109 9.13903719.995841|9.143196|10.8568645
5619.08175919.99681219.084947(10.915053 9.13994419.99582519.144121|10.85537914
579.082797 9.996797 9.08599910.914000 9.140850 9.995806 9.145044 10.85495613
1589.083832 9.9967829.087050 10.912950 9.1417549.995783 9.14596510.85 4035
5919.0848649.996766 9.088098 10.911902
9.142655 9.9957709.146885 10.853115
16019.08589419.99675119.089144110.910850 9.14255519.99575319.14780310.852197
| Co-fine | Sine Co-tang. | Tangent
Co-line Sine Co-tang (Tangenc IM
Dcgrce 83.
Degree 82.
(b)
+
1
2
1
O
1
Canon Triangulorum Logarithmicus.
Degree 8.
Degree 9.
M
M Sinc
Sine 1 Co-line Tangent Ca-tang.
Sine | Cº-fine Tangenc! Co-tang.
29.14355519.99575319.14780310.852997 7.19433219.99462019.199712|10.800287160
119.14445319.9957359.148728|10.851282 9.19512919.99460019.209529 10.799470159
29.145 349 9.9957172.149632 10.850368 19.125925 9.9945809.201345 10.798655158
319.1462432.9956999.159544 10.849456 9.196718 9.9945609.20215910.797841157
419.1471364.995681 9.15145410.848546 9.1975119.9945409.202971 10.797029 56
519.14802619.99566419.152363110.847637 9.198 30219.99451919.203782 10.79621855
619.14891519.99564619.153269|10.846731 9.1990919.994499 9.20459210.795408154
219.149801 9.995628 9.15417410.845825) 9.19987919.994479 9.205400 10.794600 53
819.1506869.995610 9.155077 10.8449231 9.2006669.994459 9.206207 10.793793152
919.151569 9.99559119.155978 10.844022 19.2014519.99443819.207013 10.791987151
1019.15245119.99557319.156877|10.843123 9.29223419.994418|9.20781710.792183150
1119.1533399.99555519.157775|19.842225 9.20301 79.99439819.20861910.79138 149
129.154208 9.9955379.158671 10.841329 9.203797 9.994377|9.209420110.790580 48
139.1550829.995519 9.159565 10.340435: 9,204577 9.994357 9.210220 10.789780 47
149.155957 9.99550119.160457 10.839543 9.205354 9.994336 9.271018 10.788982 46
1519.15683019.99548219.161347|10.838653 9.20613119.994316 9.211815710.788185145
16|9.15770019.995464!9.162236 10.837764 9.206906 9.99429519.212611110.787389144
17 9.15856912.9954469.163123 10.826877 9.207679 9.99427419.21340510.786595 43
18 9.1594359-99542,79.164008 10.835992 9.208452 9.994254 9.214198 10.98580242
1919.16030: 9.99540919.164892 10.835108 9.209222 9.9942339.214989 10.78501141
2019.16116412.99539019.165773110.834226 9,20999219.99421219.215780 10.784220 40
2119.16202519.99537219.166654 10.833346 9.21076019.99419519.216568/10.78343239
2219.16289519.9953539.167532 10.832408 9.2115269.99417119.21735610.732644/38
239.163743 3.995334 9.168409 10.831551 9.212:91 9.994150 9.218142 10.781858137
24 9.16460319.995310 2.169284110.830716 9.21305519.994129 9.218926 10.781074136
2512.16545419.99529719.170157110.829843 19.21381819.99410819.219710110.780290135
2619.16630719.0952739.171029 10.828971. 9.2145797.99408719.2 20491|10.779508134
27 9.167158.9.995260 9.171899 10.8281011 9.215338 9.994066 9.121272 10.778728 33
2819.1630089.995241 9.172767 10.827233/ 9.21609 9.994044 9.222052 16.77794832
299.1688569.9952229.1 73634 10.826356) 9.2168549.994024 9.222830 10.777170 31
3019.16970219.99520319.174499 10.825501] 9.21760919.99400319.223607110.71639332
3119.170546 9.995184.9.175362 10.8246389.21836319.99398219.22438:10.77561829
32 9.171387 9.995165 9.196224 10.823776 2.219116 9.9939609.22515610.774844 28
3312.17323019.995146 9.177084 10.822916 19.2198689.993939 9.325929 10.774071127
34 9.173070 9.9951 27 9.17794210.822057 19.220618 9.99391819.226704110.773300126
3519.173908 9.99510819.178799110.821201 2.22136719.9938 9719.227 47110.772529125
3619.17474419.99508915.179655110.820345 9.22211519.99387519.228 240|10.771760124
37 9.17557319.99507019.18050810.819992: 19.2228619.9938549.229007 10.770993 23
38 9.1764119.9950619.181360110.818640 2.223606 9.993832 9.229774 10.77022622
399.177242 9.995032/9.182211 10.817789 9.224349 9.993811 9.230539 10.769461 31
4019.178c7219.99501219.183060119.816940 19.22509219.99378919.23130210.768698 20
4119.17890019.99499319.18390710.8160939.2258339.99376819.232065|10.767935119
42 9.179726 9.994974 9.184752 10.815248) 19.226573 9.99374619.232826 10.767174.18
439.1895519.9949559.185597 10.81440319.22731119.99372519.23358610.766414 17
44 9.181374 9.9949359.186439 10.813561 9.228048 9.99370319.234345 10.765655116
4519.18219619.994916 9.18728910.812720
10.8127209.22878419.99368119.235103/10,764897115
4619.1830169.994896|4.183120 10.811380 19.23951819.99366019.235859110.764141114
4719.1838349.994876 9.188957|10.8110429.2302529.993638 9.236614 10.76338613
48 9.184651 9.994857 9.189794 10.810206 9.230984 9.993516 9.237368 19.762632/12
499.185466 9.994838 9.190629 10.809371 19.2317159.99359419.238120 10.76138011
5019.18628019.994818.9.191462|10.8085381 9.23244419.99357219.238872
2110.761128/10
5119.18709219.99479-19.192294110.807706
9.23317219.99355019,239622410.760378 9
5219.187903 9.994779 9.193124110.8068769.23389919.993528 9.24037510.
7.7596291 8
5319.188712 9.994759/9.193953 10.806047 19.2346259.9935069.24111810.75888217
$4.9.1895199.994739|9.194780 10.805220 9.2353499.9934849.24186510.758135
6
3519.19032519.99471919.195605 10.804394) 9.236073.9.99346219.242610110.7573901 s
5679.19113919.99469919.195440 10.803569 9.23679519.9934409.243354/10.756646 4
5719.1919339.994680 9.197253 10.802747 9.2375159.99341819.244097 10.755903) 3
589.192734 9.99466019.198674|10.80192€ 19.23833519.993396 9,244839 10.755161
5919.193534|9.994640 9.198894 10.8011069.23895219.99337419.245579 10.754421
6019.194332 9.99451019.199712/10,800287 9.23967019.99335119.246319110.753681/ .
+ Ca-fe | Sinc Co-tang. Tangent Co-line Sine I Co-tang | Tangent IM
Degrce 81.
+
+
1
1
Degree 8o.
A
Canon Triangulorum Logarithmicus.
Degree io.
Degree 11.
MSine 1 Co. fine. Tangent Co-tang. Sine 1 co-line Tangent Co-fang.
019.23967019.99335117.246319110.753681 4.28059919.99194719.288652|10.711348160
119.24038619.99332949.247057110.752943 19.28122919.99192219.289326110.710674159
219.2411019.9933079.247794 10.752206 9.281897 9.991897.9.28999910.710001155
3.9.24181419.993284 9.248530 10.751470 19.232544 9.991873 9.290671 10.709329157
4 9.242526 9.9932629.24926410.759736 9.2331909.991848 9.291342 10.708658156
$19.24323719.99324019.24999810.750002 9.28383614.99182319.29201310.707987155
619.24394719.99311719.250730 10.749270 9.28448019.99179919.292682|10.707318154
719.2446569.9931959.25.1461 10.74853919.285124 9.991774 9.293350 10.
0.706550153
819.2453639.99317-19.252191 10.747809 9.2857669.9917499.294017|10.705983152
919.246070 9.993149 9.252920 10.747080 19.2864089.991724 9.294684 10.705316 si
109.24677519.99312019.25364810.746352 9.28704819.99169919.295349 10.704651150
1119.24747819.9931049.25437410.745626 9.287688/9.9916749.296013119.703987149
129.2481819.9930.1 9.255200 19.744900 9.288326 9.991649 9.296677 10.70332348
139.248883 9.9950599.255824.10.7441769.2889649.99162419.297339 10.702661 47
149.2495839.9950369.256547 10.7434539.289600 9.99159919.298001|10.701999 46
1519.25023=19.99301 319.257269110.742731 2.290236 9.99159419.298662110.701338 45
1619.25093019.99299019.257990|10.742010 9.29987019.991549 9.299322110.700678144
179.25167719.99296719.253710110.7412901 9.29150419.991524 9.299980 10.700020 43
1819.252373 9.992944 9.25942910.7405719.292137 9.9914989.300638) 10.699362 42
19 9.25306719.99-9219.760146 10.7398549.29376819.99147319.301295 10.698705 41
2019.25 376119.99289819.60863 10.739137 9.2933999.99144819.301951210.698049 40
2119.25445319.992875|9.261578|10.738422 9.-949299.991422|9302607110.697393139
229.2551449.992852 9.262292 10.7377081 9.294658 9.9913979.30326110,69673938
239.2558349.992829 9.263005 10.7369959.2952869.99137219.303914 10.69608637
249.25652319.9928069.263717 10.736283 9.29591319.991346 9.30456710.69543336
2519.25721119.99-78319.264428110.735572 9.29653919.99132119.305218110.694782135
2619.25789319.99275919.265138|10.7 34862 9.29716419.99129519.305867 10.694131134
279.2585839.992736 9.265847 10.7341531229778819.991370 9.306519 10.69343133
2819.2592689.99271319.266558 10.733445 9.298412 9.991244 9.307168 10.69283232
2919.2599519.9926909.267261 10.732739 19.2990349.991219 9.30781610.692184/31
36 9.26063319.99266619.267967 10.732033 9.29965519.99119319.30846310.691537130
3119.26131419.992643|9.26867110.731329 9.30027619.99116719.309109/10.69089129
32.9.261994 9.9926199.269375 10.730625 9.3008959.99114'19.30975410.640246 28
33.9.26:67319.99259619.270778 10.729923 19.3015149.99111519.310399 10.689601 27
34 9.26335119.99257-19.271479 10.7292219.302132 9.99109019.311042 10.638958/26
3519.26402719.99254919.271479110.728521 9.30274919.99106419.311685119.688315125
3619.2647039.99252519.272178410.727322 9.30336419.99103819.312327 10.687673124
37 9.265378 9.9925019.27287610.727124 9.303979 9.991019.312968 10.68703223
38 9.266051 9.992478 9.273573 10.726427 9.304593 9.9909869.313608 10.686392 23
39 9.266723 9.992454 9.274269 10.725731 9.3052079.9909609.314247 10.685753125
4019.26739519.99243019.27496-10.725036 9.30881919.99093419.314885110.685115120
4119.26806519.99240619.275658110.724342 19.30643019.99090819.315523/10.684477119
42 9.268734 9.9923829.276351 10.723649 19.3070419.99088219.31615910.683841 18
4319.26940219.9923629.277043110.722957 19.307650 9.990855 9.316795 10.68320517
4419.2700699.992335 9.277734 10.722267 9.3082599.99982912.317430|10.68257016
459.27073519.99-3119.278424|10.721576, 9.30886719.99080319.31806410.681936115
4619.27140019.99228719.279113110.720887 9.30947419.99077719.318697 10,68130314
479.272063 9.99226319.279801 10.720199 9.310080 9.990750 9.319330 10.680670113
489.272726 9.99223919.280488 10.719512 9.3106859.9907249.319961 10.68003912
49 9.273388 9.9922149.28117410.7188269.311289 9.99069719.320592/10.67940811
Sol9.37404919.9931999.281858110.718142 9.3113999.99067119.321222 10.678778|19
5119.27470519.9921669.282542|10.717458; 19.31249519.990645 19.321851110.678149] ?
529.27536719.992142 9.28322510.716775, 9.313097 9.990618 9.32247910.6775211 8
$39.2760259.992118 9.283997 10.716093 19.313698 9.9905919.323106 10.6768947
54 9.27668119.99209319.284588 10.715412 19.31429719.9905659.323733 10.6762676
5519.27733719.9920049.28526810.714732 19.31489719.99053819.324358110.6756425
562.27799119.99204519.285946110.714053 9.395495|9.99051219.324983(10.67501914
57 9.2786859.992020 9.28662410.713376 9.316092 9.9904359.325607 10.6743933
5819.27929719.991996 9.287301 10.712699 9.3166899.990458 9.32623110.673769
599.2799489.9919719.287977 10.712023 9.31728419.99043119.326853110.673147
6019.28059919.99194719.288652110.711 348 9.33787919.99940419.327475(10.672525
1 Co finSine | Cu-tang. Tangent Co-ſine | Sine 160.tang. Tangent 1M
Degrcc 79.
Degree 78.
|
2
1
Canon Triangulorum Logarithmicus.
"
L
Degree 12.
Degree 13.
M Sine 1 Co-fine. Tangent Co-tang. Sinº | Co- (ệne |Tangent Co tang.
019.31787919.99040419.327475110.6725259.35208819.98872419.363364110.636636160
119.3184739.99037719-328095/10.671905 9.35263519.98869519.363940|10.636060159
219.319066 9.9903519.32871510.671285 9.3531819.9886669.364515 10.635485158
319.319658 9.990324 9.329334 10.670666 9.35372619.988636 9.365090 10.634910157
49.32025019.990-97 9:329953 10.6700479.354271 9.988607 9.365664 10.63433656
519-32084019.99027019-320570 10.6694309.35418519.98857819.366237 10.63376355
619.3214309.9902429.331187|10.668813 9.3553589.98954819.366810|10.633190154
79.322019 9.990215 9.331803 10.668197 9.355901 9.98851919.36738210.63261853
8 9.3226079.990188 9.332418 10.667582 19.356443 9.9884899.367953 10.632047 52
99.323194 9.9901619.333033 10.666967 19:356984 9.988460 9.368524 10.63147651
109.32378019.99013419.33364610.666354 9.35752419.98843019.36909410.630900 50
119.3243669.99010719-334259 10.665741 9.3580649.98840149.369663/10.630337149
1219.3249509.9900799.334871 10.665129 9.35860319.9883719.370232 10.629768 48
139.3255349.9900529.335482 10.664515 19.3591419.989 341 9.370799 10.629201|47
149.326117 9.9900259.336093 10.663907 19.359679 9.988312 2.371367 10.628633 46
1519.32669919.98999719.33670: 10.663298 9.35021519.98828219.371933/10.628067 45
1619.32728119.98997019.337311|10.662689 19.36075219.98825219.372499110.627501144
179.3278629.9899429.337919 10.662081 9.361287 9.98822319.37306410.626936 43
18 9.328441 9.9899159.338527 10.661473 9.36182319.98819319.37362910.626371 42
1999.32902019.989887 9.339133 10.660867. 9.362356 9.98816319.374193 10.625807/41
2019.32959919.98986019-339739/10.660261 9.36288919.98813319.374750 10,625244/40
2119.330176 9.98983219.340344110.659656 9.36342219.98810319.37531910.624681139
229.33075319.989804 9.340948 10.659052 9.36395419.98807319.375881 10.624119138
239.331328 2:989777 9.341552110.658448 9.3644859.98804319.376442 10:623558137
249-3319039.989749 9-342155 10.657845 19.365016 9.98801319.377003 10.622997/36
2519.332478|9.98972119.34275710.657243 2.36554612.98793319.377563/10.622437135
2619.33305 119.98969319.343358/10.656642 19.36607519.98795319.378122/10.621878134
27 9.33362419.9896659.34395810.656042 9.366604 9.987922 9.37863110.62131933
28 9.33419519.989637 9.344558 10.655442) 19.36713219.98789219.379239 10.620761 32
29 9.334766 2.9896099.345157 10.654843 9.3676599.987862 9.379797 10.620203/31
3019.33533719.98958119.345755 10.654245 9.36818519.98783219.380354 10.619646130
3119.3359069.98955319.346353/10.6536479.36871119.98780119.380910|10.619090129
3219.336475 9.989525 9.34694910.6530519.36923619.987771 9.381466 10.618534 28
339.337043 9.989897 9.34754510.6524551 9.369761 9.9877409.382021 10.617980127
349.337610 9.9894699.348141 10.651859 19.3702859.98771019.382575/10.617435126
3519.3381769.98944119.348735110.651265 2.37080819.987679/9.383129110.616871125
3619.33874219.98941319.349329|10.6506719.3713309.98764919.383683110.616318124
3719.33930619.9893849.349922 10.650078 19.371852 9.9876189.384234 10.61576623
3819.3398709.9893569.350514 10.6494869.37237319.987588 9.384786 10.615214 22
3919.3494349.989328 9.351106 10.648894 9.372894 9.98755719.385337 10.614663 21
4019:34099619.98929919.35169711@.648303 9.37341419.9875269.385888/10.614112120
4119.34155819.98927119.352287110.647713 9.37393319.98749619.386438|10.613562119
4219.3421199.9892439.352876 10.647124 9.3744529.9874659.38698710.613013 18
439.342679 9.989214 9.353465 10.646535 9.3749709.98743419.38753610.61246417
4419.343239 9.9891869.354053 10.645947 9.375 487 9.9874039.388084 10.611916 16
4519.34379719.9891579.354640 10.645360 9.37600319.987372 9.38863110.61136915
469.34435519.9891289.355227|10.644773 9.37631919.9873419.38917810.610822114
47 9.344912 9.989100 9.355812 10.644187119.3770359.987310 9.389724 10.610276 13
489.3454699.9890719.356398 10.643602 9.3775499.987279|9.390270|10.609730 12
499.346024 9.98904219.356982 10.643018 9.378063 9.9872489.390815 19.60918511
so 9.34657919.98901419.357566|10.642434) 9.37857719.93721719.391360 10.60864010
5 119.34713419.98898519.358149|10.6418519.37908919.9871869.391907110.60809719
$219.347687 9.9889569.358731 10.6412699.37.9601 9.987155 9.392467 10.6075538
53 9.3482409.988927 9.359313/10.64068719.380113 9.987124 9.392939 10,6070117
549.34879219.9888989.359893/10.6401079.3806249.987092 9.393531 10.6064696
5519.34934319.98886919.360474 10.639526 9.3811349.9870619.394074/10.605927
5619.34989319.988 84019.361053/10.638947 9.38164319.98703019.394614/10.605386 4
579.390443 9.9888119.361632-10.638368 9.3821529.986998 9.395154 10.6048463
589.350992 9.9887829.362210 10.637790 9.3826619.9869679.395694 10.6043062
599.3515409.9887549.36278710.637213 9.3831689.9869369.396233|10.603767
6019.35208819.98872419.363364 10.636636 9.33367519.98690419.396771110.603229) o
I Co.fone į 1 -
Sinc Co-tang. Tangent Co-sac | Sinc Co tạng Tangent M
Degree 76
Degree 77
Canon Triangulorum Logarithmicus.
1
!
Degree 14.
Degree 15.
M] Sine 1 Co-fine. [Tangent Co-tang. Sine 1 Co-fine (Tangenc! Co-tang.
019.38367519.98690419-396771|10.603229 9.41299619.98494419.428052|10.57194760
119.38418119.98687319.397369110.6026949.41346719.99491019.428557110.571442 sy
29.3846879.98684119.397846 10.6021549.4139389.98487619.42906210.57093858
319.3851929.98680919.398383|10.601617 9.4144089.9848429.429566 10.57043457
419.385697 9.9867789.398919 15.60108119.4148789.9848089.420570 10.579930156
$19.38620119.98674419.399455 10.600545 9.41534719.98477419.420573 10.579427155
619.38670419.9867149.399990|10.600910 9.4158159.9847409.431075 10.568925154
79.387207 9.9866339.40052410.599476 9.4162839.9847069.43157110.56842353
819.387709 9.9866519.401058 10.598942 9.41685019.984672 9.432079 10.567921152
9 9.388210 9.986619 9.401591 10.598409 9.41721719.98463719.432580 10.567420151
1019.38371119.98658719.402124/10.5978769.41768419.98460319.433080 10.566920150
119.33921119.98655519.402656/10.597344 19.41814919.9845699.433580119.566419149
129.3897119.9865239.433187 19.496813 9.41865519.9845359.434080 10.565920148
139.390210 9.9364919.40371819.596282 9.4193799.984500 9.434579 10.565421 47
149.3907089.9864599.404249 10.595751 9.419544 9.984466 2.439078 10.564922146
1519.391206 9.98642719.404778|10.595222 2.42000719.98443119.435576 10.564474145
169.3917039.98639519.405 30610.594692 9.42047019.98439719.436073|10.563927144
179.392199 9.9863639.405836 10.594164 9.4209339.9843639.436570 10.563430 43
18 9.3926959.986331 9.406364 10.593636 9.4213959.9843289.437067 10.56293342
1919.393190 9.9862999.406892 10.593608 9.4218569.9842939.437563 10.562437 41
2019.39368519.9862669.407419|10.5925819.42231719.9842599.43805910.561941/46
2119.39417919.93623419.40794510.5920559.42277819.98422419'438554110.561446139
229.3946739.9862019.408471 10.591529 9.423238 9.9841899.43954810.56095238
239.3951669.986169 9.403996 10.591001 9.423697 9.9841559.439543 10.560457137
24 9-3956549.9861379.409521 10.590479 9.4241569.98412019.440036 10.559964136
2319.39615019.9861049.410045 10.589954' 9.42461519.98408519.44052910.559471135
2619.39664119.98607219.410569 10.589431, 19.42507219.98405019.441022|10.558978134
279.3971319.986039 9.411092 10.588988 2:425530 9.9840159.441554 10.588486 33
2819-3976219.986907 9.411655 10.58838519.4259872.983980 9.442006 10.557994 32
2919.39811119.9859749.412137 10.587863 9.4264439.9839459.442497 10.557503 31
3019.3986009.98594219.412658/10.587343 9.4268999.9839109.442988/10.557011 30
3119.39908719.985909 9.413179|10.5.8682719.42735419.98387519.443479|10.556521129
32 9.399575 9.985876 9.413699 10.58630119.42780919.9838409.443968 10.556031 28
339.4000629.9858439.414219 10.585781 9.428264 9.9838059.444458 10.555542 27
34.9.4905499.985811 9.414738 10.5852629.4287179.983770 9.444947 10.555053/26
3519.40103519.98577819.415257|10.584743 9.42917019.98373519.445435|19.554565125
3619.40152019.98574519.415775|10.584225 9.42962319.98369919.44592310.554077114
37 9.4020059.9857129.41629310.583707 9.430075 9.9836649.446411 10.553589 23
38 9.4024899.985679 9.416810 10.5831909.430507 9.9836299.446898 10.553102 22
399.402972 9.985646 9.417326 10.582674 9.430978 9.9835939.44738410.55261621
4019.40345519.98561319.417842 10.582157 9.43142919.98355819.447870|10.552129120
+119.40393819.98558019.418357|10.581642 9.43187919.98352319.448356 10.551644119
429.4044209.9855479.418873 10.581127 9.432328 9.983487 9.448841 10.55115918
4319.404901 9.9855139.419387 10.580613 9.4327789.983452 9.449326 10.55067417
44 9.405382 9.9854309.419901 10.580099 9.4332069.9834169.449870 10.550181216
4519.40586219.98544719.420415118.579585 9.43367419.983380 9.450294 10.55970615
4619.40634119.98541419.42092710.579072 9.43412219.98334519.450777 10.541722314
47 9.406820 9.9853809.421440 10.578560 9.4345699.983309 9.451260 10.548740113)
489.407-999.985347 9.421951 10.578048 9.4350169.98327319.451743 10.5 48257 12
499.407776 9.98531419.42 246310.577537 19.435462 9.98323819.452225 10.5477751
5019.40825419.98528019.422973110.577026 19.43591819.983202 9.45:706 10.5 47294110
5119.40873119.9852479.42348410.576516 19.43635319.983166 9.453187 10.54681319
$29.40920719.985213 9.42399310.5760079.4367989.9831309.453668 10.5463328
$319.409682 9.9851809.42450310.575497 9.437242 9.983294 9.454148 10.5458527
5.49.410157 9.9851469.42501110.574989 19.4376869.983053 9.454629 10.5 45372 6
5519.41063219.98511219.425518/10.574480 9.43812919.9830229.455107 10.54489315
$619.41110619.9850799.426027|10.573973 9.43857219.9829869.455586 20.54441.
5? 9.4105799.985045 9.42653410.573466 9.439034 9.9829509.45606410.5439.36 3
5819.4120529.935011 9.427041 10.572959 9.4394569.982914 9.456542 10.54345812
5919.4125249.9849779.427547 10.572453 9.439897 9.98287819.457019 10.542980 1
609.41299619.98494319.428052 10.571947 9.44033819.982842 9.45749610.5425031
| Co-fine | Sine | Co-tang | Tangent Co-fine I Sine 1 Co-tang | TangentIM M
Degree 75.
C
1
1
Degree 74
Canon Triangulorurn Logarithmicus.
1
Degree 16.
Degree 17
ML Sine | Co-int Tang@nt Ce-tang.
Sinc 1 Co-fire Tangent | Co tang.
019.44033819.98284219.457496|10.5425039.46593519.98099619.485339110.514661160
119.44077819.98280519.457973|10.542027 9.4663489.98055819.48579110.514209159
29.44121819.93276919.458449|10.541551 19.46676119.980519 9.486242 10.513758158
319.4416589.9827339.458925/10.541075 9.4671739.980480 9.48669310.513307157
419.442096 9.9826969.459400 101540600 9.4675859.980441 9.487143 10.5 12857156
519.44253519.98266519.459875 10.5401252.4679969.98040319.48759310.5 12407155
619.44297319.93262319.460349|10.539651 9.4684279.98036419.488043 10.511957154
7 9.443416 9.982587 9.460329 10.5391779.468817 9.98032519.48849310.511507153
89.44384819.98255019.461297 10.5387039.469227 9.9802869.488941 10.511059152
919.444284 9.9825149.461770 10.5382309.469637 9.980247 9.489390 10.5 10610 5.
109.444720 9.98247719.46224210.537758 9.46044619.98020819.489338110.510162150
1119.44515519.98244119.462714/10.537285) 9.47045319.98016919.490286|10.509714149
129.4455909.982404 9.463186 10.536814 19.4718639.9801309.490733 10.509261148
13 9.4460259.982367 2.46365810.536342119.471071 9.9800919.491180 10.508820147
149.4464599.9823309.4641 29 10.535871 9.4716789.9800529.491627 10.508 373146
3519.44689319.98229419.464599110.535401 9.47208619.98001219.492073110.507928145
169.447326 9.98225719.465069110.534931 19.4724929.97997319.492519|10.507481144
17 9.447759 2.982320 9.465539 10.534461 9.47289319.9799349.492964110.507035 43
182.448199 9.9821839.46600810.533993 9.473304|9.9798949.493410 10.506599 42
199.4486239.9821469.466476 10.5335231 9.4737199.979855 9.493854|10.506145 43
2019.44905419.9821099.466945110.533055 2.4741159.97981619.494299 10.505701140
219.44948519.98207219.467413110.532587 9.47451919.979776 9.494743 10.505257/39
229.449915 9.9820359.467880 10.532120 9.4749239.9797379.49518610.50481338
239.450345 19.98199819.468347 10.531653 9.475327 9.979697 9.495630 10.504370137
249.450775 9.98196.119.468814 10.531186 9.4757309.979658 9.496073 10.503928 36
2519.45120319.98192319.469280 10.530720 9.47613319.97961819.49651510.523485135
3619.45163219.93188619.469746/10.530254 9.47653619.97957819.496957110.503043134
279.452060 9.98184919.470211 10.529789 9.47693819.9795399.49739910.502601 33
289.452488 9.981812 2.470676 10.529324 9.477340 9.97949919.497840|10.502160 32
2919.452915.9.981774 9.471141110.523859119.477741 9.97945919.49828210.501718 31
309.4533429.98173719.71605/10.528395 1964,7814219.97941919.498722 10.501278130
3119.45376819.98169919.472068110.5 27931 19.47854219.9793809:499163110.500837129
32 1.45419419.981662 9.472532 10.827468 9.4789429.979340 9.499602 10.50039828
3319.45461919.981624 9.472995 10.527005) 9.479342 9.979300 19.500042 10.49995827
3419.455044 9.981587 9.473457 10.526543 9.4797419.9792609.500481 10.499519126
3519.45546919.98154919.47391910.526081 9.48014019.97922019.500920 10.49908025
3619.45589219.98151219.474381|10.525619 9.48053819.97918019.501359 10.498641124
3719.4563169.981474 9.47484210.525158 9.4809369.9791409.501797 10.498203/23
38 9.456739 9.981436 9.475303 10.524695) 19.48133419.9790999.502234 10.49776522
13919.45716219.981358 9475763 10.524237 9.48173119.97905919.50267210.49732821
4019.45758419.98136119.476223 10.823777 2.48212819.97901919.50310910.496891 20
419.45806619.981323|9.476683|10.523317 9.48252519.978980 9.50354610.496454119
429.4584279.98128519.47714210.522858 2.482921 9.97893919.50398210.496018118
439.4588489.981247 9.47760110.5223999.483316 9.978898 9.504418 10.495582 17
4419.459268 9.981209 9.478059 10.521941 19.48371119.998858 9.504854 10.45514616
4519.45968419.98117119.478517119.5214839.48410619.97881719.505289110.494711115
4619.460108 9.93113319.478975|10.521025 19.48450119.97877719.505724110.494216|14
479.460527 9.9810959.479432 10.320568 19.48489519.978736 9.50615810.
1.493841113
48 9.4604469.981057 9.47988910.52011 9.48528919.97363619.506593|10.493407 12
499.46136419.931019 9.480345 10.519655 9.485682 9.9786559.507026 10.492973"}
5019.46198219.98098019.480801110.5191999.48607519.97861519.597459|10.492540140
5719.46219919.9809429.481257110.5187431 19.43646719.97857413.507892110.49210719
52 9.462616 9.9809-49.48171210.518288 9.486959 9.978833 9.508326 10.491674
539.463032 9.9808669.48:167 10.517833 9.4872519.97849319.508759110.4912411 7
$49.463448/9.980827 9.482621 10.517379 9.4876429.9784529.509181 10.49:3096
5519.46386419.9807899.483075|10.516925 9.4883319.973411 9.50961:10.4903775
3619.464279|9.9807509.483528/10.5164709.4384249.97837019.51004410.41.9946
579.464694 2.980712 9.48398-10.516018 9.488814 9.978329 9.510486 10.489515 3
5819.4651089.980672 9.484434 10.515505 9.489204 9.978238 9.51091610.4896842
599.465522 9.9806359.484887 10.515113 9.48959319.9782479.511346 10.4856544
6019.46593519.98059619.485339 10.514661 9.48993219.9782-619.511776115.4882251 0
I Co fine | Sine Coutang.1 Tangent Co-line / Sine 150.tang. I Tangene !
Derec;8
!
11
Degree 73.
!
Canon Triangulorum Logarithmicus
A
Deorce 18.
Degree 19.
MI Sine
Sine 1 Co-finc. [Tangent Co-tang. Sine I wo-fine Tangentt Co-tang. I
09.48998219.97820619.511776|10.488224 19.51264219.97567019.536972/10 1463926160
19.49037119-9781659.51220610.4877949.51300919.975626.!9.537382(10.462613159
29.490759 9.9781249.51263510.487365 9.5133759.9755839.537792 19.46.220858
39.491147 2.9780839.51306410.486936 9.513741 9.97553919.538202 10.46.1791157
49.4915349.9780429.51349310.486507 9.5141074.9754569.538610 10.461389156
$19.49192219.978000 9.51392761 0.486079 9:51447219.975452 9.535020 10.460980155
619.4923089.9775599.514349|10.485651 9.524837|2.97540819.5 39429|10.460571154
79.4926959.9779139.514777|10.485223 9.51520219.975364 9.539837 10.46016353
8 9.493080 9.97787719.515204 10.48479619.5155669.975321 9.540245 10.459755152
99.493466 9.977835 9.515631. 10.4843699.515930 9.975277 9.54-653 10.45934751
109.49335119.97979419.516057110.48394: 9.51629419.97523319.541061 10.458939150
1119.49423619.97775215.516484|19.4835169.5166579.9751892-541463.110.458932149
129.4946209.977711 9.516910 10.483050 9.5170209.9751459-541875 10.45512548
132-49500519.9776699.517335 10:48:665 9.517382 2.9751012-542281|19.45771947
149.495388 9.97762819.51776110.482239 9.597745 9.975057 9.542688 -0.45731346
1519.495773969775869.518185|10.481814 9.5:8107|9.975013|9.543094|10.456906145
169.49615419.97754419.518610|10.4813909.5184689.974969|9,543499|10.456501144
1719-49653719-9775039.519034 10.480966 2.5188299.97492519.5 43905 10.456095 43
1819.4969199-9774619.519458 10.480542 2.5191909.974880 9.5 44310 10.455690/42
199.497301 9.9774197.51988210.480118 9.51955119.9748369.544715 10.455.28541
2019.49768219.97737719.520305110.489695 9.51991119.97479219.545110 10.454881|40
2119.49806319.97733519.520728110.479272 2.52027119.97474719.545524/10.454476139
2:19.49844419:9772939.521151110.4788459.5206319.9747039.545927 10.454072/38
-39.4988249.9772519.52157310.478427 19.5209909.9746599.546331 10.45366937
249.499204 9.979209.521995 10.478005 9.521349 9.9746149.546735 10.453265136
2519.49958419.97716719.522417 10.4775839.52170719.9745709.547138 10.45286:135
2619.49996319.97712519.522838|10.477163 19.52206.519.97432519.547540 10.452459134
27 9.500342 9.977083 9.523259.0.47674272-5422423 9.974480 9.547943/10.453057133
289.500720 9.977041 9.52367910.4763209.522781 9.974436 9.54834510.451655132
29 9.5010995.9779999.524109 10.475 900 9.523138 9.974391 9.548747 10.451233li
3019.501476|5.97795619.5 24520110.475480 9.52349519.9743469.54914910.450851130
3119.50185419.97691419.524939|10.475060 19.52385119.9743029.549550210.450450129
3219.502231 9.276372 9.525359 10.474641 9.524208 9.97425719.549951 10.450049/28
139.5026079.976830 9.525778 10.474222 9.5245649.974272 9.55035210.449648 27
349.502584 9.97678719.526197 10.473803 9.524920 9.97416719.550752 10.449248 26
3519.50336019.97674519.52661510.473385 19.52527519.97412214.551152|10.448848 25
3619.50373519.976702\5.527033|10.473967 9.52563019.97407719.551552(10.448448124
3719.504110 9.9766609.52745110.472549 9.525984 9.974032 9.551952 10.44804823
3819.50448519.9766179.5 27868 10.472132 19.526339 9.9739879.55235110.447649 22
39 9.5048409.9765749.528285 10.471715) 19.526693 9.973942 9.552750 10.447250 11
4919.50523419.97653219.528992|19.4712989.5270469.97389719.553149|10.4.46851 20
4119.50560819.97648919.529118110.470881 9.52740019.9738529.553548(10.446452119
42 9.50598119.9764469.529535:0.4704651 9.5297539.973807 9.533946|10.446054 18
43 9.5063549.9764049-529950 10.470049 9.5281059.9737619.554344 10.44565617
449.50672712.976361 9.530366 10.469634 9.52845819.9737169.554741 10.445259 16
4519.50709919.97631819.530781|10.469219 19.5288109.97367119.555139110.444861115
4619.50747119.976275|9.531196|10.4688041 9.52916119.97362519.555536|19.444464114
4719.507843 9.9762329.531611110.468389 9.52951319.9735809.555932 10.444068 1 13
4819.508 214 9.9761854.532025 10.467975 9.5298649.973535 9.55632910.443671|12
49 9.50358519.9761469.53243610.4675619.5 303349.973489 9.556725 10.44337515
50 9.50895519.9761039.53285-10.467147 9.53056519.97344319.557121|10.442879110
5119.50932619.97606019.533266 10.466734) 19.5 3091519.97339819.557517|10.442483
529.5096969.9760179.5 33679 10.466321 9.5312659.973352 9.557912 10.44208818
$39.5100659.9759737.534092 10.465908 19.531614.9.97330719.598308|10.4416931 7
549.5104349.97593019.534504 10.465496 9.53196319.97326119.558702 10.4412986
5519.51080319.97588719.53491610.465084 9.53231219.97323519.559097110.44090315
5612-51117119.97584419.535328|10.464672 9.53266119.97316919.55949110.44050914
57 19:5115409.975800 9.535739 10.464261 9.5330099.97312319.55988510.440115
5319.5119079.97575719.536150 10.463849 9.533357 9.973078 9.560279 10.439721
599.5122759.97571314.536561 10.463439 9.5337049.97303219.560673 10.439327 i
6014.51 264219.97567 19.536972110.463028 19.53405-19.9729869.561066|10.4389341 •
| -fine | Sine | Co-tang. | Tangent Co-fine! Sine I Co-tang | TangentIM
Degree 71.
1
2
Degree 7o.
1
.
Canon Triangulorum Logarithmicus.
Degree 20.
Degree 21.
My Sine | Ca-line [Tangent} C-tang. Sinc 1 Co-fine (Tangent | Co.lang. I
09.53405 219.9729869.561066|10.438934) 9.55432919.97015219.584177110.41582216
119.53439919.97294019:561459 10.438541 9.55465819.97010319.5*84555110.415445 159
29.534746 9.97289419.561851 10.4381481 9.554987 9.9700559.584932 10.41506858
314.535091 9.972848 9.562244 10.437756 9.555315 9.970006 9.585308 10:414691157
419.5354379.972801 9.562636 10.437364 9.555643 9.9699579.585686 10.414314 56
519.53578219.99275519.563028 10.436972 9.55597119.96990919.586062110.413938155
619.53612919.97270919.563419|10.436580 9.55629919.9698609.586439 10.413561154
7 9.536474 9.9726639.563811 10.436189 9.556626 9.969811 9.58681510.413185153
819.53681919.97261719.564202 10.435798 19.55695319.9697629.587190 10.41:800152
99.5371639.9725709.564592 10.435 401 9.557279 9.96971319.587566 10.412434 51
1019.53750719.97252419.564983/10.435017 9.557606 9.96966519.587941110.412059150
1119.537851 9.97247719.565373|10.434627 9.58793219.96961619.588316|10.41168449
12 9.5381949.9724319.565763 10.434237 9.558258 9.969567 9.588691 10:411309 48
1319.5385.379.9723849.566153 10.433847 19.558583 9.969918 9.589066 10.410934 47
149.538880 9.972338 9.566542 10.43345719.558909 9.969469 9.589440 10.410560 46
1519.53922219.97229119.566932110.433068 9.5592349.96941919.589814/10.410135/45
1619.53956519.97224514.567320110.4326799.5595589.96937019.590188|10.409812 44
17 9.53990719.97919819.567709 10.432291 19.559883 9.969321 9.590561 10.409438 43
118 9.540249 9.97215119.568097 10.431902 19.5602079.969272 9.590935 10.409065142
1919.5405909.972105 9.568486 10.431514 9.560531 9.9692239.591308 10.408692 41
2019.54093119.97205819.569873110.431126 9.56085519.96917319.591681110.40831940
2119.5.41272 9.97201119.569261110.430739 9.56117819.96912419.592054110.407946139
229.541612%.971964 9:569648 10.4303511 9.56150119.9690759.592436 10.407574 38
23 9.54195319.9719179.560035 10.429964 9.561824 19.96902519.592798 10.407201 37
249-542292 2.971870 9.560422 10.429578 9.562146 9.9689769.593170 10.40682936
259.54263212:97182319.560809110.4-9191 9.56246819.9689269.593542 10.406457135
2619.54297119.97177619.571195 10.428805! 9.56279019.96987719.593914/10.406086134
27 9.5433109.971729 9.571581 10.428419 9.5631129.9688279.59428510.40571533
28 9.5436499.97168219.571967 10.428033 19.5634339.968777 9.594656 10.405 344 32
2919.543987 9.971635 9.572352 10.4276481 9.563754 9.9687289.59502710.405073131
3019.54432519.97158819.572738/10.427262 9.56407519.96867819.595397) 10.404602130
3119.5446639.9715409.57312310.426877 9.5643969.9686289.595768110.404232129
3219.54500019.97149319.573507110.426492 19.564716 9.96857819.596138 10.403862 28
3319.54533819.9714469.57389210.4261089.565036 9.963528 9.596508(10.403492 27
349.545674 9.97139319.574276 10.425724 9.565356 9.9684789.596878 10.403122 26
3519:54601119.97135119.57466c110.4253401 9.56567519.96842819.597247110.402753125
3612.54634719.971303.575044|10.424956 9.56599519.96837819.597616|10.402384124
3719.5466839.9713569.575427 10.424573 9.566314 9.968328 9.597985 10.402015123
389.547019 9.971208 9.575810 10.42418.9 9.566632 9.96827819.598354 10.401646 22
3919.54735419.97116119.57619310.423807 9.566951 9.96822319.598722 10.401 27721
4019.5476899.97111219.57657610.423424 9.56726919.968178 9.599091 10.40090920
4119.54802419.97106519.576958110.42304.1 9.56758719.96812819.599459110.400541119
429.5483589.971018 9.577341 10.422659 9.567904 2.968078 9.599827 10.400173 18
439.54869319.9709709.577723 10.422277 9.5682229.968027 9.690194 10.399806 17
4419.549026 9.97092219.578104/10.421890 9.568539 9.967977 9.60056- 10.399438 16
4519.54936019.97087419.57848619.4215.14 19.56835519.96792719.600929|10.39907.1115
4619.54969319.97082619.578867|10.4211331 19.86917219.96787619.601 296|10.398704114
47 9.5550269.9707799.579248/10.420752 9.569488 9.967820 9.601662 10.39833713
48 9.55035919.97073119.579628 10.420371 9.569804 9.96777519.603029 10.39797112
49 9.5506929.9706339.580009 10.419991 9.5701209.9697259.632395 10.39760511
5019.55102419.97363419.580389|10.416611 9.57043519.96767419.672761110.397239/10
$119.55135519.97053612.580769|10.419231 9.57095119.96762319.603127|10.3963739
$219.551687 9.970538 9.58114910.41885 11 9.571065 9.967573 9.603493 10.396507) 8
5319.552018 9.970490 9.58152810.418472 9.571380 3.96752219.603858 10.39614217
549.552349 9.970442 9.581907 10.418092 9.571695 9.967471 9.604223 10.3957776
ss19.55268019.97039419.582286 10.417713 9.57200919.96742019.604588|10.395412 s
5619.55301019.97034519.582665|10.417335 9.57232219.96737019.604953 10.3950471 4
579.5533409.970297 9.583043 10.416956 9.5726369.967319 9.605317 10.3946831 3
582.553670 4.970249 9.583422 10.416578 9.572949 9.9672689.695687|10.3943182
5 9 9.554000 9.970200 9.583800 10.416200 9.5732639.9672179.606046 10.39395-41
6019.55432919.97018219.58417710.415813 19.57357519.96716619.606409/10:39359010
+ Cơ in 1 Sine | Co-tang. Tangent | Co-sae | Sine (Co-tang/ Tangent !M
Degree 68.
Degree 69.
1
1.
-
Canon Triangulorum Logarithmicus,
.
۲۶
Dezice 22.
Degree 23.
M Sime 1 Co-fine. [Tangenel Cotang Sine 1 6s-fine Tangent! Co-tang.
ol9.57357519.96716619.(05499110.3935909.59187819.96402619.627852|10.372148160
119.57338819.96711519.606773110.393227 9.59217519.9639729.628203110.371797 159
29.574200 9.9670649.60713€ (10.392863 9.5924739.963919 9.628554 10.37144658
39.574512 9.967012 9.607500 10.392500 9.5927709.963865|9.62890 5 10.37.095 57
419.574824 9.96696119.60706-10.392137 9.5930679.963811 9.629255 10.370744 56
519.5751.3519.96691019.608225110.391774 9.59336319.96375719.62960510.370394155
619.57544719.96685919.60858810.391412 19.59365919.96370319.629956\10.370044154
719.5757589.96680719.603750 10.391050 9.59395519.963650 9.630306j10. 369694153
819.57606819.96675619.609312 10.390688 9.594251. 9.9635969.630655 10.369344 52
99.576379 9.9667051.609674 10.390326 9.594547 9.9635 42 9.631005 10.368995 51
1019.57663919.96665219.600036 10.389964 9.5948429.963498|9.631354110.368645150
119.57699919.9666029.610397|19.3876039.595137 9.963433 9.631704110.368296149
129.577309 9.0665507.61075510.
.389241 9.5954329.9633799.632053 10.367947 48
139.57761819.96649219.611119110.398889 9.595727 9.963325 9.632401|10.367598 47
149.577927 4.9664479.61!481|19.288520 9.596021 9.96327119.632750 10.367250 46
19.57823619.966395|9.611841|10.388159 2.596315|9.96321719.633098f10.366901145
11619.57854519.96634419.012201110.387799
10.387799 9.59661019.96310219.633447110.366553144
11719.57385319.06629-4.612561 10.;8743 9.59690319.9631089.633795/10.36620543
1819.5791611965662409.61 292110.387078 9.597196 9.963054|9.634143 10.365857142
99.5794659.5061889.613281|19.386719 9.597490 9.962999 9.634490 10.36551041
2012.57977719.,66136.9.6: 364111.0.386359 9.59778319.962945 19.6348 38 10.365162140
2119.58008419.966034|9.61400510.386000 9.59807519.9628929.635185 10.364815139
229.580392 19.966032 9.61435 y 10.385641 9.598 368 9.962836 9.63553010.364468 38
239.5806989.96598019.6:471810.385282 19.598667 9.962781 9.635877 10.364121 37
24 9.581095 9.96592819.61507710.384923 19.598952 9.962726 9.636226 10.363774136
25 9.5813119.9658769.615435110.384565 9.59924419.96267219.636572 10.363428/35
2617.58161019.965824|9.615793110.3842071 9.5995362.96261719.636.91810.363081]34
2719.5819239.905772 9.6.615.0-383448 9.5998 27 9.962562 9.637205 10.36273533
2812.5822299.9657209.61650910.38 3491 9.600118 9.962507 9.637610 10.362389 32
299.5825349.96566319.61686710.383133' 9.603409 9.9624539.637956 10.362044 31
309.5828409.96561519.617224110.382776 9.60070019.96239819.638302|10.361698130
3119.5831449.96556319.617581|10.382418 9.60099019.9623439.638647(10.361353129
32 9.583449 9.5655112.61793810.382061 9.6012809.962288 9.638992 10.361007 28
3319.58375319.9654539.618295 10.381705 9.601570 9.962233 9.639337 10.360662 27
349.584058 9.965406 2.618652 10.381348 9.6018609.9621789.639682 10.360318126
3519.58436119.96535319.619008|10.380992 19.60214919.96212219.640027|10.35997325
3619.58466519.96530115.619364110.380635 9.60243919.962067|9.640371 10.359629
3719.5849689.9652489.619720 10.380279 19.6027289.962012 9.640716 10.359284.23
3819.585271 9.965 195|9.620076 10.379924 19.60301719.961957 9.641060 10.358940122
3919.5855749.9651439.62043 10.3795689.6933059.9619029.641404 10.35859621
1019.585877 9.96509019.620787|19.379213 9.60359419.961846 9.641747 10.358253120
+119.5861799.96503719.621342|10.378858 9.60388219.96179119.642091 10.357909119
+29.586481 9.9649849.621497 10.3785039.6041709.961735 9.642434 10.357566|18
9.58678319.964931|4.621852|10.378148
•378148 9.604457 9.951680 9.642777 10.35722317
14 9.587085 2.96487819.62220610.377793 9.604745 9.961624 9.643120 10.35698016
+519.587386 7.96482519.622561110.377439 9.60503219.96156919.643463110.356537/15
4619.58763714.964772|4.622915110.3770859.60531919.96151319.643806 10.356194 14
+7 9.587981 9.964719 9.62326910.376731 9.605606 9.961458 9.644148 10.355852113
1819.58928919.96466*7.623623 10.376377 9.605892 9.961402 9.64449010.355510 12
+99.588589 4.964613 9.623976|10.376024 9.6061799.961346 9.644832 10.35516811
5019.583890119.96456019.624330|10.375670 9.60646519.961290 9.645174110.354826 LO
5119.58919019.96450719.624683|10.375317 19.60675019.96123519.645516|10.39448419
529.589489 9.96445419.625036 10.374964 19.607036 9.961179 9.645857 10.3541428
139.5897899.264400 3.625388 10.374612 9.6073229.961123 9.646199 10.3538017
5419.590088 9.96434719.625741 10.374259 9.607607 9.961067 9.646540 10.35.3460
1519.59038719.96429419.626093|10.373907 9.60789219.961011|9.646881|10.3531195
169.5906869.96424019.626445|10.373555 9.608 1769.96095519.64722210.352778
5719.590984 9.96418719.62679710.373203 9.60846: 9.9608999.64756210.352438 3
5812.5912829.964133 9.62714910.372850 9.6087459.96084219.647903 10.352097
5919.5915809.9640894.627501 10.372499' 9.6090299.9607869.648243 10.351757
6019.591878 9.9649:69.627852 10.3721 48 9.6093139.96073019.648583 10.351417
| Co-fine | Sine I Co-iang. Tangene Co-fine | Sine | Co-tang | Tangent LM
Degree 67:
Degree 66.
(a)
1
2
I
1
Canon Triangulorum Logarithmicus.
1
1
I
1
1
12519.6:6;35:9.459310.9.657028 10.342972; 9.6.265714.45578919.679869110.323131135
Degree 24.
Degrec 25.
M] Sinc I Co-line Tangentt Ca-tong.
Suc | Ce- tnt {Tangent Co tang.
019.60.931319.96073019.648533!10.351417 19.62594819.95727619.660672710.332327160
119.60259719.96067419.648,27110.351077 19.626219 9.9572179.669002|10.330998159
2 9.60988019.96061719.64.926; 10.350737 9.626490 9.9571589.669332110.330668 58
39.6101639.9605619.649602 10.350398 9.6267609.9570999.669.61 10.330339157
4 9.61044619.96050519.549942 10.350058 9.627039 9.95704019.669990 10.330009156
$9.61072919.96044919.650281110.349719 19.62730019.95695719.670320110.329636155
6.9.61101219.960;929.65062010.3493809.62757019.95692219.670649 10.329351154
79.611294 9.960335 9.650959.10.3490411 19.62784019.95686219.670977 10.32901253
8 9.6115769.960279 9.651297 10.3487031 9.628109 9.956803 9.671306 10.328694 52
9 9.61185819.960222 9.651636 10.348364 9.628 378 9.956744 9.671634 10.328365151
1019.61214019.960165 9.651974 10.343026 9.62864719.95668419.671963|10.328037150
1119.61242119.96010919.65231210.347638 7.6289169.95663519.672-91110.327709149
12 9.612702 9.960:529.652650 10.347350 9.6291849.956565 9.672619 10.327381 48
13 9.61298319.95949819.65298810.347013 9.62945319.9565069.672947 10.32705347
1419.613-5419.959938 9.653326 10.346674 9.6297219.9564469.673274 10.3267=5 46
1519.61354519.99988119.653663 10.346337 9.62998919.956387 9.673603/10.326343/45
169.61382519.95982419.65 4009110.345999 9.63023719.9563-719.673924110.326-70144
17 9.6141059.95976819.554337 10.345662 9.6305249.95626719.674-56.10.3:574343
18 9.614385 9.959710 9.654674 10.345325 9.63079219.956208 9.674584 10.325416 43
1919.614665 9.959653 9.555011 10.344984 9.631259 9.956148 9.674910 10.3=5089410
2019.61424419.9595969.655343|10.344652 9.6313-619.95608819.675237110.324764/45
Si
2119.6152239.95953919.655604.110.3443169.6315929.95602919.67556410.3=4436139
2:19.6155529.959482 9.65602010.3439809.631859 9.95596919.675890 10.324110 38
2319.61578119.959425 9.65635610.343643 9.63:125 9.955909 9.67621610.3²3783137
2419.6160609.959367 9.656692 :0.743393! 9.63239249.955849 9.676543 10.323457136
2619.016616 9.9592539.657363 10.34-636 9.6329-319.9557399.677!.4110.32.305134
2719.61689419.95914519.657699110.342301 9.6331899.955661 9.677520110.32:12:0133
28.9.6171729.99913819.658034 10.3419.50 9.6334549.9556099.677845 10.32:15413
29 9.657450 9.9570509.03:3699.3415312.6337:92.9555489.673173 10.32182031
3019.6177:719.979:2312.658704110.341296119.63398419.95548819.678496110.321504/30
3119.61800419.958965|9.659039110.9409201 9.63424919.9554289.578321 10.32117929
32 9.613:8:19.9529987.65937310.340627 19.63451491955367 9.679146 10.320954/28
3319.618559 9.9588500.659703 10.340292 19.6347734.955307 1.67947110.320527 27.
34 9.613334 9.658799.66004210.339958 19.61350429.955246 9.6747951:0.320:05/26
3519.619110 9.9587349.660376110.329624 9.67530619.95518619.6801:0110.919880125
369.6193869.9580779.660710710.3392909.63557919.095512519.680444110.319556124
37 9.6166629.95861919.661043 10.338957 9.6358339.9550659.68076 10.31923:12
38 9.61993819.958561 9.66137:10.339623 9.636097 9.955004 9.68104210.318909122
39 9.620213 9.9585039.661710 10.3382929.63636€14.9549449.671416 10.313584121
4919.62048812.95844519.662043/10.3379569.63662319.9548339.631740110.318260120
4119.620763 9.9583879.662376|19.337623 9.6363869.9543-30.682063|10.3179;7119
42 9.621038 9.958329 9.662709 10.337297 2.6371487.9547629.63239619.317613,18
43 9.6213139.958271 9.6630410.3369581 9.63741119.954701 9.682710110.317-9917
44 9.621587 9.958:12 9.663374 10.336625 7.637673 9.9146409.53303319.316907116
4519.62156119.95815419.663707119.3362931 9.63793519.95457919.683356110.316649115
4619.62213519.9580969.664039110.335961 19.63819719.95451819.683678|10.3.16321114
47 2.6224299.9580389.6164371110.335629 9.6324589.954457 9.654937 10.315,79113
4319.62268219.957979|9.664703110.3352971 9.6-38720 9.9543969.684724 10.31567612
499.622356 9.95792119.665035 10.334965119.63898119.95433519.6046.612.31535411
502.62322919.957862|9.665766|10.3346341 9.63924219.97427419.684968|10.315032110
5119.62350219.95780419.665697|10.334302 19.6395239.9542139.685290|10.31.710
5219.62377619.9577459.666029 10.333971 9.6397649.954153 9.685612115.712-13
53 9.624047 9.95763719.66636710.33364019.640024 9.95409-19.685934|10.3147667
54 9.6243199.957623 9.6566.110.3333091 9.6402349.954029 9.62625*10.315745.
5519.62459119.95757019.667621.10.332979 9.64054419.95496819.686577110.31342315
569.62486319.95750119.667352110.3326489.64080 14.9539069.636893|10.31310214
579.5251349.957452 9.667682110.332318 9.6410649.9538459.687-19 10. :78! 3
58.9.6=1406 9.957393|9.668012 10.331987 19.6-413239.953783 9.687540-10.311.460
5919.6:5(779.9573349.66834310.331657 9.6455039.953722 2.637661157.31:133) I
F019.6:594$19.95727619.663672110.331327! 9.64184219.95366019.688182|10.3119
1 Co fine ! Sine 1 Cotang. Tangent Co-lonel Sine | Tangent Ico-tung.im
Dcore 55
Degree 64.
2
2
Canon Triangulorum Logarithmicus.
T
Degree 26.
Degree 27.
M Sune Co-line. Tangenti Co-tang. Sine 1 Ce-line |Tangene! Co-tang.
19.64184219.95366019.638182|10.311818 19.657047|9.94988019.707166|10.292834760
19.64210119.95359819.683502|10.311498):9.65729519.94981619.707478|10.292523154
29.642360 9.9535379.688823.10.3111771 9.6575429.949752 9.707790 10.292210 58
319.642618 9.9534759.689143 10.310857 9.6577909.949687 9.703102 10.291897 57
4.9.642876 9.953413 9.68946310.3105371 9.658037 9.94962319.708414 10.291586 56
$19.64313519.95335119.68978;:0.310217 9.65828419.94959819.708726|10.291274155
619.64339319.95329219.690103|10.309897 9.65853119.94949419.709937|10.290962154
79.64365019.953228 9.697423 10.3095771 9.6587779.949429 9.709349 10.29065153
819.64390319.953166 9.692743 10,3092581 9.659024 9.949364 9.709660 10.290340152
919.64416519.953104 9.691063 10:203938 9.65927119.9493909.709971|10.290029151
1C17.64442319.95304219.69133:10.308619 9.6951719.94923519.71028210.28971650
11:9.64468019.95293019.691700110.308 300 9.6597639.9491709.710593110.289407 49
129.6449369.9529179.692019;19.307981 19.660009 9.949105 9.710904 10.289996 48
1319.6451939.9528559.692338 10.30766219.66025519.949940 9.71121410.288785142
149.645449 9.95279319.692656 10.307343 9.660500 9.9489769.711525 10.288475146
1919.6457c619.95273119.692975|10.307025 9.66074619.94891019.711836|10.288164 45
164.64596219.95266819.693293|10.306706 9.66099119.94884519.71214619.287854144
11719.6462189.952606 9.69361210.3063889.661236 9.948763 9.712456 10.237544143
187.6564737.952544 9.697930 10.306070 9.66148119.9487159.712766 10.287234/42
19 9.646729 9.95248119.694248110.305752 9.6617269.948650 9.713076 10.286924 41
2019.64698419.9;241919.69456610.305434 9.66197019.94958419.713386119.28661440
2119.64723919.95235619.694883|15.305137 4.66221419.94851919.713695110,28630539
2219.647494 9.9522949.69520110.304799 9.66245919.9484539.714005 10.285995.38
239.647749 9.95223119.695513 10.904432 9.6627029.94838819.714354 10.28563637
24 9.648004 9.95216319.695335 10.304164 9.662947 9.9483239.714624 10.2853763€
2519.6482589.952105 9.696153110.303847 9.66319019.94825719.714933110.
269.6485129.952043|9.696470|10-30353019.6634332.94619119.715.241 10.284758134
27 9.6487669.951980 9.696786 10.303213 9.6636779.948126 9.715550 10.28444933
289.648020 9.951917 9.697103 10.302897 9.66:92019.94806 -9.715859 10.18414032
29 9.649274 9.9518549.6974:0 19.302580 9.6641639.94799519.716168 10.283832/31
3019.64952719.95179119.697738|10.302264 9.66440619.94792919.716477 10.283523130
3119.64978119.95172819.69305210.301947) 9.66464319.94786319.716785110.283215129
32 9.6500349.95 1665 9.698369 10.301631 9.66439119.9477979.717093 10.282907 28
339.050287 9.9516029.698639 10.301315 9.66513319.947731 9.717401 10.282598 27
34 9.65051919.95153919.699001|10.300999 9.6653759.9476659.717709110.282290 26
3:19.65079819.95147619.699316 10.300684 9.66561719.94759919.718017110.281983/25
369.65104419.95141219.69963210.3033681 9.66585819.94753319.718325 10.281675134
37 9.6512969.9513499.699947 10.3000529.66610019.9474679.718633 10.281367 23
38.9.6516489.951286 9.700263 10.2997379.6663419.947431 9.718940 10.281060 22
3919.651800 9.951122 9.700578 10.199422
9.6665839.94733519.719248 10.280752121
4019.652052 9.95115919.70089310.299107,9.66682419.94726919.719555110.230445120
4119.65=30319.95109519.701203110.2987929.66706519.94720319.719862110.280138119
4219.6525559.9510329.705522 10.293477 9.6673059.947136 9.72016910.279831 18
4:19.652806 9.9509689.701837 10.2981631 9.667546 9.947070 9.720476 10.279524 17
44 9.653057 9.9509059.702152 10.2978489.6677869.947004 9.720783 10.279217 16
4517.653307 9.95084119.702466110.297534) 9.66802619.94693719.721089110.278911115
46.9.6;355819.95077719.702789 10.297219 19.6632669.94687119.721395110.278604 14
47,9.653608 9.9507149.703095 10.2969051 9.66350619.94680419.721702 10.27829813
48 9.65405919.9506509.703409 10.2965911 9.668746 9.946738 9.722008 10.277991112
49 9.654309.9.950536 9.703722 10.296277) 9.6689869.946671 9.72231510.277685|11
5014.65455819.95052219.704036 10.2959649.669225 9.94660419.722621110.277379110
5119.05480019.95045819.704350 10.295650 9.66946419.94653719.722927 10.2770731 9
$29.655057 9.95039419.704663 10.295337 9.6697039.946471 9.723232 10.276768
$39.65530719.9503309.704976 19.295023 19.669942 9.94640419.723538 10.276462 7
$49.6555569.9502669.705290110.294710 9.670181 9.946337 9.723843 10.2761566
5519.655805 9.950202 9.705603110.294397 9.67941919.94627019.724749|10.2758511 5
569.65605319.95013819.705915|10.29408419.67065719.94620 319.724454110.27554614
57 9.656302 9.95007419-706228 10.293771 9.670896,9.94613619.724759 10.2753401 3
589.65655019.95000919.706541.10.2934599.6711349.9463699.725065 10.2749352
599:656799 9.9499459.706853 10.293146 19.6713729.946002 9.725369110.274630 1
6019.656347 9.94988119.70716610.292834! 19.67160919.94593519.725674 10.2743201 c
Co-fine | Sine | Co-tang. Tanger Co-fine ! Sine I Co-tang | Tangent iM
Degree 63.
Degrec 6 2.
ܐ
8
1
Canon Triangulorum Logarithmicus.
ܐ
L
Degree 28.
Degrec 29.
M
M} Sine | Ca-line 1Tangcnth Ca-tang. Sine 1 Co-linc |Tangent I Co tang. I
019.67160919.94593519.725674110.274326 9.60557119.94181919.743752110.256248160
117.67184719.94586819.725979|10.274021 9.6857999.941749|9.744050 10.255950159
219.67208419.945800 9.726284110.273816 9.686027 9.941679 9.744348 10.255652158
319.672321 9.94573319.726588110.273412 9.6362549.941609 9.744645 10.255355 57
419.67255819.9456669.72689210.273107 9.6864829.9415359.744943 10.255057 56
5 9.67279519.94559819.727197 10.272803) 9.68670919.94146€17.74524c110.25476:55
69.67303219.94553119.72750110.272499 9.6369369.941 39819.745538 10.254462154
7|9.6732689.945463 9.72780510.272195 9.6871639.941328 9.74583; 10.254165 53
8 9.67350519.945396 9.728109 10.271891 9.6873849.94125719.746132 10.253868 52
9 9.6737419.945328 9.728412 10.271587 9.6-376164.94118719.746429 10.253571151
1019.67397719.94526119.728716110.271284 9.68784219.9411169.746726110.253274150
114.67421319.94519319.729020110.2704809.688669 9.94104619.747023|10.252977149
129.6744489.94512519.729323 10.270677 9.68829519.949975 9.747319 10.252680 48
139.67468419.945058 9.729626 10.270374 9.6885239.94090519.747616 10.252384/47
14 9.674919 9.9449409.72992910.270070 9.088747 9.94083419.747912 10.25208746
1519.6751549.94492219.730232110.269767 9.68997-19.94076319.748205 110.251791145
169.67538919.94485419.730535110.269464 9.68919819.4469319.748505110.251495144
17 9.67562319.9447869.730838 10.269162 9.6894219.940622 9.748801 10.25.1199 43
189.675859 9.944718 9.731141 10.268859 19.6896489.940551 9.749997 10.250992/42
1919.676094 9.944650 9.731443 10.2685569.68987319.940480 9.749393 10.250607 41
2019.6763289.94458219.731746 10.268254 9.690098/9.94940919.745684110.250311 40
2119.67656219.94451419.732048|10.267952 9.6903239.94033819.749985|10.150015139
22 9.696796 9.944446 9.732351|10.267649 9.690548 9.940267 9.750281 10.249719138
239.6770302.9443771 9.732653 10.267347 9.690772 9.940196 9.750576 10.249424137
24 9.677264 9.944309 9.732955 10.267045 9.69099019.940125 2.750872 10.249128136
259.67749719.94424119.733257/10.2667431 9.69122019.94005319.751.167 10.248833
26 9.67773119.94417219.73355810.266441 9.69144419.93998249.751462|10.248538 34
27 9.6779649.9441049.733860 10.266140 9.6916689.939911 9.751757 10.2482431331
28.9.678
1979.94401619.734162 10.265838 9.6918929.935840 9.752052 10.247943
299.678430 9.94396719.734463 10.265537 9.692115 9.939768 9.752347110.247653131
3019.67866319.94389819.734764 10.265230 9.69237919.939697 9.752642110.247358130
3119.6788959.94383019.735666|10.2649341 9.69256219.93962519.752937110.247663129
329.679128 9.9437619.735 362 10.264633119.692785 9.939554 9.753231 10.346769.28
339.679360 9.9436429.735668 10.264332 9.6930089.939482 9.753526 10.246474127
3419.6795929.943624 9.735968 10.264031 9.693231 9.939410 2.7538 20 10.24618026
3519.67582419.94355519.736269|10.263731 9.69345319.9393399.754115110.245885125
3619.68005619.94348619.736570110.263430 19.69367019.93926719.754409|10.245591424
379.68028819.9434179.736870 10.263130 9.69389819.9391959.754703 10,245297 23
389.6305 199.943348 9.73717110.262829 9.6941209.939123 9.754997 10.245903 22
39 9.680750 9.943279 9.737471 10.262529 9.694342 9.93905119.75529110.144709 21
49|9.6809829.943210 9.737771 10.262229 9.69456419.93898e9.95558410.244415 20
4:19.68121319.94314119.738071|10.261929 9.69478619.938 90819.75587810.244122/19
42|9.681443 9.9430729.738 371 10.261629 9.695007 9.93883519.756172 10.243828 18
4319.681674 9.9430039.738671 10.261329 9.695279 9.9397639.756465 10.243535117
44 9.681904 9.9429339.738971 10.261029 9.69545019.938691 4.756759110.243241-16
4519.68213519.94286419.739271119.260729 9.69567119.93861919.757052|10.24:948115
4619.68236519.94279519.739570110.260430119.69589219.93854719.757345110.242655114
47 9.68259519.942725 9.739870 10.260130 9.6961139.9384759.757638 10.24235213
489.6828259.9426569.740169 10.259831 9.6963349.938 402 9.75793110.24266912
499.6830559.942587 2.74046810.259532 9.6.965547.9383309.75822419,241776"}
5019.68328419.94251719.740767110.256233 9.69677419.93825719.758517110.241483110
5119.68351419.94244819.741066|10.258934) 9.69699519.9381859.758810|10.24119919
5219.683743 2.942378 9.741365 10.258635 9.697215 9.938112 9.75910216.2408985
$319.68397219.942308 19.741664 10.258336.9.6974359.93804:19.759395 10.2406057
54 9.6842019.942239|9.741962|10.258038 9.69765419.9379679.75968710..403136
5519.684430 9.943169 9.74226110.257739 9.697874)9.93799519.759979110.240021 |
569.68465819.94209919.742559110.257441 9.69809319.937$2219.760271110.239728 4
5719.684887 9.942029 9.74285810.2571421 9.6993139.9377499.760564119.239436 3
5819.6351159.941089 9.743156110.256344 9.6985329.9376769.760856 10.239144
59 9.685343 9.9418899.74345 4 10.2565461 9.69875119.9376039.761147-19.23885:
609.68557119.94181919.743751110.256248 !9.69397019.93753119.76143.110.238561
1 Co fine Sine 1 Cotang., Tangent Co-line Sine co tong Tangene IM
Degree 61.
Dirce 60.
+
.
I
A
Canon Triangulorum Logarithmicus.
38
Degree 30.
Degree 31.
MI Sine 1 Co-line Tangent Co-tang. Sine I Co-line Tangent | Co.tang.
019.69897019.93753119.761439110.238561 9.71183919.93306619.773774/10,221 226160
119.69918919.93745819.761731|10.2382691,9.71204519.93299019.779060|10.220940159
29.6994079.937385 19.762023 10.237977 9.7122599.93291419.779346 10.22065458
319.6996269.93731: 9.76235410.237686 9.7124699.932838 9.779632 10.220368 57
419.69984419.93723819.76260615.237394' 9.7126799.932761 9.779918 10.220082 56
$19.70006219.93716549.762897110.237103 9.7 1288919.93268519.780203110:219796155.
619.70028:319.93709219.763183110.236812 9.71309819.93260919.780489|10.219511154
7 9.70049819.9379199.763479 10.2365211 9.713308 9.932533 2.780775(10.219225153
89.70071619.9369459.763770 10.236230
10.236230' 9.7135179.932457 9.781060 10.218940152
919.7009339.93687219.764061|10.235939 9.7137269.93238019.781346.10.21865451
1019.701151 9.93679919.764352 10.235648 9.71393519.93230419.781631110.218369150
1119.7017689.93672519.764643|10.235357 9.71414419.93222719.781916110.215084149
129.70158519.93665219.764933 10.235067 9.71435-9.932151 9.782202 10.217799 48
139.7018029.9365789.765224 10.234776 9.7145619.932074 9.782486 10.217514 47
14 9.702019 9.93650519.76551420.934486 9.7147699.931998;9.782771 10.217229 46
1519.702236|9.93643119.765805|10.234199 9.7149779.931921 19.783056 10.216944/45
1619.70245219.93635719.766095110.233905 9.7151869.931845 19.78334110.216659144
17 9.7026699.936284 9.766385 10.233615 9.715394 9.931768 9.783626 10.216374 43
1819.70288519.93621019.766675 10.233325 9.71560119.931691 9.783910 10.21609042
19 9.703101 9.9361369.766965 10.233035 9.7158099.9316149.784195 10.21580541
2019.703317 9.93606219.767255 10.232745 9.71601719.93153719.784479|10.215520140
2119.70353319.93598819.767545|10.2324551 9.71622419.93146019.784764 10.21523639
2219.70374819.9359149.767834110.232166 9.716431 9.93138319.785048 10.214952
239.703964 9.93584019.768124 10.231876 9.716639|9.931306 9.785332110.214668 37
249.70417919.935766 9.768413 10.231587 19.716346 9.93122919.785616 10.214384136
25 19.70439519.93569219.76370310.231297 2.71705319.93115219.785900 10.214099135
2619.70461019.93561819.768992(10.2310081 9.71725219-931075 19.786184/10.2133 16 34
27 9.70482519.93554319.769281.0.230719 9.717466 9.930998 9.78646810.2135 32 33
289.7050409.9354699.769579 10.2304309.717672 9.9309209.786752 10.213248 32
299.7052549.9353959.769859 10.230141 9.717879 9.9308439.787036 10.213964 31
3019.7054699.93532019.770148 10.229852 9.71808519.93076619.78731910.212681130
3119.70563319.93524619.770437|10.229563 9.71829119.93068819.787653110.212397129
32 9.705897 9.93517119.770726 10.229274 9.71849719.93061119.787886 10.212114 28
33 9.706112 9.9350979.771015 10.228985 9.7187039.9305339.788170 10.211830127
349.7063269.935022 9.771303 10.228697 9.7189099.930456 9.788453 10.211547 26
3519.70653919.9349489.771592/10.228408 9.71911419.93037819.788736 10.211264/25
3619.70675319.934873 9.77188010.228120 9.7193209.93039019.789019110.210981|24
3719.706967 9.9347989.772168 10.227832 9.7595259.9302239.789302|10,210698 23
38 9.70718.919.9347-39.772456 10.-27543 9.71973019.9301459.789585 10.210415 22
39 9.70739319.934649 9.77:745 10.227255 9.71993519.930067 9.789868|10.210132 21
4019.707606 3.9345 7419.773033/10.226967 19.7201 4019.92998919.790151|10.209849120
4119.70751919.93449919.773321 10.226679 9.72034519.92991119.750433|10.20956619
429.708032 9.934424 9.773608 10.226391 9.72054919.929833 9.790716 10.20928418
4319.708245 9.934349 9.773896 10.226104 19.72075419.92975519.790999.10.20900117
44 9.708457 9.9342749.774184 10,225816 9.7209589.929677 9.790281 10.208719 16
4519.70867019.93419919.774471|10.225529 2.72116219.92959919.791563 10.20843615
4619.70838219.9341239.774759|10.225241 9.7213669.92952119.791846|10.208154114
479.709094 4 93404819.775046 10.224954 9.721570 9.929442 9.792128 10.207872113
48 9.7093069.933973 9.775333 10.224666 9.7217749.929364 9.792410|10.20759012
499.70951819.933897 9.775621 10,224379 9.72197819.929286 9.792692|10.2073081
5019,70973019.9;:8:219.775908(10.22409 9.72218119.92920719.792974110.207024110
S? 19.70494119.93374719.776195|10.223805 9.72238519.9291 2919.793256.10.2067441 9
5219.7101539.93.3671 9.77648210.223518 9.722588 9.929050 9.793538|10.206462 g
$39.710364 9.9335969.776768 10.223232 9.722791 9.928972 9.793819 10.2061807
549.710575 9.933520 9.777055 10.222945 9.722994 9.9288939.794101 10.2058996
5519.71078619.933444 9.777342 10.2226581 9.72319719.92881419.794383|10.20561115
5619.710997 9.93336919.77762810.222372 9.72340019.9287369.794664|10.20533674
$79.7112089.9332939.77791510,222085 19.7236039.9286579.794945 10.205054 3
$819.71141819.933217 9.778201 10.221799 19.7238059.92857819.795227.10.204713
59 9.711629 4.733141 9.778487 10.221513 9.7240079.928499 9.795503 10.204492
6019.7::87419.45306619.778774 10.221226 9.72421719.93842019.795789.10.204:111c
I Can I sine 1 Cotang: 1 Tangent Co-fine Sine ! Co tang. I Tangent iM
Degree 59.
Degree s.
(c)
2
1
.
1
Canon Triangulorum Logarithmicus,
O
10
1
1601
Degree 32.
Degree 33.
MSue 1 Co-line Tangent Co-tang . Sinc | Co-line |Tangent 1 Cotang,
119.72441219.9283419.796070|10.203930 9.7363099.92350919.81279410.187206159
219.724614 9.92826319.796351|10.203649 9.736497 9.923427 9.913070 10.186930 58
3 9.724816 9.9281839.796632 10.203368 9.7366929.923345 9.813347 10.186653157
419.725017 9.9281049.796913.10.203087 9.73638691923263 9.813623.10.186377 56
519.72921919.92802519.797194|10.202806 9073708019.92318019.813899.10.186101155
$69.72542019.92794619.797474110.102522 9.73727419.9230989.814175|10,185824154
7 9.7256229.927867.9.797755 10.202245 19.737467 9.923016 9.814452 10.185548153
8 9.7258239.927787 9.798036 10.201964 19.73766119.92293319.814728 10.18527252
919.726024 9.92770819.798310 10.201684 9.73785419.92285119.915004 10.184996151
1019.72622519.92762819.798595.10.201404 9.73804819.922768/9.815279|10.184720 50
1119.72642619.92754919.798877|19.201 123 9.73824119.9226869.815555110.184445 149
1219.7266269.927469 9.79915710.200843 9.7384349.9216039.815831 10.184169 48
13 9.7268279.927390 2.799437 10.200563 9.7386279.9225 2019.816107 10.183893 47
14.9.727027 9.927310 9.799717 10.200283 9.738820 9.922438 9.816382 10.18361746
1519.72722819.92723119.79999110.200003 9.7390139.92235519.816658_10.183342/45
1619.72742819.92715119.800277110.199723 9.73920519.92227219.816933 10.133066|44
17 9.727628 9.92707119.800557 10.199443 9.73939819.922189 9.817209 10.182791/43
18 9.7278289.926991 9.8008 36 10.199163 9.7395909.922106 9.817484 10.182516 42
1919.728227 9.926913 9.801116 10.198884 9.739783 9.9220239.817759 10.182240 41
2017.72322719.92683119.80135610.198604 9.73997519.921940 9.318035 10.181965140
2119.72842719.92675119.801675/10,198325 9.74016719.92185719.818310|10.181690139
22 9.728620 9.926671 9.801955 10.198045 9.740359 9.9217749.818585 10.181415138
-319.7288259.926591 2.802234 10.197766 12.74055019.921691 9.818860 10.181140137
249.7290249.926511 9.802513.10.197487 9.7407429.921607 9.819135 10.180865136
2519.7292239.92643119.80279:10.197207 9.74093419.9215249.819450/10.180590135
2619.7294229.9263519.803072|10.196928 9.74112519.92144119.819684/10.180315134
2719.729621 9.92627019.803351110.196649 19.741316 9.92135719.819959.10.180041 33
28 9.72982019.9261909.803639 10.196370 9.74150719.9212749.820234 10,179766132
29 9.730018 9.926110 9.80390810.1960911 9.741648 9.92119019.82050810.179492 31
309.73021619.92602919.804187110.195813 9.74188919.92110719.820783/10.179217130
3119.7394159.925949 9.80446610.1955341 19.74208019.92102319821957|10.178943129
32 9.730613 9.9258689.804745 10.195255 9.742271 9.92093919.82133210.178668 28
33 9.7308119.925787 9.805023 10.194977 9.7424619.920855 9.82160610.17839427
349.73100g 9.92570719.80530210.1946981 19.74265219.920772 9.821880 10.178120126
3519.731206.9.925626 9.805580110.194420 9.74284219.92068819.822154119.177846135
369.73140419.92554519.805859)10.1941411 9.74303219.92060419.822429|10.177571134
37 9.731601 9.925464 9.806137 10.193863 9.7432239.920520 9.82270310.177297 23
38 9.731799 9.925384 9.806415 10.193585 9.74341219.920436 9.822977|10.17702322
1399.731996 9.9253039.806693 10.193309 9.74360219.920352 9.823250110.176739 21
4019.73219319.92522219.80697110.193023 19.743792 19.92026819.823524110.17647620
4119.73239019.92514119.807249|10.192751 9.74398219.92018419.823798|10.176302|19
4219:73258719.92506019.897527 10.192433 9.74417119.92029919.824072 10.175928 18
439.73278419.92497819.80780510.1921959.7443619.920015 19.824345 10.17565517
449.732980 9.924897 9.808083 10.191917 19.74455019.91993119.82461910.175381 16
4519.73317719.92483619.808361 10.191639 9.7447399.91984619.824892/10.17510815
4619.73337319.92473519.808638|10.191362119.74492819.9197629.82516610.1748 34/14
4719.733569 9.92465319.808916 10.191084 9.7451179.919677 9.825439 10.17456013
48 9.7337659.924572 9.809193/10.190807 9.745 306 9.9195939.825713|10.17428752
49 9.733961 9.9244919.809471 10.190529 9.7454949.919508 9.82598610.17401411
5019.73415719.92440919.80974810.190252 19.74563319.91942419.826259.10.17374110
5719.73435319.92432819.810025|10.189975 1974587119.919339|9.326532|10.1734989
$219.734548 9.924246 9.81030210.189697 9.7460599.919254 9.826805/10.173195 Ś
339.73474419.9241649.810580 10.189420 9.7462489.919169 9.827078/10.172922 al
54 9.734939 9.924083/9.810857 10.189143 9.746436 9.9190849.827351|10.1726496
5519.73:13419.92400119.811134.10.188866 9.74663419.91899919.82762410.1723765
562.13533419.92391919.811410 10.188589 2,74681119.91891519.8 21897|10.172103 4
37 9.7355259.9238379.811687 10.18831397469999.9188309.828170 10.171830 3
5819.7357199.923755 9.811964.10.188036,:19.74718719.918744 9.828442 10.1715582
19 9.73591419.9236739.812241110.187759 9.747374 9.918659 9.82871510.171285
6019.73610919.92359119.812517110.187483) 19.74756219.91857419.828937110.171012
| Co fine Sine 1 Cotang. Tangent Co-line | Sine Ico.tang. I Tangent IM
Degree 57.
Degree so.
:
I
Canon Triangulorun Logarithmicus.
1.
1
1
Degree 34.
Degree 35.
M] Sine 1 Co-fine. (Tangencl Co-tang. Sine 1 Co-fine |Tangent! Co-tang.
09.74756219.918574!9.828987110.171912 19.75859119.91336419.845227110.154774/60
119.7477499.91848919.329260 10.170740 9.75877219.91327619.84549610.154504159
2 9.747936 9.918404 9.82953210.170463 9.758952 9.913187/9.845764 10.154235158
319.748123 9.91831819.82980510.1701951 9.7591329.913094 9.846033 10.153967157
419.748310 4.9182339.830077 10.169923 9.7593129.91301019.846302 10.153698156
$19.74849719.91814719.830349 10.169651 19.759492'9.912921 19.846570 10.153429155
69.74868319.91806219.830621 [10.169379 9.75967219.91283319.846839|10.153161154
79.748870 2.9179769.830893 10.169106 9.7598519.9.2744 9.847107 10.152892153
8 9.74995619.9178919.831165 10.168834 9.760031 9.91265519.84737610.152624152
99.7492429.91780519.831437 10.1685639.760210 9:9125609.847644.10.15235651
109.74942919.91771919.831705110.169291 9.76039019.912477/9.847913110.152087150
1119.74961519.91763419.831981|19.168019 9.7605699.912388|2.848181|10.151819149
12 9.749801 9.9175489.832253 10.167747 9.7607489.912299 9.848449|10.15155148
139.7499869.9174629.83252510.567475 9.760927 9.9122109.848717 10.151283.47
149.750172 9.917376 9.332796 10.167204 2.761106 9.9121 21 9.843985 10.1510ks 46
1519.750358 9.91729019.833068|10.166932 2.76128519.91203119.849254|10.150746/45
1619.75054319.91720419.833339|10.1666609.76146419.91194219.8495 23/10.150478144
1719.750729 9.917118 9.83361110.1663899.76164219.9118539.849789 10.15021 4143
18 19.75091419.91703219.833882 10.166118119.7618219.9117639.850057 10.149943142
1919.751099 7.9169459.83415410.165846 9.7619999.911674 9.850325 10.149675141
2019.75128419.9168599.834425|10.165575 9.76217719.91158419.850593.10.149407140
2119.75146919.916773/2.8 34696|10.165304) 19.76235619.91149519.850861 (10.149139139
2219.75165419.9166869.834967 10.165033119.762534 9.9114059.851128 10.148872138
23 9.751838 9.91660019.835238110.16476219.762712 9.91131519.851396 10.14860437
24 9.75202319.91651417.835509 10.1644919.762889 9.911226 9.851664 10,148336 36
2512.75.207 9.91642719.835780|10.1641202.763067|9.91153019.851931.10.148069135
2619.75239219.9163409.836051|10.163249.76324519.9110469.852199|10.147801134
27 9.7525762.9162549.83632210.1636781 2.763422 9.9109569.85246610.14753433
2819.7527609.916167 2.836593110.163407 19.763599 9.950366 9.852731 10.147267132
29 9.752944 9.91608019.836864 10.163136 9.763777 9.910776 9.853001 10.14699931
3019.7531289.91599419.837134|10.162866 9.76395419.9106869.85326810.146732130
3119.75331219.91590719.837405|10.162595 19.76413119.91059619.853532|10.14646512
32 9.753495 9.9158209.837675 10.162325 9.764308 9.910506 9.853802 10.146198 28
3319.753619 9.915733\9.83794610.162054 9.76448519.9104159.854069|10.145930127
3419.753862 9.915646 9.83821610,161784 19.76466219.9103259.854336 10.145664 26
3519.754046 9.91555917.838487110.161513 2.76483819.91023519.854603|10.145397125
3619.75 422919.91547219.838757110.161243 9.76501519.91014419.854870 10.145130124
3719.754412 9.91538519.839027 10.160973 9.765 1919.910054 9.855137 10.14486323
3819.754595 9.9152479.839297 10.160702 9.76536719.90996319.855404 10.144596 22
399.75 4778 9.9152109.83956810.160432 9.7655449.90987319.855671 10.144329 21
4019.75496019.91512319.839838(10.160162 9.76572019.9097829.855937110.144063|20
4119.75514319.91503519.840108|10.159892 9.76589619.90969119.856204|10.143796119
4219.75532519.914948 9.840378 10.159622 9.7660719.9096019.856471 10.143529 18
439.755508 9.91486619.840647 10.159352 9.766247 9.909510 9.856737 10.14326317
44 9.75569019.91477319.840917 10.1590831 9.7664239.909419 9.857004 10.142996 16
4519.755872 9.91468519.841187110.158813 9.76659819.90932819.857270 10.142730115
4619.75605419.91459719.841457|10.158543 9.76677419.90923719.857537|10.142463114
4719.756236 9.91451019.841726 10.158273 9.76694919.9091469.857803.10.14319713
48 9.756418 9.914422 9.84199610.158004 9.7671249.90905519.85806910.141931 12
49 9.756600 7.9143349.84226610.157734 9.7672999.908964 9.858336 10.14166411
5019.756781 9.91424619.842535110.157465 9.76747419.90837319.85860210.141398|10
5119.75696319.91415819.842804|10.157195 9.7676499.90878119.858868|10.1411329
52 9.757144. 9.914070 9.843074|10.156926 19.767824 9.908690 9.859134 10.140866 %
5319.7573269.9139822.843343 10.156657 6.767997 9.9085999.859400 10,1406007
54 9.757507 9.9138949.843612 10.156387 9.7681739.908507 9.859666 10.140334
5519.75768819.913806 9.84388-10.156118 9.768 34819.908416 9.859932 10.140068
5619.7578699.91371819.844151/10.155849 9.7685229.90832419.860199110.139302) 4
5719.7580499.9136309.844420 10.155580 9.7686969.9082339.86046410.139536 3
58 9.7582309.913541 9.844689 10.15530 9.7688719.908:419.860730 10.139270
599.7584119.913453 4.844958110.1550429.769045 9.9080499,860995 10.139095
6019.75859119.9133649.845227110.154774 9.76921419.99795819.861261 10.1387391
| Co-ſine | Sine I lo-tang. I Tangent Ca-fine | Sine | Co-tang | Tangent LM
Degrce ss.
Degree 54.
(e2)
1
2
I
O
A
1
:
.
"L
Canon Triangulorum Logarithmicus.
1
1
Degrce 36.
Degree 37;
MI Sine Co-fine. (Tangenel Co-tang. Sinel Co-line |Tangent! Co-tang.
019.769-1919.90795819.86126110.138739 9.77946319.90234919.877114|10.1 22885160
119.76939219.9078669.861527110.1384739.77963119.90225319.377377|10.122623159
219.769566 9.9077749.861792 10.138208 9.77979319.90215819.877640 10.122360158
319.769740 9.9076829.86205810.137942 9.77996519.902063 9.877903 10.122097157
419.769913 9.90759019.86232310.1376771 9.7801339.901967 9.878165 10.121834156
$19.77008719.90749819.862589|10.137411 9.78030019.901872 9.878428 10.121572155
619.77026019.90740619.862854/10.137146 9.78046719.9017769.878691|10.121309154
79.770433 9.90731419.86311910.1368801 19.7806349.90168119.878953 10.121047 53
819.770606 9.907221 9.863385110.136615 9.780801 9.90158519.879216 10.120784152
919.770779 9.90712919.863650 10.136350 9.7809689.901488 9.879478 10.120522 5I
109.77095219.90703719.863915|10.1360859.78113419.90139119.879741 10.120259150
1119.77112519.90694519.864180/10.135820 19.78130119.90129819.880003|10.119997149
1219.771298 9.906852 9.864445 10.135554 9.78 1467 9.901202 9.880265 10.119734148
139.77147019.90676019.864710 10.135289 19.7816349.901106 9.880528 10.119472 47
1419.77164319.906667 9.864975|10.135024 9.781800 9.901010 9.88079010.119210146
1519.77181519.90657419.865240 10.134759 19.78196619.90091419.881052110.118948 45
1619.77198719.90648219.865505|10.134495 9.78213219.90082819.881314|10.118686144
17 9.772159 9.9063899.865770110.134230 9.782298 9.900722 9.881576110.118424 43
18 9.77233119.9062969.866035 10.133965 19.78246419.900626 9.881839 10.11816142
1919.772503 9.906203 9.86630010.133700 19.782690 9.900529 9.88210110.117899/41
3019.77267519.90611119.866564110.133436 19.78279619.90043319.882363110.117637140
2119.77284719.90601819.866829/10.1331711 19.78296119.90033719.882625|10.117375139
229.77301812.9059259.867094 10.132906 19.783127 9.90024019.88288610,117114 38
239.7731909.905832 9.867358 10.132642 9.78329219.90014419.883148 10.116852137
249.7733619.905738 9.867623 10.1323779.783437 9.9000479.883410 10.11659036
2519.77353319.90564519.867887110.132113 9.78362319.89995119.883672|10.116328135
2619.7737049.90555219.868152110.131848 2.78378819.89985419.883934|10.11606634
2719.773875 9.9054599.868416 10.131584! 19.78395319.899757 9.884195 10.115805 33
28 9.774046 9.90536519.868680 10.1313201 19.784118 9.899660 9.884457|10.115543132
299.774217 9.90527219.868945 10.131055 9.784282 9.899563 9.884719 10.115281 31
3019.77438819.905 17919.869209|10.130791 9.78444719.89946719.884980/10.115020 30
3119.77455819.90508519.864773|10.1305 27 9.78461619.89937019.885242|10.114758729
3219.77472919.90499219.867337 10.130263 19.7847769.89927319.885503 10.114497 28
3319.774899 9.90489819.870001 10.129999 9.7849419.899175 9.885765 10.114235 27
34 9.775070 9.9048049.870265 10.129735 19.7851059.899078 9.886026 10.11397426
3519.77524019.90471119.870529 10.529471 9.78526919.89898119.83628810.11371225
3619.77541019.90461719.870793110,1292071 9.78543319.89888419.886549|10,113451134
37 9.775580 9.90453319.871057 10.1289439.78559119.898787 9.886810 10.11319023
38 9.77575019.904429 9.871321 10.128679 9.7857619.898689 9.887072 10.112928122
399.775920 9.9043359.871585 10.128415 9.785925 9.898592 9.887333/10.112667 21
4019.776090'9.9042419.871849 10.128151 9.78608819.89849419.887594110.112400 20
4119.77625919.90414719.872112 10.127888 9.78625219.89839719.887855 10.11214519
429.776429 9.90405319.872376 10.127624 9.786416 9.89829919.888116 10.111884 18
439.77659819.9039599.872640 10.127360 9.786579 9.8982019.888377 10.111623117
449.776768 9.99386419.872903 10.1 270971 9.786742 9.8931049.888638 10.111362 16
4519.77693719.90377019.873167110.126833 9.78690919.89800619.888899110.11110115
4619.7771069.90367619.873430 10.1265701 9.78706919.89790819.889160110.110840114
47 9.7772759.90358119.873694 10.126306 9.787232 9.89781019.889421 10.110579/13
489.777444 9.9034869.873957|10.126043) 9.787395 9.897112 9.889682 10.110318 12
4919.77761319.903392 9.874220 10.125780 9.7875579.89761419.889943 10.110057 11
50 9.777781 9.90329819.874484/10.125516 9.78772019.897516 9.890204 10.10979610
$119.77795019.903203|9.874747|10.125253) 9.787.88319.89741819.390465|10.10953519
52 9.778119 9.903108 9.8750ic 10.124990 | 9.78804519.897320 9.890725 10.109275) 8
5319.778287 9.9030139.875373 19.124727 9.7882089.8972229.89098610.10901417
$42.7784559.902919 9.875536 10.124464 9.788370 9.89712319.891247 10.1087536
5519.77862319.90282419.875799 10.124201 9.78853219.89702519.891507110.1084931 s
5619.77879219.90272919.876063|10.123937 9.78869419.89692619.891768110.1032321 4
57 9.778960 9.9026349.876326 10.123674 9.7888569.896828 9.89202810.107972 3
5819.77912919.902539 9.87658910.12341119.7890189.896729 9.89228910.10771112
599.7792959.902444 9.876851110.123149/ 9.789180 9.896631 9.892549 10.10745 I
6019.77946319.90234919.877114110.122886 9.78934219.89653219.892810110.1071901 o
Co-line | Sine I Co-tang. Tangent Co-finel Sine 1 Co-tang | Tangent |M
Dogrce 53.
Dcgrec 52.
I
1
I
IHL
Canon Triangulorum Logarithmicus,
Degree 38.
Degree :9.
M) Sine Co-fine. \Tangent Co-tang. Sine 1 Co-fine (Tangent! Co-tang.!
019.78934217.89653-19.892810110.1071909.79887219.89050319.90 : 369|10.091631160
119.78950419.84643319.393070|10.106930 9.79902819.890400 9.90862710.09137315y
219.7896659.896335 9.8.93330 10.106669 19.7991849.8902989.908886101091814158
319.7898272.896236 9.893591|10.106409 9.7973399.890195 9.909144 10.090850 57
419.789988 9.89613719.893851110.106149 19.799495 9.89009319.909402 10:090598156
$12.7991499.896038.9.894111 10.105889 9:79965119.88999019:993660110690140lis
619.79031019.8959399.894371|10.105628 9.79980619.889888[9.909918110.092031154
219.79047119.89584019.894632110.1053689.79996119.889785 9091o176.10.089823 53
8 9.790632 9.8957419.894892 10.105108 9.8001179.88968-19.910435 10.089565152
99.7907939.8956417.895152 10.104848 9.800272 9.8895799:910693 10.08930751
104.790954.9.895542 9.895412 10.101588 2.80042719.88947619.910951 10.089049150
119.79.112519.895443/9.89567:|19.104328 2.8005829.8893749-911209|19.088791 149
129.791275 9.875 3439.895932 10.104068 9.800737 9.889271 2.911469|10.088.5 33143
139.7914369.8952449.896192 10.103808 9.800892 9.8897699.911724 19.08827547
149.7915969.89514419.896452.0.1035481 9.8010479.8892649.91198210.088019146
1519.7917569.89504519.896713|10.103288 2.80120119.88896119:912240|10:087760
1619.791917\9.89454519.89697110.103028] 9.80135619.88885819.912498)19.087502144
1719.7920770.8948469.89723110.102769 9.Sors 10 9.88875519:91 275610.087244 43
189.7922379.894746).897491 10.102509 9.8016659.888651 9.913014 10.08698642
1919.792397 9.8946469.87775410.102249 9.8018199.888548. 9.913271 10.0861294?
20 9.79255719.8945469.89801c|io.101990 9.80197319.88844419.913529110.086471 40
2117.79271615.8944469.89827010.101730 9.802127 9.88834119.913787110.086213/39
22 9.7928769.8943469.898530 10.101470 9.802282 9.888237 9.914044 10.08595638
2319.793035 19.8942469.89878910.1012119.8024.359.888133 9.914302 10,08569837
249.793195 2.894146 2.899349|10.10995 | 9.8025899.88803019.914560 10.089440 36
2517.79335419.8940469.899308|10.1006921 9.$0,274319.8879-69,914817 10.089183135
2619.79351319.8939469.899568|10.100432 9.8.28979.88782319.915075110.0849254134
279.7936739.893845 19.899827 10:190573 2.803050 9.887718121915332 10.08466633
2819.7938329-893745 91900086 10.099913 9.8032049-8876149.915590 10.08441032
29.9.1939959893645 9.900346 10.099654 9.80335719.8875109.915847 10.08415331
3019:79114919.89354419.900605|10.099395 9.8035109.88740619.916104 10.083895130
3119.79430317.8934449.900864.10.099135 9.80356419.38730219-916362110.083633/29
3:19.744679.093343 9.901124. 10.098876 1.803817 9.8871989.91661919,083381128
33 9.7946269.897243 -.901383 10.098617 19.803970 9.887093 9.91687610.083123127
349.7947047.893141-501642 10.098358 9.8041239.889909 9917134 10:982866/26
1:7:54 14.87704119.901901|10.098099 9.8042764.88688419.917391110.082609125
36 9.79510110.8.7294|5.702160|10.097839 9.82442819.88678019.917648 10.08:352/24
5719.795:59 9.89:8399.90241910.047580 7.5045819.886675 9.917905 10.682094.23
19.7954:7 2:892753 11.302678 10.0973-19.8047349.886574 9.918162 16.08183722
395.7955741..89:637-702937 10.097062 19,8048869.8864609:91842010.081580 17
4119.7:57 1.851 26.9.923196|15.096803 9.80503819.88636119.918677|10.081.323.20
4' 19.79:$9.9.89243519.993455|10.0965.44 9.805191 4.88625719.918934/10.08 106619
429.7960499.8923349.90371410.096285 9.805343 9.88615219.91919110,08980918
439.7962c69.8922337.903973 10.096027 805495 9.886047 2.91944810,o$095217
44 2.796364 9.8921329.90423-10.09576819.8.95647 9.885942 0:919795 16.080295 16
459.79652119.89203019.904491110.095509 9.80579919:885837191919963|40.989938115
4019.7966781..89192979.904750 10.0952501 9:80595119.885934|9:970219149,079781444
4710.7965362.841827 9.99500810.094994 9.8961039:885627 992947610207952453
4819.7969939.891726 y.905767|10.094733 9.896254 9:885521.2.920733 10.979267412
49 2.79715019.8916249.90552610.094474 9.8064069.8884169.929990 10.97901011
51.79730719.87152:12.965784|10.094215 9.80655719.885311.07921247|1972.7.975:3|10
5719.79746419.8914219.9.6243(10.093957 9.80670919.585205.19.921593|10.6784961:9
52 9.7976219.891319 9.90630210.09;698 9.406860 9.8.854032.9217601.10.0782408
5319.797777 3.89121719.996560 10.093440 7.807011 9.8349941.7.9.7.20171106977983 761
549.7579349.89111519.90631910.093181 9.80716212.88488
9.80916-19.8848899.92227440227722616
5519.79899119.891013b).907071 10.092923 9.80731419.88478319.922530110,0774655
5619.79624719789991119.907336|10.092664 9.8074644.8944729922787|10.0771.374
579.798403 9.8908099.90759410.092406 19.807615 9.88457812192304410.9760xál.
5819.79856€ 9.890707 9.90735210.9921471 19.8077669-804466.7.9.23300 19.07669952
5919-7987104.890605 9.938111, 10.691889 9.8979179.8943%2:223547101876443
60 4.79387-19.8901039.908369110.0916311 9.80806719.83425414.923813119,0761860
1 Co-line | Sine I Co-tang. 1 Tangeni Co-fine | Suiça 1 Co-tang. Tangent::M
Degrce 51.
(f)
1
Degree so..
1
Canon Triangulorum Logarithmicus
+
Degree 40.
Degree 41.
M Sine 1 Co-line. (Tangenti Co-tang. Sine 1 6o-fine (Tangenc! Co-tang. I
o 9.80806719.88425419.923813|10.076186, 9.816943/4.87778019.939163|10.06083760
119.80821819.88414819.924070 10.075930 19.81708319.3776709.939418110.060582159
29.8083689.88404219.924327 10.075673119.817233 9.877560 9.93967310.060327158
39.808519 9.8839369.92458310.075417 9.8173789.877450 9.939928 10.060072157
49.8086699.8838299.924839 10.07516019.8175239.8773409.94018310.059816 56
$19.8088139.88372319.925096 10.074904 9.81766819.8772309.940438|10.059562155
69.8689699.8836179.92535210.074647 9.8178139.87712019.940693 10.05 9307154
79.8091199.8835109.925609 10.074391 9.817958 9.877009 9.940948 10.05 9052 53
819.8092699.88340419.92586510.074135 9.8181039.876899 9.94120310.05879753
919.8094199.883297 9.926121 10.073878) 19.8182479.876789 9.941453 10.058542151
109.80956919.88319119.926378|10.073622 2.81839-19.87667819.941713110.058287150
1119.8097189.88308419.926634 10.073366 9.81853619.8765689.941968|10.058032149
129.809868/9.882977 9.926890 10.073110 9.818681 9.87645719.942223 10.057777 48
1319.810017 9.88287119.927147 10.072353 9.818825 9.87634719.942478 10.057522471
149.810166 2.882764 9.927403 10.072597 9.8189699.87623619.942733 10.057167 46
1819.81031619.88265719.927659 10.07-341 9.818113|9,87612519.942988(10.057012145
169.81046519.88255019.92791510.072035 9.8192579.8760 149.943243 10.056757144
17 9.8 1061419.88244319.928171 10.071829 9.819401 9.8759049.943498/10.056502 43
1819.8 1076319.882336 9.928437 10.071573 9.819545 9.87579319.943752 10.05624842
1919.810912 9.8822289.928683 10.071317 9.31968919.87568219.944007 10.055993) 41
2019.81006119.88212119.928940110.071060 9.81983219.87557119.944262 10.055738/40
2119.8113109.88201419.929196|10.0703049.8199769.87545919.94451710.055483139
229.81135819.881907 9.92945210.070540 9.82011919.8753489.944771 10.05522938
23.9.211506 9,8817999.929703 10.070292 9.82026319.875237 9.945026 10.054974 37
14.16:81165519,88169219.929964 110.6700369.820406 9.875125 9.945281 10.054719130
2519.81180419.88158419.930219110.069781 2.81054999:87501419.945'535110.054464135
2619.81199219.8814779.93947510.069525 9.82069319.87490319.945790 10.054210134
27 9.812100 9.8813699.930731 10.069269 9.820336 9.87479119.946045 10.053955
289.8122489.881261 9.930987 19.069013 9.8209799.8746799.946299 10.053701 32
199.8123969.88115319.931243 10.068757 29.8211229.8745689.946554 10.053446 31
3019.81254419.88104519.931499110.068501 9.82126419.87445619.946808 10.053192130
3119.8126929.88093719.931755!10.068245 9.8214079.87434419.947063|10.052937129
32 9.8128409.8808299.93201010.067989 9.821550 9.8742329.947317 10.05268228
339.8129889.88072119.932266 40.067734 9.821692 9.874120 9.947572 10.052428 27
349.8131359.8866139.93252210.067478 9.8218359.874008 9.947826 10.05217326
3519.81328319,88050819.932778/19.067222 19.821977\9.81389619.94808110.os1919125
36 9.813430\9.88039719,933033|10.066967 9.82212019.87378419.94833510.051664124
379.813578 9.88028919.933289110.066711 9.3222629.873672 9.948590 10.05141023
38 9.8137259.88018019.9335 45 10.0664551 9.8224049.873560 9.948844 10.051156122
3919.813872 9.8800729.93380010.066200 9.8225469.873447 9.949099 10.050901 21
4019.814019 9.87996319,934056 10.065944 9.82268819.87333519.949353110.05064720
4119.81416619:87985519.934311 10.065688 9.8228309.87322319.949607 10.05039319
429487431319.8797469.934561 10.065433 9.822972 9.873110 9.949862 10.75013815
4319.814460 9.879637 9.934822 10.065177 9.8231149.87299819.95011610,04988419
44 9.814607 9:8795299.935078 10,064922 19.823255 9,872885 9.950370 10.049630 16
4519.8147839.87942019.935 333 10.064666 9.82339719.87277219.9506251 10.049375115
4619.814900 9.81931119.93558910.064411 9.8235389.87265919.950879 10.04912114
47 9.815046.9.879202 9.935844 10.0641569.823680 9.87:546 9.951133/10.048867113
4819.8151939.8790939.936100 10.063900 9.823821 9.872434|9.951388 10.048612 12
499.8153399.87898419.936355 10.063645 9.823962 9.87232119.951642 10.0483581
5019.81548519,87887519193661010.063389 9.82410419.87220819.95189610.048104110
571981563219.3787669.936866|10.063134 9.824245 19.87209419.952150 10.04785019
52 9.81.5797 9.8786569.937121 10.062879 9.8243869.871981 9.952404 10.047575
$319,8145.9239.8785479.937376 10.062623 9.8245279.8718699.992659 10.047341);
$19.8860699.878438 9.937632 10.062368 9.834667 9.871755 9.952913 10.047087
878328/9.937887 10.06:113 9.82430819.87164119.953167 10.046933
5619:81-636119.89821919.93814210.06185819.8249499.87152819.953421|10.0465791 4
579.8165069,8781099:938397 10.061602 9.3250909.87141419.953675 10.0453:53
589.8766529.8779999.938653 10.061347 9.8252309.8713019.953929 10.0460712
599.816797 9,87989619.938908 10.061092 9.825370 9.871187 9.954183 10.0458171 )
6c19.816943/2.87775019.93916310.060837 9.82551119.87107319.954437 10.0455621
Confine ſ Sine Ca-tang. I Tangent Co-fixe | Sine Co-tang 1 TangenciM
Degree 49:
Degree 48.
.
!
I
1
L
I
Canon Triangulorum Logarithmicus,
5619.833-414.86459819.760643110.031357 9.84124719.85742119.983826110.0161741 4.
Degree 42.
Degree 42.
MI Sine 1 Co-line Tangent Co-tang. Sine 1 Co-fine
Tangent 1 Cotang.
019.82551119.87107319.954437(10.045562 9.83378319.86412719.969656110.030344160
1 9.82565119.87096019.95469110.045 3081 14.83391919.8640109.969909110..30091159
29.8:5791 9.8703469.954945 10.045054 9.834054 9.8638929.970162 10.029338 58
39.825931 9.8707329.955199 10.044800 9.8341899.8637149.970416 10.029584157
49.82607119.870618 9.955453 10.044546 9.834324 9.863656 9.970669 10.029331 56
$19.82621119.87050419.955707|10.044292 2.83446019.8635379.970922|10.029078155
619.82635119.87039019.955961/10.044038 9.8.3459519.86341919.971175|10.028827154
79.826491 9.570275 9.95621510.043784 9.8347309.863301 5:971428|10.028571133
89.826631 9.8701619.95646910.043531 9.834865 9.86318319:97168210.028318152
99.826770 9.8700479.956723 10.043276 9.8349999.8630649.971935 10.028665151
1019.826910 9.86993319.956977110.043023. 9.83513419.8629469.972188 10.027812150
1119.82704919.86981819.957231 10.0427699.8352699.86282719.972441|10.027559149
129.8171899.86970419.957485 10.042515 9.8355039.862709 9.97269410.02730648
139.8273289.869589 9.957739 10.042261 9.8355389.8625909972948 10.027052 47
1419.8274679.869474 9.957993 -0.042007 9.835672 9.862471 2.973201 10.026799 46
1519.827606 9.86936c19.95824610.041753 2:835806|9.86235319.97345410.026546/45
1619.82774519.86924519.95850010.0415009183594119.8622349.97370710.026293|44
179.8278849.8691309.958754 10.041246 9.8360759.8631159.973960 10.026040 43
189.8280239.869015 9.95.9908 10.040991 9.8362099.8619969.974213 10.025787142
199.8281629.868900 9.959262 10.0407389.836343 9.861877 9.97446610.025533/41
2019.82830119.86878519.959515 110.040485 9.83647719.86175719.974719|10.025280 40
2119.82843919.86867019.959969|10.04023117-836611 9.86163849.974973|10.025027:39
2219.823575 9.86855519.960023 10.039977 9.8367459.8615199.975229 10.02477438
239.825716 9.868439 9.960277 10.039723 9.8368789.8613999.975479 10.624521137
3419.82885519.868324 9.960530 10.039469 9.837012 9.861280 9.97573210.024268 36
2519.82899319.36820919.960784110.039216, 9.8371469.86116119.975985 10.024015135
2619.82913119.86809319.961038 10.038962 9.8372791286104119:976238|10.023762/34
2.7 9.8292699.86797819.961291 10.03860812-8374129:86092r 9.976491 10.02350933
2819.813407 9.8673629.961545 10.0384511 2.8375.469.86080327976744 10.02325632
29 9.829545 9.8677475.961799 10.0382019.837679 2.8606829.976997 0.023003131
3613.8296839.867631 9.962052|10.0379479.8378139.86056219.977250/10.023750 30
3719.82932119.-6751519.962306 10.0376949.83794519.86944219.97750310.022497129
219.829959 9.867399 9.96256910.03744-19.8 380789.8603229.971756 10.022244 28
3319.8300969.867283 9.962813 10.0371879.838311 9.8602029.978009 10.02199127
3419.8302349.3671679.963067 10.036933 9.8383449.860082 9.978262 10.021738 26
3519.83037219.86775119.96332010.0366809.838477 9.8599629.97851510.021485 25
3619.830509 9.86693519.963574|10.0364269.8386099.8598429.978768|10.021232124
3719.830646 9.8663199.9638 17 10.036173 2.838742 9.85972119.979021 10.02097923
38 9.830784 9.8667039.964081 10.035919 19.8388759.859601 9.979274 10.02072622
399.83092119.8665869.964335 10.035665 19:839007 9.859480 9.979527 10.02047321
4019.83105819.86647019.964588110.035 412 9.83914019:85936019.979780 10.020226 20
1
4119.83119519.86635319.964842110.035158 9.83927219.859239/9.980033|10.019969119
429.831332 9.866:37 9.965095 10.034905 9.83948419,859118 9.980285 10.019714 18
439.8314699.866120 9.965348 10.0346522-8395369.858998 9,980538 10.019461 17
44 9.83160619.8660049.965692 10.0343989.839668 9.858877 9.980791 10.01920916
4519.83174219.86538714.965855110.0341449.83980019.8587569.281044 10.018956125
4619.83187919.86577019.96610910.0338919.839932 9.85863919.981297|19.01870314
47 9.832015 9.865653 9.96636210.0336381 9.840064 9.858514 9.981550 10.01845013
4819.832152 9.865536 9.966616 10.033384 9.840196 9.858398 2.981803 10.01819712
499.832288 9.865419 9.966869 10.03313119.8404389.858272 2.982056 10.017944 11
50 9.83242519.86530219.96712210.032878 2.84045919.858150 9.982309110.017691110
$119.93256119.865 185 19.967376 10.032624 9.84059119.85802919.982562|10.0174389
$219.832667 9.8650689.967629 10.032371 9.84072219.857908 9.981814 10.0171853
$39.832833 9.86495c19.96788310.032117 9.840894 9.8577869.983067 10.1169337
5419.832969 19.8648339.968136 10.031864 9.84098519.857665 9.983370 10.01668016
5519.83310519.8647 69.968389110.03161119.8411169.857543|9:983573110.016427
57 9.8333769.864480 9.968896 10.0311049.8413789.857300 9.98407910.019911
589.8335129.8643639.96914910.030851 9.841509 9.857178 9.984331 10.015668
599.8336489.864245 9.969403 10.0305979,847640 9.857056 9.984584 10.01$416
609.83378:19.86412719.96965 10.030344 9.84177119.8569349.984837 10.01316310
Co line | Sunc 1 Co-tame. Tangent Co-line | Sine !Cotong. Tangent IM
Degree 47
(+2)
1
+
1
Degree 46
1
:
-1
Canon Triangulorum Logarithmicus.
1
.
.
+
1
?
301
Degree 44.
MI Sine 1 Co-ſine 11 Tangene i Co-tang. I
019.84177119.8569349.984837110.015162160
119.84190219.8568121 | 9.985090110.014910159
2 9.842033 9.8566.90|| 9.985343 10.014657158
319.84216319.856568 9.985596 10.014404157
49.842294 9.856445 9.985848 10.014151156
519.842424 9.85632311 9.986101|10.013899155
619.84255519.856201|| 9.986354/10.013646154
7 9.84269519.8.56978| 9.986607 10.01339353
89.84281519.8559569.986859 10.013140
919.8429459.855833|| 9.987112/10.0128885
109.843076 9.855710|| 9.98736510.01 263515
1119.843206.9.855588|| 9.98761810.01238249
129.843336 9.855465|| 9.987871 10.012129 48
13.9.843465 9.8553421) 9.988123 10.011877 47
149.84359519.855215|| 9.988376 10:011634 46
1519.84372519.855096 | 9.988629110.011371145
1619.84385519.854973|| 9.988882f16.011118144
1.79.8439849.85485011 9.989134 10.010866 43
18 9.844114 9.854727 9.989387 10.010613142
1919.844243 9:854603|| 9.989640 10.010360 41
2019.844372 9.8544801] 9.9898931 10.01010740
2119.84450219.85435611 9.990145710.009855139
2219.844631 9.88423311 9.990398 10.00960238
39.84476019.854109|| 9.990651 19.00934937)
24 9,8448899,853986. 9.99090310.00909636
2519.84501819.853862 9.99115610.008844135
26/9.845147 9.853738|| 9.9:71409110.008591134
27 9.845276 9.853614) 9.921662 10.008 338 33
28 9,845 404 9.8534901| 9.991914 19.008086 32
29 9.845533 9.853366 9.992167 10.007833131
3019.84566219.853242|| 9.992420110.007580130
3119.84579019.853118119.99267210.00732829
329,84597919.852994 9.992929 10.007078 28
3319.846047 9.85 2869 9.993178 10.00682227
349.846175 9.852745 9.993430 10,00656926
3519.84630419.852620|| 9.993683119.0063171-5
3019.84643219.8524961| 9.993936110,00606424
37 9.8465609.852371|| 9.99418910.09581123
38 9.8466889.852246 9.99444110.00555922
3919.846816 9.852122 9.99469419.005 306:1
4019.84694419.851997|| 9.994947110.005053120
4119.84707119.85187211 9.995199110.004801119
4219.847199 9.851747|| 9.995452110.004548118
439.84732719.851622 9.995701 10.00429517
4419.8474549.851497 9.995957 10.00404316
4519.84758219.8513721) 9.996210 10.003790115
469.84770919.8512461) 9.99646310.00353714
47 9.847836 9.85112I 9.99671510.003285113
4819.847964 9.85099611.9.996968 10.003032112
49 9.84€09119.850870 || 9.997220|10.00277911
5019,84821819.850745|| 9.997473110.00252710
51 19.848 3459.3506191 9.99772619.00-27419
52 9.8484729.850493 9.997979 10.002021 8
53 9.848599 9.850367|| 9.998231 10.0017691 ;
5419.8487269.850242 9.998484 10.001516) 6
5519.848852 9.850116|| 9.998737110.0012631,5
5619.84897919.849990|| 9.998989/10.00101114
5719.84910619.849864|| 9.999342 10.0017581 3
5819.84923219.849737|| 9.999495 10.000505 2
59 9.849359/9.849611 9.99974710.000253) I
6c19.84948519.8494851/10.000000 10.000000
I co-fine I Siue ll Co-tang. Tangent /M
Degree 45.
1
2
t
N
!
1
!
1
CHILIADES
DE CE M
LOGARITHMORUM,
SHE WING
The LOGARITHMES of all NUMBERS
Increaſing by Natural Succeſſion from an Unite to 10000 :
Whereby the Logarithmes of all Numbers under 1000000 may
bc ſpeedily deduced.
Firſt Calculated by that Excellent Mathemacician
Mr. H E N R I B R Í GG S;
Profeſſor of Geometry in the Univerſity of Oxford.
11
And their Uſe now Amplified,
By Capt. SAMUEL STURMY.
ܐ
410.602060
+
Ni Log. IN Log. 1 INI Log. I INI Log. TINI Log.
110.000000 21(1.3222191 14111.612784.16111.785330 8741.908485
20.301030221.342422 42 1.6232491 162 1.792391
821.913814
310.477121 123 1.361728 4311.633468 16311.799340 831.919078
34/1.380251 44 1.643452
641.806180
8411.924:79
510.6989741125 11.397940 145 1.653212 16511.811913 8511.929419
680:778151 12611.414973. 1461.662758116611.8195.441 | 8611:234498
710.845098 271.431364471.672098 67. 1.826075
871..939519
80.903090 28 1.4471581 48 1.681241
68 1.837509
88 1.944483
9 0.954242 2911.462398 49 1.69019669)1.838849 89 1.949390
101.000000 1301.4771 21! Iso 1.6989701 170/1.845098 ] go 1.954242
1111.0413931 13:41.491361115111.7075701 17111.851258 911.95.9041
1211.079181 32 1.505 ISO 521.7160031 1721.857333
9211.963788
13/1.113943
3311.518514) 1:3 1.7242761 1731.863323
93.1.958483
141.149128 34 1.531479
54 1.732394 7411.869232
941:973128
15 1.1760911 13511.5 44068 5511.740362 1751.1.875061 95:11.977723
1611.2041201 1361.5563031 15611.748188] 17611.880813
961.982271
171.230449 37 1.568202 57 1.755875 77 1.886491
971.986772
18.1.255272138 1.5797831 15811.763428 7811.892094
981.991226
191.273753 39 1.5910641 59 1.7709521.79 1.897627 9911.995635
2011.301030 4011.60.0601 16011.778151 8971.903090
100 13.000000
1
!
f
F
i
I
LONDON ,
Printed by E, Cotes, Anno Domini 1669 .
(8)
The Table of Logarithmes.
.
1
illo
I
.
21 3 14 15 16 17 181.9D
100||oooooo100043400086810013010317341002166|0025981003029|003461|00389111432
10111004321 004751005181005609006038 006466006894007327 00774810081741428
10200860n|c09026009451009876010299 0107241011147/011570011993/012415 ||424
10310128371513259913679014100|O1452110149400153590157791016197 016616 1419
19411017033/?174511017868018284101870001911601953240199471020361/0207751 1416
105192118 910116931022016027428|0228411023252102366410740751024486|024896|412
1062025336 025715026125|06533 026942027349027757028164102857110289781408
107||0293841029789030195030599 03100403140803181203227632619 033021||404
108|033424 033826034227|034628035029035429 035829 036229103662940370281400
109110374261037825103822310386201039017103941 41039811040207040602040998|l396
11004139310417871042182104257610429691043362104375510441 480445 390449321/393
111||04532304.5714/0461050464950468851047275104766404805 3 04844204883011389
112||049248 049603104999305037910507660511:53 051538 JOS 1924 08 2309 08 26941386
113|05 30781083466083846 054229108 46131054996055378 085760105614205.65241382
114110569051057283 1057666058046105842610588951059185 1059563108994206032011379
115||060698061075 10614521061829706220606258210629581063333.1063709106408311376
116||06445810648321065 206 065579,0659531066326 066699 067071|067443 06781511372
117||068186 068557068928969298 069668 070238 070407070776 071145071514||369
118||071882072249 0726170729850733527073718 0740851074453 0748160751821.366
1191)0755471078912/07627610766401077004107736810777311078094|078457078819|1363
1201107918140795431079904/080266|08062608098710813471081707 0820670824261/360
1121||0827850831441083503 083861084219 0845761084934 085291 0856470860047357
122|086359 0867160870711087426 087781088136088490 083845 089198108955211355
123||089905 090288 0906100909630913151091667|092018|0923691092721109307111351
12411093427/6937721094122109447140948202095169109551810958661096275109656211349
125110969101097.25.91097604/09795110982981098644/098.98910993354099681|1000261346
12010037110071510105 9101403 101747102091 102434 102777 103119103462|343
127||103804 104146 104487 10482810S169 105510105851 106191|100537 106871340
128 |107209 1075 49 107888108227|108565108903109241 109579|109916110253338
129 11105891109:6111263111159911119341112269118260511129391113275/11360911335
130||113943[114277|114611|114944115278|115611|115943|116276116608|1169391333
131|11727111760371 179341182651.185951118926 119256119586 11991$ 120145330
132110374 129903121231 121559121888 121216122544 1 228711123198|12352511328
133|12385211240781245041114830125156129481 125806126131 126456 126781||325
134111271931127429112775311280761283991287221290451 293681296891130012 | 323
1351|130334|1306551130977|1312981131619131939|1322591132579|132899113321911321
136133539133858341771134496134814135133 1354511135769 136086136403318
137||136721137037 - 37354.137671|1379871138303 138618 138934139249113956411315
138 139879|149194 1 40508 140822 141136141449 1417631420761423891427021314
1391|14301,51143327_143639|14395 1114426314457411448851145 196 145507114581811311
1401|14612814643814674814705814736714767614798514829411486031148911309
141||149219149527 149835150142 150449 150756 151063/151369 151676 151982|1307
142|152288 152594152899 153205153599 153815154119 154423154728 555032305
143||155336 155639 155943156246 156549156352157154157457157759158061303
14411158362415866411589651159266|15956711598681160168160469|160769|161068H301
145111613691161667|161967|162766162564/163863163761|1634591163758/164055|1299
146164353164650164947165244 165541_165838166134 166430166726167023|| 292
147||167317167612167903168203 188497 168792 169086169380169674 169968295
1481702925705131108431792417.434 47370907201917017266317789511293
149||17318611734781173769|1740511174351|174641|174932175122 115sizli7580211291
1
3)
1
The Table of Logarithmes.
܊
N0.11.12 13 14 15 16
1 3 14 15 16 17 18 19 D
1150|176091|176381f176669|17695911772481775361778251178113178401|17868911289
i511178977179264.179552173839 180126180413180699 180986181272181558287
152181844 1821291182415182699 1829851183269183555 183839 1841231184407285
153184691 1849751852591855421185825 186108 186391186674186956187239283
541|1875211187803119828418336618864711889-81189209|19949011891711990511|281
15511190332f1936121190892119117111914511191730|192009192289|192567|19284611279
56|193125 193403193681193959 194237 194514194792195069195346 1956131278
1571195899 196176 196453196729|197005197281197556197832 198107 1983821276
1581198657|19893-199200 199481 99755200029209303/200577 2008502011241274
15911201397 201670|2019432022162024882027611203033|203303|203577203848272
1601|20411942043911204663 2049341205204 20547512057462060161206286206556|271
161||206826 207096 207365207364207904208173 208441208710208978 2092471:69
.6220951512099831210051|210319 210586-10853 211121 211388 211654/211921 | 267
163|1212187212454212730212986 213252 213518 213783214049214314214579266
164 | 21484412151091215373121563812159021216166216429|21669421695721722111264
1651|217484/21774721801021827312185361-187982190601219323121958512198461|262
166|220108 220369220631122089222115312214141221675221936 222190 2224561161
1672227161222976 223236 2234961223755122401512242742245331224791|2250511259
168||2253092255681225826 226084226342/226599)22685812271152273721229629||258
1691|227887228142 2284001228657 2289131229169|229426122968212299381230193||256
170|1230449|230704|230959123121512354691238724|231979|232234/8324881232742||254
1711232996 233250233504 233757-34011 2342641234517 234770235023-35276353
1723355282357811236033 236285 236537236789237041237292 237544237795 252
1731233046 238297-38548 2387991239049|239299 239549 239799124004924029911250
1741|2405491240799174104872412971241546241795124204412422932425411242789|1249
1751|24393812432862435341243782/2440291244277/244525|244772/2450192452061|248
1762455132457592460062-46252 246499246745 246991 247237247482 247728|246
1771247973124321924846412487091248954/249198 2494432496871249932125017611245
1781|250420250664250908-511511251395 25163812518812521251252368 252610243
179 1252853125309612533341-5358012538222540641254306 254548125478925503111242
180112552731255514125575512559961256237125647712567181256958125719812574381)241
181|1257679 257918 258158258398125863712588772591162503551259594259833||239
1821260071 2603091260548 2607872650251261263|261501 261739 261976 261214||238
183 262451 262688 262925263162 263399 263636 263873 264109|264346264582||237
1841126431812650541265289265525126576 1265996266232 266467 266702 266937||235
185112671721267406/267641|267875|268109|26834412685781268812 269046126927911234
186269513 26974612699792702131270446 270679|270912271144271377271609233
187271842 27,20741272306 272538 272769 273001 27323312734641273696273927||232
1882741581274389 274619|27485012750811275311 275542 275772 276002 2762321|230
1891|27646212766921276921127715112773791277609127783812780671278296]27852511229
190278754127898212792.11|279439127966712798951280123|280351/280578128080611228
191 1281033 281261 28.1488281714 281942 281169 282396 282622 282849283075227
192||283301 2835.27. 2837531283979284205284431 284656 284887 285107285332 226
1932855572857821286007 286232 286456286681 286905 287129 289354 287578||225
19411287802 2880261288249|28847312886962889191289143|289366289589|289812||223
195.||290035|29025.7|293479|290702290925|291147129136912915911291813|2920341|222
196272256 292478 2926992929202931432933631935841293804 294025294246)|i21
197 | 294466 294687 294907295127295347 295567295787 296007 296226 296446 220
1981|296665 296884 297104/297323297542 297768|297979298198298416 2986351219
1991|2988531299074/29928912995072997251299943300161 300378 300595130081311278
1
(g2)
1
1
The Table of Logarithmes.
No 11
2 13 14 15 16 17 18 19 ID
200|130102y|3012471301464130168113018983021141302331130254713027641302979|1217
201||363196 203412 303628 303844 30405913042751304491304706304921 305 136216
202|13053511305.566 305781 305996 306211 3064253066391306854 307068 307282 215
203||307496 3077093079241308137 30835 11308564308778 308991309204 309417213
2041130963013098431310056315268310481|3106931310906131111813113291311542||212
205113117541311966131217713123893126001312812131302313132341313445313656||213
2061313867 314078 314289314499 314709 314920 315130315340315551315760210
207||315970 3161801316389 3165991216809317018 317227317436317646317854/209
208 ||318063/318272 318481 318689 318898319106 319314319522319730319938||208
20911320146 3203541320562132076932097713211841321391132159813218051322012||207
2101322219 322426322633132283913230463-3252132345813236651323871|324077||206
211||324282324488 324694/3248993251051325310 325516 3257211325926 326131||205
2121326336326541326745 326949 3271551327359 3275631327767 327972 328176204
213328 379 328583328787 328991329194/329398 329601 329805 330008 330211203
21411330414 3306171330819/331022133122513314271331629)33183213320341332236|| 202
2151133243813326403328421333044133324613334471333649133385913340511334353||202
2161334454 334655 334856335057 335257 335458 335658 335 859133605913362591 201
2171336459336659 336859 337059337259337459337659337859 338058 338257200
21813384593385563388561339054] 3392531339453 339650 339849 340047|340246199
2191134044413406421340841/3410391341 237)3414351341632134183013410281342225||198
2201134222713426201342817|34301413432121343409|343606134380213439991344196||197
221||3443923445891344785 344981 345178 345737 345569345766345962 346157||196
222 3463531346549 346744 346939 347135347330 347515 347720 347915 348110|195
22311348305348499 348694 34888913490831349278 349472 349659134986013500541194
224|13502481350442|350636135082913510231351216135140913516031351796135198911193
22511352183135237513525681351765135295413531471353339135353213537241353916||193
2261354108 3543013544933546851354876 355068 3552591355452355643355834||192
227|| 356026 3562173564081356599 3567921356981357172 357363357554 357944||191
2281357935 358125358316 3585261358696358886359076359266359456359646|190
2291135983513600251360215|3604041360593|3607831360972 361161361350361539||189
23011361728361917 362105136229413624821362671|362859136304813632361363424|188
1231||3636121363799363988 364176364363 36455113647391364926 3651131365301||188
232365488365675365862 366049366236366423 366609 366796366983367169|187
233367356 3675421367729 367915 368101 368287368473|368659 3688451369030 ||186
234|| 369316136940113695871369772/36995813701431370328/370513137069813708321189
2351|37106813712531371437137162237180613719911372175137235913725441372728||184
236|372912 373096373279373464 373647 373831 374015374198 374382 374565184
237|| 374748 374932 3751151375293 3754813756641375846 376029 3762123763941|183
238376577 3761591376942 377124 377306 377488 3776703778921378034 378216182
23911378395/378579 378761137894313791241379306379487137966813798491380030||181
2401138021113803923805731380754138093413811151381296138147613816561381837||181
24111382017 382197382377382557382737/382917 3830973832771383456 383636||180
2421383815 3839953841741384353 384533 384712 384891/3850691385249 385428||179
2431385606 3857853859641386142 3863211386499 3866771386856387034/387212178
244113873891387568/3877461387923138810113882791388456388634 .3888111388989|| 178
245113891661389343/38951013896981389875/3900511390228 3904051390582139075$||177
246|390935139111: 391288 391464/391641 391817 391993 392169392345 392521.176
247|| 392697 3928733930483932241393399 393575 393751 393926 394101 394277||176
248|394452 394627 394802 394977 395152 395326395501 395676 395850 396025||175
24911396199|3963741396548/3967221396896 3970711397245/397419/3975921397766|1174
1
1
The Table of Logarithmes.
1
NI 0111?
1 3
14151612
16 17 18 19 D
Tim
269)
250113979401398134139828713584611398634|398808139898139915413993281399501||173
251||399674 399847 400019400192 400365 400538 400771|400883 401056 401228||173
252401401 401573401745 401917 402089402261 402433402605 402777 402949172
2531403121403292 403464403635403807 4039781404149 404320 404492 404663|171
2541|404834|405005|405176405346405517 405688 40585814060291406199|406369|1171
2551|406540|40671014069811407051407221|4073911407561407731407901|40807011170
256408239 408 409 408579408749.408918 409087|409257409426 409595 409764169
257|409933|410102410271410439410609 410777 4109464111141411283|411451||169
258411619 411788 411956 412124 412293412461 412629 412796 412964 41.3132||168
2591413299141346714136334138031413969141413714133051:414472 4146391414806|167
2601141497314151401415307141547414156414158081415974 4161411416308416474||167
2611416641416807 416973|417139417306 417472 4176381417804417969 4181351166
262418301 418467 418633418798 418964.419129 4192951419460419625419791165
263419956420121420286.420451 420616 4207814209454215104212754214391165
2641|42160414217681421933|4220971422261142242614225891422754 4229181423082|1164
265 ||423246423409|423574423737|42390114240651424228142439214245551424718||164
266424882425045 425208 425371425534 425697 425860 426023 426186 426349163
267||426511 426674426836 426999 427161 4273241427486 427648 427811 4279731 162
2684281351428297 428459 428621: 428783 428944 42910G 429268 429429 429591||162
42975214299141430075.430236430398/430559430714436881. 431042 43120311161
270431364|431525143568514318461432007 432167 43232814324891432649|432809||161
2911432969433129 433289 433449433609 4337694339291434034 43424914344091601
272 1434569 434729 434888 435048435207 435366 435526 435685435844 4360041159
2731|436163 436322436481 436639 436799 436957 437116 4372754374334375921159
27414377514379091438067|438226 43838414385421438701 4388591439017|4391751 158
275 || 439333|439491 4396484398c6439964 440122/440279440437 440594|440752||158
276440909 441066 441224 441381 441538 441695 441852442009 442166 442323|157
2771442479 442637 442793 442949443106 443263 443419 443596443732 443889157
2781444045444201 444357 444513444669 444825 444981 445137 445293445449|156
279|1445604|44575914459151446071|4462261446382 446537 446692 4468481447093|1155
280||4471581447313447468|447623|4477784479324480881448242 448397f448552||155
281448706443861 449015449169 449324 449478 449633 449787449941 450095 ||154
282450249 450403450557450711 450865 451018 451172451326 481479 451633|154
283451786 451939 452093452247 452399452553 452706 452859 45301245316511153
284114533181453471 4536241453777/4539291454082 454235/454387 454539145469231153
2851145 4845|454997 455149|455302 45545445560645575814559101456061145621411152
2861;456366456518456669 456821 456973457125 457276 457428 457579 457731152
287 1459889 458033 458184 458336458487 458638 45878914589391459031 459242151
288459392 459543 459694 459845 459995 460146460266 460447 460597 4607481153
2891|4608984610481461198 4613481461499|461649|4617991461948 4620981462248111so
290||462390|4625481462697|462847462997|463146463296|463445|4635941463744||150
291|1463893 464042 464191 464340 464489 464639 464787 4649364650854652341149
292465383 465532465689465829 465977 466126 466274 466423 466571466719149
2931466868 467016 467164 467312 467460 467608 467756 467904 468052468199 148
294114683471468495 468643|4687901468798 469085469233|469380469527/469675|147
2.9511469822146996914701161470263|470410 470557 470704|470851/4709981471145||147
296471292471438 471585 471732 471878 472025472171 472318 472464 472610146
2971|472756472903 473049 473195 473341 473487 4736331473779 473915 474071146
298474216 474362 474508 474653474799474944 475089475235 475381|475526146
29914756711475816 475962/476107 476252/4763971476542 +76687/4768321476976|1145
I
+
}
(h)
A
1
1
+
ܙ܀
0
The Table of Logarithmes.
No| 1 2 3 14 15 16 17 18 19 D
3004771211477266477411 4775551477699477844/47798914781331478 2781478422||145
2011|478566 478711 478855478999 4791 43 479287 479431479575 479719 4798631144
302480007 480151 480294 480438480582 480725480869431012481156 481299||144
303||481443 481586 481929481872 48 2016 482159 482302 482445 482588 482731143
304114828741483016 483159483302483445433587/48372954838724840151484157| 143
305 ||484299484442148458514847271484869 4850114851534852951485437/4855791|142
306485721 485863436005486147 486289 486430 436572 486714486855486997143
307487138487279 487421 487563487704 487845 487986 488137 488269488409141
308 ||488551 488692488833 488974489114 489255 489396 489537 489677|489818141
309||489958/4900991490239/490379149052014906611490801139094114910811491222 || 140
310|49136214915021491642 49178214919227492062492201 14923411492431/492621||140
311492760 492900493039493179 493319 493458493597 493737 493876 494015||139
312||494155 494294 494433 494572 494711 4948 50 494989495128 4952671495406|139
313 495544495683495822 495960496099 496238)4963764965154966531496791|1139
3141|496929497068497206 497344/497483497621|4997591497897/4930351498173 138
315||498311/498448149858614287241498862/4989991499137 4992754994121499549||138
3:16|4996874998.24 4999625000991500236 500374 soos 10 500648500785500922|137
317|| 501059 501196501333|501470 501607 505744 501880502017 502154 502291|137
31815924:7502564.502790 502837 502973503109 503246503382 503518 3036551136
319||503791/50392715040631504199 504335 50447115046071504743|504878150501411136
320 115081491505286|5054211505557/5056931505828150596415060991506234|5063691136
321||506505 5066401506776|506911|5070461507181 507316 5074511507586 507921135
322||07856507991 508126 508260 508395 5085291508664150879,15089341509068|1135
1323509203509337 5094711509606 509740509874 51000915101431510277 510411134
32411510545151067915108131510947151108115112151511349|51148215116161511749134
1
325||511883151201715121511512284151241815125511512684151281815129511513084||133
32611513218 31335151348415136171513750513883514016 5141491514282 514415133
327||$145481514681 514813|$14946515079 515211|51534415154765156091515741133
328515874 516006 5161391'51627151640351653515166681556799151693215170641132
329115171961517328151745915175921517724151785515179871518119151825115183821|13:
!
330||5185141318646]518777|51890951904015191775193031519434/519566151969711131
331519328 51995915200901520221520353 5204841520615520745 5208761521007 131
3325211381521269152139952153015216615217921521922 522053522183]522314||131
3331152244415225751522705 522835152-9661523096 5 2322615233561523486 5236161130
33411 52374615238761524006152413615242661524396152452615246561524785152491511130
3351152504515-517415253041525434525563152569315258231525951/526081152621011129
336 192633915264691526598 52672715268565269855271145272435273721527501|129
337527629527759 5278881528016 528145528274 528402|52853115286591528788129
338||$28916 529045 529174152930215294301529559152968715298151529943530072|128
339 53019915 30328153045615305841530712153083915 30968 53109615312231531351||128
340||5314791531607153173453186215319891532117153224515323721532499|532627||128
341||532754 532882533009 5331365332645333915335181533645 533772 5338991|127
342||$34026 33415315342801534407 534534534661 534787 534914 5350415351671|127
535294 5354211535547 53567453580015359271536053536179 53630415364321|126
34411536558)536685153681115 369375 3706315371891537315153744115 37567537693||126
345||$37819153794515380911538197538322|5384481538574153869953882515389511|126
3461439076 539202 539327 5394531539578539703, 539829 539954 54007954020411125
540320 54045515 40579 540705540829 540955541079541205 54132915414541125
34854157915417045418:9 541953/5420781542203 542327 54245215425761542701||125
349||44282=15429191543074 54319915433231543447/54357115436965438191543944_1124
:
347
+
1
1
+
The Table of Logarithmės.
N|| o | I | 2
1
3 14 15 16 17 18 19 D
35011544008154419215443161544440|5445645446881544312/544934154505915451831|124
35.111545307 545421 545555 545678 545802/545925|546049 546172 546296 546419||124
352546543 546666 5467891546913 547036 5471591547282 547405 547529 547652||123
35311547775 5478981548021 548144 548267/548389 548512 548635548758 548881||izz
3541149003154912615492491549371154949415496161549739 549861549984550106||123
1
3551155022855035115504731550595155071715508391550962|551084|5512061551328||122
35615514495515721551694 5518161551938 552059 55218115523035524251552547||122
357||55266815527891552911553033|553155553276 553398553519553640 553962121
35815538835540041554126554247 5543681554489554610554734554852 5549731121
3591155509455521515553361555457.55557815556991555819155594015560611556182) 121
3601155630315564231556544155666415567851556905155702615571465572671557387||1 20
3611557507 557627 557748 557868 557988155810815582281558349 558469 5585891|120
26211558709 55882955894815594681559188|559308 55942815595481559667559787 || 120
3631559907|560026 5601461560265560385 560504 560624 5607431560863 560982|119
3641156110115612211561339|5614591561578156169815618171561936|5620551562174119
1
365 ||562293|56241215625311562649156296915628871563006156312515632441563362||119
366|563481 563599 563718 5638375639551564074564192 5643115644291964548||119
3671564666564784 5649031565021 565139 56525756537615654941565612 565719|t 18
368|15658481565966 566084)566202 566319 566437 566555 566673 5667911566909||118
36911567026 56714415672621567379156749715696141567732156784915679671568084||118
370||568202|5683195684365685541568671568788/56899515690231569139|5692591|119
1327569374 569491 569608 569725 569842 569959570076 570193 570309590426 127
372||$70543 570659570776 570893 571009 571126 5712435713591571476571592|1171
373||571709 571825571942|572058572174.5722911572407 572523 57263915727551 116
374115728721572988157310415732191573336|573452157356815736841573799/5739151|16
1
375||57403115741 47 5742631574379574494157460915747261574841157495715750721|116
3761575188 575303 5754191575534/575649575765575880575996 57611157622611115
377576347576457 5765721576687 576802 576917 577032 577147 577262 577377||115
37811577492 5776071577722 5778365779511578066578181 578295578409 57852511115
379|157863915787541578868 578983157909715792125793261579441/579555157966910114
38011579784 579898580012580126580241|5803551580469158058315806371580811||114
38111580925 181039581153 581267581381 581495 581608 5817221581836 581949|114
332115820631582177 582291 532404582513 582631582745 158285815829725830851114
383 583199 583312583426 583539 583652583765583379 5839921584105 484218||113
3841158433115844441584587158467015847831584896585009 58512215852351585348||113
O.
38511585461 5855741585686158579915859125860241586137158624915863627586475||113
3865-865871586699 5868121586925 587037 587149 587262587374 587486587599||112
387158771115878231587935 58804715881591588272 588384 588496 588608 5887191112
383 588832 588944589056589167589279158939115895035896151589726|5898381112
389589949 590061159017315902841590396 5905071590619 5907305908425909531/ 112
1
.
1
393||5910651591176159128715913991591509591621159173215918435919551592066|111
391||592177592288 5923991592509159262115927321 592843592954593064 5931751|111
392593286593397593508 593618 593729 5938391593950 3940611594171 594282|111
393 594393 594503 594614 5947241594834/594945 3950551595165 595276 5953881|110
394/13954965956001595717159582715959371596047159615715962671596377|59648711110
1
395||596597596707159681715969-71597037159714615972563973665.97476 59758611110
396597695 5978051597914 598024598134 598243 5.98353598462598572 598681|110
397 ||598790 598399 599009599119 599228 599337 5994465995565936651599774 109
398599883599992 600109600210600319 6004286005376006466007551600864 109
399||600973 6010826011911601299|601408601517|60162516017341601843601951109
21
(h 2)
.
The-Table of Logarithmes.
NO 1 | 2 | 3 | 4 15 16 17 18 19 D
402||602059160216916022771602386602494160260316028191602819160292816030361108
4016031446032536033616934691603577603686 603794603902 604009604118 |108
40260422660433416044421604550 604658 604766 604874604982605089605197||108
4031|6053056054131605521 605 628605736605844 60595 1160605 91606166606274||108
704116963811606489606596160670416068111606919160702660713316072411607348|1107
405160745516075621607669160777716078841607991608098160820516083121608419][107
400608526 6086336087396088471608984609061609167609274609381609488107
407|6995946097011609808 609914610021 610128610234610341 610447 610554107
408||61066016107671610873610979 611086611192611248611405|6115116116171106
4096117231611829|611936|612042]612148612254612359|6124661612572612678||106
4101612784/61288916129961613102161320716133131613419161352516136301613736||106
4111613842 613947614053 6141591614264614369 614475614581614686,614792|106
4121614897 615003 615108615233615319 615424 615529615634/615.73916158451105
4:36159501616055 6161606162656163706164766165811616686 61679016168951|1os
414116170006171051617210|6173151617419|6175251617629 6177346178391617943||105
4151618048161815316182571618362161846616185711618676618780161988416189891|105
41616190936191981619302 619406 619511619615 6197196193246199281620032||104
4171610135620240620344 620448620552620650620760620864 620968621072||104
4181620176 521280162138462144816215926216951621799621902 622007 622110|104
419|162221416223171622421|62252516226286227311622835 1622939633042623146||104
47011623249162335316234561623559162366316237661623869162397316240761624179||103
471624282624385624488 624591624695624798624997 625004625107 625209103
422||625312625415625518625627 62572446258271625929626032626135626237||103
423626340 62644362654662664862675162685362695616270581627161627263]|103
424116273661627468|6275711627673162777516378981627979162808 216281851628287 || 1oz
1
42516283891628491162859316286951628797628899162900216291041629206162930811102
42662940916295126296136297151629817629919,630021630123 630224630326102
427|6304286305296306316307336308356309361631038631139631241631342102
42863144416315451631647631748|631849631951632052|632153632255632356101
14291632457|63255916326591632761|632862/632963163306416331651633266633367| 101
4301163346816335691633670163377116338721633973634075163417516342761634376|109
431|6344776345781634679634779|634880634981 635081 635182635283635383/100
43211635404635584635685 6357851635886635.98663608763618716362886363381100
433|636483 636558163668863678916368895636989637089637189637289637389|100
4341|6374891637589163768916377891637889163798916380891638189|6382891638389 99
4351|638489163858916386891638789163888816389881639088163918816392871639387|| 99
4361639436639586639686,639785639885639984 640084640183640283640382 | 99.
4371640481 64058164068064077916408796409781641077 641177 6412761641375 99
43811641475 641573 641672641771641871 6419696420691642168642267642366
99
43911642465164256316416626427616428671642959643058164315616432551643354|| 99
449|64345316435511643650164374964384716439461644044164414316442421644340|| 98
441|644439|644537 644636 644734 649,832 6449301645029 645127 645226645324 98
4421645422645521645619 645717645815 645 913 6460r1 646109646208646306
4431646404164650216465996466981646796646894646992647089 6471871647285
4441647383647481647579647676647774647872 64796964806716481656482621) 98
445|164836016484581648555164865316487501648848164894516490436491401649237|| 97
446649335649432649529649627649724649821649919650016650113650210 97
44765030816504051650502165059965069665079316508906509871651084651181
97
448691278651375 6514721651569651666651762651859651956 652053652149 97
449||6522461652343.65 2439165253665263316527291652826_65292316530197653116| 27
98
98
IH
The Table of Logarithmes.
No11
II 2 13 14 15 16 17 18 19 D
94
45011653213165330916534051653502165359816536951653791/6538881653984165408011 96
45111654177 654273)65436916544651654562165465816547546548506549466550421 S6
45211655138165523516553316554271655523655619 6557156558101655906656002
96
45311656098165619465628916563861656482656577|6566731656769/6868641656960
96
454116570561657152365724716573431657438165753416576291657725565782016579161 96
4551165801116581071658201165829816583931658488165858416586791658774165886911 95
4561658965165906016591551659250659346659441 659536659630659726659821
95
457165991666001166010666020116602966603911660486660581660676 66077111.95
45816608656609601661055661149 661245 661339 661434 661529661623 661718|| 95
459)|661813661907|66200216620961662191 1662286166238016634751662569|662663|1 95
460166275816628521662947|663041|663135 166322966332466341816635121663607
94
46116639011663795663889663983664078166417216642666643591664454664548
94
46216646426647366648296649246650181665112665206)665299 665393665487|| 94
46316695816656756657696658621665956666049166614366623716663311666424
46411666518166661116667051666799666892166698616670791667173|6672661667359|| 94
465|1667453166754616676391667733/66782666791916680136681061668199166829
4661668386663479668572166866566875966885216689451669038 6691311669224 93
467669317 669409669503166959666968966978266987516699671670060670153
93
468|670246670339670431 670524 6706171670709 670802670895 670988 671080193
4691167117367126567135816714511671543167163616717286718211671913672005|1 93
4701167209816721901672283169237516724671672559167265216727441672836167292911 92
471 673021673.113673203673297 673389673482673574673666673758 673849
921
4721673942674034 674126 674218 674309 674402674494 674586674677674769
92
4731|674861674953675045 675 137 675228675319675412675503675595 675689
92
4741167577816758691675962676053167614516762366763281676419|676511167660211 92
47511676694167678516768761676968167705916771516772421677333167743467751611 91
4761677607 6776981677789677881677972678063698154678245678335, 678427
91
4771678518678609 678700 678791 6788821678973 6790641679755 6792461679337
478|6794281679519 679609679700679791 679882 679972 680063/680154 680245|| 91
4791168033616804261680517168060716806981680789168087968096916810601681151 91
91
480116812411681332 681422468151316816031681693168178416818741681964/682055|| 90
481116821456322351682326682416 682506682596682686682777 6828671682957 90
4821|68304716831371683227683317 683407_683497683587683677|68376768385711 90
4831683947 684037 684127|684217 6843076843966844866845761484666|684756 90
48411484845684935168502516851141685204685294168538316854731685563168565211 90
485|1685742168583116859211686010168609916861891686299168636816864581686547|| 89
486|686636686726686815686904 686994 687083687172687261 687351687439|| 89
4876875291697618687707|68775616878851687975 688064 6881531688242 688331|| 89
488|688419 688509688598688687683776 688865|688953 689047 689131689220 | 89
48911689309689378689486689575168966468975316898411689930 6900196901071) 89
49011690195169028516903731690462169055016906391690728169081616909051690993|| 89
491169108116911696912586913476914351691524691612691709691789691877|| 88
4921169 1968 16930536921426922291692318 692406 692494 69258369267.5 692759 88
493|1692847692935693023693111693199/693287 6933756934636935511693639|| 88
49416937271693815169390316939916940781694166694254/6943471694429|694517|| 88
49511694605169469369478716948681694956169504469513116952191695307169539411 88
49616954826955696956576957446958326959196.96007696094696183696269 87
497696356696444696531 696618696706696793696880696968697055697142 | 89
4981697229697317697404697491697578697665 977521697839697926698014
1499|1698101628788698275|698362698449|698535/698622169870916987061698883
87
871
+
(i)
5
The Table of Logarithmes.
NI 0111213141516171819 D
500||69897069905716991441699231169931716994041699491169957816996641699751|| 89
sou699838699924700011 700098 700184 700271700358 7004441700531700617 87
so27007047007907008771700963 7010491701136701222701309701395701482 86
503||701568701654701741701827 701913|701999 702086702172 7022581702344 86
5041170243017035177026031702689|702775 7028611702947170303317031191793205 86
CO CO
1
85
85
84
7
83
83
sos||70329117033777034637035 49/703635170372170380717038931703979704065 | 86
50670415317042367043221704408 7044947045791704665 704751704837704922 86
507|705008705094705179705265 705350705436705522705607 705693705778 86
50817058637059497060351706120706206 706295706376706462 7065 47 706632185
5091|70671817068031706888/706974170705917071441707219170731517073991707485:11 85
51011707570707655170774017078261707911170799617080817081667082511708336|| 85
$117084217085061708591708676708761 708846 708931 709015709100709185) 85
$12||709269709355709439 709524 709609 709694.7097791709863709948 710033
553|7101171710202710287 710371 7104567105401710625710709710794 710879
514117109631711048171113217178171711301/7113851711469171155417116391711723!! 84
515|711807171189217119767120601712144171222917123131712397 7124811712566|| 84
$16712649|712734 7118187129021712986713070713154713238713223-713409.
517|7134917135757136591713742 713826713910713994714778714162 7142461 84
$18714329 714714 714497/714581 7146657147491 714833.714916/714999 715084184
51911715167170525*17153351715478|71530117135*51745669171575317158361715919|| 84
5201|716003|716087|71617017162541716337|7164211716504|71658817166711706754783
521|716838716921717004 7170831717171717254717338 7174211717504717587183
522|71767117177547178371717.920171800317780867181691718253718336 718419
523718502 71858517186687187511718834,718917/7189991719083719165719248
524117193311719414719497171.9579171966371974517198:8|7199111719994720077|| 33
52511720159172024272032517204077204901720573|720655172073817208 2117209031|: 33
526720986721968 721151721-233721316 7:13997314811721563 7216467217281 82
52711721801172189372197572205817.2214017222221722305722387 722469172255282
5291722634722716 722798722881 72296372304572312717232091723291723374 32
529117234567235381123619|7:23701172378417238661723948 7240291741127241941) 82
$3011724276724358724439724327246041724685172476717348497249311725013 82
13711725095725196.125.215817253397254227255037255851725667725748 725829 82
532|17:25912725993 726075 1726156 726238|7163-19|77-26401172648317265641726646
133||72672772880072689017269721727053727134/727216727297 727379727459
5:3411727541172762317.2770417277851727866727948 72802917281701728 191.72827||
835172835417284351748516172859772867817287591728841172892217290031729084
536|72916517292491729327729408429489 72956917296511729737298131729893|| $1
537172997417:300551730136 730217 730298 730378 730459|7305407306211730702 81
5381730782730863730944 731024 731105 131186 731 266 7313471731428 731508 SI
539117315891731669|731749|7318301131911/731991.732073173215317322331732313|| 88
5401173239417324"41732555173263517327151732796173287617329561733037733117 80
5:41||73319717332781733358733438 733518 733598 733679173375917338391733919
5421173399917340791734159 734239 734319734399173447917345591734639 734719 | 80
5431734799 73487917349591735039 735119 735 1997-35279735359 735439735510480
S4411735599173567917357591735838113591817359981736078173615717362371736317|| 80
5451736397173647617365561736635736715173679517368747369547370341737113|| So
546737192737292 737352737431737511737590737669737749737829737908
79
547||73796 738067738146738225.17383057383841738463738543738622738701
5482387881738859738939 739078739097 739177 739256739335739414739493
79
5491739572739651439731 739809|73988917399681740047/74012617402051740284
gt
SI
so
1
79
-79
1
!
The Table of Logarithmes.
1
N| o | 1 | 2 |
213141516171819D
791
79
78
73
55011740363174044217405 21 17405991740678174c757174083674091517409941741073
79
55111741152 741230741309741388 741467 741546741624741703 141782 741360
55211741939742018 1742096 74217517422547423327424111742489 7425681742647
553|7427257428027428827429611743039743118 743190743275 1743353743431
5541174350917435887436671743745174382317439021743979|74405817441361744215
5551174429317443711744449174452817446067446841744762 7448401744919/74-997|| 78
55674507517451.53745231 745309745387 745465745543745621 7456991745777
5571745855745933 7460117460897461671746245746323746401 746479 246556 78
55874663417467121746789 7468687469451747023747101 747179|147256747334
55911747412747489174756717476451747722174780017478 78 17479551748033748 110
78
78
!
17
77
76
5601 748188748 266748343748421194849817485761748653748731/7488081748885
771
56174896374904017491187491957492721749349 749427 74950417495821749659
77
562|1749736 7498141749891749968 750045175012317501991750377 7503541750431
77
5631750508175058617506631750739750817 7868941750971175104817511251751202 77
564l|7512791,751356/75 14331751510/751587|751664|751741/75 1818{7518958751972
77
5651175204817521251752202175227917523561752433175.2509175253617526631752739
77
56617528167528931752969753047 753123 753199175377717533531753429753506
5671753583 7536591753736753813753889753966 7540427941197541951754272
568754348 75442575450175457817546547547301754807 7548837549591255036
56911755112175518917552651755341755417175549417555697556461755722175579911 76
57011755875175595117560271756103756179175625617563327564081756484175636011 26
59111756636756712756788756864756940 757016757992757168 757244 757320
59211757396 757472757$48757627 757699 757775 75.765 1757927758093.758079
573!1758655758230 758306758382758458758533 7586091758685758761758836
574117589121758988075906317591391759214175929017593667594411759517175959211 76
5751(7596681759743|75981917598941759969|760045|76012117601967602721760347|| 75
5761760422 760498 760573/760649 7607231760799760875760949 761025761101
577|1761176 761251,761326 761402 761477 7615521761627 7617021761778761853|| 75
5781761928 76200317620787621531762228 762303 7623787624531762529 76260411 75.
579117626791962754176282911629041762978176305317631281763203176.32791763353|| 75.
76.
761
76
75
74
58011763428176350317635781763653176,37271763892176387717639521764027_764101 75;
581764176764251 764326 764400764475764549.7646247646991764774/764843
75
5827649237649987650727651471765221 765296765370765445 765519765594
5831765669765743 7658181965892 7659667660471766115, 766189 766264 76633824
584117664131766487176656276663617667107667851766859176693337670071967082 74
585117671567672307673041767379176745376752917676011767675176774917678231) 74
58617678981767972 46304676811917681941768268768342 768410 763490176856474
587768638176871217637867688607689341769008 17690821769156769329,769303 | 74
588117693771769451 7695251769599 1696731769746 769820 7.698941769968 770042
58911770115 77018917702631770336770410177048417.70557777063117707051774778|| 74
s gal|7708521770926770999|771073|7911461771219|791293177136717714401771514|| 74
59117715877716.61771734771803177188117719551772028 772102 772175772248
59211772322772395772468177-54377261577-6881772762772835 772908772981
593||773055773128773201 773274773348773421 773494 773567 773640773713
59411773786177385917739337740061774779177415217742251774298177437137744.
595177451717745891774663)774736177480977488217749551775028]7751001775173|| 73
596975246 775319775392 775465775538 775650775683 775.756775829775902
73
597 ||775974 776047 776119776193173626.5 7763387964111776483776556776629
73
5981776701776774 7768467769191776992 777064777137 7772091797282
777354
59911777427777499/77.757217776441797717177778977786217779341778006778079
73
73
73
73
73
72
(12)
1
1
-
t
The Table of Logarithmes.
NI 0 | 1 | 2 | 3 | 4 15 16 17 18 19 D
60011778151!7782241778296778368177844117785131778585177865817787291778802|| 72
60117788741778947 779019 779091 77916317792361779308 7793801779452 7795 24 72
6021779596 77966917797417798131779885 779957780029178010117801731780245 72
60317803177803897804617805331780605780677780749 7808211780893780965 || 72
6041|781037 781109|78118178125317813241781396178146817815391781612178168472
505117817551781827178189978197917820421782114178218617822557823291782401
72
606|782473|782544 7826167826881782759782831 782902 782974 78 3046783117 72
607|783189783260783332178340378 3475 783546783618 783689783761783832 || 71
608| 78390417839751784046 784118 784189 784261 78433278443784475 1784546
609||784617|784699 764759 7848311784902 78497417850451785116785137178525911 71
71
61011785329178540117854721785543178561517856861785757178582817358991785970!| 71
611|78604178611217861837862547863257863967864671786538 786609|786680 71
612117867517868327868937869647870357871067871777872487873191787389
71
6131 787460 787531 7876027876731787744 787815 787888787956788017788093
71
614178816417882391788309|78838178845217885221788593178866317881341788804|| 71
615117888751788946178901678908717891571789228178929917893697894391789510 71
6161789581 7896511789722 7897927898631789933|790004790074|7901447902151| 70
617||790285 7903561790426 790496790567790637790707790778790848790918|| jo
618||790988 791059 791129179.1199 7912691791339 791409 7914801741550791620 70
61911798691179176117918311791901179197117920411792111179238117922521792322|| 70
620||79239217924627925321792602179267217927427928121792832 17929521763022
621||793092793162 79323779330117933711793447 79351179358117936511793721 70
6221793791 793860179393017939991794069|794139 794209 794279 794349794418 10
16231794488 794558794627 794697 794767 7948361794906 794976795045 795115!! jo
62411795185179525417953241795393179546317955321795602795672795741795810
6251|795880179594917960591796088179615817962271796297179636617964361796505 || 69
636|796574,796644796713 7967827968521796921 79699017970597971291797198|| 69
627||7972687973377974067974757975457976147976837977521797821797890 69
628|79795917980291798098 798167 7982361798305798374 798443179851317985321 69
62917986511798719179878979885817989271798996|7990651799134179920317992721 69
691
68
63011799341179940917994781799547179661617996851799754179982317998921799961
6311800029 800098800167 800236 800305 800373 800442800511 8005798006481 69
632800717 800786 800854/800923800992 801061801129801198801266801335. 69
633 801404801472801541|8016091801678 801747, 801 815 80188418019521802021 69
$34||8020898021581302226]802295/8023631802432/8025001802568/802637|802705|| 68
63511802774180284218029101802979|803047803116180318418032521803321803389|| 68
636||$03457803525 803594 803662 803730 803798803867803935804003 804071 68
6378041398042088042768043441304412804480804548/80461618046851804753 68
638 804823 804889804957805025805093 805161 805229805297/805365805433
639||805501805569805637805705805773805841805908.805976806044/806112
68
64011806179180624818063161806384/806451180651918065871806655180672318067901168
641806858806926806994807061 807129807197807264807332807399807467
6421807535807603807670807738807806807873 807941808008808076 8081143 68
643 808211 808279 808 346808414808481808549808616 808684808751808858
64411808836_808953180902:180908818091561809.223.8092901809358180942518094921) 67
6451/80955918096271809694180976218098291809896809964/88003118100981810165 | 67
646|1810233810299 810367810434 810501 810569810636 810703810770810837 67
647||810904810971 811039811106811173 811239811307811374 811441 811508 67
648811975811642811709811776811843 811909811977812044)812111 812178
16491812245812312812379812445 8125121812579812646812713 8127121812847 67
68
67
67
.
I
The Table of Logarithmes
.
I
4
NI 0112 13 14 15 16 17 18 19 D
.
67
650||8129131812980181304781311418131811813347181331418133811813448181351
651813581813648813714813781 813848813914813981 814048814114814181 679
65211814248814314181438131444781451481458148146471814714814780 814847
67.
653118149133149798150468150138151291815246 815312815378815445815511 66
6541181557881564418157118157773158431815909815976816042816109816175 || 66
655|181624148163081816374/81644018165c68165731816639816705|3167711816838|| 66
6561816904816910817036817102817169817235 817301817367 817433817499 66
657817565817631 8176988177648178298178968179628180288180941818159 66
6581818226818292 8183588184241818489 818556818622818688 818754818819 66
659118188851818951181901718190831819149819215/819281819346181941281947€ 66
660118195431819609181967618197411819807181987318199398200041820070182013611 66
661||$20201 820267820333820399820464 820529820595820661820727820792 66
6621820858820924820989 8210551821129]$21186821251 821317821382821448 66
663821514821509821645182170982177518218411821906821972822037822103
65
66411822168 82223318222991822364/32242918224951822560 8226168226911822756 65
66511322822182288718229521823018182308318231481823213182327918233441823409|| 65
6668234748235391823605823669 823735 823800 823865823930823996824061 65
66718241268241911824256824321 824386824451 824510924581/824646824711 65
668824776824841 824906924971825030 825101 8251668252318252961825361165
6691182542682549118255568256218256868257511825815825880 825945826009ll 6's
67011326075182613918 26204182626918263341826399182646482652818265931826658 651
6911826723826787826852 826917 826981827046827111827175 827239 827305 65
6721827369827434327499827563827628 827697827757827822 827886827551
673828015828079828144828209 82827318283381828402828467 828531828595
67418286591828724 82878982885382891882898218290468291118291758292
65
64
+
64
64
641
67511829304182936818294321829497829561182962518296891829754182981818-5882
676 829947830011830075 830139 830204830268 83033-183039683046830525
6773305898306538.30717830781 830845 830909 8309731831037831102831166
6781183122918312948313581831422831486 5315491831614831678831741 83180664)
6791183186918319341831998832062183212615321821832253183231718323811832445|| 64
64
64
641
64
63
68011832509183257318326371832700183296418328 28.183 289218329561833019833083|| 64
6811833147833211 833275.833338 833402 833466833529833593833657833721
682833784 8338488339128339758340398341031834166 834229834294834357
6831834421 8344841834548834613834675834739834802834866 834929834993
6841183505618351191835 183835247183531018353738354371835500/3355641835627\| 63
-
6851183569113357541835817183588118359441336007183607183613413361971836261 63
6361836324 836387 836451836514836577 936641 8367048367671836830836894 63
687836957837019 8370838371468372098372731837336 8373991837462837525
6881837588837652837915837777837841837904837967838039838093 8381501 63
68918382191838282 83834513 334088384718385341838597183866018387231338786|| 63
6901183884913389121838975183903818391019839164183922718392891839352f839415
6918394788395418396041839667839729839792839855 839918839981840043
692840106 840169840232 840294340357840419|840482 8405 45840608840671
63
693840733840796840859840921 840984841049 841109841172841234 841297|| 63
6941|841359841423184148518415.4718416091841672841735184179718418551841922 63
69511841985184204718421091842172542235184229718423591342432842484184254
696842609842672842734 842796842859842921 84298 3843046 843108843170
697843233 843295 343357843419 8434828 435 44 8436068436698437314843293
699843855 8439181843979844042844104844166 84422984429184435384441562
693844477844539844601 844664/3447-26844788 844849844912844978450361 62
63
63
62
62
62
1
(k)
+
The Table of Logarithmes.
f
N 0111213141516171819 D
62
6.2
70011845098184516013452228452841845 3461845408184547084553218455941845656 62
7018457181845779|845842 8459041845956846028846089 846151846213846275
702||463378463998464618465238465851846646 846708846769846832846894 62
10318469553470171847079 8471418472023472648473268473388474491847511
1041847573847634847694347758)84781913478318479438480041348067 348128
62
7051.18481891348251134831284837+1848 +3584849734855913486208486821848743 62
706848805848866 84892884898984905 18491121849174849235849297 849358
7078494193494312495421849604 849665849726 8497881849849849911 849972 61
70813500338520911850156185021.7850279185033918504011850462 8505 24350585
7091 350646135070718507698508291850891 8509521851014851075851136851197 61
61
61
71011851258185131918513811851442/8515031851564135162518516861851747851809|| 61
7118518698519311851992 852053852114 852175852236 8522971852358 852419 61
712185247918525411852602852663852724 85278518528468529071852968853029 61
713185308918531509532118532728533338533941853455/8535168535771853637 61
71411853698853758135381918538818539418540021854063185412418541851854245 61
71511854306185436718544281854488185454918546091854670185473118547921884852 61
7161854913854974 355034355095 895156855216 855277855337 855398855459 61
71718555198555798556408557018557618558228558828559438560031856064 61
7188561241856185 856245|856306856366/85642718564871856548856608856668
60
7191856729,856789185684983691055697018570311857091185715218572121857272 60
720118873321857393155745318375131857574)8576341857694185775518578151857875
7218579358579958580568581168581761858236858297 858357858437858477
72284853785859718586578587188587781858838858898858958859018859078
723859138859198 859258859318859379859439 859499859559859619859679
724118597391059799185985918599181859978136003818609988601581860218860278
60
60l
60
60
60
60
60
72511860338186039818604581860518860576186063718606971860757/360817186c877
7268609371360996 861056861116 8611761861236 86129586135586145 861475
7271|861534 861594 861654/861714 861773861833186189336595286201 21862072
7281862131186-1918622511862310 862369862429|36248918625491862608 862663
7291,8627288627878628471862906862966863025136308518631441363204863263
60
60
60
.
59
730118633231863382863442863501186356118636208636798637391863799186385811 59
731||863917 863977 864036 864096 864155 864254 864274864333 864392,864452
59
732 864511 864570864629864689864748 864808 864867 864926864985 8650451 59
7331865104/865163 865222865282 8653418654001865459865519865578 865637 | 59
7341/86569618657551865814/865874/86593386599-866051 866110|866169866228
735118662871866346186640586646518665241366583866642186670118667598668191| 59
736866878 8669371866996 867055867114 867173 867232 867291867349867499 !
737867467 867526867585 867644 867703 867762867821 8678798679391367998 59
738 8680568681158681741868233 8682921868350 868409 868468868527 8685861 59
739||868643/8687031868762868821868879868938186899718690568691148691731 59
اور
58
58
58
740|186923218692907869349186940'518694661869525186958418696421869701186975911 59
741 8698181869877 869935869994870053870117 8701698702288702871870345 59
742870404870462870521 870579 3706381370696870755870813870872 870930
7431870989871047 871106871164871223871281 871339 871398 871450871519
74411871573871631 87168987174887180687186518719231871981/8720391872093
745872156872215187227318723311872389187244818725068725641877622372631
746872739872797872855 8729138129728730298730888731468732041873262|| 58
147|1873325 873379873437873495 873553873611873669873727 8737851873844 | 58
7481|373902 873959874018 874076 874134874192874249|374308874366874424) 58
14987448:187453987459887465687471487477487482987488818749451875003|| 58
?
58
1
1
The Table of Logarithmes.
NO
58
58
58
17591188024-8832991880356188041317080527188058518806421880699188075611 57
78139265118927071892762189281818928731892929189298518830401893096 893151
I 2
750118750611875119/87517718752351871293187535118754091875466875524187558211 58
as 137563918756983757568758134875871 8759291975987876045876102876160
75211876278 8762768763331876391 876449 376507 376564876622876679876737
753 876795 8768531876910876468 8770261377083877141 877199 8772561377314
75411877371877429187748718775 44/877602 37765987771713777741877332/67788911 58
75511877947878004,87836218781191878177187823418782921878349878407187846411 57
7568785:2878579 878637 878694 8787528788081878866 378924 878981879039 57
757879096 879153879211 879268879325879382879459879497 8795551879612||
57
758879669879726 879784879841 8798988799553800131880070880127880185
57
7601188081418308711380928188098518810428810991881156188121318812711881328 57
761881385881442 8814991888556881613881669|88172788178418818471881898 57
7621881955882012 882069882126 382183 88:2239882297882354882411 382468 57
763|8825258325318826381882695882752882809 882866882923882979883037 57
764118830931883050833207|8832641883321|88337783343418834911083548 88360sli 57
765118836611883718188377518839321383888 1883945188400218840591884115188417211 57
7601884:291384235 884342884399884455 884512884569884625 884682 384739
767884795 884852884909884965 $850221885078 885135 885192 885248885305 57
7688853611885418885474885531 885587 885644 885700385757 885813 885 869
57
76911835926885983886039|896096886152|8862098362658863211 886378/886434
36
27011886491188654713866041386659180671618867731886329188688588694118869981 56
]
771887054 887111887167887223887279887336 887392 8874491887505 887561
56
772887617 887674 887720 887786 3878428898981887955380011 8880671388123
773 888179 888236 888292888348 388404 888460 888516888573 888629888685
56
774|8887411888797_888853 883909 588965889021 8890778891348891898892461 36
775118893021889358188941413894691889526188958:188963818896941889749188980611 56
7768898621889918899974390029 890086 8901411890197 890253890309890365
777890421 890477 899533890599890645390909 890756 890812390568890924
7781890979891035891091 891147 891203 891259 891314 391370891426891482
7791189153713915931891649189170518917601891916 8918728919281891983189203911 56
780118920951892150189220618922621892317189237318934291892484189:5391892595 56
56
78289320793262893318 393373893429893484 893939893595.893651189370656
783393762 893817893873893925893984894039 894094894149 894205894261 SS
7841/8943161894371 8944271894432894530894593 3946431894704 894759894814 | 55
56
56
56
56
78511894869139492518949801895036139509118951461895 2011895257189531218953671) 35
786895423895478895533895588/895644 895699)89575489580989586418959191 55
7871895975896029896085 896140896195 8962518963068963611896416|896471|| 55
1881896526 3965 &1 8966368966929967471896802 896857896912,396967897022 55
7891397077897132 397187 8972428972978973528974073974628975171897572 55
790118976271897682/8977378977928978471897902189795718980121893067898122 SS
1918 981761898231 898 286 898341 8983961898451898506898561898615892670|| 55
792898725895789898835898339398944 8959998990548991098991641899218
551
79318992731899328189938318994378994921899547 899602 8996568997111899766
55
1941|8998 21 89987518999291899985900039300094900149900203|90035819003121 55
7951190036719304229094761900531/9005869006409006951900749|900804190085911 55
79619009131900968 9010221901077 991131195119690124090129519013491901404
797 1901458901513901567 901622901676 931731 9017851901839 9018941901945 54
798902903902057 902112 902166902221 902275902329902384902438 902492 54
799||90:547 902601 9026551902709.902764 902818190-873|9029271902981199303611 54
SS
+
1
(k 2)
1
1
1
1
The Table of Logarithmes.
.
No | | 1 | 2 |
L|و| 8 | 7 | 6 | 5 | 4 | 3
اور 781
535و4
9035 |3190330719033619034161903469
903144903199190355 و 308
30
300
26
19640669041ء 69.401
301903633903637903741903795903849190320495395
30:1 904174| 9042990428 33904337904391 19044459044999045531904607904661
202
05و 144
955
904932904286902539905094[49e4878
48 و 9476 [4716ہوا303
اور ال905742 | 68
05وه 63
905 |26905580
5
9054721905|418
4905
36
35و90531c1[256
30411955
24
54
C
24
54 ال906281|9061731906247 |8906012906c66956119
90595[904
55وا05579690,849 || 05
و90644319c649790655190665490665819c671290676690681 و638: و 335
306
306
9730419c7355و9071439071969672500 و 768ووا 1907.035
1750798 و1906874906
807
کو 7وو37841و07787 و 076859677341و97626و07573و او: so74659075 : 19074
308
اور ا|431وو [o8378
4 : 083 و 9217908275ووا163
908 اوہ 19080569081ء 309907949190800
54
4 ؟
34
08967 و 14 و 698
968 [887
58753908واووo86و 46
6
08واد و 5
19085391908
485
8101908
909396909449909503 |0909342و18 و19091819093590
8ء ، وهو [
1909074-
9095 | 3II
اور
841910037وو 30/9
8239095779099 و90 و90960990966390971690976
909556 | 312
16571و18
105
1035910411910464 و 9715251910304
9101
44
9101 | 910c91 || 313
زر ال911051911104|دووه91 [44 و10و11و108 و 84
91083
41
106.41910678910731,1078و314
ا
53
53
1
23
23 (911584911637[911535[477
11و911424[371
9I1|81,911138911211112631911317
و120631912116191216و1و100
911946191[91184991903|5911743914797و6
8161911
مر 9125949126471917|47
9124359124889145[381 و 7519113
:
31:ة 91 ==-1و 817
913178913231|2966913019913572913125
397
85991291ء اور 912753912806[818
137081913761و655; 1و136o2و45
135و133371913389913443913494و1384وااو31
53
53
53
53
olو42
91[14237و4184
191ء3
41
21 و492691407
913973191و91391[13861و32olly1384
53 و1481و [14766و914665914713
4608
39145559-450
3211191434394396191444991
347
4212 ور4191- راواو91218
361
1ر 91[912039915583[14977واء:149و1487zو11ءة 8
475ر1وا:82 راو 205191555891561191664915716913769
15و3
39991545
8391
14391629691634919164c1
191916
91916;9161[59799160331916585
1591791و8
:
41
53
53
23
0I
8591 645 491 507]91655991661 2191 666491.6717\91676991 62 22916875 916927
7453
91734891740091) 17295
0917
:
43و170339176859171389171و8و16و8a61
191797
17925و91761119176639117161917768917820
/
91773[8 و 175و8
:
26175061
و18و449
919|19345918397
18.939 و 828918030191858391813591818891824
026و1ر ;1و18و11و18واو S86 ،9188169[8764 ، 19187129ء 918555121860791865] او:8
ܐܐ
53
53
2
در او4
9444994969195 :19ء 1939و9339
91 [918319192351919287
919130191|78رو[830
2007
9679200199و491
91و91 = 975891381091986
97o691
91 | 3
1965و9601
91 | 831
عر
3
9:05 | 41
205 و 20384920436920489 و 20176902289207992033
01:39 وراء83
II14 : 9210629 و20906920959100 و 3
2069792074919
:
08o192o85 و 33311910645
52 (21634 و309215821
9
:
15|478
21 و 1426 و 1892117092132291374
83411911661
22
52
4
8
۲:
: ال154: وا1205592210aو8و491194619
:
19 و18429
:
18 و 921790 [19
:
1686921738
835
2674 و 22
226و2466922318922569
2و41
141
192ء36:وه31
932|22069258و836
: و31و23089923140و[23037و1942985
1933 و 881=دواو:227779228و[1275 وال837
9
:
34519350392355592360792365923710
43296923348923399 و 244
23 و838
51 |28
249241769
:
42
9
:
45
:
19
:
40729241 |9
:
3969 [49238651973917
381=واء1376 و 8391
ك
52
1
دک
44
1947
4693 و41
46
9
:
443419448619245389245899[1433119
:
4383وا84011924279
: ؟
26
25و209
45و157
25 و 106
425
92500392505 |1
489992495 - 48489 و 8411914796
دک
25776و25725 و 55699256
:
125673 و18
9:55 [167 و 415
22 و 53641:واء192531ء4 ک
او 26و39
262 و 26188و26034926085916137 و 321و 25و25931 واد32 و 025328||943
در 268
49
649782654891659992665191670292675 و 84all9a63429639426445
ای او 1927114927165927216927264191731ء92701192706 و 9168571969o8191695|45
اراء 783:و | 27781وو2767494772و27576917627 و 24
75وا27473واء 742و846927370
28345 و1928242918293
9191دواو3
281 و23088و80371و7986:وار7883993-8479
28857و880 و4
2875 و8703:واء 2865 و29601 واوور28و8و28 و 8396928447 و88
360
29 و 1929166929317
16392926و 11:52
29و561وووووووور و918
90
5. و{849'
I
4
.
I
دک
I
\
F
1
1
--
1
The Table of Logarithmes
.
NII
0
I
2 13 14 15 16 17 18 19 D
SI
8501192941919-9470:9-95211929572192962314296741929725192977619298271929879 SI
$511929429 929.940 930032230083930134930185 9302369302879303381930389 SI
452930439 930491 930542930592 930643 930694930745 930796930847930898 SI
8531 9309499349991931051931102931153931204931254931305931356931407
SI
354!93145819315091931559 9316101931661|93171293176319318141931865 931915 SI
355119319661932017/9320681932118932169193222019322711932322932372/93242311 51
8561932474 932524932575 93262619326771932727932778932829932879932930
857 1932981 93303119330829331339331831933234 9332851933335 933386933437 58
8581933487 933538 9335893336399336899337401933791 93384) 933892 933943
SI
85911933993193404493499419341451734195193424693429693434719343971934448
56093449819345491234599193464993470019347511934S01 93485219349021934953|| so
861 9350031935056935104 935154193520519352551935 306 9353561935426 935457 so!
8629355079355581935608935658 535709935759935809 935859 935910 935960 So
863 936011036061936111 S36162936212|936262 936313936363936413 936463 So
3641193651419365649366141936665|936715|9367651936815 9368651936916336966!
365119370161937066 937117193716719372171937267937317193736719374181937468 || 50
8669375189375689376181937668 937718 937769937819 93786919379191937969 $0
867|938019 938069 238119|938169938219938269938319 938369938419 938469 So
868193851919385691938619 938669938719938769938819 938869938919938964
8691193901919390699391191939169|939215|939269|939319|939369939419|939469|| so
8709395191939569 939619193966919397191939769939819193986993991893996811 50
871940018 940068940118940168940218 940267|9403171940367940417940467 501
1872940516940566940616940666940716 940765940315940865940915 9409641) sos
873941014941064 941154941163941213941263941313941362 941412 941462 So
8741|941501 9415611941611941660194171019417591941809194185919419091941958 so
SO
87594200819420581942107/9421571942207194225694230612423551942405942455
376119425041242554194260319426531942702 942752942801 942851 942901 942950
577 | 9429999430499430991943148 943198 943247 943297 943346 943396943445
8781194349519435449435941943643 943692943742 943791 9438411943890943939
879||943989 94403819440981944137/94418694423612442851944335 1944384944433
So
so
49
49
49
830944483194453294458119446311944680194472919447799448281944877944927|| 49
1881944976 9450251945074945124 945:73,9452221945272 9453:1945370945419 49
1882 945466 9455189455671945616945665 945715945764945813 945862 945912 49
883945961 9460099460591946108946157 946207946256 946305 94635 4946403
49
8341194645294650119465511946599946649194669819467471946796946845946894
49
885194694319469921947041194709019471399471891947238164728719473361947385|| 49
386947434 9474839475324947581 947629 947679 947728 947777947826 947875 42
887947924947973 943022 948070 948119 948168 948217 948266 948315 948 364 49
898948413948462948511 9485599486099486571948706 94875594880494885311 49
8891 948902948951 948999 949048194999719491469491951949244949292 949341|| 49
8.90119493901949439194948894953619495851949633949683949731194978019498291|
49
8911949878 949926949975 950024 950073 9541 21 9501709502191950267 950316
49
892950365 950484950462 950511950558950608950657 950706 950754950&oz
49
893||950851 950900950949950997 951046951095951143951192951240 951289
491
8941/9513331951386 95143519514831951532195158019516391 9516771951729951775 || 49
8951195182319518721951920/951969195 2017|952066952114|952163952211195225911.49
8969523089523569524059524531952502952550195259995264795269695274
897952792952841 992889 952938952986953034 953083 953137953179 953228
898853276953325953373 953421195346995351&f953566953615 953663953711
189911953759195380819538569539051953953954001195404919540991954146 954194
48
48
48
(1)
The Table of Logarithmes.
IN || 2
| 2 | 3456
D و78
48
4677
1995
46
99
458
448495453295
95[4339954387954435
95[491
43195
42 وو !ooو
4955562955110955558
49669550I
95|4918
95و486 و 4773954821
47
:
595 زو 21و
447955495955543195559295563
19553991955
35
107955559553031955 و 95 | 95
60249560729561.0
95597695
9
955 و395566895573695578415583295588 ہو
6601
956505956553195[457
956 و40
9563611956[956265956313 [256168956116
41 دو
48
20596664995 66979567459567939468 409,668895693695.698 495703957080
48
و195755
49575
46
7
95736895741695 و 131
957224957272195|56957128957176«
307195760795765593770395775 1957799957547957894 95 94-95 7990953038
8
48
4
4
16
4689585
58373958421958 و 81775832 رو او22
5o8695813495818i958ور ہو
48
41 و89
9895894695
588واه 882
25 (9587557955853 [59659958707( 959564258612 |91 در
48
! 4
:
595947
59328959375959 ورو27و5واء 23 و 13795918595 و 95 و 98 و 95
+964
1095 و
959947اوو38
299
96149596619597699597579598042598
95
66
95
1895
95
9II95
6032896537696043 و 6o280وا233
60
9651859
009960138
6
19-96004
95و9ووداو
48 اوو 12608;608و 697569608o4و وہ07او960613960661
66
605 و 18
60471960sو | 13و
47
61346961374و61279و 961184961231 [961136
61089و 641
961 اووو960 [46 وهو !!4
91
48
2
231
1801261848 اول 658196170696175
11961ه 1961
63
196151619645 و46
61و 41
61واو
62312و127
6و2179962227
96113296
961990962039962085
1943
196
61895و16و
وو627و62653962701962748و1606
96 و 255
231196) 464
136596141762
96|97
63174963221963268و26
631واو 307
6 و 63032و5و6و37و962 و8
1842962
1896 و
963693963741 [963646 وور1963ء 2193315963363196341c96345796325496355
47
47
47
47
47
47 (1
:
1 :64118964165964و4971
6397796401496واو2و63و9637581963835963882
0
9
47 ا4
468
26463796 و6458واء4
45
6 و 4449964495
440196
496
435
964307196 و 219642 و
65155 و 198
65061965 و 3
64969650i وو491
487296
2496
48
196477496
473
96اء و
624, 96|4965531965578
48
5
6 و 437
565 و38
9653431965 [296 و 96 و4
2
65واء: 965
3
2
47 ، وهنا 96|66048 و 66001
54 و 65 و 906
5
6 و او 65766965813196585,او96571 [672 و 96 ا9
:
4
47}
47
1
47 {64
65
6651796 و470
96637696642366 [966329 [
83-9661899662391966 (966142 | 95
6967533
26633996698 و459668
668 و او6679 و 66589667059667521 وو 66611واو
4967501
96745 | 408
1967
716796731496736) و دد 9671739674|27
671 او 67
:
57و27 و
47 و96787596792296796 و96759619676429676885677359677826782 | 49
9289675
47
68436 و 9683439683821 أهو: 1968 و14
68 و 02
968
56
69 واو10
1968ء 56
966
68516 ورود و
47
47
.
939|1968 48 319685299635769636:3968669968716968763|9688o9968856968902
I
و36 و394
693 و69276واود: 31969
18و6 و 136و96 و 58و 6و349689499689969695431و
35
97419697899698
96
95
96
96 و 696 و396952996955696965
45و5و6
6941واد93
47
4970300
97025|970207
19750689701141970161
289699759700
9و96 [882 و6و33 و
97o67297071976766 (33970579970626
9704969705و97039397043
47
349703 و
47
47
47
)
46
935
970812
46 {{و97118397122[971095971137
971044|1970828970904970951973997ء
9716010971647971693\4
5
2
971503971
461
2971
4
971و2769713229713
971
36 و
411097
:
157
497
206
97|1972018
97197
92
971
71739971786971832971879و37و
که او 97.57397261 [27
49724819745
43
971263972249972295972342971383972 [38 و
46 {اء 308
97|973035 و 198
4397 و1974897972
9728497285|9716669717129727.58
46
46
1
وو
940973118|97317497322097 3266]97333973359973405197345197397|973543 44
941
942|974050974097|974 143974189974235|974281 974327|974374) 744191974466
974005و1395و131و66973
9731749738209738|1973728ء 368
37363697 او358
97
4
46
46
46
46
49
:
6
4981997
3497
97469s19747429747989748 و7460497464و943974510974558
386
294975339915
481975 * 1975ء 20
9751561975و75oi897506497510واء9441197497
1975616975662975707975763975799975845 و56
47895541975
9754321975 [451 و
76304و1976
:
58-76167976و 9469758919759379759839760299760759761
5799766259766719767171976763 تا 7633397و97634997639676449764381 [47 و
91720[77176واو:770379776839771واء وو76546976ووو768واه 97685|768o8و8و
يه (77495775419775897763977678و1و977266977311977358977455197744 اووو
46
کو
48
!
مد
The Table of Logarithmes.
|
No|1| 2 |
D| و | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
.
46
I
95 39790939791 38|979184979 :
2979:75979329793669794 2979457979503
46
46
46
46
45
45
978135 و 97804297868|2977998
97790697795
977769977815977861|5o9777
:
4
46973591
550,785
4978
45
978و36397840
978|978
:
72978317 کد 978181978 | 951
647و7واء6و7و956
978911978 | 978865و97877497881|1978637978633978728ء کو
958و1و12وو97|67
119738: 979739797769798 و 3997968
94,796
489795
49795
95
198o185 980231980-769832298636798o41 و198o094198o13 و3005398004 و 955
.8087 و 80s 9498o639980685986735980776980821 و 49
8045398oso39855و56
54798129
:
19816379316839817
:
5981773
1456981501931
589313669814198و
و 8 و 2181
235198
28 و209
204593
193
1992
493
195
3190998و 4
86: 3 [31519و 59
26331981678
258898
43198
24979825
298
98245 [407 - 136298
8و16
3: 3227193و 965
وع 830859831و3039
98
94و2
8و2949
29
294
293
285
1498
28
9و و3.76 و 3-7-3و961
3591
38
36
35
98 |33356993401983446983490و23322ca63265983310 (983175
62و
403
Sو (983987 = 394
98
33897واء 380798385
3376298و (33616933671983716و63و
د 443798448
8 - 439
4392198934798
98[4257
93 | 4:12
8416798وا:12
984077193 [64 و
ک4
1
I
45
45
r
1
I
|
1
45
45
45
45
45
965
\
84527198457298461798 466 = 984757984752198 4797 | 98 48 42984887\98 4332 || 45
1
45
45
4
4 : 38
85337985و:29
85و247
2986
20
85157985واء11
85و85067و350
:
2و4977
8و66و
38,830
386
5696985741985
56198560698565198
85و6
51
471985 و 98
416
935\967
و986234286.7 [6189
198609998614498
6009986055
8596598و 10 و875985, 8و968
و ال1986727ء 98668 [936293986637 ]48
3198645899650419865
41
6
93636998 [986324و6و
1987175 و198699698704098708598712
8690698695و1986772936817996861[دو
x2
875779876واء 8753و488
9874431987 | 398
87وا3
35
87 و 4997309 :987 و 9721و 971
So68 و 87979993624و937934 و8788و57666987711997756987800997845 و 72و
45
884259884698814و8839 و 398811398815798820298824798829983336
97
که ال8896و16و88و98871و88826واء98878 |98737وا:98869
48
886 و 88559998604 و 74
45
45
45
Q0
45
44
44
44
405 و8 و316989361و4 و 172و18398922798و8و138 و 89094193وا989049 | 35 و 196
975
856و8و8c6 و8 و761و8و1989717ء967 و 628و8 و 989584
39
495و89494 و 449 و
1978
90
:
94و9025و2206وو95117995161واء207وو002 و 83و59 و 89939 و989995
977
38
307و4و9906 و9064و90561193c6o5و90472990516و9789953399903430990428
44 ||1182
99 [37 وو3و16وو 1049وو [1004وو60 و90و16وووو [937831,90327290871 و 9791
44 ال 991625 |1580 وو1536وواء و14 وول 449
91و403 اوول 1359ووا 131 ووا1270وو 126وهو
2023992667وووووو ا35ووو هووو | 175399182991846 و1713وواو 166وو:28
465992509-وو41 وو [377 ووا:33 وو 92288و 44:وواوو دوور دوو :II:وداء و
99295 [957:ووا 863: وو او 281 و429926869917309927749
26وووووو4
455
399
93
933392{3304993348 وواو 99325\99995993539199308399312799317993216\984
44
44
44
44
I
93833و937891 و 343699348o993524993568993613993657993701993745و519و
994273او:4 وور418وو 869938779939219939659940c9994053994097994141و
94669994713و4625 وو94581و37
944499944939945و495
994 [94361د(394317 || 7» و
44
21
15
109995
95و5061وو394933994977995521 و99484599488
42oi
699 ;947
98
95547995591 و 504
995 أو 45
95و416
2995
37
995 ،99532
241 کو 95:40
96
95وووو
44
44
44
44
44
44
44
ه 4 ||996019
986
95و [942
1995
98
958و41
995811199585 [672995723995764
1995
35
56وورادوو
9638o9964-4996468و96337و3و62ووو4: 6 زو05-996 | 6117996161 وو 24
6
9999
967749968r8
]
99686 2969e6و(6731وو(99996643996687
65551965ووا:1
2965 (922
4997168997112997455597299997343: 93997037997o8o9971و9231799949996
;199773699777ء 17997569976599764899769
49975
47
97و994
/
19913861597430
44 اإ8216 وواء9817و129
98و9841998o95و8و979و41
9795و979ro1و1997823997867
95و
2
265و99852998564998608
477
998 |98390999434و969992299983531,98347و
ووووو ا999043 و99
998
399895
S91 وو او852699886ووا 98783 و 98739و99695 و 97و
: وووواو47ووو61999305999348999392999435 ووو 149992181ووو ا133وووووو
57وووووووووو2699986
0997839998
39
97ووووووووء 965وواو960وو [65 وووواووو
I
44
44
44
43
(12)
+
1
-
一
​1
4
*
{
-
十
​{
}
A TABLE OF PROPORTIONAL PARTS,
Whereby the Intermediate Logarithmesof all Numbers,
and the Numbers of all Logarithmes, from 10000 to
100000, may more readily be found out by the fore-
going Table of Logarithmes.
D111213141516171819 D111213141516171819
22
22
46
20
21
113
II 22
21
22
22
SII
171 22
18 24 30 36 42 43
431 4| 8| 121 171 21 251 301 341 38
44 41 8 13 17 26
30 351 39
451 41 91 131 18 271 31 36 40
4 91 13 18 23 27 321 36 41
471 41 9 14 18 23) 281-321 37 42
481 41 91 141 19 241 281 331 381 43
49 41 9 141 19 241 29 341 39 44
SOSIO IS 20 25 30 35 40 45
Sil: 510 IS
20
251 30 35 40 45
521 5/10 is 26 311 36 41| 46
53! 5 10 15 261 311 371 42 | 47
54 S10 16 27 321 37 43 48
551 SII 16 27 33 33 44 49
565
II 16 281 331 391 44
so
570 5
281 341 391 451 si
581.5|11|17| 23 29 341 40 461 52
59 siis 17 231 29 351 41 47 53
601 612
54
61) 6/12| 18 24 30l 36] 42| 481 54
621 6112 18 24 311 37) 431 49 55
631 61131 181 251 311 371 44 501 56
64 612 19 25 32 38 44 511 57
6.5 1673 191.26 32 391 451 52 58
661, 6113 19 26 33 391 46 521 59
67 613) 20 26 331 401 46 53 60
68.4 61131.20 27 341 401 471 54 61
6916131 20 271 341 41 481 55 62
70 7 14 21 281 351 421 491 56 63
711 714 21 281 351 421 491 56 63
72 714 21 281 36 431 50 57 64
731 71141 21° 29 361 431 SIT 581 65
741 714 221 29 37 44 511 591 66
751 7115 22.30 37 451 521 60 67
761 715 30 381 45 53 601 68
771 71151 23 30 38 46 53 611 69
781 71151 231 311 391 461 541,621 70
79 7 151 231 311 391 471 55 631 71
: 801 8 16 24 321, 401 481 56 64 72
81 816241 321 40 48 56 64 72
82 81161 24 32 41| 49 571 651 73
831.8116 241 331 411 491 58 661 74
841 816 251 33 42 50 58 67/ 75
858171 251 341 42 si 591 68 76
86 81171 251 34) 431 si
34 43 51 60 681 77
87 81171 26 341 43 52 601 6978
88] 8117 26 351 441 521 611 70 79
891 8117 26 35 441 53 62 711 80
90.9 18 27 36 45 54 63 72 81
91 918 27 36.45 54 63 72 81
92 118271 361 461 55 641 731 82
93 918 271 37 46 551 651 741 83
941,9|18 281 371. 471 56 65 75 84
95 | 919 28 39: 47 57 661 76 85
96 9 19 281 381: 48.57 67 761&6
97! 9119! 291 381 481 581.67 771 87
981 91191 291 391 491 581 681 781 88
99 919 29 394959697989
2010 20 301 40150 60 70 80 90
101 10 20 30 40 50 60 70 80 90
102/10/20
101201 30 401 51 611 71 811 91
103/10/201 301 41 SI 6172182192
104 1020
31 41 52] 62 721 831 93
IOS 1021
31 4252631 731 841 94
106 1021 31 421 53 631 74) 84 95
107 101211 32 421 531 641 741 851 96
1081101211 321 431 541 641 751 861 97
109 10 21
32 431 541 651 76 87 98
1101122 33 44 551 66 77 88 99
111 11221 33.44 551 66 77 88 99
112111 22 331 441 56 671 78 89100
331 451 571 671 78; 901101
114 11 22
34 45 57 68 791 91102
115 11 23 34 46 57 69 80 92 103
116 11231 34 46 581 691 81 92 104
1171111231 351 461 581 701 81) 93105
1181111231 351 471 591 701 821 941106
119|11 231 351 47 591 71 831.95 107
120 12 24 36 481 60 72 84 96 108
12112124 36 48 601 72 841 96 108
Iz2121241 36 48 611 731 851 97 109
1231121241 361 481611 731 861 981110
124 12 24 37| 491 621 741 86 991111
125 12:51 371 50 621 75 871100 112
126 12 25 37 50163) 75 88 100 113
127 12 251 381 501 631 76 881011114
I 281121251 381 51 641 761 8911021115
129 13 25 38 51 641 77 90 103 116
130 13 26 39 52 65 781 91 104 117
13113 26 39 5265 78 91 104117
132/13/26 391 52 66 791 92 1081118
133113126 391 531 661 79 93106119
13413 26 40 53 67 801 931107120
135 13 271 40 541 67 814 941108121
136 13 271 40 54 68 811 95 108 122
137113 271 411 54 68 821 951109|123
138131271 411 55 691 821 96110|124
1391327 411 55 691 83) 971111125
140 14 28 421 56 70 841 98 112126
141 1428 42 56 70 84 98 112 126
142 141281 42 56 711 851 9911131127
143/141281 421 571 711 85110011141128
144 14 28 431 57 72 86 100 115129
145 14 28 43 58 721 87 1011161130
146141291 43 5873 87 102116131
147 141291 441 58 731 881102|1171132
148141291 441 591 741 881031181133
149 14 291 44.591 74 89 104 119134
150115 30 45 60 75 901105120-135
ISI115301 45 60 75 gopros 1201135
15211530 45 601 761 9111061211136
1531513o 45! 601 761 9110711221137
IS 411513048 61 77
15515311 46 62 77 93108124139
156 15 311 481 62 78 93-109 124140
157115!311 47! 62! 78 941109|1251141
1581151318 47 63 791 9411101126 142
15911511 471 631 79 95T1127143
160116 32 481 64 8096112128144
161 1632 48 64 80 96112128 144
162116
97111311291145
22
11
92/107|123|138
6132
48
641 81
(m)
1
F
|
|
The Table, of Proportional
Parts.
|12 3
\8\7\6\راوا3
Dli al
و
6171819 آر او ادا2{1}D
321346 | 69| 9116/1391621 185 1:08
|
او
163 16132, 48 65 52 98 | 114 | 1351146
16:16 32 49 668 : 98114} : 31:47
165 1633 49 66 82 99112132148
633
167\113) 20 6 9312016\13 3:30
و20 (186|163
116139|93 |69
23312
:
46
10: {3117
/
140
/
163187و 10
2341
:
3146
51II
48
6
1171411اوو 1 |235123147
148
116132149|99 |83 |66 او4
1633
166
3350
Iooji171134
:
51 |84 |67 |20 أ168116133
در: 101118135|84 |67 اور 1633 و16
3
1362 أو ioi
|
r1|85 |68
136153 و 10:11 ,8
681
51
34[17ilia
3623147| 70| 94| 118|14116518] 212
23723147| 71| 94|1181421651189|213
238 2347
18135 |
17017134 21
)
]
172 1734) 51| 65| 86| 103 10/137154
131734 51 691 86|1031211;855
14:اوود 166
142و9611
11
1382347
15:بودا167
143 [95/119
7
392347
216 = و:{165
144
20
961 |72 |2448
240
ا
156|39
871041
:
11 وه 52 |1734
174
41 | 44| 48| 72| 96| 120| 144 :68} 192.16
4:{24} 49 71 96111145169|193) 17
243 24481 72 97121146110: 94-18
88105123140128
70 در 35 |17
176
و1-95 1701 | 146 |9742
73
24491
244
10 : 6و12
:
1471711 |98 |73 |49 | 4524
اد : 195
172 [47
1|123 |98 |73 و4
24
246
مدد (148/1721197
143 [s8
74 أو44 ;247
23 تا 98
1241481734 و 14
2449
248
177 1783 53 70 88| 106| 123| 141159
89|
53
|
4160
411 |106اوه {71 |53
178/1735
164 | 143 | 890715 ا ا ار173 |179
144162
126
08: 951
72 ا4
5
1836 و18
249|24 | 49| 74| 99| 124 | 149 | 174 | 1994
18118136| 54 74 95| 108 | 146 | 144162
36
18 | 36| 54| 73| 91 | 109| 128|14 |164
184 1836) 55 73| 92 | 110|128147165
1163
17:45 |109 |91 |72 |34 |18136ء18
|183
کد اه20 (475
50
35
/
1oo1251 او 255
250
کد اه: 175
130
129 |7500 اور 1:51
25
251 2550
| ]
25 | 25 || 75 | 105| 126| 151476/201226
25 30 25 50 15151 | 126 12:177) 2017
166( 48
1 و1112+91
74
54
1851837
14867|130 |111 |93
74
55 أ1861837
I
|
187337| 56| 74| 93|112 | 130| 49| 168
177
:
53
:
28ء 16
/
1o111715 اه؟ اره ا4
45
و2 : 10
:
1
:
7153178
:
04 | 76 ادوا;1ء 25
41
:
36د: 26to228153179
56:51
إ
7610 |
2572551 | 77|10|128154179|206 231
]
5 6 75
7594113132|15170
و2113112016
94/11 [15
56
1837
188
56 (1837و18
133152171 | 114 |95 | 76 |57 |38ود وود
4133152171
II
95
76
57
138و11و1
32=06ة 4180
15و11|77163 (51( 15/ 258
297.33 |155181او771031
134 | 8و18
:
12
136
135 |104 |78 |1651652
2591551
|
261 261,2| 78| 104|130156182208234
| |
192 1938| 57| 70| 96 |115 11 34,153172
1931 1938 : 57| 77| 96|115 11351154173
194|19| 38| 58| 77| 97| 116 | 134 135 174
175
: أو20 (1041131156183[ 78 آء رأ26
:
16
236
131157184210
78،05 ء۶{26
263
264106| 52| 79| 105 | 132|1581184-11) 37
38=ع 1: 13
:
159185|106 |79 |53 |26 ,26
و1223*159186|33 |106 |79 |53
6: 266
13
:
40ء 8oli06133160,186
3
5\6\267
8
156175 (17:36 |97 |78 |58 اور اورارو !
6176
15 (117
:
36[ 98 [78 ور او3 او 1961
137157177 |118 |99 |78
59 اور و1971
138178 | 138 |118 |99 |79 اور 139و1981
و11و12|39
991191
79 اور 1939|199
80 ، 160 اه80
/100
/
12014 و60 امها مع 200
8oltool1201140
]
160 i8o
601 (40و2011
11
/
141161181/8o
/
1oi ]60 ام4 اء اء20
60 امه (20910
8
268 2653| 85| 107|1341160] 1871 4 241
1:15:42
188 | 61:(8or07134
1653 او26
43: اه 1: ار18 |135162 |108 |81
2701754
275 27 5 4 31/1o135162/18921 6:43
27: :254 8110811361631 90 217 248
8:11o11111142162|18:
20410140| 61| 8102122142163183
255 (1041| 61| 82| 102|1 23 143 | 164| 184
| 41
07:0141 62 82،03/14144165186
124
36163491218 داوود | 81
1734754
1641912946 |37 |109 |82 او 275
274
247 مه 82111013716592
2755
275
185
4
6
441
31ة ro31: 61 تهامة (206
و
176 17551 8211o13816519322043
\\
وداع: 193 [138166و11|83
2755
277
اور222: 166194|11
:
39: 83 او 275
275
279-755 83111139 167195 | 223 224
187 | 166 | 45
2411
8311041 |62 |41 اه: {208
167188
83/16412146 :6 :واهد او20
و18[ 147،68 |126 |105 |84
63
.10
:
14
و14716918 |126 | 105 |84 |63 |42
21
211
190 او16
148
127
106|84 |63
42 ادعاء1
1701191او14
147
106
85
63
42 |21321
17119 و14 |128 |07:{85
64
.14
:
14
193ء15017 و12 |86to7
64
143: ارد2
و17
:
1
131و86
/
08
/
12 |64 |21621143
2
25
4: 68,196
1401ء ا84
856: اه28
2
25 (401681796224
1|12: ل84
28.2356
28 228i5 6) 34 11 | 1411091197\125:53
283 28/56 84:13:41:69|1 98226 224
284 2856 851113142 | 170 | 198 | 227455
28
286 28]57[ 85114 143| 171 | 500 :48] 257
22856وو14
/
171
/
1
114|85 |57 |28
285
| | 170 |
421
172
16143
217 2:43 65| 86| 108 | 130| 151/173/19
43
219| 143 6 971091311 3175/197
2011
221) 22 44 66| 88/10/13415 41761 98
222 22| 44| 66| 38| 111|133|152177199
423 421 441 66| 59| 111|1331 26 11781100
174196ء 13015|109 |87 أ65 أ143: 218
258 وooja2ء (%17أ14
:
43 |86
8is7i:[287
و کد 17:01:35
144 | 15 | 86
6o: 17340223 ا44 دارد 96 [2857او28
288 2857
176,198
4
15
132ه88
/
11
66 اهواء: 220
:
290 291591 87}ri 6145174-031 232 261
1912958| 87116145174) 2031 232 261
:
.
:26 أ33 (175204
146
8i sai116ء اوعjء وه
1791201
6
13415ه 89/11
67
22144
224
46175452341263
87117/1
s8 او:93
202
1351571801|2
lilدو |67 |2245 ( 5دة
264
176
:
05
:
35 |88117147 [58 و42وء
103
14313558186 هو 671 |2245
226
205
2
r8 و13
4136
11|91 |68 |8j2245)
267(17807237(148ر118|88 |59|129
297
3:06
18
137165| 14|91 |68
451 داود
238468
288 (178|149و11و8 و 29/5
298
11
:
38i61184207 |92 |69
46 ادء ا230
و109
:
396 (149179و11|89
29
/
59اوو
4:07
i6iii8/38, دايو 69
46 (a3113
دعامه |adjr alr8olino او او6|30035
67
:27 | 45| 68| 9011313618181204
295 29|59| 85118147}177|206) 236363
296 29|59| 88| 118| 148 | 17720723666
]
|
.
1
819
309/3061
1
The Table of Proportional Parts.
D111213141516171 D111213141516171819
301 3016119711201150, 1801210;2401290 369136:7311101147|1841221125842951332
302 30 60 90112015111811211241 271 370 37,74111 1481185122212591296 333
303 30 60 90121151181121212421272 37137744111481185 -221259 296 333
304 301601 9112111521182121212434273 372 37 74 11 14811861223260297334
30513016119111221152183121312441274 373137174111 1491
86 22312611-931335
3061301011 q11122315315321412441275 3741371747112114911871224126112991336
307 30161 9211221153 184 214 245 276 3751371751112150 1871225262 300 337
308 30161
92|123[154|184/215/246|277 376 377511315018822526313001338
9212311541852162471278 377137175113150118812261263300 339
(310/31/621 93/124|155/186/217/245/279 378137175111311511189122626413021340
3111311629312411551186/21712481279 379137175|113115111891227126513031341
312 3162 93112411561871218 249 280 38013876 1141521901228 266 304 342
3131311621 931125156187 219:50 281 381 38|76|1141152190228 266 304 342
314 31162 94 125 11971188 219 251 282
382 33 7611411521911229267 305343
3151311631 9411201157118912201252,283
3831381761141153119112291268/3061344
316131631 94112611881189122112521284 38413817611151153119223012683071345
317 311631 981126 1581190 2:1-53285 38538177 115115411922312693081346
31831631 95|127|159190222254286 386 3877115154193-311270303347
319311631 9512711591912232551287
387 38 77116|154193-322701309 348
320/32/64 961128 16019212241256288 388 3877 11611551194123212711310349
3211321641 9612811601192122412561288 3891381771161155194/23327213111350
32232164) 96128 16119322%257 299 390 391781171156195123327313121351
323132641 9612
9612911611193226 25 8290 3913978117156 195 233127313121351
3241321641 97129162/19412262591291 39:39178 117 156196234127413131352
325132165) 9711301162|195122712601292 393 39178117115711961235127513141353
3261321651 9715301163|1951228|2601293 394)3917811181157|1971236127513151354
327 32165 98 130163 196 228 2611294 395)39 791118115811971237 276 316355
323 32 651 9811311163 196229 2621295 396139179 11811981198 237 2771316356
329321651 981137164/1972301263/296 397 3979 19 15 8 1981238 277/3171357
330133166 99113211651198/235/2641297 39813911911191159|199123812783181358
331!331661 991132,1681198123112641297
39913917911191159|1991239127913191359
332 33166 99132/166199 232 265 298
400 40 801120,1601200240280 320 360
33333166 99113311661199233 266199 401 408011201601200240128013201360
334 33 66100133 167 200233 267 300 402 401801201601201241281/321/361
335133167 1001134|1671201123412681301 403|4080|120116112011241128213221362
3361331671100113411681201123512681303 404|401801121116112021242128213231363
337 33 67101 1341168 202 235 269 303 405 4018112111622021243 -83 324 364
3383367 101 135 1691202236 270 304 406 408112116212031 243 284 324 365!
33913367 101 135169 203 23712711305 407 4081 122 162/203 244 284 325 366
340 34168 10211361170120412381272 306 408 4018112216312041244/285132613671
341134168 102 13611701204123812721306 4091401811122116312041245128613271368
342 3468|1021361712052391273307 410 411821231164 205 246 287 328 369
343) 34 68 1021371711205240274 303 411 411821231164) 205 246 287 3281369
344 34 68 103 137 172 206 240 275 309 41241 32 12311641206247285132913701
34513416910313817220712411-76310 413 41182|12311651206 247 28913301311
3461341691103113811731207124212701311 4141411821124|16513071248128913311372
347134 6911041138 173/208242 277 312 415 41 83 124 166 207 249 290 332 373
348 34169 1041:39 174 2082432731313 4164133 124 166 208 249 291 332 374
349134|69|1041391741209 244/279314 417141 B3 125 166 208 290 291 3331375
35.0134170) 1o5li4?11731210124512801315 418 41 8311251167/2091250129213341376
$7135170105140117512101245|2801315 4191411831125116712091251129313351377
352 35170105140|1761211246 281 316 420 42 84 126 168121012521294 3361378
353 35 70 105 141 176211247 282317 421 42184 1261682102521294 336378
3541351701061411177 212 247 283318 422 421841126 1682112531.95 337379
35513517111061421771213124812841319 423 42/841126 169 211 25312961338 380
356135171710614211781213124912841320 4241421841127116912121254129613391381
35735171 107142178 2141249285321 425 421351127170 2121255129713401382
358 35171107 143 179/21412901286 322 426 42851271702131255,298 340383
3593571107143 1791215 2511287323 427 42851128170213256 398 341 384
36013672 108 144 180216/25212881324 -428 42851128171 77412561:9913427385
361\361721108|144 1801216125212881324 42934208511281171|2141257130013431386
362 36172 1081441181 217253 2891325
430 43186129172215258301344 587
36313672 108 145 181217125412901326
431 43 86129172 215 258 301 3441387
364/36172/109 145 182 2181254,29,377
432 4386 129|172 216 259 302 3451388
365136173110914611827219 255129213 433143186 129 173 216125913033461389
366/3617311091146|182/219125612921329
434|43|8611301731217 26013041347/390
369 361731110|1461183122010561293/330 4351431871130117412171261130413481391
3681361731110114711841220125712941331
3 28
(m 2)
1
'
The Uſe of the Table of Proportion, for the more ready find-
ing out of any Logarithme, from 10000 to 100000.
1
W"
Hen you have any Logarithmse or Number above 10000, you may find it
out as before, by the Differences which are in the laſt Column of the
Tables: But for your more eaſie and ready perforining it, this Table is
of
great uſe; wherein
you
have all thoſe Differences ready divided, and caſt into 10
parts: So that becween cach of the roooo Logarithmes in the Table, you may eaſily
know the cen Intermediate Logarithmes, by the Proporcional Parc of the Difference
for any of them.
Thus in the Table the Logarithose of 2000 is
3.301029
The next Ligarithme, bciug the Logarithme of 2001, is 3.301147
Alter the Characteriſticks of cheſe Logarithmes,
So have you the Logarithme of 20000,
4.301029
And the Logarithme of 20010,
4.301 247
The Difference between theſe two Numbers is. 218; which for the ten Intermediate
Logarithmes muſt be divided into 10 Equal Parts, which is ready douc in the Table
of Proportion, after this manner.
1
*
D
I 2
3 4 5
6
7
8
9
318 31 63 95 137 159 '190 232 254 286
1
So that the Logarithme of 20000 being
4.301029
The Logarithme of 20001, by adding 31, is
4.301060
The Logarithme of 20002, by adding 63, is
4.301092
And ſo for the reſt, to 20010.
Or, on the other ſide, Let your Logarithme given be 4.301251, and you deſire
to know what Number anſwers to it; the next Nuinber leſs in the Tables is 301019,
which is the Logarithme of 20000: but this is 222 more, and the Common Diffe.
rence in the Table is about 218; turn therefore to this Difference in the Table of
Proportion, and there you shall ſce chat 222 makes your Number 7 more: So that
4.301251 is the Logarithme of 20007.
And thus you, ſave the Labour of mulciplying and dividing the Differences in the
Table of Logarithmes, they being here ready done to your hand.
1
.
1
FINIS
.
.
A
SU M M AR Y
OF SUCH
PENALTIES and FORFEITURES
!
As are Limited and Appointed by ſeveral
4
ACTS
OF
PARLIAMENT
Relating to the
1
Cuſtoms & Navigation
1
1
7
AS ALSO,
For the EXPORTING and IMPORTING of
PROHIBITED GOODS.
TOGETHER WITH
The ſeveral STATUTE S whereupon they are
Grounded : Being duly Compared with the Statuces at Large,
and the Abridgment to this preſent Year, 1664.
USEFUL FOR
Merchants, Factors, for all Officers belonging to
the Cuſtoms, Maſters of Ships, Purſers, and Boatſwains,
Mariners, Wharfingers, Lighcermen, and Watermen.
In what caſe both Ship and Goods are Forfeited
upon Importatione
1.
1
LONDON
Printed by E. Cotes, Anno Domini 1669.
(n)
4
1
身
​4
1
1
*
-
:
1
1
1
}
{
}
}
{
1
长​,
主
​1
}
生
​a
1
了
​1
1
“%,
4
f
1
f
{
4
1
:
4
}
1
1
1
1
TO ALL
Merchants and Factors, and Comman-
ders or Maſters of Ships;
AND
To all other Officers and Mariners: And to att
other Honeft-minded Men whom this
may Concern.
SA MUEL STURMʻr
Wilheth Proſperity,Courage, and Wiſdom in
all Your Lawful Undertakings
}
ani
G
ENTLEMEN,
OR Your ſakes I have 'inſerted this
following Abridgment or Summary
of the Laws, and Penalties, and
Forfeitures, as dre limited and
ap-
pointed by ſeveral Acts of Parlia-
ment, relating to the Cuſtoms and Navigation ;
and in what Caſes both Ships and Goods are forfei-
ted, upon Exporting or Iinportation of Prohibited
Goods; together with the ſeveral Statutes where-
upon they are grounded, being duly compared with
the Statutes at large, and the Abridgment to
1664.
I have been provoked to annex this Summary
of Cuſtom-Houſe Laws, the more out of a Princi-
ple of good will I have to you, and by knowing
ſome of your defects, or want of knowledge in thelė
(12)
.
things.
1
+
1
-
1
1
A
1
1
The Epiſtle Dedicatory.
things, by my own in times paſt, when I was a Com-
mander my ſelf
. Through ignorance of theſe Laws
your Goods have been Teiled, and loft, and Ships
Roppd and bindred in their Voyages, to the gredt
loſs and damage of the Merchants, and Owners,
and Mariners: Whereas if all concerned had
but the knowledge of what they ſhould know, they
imight prevent this loſⓇand damage, and walk ſafe
ly,
without any detriment to themſelves or Goods,
by the Officers; whereas otherwiſe, without this
knowledge, your Ignorance is the Officers Advan-
tage, and he will make you pay for it. And as I
mould adviſés every man to follow on Saviour's
Counſel Matthew 22, 21, Render therefore unto
Cæfar the things that are Cæſar's, and unto God
the things which are God's: as likewiſe Rom: 3.
6,7. Render therefore to all their dues, Tribute
to whom Tribute is due, Cuſtom to whom (u-,
(tom, Fear to whom Fear, Honour to whom Ho-
nour: So likewiſe I do adviſe all Officers to go
diſcreetly on in their Buſineſs with all men, and
not hinder, lett, or abufe, in word or action, any
one with whom they have Buſineſs, without juſt
Çauſe; nor thoſe that fall into their Hands ; nor to
take the juſt rigour of the Law of England, left
the Univerſal God ſhould take the Law of Hea-
ven upon us for
our Errors and Failings : but take
reaſonable Satisfaction. This is my advice to both
Merchants and Officers
and all others concerned.
I am their W'ell-wiſher, and ſo remain to be,
í
1
+
Pill, (Septemb. 2.
1 66 7.
SAMUEL STURMY
N
1
:
5
E
+
A
SU M M
A
A RY
OF SUCH
PENALTIES and FORFEITURES
As are Limited and Appointed by Several
ACTS of PARLIAMENT
Relating to the
CUSTOMS and NAVIGATION
1 +
First
1
LL manner of Goods Imported into his Majeſties Plan-
tacions, or Exporced out of his Majeſties Plantacions,
in Forreign Shipping, both Ship and Goods are forfeited.
Vide Statute of Navigation, I 2 Caroli 2. 28.
Secondly, All Goods that are of the growth of Afidz
Africa, and America, Imported in Forreign Shipping
are forfeit, per id. Stat.
T'hirdly, All Goods of the growth, production, and manufacture of Aſia, Africa,
and America, (hall bc Imported from the place of their growth, production, or ma-
nufacture; otherwiſe both ship and Goods are forfeiced, per id. Stat.
Except the Goods of the Spaniſh Plantations may be brought from Spain, and the
Goods of the Portugal Plantations may be brought from Portugal, and Eaſt India
Commodities may be brought from any Port on the Southward or Eaſtward of Cape
Bona Speranza, and the Commodities of the Levant Seas may be brought from any
Port within the Straights; Provided that all theſe Goods may be Imported in Engliſh
Shipping, otherwiſe both Ship and Goods are forfcited, per id. Stat.
Fourthly, All Goods of Foreign growth, production, or manufactarc, ſhall be
Imported from chc place of their growth, production, or manufacturc, or from ſuch Ixebis cafea
place where they are uſually firſt Shipp'd for Tranſportation only, and only in Engliſh called the
Ships, or in Ships truly belonging to ſuch place where ſuch Goods are lawful to be Ship- Grals Mower
,
ped; otherwiſ both Ship and Goods are forfeired, per id. Stat.
Salt and Brana
dy, was Seifed by me, for bringing Goods from Rochel to a Briſtol-Merchant į and the ship and Goods was bought
again for 2201. by A Letter of Licence:
Fifthly, All Goods carried from Port to Port (in England, Ireland, Wales, cí
Berwick) in Forreign Shipping, whercof the Owners or Part-owners are not all Éng-
liſh, as alſo the Maſter and three fourchs of the Mariners, both Ship and Goods are
forfeited, per id. Stat.
Sixthly, All Goods of the growth, production, or manufacture of any of his Ma-
jefties Plantations, ſhall be firſt landed in England, Ireland, Wales, or Berwick, before
they can be tranſported; otherwiſe both Ship and Goods are forfeired, per id. Stat.
Seventhly,
laden will
+
1
2
Penalties and Forfeitures relating to
ܪ
Seventhly, All manner of Wines, except Rheniſh; all Spicery and Grocery, Tobac-
co, Poc-aſhes, Pirch, Tar, Rofin, Salt, Dbal Boards, Fir Timber, or Olive Oyl, that
shall be imported from the Netherlands or Germany, are forfeited,
as alſo the Ship in
which they are Imported. Vide Stat 14.Car. 2. 1 intituled, Ant Act to prevenc
Frandš; &c. in his Majefties Cuſtoms. ;
.
Eighthly, All Freſh Herring i Freſh: Cad or Haddock, Cole-fiſh or Gul-fith, that
ſhall be Imported into England or Wales in Forreign Shipping, both Ship and Goods
așe forfeited. Vide State is Car. t. 5: jõtituled, an At For Encorráging of Trade.
1
ከ 41
ni
Goods forfeited for being Imported inta
England or Wales, with-
out any Penalty upon the Shib; - theſe Goods being all Engliſh
Manufactures.
15:
LL 'manner of Tinand Pewter Manufactures made in Forreign Parts are for
:
feited. Vide Stat. 25 Hen. 8. 14.
Officers may ſearch and ſeiſe Wares broughc into the. Realm contrary to the Täid Act,
and none ſhall withſtand the ſearch of Braſs, Tin, and Pewter, on the forfeit of five
pounds, per id. Stat.
Tin and Pewter
Braſs.
1
}
Several com-
1
1
1
Woolen
Clothes, Woulen Caps, Ribbons, Fringes of Silk and of Thred, Laces of
modities for. Silk and of Thred, Silk Twinc, Embroidered Laces of Silk or Gold, Saddles, Stir-
feited.
Tops, or any Harneſs belonging to Saddles, Spurs, Boſſes for Bridles, Andirons, Grid-
irons, any manner of Locks, Hammers, Pincers, Fire-tongs, Dripping-pans, Dice,
Tennis-balls
, Points, Purſes, Girdles, Gloves, Harneſs for Girdles, Iron, Latten,
Stecl, Tin, or Alchyny, or any Wrought or any Tawed Leather, any Tawed Furs,
Biskin Shoos, Galloſhes, or Cork, Knives, Daggers, Wood-knives, Bodkins, Sheers
for Taylors, Sciſſors, Razors, Cheſs-men, Playing Cards, Combs, Pacrens, Pack-
needles, any Painted Wares, Forſers, Caskers, Kings of Copper or of Latten, guilo
Chafingdiſhes, Hanging Candleſticks, Caffing Balls, Sacring Bells, Rings for Cur-
tains, Ladles, Scummers, counterfeit Baſons, Ewers, Hats, and Bruſhes, Cards for
Wooll, black Iron, Thred called Iron Wyre, or whited Wyre, are forfeited if any. ,
ſuch be Imported into England or Wales. Vide Stat.4 Edw. 4.
Prohibited All Iron Wyre, Card-wyre, or Wool-cards, chat ſhall be Imported into England
Goods forfeited. or Walès, are forfeited. per Stat. 39 Eliz. 14. 14 Car. 2. 19.
Probibited more
All manner of Girdles, Harneſs for Girdles; Poirits, Leather, Laces, Purſes,
forfeited.
Pouches, Pins, Gloves, Knives, Hangers, Taylors Sheers, Sciſſors, Andirons, cob-
bards, Tongs, Fire-locks, Gridirons, Stock-locks, Keys, Hinges and Garnets, Spurs,
painted Glaſſes, painted Papers, painted Forcers, paisited Images, painted Clothes,
bearen Gold or Silver wrought in Papers for Painters, Saddles, Saddle-crees, Horſe-
Harneſs; Boots, Bits, Stirrops, Chains, Buckles, Lacten Nails with Iron Shanks,
Curvets, Hanging-candleſticks, Holy-water, Stops, Chafing-diſhes, Hanging Lavers,
Curtain-rings, Cards for Wool, Roan Cards, sheers, Buckles for Shoos, Broaches
for Spits, Belés, Hawk-bells, Tin and Leaden Spoons, Wyre of Latten and Iron,
Candleſticks, Graces, Horns for Larithorns, or any of theſe, bcing Imported into
England, are forfcitéd, or the value thereof, betwixt the King and the Proſecutor,
Theſe may be ſued for in any Corporation where they are. Vide Stat. T R. 3. 12.
All Girdles, Harneſs for Girdles, Rapiers, Daggers, Knives, Hilts, Pummels,
Lockets, Blades, Handles, Scabbards, Sheaths for Knives, Saddles, Horſe-Harneſs
,
Stirrops,
F
1
1
1
1
+
-illo
1
1
1
ted.
the Cuſtoms and Navigation.
3
Scirrops, Bits, Gloves, or Points, Lcather Laces, or Pins, chat Thall b: Imported in. Goods prohibi
,
to England or Wales, ſhall be forfeit. 5 Eliz. 7
All manner of Silk wrought by it felf
, or with any other Stuff, in any place out of
the Realm, Ribbons, Laces, Girdles, Corſes called Corſes of Tiſſue, or Points, ſhall
be forfeited, per Stat. 19 Hen. 7. 21.
All Forrcign Bone-lace, Cur-work, Frinige; Embroidery, Bandftrings, Buttons, or Goeds probibi-
Needle work, made of silk or Thred, or either of them, being Imported into Eng- ted 100 l.
land, Wales, or Berwick, ihall be forfeited, beſides the Forfcicure of one hundred
pound. 14 Car. 2. 13.
All manner of Woollen Cloth that ſhall be Imported into England, Irgland, or
Wales, from beyond the Sca,i thall be forfeited. Vide Stat. 2 Edv. 3. 3. & 4' Ed. 4.1.
?
L.
--
In what Caſes Goods arë forfeited for Undue Shipping or
Landing
4
!
A
IL Goods that chall be shipped or Landed before the Cuſtom paid or agreed for Goods.fsipped
in che Cuſtom-houſe, are forfeited. Tide Stat. 12 Car. 2. 4. intituled, The AEE or landed before.
for the Tonnage and Poundage.
Cuſtom paid arc
(All Goods that ſhall be shipped or Landed, or put into any other Veſſel to be Ship- unlavoſ ul time
ped or Landed, at any unlawful time or place, are forfeit, or the value of them. or place loſe.
1 Eliz. 2. 14 Car. 2. II.
forfeir.
inward bound
without irar-
1
ty.
All Goods that ſhall be pue into any Lighter, Boat, or any other Veſſel, to be
Shipped or Landed, without Warrant from the Cuſtom-houſe, and the preſence of
one or more Cuſtom-houſc Officer, are forfcited, as alſo the Lighter or other Veſſel
in which they are found to be shipped or Landed. Vide Star. 14 Čar. 2. II.
If any Maſter of Ship, Purſer, Boatſwain, or other Mariner , knowing or con- Perſons confexca
ſenting to the diſcharge of Goods inward bound, without Warrant from the Cuſtom- ing to the dif-
houſe, or the preſence of one or more Cuſtom-houſe Officer, lhall forfcit the value of charge of Goods
the ſaid Goods ſo unſhipped. Vide Stat. 14 Car. 2. 11.
Tant, the penal-
Every Cuſtomer, Collector, and Comptroller, that doth conceal his Majeſties Cu-
ſtoms; being duly Entred, ſhall forfeic treble the value thereof, per Stat. 3 H.6. 3. Officers concea-
ling Cuftom, the
If any Goods having paid Cuſtom at the Importation, and ought to have allow- penalıy
.
ance at the Exportation; if the Merchant Ship our leſs in quantity than is expreſſed ſhip out cler
in liis Cercificate, thall be forfeited, or the value of thein. 14 Car. 2. 11.
iban is express
sed.
If the faid Goods be Landed again in England, wales, or Berwick, except they be Landing Goods,
made known in the Cuſtom-houſe, ſhall be forfeit, per id. Stat.
unleſs made
known in C160
If any Goods be put on board a Ship to be carried from Port to Port, without em boufe, for-
Warrant from the Cuſtom-louſe, all ſuch Goods ſhall be forfeit, per id. Stat. Goods carried
from Port to Port
If the crue content of Quaratity and Qulalicy be not mentioned in the Certificate, without war-
under the Cuſtomers Hand in the Port
where they are shipped firſt to paſs for rant forfeited,
another Port, all ſuch Goods not certified or diſcharged before the ſaid Certificate Quantity and
delivered, and the Goods viewed, ſhall be forfeit, per Stat. 3 H. 7.7.
quality muft be
expreſſed.
Quere, whether this Statiste be in force or not?
All
3.
.
.
I
1
1
4
Penalties and Forfeitures relating to
***
Goods èxported, All manner of Goods, Wares, or Merchandize, that ſhall be Exported, and eſcape
and not diſcove- undiſcovered unto the Officers of the Cuſtoms, the Owner or Propriсtor ſhall forfeit
red onto the of- double the Value, according to the Book of Rates; Except for Coals, for which they
fecer, forfcit ihall forfeit double the Cuſtom. Vide Stat. 14 Car. 2.
double the on-
England and All Goods, Wares, and Merchandize, that shall paſs by Land betwixt England
Scotland, Goods and Scotland, ſhall paſs by and through Berwick and Carlife, and pay Cuſtom at
that paſsi
one of thoſe Ports, otherwiſe be forfeited, per Stat. 14 Car. 2. 11.
Luc.
In what Caſes Ship and Goods are forfeited upon Exportation
of Goods.
Licence to be
Leather, Tal-
F any Woman, or other perſon under the Age of twenty one years, except Ship-
granted for Paf boys, Saylors
, or Merchants, Apprentices, or Factors, ſhall paſs over the Sea,
ſengers to paſs without Licence from the King, or 6 of the Privy Council, the Ship in which ſuch
beyond the Set. Perſon ſhall ſo paſs thall be forfeit. Vide Stat. I Jac. 4.
If any Perſon ſhall Tranſport, or Ship to be Tranſported, Leather, Tallowor
Low,&c. Raw Hides, to any Place beyond the Sca, all ſuch Goods ſhall be forfeit, as allo the
Ship wherein they are Exported. Vide Stat. 18 Eliz. 9.
Hoys or Plats If any Hoy or Plac croſs the Seas beyond Norway Eaſtward, or Caen in Normandy
Southward, they ſhall be forfeit. Vide Stat. 1. Eliz. 13. s Eliz. 5. 13 Eliz. 15.
&c.
Cora oy Vianal. If any Corn, or other Viftual, be Tranſported, exceeding the Prices mentioned
in the Act for Encouraging of Trade; or if any Wood ſhall be Traſporrcd, they ſhall
forfeit the Veſſel in which ic ſhall be Exported, and alſo double the value of the
Goods. Vide Stat. 1, 2 Phil. Mar. s. the Maſters and Mariners all cheir Goods,
and a years Impriſonment.
Engliſh Manu If any Goods of the growth, production, or manufacture of Europe be tranſported
factures to be into his Majeſties Plantations, except from England, and in Engliſh built Shipping,
exported in Eng- both Ship and Goods are forfeited. Vide Stat, 15 Car. 2.5.
liſh Shipping.
Due entry to be If the Maſter shall ſuffer any Goodsco be Landed before a due Entry made with-
made in 24 bo. in twenty four Hours after arrival in the ſaid Plantations, both Ship and Goods are
forfeited, per id. Stat.
.
Ships for fifb If any Ship thall ſet out to Fiſhing, or other Veſſel ſhall ſet out for the Weſt
ing, the time to Country or Iſèland Fiſhing, before the tenth day of March in any Year, ſuch Veſick
Set ont.
ſhall be forfeit. Vide Stat. 15 Car. 2. 14. intituled, An Act for the Fiſhing Trade.
If
Sheep,flool,&c.
any Sheep or Wooll, Wooll-fells, Wooll-flocks, Mortlings, Shorlings, Yarn
made of Woolly
, Fullers Earth, Fulling Clay, ſhall be Exportéd, all ſuch Goods are
forfeit, as alſo the Ship wherein they are Exported. Vide Stat. 12 Cor. 2. 32.
Silver Gold,
If any Silver or Gold be Exported without Licence, it ſhall be forfeited. Vide Stat:
SR. 2. 2. 6 9 Edw. 3.1.& 2 Hen. 4.5.0, 2 Hen. 6.6.
None but Merchant Strangers ſhall Tranſport Wooll, Wooll-fells, Leather, and
Lcad beyond the Seas, upon the Forfeiture of the ſaid Goods. Vide Stat, 27 Ed.3.3.
14 Rich. 2.5.
IF
&c.
s
11
1
4
01 priseOnfamound Navigation
:/:!!) 19981:1 JXiv:
is
is
per id. Stat.
..g? ain
1
7
i
1.
iment
.title
Adem Stat...100.15 m2!1d...cil puis 2001:21: !..
made
IF
Skin, cantor untažin?a; öfany Oxil Stecr; Bill; Cotv, or Calf (except skins tand or
any
Calve-skins of four pound weight apicce, or under) and Sheeps Skins dreſſed without
untana'd.
the Woolf of Cuci Skirls ofHides, Awhich are for élie Ships iieceffary Proviçon, ſhall.),
paſs out of Englandibeydnatlid Sess, or into Ireland or scotland, or tļic Iands be-
longing to England, fliah' 5c-forféicédoide Stat. 14 Car. 2.7
illaup of cHe Hidesio7 Skins áforėlard," that ſnall be caken off of any Beaſt in any hides traitzen
.
-oficlic-Illähas belongina to Engibrid, except freland, ſhall be Transported into any ted, except from
Place cxculpt2England, chit Oftendcrtani forfeit-double de valab for every Offenice, Ireland, "th:
penalty.
2500!!!::1107 liesoorl.z.?. i.'
All manner of Ammunition may be prohibited at his Majeſties pleature: -12 Cai. Ammunition
2.4.
may be probibi-
c.1.W Yviq no 1993 :: .:/?. 1o milo
1. teden
Ifrâny:SheepfHallbExported!delle?Offender fidll
. forfeić 2097 fotiztery Shëep. Sheep exported,
zriye 70.!. Vie to the penalis.
Vide 12 Cars2 3 27M:1 to
bris. 578
976 902 :: 5r... Y om bij
If any WoollyWollfells; Wböl-fockis; Moltlings; StörlingskaYarñ made of Fillers Earelo
Wooll, Fullers Earth, Fulling Clay, ſhall be ſhipp'd to be Exported, the Offender & Fulling clay.
ſhall forfcit three ſhillings for every pound weight.
Cinsi 2015-2:11. for
If any Maſter of a Ship, or other Mariner, We knowhigland conſenting to the Ex= The penalty of
the shipping of
csno od 9.15
LEO..!!L...? 303964il Jald Goods
Weather-fhçep, Wooll,,or Wooll-flocks, as are før neceſāry' proportion for the Sheep.
Ships uſea":
If any Wooll, Wooll Aocks, or Yarn made of Wooll
, shall be preſſed with
any wooll, &c. pret
Enginc into any Sack; Pack, or other Wrapper, or Thall puţ, preſt;' or ftecve Wooll with any times
or Woollen Yårn into any. Pipc, But; or Hogſhead, Cheſt, or other Cask or Vefici, gine,&c.penulij
.
or cafry or lay any
ſuch Wooll
, Wpoll-flocks,
or Yarri made of Wohl, neer to "thlé -..
Sea
or any Navigable River, all ſuch Wooll, Wooll-flocks, änd-Yařk hade of Wooll
ſhall be forfeited. Vide Stat, 13 % 14.Car. 2. 18.
:: boca )
If any Wooll, Wooll-fells
, Mortlings, Shorlings, Yårn made of Wooll, Wcolli Fullers earli,
Rocks, Fullers-Earth, Fulling-clay, or Tobaccopipe-clay, being in any Pack, Sack, &c. Tobacco-
pipe clay.
Bag of Cašk, ihall be carried upon any Horſe, Carė, or other Carriage; except in
the day'titne, viz. from the first of March to the twenty ninth of September betwixo
the Hours of four in the morning and eighie at night, and from the twenty nincli of
September until the firft of March between the Hours of ſeven in the morning and
five at hight otherwiſe to be forfeited; per id. Stat.
If any Tobaccopipe-clay be Exported beyond the Sea, the Officer ſhall forfeit darčė Tobacco pipe
Shillings for every pound weight, per id. Stat.
clay.
:)!i.:
If- any manner öfshidépéskîns; Widoll-Fells
, Mortliigs, Shorlings, or the Skins shtep' skins, en
of any Stag, Buck, Hind, Doe, Grat, Fawn, or Kid, or the Peſcs or Skins of the Leather,&c.
any of them, or the Leather made of any of them, be pur on board any Veſſel to perialty.
be exported, they fiallobé forfeited; tasallo tivo fhillings fix
pence for every
Shorling, Mortling; Peles or Skin, ſo Shipped to be Exported. Vide Stat. 5 Elizzk.
All
grcat
Carrel, except of Scotland, that ſhall be Imported into England or Great Cartel
Walis berwise the firth of July and the riventieth of September in anyycar; and all imported, thie
great Carcel of Scotland that shall be brought int becwixt che twenty fourth of Awa penalty.
gaſt and clie ewentieth of December in any year, ſhall forfeit for every Head forty
(0)
fhillings;
11
0. Via
4
...
,).
.
'!!!!!!!
1
L
2
.
1
..fio !!
31
iW.....
every Féli,
+
1
/
another mans
name.
Butlerage of
Wines.
1
1
Tiit
Penalties and Forfeitures relating to
Shillings; and for every Sheep brought in becwixt the one and twenticch of Auguſt
and twentieth of December, ten ſhillings, per-Stat. 15. Car.. 20:56
Gesods entred in: If any Goods be entered in any other mans Name than she mue Ownct and Pro-
prietor, they fhall be forfeit: And if the Officcr conceal any Offence in the ſaid Act,
lic ſhall forfeit one hundred pounds. Vide Stat. 1 Eliz. II.
Prifage and If any man, being free of the Priſage or Butlerage of Wine, fhall Enter another
mans Wines in his Name, whereby the King loſeth his Butlerage, all Wines ſo En-
: fred arc co forfeit double clie valuc of the Cuſtoms chereof, Vijde 'r Hen, 8.5..
If any man offend contrary to the Stat. I Hexs. 8. . he ſhall forfeir all his Goods.
Vide Stat, 2:3 Edw. 6. 2?.
Sugar, Tobacte,
If any Officer of the Cuſtoms ſhall ſuffer or give any warrant for any Sugar,
GingerTobacco, Ginger Cotton-Wooll; Indicos Speckle-wood, Jamaica-wood, Fuſtick,
or any other Dying-wood, of chic growth of any of his Majeſtięs Plantations, to be
conveyed into any Parts beyond the Seas , before they are Landed in England or
Wales, for every Offenco hic, ſhall forfcic the value of the ſaid Goods. Vide Stat.
15 Car. 2.5.,'
.
Goods of Alien All che Goods of an Alien Merchant or Factor in any of his Majeſties Plantations
Merchants. are forfçired. Vide Stat, 12 CAT. 2. 18.
qiri? a
>
Copper, Rrafsi IF
any manner of Copper, Braſs
, Latten, Bell-mercle, Pan-metcle, Gun-medle,
Bell-inettle,&c., or Shroof-mectle, Thall be put on board any Veſſel to be tranſported, the Offender
Thall forfeic double chc values to be divided batwise the King and the Proſecutor.
Vide State 33 Hen. 8.7.
101. penaliy.
And alſo ten pounds more for every thouſand pound weight, per Stasi
2&fo 3 Edw. 6.37.
از...
ܢܢܐ܀ ܟ݂||
1
1.
tj.
The penalty of
The Cupomer
The Cuſtomer ſhall take Bond in double the value of the ſaid Goods, when
ust performing they ſhall be tranſported from Port to Port, and alſo rol. over and above for every
duty
, the penats chouland pounds weight, and give Bond; which Bond, if it want a Date, the
Cuſtomer ſhall forfeit the value of the ſaid Goods, and alſo his Place, per id. Stat.
To grant a false If any Cuſtomer grant a falſe Certificate for chc ſaid Goods, he ſhall forfeit his
Certificate, lbe, Place, and the value of the Goods ſo concealed. 33 Her. 8.7.
penally.
If any Maſter of a Ship, Owner, Purſer, or Boatſwain, knowing ſuch Mercles to
the not diſco- be Shipp'd, and do not diſcloſe it within three days, he ſhall forfcit double the value
vering
of it. Vide Stat. 2 & 3 Edw.6. 37.
Not ſeifing ebe If any Officer of the Cuſtom-houſe, knowing ſuch Meccles to be shipp'd to be
Said Goods, the Tranſported, do not ſeiſe it, he ſhall loſe his Office, and the value of the Goods ſo
pexally.
Shippd. Vide Stat. 2 & 3 Edw. 6.37.
The Said Goods IF
any Perſon Ship any of the ſaid Mercles at any Place, except where there is a
not to be shippid Cuſtomer, he ſhall forfeir the value of the Goods, and alſo ten pounds for Cvery
but where there 1000 pounds weight, per id. Stat.
is a Cuftomti.
Penalty 1000 l.
If the Governour of any Plantation belonging to his Majeſty, do not his duty
to a Governoar juſtly, according to the Alt for Encouragement of Trade, he ihall forfeit his Place
not doing his and 1000 l. per Stat. 15 Car. 2. 15.
duty.
Transporting
Every perſon that ſhall be found guilty of Tranſporting of Leather, Shall for every
Leather. Offence forfeit sool. Vide Stat. 14 Car. 2.7.
Every
ir
1
.
gishieCuftoms and Navigation
doci
19:41
Every Cuſtomer, or other: Officeryl.thas ſhall negleet his duży, or.corinįve ät che
Oficentage-
Tranſportation of Leather, ſhall før every Offence forfeic 109.pounds. Vide Stato ing bis helyen
+ Fac. 22
Balty 100 ,
Every Cuſtomers, or other Officeri that (haļl make a fallc Certificate of the Land- To make fase
ing of Leather, ſhall forfeit 100 l. per id. Stat.
Certificates of
Landing Leatberg
perally 1996
If any Goodsor. Merchandiſe ſhall be shipped or Landed at any unlawful time or Goods Shipping
place, for every Ofence che Maſters Owners. :or: Purſer ſhall forfeic 100l. 1 Eliz. ang unlarsful
17,6 14 Car. 2. 11.
ty 100 l.
Laxful times are only from the firſt of March until the firſt of Septemberzbetwixt What läroful
Sun-riſing and Sun-Settings and from the forff of September antil the times are by
firſt of March betwixt seven-a clock in the Morning, and for a clock Ordera tako
in the afternoon. The Port of Hull is here excepted.
hour, the persals
ان . .
before entry
made 100:
11
1
܆jiܙ܆
1
with as much
1
If the Captain, Maſter of a Ship, or Purſer outward bound, Chall take in any Goods taken is
Goods before Entry, he ſhall forfeit 100 l Vide Stab. 14 Car. 2. II.
.
if hè go away before cleared on Oath in the Cuſtom-houſe, giving 4 truc Accompt to go before
of his Lading, & c. he lhall forfcic 100 l. per id. Stat.
cleared morath
perally 100 1.
Li-IF
If any Captain, Maſter of a ship, or Purſer, do not bring. his:Ship.co.tbc Pors, Maſter of Ship
and make Entry with as much ſpeed as Wind and Water will permic, for every Of to bring his ship
fence he ſhall forfeit 100 l. Vide Stat. 14 Car. 2.
convenience,
,&ς,
If he permit any Goods to be taken out of the Ship, to be Landed, before he hath To ſuffer Gåods
inade his general Entry upon Oath in the Cuſtom-kouſc, for every Offence he ſhall taken
out before
forfeit 1001. Vide Stat. 1 Eliz. 11.
penalty 100
If any Captain, Maſter of a ship, Purſer, or Boat-ſain, or other Perſon taking Goods imbeze-
Charge of the Ship, ſhall permit any ſort of the package cherein co be opened, im- ledy &cJoeyro
bezeled, or altered; for every Offence he shall forfeic 100 l. Vide Sta. 14. Car. 2, 11.
I
Men of War to be liable to the Rules that Merchant Ships are ſubject to. Liberty
10. go on board and take out Prohibited and Uncüſtomed Goods. The Commiffi-
oners and their Deputies to enter on board, and bring on ſhore Goods outward and
inward bound. The Officers may ſtay on board until the Goods be diſcharged.
14 Car. 2.
A general entry)
1.
If any Goods be found conccaled aboard the Ship, when the Officers of the Cu- Goods concealed
ſtoms have cleared the Ship, the Maſter, or other perſon, ſhall for every Offence for. 100l.
fčit 100l. per id. Stat.
If any Wharfinger, Crane-keeper, Searcher, „Lighterman, or other Officer, know- Wharfinger, &c: -
ing
any Offence contrary to the Statute, do not diſcloſe it to the Cuftomer, he thall not diſcloſing
foifcit 100 l. Vide Stat. I Eliz. 11,
&c. penalty
If any Wharfinger or Crane-keeper ſhall take up.or Land, or ſuffer to be Landed, To ſuffer Goodser
or Ship off, or ſuffer to be Water:bound, any. Warts or Merchandiſes, at any unlaw- to be taken up or
fül cimc, or without the preſence of, or nocice given to an Officer at the Cuſtom- landed as
houſe
, he shall forfcic for every Offence 1001. per, Stat, 14 Car.2, 11. The Porr of laxofal times
Hull excepted.
(02)
If
Iool.
+
8
to
Penalties and Forfeitures delating i
- Yotarating
Bathua
2001. perally. If any Oficers of the Cuſtoms ſhall directly or indirectly receive any Bribe, Be-
;
ifamprotſe, or Reward, or (hall connive at any falle Entry of Goods, he ſhall forfeit
100l. per id. Stas.
50d. penalty to If any Merchaisc, ôr other Perſon, ſhall give fuchi Bribes, for every Officace he
Thall forfeit sol. per
id. Stat.
Frük a Bribe.
87.
1002
in
If any Packet Boat, or other
Voffet appointed to carry Letters, ſhall Import or
Expore any Goods or Merchandiſe, for every' Orience the Maſter Thall forfeit. 100 ).
qad: ::44 per id. Stat.
Foreign Bacem fa Perſop offer to fale any Forraign Bonc-lace'; Cut-work, Imbroidery, Fringe,
lave Spil-pamerite or Needle-work made of Silk or Thred, for every Offencë he ſhall forfeit sol.
penalty soubor be pat jidSfat.
lool.
IF
any of the ſaid Goods be Imported into England or Wales, the Offender ſhall
forfeig 100%; por id. Seat.
s 500
.ii.ee
i
r
7
!
Falle Certif-
If any Officer of any Port ſhall make a falſe Certificate, he ſhall forfeir soli
cate,
perally
per, id. Stat!?
sola
To'false'ang If any perſon ſhall falſifie any Cuſtom-honſe Warrant, he ſhall forfeic for syery
WATAN 1006. Offerre 100 l. per 'ida Stai.
i
ii
I
1
The Office? 10 Every Officer appointed to perfect an Ency, ſhall make report thereof under his
make atrne re. Hand, unto the Chief Officers of the Cuſtoms, che next day, upon cae Penalty of
parts ont penalty
rool. unleſs there
be cauſe of longer time to be allowed by the Chief Officers of che
of 100...
Cuſtoms, per id. Stat.
.
Pilot or I'atti-
French VrNels
If any French Vefſel put any Goods or Paſſengers on Shore, or into any Boat, to
not copiatGoods be conveyed on Shore, and not pay the five ſhillings.per Tonnage duc upon French
on flore mot par - Venels, upon their return ehey ſhall forfeit ten pounds, and pay all the former
ing 5 s. per Tube
Duty, per id. Stat.
the penaliy tola
If any Pilot or Waterman Shall go out from any Port, to bring in'any Goods or
mais going out of Paliengers from aboard of any French Veſſels, for every Offence he ſhall forfeit 401,
auf Port brings
Goods,&c. per
per id. Stat,
nalig 401.
officers not gi. If
any Cuſtomer, Comptroller, or Searcher, or their Deputies, do not give their
ving attendance atecadańce at the Cuſtom houſe at ſuch time and places as are appointed by Law,
pinaliy 1oole and allo do not cheir utmoſt diligence in their reſpective Places, for every Offence
the Offender ſhall forfeit 1ool. Vide Stat. 1 Eliz. i 1,
Nat refdent on
IF
any Cuſtomer, Comptroller, or Searcher be nof reſident upon his Place and
kis places pe Office for every:Offerice he ſhall forfeit 100l. Vide Staf. i Hen. 4.13. 4 Hen. 4.
naliy 1002:
20,6 21, 13 Hen. 4.5.
No Ciftom offi-
If any Cuſtom-houſe Officer fraighç any Slip, or uſe aay Merchandiſe, or keep
cer to be Owner any Wharfe, or hold any Hoſtery, or Tavern, or be Factor, or Attorney, or Holt
of a shiptor wife to any Merchant, for every Offence he hall forfeit 40 l. for every ſix Months,
Merchandiſes to be divided betwixt the King and the Proſecutor. Vide Stat. 20 Hen. 6. 5.
&c.penalty 40 1.
IF
the Cuſtoms and Navigation.
9
IF
any
Cuſtomer, Comptroller, or Searcher, be a Common Officer, or Deputy to No Cuftom o ffon
a Common Officer, in any City, Burrough, or Town, upon the penalty of 40 l. cer to be a coms.
for every fix Months he ſhall ſo officiate boch Offices cogecher. Vide Stat. 3 Hen. men oficer, pes
nally qol
7.1.
Quere, whether this Statuse be in force, or Repealed by the Statute of
į Hen. 8. 5.
Engliſh Shipping is either Engliſh builc, or bought bona fide by Engliſh Money, English built
whereof every Owner or Part-owner are Engliffen Iriſh, Welcb, or of his Majefties shipping, was
Plantations.
C
is mean
op
1
---
tait
18
72 55
M
38
67
62 55
4
4
+
isini
A TABLE
A
010
A
Τ Α Β L E
OF THE
STATUTES
1
TO
Rclating to the
Customs, and NAVIGATION, and TRADE,
Made in the Reign of
King CHARLES the Second, .
Onnage and Poundage
.
.
Preventing Frauds
14 Car. II. 11.
Navigation Encouraged
12 Car. II. 18.
Wooll, Sheep, &c.
12 Car. II. 32.
Leather, &c.
14 Car. 11.7.
Bonelace, &c.
14 Car. II. 13.
Wooll-fells, &c.
14. Car. II. 185
Card-Wyre and Iron Wyre, &c.
. .
Encouragement of Trade, &c.
15 Car. II. 5
Encouragement of Fifing
15 Car. II. 14
i
1
THE
L
!
II
!
THE.
A U THOR
1
j
Τ Ο
His Books.
SH
Ince now, my Book, thou art ſo far gone on,
Abroad on Gods Name ,, and be better known :
But had there been now but one quarter done,
That, nor the rest, ſhould n’er have ſeen the Sun.
To Friends be free, ope them thy Treaſures Store :
But carping Scoffers, let them have no more
But Scraps; for that's enough, and good for ſuch
As poyſon all they fee, foul all they touch,
And on Mechanick Scapes forge Arts detraction,
Ere they will wink, or mend the faultier action
Tb' Errata's made. I never did intend it
For ſuch as not commend, nor can come mend it;
Not I: And ſo I end it.
1
1
5
ERRAT 4.
{
!
f
|
1
ܪ
+
A
COMPENDIUM
A
OF
+
FORTIFICATION,
BOTH
4
Geometrically and Inſtrumentally,
1
BY A
SC A L E
+
The Making whereof is ſhewed by the Tables,
and their Uſe, both of the Tables and the Scale,
for ſpeedy Protracting
of any Fort conſiſting of
8 Bulwarks, whoſe Baſtion-Angles ſhall not
exceed 90 Degrees; and ſo the like for Baſtion-
Angles of 12 Bulwarks,
:
WRITTEN BY
PHILIPS T A Y N RED
Profeffor and Teacher of the MATHEMATICKS in the
City of BRISTOL.
/
JE R. XVI. 19.
The Lord is my strength, and my fortreſs ,' and my refuge in the
day of affliction.
1
1
LONDON,
Printed by E. Cotes, Anno Domini 1669.
4
1
-
,
1
1
1
1
+
A
COMPENDIUM
OF
FORTIFICATION.
Irfts
of the sides and Angles thereof, as in the Figure following.
the Names
1
A
1
3
+
.
H
>i
F
M
...
UTY II IIIIIII
XT
I10
JI1
0:-
111111
min
A
E
*
HI
4
Names of the sides of # Forf.
A B the Ourward side of the Polygon, and D E che Inward Sidc.
CA the Semidiameter of the Outward Polygon, and CD the Inward.
IHAGF the Bulwark or Baſtion.
AG, or A H, the Front or Face of the Baſtion:
280 Foot.
A D the Capital or Head Line,
D I, or D F, the Gorge Line..
IH, or FG, the Flank; and FM the ſecond Flank, :
D G the Eſpaule, or Shoulder.
FK the Curtain-
A K the longeſt Line of Defence
AL the ſhorteſt Linc of Defence.
Names of the Angles.
N A B the Angle of the Polygon, and N A C the half Angle.
HA G che Flanked Angle of the Bulwark,
(p2)
AGR
-420 Foot.
720 Foot.
1
A
2 स
Fortification.
L
A GF the Angle of the Shoulder.
FDG the Angle forming the Flank, commonly 40 Degrees.
GA O the Inward Flanking Angle.
APB the Outward Flanking Angle, and APO half the ſame.
Note, That_thc Baſtion or Flanked. AngletH A G muſt never be leſs than 60
Degrees neither above 90 Degrees; but as neer as you can toán Angle' of 90 De-
grees : So that it may be defended from the Flank and Curtain on either side.
The longeſt Line of Defence K A not to be above 12 ſcore Yards, that is, 720 Foot,
being within Musket-hot; and the bredder of the Rampire to reſiſt the Barcery
100 Foot.
***
como
.
any Deſcribe a Fort of Five Pulmarks, or any other; ſo that the
Baſtion, of Flanked Angle of 8 Baſtions or Bulwarks exceed
not go Degrees by the Line of Chords.
Irft, Draw an Oblaire Line, as A.B; and upon A, as a Center, with the Chord
of 60 deg, deſcribe an Arch, as CDE; and from Clay down half the Polygon
Angle (which in the Table following the Pigure you ſhall find to be) 54 deg. as
CD; alſo the ſame again from Dto E, and draw the Line A E. Now divide the
half Polygon Arches CD and D E each into three equal parts, as in FHIG, and
from two of thoſe parts from D, as Fand G, draw Lines unto the Baſtion Point ac
A. Then take any convenient Diſtance, and lay thie ſame on thoſc Lines from A
unco K and L, which ſhall make the Front or Face of the Bulwark. Next, from
the ſhoulder at K let fall a Perpendicular to AB, as K M; and on the Center at K
deſcribe an Arch of 60 deg. from M towards N, and from M lay down on the ſame
Arch sodeg. or more exact 49 deg. 24 min. and ſo dray KN, which will cut the Sea
midiameter of the Polygon in the Point O ; fo fhall A O be the Capital Line of the
Baſtion. Then fom o draw a Linc parallel to AB, as OP; ſo ſhall you have OR
for the Gorge Ling, and R K the Flank. Now for the Currain, take half the Front
A K, as AT, and lay ic down three cimes from R cowards P, which will fall in S;
lois R S the Curtain. Then on the Point at Screct a Perpendicular, as SV, equal
to R K, which ſhall be the Flank of another Baſtion; and ſo the Front K A being
laid from V, ſhall cut the firſt Linc AB in B ; ſo drawing V B, you have the Front
of the ſame Baltion.
Láhly, Divide A B in the middle, as in W, and from W let fall a Perpendicular
to A B, which will cut the Semidiameter of the Polygon in the Point D; lo is D the
Center of the Polygon. And with the ſame Semidiameter D A you muſt deſcribe
a wholc Circle, of che whſchi A B is the part chereof, which Diſtance will reach
from B unto X, and from Xunto Y, and ſo to Z, and your firſt Point at A, where
you begun your Work.
For the other Baſtions, they may be eaſily tranſported from the firſt Baſtion. And
norç, That if your Fort exceed 8 Bulyarks, you muſt add 15 deg, to half the Poly-
gon Angle, lo have you the Baſtion Angle; and then work as before. Bur in the
Forts that exceed not 8 Bulwarks, where the Baltion Angle will not be above 90 deg.
you muſt take the part of the Angle of the Polygon.
The longeſt Line of Defence is from A unto , and ſhould not exceed 720 Foot
(becauſe of being-szichin Musket-Shot) the Currain RS about-420 Foot, and the
Frone A K 280 Foor:. And for the Flank R Ky and Gorge R O, their proportion
commonly is as 6 to 7: but the Angie KOR forming the Flank is about 40 gr. by
which the Proportion is neer as 5 to 6.
1
1
A Table
LI
Fortification, .
3
T
C
B
K.
V
R
IR
5.
1
F:
H
1
IN
N
M
A Table for 8. Baftions.
1
A
60
64
9
5
36
A Table for 12 Baftions.
Polygoms. i Angle of 15, Angie of
thie Polycon. the Baftion.
Degrees, Degrees .
4 Tetragon
45
30.
5 Pentagon
54 34
6 Hexagon
37
7 Heptagon
39
8 Octagon 67
41
9 Enucagon 70
42
10 Decagon
72
43
11 Undecagon 73 77
4475
12 Dodecagon 75
The Baſtion Angle is here found
by adding 15 d. to che Polygon
Angle, and take the thereof: So
che Baſtion Angle will be an An-
gle of 90 deg. in a Dedecagon.
of
Angle of Angle of
Polygons. the Polygons the Baſtions
Degrees. Degrees.
4 Terragon 45
30
5 Pentagon
54
6 Hexagon
60
40
7 Heptagon
8 Oeagon
45
The Baſtion Angle is here found
by caking the of the Polygon
i
Angle: So the Baſtion Angle will
be an Angle of go degr. in the
OEtagon. And no more muſt the
Baſtion Angle be in any Polygon.
I
64
42
7
1
67
45
1
?
.
4
Fortification.
Of the works that are in or about Forts of moſt
Importance.
41
Catal
1
SA
I.
H
VIUNTITWINTHRIINUNLIMITATITANJE
ID
HOINNTNIMIT
Ufufomunid
AMIDINI
QUIMITOLITICULATE
B
A
TUNNINHUUMIA Wimno
HWA
TINDINI
HE
AB the Breadth or Walk on the Rampire
40 Feer.
BC the Breadth of the Parapet of the Rampire, with the Fauſſe-
bray, and Parapet chercof, cach 20 Foot ; in all
a
}
DE the Breadth of the Moat, Ditch, or Trench
-120
E F the Coridon,
or Covere-way of the Counterſcarp
F G the Argin or Paraper thercof, being
-go or 6o
H chc Ravelines; I the Half-moons, with their Parapets---
20
20
T
!
}
There may be ſometimes an occaſion in Forts to raiſe Mounts, Cavaliers, Plat-
formis, or Batteries, to command all the other Works, and to view the Country
about; which may be raiſed upon the Baſtions, if yon have room withal to make uſe
of the Flanks: Ocherwiſe let them be raiſed on the Curtains, a little within the
Rampire, ſo that you may have room left for the Walk.
>
.
TO
L
j
9
1
[]
r
Fortification.
5
+
To Draw the Platform of a Fort, beginning with the capital (or Hend)
Line ; And alſo to draw the Horn-works.
1
P
1
OR
L
T
con
1
.
1
1
.
12
1
1
llimine
H
E
M
Kos
1
4
D
5
um
beli
!
Et thc Fort bean Hexagon, that is, of fix Baſtions or Bulwarks. Firſt draw the
Line A B, and upon X deſcribe an Arch of a Circle, as BDC, whereon lay
down half the Polygon-Angle, which in che former Table you ſhall find to be
6o deg, as from B unto D, and theace to C; and draw A Cand AD. Now the
part of the half Polygon Angle is BG and C F; then draw the Obſcure Line A Fand
AG. Next you ſhall make choice of the Capical Line, of any ſufficient length,
which let be A E, and from E draw a Line parallel to AB, as EH, continued ;
and upon the Point E, as a Center, deſcribe K l, making it an Angle of 40 deg, as
KEI; ſo ſhall EI cut out the Fronc in L, as A L: So from L lec fall the perpendi-
cular LM, which ſhall be che Flank; and MN the Curtain ſhall be as formerly che
whole length of the Front, and a half more. For the reſt of the Work, you muſt
proceed as formerly.
Pop
1
1
1
6
Fortification.
1
1
For the Horn-works.
you
and
Y Qu muſt continue the "Flárikers' M L and. No anco- P and Q; then take the
longeſt Line of Defence AN, and lay it thereon from the ſhoulders at L and
O, unto P and Q; drawing the Line PĆ, dividing it into three equal parts;
from thoſe parts 1 and 2, draw parallels unto PL and Qo: alſo from thoſe Points
P and Qdraw parallels to the Fronts A L apd. B O, thoſe will cut the former Pa-
rallels in R, S, T, and V, which Inçerfections will limit the Fronts, Flanks, and
Curtains, as you may caſily perccivc; unto which you muſt make che Rampire, Para-
pet, fc. as in the former Works.
/
*
$
Now follow two Tables; the one for 12 Bastions, and the other for Forts
of 8 Baftions"Whereby you may trace out any Fort by help of a Line of
Equal Partss which fall divide the side of the Outer Polygon
into 10000 parts.
The Firſt Table for 12 Sides.
Number of Sides 4
Ś
6
7
8
9
10
II
1 2
)
The Side of the
10000 10000 10000 10000 10000 10000 10000 10000 10000
Outer Polygon
The Capital Line 2428 2592 2756 2926 3086 3136 3148 3180, 3204
The Gorge
1088 1264' 1378 1470 1538 1640 1722 1793 1842
The Front 2914 1953 2986 -3014 3024 3054 3070 3083 3094
The Flank
970 1128 1246' 1360 1516 15261 1536 1542 1546
1
The Second Table, for 8 Sides, whoſe Baftion Angle
then fall make 90 Degrees.
kom
Number of-Sides
4*
** 5
6
7
oo
The Sidc of che
Outer Polygon
10000 10000 10000
10000 10000
"
The Capital Line
2778
The Gorge
2396 2498 2602 2695
T120
1327, 1480
1592
1698
2939 2959 297.5 2987
940
II13.11242 1:1342)' 1423
'
The Front
The Flank
2914
:
The uſe of theſe Tables.
I
Et it be required to draw the Proporcional Dimenſion of a Regular. Fort of
6 Sides: As for Example, in the fourth Figure, whoſe Side A B muſt be divi-
ded into 10o equal parts, and cach part ſuppoſed to be ſubdivided into 10 parts ; ſo
have you 1000 parts, which shall ſuffice. Now proceeding according to former Di-
rections, until you come to make choice of your Capical Line, you ſhall here find
in the ſecond Table, which is beſt for the purpoſe, under the Figure 6, and right
againſt the word Capital in the firſt Column, 2602, but 260 will ſerve : Take the
ſame
mit
1
Fortification. "
+
R
fame from the Scale of Equal parts, and lay it from the Baſtion Point at A, and it
falls in the Point E, which will be the Center of the Baſtion. From thence you may
lay down the Gorge Line out of the Table, which is 148 unto M: So will che Front
Al be a96, and the Flank ML. The Curtain, being once and a half che length
of the Front, will be MN 444. Thus you may do for any of the reſt. Theſe Ta-
bles are uſeful for Irregular Forts; But firſt I will ſhew you the Height, Broadch,
and Scarpings of the Rampire, Parapec, Ditch; for of theſe Sconces, as they are
repreſented in the Profile, or Section, as followch.
H
M
7
10 4/7
7.
E F G
7.
?
I
6
2.3..
Terra plana
6.30
3
К во
A
6
"6
པརབ་
6
20
B
ន
1
The Breadth of the Rampire may be 24, 30, or 40 Foor; blir here A B is büca
The Inward Scarp AC-
The Height of the Rampire CD
· The Breadth of the Walk of the Rampire D E
Ig
The Breadth of the Bank or Foot-pace of the Parapet EF
and the Height of the ſame Foot-pace.
The Inward Scarp of the Parapet FG-
The Inward Height of the Parapec GH
The Brcadth of the Parapet at the Foot FI__
The oucward Scarp of thc Rampire BK
Tlie Inward Scarp of the Parapet I L-
The Outward Height of the Parapet L M-
The Thickneſs of the Parapet ac the
top
MN-
The Brim of the Ditch BOL
The Breadth of the Ditch at the top OP.
33
The Scarp of the Dicchi o Q
The Depch of the Ditch on-
The Breadth of the Ditch ac the Bottom RS
The Profile or Section of A Fort with a Fauſſe-Bray and Counterſcarp;
alfo Subtrenched.
1
6
NAILINIIBEIIIII
D32
2:M
6
3 6
20 K
1
B to
:6
10
30
6I
wwrivom
A Terka plana
6 15 27 JH 150
To
CD is che Fauffc-bray, and D M his Parapet: EFGH is the Suberench, and
I K che Coridor, or Coverc-way. Laſtly, KL is the Argin or Paraper of the
(9)
Counter-
32
$
1
1
8
Fortification.
1
Counterſcarp.' Nore, That the Height of the Rampire A Bought.co be raiſed is or
18.Fooe above the Terra Plana, although here it is bác 12 Foor, which is ſomewhac
too low to command the Trencil of Dicch: Butlif the Trench bewade broader, then
is will command the bottomh thereof,
.
of Irregular Fortification.
IN
N the ſeventh Figure following let A B CD E be an Irregular Fort, containing
5 Baſtions, or 'Bulwarks. Firſt we will make a Baſtion on che Angle ac Az
which do thus : Divide the Polygon Angle in half with the Line A F, and draw:
the Baſtion-Angle as formerly, beings of the Polygon-Angle, as AH, and A I con-
cinued, being the Sides wherсon the Fronts muſt be laid down.
Now apon ſome ſpare Paper you ſhall make the half Polygon-Angle G AF, as
you may ſee underneath this ſeventh Figure, as LKM : Then make choice of the Ca
pital Line, as before ; let it be of any convenient length (larger than you think your
Baſtion will be in the ſeventh Figure) as underncach KN, and from N the Center of
the Gorge draw a Parallel to KL, continued to O, as NP; and ſo proceeding as be-
fore, you ſhall find the point of the ſecond Baſtion at O: So have you the Propor-
tion of your Baſtions, whereby you may gain thoſe in the ſeventh Figure.
Now co reduce it from the Baſtion Poinţat Az you muſt take. A B che fhorrect
Gde, and lay it from O unto Q and from Q draw a Lino parallel to the Capital
Line KN continued as Q R. Laſtly, drawing a Line from N to C, it ſhall cit QR
in the Point S; To is Qs the length of the Capical Line fouglıç for, which muſt be
faid down on the ſeventh Figure from A unto T; fo is I the Center of the Gorge
Then for the Front take KV, and lay it on the Capital Line fram K to W:fo.a Ru-
ter being laid from W to o, it ſhall cut the Line QR in the Point at X; ſo is
Q X the
length of the Front, to be laid down in the ſevent, Figurç from. A unto
and 1. Thus ſhall you finiſh your Baltion, when you have let fallen . your
Hanks perpendicular on the ends of your Gurrains as you ſeç. The like Method
you are to obſerve for the other Baltions,
And when you have finiſhed your Fors, 'yor mult.obferve whether your; Çurraiti
Enes (that is
, from the Centers of the Gorge) be parallel to the Qurward Sides
& B, o c. which if they are not, you muſt correct them; and, by your Judgment,
Hy help of the Lincs of Defence, you may as you ſcc occafior widen the Necks: of
She Gorges, and alſo the Baltion Angles; but not above go Degreas: And fo lectie
Flankers be as nicer proporcional as the Rules (or Oculan Demonſtration dirotech).
which commonly the Gorge Line to the Flank bcare proportion as 7 to 6
Much more could I write of Irregular Fortification: bac my purpoſe at this time
is but to make a ſmall Treaciſeor-an Epicomechereof.
Tiba ง
1
}
1
I
YVYYTTYY
H
A R K L A
40
Fortification
The ſeventh Fizure, of an Irregular Fort, containing 5 Baſtions ; being the
Platform of the Royal Fort ſometimes on St. Michací's Hill, on the North
wejt side of the City of Briſtol.
E
D
M
Z
1
i
<
DIM
N.
A
Y
dows
I
R
+
1
P
(ga)
lo
IO
Fortification.
To make a Scale for Fortification by the
Tables.
А
D
Equall
parts
o
986
75
100
Ont Pot F
His may be performed Geometrically by obſerving
che former Inſtructions, whereby you may gain
che length of every Line: but it will be ſooner done,
and more caſie, by thcſc Tables following.
१०
M
80
1
in Pol:
L
K
70
I
Firf Table, for 12 Sides.
H
60
10
Numb.or Sidesi 4
S
7
8
9
10
1
IZ
1
4
50
ИМ
40
Erant
P
Semi.Out.Pal. 1000 1000100010001000 1000 1000 10001000
Semi. Inn.Pol. 661 700 7311 756 777 795 810 823 834
The Front 412 3471 299 261 232 209 193 174 160
The Gorge 158 151 1411 132 133 115 108 10196
The Flank
133 127) 1191 111 1031 971 90 PS
80
30
20
Gorg
Second Table for & Sides.
Flar
12
Nnmbct of sides
1
S
7
4
6 7 8 9 272
90
4
Semi. Outer Polyg. 1000 1000 1000 1000 1000
Semi. loner Polyg. 96117061740 766 787
The Front
412 346.296 259 229
158 156 1481.5381 130
The Flagler
133 1311,121 116 109
The Scales
of Forti -
fication.
80
The Gorge
5
70
too
Out Pol:
60.
6.
90
7
50
30
+
Inpol
70
HO
9
10
4
5 6 7 8
11
12
30
40
Eront
IP
Make your-Scale afe" a ſufficient'length, that may
hold both Liries, che one for 12 Sides, and the other
for 8. Make choice wichin the brcadch of the Scals,
between the Border's, any fufficient breadth, as
CD; froin whence draw_Parallels to the Sides,
and djwide. CD into 30 Equaſ parcs, and begin
: your Account from C wich 45 : lo ſhall the end
; a D be 75 degr.
Now make choice of the length of the outer Po-
lygon, which here I make three Inches apped di-
vide á Line by the Side thereof, cquial. chcreto,
in 100 cqual parts, and ſuppoſe each part into 4o;
ſo have you iooo paros agreeablçı to: the Tables
The next thing is co draw Parallels to CE, ac="?
cording to the Polygon halt Ingtes, : as you may
ſee in the Tables under che Pencigog. Fort, being
che ſecond Figure: So from the Scale CĐ you
have for the half Angle of a Pentagon 54 Degrees,
whereby you may dray the Psabagoir parallel FG;
and ſo in the lower. Stále of 8 Baſtions. In the
like manner you may do for all the reſt.
Now to draw the croſs Lines for the Semidiame-
ters of the Ingér Polygons, as alſo the Lines for the
Froncs, Gorges, and Flanks, you ſhall work chus.
Firſt, you muſt noce, That the Semidiameter of the
Outer Dolygon is the Radius of whole Line of 1000
equal
20
30
2o
Gorg
10
Flank
10
10
N
G
5 6 7 8
4
Chords
Poligons
Sides
B
+
+
+
Fortification.
/
1
។
equal parts, and that is drawn at Right Angles, or a croſs at F: Buc for che Semi-
diameter of the Inner Polygon, look in the Tablc of 12 Sides in the ſecond Ccm
lumn under 4, you have 661. Take the ſame Number off your Scale of Equal
Parts, and lay it from Eco H: Then in che third Column under 5 you have
700 parts ; lay the ſame down from G to I, and make chcre a prick or point :
Do the like for the Hexagon and Heptagon, as at I and K; proceeding along with
all the reſt, unto the Dodecagon. And laſtly, draw a Line through all thoſe Points:
So have you the Arch Line H M for the Scmidiameters of the Inner Polygon. In
the ſame manner work for the Front, Gorge, and Flank Lines. The scale of 8
Sides is the ſame Method.
I have allo inſerted on the left ſide of the Scale a Linc of Chords, whoſe Radius .
(or Arch of 60 Degrees) is chrec Inches ; and on the left ſide, a Line for the sides
of the Polygons. The Hex!gon, or ſix Sides, is equal to the Radius: And for
che Tetragon, or four sides, it is cqual to che Chord of 90 Degrces. So having
defcribed a whole Circle with the Chord of 60 Degrecs, you thall find that if
you cake from the Cen:er Nunto 5; it ſhall divide the Circle inco s.equal parts, fok-
drawing the Figure of a Pentagon, which in the ſecond Figure of a Pentagon Fort
will reach from A to B, and lo co X, Y, Z, and A. Now D A in the ſame le-
cond Figure you ſhall find to be che Semidiameter of the Oucer Polygon, which
in the Scale is a F; and taking GI off che Scale for 12 Baſtions, or GO on the
Scale of 8 Baſtions, it will give the Semidiameter of the Inner Polygon, as DE
in the ſame Figure. So likewile G P on the Scale will be equal to the Frons
A K in the Pentagin För:. The like you may underſtand for laying down the
Gorge and the Flank. And for the Curtain, as before, you muſt make R$i; che
length of the Front A.K.
This Scale will alſo be of good llle in Irregular Forcification. As for Inſtancea
In the Irregular Fort, the ſeventh Figure, you thall find the halk Baſtion Angle
GAF to be 58 Degrees, which falls on the Scale between the pentagon and
Hexagon, from whence you may draw a Baltion on ſome ſpare, place, and from
thence proportionate the ſame unto the outer ſide of the ſeventi Figure A B. The
reft I lcave to your own practice.
!
-
A
+
.
Hong
:
l
--
*
112
Fortification.
::
.
How to Fortifie a long Curtain with Bulwarks, or a ſtrait /
Town wall.
.:
.
1
:
:
5.
3
%.
72
H
3
K
В,
A
D.
420
420
10
,
Er the Curtain be A B. Firſt cake 200 Foot from th: Scale of Fortification,
accounting to for 100; and lay the ſame from A unto C, and from C unco
D: So ſhall A D be the breadth of the Neck of the Gorge: and upon the Polit C
creet a Perpendicular, as CF. Then take 420 Foot off your Scale, and layıthe
fame from D ro E, which ſhall be the length of the Curtain. Next you muſt take
720 Foot, and lay the ſame from E, to cut off the Capital Line at F; fo fhall EF
be the longeſt Line of Defence, and CF the Capital Line; which Line F muſt
be laid down from Cunto G: and draw G F for the ſhorteft Line of Defence
Laſtly, upon D erect a Perpendicular, which will cut the fame Line in H ; ſo have
you
DH for che Flank, and HF the Front. Thus have you finished half the Ba-
ftion, from whence you may tranſport all the reſt of the Baſtions, were they ever fa
many.
Nore, That theſe Baſtions muſt not exceed 720 Foot, that ſo it may not be with-
out Musket-ſhot. Bur if you will defend the Front with Cannon, you may make
the Line of Defence almoſt twice ſo much. The like for the Curcain, which inay
be 8oo Foor; and in the middle of the Curtain you may make a Spur, or Point of
a Baſtion, as at K, which will be neceſſary for Musket-ſhor, beſide the Cannon;
which in the Linc drawn about the City of Bridal I have ſeen many of them.
:
-
.
How
to
Role faire. José too
:
ullos hos
two
noworly
of cua dagoenote galben annyire a) a luat
qe as tirg frumger. Har ing you must enables
harro ou
3 or 4 times reals
bo oust of itt
T'lishiguien
would land allow .
o this will maks
amfish will acesta Chango y
voel
mi
top
may
1
Groc
24 houros
greatest
.:
::
i
Fortification
Theſe five following Pieces are taken out of Malthus ; The Proportions you may
find by the Scale, and the Rales before ſhered.
Tetragon
pentagory
Hexagon
ALAMASINI
ht
Heptagon
7
octagoni
-
FINIS.
2:
*
,
.
... ,
1
P
* *
1
FA
•!