m\GE EVALUATION
TtST TARGET (MT-3)
7
A
O
¥/
t^
f/j
1.0
I.I
*"IIIIM IM
S IM iiiii'z^
!^
IM
114
2.0
1.8
1.25
1.4
1.6
^
6" —
►
%
<^
/}.
4.
^
'c>l
e
/^
^# .1
'^s
%
^
^
O^A
Photographic
Sciences
Corporation
23 WEST MAtN STREET
WEBSTER, NY. 14580
(716) 872-4503
<" c^^
?
CIHM/ICMH
Microfiche
Series.
CIHM/ICMH
Collection de
microfiches.
Canadian Institute for Historical Microreproductions / Institut canadien de microreproductions historiques
'^
iOAT
I
Technical and Bibliographic Notes/Notes techniques et bibiiographiques
The Institute has attempted to obtain the best
original jopy available for filming. Features of this
copy which may be bibliographical / unique,
which may alter any of the images in the
reproduction, or which may significantly change
the usual method of filming, are checked below.
L'Instltut a microfilme le meilleur exemplaire
qu'il lui a ete possible de se procurer. Les details
OB cet exemplaice qui sont peut-^tre uniques du
point de vue bibliographique, qui peuvent modifier
une image reproduite. cu qui peuvent exiger une
modification dans la m^thcde normale de filmage
sont indiqu^s ci-dessous.
D
Coloured covers/
Couverture de couleur
□ Coloured pages/
Pages de couleur
□
Covers damaged/
Couverture endommagee
□ Pages damaged/
Pages erdommagees
I \ Covers restored and/or laminated/
I I Couverture restaur^e et/ou pelliculde
D
Pages restored and/or laminated/
Pages restaurees et/ou pelliculees
p Cover title missing/
L_ J Le titre de couverture manque
Pages discoloured, stained or foxed/
^lJ Pages d6colorees, tachet^es ou piquees
D
Coloured maps/
Cartes giographiques an couleur
□ Pages detached/
Pages detachees
D
Coloured ink (i.e. other than blue or black)/
Encre de couleur (i.e. autre oue bleue ou noire)
v/
Showthrough/
Transparence
n
Coloured plates and/or illustrations/
Planches et/ou illustrations en couleur
□ Quality of pri
Qualite in^ga
nt varies/
6gale de I'impression
n
n
Bound with other material/
Rali^ avsc d'autres documents
Tight binding may cause shadows or distortion
along interior margin/
Lareliure serree peut causer de I'ombre ou de la
distorsion lo long de la marge interieure
Blank leaves added during restoration may
appear within the text. Whenever possible, these
have been omitted from filming/
II se peut que certaines pages blanches ajout^es
lors dune restauration apparaissent dans la texte,
mais, lorsque cela dtait possible, ces pages n'ont
pas m film^es.
□ Includes supplementary
Comprend du materiel s
r~~] Only edition available/
n
material/
supplementaire
Seule Edition disponible
Pages wholly or ^iartially obscured by errata
slips, tissues, etc., have been refilmed to
ensure the best possib<e image/
Les pages totalement ou partiellement
obscurcies par un feuillet d'errata. une pelure.
etc.. cnt ^t6 film^es d nouveau de facon a
obtenir la meilleure <mage possible.
Q
Additional comments:/
Commentaires supplementaires:
Various pagings.
This item is filmed at the reduction ratio checked below/
Ce document est filmd au taux de reduction indiqud ci-dessous.
10X 14X 18X 22X
26X
30X
J—
J
J
12X
16X
20X
24X
28X
32X
Th« copy filmed here has been reproduced thanks
to the generosity of:
New Brunswicl( l\Auseum
Saint John
L'exempiaire fltm6 fut reproduit grdce it
g^nirositi de:
New Brunswick IVIuseum
Saint John
The images appearing here are ;he best quality
possible considering the condition and legibility
of the original copy and in keeping with the
filming contract specifications.
Original copies in printed paper covers are filmed
beginning with tha front cover and ending on
the last page with a printed or illustrated impres-
sion, or the back cover when appropriate. All
other original copies are filmed beginning on the
first page with a printed or illustrated impres-
sion, and ending on the last page with a printed
or illustrated iiy?pression.
The last recorded frame on each microfiche
shall contain the symbol ^^ (meaning "CON-
TINUED"), or the symbol y (meaning "END"),
whichever applies.
Maps, plates, charts, etc., may be filmed at
different reduction ratios. Those too large to be
entirely included in one exposure are filmed
beginning in the upper left hand corner, left to
right and top to Lottom, as many frames as
required. The following diagrams illustrate the
method:
Lea images suivantes ont 6x6 reproduites avec le
plus grand soin. compte tanu de la condition et
de la nettet6 de l'exempiaire film^, et an
conformity avec les conditions du contrat de
filmage.
Les sxemplaires originaux dont la couverture an
papier est iniprimde sont film^s an commenpant
par le premier plat at en terminant soit par la
derni^re page qui comporte une empreinta
d'imprnssion ou d'illustration, soit par le second
plat, salon le cas. Tous les autres sxemplaires
originaux sont filmds en commen^ant par la
premiere page qui cdmporte una empreinte
d'impression ou d'illustration et an terminant par
la dernidre page qui comporte une telle
empreirte.
Un des symboles suivants apparaitra &ur la
derniAre image de cheque microfiche, salon le
cas: le symbole — *> signifie "A SUIVRE", le
symbols V signifie "FIN ".
Les cartes, planches, tableaux, etc., peuvant §tre
filmte A des taux de rMuction diff^rents.
Lorsque le document est trop grand pour Atre
reproduit en un seul cliche, il est film6 A partir
de I'angle sup^rieur gauche, de gauche d droite,
et de haut en bas, en prenant le nombre
d''mages n^cessaire. Las diagrammes suivants
illustrent la m^thode.
1
2
3
1
2
3
4
5
6
I
♦
sng if
I? IB IS A ^ S ^ I^
OH
THEORETICAL AND PRACTICAL
I
c^
li.ir|jf^fji0j^
SUnVLVi ;
I
TO WHICH IS ADDEt
GIVEN IN THE WORIT,
With all the xNecessary tables.
Br
ALEXANDER MONRO,
LAND SURVEYOR.
PICTOU, N0VA-SC0TL\:
ftlnm hj Gcldert & Pattsrsoii, Eastern Chronicle Off!.?,
. Por the Aiitbor,
WDCCCXLIT,
the
I
DEDICATION.
TO THE
HON. AMOS EDWIN BOTSFORD,
Membeu ok Her Majesty's Legislative Council,
Memleu op the Board op Education,
Lieutenant-Coeonel,
&c. &c.
Si.1, —
Of tl,e .ncrits of tlie following Ti-eatiso it is not
for „,e to judge. That poiut „,u.t he left to the cleci i™ "f
.n.partK,l reader. I „,„j, however, be permitted to ex-
1 -.,n,y r,^re.thatitis„ot n.ore worthy of your patrt
. tc U to one ,0 whose .nhul the urgent necessity which ex-
L^l vl.;™ „ •'" "f '""" ""^-^ «o,ue„.ly s„.ges"a
.K-ut:,W ■'■":'"=" "'° '■'^"'■^"f ■ho public, and
■I , . r •,• '"■"'-•""^'"S^ in <""■ Courts of Law,-to one
o,,e lanuhar a„|ua,n.ance with the subjec: discussed, af!
1^,1 M "'",?"'"■'"""' "■"' "■" "■"'* "«<="• '^ "<" »lt"SC l,or
u u , y „, notae, and favourab.e accep.ance,-to one from
'Mioni „s numerous in.i.erfcctious are sure to meet n-ith
mos parental in.lu,ge,,ce.-and to one whose pa.",:.;'
coLua of the „b,cun.y, and want of personal or relative ia-
r
ir
DEOrCATIOW.
fluenre on the part of the author, it might otherwise bo ex-
posed,
Oihor considerations have likewise induced mo to solicit
the honour ofdedicating to yon this, my first attempt at au-
thorsh.]). I wa,^ e.vcecdingiy desirous to avail myself of this
opportunity puhliely to express my grateful sense of tho
courtesy, kindness, and attention, which you have so gene-
rously oxtMidcd towards the humble author of the following
work, ever since he had first the happiness of being intro-
duced to your notice. ]Jc assured, Honoured Sir, I am not
ungrateful. Your name will ever be associated in my
recollection with the most lively emotions of esteem and
respect.
lieside.s, the countenance and cncoura-emcnt which you
have uniformly extended to the industrious and enterprising
youth, and the interest which you have ever manifested in
tlie cause of Education, and in every movement in which tho
prosperity of British North America is involved, justly en-
title you to this public expression of grateful approval.
Hopuig- that tho work itself may not altogether disappoint
your exj)ectation,-hoping even that it may meet with some
degree of approbation,~.and praying that you may be Ion-
spared to enjoy the confidence and respectful esteem of thoso
wJio may be honoured with your acquaintancc,~an(l to wit-
»ess w.th delight tho rapid progress of intoHectual improve-
ment, unci tiie growing prosperity of your native country.
I remain, Sir,
With nmch esteem and respect,
Your Most Obedient,
And Very Humble Servant,
^ ,, ,. ALEXANDER MONRO.
liRv De \ erf«, N. B., October, 1844,
'"%
iso bo ex*
to solicit
ipt at au-
Llfofthis
sc of tho
so gone-
rullowing
ng intro-
I am not
d in my
eein and
lich you
jrprising
fosted in
'hich tlio
ustly en-
val.
sappoint
ith somo
be long
ofthoso
[1 to wil-
nprovo-
Lintry.
TOTHKL.vxD«:uvi.:vousormt.T,.sn.vouru
AMERICA.
GE.VTLnMEX,
.ho following TrJle'rih' "'"''• ''''"''"^'" ""' ' '*'"'"•
ence, !,„v.„vcr, I cSf" r, , '"""'• '''" ™'"-<-M.cri-
^<'" very we,, l r,^' 7,:7 r'"'', "^ ""'■' "''■'■'' '— •
work, upo„ „„,, .«1,:\ '",;'';«'^ "''»l"«' <'-«n,i„,
Lm,l Surveying _„;,",?, "!" l""-i>o^*'=« <"' Colonial
--.raein.; of,),,-; „';,'"' '" ""■■ '""""'"• '"^'■"S. "■"'i
-nts of i,,o p,,.,;, ,I:-'«™; of I.and.,_,„o £pr.n.
-tood by,,,,, Colo,, i, V r«"""^'^' ""'■"'■'"'>■'"'"' '"•'■'"-
investigation of ZLuorT-''" ' '""'" B'""" ■'>»
•he reaso,,,, of.,, ,°':';'''»'t'l^v tl"M.or,>.al„f „-,,,„,,
'- -:^iiy «n,,c,. o" ' f , "'" 'n ,'","•' "'■ "■" "-■'^ '"»^
worl< a„ ,1,0 TalivT'n " *"' ='"'' '"' "'<' <■"<' o'' tho
' have ,ho H'o„;:";.';t; '^" "'■'"'■"' '"'^I'"-'-
^*cntIonicn,
Vour Obodirnc,
And Very Humble Servant.
JBayDeVcrt. Nn ,, /^^^^'-'^•^™^i^ -MONRO,
'-'t.,, ,>, 1^,, October, 16 ij.
I
til
o
I
TlIR xnn
the nieasin
n('f'0|)£arIoi
tlir extcnsi
of hndic.?.
tiulc, distill
Its of baflio-
Respcotii
liavc Ijooii (
Kjryptians t
inotry. Ti
their landm
tlioir houiK
origin of L
Achiilo.s ':
that the P:.r.
and tlie eart
ledge of tlie
^'"rodotu:
to he the SI
war up 0)1 ]
reported to ]
tlivided the
lotinent, for
Aristotle h
ii
PHE/'ACK.
— "" »
T
'"-• v„.svrn.,r.l ofl,,c K,„-th, or cf Land. .„ i,, ,,,,i„,,^:^
Kl ;• " "'',''''"' '" "'" ""='•'■■''"■<""'=•" of tl.n ,na...i.
.1.S of l,n,l,c.s on or „c,il' llic earth's si.rfa.'o
Rc.,pccli„g ,1,0 oris^, of this „,,.:„„„, ,-„;,„„ i,,,.
have heen entertained. The ancients n^ree in .min ■ t t e
J.,,-p.,an, the cre<„t of heln. .he earhe;; .^^::::ZZ
lo.., , he annual overflowings „f the Nih, ,Iis,n,Ci„..
on-h,n.h„arl„ren.,ere.nt necessary fre^nentlv to '^t
X^^t::^i:^-r~- "^- -^ "•---
tlnt'the'rlT";.'"'' " M..hen,ati,.ian ef Greece, inf„r,„s „,,
m tiieeaith. Moses .sevensaid to hnvcac,,„ire,l „ l<„,„v-
l«lg of the se,e„ee, ,vhen he reside,! a. the l^vptia t 't
enor;;' '"'"'"'"' "'" ■^''" "f S' .m.,. This King is
1 fide, , '■"■'; "'"•■'■■*""'' ''■^'J'l" •'-'' C-'.-.!^, and ,„ have
''>"lod the hind among his suhjeets, tjivin- to e.ch tn •,!
.Ar,stotIe has attributed its origin to the Egyptian priests.
viii
W
Hi
PBRFACC.
who nw„, ,co>u,.oa f,.„,„ .he .„.,„, h.U .,.„„ , ,.„„„
TJjc antiquity of this soionco nCToviU sn-.in ..: »
r« uie o..s.^ of Land and Marino Sui vovin- ThMt. ,i-
nil tie (nlcuhtinn. ;,. V • • '^"'^•'^'"o- -^ ncy direct
\nude fhn . I >^'^Viivat.on and Astronomy. Thcv
ocean, and the Minnr t^ . ^'^^■^''■^" ^''^ P'-^^'^lcs.
rho .ul.jeot of ,ho p,-o,,o„t tre..i.e i, /.™,; ,%.„e,„-„.
i'ns constitutes on«> of thn m«.-. • ^- ^t^ym^.
branches of tlm M !l '^nportant and useful
dill of ti. f ^^^.^'^'"f ^'«^- 'i^ho Surveyor, in the dis-
1 To ^^'•"^^•^^''^"^^' ^'"ti^^> ^i'-ects hi. attention '
Ji- 10 the tracing and measurement of lines:
m1 ''V'", '''I'm'"'"" "'■'" ''»^""l'"i« "Pon« plan or
.hr';„, ":;'"", ° """ "'r' "■"■" '■^^"'""''" "«"■-""
™,e rf hi ' '■'='"•■"■'"'••'<' "''J-^"^ «i'lm.<'.' near ,ho
h^!h. , "P"'-" »"».-to ...csun, their distances a,.I
;usht..-to a=.crta,„ .ho van,,;ion of ,ho compa.,,,-.ho la-
mu a,,,, lo„.i.,.d„ „f p„.,i„,„,^^ __,^^, • -
- o loh,...,.o „„, o,.lj' the boumlarie, c • an onlin,.,-y ^ :;'
but also of coast, and harhour.,,_„r .o ?ivc a corroc. rlDro*
=o„.a„„„ ,.,-.hc i„„,„ali.ic., of the earth'. JC "'"""
Anions the ancicn..,. Arohin,edc,and Tan.agli, inade coi-
pnirACK.
nidcral)!(
ix
progress in evolving the pritirlnlea bv whi^^K
are dct*innincd Tho r *" '"'P'''* oy «nich areas
.00 much ;;: iio::,,'"' ,. r;z:^> - - f 7-. - -een
1. St, I more just ,„ ii, „pp|ica,i„„ ,„ ,, ,j,:,. /'""''
American Colonies W„<,d.L„„d Survey ,!» nftl', i "'
For th,s re„so„ „.,„ o/'^: „: Lef;::!: d"; ""'"• ^•
whh which I have „,e,, arcs,, s„i.ed X ,lhf r^'lh?
vmces. I he necessity f„r so.ne „orI< on thesninec. ada r;
to our condition, appear, to mo to bo trrea. 2: '
Fron, the frec,uency with which disp^e , t, t L'^'f'"'-
are introduced into our Courts of Lav fcr I r- ""''"''
«ic.ent that son.o .-.e.uaintance wi^"l!:d t: ;;• rj:!^:
tlio i.gal profession. Scarcely less necessary i, an ,c
ri::rr:' ;,;:;:— 'rrir^r '■-----
o.,,ht to have son,e Unowi^e 7 :t.':Zr7 :tX
tie aequanuance with Land Surveying would hav. ,
.nany an individual from minors liticaZ I ^"^
only his property has been sauntfd ,:,;',:""" ""i
■mnd disturbcl, and strife and contonti „ t'pr d thrCh !:
commun tv. Im')rcs=!Pr1 xviti. . i ' l"^eaa tnrough the
<Icrlako Ihofollowtng treatise, which I now offer to the ac-
PREFACE.
cq,l„„«of„„,„,]„l.,o,«jn,Mic. H„„-n,rIhavc.uc..cc,ln,l
.t .» no, fur ,„e t,. ,lot=mn„o. Ti,o ,leci.i„„ i, u.ft , , , .l
u,l,,„e„. .,f ,1.0 ,li»c-,i„,;..a.i„, .cder. I l,ave on , o
mark ,„ condu.Mun, .l,.t I make „o „reten.i„,„ to our ,V, r
Sreat uo-ura-y of .„,„p„,iti„,, „■ , ,,,,, ,„e,3eUe , • fe
-lonug u.y ,ncauu,s intolligil,!., r have arrived at the .u -
not of my ainijition on this point.
^VwCrun3lvick, Oetoher, !811.
C'Oi\TEr>TS,
Befinial Fractions, . . _
Kxtraction of the Square Root,
GKOMETRY^Definitions,
Contractions, - _ _ _
Explanation of Signs, - . _
Geometrical Problems,
Concerning Scales of Equal Parts,
Logarithms, - . _ _'
Trigonometry, _ . _
Mensuration of Heights and Distances,
l^ANo SuRvEYmG-Instrument employed,
L-sc of the Chain, . . . _ '
Circiimferentor, - . .
Theodolite, - . .
S3 r*rotractor,
Field Book, --..""
Variation of the Compass,
Running of Lines, - _ _
Mensuration of Land, - _
Division of Land, - ,
Location of Land, - _ _
ApPENDix-Demonstration of Problen.s, '
1 romiscuous Probleins,
Levelling, - - .
33
33
PAGE.
1
9
11
16
17
20
29
31
452
50
54
61
66
69
70
71
74
85
98
117
128
IS!)
153
17;>
4
Xll
M
CONTEXTg.
PXrrZ.
msc>:r.r.AJ.Eotrs^Re-e8tablish^cntoflost boundaries, 176
I- acts concerning Magnetiiim, . . . ,--
Meridian Lines,
Concluding Hemarks,
179
180
I:
itiaries, 176
- 177
179
- ISO
DECIMAL FRACTIONS.
IHE TERM FractioiV, literally denotes something broken
To form a distinct conception of the nature of fraction's
employed in calculations, let the Student suppose any object
or quantity broken, or divided, into several equal parts. Any
number of these parts, considered in their relation to tho
whole objector quantity, constitutes a fraction.
A fraction is expressed by two numbers, placed the one
above the other, with a line between them, thus: — .
5
The figure below the line (5), called the denominator, ex-
presses the number of equal parts into which any obje;'t or
quantity is supposed to be livided; and the figure above the
line (3), called the numerator, specifies the number of these
parts which the fraction represents.
Decimal Fractions are such as he . e for their dunomin.i-
tors, 10, or some multiple of 10, that is, 10 multiplied into
itself a certain number of times, as 100, 1,000, 10,000.. ix.c.
Expressed in the common form, they appear thus: — ~
1 000 ^^' wecimal form, the denominator, btini? etisi'
— r-
CI
DECIMAL FRACTIONS,
ly ascertained, is omitted; and its pKace is supplied hy a dot
or decimal point, (.) prefixed to the numerator, thus: .3,
.15, .261, &c.
To ascertain the denominator of a decimal fraction, it is
only necessary to write doAvn as many cyphers as there are
iigures in the fraction; and then to place the figure 1 before
them.
Cyphers on the riglit hand of decimal fractions, do not
ailect their value; but every cypher on the left hand dimi-
nishes their value tenfold. *
The value of figures in decimals as in whole numbers is
determined by their position. The following table, in which
fhe figures on the left hand of the decimal point are Avholc
numbers, and those on the right are decimals, will illustrate
the influence of position in determining their value: —
Integers
Deci
mals
8
1
3 4
1^
4
6 .
4
1
^*0
3
5
7
X
H H
ffi
H
g
H
ffi
H
»^
X
Is:
o
3
5
ST
housand:
ens of T
5
o
57
2
2
Cm
c
c
2
O
c
-T" \J*
J.
r^
cc
c
^
•-»
o
\f
7J
p
H^
<ri
&.
X
O-d
—
!Z
zz^
^'
•11
or.
S
"■•
'Jk
The notation and numeration of decimals will be obvious
from the following examples: —
4. 7 signifies four, and seven tenth parts.
47 " four tenth parts, and seven hundredth parts,
or 47 hundredth parts.
• 047 " fom- lumdredth parts, and seven thousandth
])arts. or 47 thousandth parts.
4.07 " four, siufl seven hundredth i>urts.
4.O07 '• i'our and seven thousandth parts.
1 by a dot
thus: .3,
tion, it is
tliere are
3 1 before
s, do not
md diiTii-
iiiribcrs is
in which
ire Avholc
illustrate
Z^ECIMAL TRACTIONS (
ADDITION.
RULE.
Place the figure? dh-ectly underneath'those of the same va-
lue, whether they be mixed numbers, or pure decimals, pay-
ini]f particular attention to the separating points. These
should always appear in a direct line, one under another.
Then add as in whole numbers.
EXAMPLES.
1. Add 2.81, 5.50, 1.6, 4.334, 6.3431, together.
2.81
5.50
1.6
4.334
6.2431
*&ns. 20.4871.
7
2.
Add4. 28,3. 2187, .0024,342.501, .223, and 1 .2324101
5
together.
X
)^
4.28
*— •
3.2187
r.
n02-4
H>
3.
842.501
tr
7j
%
.223
c
i
1.2324101
'Mns. 351.4375101.
obvious
th })arts,
)usandth
SUBTRACTION.
RtTLE.
Place the figures as directed in Addition, then deduct as
in wJiole numbers.
EXAMPLES*
! From 28.4 take 24.35.
28.4
24.35
*^ns, 4.05.
DECIMAL FRACTIONS,
2. From 70,38 take .829.
70.38
.829
*fins. 69.551.
MULTIPLICATION.
KULE.
m^frtn h. T ."''; P""" "^ "^ "^^"y PJ^<^^« for <^eci.
r toTh r Tf th "' " ''"^ are decimals in both Fao-
ductSt^ 1 T "'" ""' ^^ "^^"y %»»-^« ^" the pro.
prcfo- ' r^' "' '^' '"'^ '^^"^^^^^ «WJy the defect by
prehAing cyphers on the left hand. *
EXAMPLES.
1. Multiply 3.141592 by 52.7438.
3.141592
52.7438
25189736
9424776
12566368
21991144
6283184
15707960
^ns. 165.6995001296,
2. Multiply .15 by .3.
.15
t^ns. .045
DIVISION.
RULE.
Divide as in whole numbers, annexing cyphers to the di-
vidend when necessary, observing that the divisor and nl
tienc musi together contahi as many decimal figures "as are
M
numbers.
3 for deci-
both Fao-
i the pro-
defect by
DKCIMAL FRACTIONS. 5
tontained in the dividend. If at the conclusion of the work
the divisor and quotient do not contain as many decimal
figures as are contained in the dividend, the deficiency must
be supplied by prefixing cyphers to the quotient.
EXAMPLES.
1 . Divide 66 , 993548 by 27 . 4.
87. 4)66. 99354»(2. 44602 w3ns.
548
1219
1096
1233
1696
1375
1370
548
548
«. Divide .45695 by 12.5.
13.5).45695Qa{.0S6ii56»an».
375
819
750
695
635
700
625
^50
760
ft
m
f
'
W W ?
I
<.
f
® J^ECIMAL FRACTIONS.
REDUCTION.
To reduce a Vulgar Fraction to a Decimal of the same value.
RULE.
Annex cyphers to the numerator, and .livide hv the Ao
EXAMPLES.
8
1. Reduce — to a Decimal Fraction.
4
4)3.00
. 75 ^ns.
5
2. Reduce _ to a Decimal Fraction.
64
64)5. 00(. 078125. ^W5.
448
520
512
80, &c.
Every quantity may be considered as a fraction of a lar^
ger quantity of the sam. kind; as an inch is the 1 of afoot,
a pole or perch is 1 of a rood, or -i- of an acre! ^c, and
may be reduced to a decimal fraction by the preceding rule,
observmg that the given quantity is the numerator ff the
fractK>n and the number of that denomination contained in
the higher denomination is its denominator.
EXAMPLES.
1. Reduce 9 inches to the decimal of a foot
In this example 9 is the numerator, and 12,'the number of
inches in a foot, is the denominator; thus : -. Tho opera-
tion is as follows ;— ^^
1^)9.00
. 75 dnsi
M
DECIMAL fractions/ 7
2. Reduce 20 perches to the decimal of an acre.
In this example 20 is the numemtor, and 160, the number
of perches in an acre, is the denominator. Then
160)20. 0(. 125 Ans.
160
400
S20
800
800
When the given quantity is of different denominations,
reduce them to the lowest denomination for a numerator.
The number of the same denomination contained in the in-
teger will be the denominator. Then proceed as above.
KXAMPX.ES.
1. Reduce 1 rood 14 perches to the decimal of an acre.
r. p.
1 14
40
rr,, , ^, H perches, Numerator.
Then 160)54.0(.3375 w2n».
48
600
480
1200
1120
800
800
2. Reduce 21 min. 54 sec. to the decimal of ft dejiree.
21' 54"
60
60
3600) 1 31 4. 0(. 365 .^n*:
1080
13M Numerator. 60
60
%wQ JDtuoiiiinator*
23400
21600
18000
18000
3
!
DECIMAL PRACTIOKS,
To determine the value of a Decimal.
KVLZ.
Multiply the decimal by the numhpr «<•
inferior denomination contaTned r^h! /'"' '^ ^^' "^^*
in the product as many pllce' for d '"T' ^^'"^'"^ ««*
hand, as tho given decfm^rc J«ts o ".t' '^ *'' ^'^'^
gures on the left hand of thp .W , • ^ ^^"'^ «>• fi"
ger number, xvhile the fi^ur.?'T ?"'"* ^'" ^« «» '"de-
cimals. Then muldnfv Z^^ 'f ' ^'^"^ ^^'" ^^ de-
parts c.mtained in tKLinl?"'' '^ ^^^ ""-'^er of
off as bef,re. Pro e d" h tuUtt^ ^"^' "^^^
tlenomination. ^ " '' ^^^"^^^ to the lowest
EXAMPLES.
i What is the value of .6 of
an acre?
• 6
4
r. p.
^ins. 2 16.
2.4
40
16.0
«. What is the value of. 175 of a Pound?
9, d.
*^ns, s 6
-175
20
3.500
12
6.000
«• What is the value of .« of a degree?
.42
60
25.20
60
vJA*. 25' 13'
13.00
^f the next
pointing off
J the right
gure or fi-
56 an inte-
viH be de-
lumber of
and point
the lowest
THE EXTRACTION OF THE SQUARE ROOT.
The Square Root of any number is the quantity or num-
ber, which, when squared or mulfiplied by itself, will yield
the given number as its product. Thu.s, 4 is the square
root of 16, as 4 squared or multiplied by itself will yield 16
aa it3 product. 16 is also the square of 4.
To Extract the Square Root.
RULE,
Point the given number into periods of two figures each,
beginning at the units place; then find the {greatest number
the square of which shall be equal to or les than the first
period, or the quantity before the first point towards the ^eft
band. Place that number in the quotient. Write the square
of that number under the first period, and subtract. To the
remainder bring down the second period, and call the whole
quantity the resolvend. On the left hand of the resolvend
write the double of the figure placed in the quotient,
after the manner of a divisor. Enquire how often this
divisor is contained in the resolvend, omitting the figure in
the units place of the resolvend. Write that number in the
quotient and also on the right hand of the divisor. Multiply
this divisor by the figure last placed in the quotient, and sub-
tract the product from the resolvend. To the remainder
bring down the third period for a new resolvend. To the
last divisor add the figure last placed in the quotient and
write the sum on the left hand of the resolvend. Then pro-
ceed as before until all the periods are brought down. The
quotient will be the square root required.
iVofe.— When there is a remainder at the termination of
the process after the last period has been brought down the
operation may i,e continued at pleasure by annexing periods
of cyphers for the formation of new resolvends: remember-
ing always that all Inu figures placed in the quotient after
the annexation of the first period of cyphers, are decimals.
10
■ if
THE EXTRACTION OF THE
EXAMPLES.
i' VVhat is the square root of the 5
S«ll/ABE HOOT.
square numl>cr i(H5 ^
20.35 (45 ^ns.
16
35)425
425
2. What is the
square root of 22071 204.?
22.07.12.04 (4608 »^,w.
10
86)607
6 516
929)!)II>
9 8361
9388) 75104
75104
^Vhat is the square root of 180000000?
1.80.00.00.00 (13416 ^rw.
23) 80
8 69
264)1100
4 1056
2681) 4400
I 2681
26826)1 71 900~
160956
10944
'"■^
GEOMETRY.
DEFINITIONS,
1. Geometry is that science which treats of the proper-
ties and rehitions of inagnitu(le.s.
2. A Point is that Avhich has posi^ion, but not inafrnifiule.
3. A Line id that which 1ms lengtJi, without brcadtu or
thickness.
^' B- — The extremities of a line are points.
4. A Straight or Right Line is the shortest line which
can be drawn betweeen two points,
5. Every line which is neither strai<rht nor composed of
«trai<,'ht lines is a Curve Line; as A B. (Fig. ].)
6. A Si'PERFiciEs or Surface is that which has lengnh
and breadth, without thickness; as A B C D. ( Fig. i.)
". Convexity, when applied in reference to a curve line,
•lenotes its exterior or outward part; as A B C; and Con-
cavity, its interior or inner part; as D E F. {Fie:, 3.)
S. An Angle is the inclination of two sirai<,dit lim s to-
ward each other, which meet in a point; as A B C. (Fig-,
i.) The point in which the straight lines meet is called the
<in-i^uhr point.
JSate.-AVhcn an angle is expressed by three letters, ihe
tetter denotinj; the annular point is alwavs placed in the jrud-
(lit, between the other two; as A B C. *An unule. however.
13
I
GEOMETRY.
is frequently expressed by one letter, which in the figure ia
always placed at the angular point; as B. ^Fig 5 )
9 When a straight line, standing on another straight line,
makes the adjacent angles equal to each other, each of the
angles ,s called a right angle; and the straight line whch
Tt^. 60 ''"'''' "" '' '''''' ^ Jyendicul^t
10. A Mixed Angle is an angle formed by one straight
line and one curved Ime. (Fig 7 ) « J»"djgnc
gie '' (^'^'o'r '^''''' '' '^'' ''^''*' ^' ^''' '^''" " ''^^' ^-
13. A Figure is that which is enclosed by one or more
boundaries. Two straight lines cannot enclose a space!
1 he space contained within the boundary, or boundaries ia
called the area of the Figure. '"^nes, la
14 A Circle is a plane figure contained by one line,
which IS called the circumference, and is such that all straight
Imes drawn xrora a certain point within the figure to the 5r.
cumference are equal to one another. This point is called
the c../.e of the circle. Thus A B C D E, ijthe circumfe-
rence, and F, the centre; and the lines F A, F C, F D, and
t iL, are all equal to each other, (Fig. 10.)
15, A Diameter of a circle is a straight line drawn through
tlie^centre, and terminated both ways by the circumferenct^j
16, A Radius or Semidiameter is a straight line drawn
from the centre, and terminated by the circumference; as F A
17, An Arc or Arch of a circle is any part of its circum^
ference; as C D. The chord of an arc Is the straight "e
whicii joins Its extremities; as C « D.
18 A Semicircle is the figure contained by a diameter
and the part of the circumference cut off by ii. (Fig in
10 The circumference of every circle is supposed to be
a yided into 3 equal parts, called Degrees; eacdi degree ^
ZT' ? '-^ ''^'''''''' ^"^^ '' -i--^! parts, callfd M !
^'UTEs; each minute into 60 cquul parts, called seconds; and
1 angle.
I the figure is
r. 5.)
straight line,
each of the
^t line which
^endicular to
one straight
than a right
n a right an-
iie or more
se a space,
undaries, ia
^y one line,
t all straight
3 to the cir-
nt is called
e circumfe-
, F D, and
m\ through
liuiferencej
ine drawn
3e;asF A.
its circuin-
raxght line
diameter,
{Fig. 11,)
3sed to be
degree ig
ailed Mi-
ff
f'KOMETRV. jy
.0 „„ ir ,i,<.,-<.,b>-.. „ „i,vi,., „,. „ p„,, .,,. „ ,,„,,
cr,l,e,l <„,„, ,l,e ver.c;. of „„y „„^,„ ,„ ;„ „,^ ^
be.- of degree.,. ,„,„„,«, &e., contained in ,l,c arc of .1,^,
••.rcle, ,„.ercep.e,l between the line, forming the angle, i, he
measure of that angh.. Thus in the figure at definition R
ho number of degree., minutes, &c., containe,! in the arc
^ U, IS the measure of the angle C F D.
I 20 I^^'^ALL^i^ or Collateral Lines are lines equi-dis-
J tant from each other in all their parts, or lines which, being
. .n the same plane and produced ever so far both way. wiU
never meet. • '
\ bytatr,;:::."" """"" '-' "■■'- -'-^ - "<-<-«''
22. MixT,L.»EA.. F,o„BEs are those which are bounded
partly by straight and partly hy curved lines.
23 Thilaterai. Figures or Triancrs ,re those
which are contained by three straight lines
24. (JcADRiLATERAL FiouRE, arc thosc wlfich arc bound-
ed by four straight lines.
25. Mur,TiLATERAr. FiouREs or P„,.vgom are those
which are bounded by more that four straight lines 4T
.ure of five sides is .sometimes called a ittg:; olsf,"
sides .Hexago,,, &c. If their si.lcs are all e.,mil they a '
.lulil'-^X^. I:;^-™" -'-»'"'■'■•■••-'.- -e.
A B is called the t." ™i cT'Z"'"'' '" V- "' ■'''"• -''"
angle
nbt
SO. An Obtuse-
usf anfflo.
(P
ANGLED Trj^n
lil\
Ui.)
J>erpen(licular.
i-r.K is that whioh h
as an
'I
; *i^ :
^1^
f4
GEOMETRT.
31. An Acute-angled Triangle is that uhich has thre«
acute angles. (Fi^. 17.) ^^^**
32 Of four^sided figures, a square is that which has all
sfl equal, and all it. angles right angles. (Fig , V"
34. A Rhombus is that which has all its sides equal, hut
•t« an^Ues are not right angles. (Fig. 20 ) ^ ' ■"
35.. A Rhomboid is that which has its opposite sides equal
(«^. ily"'" '"""'""''*'' """"^^ °™ ^■•'"'^'^ T„.„..„„».
_37. A srraishtn„c, joining the opposite point., or angle,
of a quudnlatoral %„rc, is called a D,Aoo^ „
Sr.a.e^thanr.vo'Ag ft al "te "i,t ^ /" f ^ """ «'''? '' ''
o'"^ aiij^fcs, ir IS said to be rc-entraiit.
';'• ;^"^ ^^^•^ f ^ yocXa\no^.,l figure n:ay be called the base
>1.K 1 he angular point, opposite to the base of a triangle
40. -rHE Altitude of any triangle or parallelogram, i« a
I J 'it
POSTULATES.
1- Let it be granted that a right line .na.v be drawn from
any one point to any other point;
9. That a terminated straight line may be produced or
'-•ont.nucd in a straight line at pleasure; and
»i.h any radiu,,. '' "" "'"" ""'' """■<•■ ->*
^ withou
I in the t
7 t5. A
' or inon
'' 7.. A
^
ch has rhre«
^hich has all
i^ig. 18.)
has all itt«
' all equal.
equal, bur
sides equal
r its angles
an inclined
5 are paral-
lAPEZIUMfr.
or angle*
s less thair
when it 't»
drant.
1 the BASE.
a triangle.
called the
grarn, it) a
side upon
iwn trom
duced or
It re. and
OEOMETRT, jj
AXIOMS.
2. If equals be added to equals the wholes are equal
equal ''" ''"'" '"" ^^"^^^' ^^« ---nders are
4. If equals be added to unequals, the wholes are unequal
^^^5.^ If^equals be ta.en fro. unequals, the re.aindrar.
r^"^:""^ ""'' ''''''' ^' ''^ -- *^-^' are equal
ro Le™X:.''^' ^" '^"'^^^ ^^ ^^^ -- ^^-^' - equal
role'Sf '''''''' ^^^"'^ '" ^^^ -- ^P- - equal
9. The whole is greater than its part.
0. All nght angles are equal to one another,
i. 1 wo straight lines cannot be drawn throimh th«
, A rrr Explanation of Terms.
<e ; of I '^ '" ' *'"'^ ^^''^ ^^^«'«^« evident by a pro-
T f ^"'^;«»'"g called demonstration. ^ ^
'■ ^*''^'"''"»'- -mark n.Uo .p„„ ,«■„. preceding
J'^
.k
18
GEOMETRY,
proposition or propositions, for the purpose of illustrating
their connexion, their restriction, their extension, or the
nmnner of their application,
S, An Investigation is a process employed for the (lisc(,-
very of unknown truths,
J». The Cow.s/rMc^jon of a fiirure is an operation in which
lines are drawn and points determined, according to certain
specified conditions.
10, The Data, or Premises of n proposition, are the majj-
nitudes, quantities, relations, and conditions stated oririven.
from which ne^v relations, &c., arc to be deduced, or' from
which a figure is to be constructed.
signifies
Contractions employed in the following part or this
Work.
Problem.
Geometry, Geometrical,
Trigonometry, Trigonometrical,
Logarithm, Logarithmic.
Euclid.
Theorem.
Hypotenuse.
Perpendicular,
Mensuration.
Division.
Location,
Tangent.
Secant,
Natural.
Scholium,
Radius.
Appendix.
Amplitude.
Difference of Jjatitude.
Departure.
Example.
Prob.
Geo.
Trig.
Leg.
Euc.
Theo.
Hyp.
Per.
Men.
Div.
tiOC.
Tan.
Sec.
Nat.
Scho.
Rad.
App.
Amp.
Diff. Lat.
Dep.
(C
(
((
cc
Si
((
a
cc
(C
cc
((
'fir
Ch.
(I
Ch
un.
^nsion, or the
for the disco-
I
GEOMETRV.
^^ signifies
/ er ^s.
tO", 3(y, 20" ''
Link.
Ang-le or Angles.
Triangle.
11
S 20<^ W.
^ _.^^6...^, xiuny minutes,
I wenty seconds.
South, Twenty degrees West.
PART or THIfl
^ Explanation op Signs.
Thus A = B "ills h,! , " "''""'' "'■ '"■ ^'"'='' ™'"''.
sentccl by B. ' *" ••"""'"y °'- magnilude repr,,-
'he quantities between which' it r„,. ""'"='"es that
«e.her, and the vvhoinin ".'''' '^ '''' "'''''''l «»-
o-antitie, between ;v;;::h?ir;red'T„'!v:: ;- °; ••'•'
~— (read minus) is the siJ<rn «r „, u.
that the latter of the uT ^ subtraction. It denotes
- '^. .0 he .:c:rzr r„r r;^;:f '; t-'-
e<i by the let't^Aabol 1";,""" "' "" "-"'"y «P-se„,:
ler B. "^^ "'" 1"="""^ --epresented by the let-
each other; and he wL,!. 'I'r'' "" '" '« multiplied i„,„
whieh re,,„ ,., f™, uhei * ,h 7'""" *"'""'' "'« I»-»''"«
A X B XCde„:e;,hTp:duc^wH'T''"'r^'''' ''''■"'''■•-
ing .he quantity r„pro3„me,l bv I ' T""' '''"'"" '™'"P'>-
bythe letter B, a, dthTL{ ,■ '""""'''^ '*''"<''''''
<l""...ityrepre,;„,"t,!!; C ™ '""'"'"'"'' "«''"' l-^ ">«
-r- is the sign of division r* -
the two quantities betw:;" wh h'T''''. '''''' '''' '«"»-• «*'
<^-' by the latter, and he "^ot '' ^''^''^ '^' ^« ^^ ^'^•-
' ^''^ ''^^'•^ expression denotes the quo-
^
IB
OEOMETRF.
..;^i
ii
i II
^t
m
■ -I'l
t.ent ^vli,ch will result from the division of the former quan-
tity by the latter, thus A ~ B indicates that the quantity re-
presented r>y A, is to be divided by the quantity represented
by C. Divjsion is also frequently expressed in the form of
a fraction, by writing the quantity to be divided above the
quantity by which it Is. to be divided, with a line between
them; thus — expresses the quotient of A divided by B.
: :: : is the sign of Proportion. Thus, A: B:: C- D
denotes that A bears the same proportion to B, which C
beai-s to D, and is read thus, as A is to B so is C to D.
A= denotes the square described on a line A. If the line
IS expressed by tAvo letters, A B, then the square described
upon It IS denoted by the sign A B^.
The principal signs employed in Algebra are the same
with those exi,lained above. It may be observed, however,
that Algebraists generaUy employ the small letters of the al-
phabct in their calculations. Instead of X the sign of multi-
plication, a dot ( . ) is frequently employed in Algebra, or the
letters are written together without any sign between them.
1 hus aXb,a.b, and 06, all express the product of a mul-
a-\-b — c
tiplied by b,
ax
expresses the quotient which re-
sults from the division of the excess of the sum of a added
to 6 above c, by a multiplied into x. It is read thu.^, a plui
6 minus c, divided by a multiplied by x.
x = a~{-b — c shews that the quantity represented by x is
equa. to the excess of tlie sum of a added to b above c. It
IS read x equal to a plus b minus c.
y is called the radioal sign, and. denotes that some root
of the quantity before which it is placed is to be extracted.
1 hus, Va denotes the square root of a. Wa denotes the
cube root of a. Instead of the radical sign, a fraction is
aom.etime.s employed; thus, ffi denotes the square root of a..
<?' denotes the cube root of the square of a.
■Si
ceCMlTRT,
V»
former quan-
e quantity re-
y represented
n the form of
led above the
line between
nded by B.
i: B:: C: D
B, which C
C toD.
If the line
ire described
re the same
ed, however,
;ers of the al-
ygnof multi-
Igebra, or the
3twccn thenx.
ict of « mul-
■0
Positive or affirmative quantities are those which aro to be
i<<ded, or which have the sign -f before them.
Negative quantities are those which have the sign - be-
fore them.
A co-efficient is a letter or number prefixed to any quanti-
ty into which it is to be multiplied. In the expressions ax.
Sx, a and 3 are the co-eificient^ of x. When a quantity ap-
pears without any co-efficient unity or 1 is understood as
being its co-efticient.
A Vinculum is a line drawn over several quantities, for
ihe purpose of collecting them into one, Thus a -f 6 X e
denotes that the compound quantity « -|- 6 is to be multiplied
by «, So in like manner V ab -\- c^ denotes the square roof
of the compound quantity ab -\- c=. Instead of the vincu-
lum, parentheses are frequently employed, thus (a-\-b)X <^
or (a -\- b) c.
A quantity without any sign prefixed to it is a positirc
quantity, th^ sign + being understood as placed before it.
sVl
'r.i
I
It which re-
n of rt added
thu9, a plui
cnted by a; is
above c. It
It some root
>e extracted.
denotes the
a fraction is
re root of a
GEOMETfilCAL PROBLEMS,
fVhi
PROBLEM I,
To binect a right line A B, (^Fig. 23.)
From A and B, as centres, with any distance A6, Ba .roa,--
er than hal the line A B. describe two arcs, c a Ln/cZ
cutting each other zn c and d. Through the points of inter-
section . and rf, draw the line, . . d, cutling A B in .
I hen \e =; eh.
PROBLEM II.
To raise a perpendicular from a given point C, in « ^<v.n
right line A C. {Fig. 24.)
CASE i.
When the point C is at the end of the line.
From aay point a, out of the lino with the radiu, «C
araw the arc c C b, cutting A C in b; from the point of n.^
tersect.oni, through the central point «, draw the .straight
fine /. a c, cutting the arc c C b iu e; join c C and it wdl be
the perpendicular required.
CASE II.
iVhm the point is near the middle of the line.
""i ^ke the points a and b, (Fie ^b > ,» ^«,.oj i- .
</, \^ig, -.0,; dt equal distances
GEOMETRY.
21
froni C, and from them as centres, with jiny radius greater
than aC, describe arcs cutting each other in n; then draw h
straight line from C, through n and it will be the perpendi-
cular required.
IMS,
^a great -
and c b d.
s of inter-
A B in e.
I a ec'ven
PROBLEM III.
From a given point D, (F/g-. 2G,) to let fall a perpendicular
on a given right line A B.
CASE I.
IVhen the given point is nearly opposite to the middle of
the line.
On D, as a centre with a radius sufficiently great de-
scribe an arc intersecting the line A B in m and n; then on
m and n, as centres, with a radius greater than half of m n,
fiesciibe arcs cutting each other in C: then draw a straight
line through the points C and D, intersecting A B m e; the
line e D is the perpendicular required.
CASE II.
IVhen the given point is nearly opposite the end of the line.
Draw a straight line from D, {Fig. 27,) to any point wi in
The line A B; bisect the line D m; from the point of bisec-
tion n, with a radius n m, or fi D describe the arc D C f«
intersecting A B in C; then join D C and the line of junc-
tion will be the perpendicular required.
n
dins rtC.
It of in-
straight
will be
stances
PROBLEM IV.
.i/ a given point A, (Fig. 28,) in a gi>:^n line A B, to make
an angle equal to a given angle E.
From the point E, with any radius describe an arc meet-
ing the lines containing the angle E in a and h; with the same
radius on the ])oint A as a centre describe the arc c d: apply
!he distance a b on the former arc f. the arc c d, from d to c;
then through the points Ac draw the line A D, which will
form an angle with the line A B, equal to the angle at E.
^
•EOMKTar.
PROBLEM V.
To Hra. a .trm.kt line through a ,i,en point parallel to a
^'ven s'-aighl lijic.
l,M A (Fig. .29,) bo the given point nn,l n r ,1. ■
Fron, the point A draw a stmij;ht line meeting the li„„
t ar.rra'T.^'i'o'c' "t" t •■""" - -' ■- -^ -' ^"
I
PROBLEM Vf
f.ct At (Rff. 30,) be the given .straight line
A C a„U C B td' h '«;';: r Bc'^wi,;?™,,'""'' ""' """
quired. oure A B c will be the triangle re^
M
PROBLEM VII
Vo construct a triangle, the sides of y^hich may he eaual to
three g^.en straight lines A, B, and C. /i^. T '
I-'ay ofl'astrai'^ht lino DP « i '^
-aigh. ,i„e, aI'Ld a^fere'lwtr: °',""' ^'™"
'o another of the, iven line. B, descHh "'a:e™rET°'
.iv„';:'trhT^r:;;'eTa*^L'r''a,iL":"^''"^ - "f--'
PROBLEM VIII
"" ^'^ ^^'"^ -^ ^-- '^ D perpendicular and equal to
|i:K;
OEOMKTnr,
e«
parallel to a
3 C the given
'ing the lino
I angle D A K
straight line
o the straight
straight line.
t
^ B dejcril>e
'aw the lines
triangle re«-
' be equal to
Fig. SI.)
f the given
idius equal
To/n E as a
ven line C,
; then join
luired.
two of the
d equal to
A B: tli<..n from the points A and I) with the di.stanco A B
or A D, describe arcs intersecting each other in E Draw
the lines D E an.l B E, and the figure ABED will Ih- thf,
«qui,re required.
PHOBLKM IX..
I ToJinJa third line proportional to ttoo given straight lims
I A and B. {Fig. 33.)
I Froui any point C, draw two straight lines, tHe one C D
-equal u> A, iho other C E equal to B running any ang:e;.
.|.nnD E: produce C D and C E; lay oif D F equal to B.
or C E, then draw F K paraMel to I) E, meeting C E pro-
duced .n .., the line E K will be a third proportLnal. '
PROBLEM X.
To find a fourth proportional to three given right line, A ..
B, and C. {Fig. 34 ),
Draw two straight linos D R, and D F fonnin. an> an^r
K Bl, upon the line D E lay off D G equal to A, and GE
jojn G H: from the point E, draw E F parallel to G II
meeting the line D F in F; then H F will be a fourth pro:
portional to the lines A, B, and C.f ^
PROBLEM XL
To find a mean proportional between ttoo given strairht line.
A and B. (Fig. 35).
Draw any right line C D, on itlay off C P := A and P D ^.
I {.e tru.„.le C F K. according to Euc^;" , 'r d • ^.' " r v'
J^. K; '^at according to th. construction CD ~ k u' H r ' ^.^^
a|=K; therefore A : B : : B • E K " " ^
^ut --r<ling to tho;co„Btr«ction of the' F
J^' I. -~ K. and I) U =^ C; therefore A: B: : C
of ihR sidosol'sh*
<ng to the 'construction "of'therTg^rV. D "'i'!i',
Hi!
hi
• I)
<i4
CiEOMRTRV.
B. Bisect C D in o, and with o C o,- o I) ns i-ndiiis, de-
scribe the semicircle C F D. Again from the; point 1', draw
V F perpendicular to C D: P F will bo a mean proportional
between A and B: i. e., C P (A) : P F :: 1» F : P D (B).
{Enc. vi. 13.)
PROBLEM XII.
Ma given point D, (Fig. 36,) to make an angle eiiual to a
given rectilineal angle A B C.
From the points B and D, as ccMitres, describe two arcs,
a b, and m n; make m n=ab; then throiiirh the points D, «,
draw the straight Ymv. D R, and through the points I), m,
draw the straight line D F: the angle K D F will be oqual
to the anirle ABC.
PROBLEM XIII.
To make an angle of any proposed number of degrees.
Draw any straight line A B, {Fig. 37,) take the first 60
degrees from the scale of chords,* and with this distance as
a radius, describe the arc ,n n. From the same s-ale, take
the chord of the proposed number of degrees and apply it to
the arc from n to m; then from the point A draw the line A
C through thp noint w, the angle CAB will be the angle
required.
r ^' ^' u'L*'*^ proposed angle exceed 90 degrees, lav oft-
first one half, and then the other: e. g. if the pi^oposed nnm-
ber of degree be 130, from the point n, towards m, lav off
iirst 6.5 , to 0, then trom o, towards m lay off 65° more,' and
It will give the measure of an angle of 130^
To Jim
Fron
the arc
Then t
chords
the anjj
N. H
gre«.'s V
inents.
To bise
Froir
arc A ]
scribe a
C n, dri
A C B,
i insc
Divid
number
'M'lrcle n
inoasurt
fho poii
re nee, f
polygon
A line of chords, adapted to 90 o , or the fourth part of
circ e. IS commonly put upon the plain scale, which will befoun
n almost every portable case of Mathem.ntical Instrument^
To desc
be r
GEOMETRY.
lib
radius, dv-
nt I*, dinw
roportional
: P D (H).
equal fo a
two arcs,
oints D, w,
lints J), m,
II be oqual
legreeit.
lio first 60
listaiire as
■■"ale, tako
apply it to
the line A
the anerle
es, lay oft'
osed npm-
m, lay off
more, and
1
PROBLEM XIV.
To find the number of degrees contained in any given angle
CAB. {Fig, 37).
From the angular point A, with the chord ot'GO^ descrihf
the arc m n, intersecting the lines A C and A B, in m aiut n.
Then take the distance m n, and apply it to the same line of
chords and it will show the number of degrees contained in
the angFe CAB.
N. B. If the distance m n exceed 90*^, the number of de-
grees which it contains must be ascertained bv two measure-
ments.
PROBLEM XV.
To bisect a give\. angle A C B, (Fig, 38,) i. e., to divide il
into two equal parts.
From the angular point C, with any jUstance describe the
arc A B, and from the points A, B, with any distance de-
scribe arcs cutting each other in n; then through the points
C n, draw the straight line C n, and it will bisect the angle
A C B, as was required.
PROBLEM XVI.
To inscribe, in a given circle, a regular polygon of any pm^
posed number (5,) of sides.
Divide 360 (the number of degrees in a circle,) by the
[number of sides, (5,) and at the centre O (Fi^; 39,) of the
■circle make an angle A B, the number of degrees in the
f measure of which shall be ecjual to the quotient, (72;) join
Uhc points A B, and apply the chord A B to the circumfe-
rence, the number of times that there are to be sides to the
polygon, (5j) and they will form the figure reciuired.
part of a
II be foun<}
lent?.
PROBLEM XVII.
To describe a parallelogram lohose area and perimeter shall
M be respectively eoual to the area and perimeter of
a given triangle \ Bi^. (Fig. 40.)
Produce A B to D, making B D = B C. Bisect A D in fcJ,
G
y^-^-^J^:
I
iii
OEOVIETRT,
and draw IJ F parallel to A C. With the radius A E, and'
centre A, (lescrihe a eircle intersecting B F in ^r, then join
A G. Hiseet A C in H, and draw H F parallel to A G.
The parallelogram A G F H will be equal to the triangle
A B C, both in area and perimeter.
PROBLEM XA'in.
To dr.sr.rihe. a circle about a triangle A B C (^Vg*. 41.)
Binect the line* A C by the perpendicular D E: bisect also 1
the li?ie C B by the perj)endiciilar F G, intersecting the por- i
pendicular D E in H. On H, the point of intersection, as 1
a cfintrc; with any of the distances H A, H B, or H C, as a ^
radius describe the circle ABC, passing through the points
A, B, and C, and it will be the circle required.
PROBLEM XIX.
V'o ronatmrt a triangle that shall he equal to a g-iren trapc-
zi^im ABC D. (Fig. 4^i).
Draw the diagonal D B, and niakc; C E parallel to it, 1
meeting the sid(; A B ,»roduccd in E. Join the points D, E, 1
and A D E will be the triangle recjiiired.
PROBLEM XX,
To describe a triangle that shall be equal to a given recti-
lineal figure A B C D E A. {Fig. 43.)
Produce the side A R both ways. Join D B^ and from C |
draw C G parallel to D B. Join alsA D A, and through K
draw E F parallel to D A. Then join D G, and D F,
and the triangle F I) G will be equal to the figure A B C 1)
E A.
PROBLEM XXL
To draxc a square equal to a given Rectangular Parallelo-^
gram Mi CD. {Fig. 14.) ^
Produce the line D A. und on the part thus produced, lay -gfor the mos
off the dis
on G, as t
DF. Pr
area is eq
To discri
Take ai
On A and
scribe circ
On the lin
the anguh
or D B, d
circles in t
as was re(i
To descril
A B, at
{Fig. 4(j
On the t
diameters
V and G,
the extrem
irUerscctioi
bisect the t
lar to iti
fected in F
in the righi
be placed i
passing thi
[conjugate i
G A F in
I' D s, and
'U'hf'n tl\<! Ifjrtn hue f employ
I't'tK a strain-ht !i»e i« •.tlvvuT* lu,
iilt'd,
Hiut'sst t!i»' contrary i^ e\j»tc
fetnl ,
Mr accurao
f .'
'^i
.s A E, and
, then join
el to A G.
;ie triangle
(Fig, 41.)
: bisect also
ng the por-
rsection, as
r H C, as a
k the points
;iven trape-
rallel to it,
oints D, K,
^iven recti-
and froni (.'
through E
and D V.
re A BCD
r Vuralhlo-
•oduced, lay
i« •.iKvuT* in,'
GEOMETRr. IJ7
ofTthe distance A B from A; biseet the whole line in (;, find
on G, as a centre with the radius G D describe a semicircle
I) F. Produce B A to F; A F is the side of a square whose
area is equal to the Rectangle A C.
PROBLEM XXII.
To describe three equal circles ivhich shall touch, withotU in-
tersecting each other.
Take any straig^it line A B, (Fig. 45,) and bisect it in D :
On A and B as centres, Avith the distance A D, or D B, de-
scribe circles, and they will touch each other in the point D.
On the line A B, draw an equilateral triangle A B C. On
the angular point C, as a centre with the same distance A D
or D B, describe another circle, and it will touch the other
circles in the points E and F, and will also be equal to them,
as was required.
PROBLEM XXill.
To describe an ellipse, the transverse diameter or major axis
A B, and the conjugate diameter or minor axis C D
(Fig. 4(i,) being given.
On the transverse axis A B, describe two circles of such
diameters that while they intersect each other in the points
F and G, they will also pass through the points A and B,
the extremities of the transverse axis: through the points of
intersection F and G, draw the straight line O P which will
l>isect the transverse axis A B in E, and also be perpendicu-
lar to it.. On O P lay off the conjugate diameter C D bi-
sected in E by the transverse axis A B; then find two points
HI the right line O P such that if one foot of the compasse*
jbe placed in them successively, the other will describe arcs
|passu.g through the points C and D, the extremities of the
jconjugate axis C D, and also touch the circles G B F and
|(i A F in the points r, s, t, u. Draw the arcs t* C r, and
HJ D s, and a figure will be tbrmed sufficiently near an ellipse
Jfor the mo«t of practical purposes. Where greater nicety
pr accuracy is required the following method may be adopt-
I
v'^
■I 1*1
■i^
GEOMETRY.
I ^
m-
Another method to describe an Ellipse.
Draw the transverse and conjugate axis A B, C D, {Fig.
47,) bisecting each other perpendicularly, then with half th»^
longest diameter as a radius and centre C, describe arcs
cutting A B in F, G; the points F, G, will be the foci of the
ellipse. Then take two pins and fasten a thread upon them
in such a way that when the thread is stretched the dis-
tances between the pins shall be equal to the length of the
transverse axis A B. Fasten the pins in the foci F, G, then
by moving a pin or pencil round within the thread and keep-
ing the thread always stretched by it, a curve will be traced
out forming the ellipse required.
PROBI \U XXIV.
To project lines of Chords, Signs, Tangents, Secants, ^'C,
to any Radius.
On the line A B, (Fig. 48,) describe the semicircle A D B,
Upon the centre C, erect the perpendiculr.r C D, continued
at pleasure to F; through B draw B E parallel to C F, and
consequently perpendicular to A B; and draw the right line
D B. Divide the quadrant D B into 9 equal parts, and with
one foot of the compasses in B, and the distances B 10, B 20,
B 30, Stc, on the curve line B D transfer them to the right
line D B, and it will be a Lvne op Chords.
From the points 10, 20, 30, &c., on the arc B D draw a line
parallel to D C, and it will divide the radius C B into a line
OF Sines, reckoning from C to B, or of "Versed Sines, if
reckoned from B to C.
From the centre C through the several divisions of
quadrants D B, viz: 10, 20, 30, Stc, draw right lines, until
they meet the line B E, and it will be a line op Tangents.
Transfer the distances between the centre C and the di-
visions on the line of Tangents to the line D F, and it will
give a LINE OF Secants which nmst be numbered from D
to F.
In the figure the divisions are only given to every tcntli
degree; but by subdividing each of the 9 divisions, we niav
'i'lvo a ii
-rt;e.s; a:
'i'lve sea
Scales
fiiiniatur(
known m
"'•ale an,si
A ijl bf la
■1^ iiivator
llicll he f;
'-Pf't'ified
ti'iucd oni
fiif'la.st (Ii
i first or pr
;«*is the first
i'arts. ']']
^0. -10, &.CV
.4, &,c.; or
than the s
they may r
Will stand 1
.? Of scales
Ipery case
C* ''Jiles of e(
^ f equal pa
|d into Urn
pressing the
#ided on th?
* usually cl
**t the letter
laces of fig
determined j
ide.
The most
inches, or
^'J having o
C D, (Fig.
nth half thr
;scribe arcs
B foci of the
1 upon them
led the dis-
ngth of tho
i F, G, then
id and keep-
ill be traced
kcanls, fyc,
ircle A D B,
), continued
to C F, and
tie right line
rts, and with
5 B 10, B 20,
to the right
) draw a line
I into a LINE
SD Sines, if
iions of i,
t lines, until
Tangents,
and the di-
', and it will
ered from D
every tentli
ons, we mav
GEOMETRY.
C<,xcERN-,s„ Soles op E,ial P„.ts.
fCiil-s of pq,j„| p„rts am nothino- more fh„.
™n,a..„, e,„p,„,„., r,. laying doC;™':"''"-^ '"
Known measure ,„■ chains, var,l fee, &e , L "'
-«le .uiswering to one el ,in „„ / '^ '"'" ™ ""'
' -■'.•..,.■ number of par s in „° teh " "'""'^ " ™»"'"- -
""•'«! <.nw,.r,l i„ ,he"a^ ' " ■• T'- "^ """* P""'' ''« ■•"-
'"-on.Hn,arvaiv^:itt,c "/;'""' r'r^^
'•'■< fh<. first line has liPm ,. *• ™ ^' 2. 3, 4, &c., as far
i«-. T,,;, :,!::,;" :" ":t:' "'" "^ " ^""'^ '"■«-'
».J., .., an., .„e„ ..e'^n^-.tt:; 'S I~ /"; ^«-
"• «-c.i or fhey may represent 100 oflO Snn Tor, . ' ' '
•■•ale; of equaloar ,""■"""'""' »'•« '<> l« '"und
o.;..a«a, pX '•::":■;;: ,:;'s™' -'« -'%' '^ « -'^
e'l into lines comn encinn^ w!. ..' ^ °"^ ''^^ ^'^ ^''^^^J-
^i'Jecl on that scale or 11^^^. T '"^'"^ "" '"^^ ^''^ ^'"
fc usually changed in to a I'ine of hf f ^'^^ "P^'^^ '"-
It the letter C When thl ^''"^'' '''^''^ oommences
i-s of figures t^Ce if triro'^rf "' ^^'^^^
etermined accurately hv th. V " '^' P'"^*' '""^ ^^
^^^ ately by the diagonal scale on the opposite
1 The most useftil scales for « «J,...
inches, one of who e si •7"'''''' ''' '^"'^^ «^ ^2 or
''^''-'ingonthe t;^^^^^^^^^^ '/ -'^ ^'- other convex.
^ ^'tk th divisions and subdivisions
^0
GEOMETRY.
marked on the edges, nnd continued to the end of the t^calo.
On the centre Avill be found the numbers 20 and 40, 25 and
50, &,c., distinctly marked. (See Plate.)
By means of the scale of equal parts it is easy to measure
any line laid down upon a plan if we only know the scale
by which the plan has been drawn, and also to lay down any
distance upon any giver, scale.
f the BC.alo.
40, Q 5 and
to measure
iW the scale
y down any
-H-
I i i
&P
v,'-~-
2^
-^
^
Pi
I I
^
S2
:- I
J J-
LOGARITPMS.
.1.
About the end of the sixteenth century and the bopfinningf
of the seventeenth, several Mathematicians began to con-
eider by what means they might simplify the arifhmetic^il
operations of multiplication, division, and the exLraction ot
roots, which formed no inconsider.ible obstacles to the im-
provement of those branches of knowledge; in the prosecu-
tion of which tedious calculations were indispensable. For
abridf'ing these calculations several ingenious expedient*
were suggested. Of these by far the most complete, was
the system of numbers called Logarilhms, invented by John
Napier, Baron of Merchis^on, in Scotland, and afterwards
improved and extended by Mr. Briggs, and others; forming,
doubtless, one of the happiest and most useful contrivance!*,
of modern times.
Let two series of numbers be formed, the one in geome-
trical progression, whose first term is unity or one, and the
common ratio, 2; and the other in arithmetical progression,
whose first term is 0, and the common difference 1, thus:— ^
Geo. Prog. Jlrith. Prop;.
1
2 1
4 2
8 S
16 , A
82 5
64 .6
J 28 7
256, &.C., . S. &c.
.m
•rilaM
4fi
I
32
LOGARITHMS,
The t.nm ,n the arithmetical series will l,e the lo.ar.th.ns
of the corresporuhng terms in the geometrical series; that is
.s he loganthm of I, and 1 is the logarithm of,, J ^Z
the logarithm of 4, and 3 is the logarithm of 8, &<• '
s«mrf?V'"'T' ^'•'^'" ^ "^«^»««t''^ inspection,' that the
sum of he logarithms of any two nu.nbers i.i the fo.e.oin.
ser.es, .s ec^ual to the product of the numbei-.s themsSv "^
f^ or exan,ple the product of 4 multiplied by 32 is 1 28. Now
he number m the arithnietical series coiTesponding to the
ei-m 4 ,n the geometi-ical series, or in othe'r words
oganthm of 4 is 2. In like manner the logarithin of V] is
^' Again 2 added to 5 is equal to 7, and the term in the
I'lTr'tol '"'"'u "°''''^'^P'^»<l'»g ^o 7 in the arithmetical
series ,s 128, or the product of 4 multiplied by 32. In lik.
manner it is evident that the difference of the logarithms of
any vvomunbers^ ^elogai-ithm of the quotient arising from
the divisioii of the one nuinber by the other
From these statenients it appears that multiplication of
natura. numbers ca.i be effected by the addition of their
logarithms and that division of natural nmnbers may be ef-
tected by the subtraction of their logarithms " '
In the logarithmic tables usualy employed the series in
geometrical progression is
h 10, 100, 1,000, J 0,000, &c„
and the corresponding series in a.-ithmeticul progi-ession is
0, 1, 2, 3, 4, &c.,
that is the logarithm of 1 is 0, the logai-ithm of 10 is 1, the
logarithm of 100 is 2, and so on.
The logarithms of the terms of the progression 1. 10,100,
»,000, &c bemg thus determined; in order to find the logar-
thms of the numbers between 1 and 10, and between 10 and
100, ice, we must conceive a g,-eat number of geometrical
means to be interposed between each two adjoining ter.iis of
mearb!f"^'T'^'""' -ries, and as many arhhmetical
means betvveen the corresponding tenns of the arithmetical
^er,es. Then as the terms of the arithmetical series 0, 1. 2
!;nn!*:^'^ ^^^^^^''^^^^^ ^^^^^ coricspondin^ terms of th.
geometrical series 1, H
100, 1,000, &c., the
jnterposed
LOGARITHMS.
Sfl
terms of the former will also be the logarithms of the cor-
responding interposed terms of the latter.
The integral part of any logarithm, usually called its »n-
dex or characteristic , is always less by 1, than the number
of integers of which the natural number consists. In thr
logarithm of a decimal i* '^ the integral part also, or the in-
dex, which determines tht, distance of the first significant
figure from the decimal point. Thus, the logarithm of the
Nat. Num. 2651 .
265.1
26.51
2.651
.2651
.02651
.002651
IS
.S. 423410
2.423410
1.423410
0.423410
•1.423410 or 9.423410
■2.423410 or 8.423410
•3.423410 or 7.423410, &c.
N. B. The Negative sign ( — ) is frequently written over
the index, instead of before it; thus, 1.423410.
series m
EXPLAN.\TION OF THE TaBLE OP LOGARITHMS OP NuMBERS.
I. Tojind the logarithm of any whole number, under 100
On the first page of the logarithmic Table, in the column
marked N, or No. look for the given number; immediately
to the right of it and in the same line with it, in the column
marked Log. you will find the logarithm sought, with its
proper index prefixed. Thus, the logarithm of 63 is 1 . 799-
341 ; and the log. of 74 is 1 . 869232.
n. Tojind the logarithm of any whole number between 100
and 1 ,000.
Fnd the given number in the left hand column marked N,
or No. and immediately opposite to it in the column marked
at the top and bottom you will find the decimal part of the
logarithm, to which prefix the proper index and you have
the logarithm required. Thus, the log. of 364 is 2.561101,
and the log. of 333 is 2 . 522444.
III. 10 find the logarithm of any number consisting of four
figures.
Find the first three figures as before, and opposite to it iu
r"
54
l-OrjAUrTHM.*.
.,' f
! ri'
.»!
niarkwl I). „r DifV Z^, '""'«'"'"""' '•"I"'"",
-ho ,Iiff,.„.„,.., b«^';.i ;:'!"' '"' "*,'"" '"ff"™'-. .ha. i,
'^■- '■ - -y .h:;:r ,;',:'^;:;.:: ;^.;;"""-^)- ,Mu„ip„
(riiros a.s are pontahid in tl,„ '■ ■ "' '""">' «"
quired. * "^'iAta ^\iii by the Log. re-
Kv. Required the Log. of 36548.
quired. "= i-5028Gi, the logarithm re-
». - aiiy j;;,:!!: f-^-^. -j'-^ ««. of ,,,„ , ,„. „^ „^
"aw ol'lhc rcnminilor. '""' '° "''" I to tl,e unit
■V. To Ma ae U,. of a Vulgar Fra.,Un, or „/„„.,„,
number,
o.ot':i;':;:;'S::™''''''' "" " '^-"""'' ">" "■- p™-
V. ToMUme natural nu.,^,r corresponding ,o a ,f,en
t( "ithm
*'Ook for tJie jriven ^ mr u r
'he o.-s„c( i,„M.i,i„., ,i,. ' ""^ '" "'" '"'I'-"- It vou fin.) ■
lett hand of the page mafked \,
I^.OOARITIIMS.
.11?
6r No. TUni i{ thr index of the iriven Los?, be los-i than .%
cutotr froi,. thori-lit haiul of the iiuiribcr ': ..,d nn many
fif,'uro.^ as the index \h h>.ss than 3, the rigun . . c.it off uill
he deeimals and the reniaiiuh'r a Nat. No. or Nut. Nos.
Thus, if the Nat. No. eorrespoitdinir to the logurithin -2 S^Hi-
950 he required: the K.^.3i(i<>.50 being found in the taWes,
opposite to it in the left column is 21i2, and at the top and
bottom of the cohnnn in whieh it is found is 3, whieh, plaeed
nfter '212, gives 2)ilS: and since the iiuh-x 2 is hss bv 1 than
S, one figure is to be cutoff from the right hand as a'ch-eimaJ
wh.eh wdl give 312.3 a^ the Nat. No. corresponding to the
given h.g, 2.3-:J6950.. If however the in,h3x exceed 3, annex
Its many cyphers to the number found as the index exceeds*
•1, and you have the Nat. i\o. retjuired.
But if a Logarithm exactly corresponding to the siven
Log. cannot be found in the table, take; the Log.- next to the
given one an(J less than it: then take the .litferonce between
that arid the given Log., to which annex cvjihers and divide
it by the tabular difference found opposite to tliteLo-. which
you have taken iVom the table. Annex the quotient t.vthe
Nat. Nos. corresponding to the F.og. taken from the tabic
and place the decimal point wherever the index points c»ut
and you have the Nat.- No. required, For example :-To
find the Nat. No. corresponding to the Log. 4. 478309 • the
Log^nearest to it, and less than it, is 478^78, answering to
the Nat, No. .3008. The difference between 478278 and^he
g.ven I og. 478309 is 31 . By annexing cyphers to this num-
ber ai.d d.v.dmg by 145, the tab. diff. f„und opposite to th«
Log. 478278 you have for a quotient ,:3 +, JLu an led
o figures 300,3 mak,, 3008.132 +. 13nt the index 4 shows
hat th.^e can only be five places of whole numbers. Tul
decnnal ) mt bemg therefore placed after the fifth figure
.v.s3.^2.3-Ha
Jf the nmnber acquired is to consist altogether of decin.al
figures, the sn?
ne
method must be used to obt
Jiin it as direct-
od above; only observe, that 9 cyphers, less the index
»>4? prefixed to the No. found. Tb
are to
li
us, to find the decimui
I:i.
11 .
.Sti
LOGARITHMS.
So. corrospomliiifr to the Log. 7.819083; look in th« Table
<"or th«' Log. 819083, nnd you will find the corrcHporulinf?
Nat. No. to he 6593 Now 9 cyphers les.s the inde.x 7, leave
two cyphers to he prefi.xed to the No. found, giving .00()59.S
HH the decimal number answering to the Log. 7.819083.
VI J. To find the Arithmetical complement of a L „nnthm.
The arithmetical complement of a Logarithm is the loga-
rithm of the recii)rocal of the corresijonding natural number,
or it is the number it wants of 10.000000 or 20.000000. To
find it, begin at the left hand and subtract every figure from
9 except the la.st significant figure, which is to be subtracted
tVom 10. If Uic index exceed 9 it is to be subtracted
from 19, or if it be negative, it is to be added to 9, and the
rest subtracted as bei;;re. In taking the sum of the Loga-
rithms, observe that for every arithmetical complement em-
ployed, 10 must be subtracted from the sum of the indices, in
order to obtain the proper index of the result. The arith-
metical comi)lement is frequently used in proportions, and
in trigonometrical calculations, to change subtractions into
additions.
iMDLTIPLICATION BY LOGAllITHMS.
RULE.
Add the Logarithms of the numbers to be multiplied and
Iheir sum will be the product in Logarithms. If there be
negative and affirmative indices, their difference, with the
proper sign prefixed will be the index of the Log. of the
product. If, in any consequence of either of the factors or
of both of them beinfe decimals, the index of the sum ex-
ceed 10, reject the 10, and the remainder will be the index
^>f the Logarithm of the product.
EXAMPLES.
I. Required the product of 23.14 multiplied by 5 062
J he Log. of 23.14 is 1.364363 J' o*.
1 he Log. of 5.062 is 0.704322
Hroduct,
U7.134. 2.068685 Log. ofthe ^roduct.
A?
Log. (
Log. o]
Quo.
Here
▼isor i^
I in thn Tabh;
^orrt'Mporulinjj
index 7, leav«;
iviiig .00()59.-{
7.819083.
!» L- „anthm.
II is the loga-
urul number,
000000. To
y figure t'roin
)e Hubtraeted
e subtracted
to 9, and the
f the Loga-
pleinent oni-
ic indices, in
The arith-
ortion.s, and
'actions int(»
IMS.
iltiplied and
If there be
;ej with the
Liog. of the
e factors or
lie sum ex-
5 the index
LOOtRITHMS.
37
y 3.06-2.
Product.
2. What is the continued product of 9. 9n»J, 597 16 and
081173."
The Log. of 3,002 is 0.. 591 ^287 or 0.50 1 -28"
1 he Log. of .':.:)7. IG i.s 2. 770091 or 2.776091
Tho Log. of .O.JM7.i i.s 2. 497938 or 8.4979.^8
Troduct, 73.3357 1.865316 1 .865316. rejecting 10
from the in«lex. ,
DIVISION DY LOGARITIP'S.
RULE.
From tho Logarithm of the Dividend sub;. act th<^ Loga-
rithm of the Divi.sor, and the natural number answering to
the remainder will bo the Quotient required.
If tlic Log. of the divi.sor exceed the Log. of the divi-
dend, proceed as before until you come to the index. If the
decimal jiart of the Log. of the divisor exceed the decimal
part of the Log. of tiie dividend add 1 to the index of tlie
r.og. of the divi.sor. Change the .sign of the index of the
r.og. of tho <livi> ,.', add the indox of the Lo-. of ;he divi-
dend to it, and with its i)ropcr .«ign prelixcd it will be tho
index of the Log. of tho quotient; or, when the index of the
Log. of the divisor exceeds; the index of the Log. of the di-
vidend, borrow 10, and the remainder will be the index of
the Log, of the quotient.
EXAMPLES..
1. Divide 4768.2 by 36 954.
Tho Log. of the dividend 4768.2 is 3.678.«J54
I he Log. of the divisor 36 . 954 is 1 . 567661
Quotient, 129.0307
2. Divide 4.6257 by .17608.
Log. of 4.6257 is jO^^ 665 177 0.665177
of .17603 is 1.245710 or 9.24.5710
2. 110693, Log. of Quo.
Log
Quo. 26.2704 1.419467 1.419467, Lo^. ofQuotient.
Here, in thr ^ st case, the J^ndex of the Log. of^ the di-
Ti8orischangeuf»om-~l,or l,to-j-l,or 1, and the index of
I.0GARITH5fS«
the Log. of tho dividend, 0, being added to it gives Las the
index of tho result. In the second case, 10 is borrowed for
the nidex of the dividend, and the index of the divisor being
Hubtractcd from it leaves the same result.
3. Divide .19876 by ,0012345.
Log- of . 19876 isT.i>98329 or 9.298329
Log. of .0012345 is~3.091491 or 7.091491
Quo. l(U.O044 2.206833 2.206838. Log. of Quo ■
Hero again, the index of the Log. of the divisor is chang-
ed fi-om - 3, to + 3, and this added to - 1, the index of the
dividend, gives -t 2, or 2, as the index of the Log. of the
Quotient.
RULE OF TF!IEE, OR PROPORTION BY LOGA-
RITHMS.
nuLE.
From the sitm of the logarithms of the second and third
terms, subtract the lo-jfarithm of tho first term; the remain-
der vviir be the logarithm of the fourth term: or, add the
amhmeticar complement of the first term to the lorarithins
of the second and third tc is, and the sum, after subtracting
10 from the index will b( he logarithm of the fourth term
In any Compound Proportion the term sought may be
fomid by subtracting tho sum of the logarithms of all those
tcrnis which, when multiplied into each other, are to form
the divisor, from the sum of the logarithms of all the term«
which, when multiplied into each other, form the dividend
»he remainder is the logarithm of the term required
Instead of subtracting one logarithm from another, yoo
may add the arithmetical complement of the subtrahend to
the logarithm of the n.inucnd, and reject 10 from the index
of the sum.
EXAMPLES,
SoiB 857 486 og* 2 55 259 Z l'""^' SSi^^a
jiui,, ^.ojozjj or Los:, 2.553259
To 1 a, 1483
1 095103
1.095108
I
\,
BY LOGA:
rOGARITHMS.
8. ^ind a third proportional to 12.796 and 8.24718.
As 12.796 Arith. Coinn. 8.892926
3.24718 Log. 0-51I5C6
39
Is to
So is 3.24713
Log.
0,511506
'^° r,. fT\ 1.915938, third propor.
3. I* md a fourth proportional to the three numbers, 36 48
and 6Q.
Multiply 48 Log. 1.681241
by 66 Log. 1.819544
Divide the Pro. 3168 3-500785
l)y S6 Log. 1.556303
Quotient, 88 ,.944482, fourth proportional.
INVOLUTION BY LOGARITHMS.
nuLE. '
Multiply the logarithm of the given number by the index
of the power, and the product will be the logarithm of the
f ower sought.
ga^TC l,v7„'"aSnn.f,'^'"'*' " \'>e^'''!'>^ "hose index is ne-
«ne, as the ind^KTlh^^Tj',''''^ as many cyphers less
EXAMPLES.
J. Required the^square or second power of 25.791.
25.791 Log. 1.411468
25.791 Index. g
665.175
2.822936
^XlHr *'r "^"^'^ "' *^'^^ Po^^er of 30.7146.
30.7146 Log. 1.487345
30.7146 Index q
Cube, 28975.7
4.462035, Log. of Cube or Srd ptmer.
■
) !
; i
w'.
i is v^i i-^
40
LOGARITHMS.
5. Required the cube or third power of .008.
.008 Log. y. 903090
.008
Index
.000000512 77709270
Here, the index of the Log. multiplied by the index of
the power gives — 9, but as the number 2 is to be carried
from the decimal part of the Log. this reduces it to ~ 7, as
above.
4. Required the fifth power of .2.
.2 Log. 9.S01030
Index 6
.2
.00032 46.505150.
In this example the affirmative Log. for the decimal frac-
tion is used. The excess of the product of 10 multiplied by
5, the index of the power, above 46 the index of the Log. is
4. This number, less one, that is 3 is the number of cy-
phers which must be fixed to the natural quantity corres-
ponding to the Log. .301030.
EVOLUTION, OR THE EXTRACTION OF ROOTS
BY LOGARITHMS.
RULE.
Divide the Logarithm of the given number by the index
of the power, and the quotient is the root required.
Note l.—When the index of the logarithm is negative,
and the divisor is not exactly contained in it, increase the
index by the smallest number that will make it exactly di-
visible. Carry this l)orrowcd number as so many tens to the
left hand figure of the decimal part of the Logarithm. Then
proceed with the division as usual.
Note 2.— When alHnnative indices are used for the loga-
rithms of .decimal fnictions, prefix to the index of the Log.
a figure less by 1 than the index of the power; then divide
the whole by the index of the power.
EXAMPLES.
1. Required the square root of 365.
Index of the power 2)52,562293 Log. of 365.
The root required 1.281146 Log. of 19.105 ^fi».
the index of
to be carried
it to — 7, as
[ecimal frae-
nultiplied by
f the Log. is
imber of cy-
ntity corres-
IF ROOTS
jy the index
red,
is negative,
increase the
t exactly di-
y tens to the
;hmi. Then
'or the loga-
of the Log.
then divide
05 Jlnt,
LOOlRITHKfl, 41
5. Required the Cube Root of 12345.
Index of the power 3)4.091491 Log. of I2S45
1.363830 Log. of 23. 1116 the rwot
required,
9. Required the Cube_Rootof .000000512.
Index of the power 3)7.709270 Log. of .000000512
3. 903090 Lo^ of .008 the root re-
quired.
Here the index of the Log. is not exactly divisible by the
index of the Power. Two, the smallest figure rvhich will
render it exactly divisible, is added to it, making it 9. Thi«
two IS then carried forward as so many tens to the decimal
part of the Log. Say 3 is into 27, &c.
4. Required the fifth root of . 00032.
Index of the power 5)46.505150 Log. of .00032
9.301030 Log. of .2 the root re-
quired.
Here the affirmative index to the Log. of .00032 is t) to
which a figure less by 1 than the index of the power, that
« 5 — 1 = 4 is to be prefixed, making AQ as above.
Explanation op the^ Tables op Logarithmic Sines,
Tangents, &c.
From the manner in which Lines of Chords, Sines, Tan-
gents. &c., .-ire projected, {See Prob. 23 of Geometry,) it is.
evident that if the Radius consist of any number of equal
parts the Sine, Tangent, Secant, &c., of every arc described
on that Radius, bearing a determinate proportion to it,
must also consist of a determinate proportional number
of these equal parts. The computation of the number of
these parts in the Sines, Tangents, &c., contained in every
arc of the Quadrant, form Tables of Sines, Tangents, &c.
In this lorm thnv nm n..n^.i Tvr„f i o- ^n, '
---^. — ,„,!^vt iiatuiu! oincs, langents. So-
cants, &c., and the Logarithms of these numbers give u^
1 ables ot Logarithmic Sines, Tangents, Secants, &c...
&
s
41^
LOQAUITHMS.
|!
I
To find the Logarithmic Sine, Tangent, ^e., of any num-
ber of Degrees and Minutes,
If the number of degrees given be less than 45, look for
them at the top of the page, then look for the number of
given minutes, in the left hand column; opposite to which,
in the column marked Sine, Tangent, &c., you will find the
Logarithmic Sine, Tangent, &c., of the arc proposed.
If the number of degrees exceed 45, and less than 90, look
for the given number of degrees at the bottom of the page,
and for the minutes in the right hand column ; opposite to
which, in the column marked at the foot, Sine, Tangent,
&c., you have the Logarithmic Sine, Tangent, &c., of the
arc of the specified number of degrees.
If the number of degrees exceed 90, take out the Loga-
rithmic Sine, Tangent, &c., of its supplement, that is of an
arc consisting of the number of degrees contained in the re-
mainder, wliicli will result from the subtraction of the given
number of degrees and minutes from 180''.
EXAMPLES.
Arcs. Sine. Co-Sine. Tangents. Co-Tang. Secant.
18° 15' 9.495772 9. 97758G 9.518185 10.481815 10.022414
64° 5^' 9.957040 9.627030 10.330009 9.66999110.372970
The Natural Sine for any number of degrees and minutes
will be found most readily from a Table of Natural Sines,
the arrangement and uses- of which must be sufliciently ob-
vious from the explanation e*" the Table of Logarithmic
Sines already given. When the Natural Sine and Co-sine
are known, the Natural Tangent, Secc-y^t, &c., are easily
calculated.
caiit.
EXAMPLES.
1. Required the Nat. Sine of an arc of 23"* 20'.
. 396080.
4. Required the Nat. Co-sine of an arc of 87^ 15'. ^n^
047878.
^n*.
V
o
f any nutn-
15, look for
number of
e to which,
rvill find the
posed,
lan 90, look
r the page,
opposite to
!, Tangent,
&c., of the
TRIGONOMETRY
the Loga-
hat is of an
)d in the re-
[)f the given
PbAWE Trigonometrv treats of the relations sutwiBting-
between the sides and angles of plane triangles. The prin-
ftipal parts of a triangle are, the three sides and the three
angles. The main object of Plane Trigonometry is to give
rules by which, when some of these parta arc known, tb*
others may be determined.
Triangles are either right-angled or obliqut angled. Plane
Trigonometry is therefore very naturally divided into twa
parts. The first treats of right-angled triangles; and th« »«-,
cond, of oblique-angled triangles.
Paet I..
OF RIGHT-ANGLED PLANE TRL\NGLES,
DEFINITIONS.
1. In a right-angled triangle the Hypotenuse is the sitl*
opposite to tlic right angle.
2. The Base is the side opposite to the vertical angle.
3. The Perjjcndicular is the side which forms a right an-
gle with the base.
Notej,— The base and perpendicular are sometimes called
Corq^ary. — in a right angled triangle, if one of the acute
angles is given the other angle is given also.
Dem^slration.— The three angles of any plane trianf?l»
44
TBIOONOMKTRT.
If)
H
l»l H
1i ■ I
isl3^n ^h. .f •• N«^^> when one of the acute angles
IS known, the other is at once ascertained by subtractinir the
known angle from 90° and the remainder is^ the rnea "f e of
the other angle being the complement of the given angi;
a e of'fhpth^'"^'r"^'^'/"/ ^^« «f th^ Principal %nn
in/* r *^*'*' ^"^''^ ^^'^ «^ the three angles) beiL elven
and one of these given parts being a side! the other S
Sfth^oi^msfi::'^ '^ """^"^ easily iduced TromThe foC
CASE I.
t^hen a leg and the angle opposite to it, or when two sidet
are given, to find the other part:
rie^f^tTririh """^ f '^^ 'I ^. *^^ ^^^"^ «f ''' opposite an-
gle, so is any other side to the sine of its opposite aM^le- «nH
lots z'leV''' ^'"r' ^">; ^"^»« ^^^ to^^cs^«dt
•0 IS tlw sine of any other angle to its opposite side.
CASE 11.
IVhen the legs, that] is the sides about the right ahgle are
given, to find the angles and the hypotenuse :
J^heorem.-^As one of the given sides is to the other ffiven
Tn^'/?,'" "f'^'"' *^ '^^ tangent of the acute an^le ft the
end of the side at which the proportion commenced!
PROBLEM I.
Oiven the angles and hypotenuse of a right-angled plant
triangle to find the base and perpendicular : >
EXAMPLE.
In the triangle ABC, (Fig. 49,) right-angled at B, given
the angle at C, 55'^ 30', and the hypotenuse A C 121 yard.-^
required the sides A B and B C. ' '
According to the preceding corollary / C 56° 80' — / B
90^ === Z A =. S4« 30'. , ,
To find the side A B. ' • i'
As radius, (/. e. the sine of ^ B,) = to 00000
Is to the side AC 121 ^ '^ == "oS
So IS the sine / C. fifio Qo' Z I^^i^
To the perpendicular A B 99. 72 yds. =. 7^99871^
.1!
1
TRIGONOMETRT.
46
180". A right
! must together
! acute angles
mbtracting the
he rneasuie of
given angle.
>rincipal parts
) being given,
le other partsi
)m the foilow-
vhcn ttoo sidet
i opposite an-
ite angle : and
>pposite side,
3 side.
J^hf ahgle are
•nuse :
le other given
e angle at the
3nced.
angled plant
ulitr :
%
>.
d at B, given
C 121 yards ^
'80' — ^ B
0.00000
2 . 08278
9.91599
1.9987X i
Here we add the logarithms of the second and third term
of the proportion, and subtract the logarithm of the first
term from that sum. The remainder is the logarithm of tho
fourth term, or the answer.
To find the side C B.
As radius 10.00000
Is to hyp. A C 121 2.08278
So is sine / A 34° 30' 9.75312
To the base C B 68.54 yds. 1.83590.
PROBLEM 11.
Given the angles and one side, to find the hypotenuse and
the other side.
EXAMPLE.
In the right angled triangle ABC (Fig. 50,) right angled
at B, given the angle at A 85° 30', and the side A B 294
chains, required the base C B and the hypotenuse A C.
90° — / A 35° 30' = / C 54° 30'.
To find the hypotenuse A C. To find the base C B.
As sine Z C 54° 30' 9.91068 As sine / C 54° 30' 9 91068
Js to per A B 294 ch. 2.46834 Is to per. A B 294 ch. 2.46884
bo isradms 10.00000 So is sine /A 35° 30' 9.7639&
To hyp. AC 361. Ich. 2.55766 To C B 209.7 ch. 2.32161
PROBLEM III.
Given the hypotenuse and one side, to find the angles and
the other side.
EXAMPLE.
In the right angled triangle ABC, (Fig. 51,) right angled
at B, given the hypotenuse A C 3 chains and 50 links, and
the perpendicular A B 2 chains and 45 links; required tho
angles A and C, and the base B C.
To find the angle C. To find the side B C.
As hyp. A C 8.50 0.54407 As radius lo 00000
Is to radius ^ 10.00000 Is to hyp. A C 3.50 o:54407
oc IS sme a jj a. 45 0.3891 / So is sine /A 45° 35' 9.85386
f I
To sine ^ C 44° 25' 9.84510 To base B C 2.499 0.8979S
ir
46
TRIOONOMETRY.
The hypotenuse may befoiind independently of the angles;
for, according to Euc.
= V A B (A B -f
BC
. 47. we have A C = VA B'-f-BC*
This latter form of expression
A B ^
A C is by far the most convenient for logarhhmic calcula-
tion.
From the same property of aright angled triangle, viz:
that the square of the hypotenuse is equal to the sum of
the squares of the other two sido^ anyone of the two sidea
about the right angle may he found independently of the
angles, if the hypotenuse and the other side are given or as-
certained. For since A C^ = A 13^ + B C=, it follows that
B C^ = A C= - A B'=(A C + AB ). (A C - A B); and
therefore B C = V (A C + A B). (A C ^ A B), from
which expression B C is easily determined; or, Let H de-
note the hypotenuse; B, the base; and P, the perpendicular-
then, PP = B' -f- P2; and H= — B= = P'; and H^ — ps = B"
Part II.
OF OBLIQUE-ANGLED PLANE TRIANGLES.
In an oblique-angled plane triangle, a side and any other
two of the principal parts being given, the other principal
parts may be ascertained.
In every plane triangle the sum of the three interior angles
rs equal to two right angles, or to 180°.
Corollary 1. -.Two angles of any plane triangle being
given, the thn-d is also given; for it is the supplement of the
rtther two, and may be found by subtracting their sum from
Corollary ^.—Onc angle of any plane triangle being given,
the sum ot the other two is also given, and may be found by
subtracting the given angle from 180°.
The principles or rules by which unknown parts of ol>*.
Uque-angled triangles may be determined from those which
Are known, are evolved in the following theorems.
TEIGONOMETRT.
47
CASE I.
re ffiven or as-
nterior angles
IFhcn a side and two angles, or when two sides and the
angle opposite to one of them, are given:
rAeorem.--Tlie sides of a plane triangle are to one an-
other as the sines oi their opposite angles, and vice versa,
CASE ir.
fVhcn two sides and the included angle are given:
TA^omn --The sum of any two sides of a plane triangle
IS to the dilirronco between then,, as the tangent of half th«
Ibmue ^' "l'I»>«'te angles is to tlie tangent of half their dif-
.SWio/mm --Having ascertained half the sum an.l half the
difference of tlu^ unknown angle., the angles are easilv de-
termined; for half the difll-rencS being added to half the sum
?rn''.'^H-.r"'''" ""^^'^^ Hnd half tlfe difference subtracte™
from half the sum gives the less..
CASE HI.
When the three sides are given:
Theorem.~M the base of any plane triangle is to th«
«umof the other two skies, so is\he differ^.ce between
these tvvo sides to the difference between the segments into
which the base IS divided by a perpendicular let fall upon it
from the opposite angle. *^
»SVAo/mm— Slaving obtained by the preceding theorem
half the d.flerence of the segments, the segments fheiST.
are easily found; for half the difference added to half the
'^^Zf^T I ^H^T' ^^'«'"?"t, and half the difference sub-
tracted from halt the sum gives the less.f
PROBLEM I.
Given the angles and one side of an oblique-angled. triofigU
to find the other sides.
EXAMPLE.
In the oblique-angled triangle A B C, (Fig. 52,) given th*.
angle at B 46^ 22', the angle at C 54= 15', and consequently
the angle at A 79- 23'; and the side B C 1 ch.. 35/: required^
the sides A B and AC.
^_* Tor the demonelratiou of these theorems, ae« the App«...
41 TRIOONOMRTRT.
To find the side A B. To find the side A C.
As Sine / A 79° 23' 0.99250 As Sino / A 79° 28' 9.99250
.s to side li C 1.35 /. 2.13033 h to side B C 1.35 I. 2.13033
So is Sine Z C 54° 15' 9.90932 So is Sine / B 4G" 22' 9.«5960
To side A B lllJ /. 2.04715 To side A C 99.4 /. 1.99743
PROBLEM II.
Given ttco sides and an angle opposite to one of them, to
find the other angles and the remaining side. -
EXAMPLE.
In the oblifiuc-angled triangle ABC, (Fig. 53,) obtuse at
B, given the side A C S ch. 18 /. the side B C 1 cA. 95 ^ and
the angle at A 32° 40'; required the angles at B and C, and
the side A B.
To find the angle at B. To find the side A B.
As the side B C 195 /. 2.29003 As Sine / A 32° 40' 9.73219
h to Sine Z A 32° 40' 9.73219 Is to side B C 195 /. 2.2900.'5
So is side AC 318 /. 2.50242 So is Sine Z C 29° 9.68557
To Sine of Z B 61° 40' 9.94458 To side A B 175.1 1. 2.24341
But by the data, B is an obtuse angle. It is therefore the
eupplement of an angle of 61° 40', or 180° — 61° 40' = 118^
ao'=ZB.
PROBLEM III.
Given two sides and the included angle to find the other
angles and the remaining side.
EXAMPLE.
In the triangle A B C, (F?;g-. 54,) given the side AC
919.95 I. the side xY B 500 /. and the included angle at A |
a6° 52'; required the angles at B and C, and the side B C.
To find the angles at B and C.
3.15227
2.62319
10.47716
A8AC + AB = 1419.95
IstoAC — AB = 419.95
So is Tan. i Z s B-t-C = 71°34'
To Tan. A Z 8 B — C 41° 35' 9.94808
Then 71° 34' + 41° 35' = 113° 9' = Z B; and 71° 34' ~ 41«
S5' = '29°59'=Z C.
TRItiONOMETRY.
49
iide A C.
= 23' 9.99250
SrW, 2.1303S
IG" 22' 9.85960
,4/. 1.99743
To find the side B C.
As Sine Z C 29^ 59' 9.69875
Is to side A B 500 /.
So is Sine ^ A 36"^ 52'
To side B C 000.26
2.69897
9.77811
2.77833
rie of them, to
<^ side.
53,) obtuse at
1 ch. 95 ^ and
B and C, and
side A B. ,
32° 40' 9.73219
195 /. 2.2900.S
C 29*^ 9.68557
75.11.2.24.341
5 therefore the
6^40' = 118^
find the other
the side A C
3d angle at A
^e side B C.
3.15227
2.62319
0.47716
9.94808
I 71° 34'
— 41''
PROBLEM IV.
Given the three sides of an oblique-angled plane triangle to
find the angles.
EXAMPLE.
In the triangle ABC, {Fig. 55,) given A B 5 c/i. 62/.,
A C 8 ch., and B C 3 cA. 20 /.; required the angles.
To find the segments into xohich the base A C is divided by
a perpendicular D B let fall upon it from the opposite
angle B.
As the base A C 800 /. 2 . 90309
Is to A B + BC = 562-1-320 = 882 2.94546
So is A B — B C = 562 — 320 = 242 2 . 4261^
77 -*<? ■'y
9,
To AD — DC 266.8 2.46756
Now i (A D -t D ( ) = 400, and i (A D — D C) = 133 . 4;
therefore 400 -f- 133 . 4 = 533 . 4 = A D greater segment, and
400 — 1 33 . 4 = 266 . 6 = D C less segnnent.
The segments of the base may also be obtained by the fol-
lowing rule :
From the sum of the squares of the two greatest sides
subtract the square of the least side, and divide the remain-
der by twice the greatest side-, the quotient will be the great-
est segment.
Thus:
AC
A C2 =
BC'^ =
BC2 =
A B2 =
640000
102400
742400
315844
Then A C = 800
DC = 266.6
AD
bSS.
i
(A C»4- B C^)— A b'= 426556
2 A C 1600
= 266, G=DX
in
I' <!«
■Si!
^
MEHSURAXrON OP HEIGHTS AND DISTAftCtS.
To find the angle at C.
As side BC 320 2.50515
IstoRad 10.00000
oo IS side D C 266.6 2.4258G
To Sine ^ C B D 56° 26' ! , 92071
Then 90'^ — 56" 26'= 33^^ 34' = / at C.
To find the anifle at A.
As side A B 562 2, 74973
IstoRad, 10.00000
bo IS side A D 533.4 2.72705
To Sine ^ A B D 71° 39' 9.97732
Then 90"— 7F S9'= 18° 21' = / A.
Lastly, / C B D 56° 26' -f- / A B D 71° 39'= 120° 5' =
MFiNSURATION OF HEIGHTS AND DISTANCES.
Any of the instrumei.^ts employed in surveying may be
nsedto determine lines and angles which are inaccessible.
h or determming vertical angles, the Quadrant is the least
expensive. It is the fourth part of a circle divided into de-
grees, Sec, and furnished with a plummet suspended from
the centre, and with sights fastened upon one of its radii.
PROBLEM 1.
At the distance of 3 ch. 10 I. (Fig. 56,) from a wall, and
on a level with its foundation, the angle of elevation is ob-
served to be 15° 40'j required the height of the wall.
As Sine Z C 74° 20' 9.98355
Is to the base A B 310 /. 2 49136
So IS Sine / A 15° 40' 9.43142
To height of wall B C 86 94 ^. 1 . 93923
P. .OBLEM. II. (Fig. 57.)
Standing on the top of a tower I3Gi feel in height, i
ob..erved a tree ut a distance on the plane, a straight line lo
) DISTAliCtfi.
c.
2.50515
10.00000
2.4258G
9.92071
A.
2.74973
10.00000
2.72705
9.977S2
> 71° 39'= 120° 5' =
VD DISTANCES.
I surveying may be
ch fire inaccesaible.
Liadrant is the least
■cle divided into de-
iet suspended from
ii: one of its radii.
,) from a wall, and
of elevation is ob-
of the wall.
9.98355
2. 19136
9.43142
1 . 93923
57.)
i feel in height, 1
e, a straight line lo
MtNSURATION OP HEIGHTS AND DISTANCES.
l\
which from the top of the tower makes with the wall an an-
gle of 67° 20'; required the distance of the tree from the bot-
^ torn of the tower.
As Sine / A 22° 40' 9.58587
Is to the height of the wall 136.5/if, 2. 13513
So is Sine Z C 67° 20' 9.96509
To the distance A B 326.8^3?.
2..^;4^S5
PROBLEM III.
Wishing to know the breadth of a river, (Fig. 58,) I mea-
sured for a base a straight line, 250 links in length, close to
the bank. From each end of this base line I found the an-
gles subtended hy it and a tree at the brink of the river on
the opposite side to be respectively 53° and 79° 12'; required
the breadth of the river.
Lot a perpendicular fall on the base from the opposite
angle at A, the length of that perpendicular is the breadth
of the river. Firet find the length of the side A B. Now
we have / B 53°, and ^ C 79° 12', consequent! v ^ A is 47°
48' and the side B C 250 links. Then,
To find the length of the side A B:
As Sine Z A 47° 48' 9.86970
Is to side B C 250 I. 2.39794
So is Skie / C 79° 12' 9.99223
To side A B 33i .5 2.52047
Now we have the right-angled triangle B A D, in which
;»re knovv-li the side A B 331.5 links and the angle at B 53°
Then As Rad. 10.00000
Is to side A B 331 . 5 ;. 2 52048
So is Sine / B 53° 9.90234
To sfdc A D 264 . 7 2 . 42282
The perpendicular breadth of the river accu.dingly in
iu.li.
PRORT.FIU 1X7 /I7»J~ rn V
Wishing tu Know the height of, and my distance from
an ol^ect apparently on a tevel with the place on which I
1,
■1 ^;
hi
MENSURATION OF HEIGHTS AND DISTANCES.
Stood, oti the opposite side of a river; and being unable
to mctisure backward on the same plane on account of the
immediate rise of the bank, I placed a mark at the place
on which I stood. I then measured a distance of 264
Imks up the ascendinnf ground in a straight direction from
the object. At this station it was evident that I was above
the level of the object. Looking through the sights of the
quadrant first to the mark at my first station, I found the
angle of depression (A E D) 42\ Looking in the same
way to the bottoni of the object, I found the angle of de-
pression (D h A) to be 27^ Directing the instrument in
like manner to the top of the object, the angle of depression
(D C F) was found to be 19°; required the height of the
object and the distance between it and the mark placed at
the first station.
Let fall the perpendicular D A on the straight line A B,
the angle at A will be a right angle. Find fu-st the length
of the sides A D and A E.
In the triangle A E D, right-angled at A, we have the hy-
potenuse D E 264 links, and the / E 42°; rnd consequently
the Z E D A 48°.
Therefore As Rad. 10.00000
Is to hyp. 264 /. 2.42160
So is Sine / E 42° 9.8.2551
To side AD 176.7
And As Rad.
is to hvp. 264 /.
So is Sine / A D E 48°
2.24711.
10.00000
2.42160
9.87107
To side A E 1 96 . 2 2 . 29267.
Find next the length of the line A B. Now we have in
the right-angled triangle ADB the side A D 176.7 L,
and the / B 27°, and consequently the / A D B 63°.
Therefore As Sine / B 27° 9 . 65704
Is to side A D 176.7/. 2.24711
So is Sine / A D B 63° 9 . 94C38
To side A B 346 . 7 /. 2 . 539i«5
We found before A E 196. 2 I., tind now find A B 346. 7 /.,
NCES.
MENSirnATlON OF HF.IGIITS AND DISTANCF.S.
.'>;i
being unable
ccount of the
at the place
tancc of 264
rection from
I was above
sights of the
I, I found the
in the same
angle of de-
istrunient in
)f depression
might of the
rk placed at
fht line A B,
it the length
have the hy-
;onsequently
7
7.
we have
in
D 176.7
/.,
3 63°.
B 346.7/.,
therefore BE must be 1.50.5 /., vvhicii is the distance be-
tween the first station and the bottom of the object.
We have still to find the height of the object, or the lengtli
of the side C B. Draw C F parallel to A B, and because
D A ana C B are perpendicular to A B, they must be paral-
lel to each other; and because F C is parallel to A B, F A
and C B must be equal. Having already found the lengtli
of A D to be 176.7/. if we can ascertain the length of
][) F, the length of F A, and consequently of C B will be
easily determined. Now in the triangle F D C riglit-angled
at F, we have the base F C = A B, which we h e found to
be 346. 7 /., and the angle at C 19°, and consequently the an-
gle F D C 71°, to find the side D F.
As Sine Z C D F 7P 9.97567
Is to side F Cor A B 346. 1 /. 2.53995
So is Sine / C 19<^ 9.51264
To .side DF 119.4 Z. 2.07692
It was before ascertained that A D is 176.7 /., and now
that D F is 119.4 / in length, A F must therefore be 57.3 /.
But A F and C B are equal: C B is therefore 57.3 /., which
is the height of the object.
The methods of ascertaining the heights and distances of
inaccessible objects are numerous, if, however, the stu-
dent fully comprehends the preceding examples, :^e will
have r.o difficulty in applying the principles of Trigonome-
try for the solution of any problem that may occur.
Wherever sufficient level space can be obtained, the fol-
lowing rule is applicable and expeditious:
Let the observer retire from the object until the angle o
elevation is 45°, the distance between the place of observa-
tion and the object is equal to the height of the object. But
if the object be inaccessible, let the observer find the point
at which the angle of elevation is 26° 34'. Having marked
the spot let him advance towards the object until he finds
the angle of elevation is 45°; or let him retire from it until
the angle is 18° 26', in either case the distance between the
first and second places of observation will be equal to the
height of the object.
.*■
m
I
t i,
LAND SURVEYlNrx.
INSTRUMENTS EMPLOYED,
I. The Chain.
The Chain is a measure consisting of a certain nunib«^r
pf links of strong iron wire, very generally employed in
surveying for the purpose of measuring lines or distances.
It is in length four poles, or sixty-six feet, or one hundred
links. A link is therefore 7.92 inche:5 long. Hence it is
easy to reduce any number of links to feet, and vice ver&a.
To assist in counting the number of links, when any dis-
tance does not amount to an exact number of chains, a'small
piece of brass, generally marked, is attached to the end of
every tenth link, dividing the chain into ten equal parts,
Instead of the chain, a half chain, or, as it is often called,
a two-pole chain is very frccjuently employed. For mea-
suring lines rji cleared level land, such as marshes, inter-
vales, &c., the whole chain or four-pole chair is the more
convenient. In British North America, however, for which
this treatise is principally designed, us a great part of the
business of a Surveyor consists in running lines, and making
surveys mi the forest, and on every variety of surface, the
half chain is generally to be preferred. The reason is obvi-
ous. It is much easier to keep a two-pole chain in a straight
and horizontal position, than one of t^vice its length.
By frequent use the rings which connect the links of tho
chain are apt to open, and thus the length of the chain is in-
creased. Before proceeding to measure any line, tho Sur-
veyor should therefore carefully examine and measure his
Cham. To this point too much attention cannct be paid.
Two chain-men or chain-bearers are generally employed
to Carrv tho. rha'in TTno" ♦*>"!»• ^"V"^'-l-^ r-. i -^-"--
^. — _„ii-.i, ^^}i,,, ti£vir v.-.rtriuiiiusa una Kirici com-
pliance with the directions of the Surveyor, the correctness
LAND SURVEYING.
55
of ihe survey in a great measure dependg. Over their con-
duct therefore it is of great importance that he exercise n
careful supervision.
Before commencing the measurement of a stationary dis-
tance, an object easily seen is to be placed at one extremity
of the line to be measured. The hindmost chain-man then
takes up his position at the other extremity, holding the end
of the chain exactly at the end of the line. The other
cham-bearer being previously furnished with, and now car>-
rying ten pins, sharpened at one end, about ten inches in
length, if the surface to be measured is smooth, and about
eighteen inches in length if the surface be uneven, or if it
be 1,1 the woods, and holding the other end of the chain,
proceeds towards the object placed at the farthest extremity
of the line. It is now the duty of the hindmost chain-bear-
er to direct the course of the foremost. Having advanced
until the chain is stretcliQH, the latter is directed, if necessa-
ry, by the former to move to the right hand or to the iei\
until he is in a direct line with the object toward which they
are advancing. When he covers that object from the sight
of the hindmost chain-bearer, the latter knows that he is in
the proper course, and with a motion of the hand, or other-,
wise, directs his companion to stick one of his pins exactly
at the end of the chain. Both chain-bearers then advance
simultaneously toward the object at the end of the line, un-,
til the hindmost arrives at the place where the pin was de-
posited, lie then directs the foremost chain-bearer as foi-s.
merly, and pulling up the pin carries it carefully with him.
Thus they proceed until either the whole line is measured
or until all the pins carried by the foremost chain-man are
exhau.«\d. In the former case, if the line do not contain
an exact number of chains, the distance between the last
pin and the object at the end of the line is measured in links.
The number of pins held by the hindmast chain-bearer ex-
presses the number of chains or half chains measured. This,
with the odd links (if any) added to it, is the length of the
line. lii iho hitter case, at the end of the first ten chAins,
the hindmost chain-man returns all the pins tatihe foremo^t,.^
.)(]
'\ 't
1% H
1 IS , :*
iii
lifl
n
w ''H
1 v
1
LAND SURVEVING,
a note of the transfer is taken, which is sometimes called
keepins; tally,— and the chain-bearers proceed as before, un-
til the whole line is measured. Then the number of trans-
fers, or tallies, each counting ten chains,-— the number of pins
held by the hindmost chain-bearer, counting each a chain,
and the number of odd links (if any) shew the length of the
line.
It must be very evident that, in a survey, much depends
upon the hindmost chain-bearer. Inaccuracies very fre-
quently occur in consequence of bad chaining. If, there-
fore, a Surveyor cannot procure, for a hindmost chain-bear-
er, an individual upon whom he can rely, he ought to act in
that capacity himself. This is another point on which he
cannot be too careful.
The surveyor will requi-e likewise to caution the chain-
bearers against losing any of their pins, and also to teach
them that inclined planes, such a^he sides of hills are to be
measured horizontally, and not along the incliiied plane.
This subject however, we will discuss more particularly
when we come to treat of the running of lines.
n. The Circumferentor.
This instrument is employed by surveyors for taking an-
gles. It consists of several parts :
1. A brass box, about five or six inches in diameter.
Within this box are; 1st, a brass graduated circle, the upper
surface of which is divided into 360 degrees, and numbered
10, 20, 30, &c., to .360. The bottom of the box is divi-
ded into four parts or quadrants, each of which is subdi-
vided into 90 degrees numbered from the meridian, each
way to the east and west points. And ^ndly. a steel pin in
the centre, called a centre pin or pivot, finely pointed, upon
which IS nicely balanced a needle, touched bv a loadstone,
which, when at rest, and when the box contahiing it is in a
horizontal position, always points in a North and South di-
rection nearly. To the bottom of this box is also attached
a Plidc, by means of which the nee.llo may be raised from
the centre pin, or pivot, to prevent the fine point of the jat-
'*■,!
LAND St'RVETING.
57
mos called
before, un-
r of trans-
ber of pins
h a chain,
igth of the
h dependu
very fre-
If, there-
hain-bear-
it to act in
which he
the chain-
to teach
1 are to be
ed plane,
irticularly
aking an-
diameter.
the upper
lumbered
X is divi-
is subdi-
ian. each
el pin in
ted, upon
aadstone,
it is in a
South di-
attached
ied from
'the lat-
ter from being blunted when the instrument is carried from
one place to another. The box is also covered with a glass
lid to preserve the needle from being disturbed by wind or
injured by rain at the time of using. When the instrument
is to be carried from one place to another a brass lid or co-
ver is placed over the glass to protect it from accident.
•2. A brass index or ruler, about 11 or 12 inches in length,
to the ends of which, and perpendicular to it, sights are at-
tached, and to which the box above described is fastened by
screws. In each of these sights are two apertures, or slits,
the one above the other, and the one much wider than the
other. In one of the sights the wide slit is uppermost; in
the other, it is below. Through the widest of these aper-
tures is placed longitudinally, a horse hair or fine silk threa<I,
to assist in taking an observation with greater exactness.
3. Two levels at right angles to each other are attached
to the bottom of the instrument, by the aid of which it may
be readily levelled.
4. A ball and socket on which the instrument may be
placed, and by which, with the assistance of a screw, it may
easily be adjusted horizontally in any direction. The wholo
is supported, when used, by a conmion surveyor's staff.
, ^lh~Jr^^- levelling of instruments is generally done by
the Spirit Level or Magnetic Needle. '= ■^ J
By the former, as the Needle is subject to what is called
the dip, m consequence of which, there will be a difference
between an instrument levelled by the Spirit Level and ono
leveled by the Needle; hence, the latter way will be pre-
tcrab e, which is done by placing the instrument so that the
Needle will play freely and parallel to the bottom of the
Compass box.
III. The Theodolite.
The Theodolite is a complex, but most convenient
and valuable instrument. On account of its expense few
Surveyors in British North America are able or disposed to
purchase it. In consequence of its cnniplevity it is difficult
to give a description of it which would be at all intelligible,
without an inspection and examination of the instrument it-
•If nil
58
LAND SURVEYING.
h m
self. The following remarks, however, may serve to give
some idea of its intricacy and importance. Its principal
parts are,
1. The Horizontal Limb.— This consists of two circular
plates, the upper or verniei plate, and the graduated limb.
The former moves freely above the latter, without actual
contact, and both have a horizontal motion about a vertical
axis, consisting of two parts, the one external, fixed to the
graduated limb, the other internal, fastened to the vernier
plate.
2. The VERNiERS.—These are short scales on the upper
plate, and on ofpposite sides of it, or 180° asunder. They
are minutely graduated, and so placed as to subdivide th'e
divisions of the lower plate into minutes. By the assistance
•of microscepes frequently placed over the verniers, the hal<^
or even the fourth of a minute may be estimated.
3. The Parallel Plates, which serve for levelling the
instrument, and arc held together by a ball and socket.
4. Spirit Levels.— Two of these are placed at right an-
gles to each other, with adjusting screws on the plane of the
vernier plate, to determine Avhen the horizontal limb is truly
level,
5. A Compass or Circumperentor.— Thisis placed up-
on the vernier plate, and is very useful in pointing out the
meridional line, or the situation of the land.
6. The Vertical Arc and Telescope. The arc is
placed on a horizontal axis, the ends of which arc supported
by two frames. One side of it by means of a vernier, may
be read off to single minutes; the other side shews the differ-
vnce between the hypotenuse and the base of a right angled
triangle, or the number of links to be deducted from each
chain length to reduce hypotenusal to horizontal lines.— The
level, which is under and parallel to the telescope, is fasten-
«d to it at one end by a joint, and at the other end by a
screw, by which that end may he elevated or depressed.
There is also a screw at the jointed end for lateral adjust-
mont. By their assistance, the level may be placed parallel
*o fhe axis or line of collimution of the telescope.
PH"*
LaKd surveying.
59
erve to give
ts principal
two circular
'uated limb,
hout actual
Lit a vertical
fixed to the
the vernier
n the upper
der. They
bdivide the
e assistancj*
!rs, the half
svelllng the
locket,
at right an-
planeof the
mb is truly
placed up*
ing out the
rhe arc is
! supported
srnier, may
5 the differ*
ight angled
from each
nes. — The
5 is fasten-
end by a
depressed*
•al adjust-
ed parallel
By the Theodolite angles whether vertical or horizontal
may be measured with great accur.-icy. It will give the an-
gles of a field, and the bearing of any stationary distance
line from the meridian, in the b.wne manner in which these
may be obtained ty the Circumferentor, and Quartered Com-
pass. Before this insirument can be applied to practical
purposes with accuracy, its parts must be adjusted to each
other by means of the screws and levels. The first adjust-
ment is that of the line of collimation. The second places
the level utti. -hed to the telescope parallel to the rectified
line of collimation. The third makes the axis of the hori-
zontal limb truly vertical by means of the telescope level,
which is most to be depended upon. Then the levels on the
vernier plate ;,re adjusted jby thoir screws, so that their air
bubbles may remain stationary in the middle of their tubes,
while it makes a eomjjlete rev^' tion on its axis. When
these adjustments are perfect, tue vernier of the vertical arc
must be so set that its index will point to 0, or zero on the
arc, or else its deviation from zero must be marked and ap-
pli ed as an index error.
Several other instruments may be employed in measuring
surface, such as th(! Semicircle, Plane Table, and Cross
Staff, which do not require an extended notice.
The Semicircle may be employed for taking angles. It
will be observed, however, that in using this instrument
only one end of the index rests upon it. The number at'
degrees however^ marked by the other end may be obtaineti
from the end resting upon the semicircle, by substituting
181 for 1, 182 for 2, &c., proceeding onwards to 360.
The Plane Table may also bo employed for taking angles^
and in numy r(jspects serves the purposes of a Theodolite or
Semicircle. It is indeed a very valuable instrument, but in
eonse(iuence of its clumsiness it is not at all adapted to the
.survey of wilderness land.
The Cross Staff is a very simple instrument, used princi-
pally tor laying off perpendiculars, or offsets, on or from a
60
LAND SLRVEVrNG.
.1
'I
strai^'ht line. In measuring fields whore no obstructions
lie in the way, it is very useful.
The Protractor is an instrument for laying down and
measuring, with accuracy and despatch, angles upon paper,
by which the use of the line of chords is superseded. It is
principally emj)loyed in delineating or drawing a plan from
Field Notes. It is variously formed and constructed of dif-
ferent materials. It usually contains three concentric cir-
cles, at such distance from each other that figures may be
contained between them. The outward circle is numbered
from the right to theTleft hand, with 10, 20, 30, &c., to 180'^;
the middle circle is numbered in the same direction from
130^ to 360^; and the inner, from the upper edge both ways,
with 10, 20, 30, &e., to 90*^.
Instruments useful to a Land ??jRVEyoR.
A Case of good Pocket or Mathematical Instruments.
A Set of Feather-edged Plotting Scales.
Two or three Parallel Rulers.
A Cross Stafl^
A Circumferentor.
A Sextant.
A Theodolite.
A Surveying Compass.
Measuring Chains.
A Spirit Level with a Telescope.
A Protractor.
A Quadrant.
A Copying-*^?iis3.
An Azimuth Compass.
In selecting Mathematical Instruments particular attention
should be paid to their co-aptation and adjustments. Reject
those whose principal parts are immovable, for they cannot,
at least without difficulty and expense, be rectified or ad-
justed. Whatever pains may have been bestowed upon their
original fr)nstruction, it need not bo fixpnctod that they will
continue as correct as when they came from the hand of /he
Mathemati' -d Instrument Maker.
1-
ohstriictions
T down and
upon paper,
edcd. It is
a plan from
ucted of dif-
icentric cir-
ires may be
s numbered
ic, to 180'^;
cction from
both ways,
EVOR.
uments.
Si
land survktinq.
Use op the Chain.
«)
PROBLEM I.
To reduce half chains or two-pole chains and links to whole
or four-pole chains and links.
RULE I..
If the given number of half or two-pole chains be even
divide it by 2, and the quotient with the given number oi
links annexed will be the number of chains and links re»
quired.
EXAMPLE.
In 18 half chains and 40 links how many chains and links ('
2)18 40
ch. 9 40 /. j3n«..
RULE II..
But if the given number of half chains be odd, divide as
before, and add the remainder which is equal to 50 links to
the given number of links, and it will give the number of
chuins and links required.
EXAMPLE.
In 15 half chains and 20 links, how many chains and
links ?
2)15 20
ch. 7 70 I. Am.
lar attention
Its. Reject
hey cannot,
ified or ad-
1 upon their
[It they will
hand of ^he
PROBLEM II.
To reduce chains md links to half or two-pole chains and
lii^s.
RULE.
Reduce the whole to links, and divide by 50; the quotient
will be the numberof half chains, and the remainder will bt
links.
CI
LAKD SURVETING.
EXAMPI.F..
InB ch. 8? /. how many half chains and links?
ch. I.
8' 82
100
50)882 /.
halfchs. 17 32/. Ans.
PROBLEM [II.
To reduce chains and links to perches and decimal p arts of
a perch.
RULE.
Write down the chains as whole numbers, and the links
as decimals; then multiply by 4, and the product will be the
number of perches and decimal parts of a perch.
^f^^y^'~'^^^'' u^'^'''" ""S *'''" '""'^^ '^ obvious. As there are
100 Imks m a cham, a Imk i.^the hundredth part of a chain
By writing them with the decimal p.Mut before them, thev
become decimal parts of a chain, observing only that if the
number ot Imks do not exceed 9, a cypher must be writtea
before it, in order that it may express hundredth parts.
EXAMPLE.
In iO ch. 64 /.. how many perches?
ch.
10.64
4
In
41. b^ per. Ms.
PROBLEM IV.
To reduce half or txoo-pole chains and links to perches mid
decimal parts of a perch.
RULE.
Reduce the given number of half chains and links to
chams and links by Prob. I; thenreduce them tn n.v.h«- ...jd
docnnal parts of a perch by the preceding rule. '
LAND SURVETINC.
OS
EXAMPLE.
In 11 half chains and 21 links how many perches?
ch.
2)11 21 5.71
4
ch$. 5 71 I.
22.84 per. ^ns.
mal parts of
md the linlcji
t will be the
As there are
t of a chain,
them, they
f that it' the
It be written
h parts.
PROBLEM V.
To reduce perches and decimals of a perch to chains and
links.
RULE.
Divide by 4 so as ta have at least two places of decimals,
the whole numbers in the quotient will be chains, the first
two places of decimals will be links, and the remainder will
'be decimals of a link.
EXAMPCE.
In 22.32 perches how many chains and links.-'
chs. 5 58 /. ^ns. A *t
perches and
tid links to
perches and
PROBLEM VL
To reduce perches and decimals of a perch to half or two-
pole chains and links.
■RULE^
Divide the whole rumbers by 2, the quotient will be the
number of half cha,;us. To the remainder annex the deci-
mals and divide by 4, the quotient wiH be the number of
links.
EXAMPLE.
In 27. 52 perches how many half chains and links'
2)27.
half chs. 13.152
= 38 7. Ans. i3 half chs. 38 Y.
«'
M
LAiro smvETiifo.
PROBLEM VII.
To reduce chains and links to feet and decimal part, of a
foot.
RULE.
Write down the chains as whole numbers, and the links
08 decimals; then multiply by the number of feet in a chain
and the product will be the number of feet and decimal parts
of a foot required.
EXAMPLE.
In 7 chs. and 21 /. how many feet and decimal parts of a
foot? '
7 21
66 =/(?cf in a chain.
508.86//. Ms.
PROBLEM VIII.
To reduce feet and inches to chains and links,
nuLE.
^ Reduce the inches to the decimal of a foot, and annex
It to the gnen number of feet, then divide by 66. Continue
the division by annexing cyphers to the dividend until you
have two places of decimals in the quotient. Then the
whole numbers in the quotient will be the chains, and the
decimals Avill be the links required,
EXAMPLE.
In SlOyjf. 6 in. how many chains and links.**
60 210.50
= 5 /•/., then ■ = 3 . 18 or 3 chs. 18 /. Jns,
12
66
u
PROBLEM IX.
To take a survey by the chain only.
EXAMPLE.
Let A B C D E A {Fig. CO,) be a field. It is required to
survey u and to lay off the angles with the chain onlv.
r^
-^ \
LAND SlfRVEYi.ViJ.
65
nl parts of a
md the links
!t in a chain,
ecimal parts
al parts of a
links,
and annex
Continue
il until you
Then the
IS, and the
/. ^n».
cquired to
onlv.
RULE.
Commences mij r ngular point A, and on the straight
line A E n-enuro irom the point A own .'hain towards E,
and set up a ^ tJ,o or other niark exactly at the end of the
Irst chain rs at f in the same manner measure from the
point A in I. dire ' line towards 13, one chain, at the end of
which set up ^r,'.n . stake as at g. Then measure the dis-
tance het ween /and -, and enter the same in vour field
book. Proceed in the measurement of the line" A B, and
enter the result in your field hook. Measure the angle at li
as you measured the angle at A. Proceed in the same way
to measure ail the lines which enclose the field, an.l also the
nnglcs, ohservin- to take the external anijje at D. From
the rfcta whir-ii your field notes will afford, according to the
principles of Practical Geometry, already laid down in a
preceding part of this work, with the aid of a scale of equal
parts a figure may bo constructed in which the angles will
be hud off, and which will be a correct plan or map of the
hehJ Or : divide the field into three triangles A B D.
C B D, and A E D. Then measure the sides of the trian-
gles, in succession, and yon have data by which to construct
figures and lay .off the angles as before.
PROBLEM X.
To find the dist tnce of an inaccessible object by the chain
only.
Let A, (.^,;g-. 61,) be the position of an inaccessible object,
it IS required to determine its distance from the point B by
the chain only. ''
Place any conspicuous object for a mark at B, and from
It measure backwards in a straight line with A B, any con-
venient distance C. From B, at right angles and equal to
« ^, lay otl B E. Complete the square B E D C. Stand-
ing at D cause a pole to be set up ai F, the point in which
a straight Ime drawn from D to A would intersect B E
Measure the distance between E and F, the distance be-
tween I and B, will then also be known. Then, as the trU
"^^
tm;.
LAWD SUaVKYISG.
angles I) E r and A B F arc similar: As E F is t> E D.
so is F U to B A.
Note. — Tho above pro])ortion holds good in any paral-
lelogram.
Jlnother Method.
Let t][ie distance between A and B {Fig. G2,) represent
the width of a river, it is required to ascertain that distance
by the chain only.
Make A D perpendicular to A B. Bisect A D in C
Draw D E perpendicular to A D. Measure along the line
D E until you arrive at a point P2, in a direct line with C B.^
The distance between Dand E vyill bg e(]ual to the disUnce
between A and B, the width of the river.
Because A C and C D are equal, and the angles at A and
D right angles, it is evident that the triangles ABC and
DEC are not only similar but equal, and therefore D E is
equal to A B.
Note. — Tt J not neceiisary that the station A should be ex-
actly on the brink of the river. It nuiy be taken at any con-
venient distance from it. Hiiving deter. (Uiunl the distance
between the station and tho inaccessible o])ject at the oppon
site vA2,ii of the river, measure carefully the distance be-
tween the station and the river's i)riid<. 'i'li .t distance being
iiubtracted from the whole distance, tho remainder will be
the breadth of the river.
r
I .. i,
i if
^ >
Use op the ClRCUMPERV.yTOE.
To take field notes by the Circumfercnior.
1. By fore-sights.
Tiace the instrument at any angle A, (Fig. 03,) as your
first station; cause a otaff or ])ole to be erected perpendicuf
larly at the next angle B: having levelled the Circumferen-
tv)r, turn the flow^er-de-luce^ or nortii part of the box, to
your eye. Looking through the small aperture, or slit in
the sight, turn the index until you can see the staff at B,
through the large slit in the opposite sight, and until the
thread or hair which is in it divides or cuts that obicct; tli«
i
LIWD St'RVEYINO.
67-
degrees pointed out by the south end uf tlic needle * n iir
shew the number of degrees by wliich the stationary line i«
distant from a north course, counting quite round witli the
sun.
Having entered the number of degrees, or tlie l)earing of
ihc line A 13 in your field book, measure the line, and insert
irs length in chains and links in the fiehl book likewise, un-
der the title "distance.'*
Being arrived now at your second station B, cause a pole
to be erected at the next angle C. Place the instrument in
ft horizontal positio". over the spot on which the object at B
stood. Direct the sights to the object at C, with the north
of the box to jour eye. When the instrument is in such a
position, that, looking through the signts the thread in the
large slit of the opposite sight exactly cuts the pole at C,
count your degrees to the south end of the needle, and Re-
gister them in your field book as before. Measure the line
B C, and make the necessary entry in your field book. Pro-
ceed thus from angle to angle, until you arrive at the place
of beginning, noiing as you proceed the names of the own-
ers of the contiguous lands, and the names of the roads, riv-
ers, &,c., which bound the lot, or intersect the i)oundarie8
>vhich you are runninir.
'i. By back a7id fore-sight ft.
Proceeding as directed above, before you leave tiie st:ition
at A, set up a stake or pole in the spot over m hich the cir-
cumferentor stood. Having arrived ?.t the second station
B, and levelled your instrument, with the south part of
the box to your eye, direct the sights to the obj(<ct at A.
When the thread hi the opposite sight cuts the object at A,
count the degrees to the south end of the needle. If both
■ Some needles are pointed at the south end, and have a smalj
ring or croas at or near the north end; while others are pointed at
lM»th ends. The luttei kind i« to be preferred, as it enables iha
Purveyor to count the surplus numbers with greater accuracy.
Ihe iijscrlion of an agate i-ito the cap of the needle, that it may
rest on the pivot or centre pin, is a greai improvement, as it cauaei
Ujo needle .to move or play with greater beedjm.
as
X.AND SURVEYING.
1 V
observat.ons have boon correct, the number of degrees will
b the same n, h the number reckoned at the first station
the direction oi the index bein- the same. Then direct the
81-hts to the next station, a.ul ,,roreod as formerly. At thi.s
fitat.on leave also an object, and when you arrive at the nexf
station C, })rocee<l in evory res})ort as at the station B. In
this manner proceed until you ret.n-n to the fir.t station.
hnv nr* rh * p''" ••^marked befo.-e that the brass ring in the
box ot the CircumbM-ontor is divided into 360\ It Ls num-
bered f.-om the North to the West 10, '20, 80, &c , to "qo-
from the North to the East, from the South to the West'
and i-om the South to the East in the sa.no nnumen '
wards the iUisi, it is to be entered thus: N. 10^ E.
To find the number of degrees contained in any angle form^
ed bxj the two adjacent lines that bound afield.
Place tiic instrun.ent on the angular point, and direct the
Sights aloiu, the lines or legs of the triangle. Note down
their respective bearings. The number of degrees marked
upon the brass ring between tiie points cut by the end of the
needle will be the measure of the angle required, 'JMius in
the angle A B C, (Fig. G4,) having placed the circun.ferontor
on B, and having directed the sights to A, the bearing is found
to be N. 30^ W. Then turning the instrument abcmt on its
^und and directing the sights to C, the bearing is S. .55° W
1 he number of degrees on the brass ring between N 30^ W
and S 55^ W. is 95 ', which is tho measui-e of the Lnglc
A. a \j ,
To measure an angle of altitude by the rircumferentor.
Let the glass lid l)e taken ofl; and the needle removed.
1 hen turn the instrument on one side with the stein of the m
ball in the notch of the socket, so that the circle may be per- i
pendicular to the plane of the hori/on. Having suspended |
a pkimmet from the centre pin, direct the sights to the top f
Of the object, and the complement of the number of degrees i
ip
All til
as nianj
Let tl
from ar
F A, F
triangle
it is evi'
this figi,
the figui
■angles i
^%
LAND SURVEtlNG.
69
' degroea will
i first station,
lion direct the
crly. At thi8
ve at the next
itation B. In
^t station.
S3 ring in the
'. It is num-
', &('.., to 1)0°;
to the West,
.niiei-.
the cardinal
. When the
m in the field
(lie ])oints to
le North to-
E.
' angle form^
'field,
nd direct the
Note down
rees rnarkeJ
le end of tho
1, Thus, in
(MMuforentor
nx\^ is found
about on its
s S. 55° W..
1 N. 30^ W,
f the tngle
iferentor.
removed,
item of the
nay be per-
\ suspended
s to the top
of degreea
A
.*
betiveen the thread of the plummet and that part of the in-
strument which is next your eye will be the angle of alti-
tude.
Use of the Theodolite.
To talce the angles of afield by the Theodolite. (Fig. 65.)
Set the instrument on some angular point of the field as at
A, then lay one end of the index to 3G0^, when the other end
■•vill cut ISO'^; turn the whole about, until the part marked
•« SCO" is from you. Direct the sights from A to E, and
screw the instrument fast. Direct them then from A to B,
and the degree cut by the end of the index opposite to you
will be the quantity of the angle E A E; which note in your
field book, with the distance A B in chains and links. Pro-
ceed to the next station at B, and place the Theodolite on
the angular point, and unscrew it. Then lay the moveable
index so that it shall coincide with SCO and 180, with 180
next you as before, causing the thread or hair in the sight
to cut the object at A. Screw the instrument fast and di-
rect the sights to the object at C. The degree cut by the
end of the index opposite to you will be the quantity of the
angle A B C. Enter it in your field book, witli the distance
B C. Proceed thus from station to station, until you return
■to the place of beginning, or first station.
LEMftfA.
All the angles of i ly polygon are together equal to twice
as many right . igles as the figure has sides, less four.
Let the poiy , >n hr, iajd ofi' into triangles by lines drawn
from ary assigiie(! ptat F within the figure, as by the lines
FA, F T], F C, &c. Now since the three angles of every
triangle nrr togeth r equal to two right angles, (Euc. i. S2,')
It is evi-. . - that the angles in all the triangles coatuined in
this figure must be eq-inl to twice as many right andes as
the figure 1 i^ side. But, according to Euc. i. IS, all the
^gles atout ,J- point F are together qual '-four right
70
Land SURVETINO.
f
5.1
I
angles. Therefore the remaining angles are equal to twice
as many right angles as the figure has sides, less four. If,
then, the angles of a field be correctly taken in any survey!
their sum will be equal to twice as many right angles, less
four, as there have been stations taken in making the survey.
Use op the Protiiactor.
To protract or draw a plan from a field book.
It is required to protract, or draw a plan from the follow-
ing notes in a field book:
Commencing at the point F, (Pig. 66,) running thence
K. 8 chains, thence N. 15° E. 8i chains, thence S. 80° E,
7i chains, thence S, 15° E. 7 chains and 90 links, thence
S. 20' W. 10 chains, thence N. 75° W. 8 chains and 45
links, to the firs^ station at F, or place of beginning.
Consider what part of your paper will suit best for the
first station, aa at F. Frem that point draw a meridional,
or a North and South line. Now from the field book it ap-
I>ears that the first course is due North, it is only necessary
that, on that line from F towards A, the given distance 8
chains be laid off, from a scale -qual parts. Lay now
your protractor on the meridior * lii. , so that the centre may
be cxa-tly on the point A. As jxt course is N. 15° E.,
from the meridional Jine prick oflf towards the East or right
liand 15°, as ])()inted out by the chamfered edge of the pro-
tractor, Through this point from the point A, draw a
blank line A 13, formiii'r, of cmirse, with the meridional
line, an annfle of 15°. On that blank lino 4 ay oft' the given
distance 8^ chains to B. Next, through the point B draw
another meiidian parallel to the neridionai line F A. Lay
the centre of tiic ])rotractor on the ])oint B, .-tud prick off the
next course S. 80° E. Through that mark or dot fr)m B,
draw tlie blank line B C, on which lay ofl^'the given distance
7.i ciiains. Proceed in the same way until you have com-
pleted the figure.
Though it is the best way, in all protractions when the
.ionrses arc given, to draw meridians parallel to eacJj otJier
LA^fD SHRVKTING.
7r
lal to twice
four. If,'
my survey,
ingles, less
the survey.
through every angle of the field, as above; yet where the
angle at each station is given, the centre of the protractor
may be placed on the extremity of the distances when laid
off, and the number of degrees contained in the given ano-le
pricked off for the direction of the next line..
The method of using the Protractor is obvious, from the
nature and use of the Circumferentor and Theodolite.
ook,
the follow-
ing thence
) S. 80° E,
ks, thence
ns and 45
est for the
leridional,
look it ap-
necessary
distance 8
Lay now
entre may
N. 15° E.,
St or right
f the pro-
L, draw a
iioridional
the given
It B draw
^ A. Lay
ick off the
)t fr')m B,
n distance
lavG coni-
Of the Field Book.
The accuracy of every survey, and the ease with which
plans may be drawn depend, to a great extent, on the man-
ner in which the field book is kept. The adoption of a con-
venient and perspicuous method of keeping field notes is
therefore a matter of great importance The subjoined form
is simple, concise, and plain.
JVo^^.—ln keeping the field book it is customarv and use-
ful, where the distance line crosses a brook, lake, &c., to
enter the same at the projjcr place by a line draAvn between
the respective entries, representing the course of such stream,
fccc, as between 35 and 40 in the 4th or N. W. course, and
between 40 and 45 in the last course, in the adjoining field
book, in which they are represented bv straight lines, in
consequence of the Printer iwt having lines of the proper
clirecticn. if
when the
;ach otlier
i i
T*
land 8umveving.
Form of a Field Book.
3
38 ch
. to the place of be$;innin$r.
35
70
70 ch. and 7 /.
SO
•
C5
25
60
20
71
55
15
W
50
10
5
45
■ 40 ch.
•
<
o
40
35
30
NarrowLake
Thence
N 50^ E
i
40
tJ
35
"Cross
ro
25
s
30
Lake
CO
o
20
^
25
tf
15
t^
20
10
5
bo
o
15
10
Thence
S3^54'E
5
30 ch.
4
30
25
34 cA.
Thence
N45^W
i
30
i'
25
20
1
20
15
i
15
10
.5'
10
5
25 ch.
5
35 ch.
f
Thence
S 81° E
Thence
S85^ W
35
30
5
20
15
10
5
25
20
15
10
5
Thence
South
Thence
V 60^ E
■s^-
4 c
20
24 ch.
3 i
J8 eh.
4> .
15
35
^ u
10
80
Wg
5
25
20
15
10
Thence
West
efD.
Parish of C. , and County
5
tate of A B., Esq., in the
Thence I
^25°W
easterly angle of the Es-
1
Cominei
iced at the
most 11
(COK
TINUBD.)
LAND SURVKYINO.
■;;{
fining.
ro ck. and 7 /.
NarrowLake
4 cL
t ck.
eh.
Remarks o.n the preceding Form of a Field Book,
Tim procctliiig uotf>s urn sii])];osed to be entered in the
field book, wliile surveying the Estate of A. B., Es((,, (.S'.v-
Fig. ()7,) the counscs being taken from the meridian l>y ;i
'•In-umferentor, and the .stationary distances mea-^ured by a
half or two-pole chain.
Determine rirst at what part of the Estate it will be most
eonvenient to co/nmence the survey. Having conclnded to
begin at the angle No. 1., which is the m,)st Easterly corner
in the Estate, you insert in the field book the place of be-
gimiing, thus: " Commenced at the most Easterly corner
of," ike. In keeping a field book it is found most conveni-
ent to begin at the bottom of the page, and to write .ip-
ward.s. Set the instrument on the angidar point, and take
the course to the second station at tiie angle No. 2, which is
seen to be due West. Write in your field book " Thence
West," or " Thence W\" Proceed to measure the distance
between / 1 and Z % and every tin.e that the ten pins cur-
ried by the foremost chain-Jjoarer are all transferred to the
hindmost chain-man, insert 5 in your field notes, writing
upwards; for 10 half or tw^o--pole chains, only tnake 5 chains^
When you have arrived at the station write down the odd
chains and links, if any, and place the sum of the whole in
the right hand column, thus: «' 24 cA." Observe whoso
land lies to the left of the land which you are surveying,
and enter a note of it in your field book, in the left haml
colunm, thus: " Estate of Mr. A W." Observe also if any
particular objects a])j)ear in the immediate viciinty ai.'l ad-
joining the stationary line, such as stream, fence, road, &e.,
and insert it also in the left hand colmnn of your field jmok'
thus: '-Kryad." Proceed in this nmnncr from station to
station, uniil you leturn to the place of beginning, always
••arefully noting to whom tl^e adjoining land belongs, tluv
r..ads, waters, &c., which bound, the Field or Estate'^whicli
you are surveying, and also the streams, lakes, Lr., which
you cross in running the lines, and at what part of the line
you cro.ss them; as all these particulars must be expressed
T!
74
LAND SUKVEYING.
and iiccurately laid down upon the plan which is afterward^
to be drawn.
I'Jik'
ii
H r ■!
{ I 1 ■
VARIATION OF THE COMPASS.
The natural magnet or loadstone was for a long time sni.-
posed to be the only body which possessed inagnetic proper-
ties. It ,3 an ore of iron, whose specific gravity is about
five times that of water. Its colour is iron-black, a.id its
lustre metallic. It is found in almost every part of the
world, and occin-s in beds, often of vast thickness, and of
great extent. Its attractive power over small piece, of iron
has been known from the remotest antiquity. It is sai.l to
be distinctly referred to by Hon.er, Pythagoras, Aristotle,
i'lmy, !kc. Iv has been asserted that its directive i)ow.M- or
polarity was known to the Chinese in the earliest ages, a.Kl
that the needle was employed to guide travellers by lan.l a
thousand years before the commencement of the Christian
Ji.Ba.
To whatever amount of importance these statements may
be entitled, it may nevertheless be confidently asserted, that
nearly all our knowledge of the magnetic virtue, and nearly
all Its ai,plications to practical purposes, are discoveries and
mvontions of comparatively modern date. Notwithstand-
mg, however, of the numerous and valuable additions, which
have recently been made to our acquaintance with the laws
by which magnetic influences are regulated, many interestin-r
and important points still remain to be determined. To the
dotermmation of some of these points the attention of scien-
tific men has recently been directed. Magnetic observatories
have been erected, and powerful and delicate instruments
hav-e been constructed for the advancement of this brand.
ot Science. Members of the British Government were re-
questeu some years ago, to establish magnetic observatories
not only in Lntain but also in these Colonies. About the
same ime Baron Humbolt addressed an interesting letter to
the late Duke ot Sussex, President of the Royal Society,
VARIATION OP THE COMPASS.
75
1 is afterwards
long time sup-
ijnctic proj)er-
tivity is iiho'.n
hlack, and its
y part oC th(;
kncss, and of
pif3cc,> of iron
It is said to
•as, Aristotle,
tive j)(nv<M- or
iest age?, am!
ers l»y land a
the Christian
iitements may
asserted, that
le, and nearly
scoveries and
^otwithstand-
litions, which
vith the laws
ny interesting
lied. To the
tion of scien-
ohservatories
' instruments
i' this branch
lent were re-
observatories
About the
^ting letter to
»yal Society,
soliciting that learned body to extend in the colonies of
I Great Britain, the line of simultaneous observations, and to
' establish permanent magnetic stations either in the tropical
regions cm each side of the magnetic Equator, or in the high
Latitudes of the Southern Hemisi)here, and in Canada. I
mention these facts for the purpose of attracting the atten-
rion of the North American Colonist to this interesting and
important subject.
The property of magnets or magnetized bodies upon which
nearly all their value depends, is their polarity or directive
power. By these terms, is intended to be expressed tho
tendency of such bodies when suspended, or made to float
on water or mercury by being placed on a thin piece o^ wood
or cork, to assume a position in which the one end will be
tlirected to the North and the other to the South, nearly.
In consequence of their possessing this property, by their
assistance, we can at any time ascertain the direction of the
meridian at any given place, and the bearings of other ob-
jects in relation to that line. The magnetic instrument
<ommonly employed for this purpose is the Compass. It is
a matter of i egret that the name of the inventor of this curi-
ous and invaluable instrument is unknown. It deserves to
be written in letter^ of gold and to be handed down with
honour to the latest ages.
Though magnetic instruments are useful in determining se-
veral curious and important points, their principal value con-
sists in their application to the arts of Navigation and Land
Surveying. By the aid of the compass the mariner guides
his vessel through the trackless ocean, and establishes an
intercourse with the most distant nations. By the assist-
ance of the compass the surveyor divides large tracts of
country covered with dense forests, assigns to the future oc-
cupant his portion, and determines the bounds of his habi-
tation, or assigns to the respective claimants their just pro-
portion of an improved and valuable estate.
It nmst not however be supposed that the compass, how-
ever useful, aftbrds to the surveyor infallible direction in
his operations. The magnetic effects are liable to various
'SI
I
It
'l* {
7»>
T-\NI) .sun\ EYrwo,
ami varying innuiMicrs and (lisfniliiinccs, uhich tl
or nnnt Im« iil»Ic to diMcct inul cstiniiitp. 'rhn^duivoof tl><«h'i>
u' siirvov
influotn'cs is still invi)|\M!il in olwcnritv, hut tlir cd'c^ts
ar<?
Avcll known nnd fnllv cstiilrlish
Of ilicsp on«' of till
niosi iinportiint and tlicrcfoi-c first dcsorvin"- nllcnt
Til
ion is:
c I
(tiialinn of the Nrcdir.—h Iiuh alr^udy lic<<ti ol»-
o)>s not point Nonli mid South cx-
srrvi'd th.il llir iimj'nrt d
acliv : or, in other words, that the niiinnrt
terrestrial meridian seldoi
ma-fnetie needle from a true North and Sontli I
a true meridian, is called its vnrinli
eontinnally varyinjr, Aecoidiiiirlv, the va; .at
in (
lill
les.
on.
ie meridian and the
n eoineide. Th,' deviation of the
ine, or iront
on. This deviation is
ion is (lili'erent
brent jdaees, and in the same plare at <litferent tin
In some phiee.s th(>ro is little or no pereeptihh^ variati
In otluM- places the variation is great. At some times it ap-
pears to h(> stafi^mary at a particnlar place, at otiier times
in the same j)hice it increases or diminishes with <?reat rapi-
o variation was 10'
arrived at its maximnin, or
>ein-U> 17'K. The Easterly
rom 1()57 to 1G(>2, no
dity. At Lomhm, in the year 1 ;')?(), th
15' Kasterly. In 1580, it had
jjreat«>st Easterly deviation, 1
\ariation then be<ran to decreas(>. I
variation was percejuible. In KUHs it >vas S-l' V\ Csterh
The Westerly variat
ion continnetl to increase until 1815,
when it arrived at its maximum or ^n-eatest Westerlv d
ation, heinj; then '24'' i27' 18" W. It 1
V tievi-
1112
In 18;>1 it was ^21 \V
las since been decrcas-
At Paris, in 1541, the north end or pole of th
die pointed 7^ to the East of iNortlr/ In 1580 it had
its maximum of Easterly variat
c nni'nietfc nee-
attained
ion, Avhich was 11^. SO'. 'J'l
le
easterly variation declined ffrad
about Hj[)6, when it became
variation was 15' W, The Westerl
Ji-radualiy until 1814, when it had
since been decreasiui;. In 18'iO it
frradually from that time until
imperceptible. In nm thtt
Accordino- to the report of Dr. (Jesner I-
y variation increased
iirrived at 2\>^' 54'. It has
wasi2\>-^ Ui' W.
fist, made to His Ex
rovincial (ieolo-
Colebrooke,K.H., Lientenunt-C
celleucy Sir ^Villiam MacLean (
leorife
he
Governor of Nevv-Brunswicl<
ariation iu that Province ranges from 17-to ^IV ^V
V Mir \TTON or Trn. roMrx-
1 1
An.l on tlK- lirli nCN.pf., |h|.{, I (oiiixl lli<' variniion nt lluv
(If VcltO to l)«! IS' 10' W., lU'Jlllv.
IN
siflfs tlipsi' pn^'iTssivp cFmnj^'cs in tl
10 v.'iiiiiiinn (I
-•"mpri-^, tlir ii.TiMr is ,il
nr<'rial»||. osrilhitioiis, nt ijiti;
-ill
tf .nn
«M-r-iit timrs of the flay jiikI iii-Wif. 'I'Im; oI
\\\o('t \n inorp iriiriiiti> yif. ii|»-
rent soasfUH «»r ilirycMr. ami
i-^crvailMii-
whi.-ii havo Urvn ina<l<* h. (h-tcnniiin tho prcciyi! p.-riod
llic your in wliirli this annual varialion attains it
S (ll
s niavitniuii
111(1 niinitnnni ilo not. cvacfly r-ornspnnd. I'roliahly tl
ai-p (lifUM-ciif in (litrcrcnt parts of il
liowcvc.r, it may ho sfatrd to l)(! loiist
f<i ahont 7 jninnti's, and jfrcatpst
IP \vorld. In L't'ni'r-al,
ill \\ ii.tr-r, anioinititu
•> nnnntr-
T
111 s.ininuT, wlioii it is aliuii.
anic discropanpy o\ist,s in the ruiiiicron-!
-•'ivations niaiK! tu dptorniino tlio
mnni of tlip daily variati(»n. hi f.ondon tl
niax'itnntn and mini-
daily variation is «,(>' -M'
In (
iV'lco
t>. :»
ajpf it
IS only 10' .W.. I", in Ka-a.i it
10 ,',M'nf'ral moan
ioiiova it is not rpiito !(/. In
amounts to I ' 10'
'I'iic lollowini; diurnal
variations liavo hoon obsorvcd
Paris. Durinsf tlio ni^rht it was noarly stationary. At
m
rise its North cvtromit
SIUI-
y movod to tlio VVostwanI, as if uvoid-
in-,Mlio <;(dar indup.n.'o. Towards noon, or j
trom nor)!! to A o"c|
iioi'o ironorallv^
dcvialion.
Tl
ock, it attainod its maxinii
o'clock, when it had r
H'li it roturnod Kastward, till i», lo.
im VVo.st»;i-|y
ur n
'fMP
liiH-d its orijjMnal jxtsition, whoro
'•'•"iaino.1 until moi-nin- In April, May, .Tuno, Jnlv, A
:-nist, and Soptomhcr, the <laily variation v
Diirin-; the othor six months it was from 8
ir:
ion was from IS' to 1;
<lays it amountod to '25', whil
to 10'. On'somo
or ♦.
i<J oil othoi-s it did not oxcood
H.Miro it will follow that a lino run l)y tl;
•dM)ut mid-day and partly lato in the aft«
wi
II not
l)o a strai'dit liiK
compass, partly
rnoon or ovotiiiiir.
Si
\ortI
IlCC
th
on tho mairnotic; noodlo doos not al
1 and South, — sinoo tl
ways point duo
us
foront j)larps, and in tl
variation differs \vi<lely in dif-
^'n\re this variation is not
10 same placo at diffoi-oiit timos,— and
govorned hy any rule hitherto d
IS-
V
<^
/y.
I
■el
<p
>^
IMAGE EVALUATION
TEST TARGET (MT-S)
1.0
1 5 '""^™
«" lllitt
'- 1.
I.I
i.25
1.4
2.5
2,2
12.0
1.8
1.6
Photographic
Sciences
Corporation
^
<^>^>
c?-*
«^^
^<^A
"%
V
«^ >^ '<9^rN\
23 WEST MAIN STREET
WEBSTER, N.Y. 14580
(716) 872-4503
"-b'
«i*
<i.
78
LAUD SURVEYING,
covered, by which, independent of observution, it may ex-
actly be ascertained,— it becomes a matter of great iin})«)r-
tance to the Surveyor, as well as to the Mariner, to under-
stand, and to be able to employ the expedients necessary to
determine the amount of this deviation, at any particular
place, and at any specified time. In order to understand
these methods, the student should make himself familiar with
the meaning of the following geographical and astronomical
terms : —
The Equator is a great circle, equally distant in every part
of its circumference from the poles, and dividing the globe
into two equal parts or hemispheres. It is called olso The
Equinoctial Line. Its distance from the pole is 90°. If the
plane of the Terrestrial Equator be produced to the hea-
vens, it will describe The Celestial Equator.
The Latitude of any place is its distance from the Equa-
tor reckoned in degrees and minutes. If the place is North
of the Equator it is said to be in North Latitude. If it be
South of the Equator it is in South Latitude. The Lati-
tude of any place cannot exceed 90- degrees.
The Complement of the Latitude of any place is the dif-
ference between the Latitude of that place and 90^
The Horizon is a great circle of the sphere cutting the
Equator at right angles, and ilividing the world into two
parts.
The Rational Horizon is a great circle, cutting the Equa-
tor at right angles, and dividing the world into two equal
parts, the plane of which passes through the centre of the
earth.
The Sensible Horizon is a less circle, likewise at right
angles to the Equator, and dividing the globe into two un-
equal parts. It is the circle which bounds our vision, and
of which our eye is the centre.
The Declination of the sun or star is its distance from the
Celestial Equator, reckoned in degrees, minutes, &c. It is
North or South, according us the sun or star is to the North
or South of the Equator. Declination cannot exceed 90°,
The Complement of the declination of the sun or of a star
VAniATION OF TiIE COMPASS.
79
i.sthe distance between it and the pole, and may be ascer-
tained by sul)tracting the declination from 90°.
The Jlzimuth of the sun cr of u star is an arc of the
horizon, which measures the distance of the sun or a Ktar at
its rising or sitting, from the North or South cardinal points.
Azimuth distances accordingly arc measured on the horizon
frc in the North or South points.
Amplitude is the complement of the Azimuth, or its de-
fect from a right angle; or it is the distance of the sun or
star at its rising or setting,' from the East or West points,
in degrees, minutes, &c., measured on the horizontal circle.
The Magnetic Amplitude is the amplitude indicate.i l>y
the circumferentor, or quartered compass; or it is the num-
l)er of degrees, &c., which the sun or star rises or sets to
the North or South of the magnetic East or West points.
To find the variation of the Compass by Amplitudes.
The Latitude of the place, the Sun's Declination, and
the magnetic amplitude bemg given, find first the true ampli-
tude by the following rule :
As the Co-Sine of the Latitude
Is to the Sine of the Sun's Declination,*
So IS radius
To the Sine of the true Amplitude.
Then, if both the magnetic and the true amplitudes be
either North or South, their difference is the variation; but
If the one is North and the other South, their sum is the
variation.
To know whether the variation is E. or IV.
Let the observer's face be turned towards the suii, and if
the true amplitude is on the right-hand side of the mao-uetic
<n- observed amplitude, the variation is E„ but if to the left.
It is W.
* The sun's declination must be taken from an Ephcmeris It
.s a ways contained in the JVautical Almanack and Jistronomical
s ONERS Of the Admiralty. It will be found also in some oi"
the Almanacks pubhshed in the Colonies.
UM.
•so
r,AND sunvEYiNr:,
EXAMPLES.
!. On tholOtli of Aurrust, 184->, in Lat. ^}(r N.. r)„; brarinj.
of tho sun at its rising was ohsprvod to be N. 85^ K. by the
rornpass; roqiiirod tho true uniplitwlo and tho variation ot
rlif? r)(.'0{|lo.
To find the true ^mplUude.
A.s tho Co-sino of tho Lat. 46^
Is fo tho Sino of tho .Sun's Doc. 15^ .S.V
So is Radius
0.81177
i>.4-l!)17
lo.oonrij
tho
To tho Sine of the truo Amplitude 22^ 45' 9.58740
Tiio bearing of tho .sun by the conipass being N. 85^ E..
fMagnetio amplitude is flO^ — 85^ = 5'^ N. Thoroforo,
Truo Amplitude E. 2-r 45' N., for the Do.r. is N.
Mag. Amplitude E. 5'^ 0' N.
Variation 17*^45';
whioh is V/e.st, because the observation beijig talvoa in tho
morning, and the observer's face being turned towards tin.
sun, tho truo Amplitude is on tho left of tho ma-neti<' ..r
oljsorved Amplitude.
^^l. Suppo.so the bearing of the sun at setting to bo S. 80'
V/., and con.scquently tho magnetic Amplitude W. 10^ S
nhde the truo Amplitude is found to be VV. 25^ S. ^Vhat
is tin; variation?
True Amp. ^Y . 2-P S.
Mag. Amp. W. IQ^ S.
Variation 15- W.,
becausothe true Amplitude is to the left of tho Magnetic.
'"5. At the time of rising, suppo.se the .sun's bearing to bo
> . 7;)^ E. by the compa.ss, and consequently tho, Mag. Amp.
h. 1.)^ S., while the true Amp. is found to be E. S' 15' N •
required the variation. ''
True Amp. E. 8^ 15' N.
Mag. Amp. E. 15^ 0' S.
Variation -25^1.5'W,,
because the truo AinplitudG is to the left of the Marrnetic.
VARIATFON OF THE COMPASS. g]
4. ^^"FPOSO the sun':^ honring at setting to 1)0 S 83 W
Ml.en the true Amp. is found to be W. 7^ .'50' N.; re„nir.Mj
the variation-. i ■'<«
True Amp. W. 7° 3(7 N
Mag. Amp. W. 7= 0' S.
Variation 14°30'K
1.ecau.se the true Amplitude i. to the right" of the Mag.oti..
Tofmd the variation by Concentric Circles
mu^loTZ^ '""', ''' '"""^'"^^ ^^^^' (about a foot
M naie,) diaw several concentric circle.. In the centre
p aTe "^r'tV^V ''"^'^^-^ ^""«^' PerpendicuL t , J
on any of the crdes, upon ^vhich the shadow of tiie heac
of the pm rests, an. ,„ark it carefully. Observe aNo in he
rmke a ma.k. A right line joining these snot.s or mark, nil!
be m East and West linp k;.o^: .u- .- "laiKs, mil
ne an,l u win be a iVo.W and South line. 7'ZZtl
n fron, .„o centre, and in i,« p,„ec .,„„,:„ ,e „„ X ■ r
t^ uit, ^tne needle of a compass suits verv uoli ^ n» i i
nhus.„,ea„.,ea™„H.„i,e,.,,eNr.i::::<^
By tins method, with care and a little experience the v.
-^^'-' '"ay be detennined with consiclerable "euSc'
Tofnd the variation by the Nokth Sxah and Aooxn
1 be constellation Ursa Major, or the Gr.^f «
known by the name of Charles^ vLn , '"' ''''"
i'4
82
LAND SURVEYING,
Ming a rectangle, or square, which is considered as forming
the body of the wain. The two hindmost of these are call-
ed pointers, because a right line drawn from them towards
the North will pass near the polfe star in the tail of the Lit-
tle Bear. Following this square arc three other stars of
about the same size, forming the tail of the Great Bear, or
the handle of the jdough, visible every clear night. The
one next the body of the bear is Mioth. Now it so hap-
l)ens that the pole star and Alioth come to the meridian at
the same time. This star is accordingly often employed by
Navigators to determine the latitude. By its assistance also
we may ascertain the variation of the compass with consi-
derable accuracy, by the following method:—
About midnight, in the beginning of October, or about 8
o'clock, P. M., towards the end of November, the pole star
comes to the nuoidian, and it the same time Alioth is at the
meridian below the pole, and consequently both stars are
due North. Alioth is also exactly perpendicular to the pole
star, or directly below it. While Alioth is still to the West-
ward, let the compass be duly levelled and prepared to take
its bearing. Then suspend a plummet by a white thread,
in some convenient place, so that without moving the body
much the plummet and compass may both be attended to.
Bring the thread to cut the North star. Watch carefully
until Alioth comes to the thread, and is cut by it. Then
take the bearing of A ,th by the compass, and its diverg-
ence from the North wdl be the variation.
Note.— By causing the plununet to fall into a vessel of
water, the wind (if there should be any at the time) will be
prevented from moving it. In this way the object may be
gained with greater case and accuracy.
Besides these progressive, annual, and diurnal variations,
the magnetic needle is liable to disturbances from local and
from temporary causes. Of these the most important are,
the existenceofniasses of ferruginous matter in the vicinity,
the Aurora Borealis, and thunder storms.
It is well known that the magnet is powerfully affected
VVniATlON OP THE COMPASS. gg
•ho whole sui.: „";r^ V ,: ;:,r"7^ '■••™'^' ?~"^"-"
-no., are wi.hou, „„ .ZZT^r ^r^'r, .'r!''?';
>n clays, and sands, and even in n-,v, ■ ,.(■ , ''""''•""'
-1 suManee., and in the a.J ,^C"' ^S" ^ ""'V','"-
».se, ,„ eertain conditions aflc« tl e' ne^dt W ' T
therelore rcasoimhlv evnee. ,h., .. ^"-' '""''"
""•"once. DomC " '"'^"'""°»"" "f *■".,,„„„ oc-
vonenee orsu^e "r. Z ^u" ""',"""""■"' """ "'" '"'"1-
whena survey IXa.ie l.^^ "'"' '"'""'" «'■""'"'' ■""'
"thor .notallic's ,i" 7; d'' TT " '"' "'' "'"" "'•'•■ '"■
■■aWe, amounting oft«:::;e'^:Te:it!:'" T, V '"'"""-
of (loteotinff denarfiirn . r. '" "^fe'^^''- ^ iie bast inetJ„„|
..";; i.;>eed!,. r:r:i«r rjitrr ""-^ --
affect the n,„ ""e Su, v """='• '™'»'"'^ -'•■^'ances, which
-"«ioninthe;:;hLro? :i;Znrf'7''T;''^"'
'.i-"oney i„r™rtlr;:r;:^^:e*-" ''""'"'''"'•''
try it hy the following. ,cs, « '""*"""= "wtrumcnt, to
in»t">,nent, and ,11^1,! T"™ ""^ ""<''"'= *■'•"■" "'»
in a hoard. Wl e t ,he ne'T ° f '"^ "'""""' P'»' '-""-'I
vibrate, then W ,t ins r ' ""'"™'"'' ''"■' ^<^"-'' "'
«he presence of "he ittn.n T", ''"'^'"'^'' "'"' "> "• "•
".» «.nner i^ in tti:": ^/^r^^f .-'>o "r""-'
appear that the presence of th„ i„ «<« if >t evidently
'i.e instrnnien. sLuld 'i: :I,:d:,re:;""""' '"'''°" ""^ "^•""'•'
'-le ■■'«^"": "r::;- ':;:: i:::;rzr"'r ""'-" ""
"ary »-ove,„ents, to which 5. ^ H ™S T ""'""■"''-
'■apncious movements th„„„ i """"« «"''■■<'' apparently
"'otion, and fre!,"";, "".'""'= "--crseswith „ shiveri
frequently
oscillates several degrees
on each
84
LAND SURVEYING.
side ot' its moan position. Whon the Aurora only rise?? a
lew degrees' above the, horizon, the disturbance is Hniaii und
ot'ten inappreciable. But when the Aurora rises to the ze-
nith, the di.sturban(!e is generally very considerable M. Ara-
go, wlio has stuilied the influence of the Northern liights
upon the needle with particular care, states that the part ot'
the heavens at which all the beams or radiations of an Au-
rora meet and unite, is precisely the })oint to which a nuig-
wetic: needle directs itself when suspended by its centre of
gravity.
The needle may be affected by an Aurora which is iiivisi-
ble at the place of disturbance. M. Arago assures us that
Auroras visible oidy in America, at St. Petersbnrg, and in
Si])eria, produced very perceptible derangements of the nee-
dle at Paris. It seems however, that some kinds of Aurora,
though exceedingly brilliant and rapid in their movements,
.scarcely affect the ntagnetic needle; producing oidy at most,
a slight trenmlous motion.
The needle of the compass may lie afiected also by the
electri(r fluid, previous to or during a thunder-storm. The
effect of an electric shock is i)eculiar. Soinetinies it com-
nuinicat'^s the nuignetic virtue to unnnignetised iron or steel.
Sometiines it entirely destroys the m^ignetic virtue o|' u
magnetised body. And sometimes it reverses the poles of a
magnet. It may therefore be worth while to examine mag-
netic instruments after violent thunder-storms.
There is still another affection of the magnet, called tht
dip of Ihe needle, to which we nmst shortly advert. This
expression denotes the angle which a well l)alanced needk;
forms with the horixon^ after it has been mugnetiseo, and
w hen it is allowed to move freely in the plane of the mag-
netic meridian. This angle, like the angle of variation, has
different values in difierent j)U\ces; being, generally speak-
ing, very small at the Equator, and increasing towards the
j>oles. At the magnetic pole, which Connuander Ross found
to be situated in North Latitude 70° 5' il", and We.-;t Lon-
gitude 9G^ 45' 48'', the dip was 89' 59', or within one mi-
nute of being perpendicular. There is much reason to sup-
THE RUNNING OP LINES gjj,
poso that every placo Iims itij own magnetic axis, with its
own polo, and 0(iuut()r.
Like the variation, the tlij) of the needle also undergoes a
continual chamro, increasing in .some places and dindnish-
ing in others. At Lon.lon, in 17^?0, the okscrved dip wa«
74" 4-1', and the computed dip was 7b' 27, while in 1830
the observed dip was GO- 38', and ir. 1833 the computed dir.
was 69'^ 21'.
Whenever the needle of a compass is perceived to vary
from the horizontal position Avhile resting freely on the
pivot, it becomes unf.t for service until it is corrected. For
this purpose, make the instrument perfectly h>vel by mcan.s
of the spirit-level, which will bo indicated by the air bub-
bles remaining in the centre. Then supjily the end of the
needle with an additional quantity of magnetism, until it re-
sumes Its horizontal position. Then both ends ot the nee-
dle Will bo equally distant from the bottomof the instrument. .
Then too, if the compass be a good one, both ends will point
to the same degree on the graduated brass circle within
which It revolves. This operation should be repeated every
three or four years.
THE RUNNING OF LINES.
, In the runningof lines fourthing.s are to bo observed, viz-
—Course, Distance, Difference of Latitude, and Departure.
1. The Course is the angle which the lino run forms
with the meridian of the place from which you started. *
2 The Distance is the length of the line run, reckoned
m chams and links, rods, &c.
3. The Difference of Latitude is the distance of the
one end of a line from the other end. North or South, and
IS reckoned on a meridian,
4. The Departure is the distance between one end of a
hne, and the meridian passing through its other end. It i.s
East or West, and is measured on a parallel of Latitude
H
Land scrvkviwo.
PROBLEM I.
The Course and Distance of any line being given, to Jind
the Difference of Latitude and the Departure.
The Distance the Difference of Latitude, and the Dcpar-
ture, from a right-angled triangle. Therefore,
To find the Difference of Latitude.
As Radius
Is to the Distance,
I" ^;V''^;^?°-S'"^ of t'je Course
i o the Difference of Latitude. And,
2^0 find the Departure.
As Radius
Is to the Distance,
So is the Sine of the Course
1 o the Departure.
N. 2. \\ * the Distance, or length of the line 71 cA 20/ •
required the Difference of Latitude or Northing V^t
Departure or Westing. ^
To find S. N. the Difference of Latitude.
As Ra<iiU3 GO'' .^ nnn/w.
IstpDist.SA7loo/. 'IfX
So IS the Co-Sine of the Course 25° 9:95709
To the Diff. Lat. S N 6452 /. T7o"976 >
To find A N, the Departure.
As Radius S0° ,^ .^^„^
l8tptheDist.SA7l20/. '^^8
So IS the Sine of the Course 25" 9.G2595
To the Departure A N 3009 / 3 473^3"
Thus the Difference of Latitude, or the Northing i« a.
curtained to be G152 /., or (M ./. «,a 5. I., a„u t e D p .
t«re, or W estmg, to be 3009 /. , or 30 eh. and 9 I ^
By the same i.ethod, the Difference of Latitude and the
^^•Tbo Course of the line S. A. «« laid don-,, on diagram 68,.
Tilt nuifNINO OP Lll-iEf.
17
Departure may bo Tound af^er running any line, when the
Course and Distance arc known.
Note.--\Vhm the course is tlue No-th or South the rjir
vn^^'lii^^'lifT T'-^'^"^'^^ Distanco'^rand he ot
artuicisO; and when the Course is due Ka.st or WesT
PROBLEM H.
The Difference of Latitude and the Departure being given
to fmd the Course and Distance.
To find the Course. To fmd the Distance.
fs^r^'h?n ^^I* -^"^ ^' the Sine of the Counje
Sottcm^^"'^""' IH to the Departure
Tr.tv, 'r ^. ^ So IS Radius
1 o the Tangent of the Course. To the Distance.
EXAMPLE.
Suppose I run in the North East quarter until my Diff. of
Lat. IS C ch. 83 /., (Fig, c,9,) and my Dcp. 4 ch. 76 /., wh"'
was the Course and Dist.
To find the Course.
As the Diff. Lat. C85 I.
Is to tlie Dep. 476 1.
'So is Racijua C0°
2.8S569
2.67661
10.00000
To the Tan. of the Course 34° 44'
To fmd the Distance.
As the Sine of the Course 34° 44'
IS to the Dep. 476/.
So is Radius 90°
9.84092
9.75569
2.67661
10,00000
2.92092
dls^
To the Dist. 833. 5 I.
Hence the course is N. 34° 44' E., and the strtionar
tance, or length of the line is 8 t/i. 83 . 5 /.
PROBLEM IIL
To find the direct Course and Distance made good in
Traverse Running.
Make a Table divided into six colums. In the first colunm
m
TAN') 8URVCYINQ.
! >«■
*et down the acvcml rour.sos, and opposito tofliem set down
in the hccoiuI roluiim tlit;ir (•orrosjxjtidin;? distfiiice?. The
rhiid and tourth cohDiuiH arc to -rontain the Difibrenco of
l.atiiiidc, th« third being marked N., and thn fourth, S.
Tho fifth and nixth cnlunni s are to contain tho Departure,
the fifili Ueiug niark(;.l E., and tho sixth, W.
Having' entered tho Course and Distance in their rcspec-
live colunniK. find l)y thoprecodinj]^ problems, the Diflerence
uf Latitude and the Dejjarlure, for each Course and Dis-
tance, and insert them in the Table, in their prope. columns
opposite to the Course and Distance; observing that the
Diff. of Lat. must be placed in the colmun marked N. when
the course is Northerly, and in he column marked S. when
it is Southerly; and (Uat the Departure must be inserted in
the column marked E. if the Course i.- Easterly, and in the
column marked W. if it is Westerly. Add up'tho columns
of Northing and Southing, and of Easting and Westing.
The Diflerence between the sums of the N. and of the S.
columns will be the whole Diff. of Lat. made good, and will
be of the same name with tho greater: and the diflerence be-
tween the sums in the E. and W. colutnns will be the whole
Departure made good, of the same name with the larger
sum. With this Diff. of Lat. anc; Departure find a corres-
ponding Course and Distance. The defective columns of
the Table will show the direction of the home Course.
EXAMPLE.
A surveyor on an exploration, run the following Courses
and Distances, viz:-~S. 2'4° W. 54 ch., (Fig. 70,) thGnco
ti. 783° W. 39 cL, thence N. SSr W. 40 ch., thence N. 56^
E. 69 cA.,* thence N. 22^° W. CO cA.; required the direct
Course and Distance to the place of beginning.
* The distance on iho 4iti slatiora.y line of the diagram belojw-
lug to Ftg. 70, is wrong. " ^
*lia UOKHlH.i or LINKA.
TRAVERoF TABLE.
Courses.
S. 7Hr W,
N. 33.^ W.
N. 5G.1° K.
N.2-2i«W.
Oopnrturr.
I Diff. of Lat. 09.G'lN
104.09
57.30
To find the Course,
Astho DiT. of Lat. G952 /.
Is to the Dep. 4673 1.
So is Radius 90°
To Tan. of Course 33° 55'
2'o find the Distance.
As Sine of Course 33^ 55'
lis to the Dep. 4G73 t.
So ia Radius 00^
3.84204
3.6(i959
10.00000
9.82755
9.746G2
3.66959
10.00000
To the Dist. 83G1 /. S.r.l2&7
As the dcf eiency in the Table is in the S. and E. column.s,
rhc home course is S. 33° 55' E., and the di^^tance to the place
of beginning 83 c/i. 61 L
iingrarn belong-
PROBLEM ly.
To lay out a straight road from A !o D, (Fig 71,) instead
of the crooked line iV B C D.
Set your compass at A , and take the course anct dis-
tance to B, wiiich, insert in a Traverse Table as below, ac-
cording to the preceding problem. In the same manner
tAke tlie courses and distances from B to C, and froui C to
D. Then find the course and distance from D to A, as di>
rected in the foregoing example, and run it accordingly.
'I'k''
iQ
LAND SUUVEYINO.
Couives.
Dist.
Diff. of Lat.
N. ) S. ~
Departure.
E. ! W.
y. 80° E.
jP. 15^ E.
ih. 20° W.
7.ft0
7.00
10.00
1 . 80
7.63
9.40
7.3!^
2.05
1
1
5J.42
18.33
9.41
3.42
1 1
-. l'^'^2
Diff. of Lat. S.'
18.33 1
5.02Dep.E.I
■fhe course from D to A is N. IS^ G' W., and the distance
19 cA. S9/,
By traverse runnin^ir, a Surveyor being furnished with a
correct outline of the boundaries of any County or Province,
may lay off the courses and distances of roads through the
forest, or explore a body of Wilderness Land. Traverse
running may also be applied to the determination of inac-
cessible distances.
PROBLEM V.
Tofmd the Diference of variation* on an old line.
_ ff the mark.s made at the time of the first survey remain
Visible, c/ace the original line with your compass. Then
the Difference between the Course now given bv the compass,
and the Com o laid i\o^^■n ui)on the plan or ip.ap is the Dif-
fcircnee of variation, since the time at which the fr /mer sur-
vey was made,
Tut if only tiio cvtrcmitio: of the old line, or if only twa
pomts in tlie line not visible from each othe-, can be satis-
fdctordy determined, .;et your compass to tiie course laid
oown on the jilan, or mentioned in the grant or deed, and
run out the distance. Tills course and distance will gene-
rally bring you out in sight of the end of tlie original ^line.
From the end of the old line> and at a rigjit angle thereto!
measure the e\uct distance to tijc line which yo»i have just
run. Then,
* DiJJh-cncc of Variation denotes the diflereiico between the
variiitioij of tl)e compass at tlie lime when the line w.-i, fir-. r„,.
«ndil,e l.meuhenit beco.r.s necessary to run it ngain, or ra-
■#
THB RCKWIKO OP LIWEfl.
d the distance
• ngain, or ra-
As the length of the old line
Is to the Distance,
SI
So is 57.3
1 o the degrees, minutes, &c., in the Diff. of Variation.
it the course bring you out to the left of the old line, the
difference of variation is Westerly; if to the right, it is East-
erly^, and must be allowed accordingly.,
iVo/c—In changing the course by the compass to suit th«
d fferenee of variation, remend)er to add when the cou se of
the line to he run s n the N P or «; "ur V ^°"V^® ?*
tract when it is n tlio N n^' S I' ' ^"^"e'-, and sub-
EXAMPLES.
1. The original course of the old line A B, (Fj> 72 >
was N. 550 W ,„d length, 100 ch. Running that^course
and distance I eame out 6 ch. to the left of B. Required the
difference of variation, and the home course.
a/' it '^'k ''MZ '■ ''''^ = ''' ~^' + ^^'^ I^iff- of Var.
S 410 s/e r r-'° ^^'^^• = '^^^ S4'. Therefore,
o. '11 34 L., 13 the home course.
The difference of variation may also be found by the 2nd
case of right angled Trigonometry.
Thus, As the distance A B 100 ch. o noooc
Is to the Distance BCCcA. 5 77815
bo IS Radius 90 o j^-^^^^
ToTan. of/BACS=26' V^TsIs
giving 3 o 26' for the difference of variation as before
2. Uuiming the old line A B, (Fig. 73,) by its orij^iral
required the correct present course from B to A
ch. ch.
As 50 : 4 : : 57. S'^ : 4^^ 34' = Diff. of Var W Tf,n« v
10*^ R ~U Ao QA' wr XT ^'"« "I var. ». lhenls<.
A iU^^lc • a \. '• -^ ne course from B to
•a, tnereioro is .9 i.ic c></ iir ••" iv»
3. Being craployed to find the present course of an nl,l
lino, or,g,„ally run N. QO«E, (Ag. ?..,) one .."ue! B°" uo-.
93
LAND SDaV£VIRa<
I
ning that course and distance I came out 8 r/b. to ibo right;
required the correct home course.
ck. ch.
As 80 : 8:: 57.3° : 5° 43' E. Diff. of Var. Then N. 20
E. — 5° 43' E = 14 " 17'. The home course therefore ia S.
140 17 w.
4. Being employed to trace an old line A B, {Fig.
75,) formerly run N. 15° W, (30 ck,, by running that course
and distance I came out 5^ efts, to the right; required tho
present course from A to B.
ch, ch.
As 60 : 5.5 : : 57.3= : 5]° Diff. of Var. Then 15« +
6 ° 1 5' = 20 = 1 5', Hence ihc present course of A B is N,
eO= 15' W,
' PROBLEM VI.
To find the difference of variation on coeval or eontemporor'
ry lines.
Ascertain by the compass the courses of a number of old
linos, the more the better, then the difference between their
mean and the original course is the difference of variation
sought, nearly.
EXAMPLE.
In tracing the several lines of an old grant, in which tho
courses were laid down us running N. 20= E., I found them
to be as follows, viz: N. 21 = SO' K., N. 21 = 15' E., N. 21 °
40' E., N. 21 = 50' E., N. 21 = 45' E., N. 21 = 30' E., and N
21 ° 15' E.; required the difference of variation.
21 ° 30' -|- 21 = 1 5' -[- 21 ° 40' -f 21 ° 50' -|- 21 ° 45' +
21 ° 30' 4- 21 = 15' = 1.50= 45'
Then 150® 45 —7 = 21= 32' mean course. Then 21 °
32' — 20= = 1 = 32' = diff. of var. E.
Directions for blazing, and for running lines when the
course is obstructed by trees, houses, hills, ravines, <^c.
1. Corner trees, or bounds, are generally blazed on four
sides J and the initials of the Surveyor's name, the initials
kn, to tbo right;
Then N. 20
i therefore ia S.
le A B, (Fig.
ling that course
;•, required tbo
Then 15« +
56 of A B is N,
THE RDNNING OF LINES.
98
or contempora."
number of old
3 between their
ce of variation
;, in which tho
1., I found them
15' E., N. 21°
° 30'E., andN
ion.
-\- 21 ° 4i3' +
.sc. Then 21 °
lines when the
, ratrincS, ^c.
biazoct on four
me, the initials
of the owner's name, and the year on which the Burvey was
made, should be impressed on them with a marking iron.
2. Trees standing on the line are generally blazed and
marked with three notches, made by striking the axe up^
wards.
3. Lines should be well cleared out and bushed; and all
trees standing within four feet of the line, on each side,
should bo blazed on two places in the direction of the line.'
Large trees that obstruct the course, ought to be blazed and
hacked exactly on the part of the tree which is cut by the
ourse. They should likewise be blazed and hacked in the
same wr.y on the opposite side.
Suppose you are employed to run the line A B, (Fi^ 70 )
commencing at, or bounded by the spruce tree at a"* Ha-
ving marked the tree as may be necessary, place your com-
pass between A and C, as close to A as may be convenient
and set it to the given course. Let the Eushman go forward
and cut away all the bushes that obstruct the sight, .mtil
they arnve at the tree at C, standing exactly on the line.
1 h,s tree is therefore to be blazed and hacked in the cen-
tre. It is to be blazed and hacked in the centre like-
wise on the opposite side. Remove the coirpass and
place It between C and D, as near to C as may be conveni-
ent, and set u to the original course. Be cardul that your
compass is level and that the back-sights cut the tree at C,
on he centre. Direct the bushmen again until they arrive
at the tree at D, vhich does not stand exactly on the line
Cause It to be blazed and hacked, quartering on the left sic'e'
Again remove your compass, and place it a little beyond d"
and set it to the course. Let C. bushn.en proceed as before
to the tree at F, which stands on the left side of the line
Cause It to be blazed and hacked on the right side. Proceed
"1 the same manner until you arrive at the end of the dis-
tance at B. A little practice will render the whole familiar
Again suppose you are called* to run the line A B, (Fit,'
77.,) the course of which is obstructed at c by a chuiih ami
Its enclosure Proceed as before until you'arriv: a
Pomt c, within a little distance of the church. Then strike
94
LAWD SURVEnilG*
off to the right or left, as may be most convenient, exactly at
a right angle with your course. Run in this direction until
you are fairly clear of the obstructions, as at d. Then ro-
Bumc your original course, until you can, without any ob-
struction, return to the first line, as at e. Then strike off
towards the original line, at a right angle with your last
course, and run the same distance towards the firfvt line as
you formerly run from it, making r/ equal to cd. Add the
distance de to the distance Ac. Place your compass at/, set
it to the course and proceed as before
4. Allowance must also be made in running lines for ine-
qualities of surface. A line run over steep hills, or across
streams with high banks,, will evidently be longer than it
would be were the line throughout its whole extent perfectly
level or horizontal. In a survey it is generally this level or
base line, and' not the surface line, that is required. Now
as this base line cannot* be determined by direct measure-
ment, other expedients must be adopted for its determina-
tion. For this purpose different methods may be employ-
ed. Were it always practicable to measure the ascents and
descents in direct lines, it would be easy to determine the
iS-ase line by Trigonometry. Let the direct distance from A
to B, (Pi^r, 73,) up a hill, be S8 cL, and from B to C, down
its opposite side, be 51 ch., and the angle CAB 23° 4&', and
the angle A C B 54° 8', then the horizontal distance from A
to C will be found to be 103 ch. But it is seldom possible to
measure acclivities or declivities in a straight surface lino
liy the chain. In certain cases the principles laid down in
the mensuration of heights and distances might be applied
to ascertain its direct length. It is, however, seldom convc-
tiient to employ this method. The usual way is, in ascend-
ing a hill to direct the hindmost chain-man to hold the chain
directly over the pin left by the leader, at such an height
that when the chain is stretched it will he horizontal; and in
descending to direct the leader to hold his end so high that
the chain, when straight, will be level, and then stick his pin
perpindiculariy below the end of it. To determine when
tho end of the chain is directly over the pin in ascending a
THE BUKKIIIO OF LUES.
M
h.ll, and the exact spot in whici, the ph. shouW be placed in
descendnig ., ,s customary to recommond to let a stone fall
from the end of the chain. The use of a Ime and llm"
s an nnprovement npon .hi., method. In ovdina y caTet
the eye can determine «hen the chain is horizontal, men
greater accuracy i., required, the Quadrant n>.ay be employ!
In running line, over hilly ground, a two-pole or half chain
s preferrable to a ^hole chain, i: i, much easier to stretch
he former, until it becomes nearly s-raight, than thettter
meZr"" """•»••>'' P^OP" '0 employ even a shorter
Jh ^'""r """""^ "f "="'•"« 'he boundaries of land, upon ,
ac
1«'
160
155
150
145
140
135
ISO
125
120
Ji
115
4->
110
a>
105
100
95
CO
90
'3
,85
X!
80
CO
75
f^
70
.2
G5
60
J3
55
s
50
W
45
40
35
80
25
20
15
10
5
Thence N.
land surveying..
Example of a Field Book.
and 1840.
160 chs. to a juniper trco marked H, W.
on four .sidej'.
144 chs. to a maple squared and hacked
128 chs. to a pine s(piared.
112 cAs. to a birch squared.
96 chs. to a spruce squared.
80 chs. to a pine do.
G4 chs. to an ash marked do,
43 chs. and hounded by a spruco stake
32 chs. to a hemlock marked S. B.
16 chs. to a fir marked T. D. .
W.
20'
Commenced at a spruce tree marked H. W, and 1840 .
THE RUNNING OF LINES,
97
!0 murkod il. W.
W, and 1840
Additional Remarks on the Running of Lines.
The great secret, the grand security of success in survey-
ing, is to run the correct course, to run the lines exactly
straight from bound to bound, and to measure with accuracy
the distance between them. To obtain these results, every
thing ^depends upon the quality of the surveyor's instru-
ments, and on his skill and dexterity in using theiiL
As lands are frequently bounded by curved, as well as by
straight lines, it is the duty of every surveyor to make him-
self thoroughly acquainted with the properties of curves.
It is customary to bound lands on rivers, roads, &c.
When lands are to be bounded by rivers, great care should
be taken to place the boujids at a proper distance from the
edge of the stream or shore. Banks are liable to be under-
mined, and if the bound bo too near the brink it may fall
in, and leave its exact position in uncertainty. From this
source disputes frequently arise, leading to lawsuits, and
resulting in the loss of the property, peace, and character
of the parties, and the reputation of the surveyor.
Difficulties likewise frequently arise from bounding lands
upon roads. The road is liable to be changed for the pur-
pose either of straightening or levelling it, and their boun-
daries are in danger of being removed. Surveyors therefore
should, as far as possible, measure from known and well esta-
blished boundaries, and run in straight lines. It is particu-
larly desirable that the place of beginning be tlistinctly
marked, and not liable to be removed. Lines then would
be easily retraced either by the surveyor by whom they were
run, or by his successors.
In the tracing and retracing of lines, great care sJiould al-
so be taken to follow closely the original line, especially if
it be straight. No trees should be blazed except those
which were blazed before. When individuals unacquaint-
»'d with the properties of lines, mark trees which stand per-
haps two rods to the right or left of the original line, inte-
I
r
I
08
LAND SLRVEYING.
rested parties may be deceived, or an opportunity afforded
to the litigious to enibroif his peaceful neighbour in the
anx.eties and lo.ses of a lawsuit. The surveyor too comc3
in for a full share of the blame. No unqualified or unau-
thorized person has any right to take such liberties. By
such improper conduct peaceful settlements are frequently
thrown into confusion, and evUs of incalculable magnitude
have been produced.
MENSURATION OF LANDS.
The area or contents of any figure is the measure of the
surface contained within its lines or bouiKlaries.
In Land Measuke.
m feet, or 25 links make , Rod, Po,e, or Perch.
4 Hods, or 66/^, or 100 1., or 22yds. 1 Chain.
10 square poles or perches i Rood
4Roods.o. 10sq,chs.,orI60sq.poles 1 Acre.*
10,000 Square Links i Sn..n,.« ri •
,.1- t; T- , i^quare Cham.
():Jo square Links it- r. .
or. AftftQ T , 1 Square Perch.
iij,000 Square Lmks i s^„^,.„ p ,
innnnna t • , Square Rood.
100,000 Square Lmks i c^......^ *
nn u A ^ Square Acre.
«i4n Square Acres i «„., iv,.,
un r'l • • , . 1 Square Mile.
HO Chains in length j j^jjj^,
ent^ft;ro"irr^'''"^ Umd itwmbe found most convrni-
tnt to take all the measures m four-pole chains and link.
PROBLEM r.
To Ml the area of a Parallelogram, whether it be a Hanare
a hectangle, a lihomhus, or a Rhomboid.
Mnliinly the length I
the product v.-ill be th" -
RULE J.
>y the per[)endicular breadth,
arul
e area.
ortunity afTortlcd
icighbour in the
veyor too comes
lalified or unau-
h libertioss. By
s are frequently
la])Iy magnitude
MENSURATION OP LANDS.
99
DS.
measure of
the
rie-
'•■
,Pfi
lie, or Perch.
in.
xl.
1
are
Chain.
are
Perch.
are
Rood.
ire
Acre.
ire
Mile.
1 n)o.st conveni-
ins
and l\u\i!i
.
it be a Sqnare.
mboid.
r breadth, an»l
RtJLX U.
As radius
Is to the sine of any angle of the parallelogram,
So is the i)roduct of the sides containing the angle
To the area of the parallelo^nam.
' nVLE. III.
Multiply the product of any two adjacent sides of the
parallelogram by thejiatural sine of the included angJe.
EXAMPLES.
1. A square tract of land, B A C D, {Fig. 79,) fronts on
the road B A, which runs N, 80^ W., and the length of its
side 44 cA. 72 /.; required the area, the courses o/the other
sides, and a plan of the same.
AB= = 44.72 X 44.72 = 1999.8784 ch. ~ 10 = 199
.98784 ac. = 199 .?c. 3 r. 38;). = area.
The course of A C is N. 10° E.— of C D, S. 80^ E - nd
ofDB, S. 10°W.
Draw the road B A. Take any point on that road as at
B, and lay off 44 ch. 72 I. towards A, according to the given
scale. At A raise the perpendicular A C, and according
to the same scale lay off 44 ch. 72 /. Draw C D equal and
parallel to C A. Then join D B, and the plan is drawn.
2. Requn-ed the area of the Rectangle A G F E, (F»g- 80 )
whose length A G is 80 ch., and breadth A E 30 cA '
AGXAE = 80XS0= 2400 ch. = 240 ac. = area.
3. Required the area of a Rhombus A B C D, (Fig 81 )
whose front A B runs E. 15 cA., and the side B C, N. 10- E.
and the perpendicular D K is 14 ck. 77 I. '
1477Z. = DK.
1500Z. = AB = Ba
22.15500
4
. 62000
40
24.80000 ^n«. 22 ac. r. 24/).
4, Requ^ ' the area of a Rhomboid A B E F, {Fig.
82,)
m
100
LAND SURVEYING.
•n
\vh')>v front A B is 15 ch., the side B E, 50 ch. 86 /., and
the per|)eijdicular, 50 ch. 8 /.
5008 /. X 1500 /. = 7512000 /. = 75 ac. r. VJ p. =nrea.
5. Required the area of the above Rhomboid the angle
B A F being 80°.
By Rule 2: As Radius 90° 10.00000
Is to Sine / B A F 80° 9 . 99S.S5
Sois B A X A F 762.9cA. 2.88246
751.3 cA. 2.87581
751 .3 ch. = 75 ac. Or. 21 p. = area. Or, by Rule 3:
B A X A F X Nat. Sine / 80° == 762.9 cA. X 98481 =
751. SI 1549 ch. = 75 ac. Or. 20.98;>.j or 75 ac. r. 21 p.,
nearly.
PROBLEM 11.
To find the area of a Triangle.
RULE 1.
Multiply one of its sides by a perpendicular let fall upon
it from the opposite angle, and half the product will be the
area.
RULE n.
Multiply the product of any two of its Bides by the natu-
ral sine of their included angle and half the product will be
the area,
EXAMPLES.
I. In the triangle ABC, (Fig. 83,) A B is 15 cA., and
the perpendicular C D let fall upon it from the opposite an-
gle at C is 14 ch. 77 l.-, required the area.
A B = 1500 Z.
CD = 1477/.
2)2215500
A rea = 1 1 . 07700 I.
4
.30800
40
12.35000
X
1 ac. Or. 12
p.
MCN8URATI0N OP LANDS.
101
^.»u t. 8 cA., and A C runs N. 20*^ E. 26 ch. (J5 /.• lo-
quired the area.
S. 80^ E.
N. 20^ E.
180'
100'
100 Sum, 80° = ^ B A C
Then by Rule 2, A C X A B X Nut. Sine / 80' ^
,HnZn ^ ' '^- ^ ■'^'^' = 209.9614920 cA. ^ 2 = lO-i
.9807460 ch. = 10.4980746 ac. = 10 ac 2 r. nearly, ^ urea.
I he area may al«o be determined by Logarithn.s, accord-
nig to the following rule:
As Radius
I^ lo the sine of any antjle of a i ian-rle
So is the product of the sides containing the anglo
1 o twice the area of the triangle.
Then, As Radius 90°
Is to the Sine of / B A C 80'
.So is ABXAC= 26.65X8= 213.2
10.00000
9 99.335
2 . 32878
To twice the area 210. ch.
2.32213
Then 210 ^ 2 =: 105 cA. = 10 ac. 2 r. = area.
PROBLEM III.
The three sides of a Mangle being given to find the arm.
RULE.
From half the sum of the three sides subtract each sido
separately. Then multiply the half of the sun. of the sides
by .he three remainders successively, and the ..nuare root
oi the product will be the area. .
EXAMPLE.
..^T'ri'}^ ^'"^ °^ "" '"^"'^^^ ^^^°«« three «ide.. arc re>
spcctively 20, 30, and 40 ch. (Fig. 85.)
10:^
LAND SURVEYING,
eh.
30
80
40
2)90
sum,
45
half-surn,
25
1st rem.
1125
15
2nd rem.
16875
5
3rd rem.
45 — 20 = 25 l8f rem.
45 — 30= 15 2nd rem,
45 — 40 = 5 3ni rem.
V 84375 = 290.47 ch. = 29.047 ac. == 29 ac. r. 7 p.
area.
By L0UARITUM.S.
Half-.suni 45
1st rem. 25
2nd rem. 15
3rd rem. 5
Log. 1.G5.S21
1.39794
1.17G09
0.69897
2)4.92021 sum.
290.4 cA. 2.463105 square root.
Now 290.4 ch. = 29.04 ac. — 29 ac. r. Gj3. 5= area.
PROBLE ' IV.
Tcjind ih:- irea of v Trapczoii^.
nuLE.
Add together the two parallel sides, and multiir- their
sum by the perpendicular breadth, or by the distan be-
tween them. Then half the product will be the area.
EXAMPLE.
Required the area of a Tr.npnzoid A BCD, {Fig. 86,)
whose parallel sides are 12 ch. 41 I. and 8 ch. 22 /., and
whose perpcndicuicir distance is b ch. ID I.
MENSURATION OF LAMDS.
A B= 12.41
CD= 8 22
lOA
Sum of parallel sides 20. G3
And rectangular breadth =: 5.15
2)100.2445 = product.
Half prod. =: 53. 12225 cA. = 5.S12225 »c.
10 p. nearly.
= '» uc. I »•
PROBLEM V.
To find the area of a Megular Polygon.
RULE.
Multiply the perimeter of the polygon or the sum of its
Bides, by the perpendicular let fall from the centre ui)ori one;
of the sides, and half the product will be tho area.
EXAMPLE.
The sides of the regular pentagon A B C D P: A, {Fig.
87,) measure each 25 ch., and the perpendicular O P mea-
Burcs 17 ch. 2 /.; required the area.
25 cA. X 5= 125 ch. := perimeter. Then 125 ch. X 17-
.02 ch. = 2127.50 cA. -^ 2 = 1063.75 cA.= 10G.S75ar,
= 100 Of. 1 r. 20 j9. = area.
PROBLEM VL
To find the area of a Trapezium.
RULE.
Draw a diagonal dividing the trapezium into two trian-
jrles; then find tho areas of these triangles, and their sum
will he the area required.
iYo^e. -If two perpendiculars be let fall on the diagonal
Jrom the opposite angles, and their sum multiplied bv the
diagonal, halt the product will be the area.
EXAMPLE.
-n the trapezium A B C D, {Fig. 88.) the diagonal A C
is 13cA. 50/., the perpendicular Da 6 cli. 50 /., and B6 5 -/i.
70 /.; required the area.
J 04
LAND SURVEYING,
5.70 + 6.50= 12.20 X 13.50 = 164.7000 -
3500 ch. = 8.235 ac. = 8 «c. r. 37 p. = area.
•2 = S2.
ffi!
I, '
PROBLEM VII.
To find the area of any Rectilineal Figure.
RULE.
Divide the figure into triangles, find their areas separately,
then the sum of their areas will be the area required.
EXAMPLE.
Required the number of acres contained in the farm, thr,
field notes of which are contained in the following Feld
Book. (Fig. 89.)
No. OF Triangle.
Bases
in links.
Perpend.
in links.
1
o
3
4
5
Double Areas
in links.
5210
6400
6100
6100
4500
1700
2500
2900
2500
1450
8857000
16000000
17690000
15250000
6525000
Sum of douiile areas
Area in links
64333000
321
61000
4
2.44000
40
.ins. 321 ac. 2 r. 17;?. 17.60000
In running the J)ase line, the perpendicular may be taken
by the cross stafl^l Set up the cross staflTat that point in the
base at which, while one of the lines c ' the cross ranges
with the base, the other points exactly to the opposite angle.
Measure the distance from that point to the opposite angle,
and enter in the Fold Book in the column marked perpen-
diculars. Fini.«h the running of the base line, and insert its
entire length in the column marked bases. Multiply the
base by the perpendicular, and insert the product in the
column marked double areas. Proceed in this manner until
all the triangles of which the figure i.s frjinnoRfid are mea^
.sured and computed. Divide the sum of the double areas
by 2, and you have the area of the whole figure or farm,
00^0 = Si,
•ea.
Fissure,
MENSURATION OF LANDS.
Vf)h
In order to plot tlio survey it will be necessary to note the
distance of the point at whicJ^ the perpendicular was taken
from the end of the base line.
i^"i^r~'^ *^'°"^^ staff with sights is to be preferred to one
whicli has only small points at the extremities of the cr<)-<s
Imes.
reas separately,
required.
11 the farm, the
rollowing Feld
'ouBLE Areas
in link s.
8857000
16000000
17690000
15250000
6525000
643 33000
321.61000
4
2.44000
40
17.60000
may be taken
lat point in the
! cross ranges
opposite angle.
opposite angle,
larked perpen-
, and insert its
Multiply the
)roduct in the
s manner until
>sed are mea-
D double areas
ire or farm..
PROBLEM VIII.
To find the area of a Mixtilineal Figure, hy means of eqiii-
distant ordinates.
RULE.
Measure the perpendicular breadths of the figure in seve-
ral places, equidistant from each other, then divide the sum
of these perpendicular breadths by their number, the ((uo-
tient multiplied by the whole length of the line will give a
near approximation to the area of the figure.
example.
The length of the base of a field, curvilinear on one side,
is 7 ch. 20 /., {Fig. 90.); and the lengths of seven equidi?:-
tant ordinates erected upon it are respectively 200/., 225/.,
230/., 248/., 260/., 280/., and 300/.; required the area of the
field.
. . ^inks. Then, 249 mean breadth.
A I = 200 7C70
D A = 225 L
fjf = 230 1.79280 area in link*.
F/ = 248 4
G e = 260
Hdr== 280 3.17120
(3 C = 300 40
7)1743
6.84800
249 mean breadth. Ans. lac. 3 r. 6/?. nearly.
If greater accuracy l)e re(|uired, take half the sum of the
two extreme breadths for one of the ordinates, and add it to
Uie others as before; then divide the sum by the number of
parts in the base, instead of bv the mimbRv nf ni.Hin,.t«.
'•Hid this mean breadt
lultiplied by the length of the ba
so
ii
106
LAND SURVEriNG.
will give the area. It is still, however, only an approxima-
tion, but sufficiently near the truth in ordinary circumstances.
It may be observrl, farther, that the greater the number of
onlinates employed, the nearer the result will be to the ex-
act area.
When the curved boundary meets the base, as in often the
case in surveying, the area is found by dividing the sum
of all the ordinates by the number of parts in the base, and
then multiplying the quotient by the length.
If it is particularly inconvenient or imiu-acticablc to erect
ordinates at equal distances, perpendiculars may be raised
at unequal distances, and the parts into which the figure is
then divided may be computed as so many trapezoids, and
the^sum of their areas taken as the area of the whole.
PROBLEM IX.
To find the diameter of a circle whose circumference is given,
or the circumference when the diameter is given.
RULE I.
As 7 is to 22, so is the diameter to the circumference.
As 22 is to 7, so is the circumference to the diameter.
RULI^II.
As 1 is to 3. 1416, so is the diameter to the circumference.
As 3. 1416 is to 1, so is the circumference to the diameter.
EXAMPLES.
1. If the diameter of a circle be 1 ch. 12.68 /., what is the
length of the circumference.
Asl : 3.1416 : : 112.68/. : 354 /., or 3 ch. 54 L, Jlns.
I. If the circumference of a circle is 3 ch. 54 /., what is
the diameter.
As 3.14ir> : 1 : : 354 I. : 112.68 l, or 1 ch. 12.68 /. Jns.
By this problem the number of degrees contained in the
radius of a circle may be determined. For since the radius
is half the diameter, and the cirenmference cnntaing 360° it
follows that half the quotient of 360° divided by 3. Hie^ill
cive the number of degrees contained in the radius.
MENSURATION OF LANDS,
PROBLEM X.
To find the area of a circle whose diamcicr is given.
!07
RULE,
Multiply the s<iuare of the diarii
product will bo the urea.
ctcr by .7854, and the
EXAMPLE.
llequirpd the area of a circle whose diameter is 10 n
10= X • 7851 = 78. 5400 c/^. = 7.85^
= area.
ro40Qac. = 7ac.Sr. Uip..
PROBLEM XL
To find the area of a circle whose circumference is ^i,en.
RULE.
Multiply the square Of the circumference by 07Q58 md
the product will be tire area. '
EXAMPLE.
JU<>~- X .07»53 = 78«.6.9C8O0 L == 7 „.. 3 ,, ,3^. ^
/., what is the
PROBLEM XIL
To find the circyference of an Ellipse, the transverse and
conjugate diameters being given.
RULE.
Multiply the square root of half the sum of thn
the two diameter. 1^ 3.14,(i, and the ^h ■ villT"?'
'•ircumfereuee, nearly. l"-0(ni( t udl be the
EXAMPLE.
Rrquired the circumference of -.n Vii:.
trans-
v^^L±_Ji'l
^ X 3.1410'= V 5iljf_36^
xa. Mifi==.
108
LAND SURVEYING.
V58.5X 3.141G = 7.648 X 3.1416= 23.9269= the
vircumfVionce.
PROBLEM XIII.
To determine the area of an Ellipse, the transverse and
conjugate diameters being given.
RULE.
Multiply the product of the transverse and conjugate di-
junolers by . 7854, und the result will give the area.
EXAMPLE.
Required the area of an Ellipse whose transverse diame-
ter is 9 eh., and conjugate 6 ch.
9 X 6 X .7854= 42. 4116 cA. = 4ac. Or. 3B p. Ans.
Note. — In practice, a surveyor is seldom required to mea-
sure a circle or ellipse.
PROBLEM XIV.
To find the area of a farm by drawing apian of il from a
scale of equal parts.
RULE.
On good paper, draw a correct plan by a scale of equal
parts, (say a scale of 2 ch. to an inch, — the larger the scale
employed the more accurate the work will be). Draw large
lines dividing the figure into triangles, on these bases let
perpendiculars fall from the opposite angles. Measure the
length of these bases and perpendiculars by the same scale
by which the plan was drawn, and enter them into their re-
spective columns in the Calculation Table. Multiply the
several bases by their corresponding perpendiculars, and
insert their products in the column of double areas. Then
half the sum of this column will be the area of the field,
nearly.
EXAMPLE.
Required the area of the farm, the field notes of which
are contained in the following Field Book. (Fig. 91.)
MENSUR.VTION OP LANDS.
109
23.9269 = the
transverse and
?n.
[1 conjugate di-
e area.
lasverse diame-
3Sp. Jlns.
jquired to meu-
un of it from a
scale of equal
larger the scale
). Draw largo
these bases let
. Measure the
the same scale
m into their re-
Multiply the
mdiculars, and
3 areas. Then
ea of the field,
lotes of which
(Fis.dl.)
Fie
i-i) Book.
Nu.
OF S
TATION.
Course. |D
ISTANCE.
1
at
A B
N.18'^34'E.
4892
2
BC
N. 20^ E.
5000
3
CD
East
4000
4
dp:
S. 20^^ E.
5100
5
EF
N. 80^ VV.
4500
6
FG
S. 15° E.
5500
7
GA
West.
6000
Calculation Table.
No
OF Trian.
Base.
|Perp.
jDouBLE Area.
1
2
3
4
5
CED .
C K F
C B F
B G F
A GB j
7503
7503
5000
6404
6000 1
2506
2300
2600
2603
4609
18802518
1 7256900
13000000
16669612
27654000
2)93383030
466.91515
4
3.66060
40
t ... ^ 26.42400
^'Ins. 466 ac. 3 r. 26 p..
Kemarks.-.ln the preceding Field Book the first column
contain.s the No. ot the stations, and consequentiv the num-
ber ot the sides of the farm; the second, the bearings of the
Imes from the meridian; the third, the stationary (fistances
m links. 1 he second and third columns contain all the data
necessary to protract the plan of the farm, A BCD E F G A
i he first cokmmof the Calculation Table contains the
iNos. of the triangles into which the lignre is divided- the
second the base of caclMrianglo; the third, the porpemJicu-
ur; and the l.)urth, the product of the base and perpendicu-
lar, mid IS called the column of double areas, half the sum of
which gives the area, nearly.
This method, however, will only give an approximation
lil. IV''"'- 1^^ '^ ^^^\poss\h\c by it to determine the
length, ot base and perpendicular lines within several links,
tvspocially It the-e lines arc of considerable len-rh
110
LAND SURVEYING.
PROBLKM XV.
1 find Ihc'avca of am, Rectilinear Fis;urc (the courses and
distances roimd the same beinu; friven,) without the assist,
ance of a plan, by Rcctanii;ulur Surveying-; i. e. by calcu-
lation from Tables of Northing, Souihing; Easting, and
il esting.
KULE.
Prepare a Tablcj with ten columns. In the first, lioaded
'' No. of Stilt.," write the nimiber of the staiioii, 0, 1, 2, 3,
&c. Ill the second, headed " IJeurinir," write the eour/ie,'
In the third, n.arked " Di^st. in ch. /.," insert the distanec hi
chaniH and links. In the fourth and fifth eohunns, headed
" Difi". Lat.," insert the Difierence of LutitiKh.-, a.-eordin.r
to the directiojjw contained in the HI. and IV. Prob. ol"
Transverse n^tniinjr of lines.
Fill up likewise, ae(;ording to tlu' same directions, the
sixth and seventh colunms, heu(hMl " Half Dep.," observing I
only that inste-ad of the whole, the half of the Departure is
to be entered. Place the si-.i -|- before the Hastings, and
the .si-n- before the Westi..-s. W the field notes have
been ('orrectly taken, and the entries of Difi". of I.at. and of
Half Dep. been accurately nuule, the sum of the North and
S()uth C(dunuis will be equal. The sum of the East and
^V est colunuis will also be etiual.
In the eighth column headed " Mer. Dist.," and oppositu
U) in the first colunm, insert the whole Departure, or doti-
l>le the sum uf the Half-Kastin-s, or Half-VVestin-^ con-
tained in the sixth and seventh columns. Obscrre what
sum has been entered in thefi'rst lineof eitiierof the columns
headed "Half Dep.," opposite to the fi-ure 1 in the fast
column. Observe also whether the v.niry has been nutde in
the E. or W. cohunn. If the entry Jias been made in tlw
r:. colunm, add the smn to the whole Departure. If the
entry has been made in the W. column, snbtract'the sun.
irom the whole Dej.arture. Insert the smn or difierence in
the ei-dith culuiim in,
iik.ul " Mer. Dist." 01
kiriy, that the sun) now iu.serte<! is the
)serve jiarticu-
vicridional dislaua
MENSUnATTON OF LANDS.
Ill
i>f the. middle of the first line. To this sum add or subtract
accordinjr as it is P^ast or W(^st the same sum or Hjslf De-
parture, and you have the meridional distance at the end of
the fi-st line. Observe a,i,rain the sum that i'* entered in the
column of Half Dep., oj)j)osite to the figiii-.' 0. in the iirst
column ^f the Table; and accordinpr as it Ici- !he sign 4- or
— pr-^fixcd, add it to or subtract it from the in. ildional dis-
tance at the end of the first line. This will ive the meri-
dional distance of the wuV/r/Zt' of the secojid i!;)f. Proceed
in this manner until the column is completeh :i!l, d. if the
operation has been correctly performed, the last sum will be
equal to the sum at the head of the column.
Next, multiply the several meridional distances at the
middle of each line into the Northing or Southing, which
will be found opposite to that meridionnl distance hi one of
the columns marked Dim Lat. in the Table. Y\heu the
sum has been taken from the column marked " N." the pro-
duct is to be inserted in the ninth colunm, marked '' North
Areas;" but if it have been taken from the column mnrKed
"S." the product must be entered in the tenth coiinim,
headed "South Areas.^' Then the difference between the
sum of the products contained in the column of North Areas
and the sum o" the products in the column of South Areas
will be the area of the Figure.
t.
Idionul dislauGi
EXAMPLES.
1- Required the aroa of a farm whose field notes are as
follows, viz:— N. 20° E. 50 ch., P^ast 10 ch., S. 20° E. 51 eh.,
N. 80° W. 45 ch., S. 15^ E. 55 ch., West 60 ch., and N. 18°
34' E. 48 ch. 92 /.
Having prepared your Table, and entered your stations,
courses and distances in their resjjective columns, by the di-
rections and principles laid down for the running of linos,
or from the annexed Tables, find the DitT. Lat., and the
Half Dep., and insert them in their proper i.dace in the Tn-
ble. Thus the first course is N. 20^^ E., and distance 50 ch.,
the Northing is 40 ch. 98 /., and the Half Easting 8 ch. 55 /.
Insert the former in the column marked N.; and the latter
Ml
l.KTil) SURVEYING.
n the column marked E., and place the sign + before U.
Proceed in this manner until the columns of DifT. Lat. and
Half Dep. are filled u]).
Add up the colunm.s of DifT. Lat. and of Half Dep.
The sum of the Northings is 101.05. The sum of the
Southings is also 101 .05. The Eastings and Westiygs arc
likewise equal. These agreements shew that the survey
'las been correctly taken.
Proceed next to fill np the column headed " Mer. Dist,"
In this column, in the same line with in the first column,
write 104 ch. 32 /., the whole departure or double the sum
of the Half Eastings or Half Westings. Under the head-
ing «' Half Dep.," and in the column marked E., you will
find the sum 4-8.55 has been entered. Then 104.32-4-
8.55 = 112.87. This sum is the meridional distance at the
middle of the first line. Insert it in the column of Mer.
Dist., opposite to 1 in the first column. Then to this sum
add 8.55, and you have 121.42, the meridional distance at
the end of the first line, or at the beginning of the second
line. Place this sum in the column of Mer. Dist., perpen-
dicularly below the Mer. Dist. at the middle of the first line.
Again to the sum last entered add 20.00, which you will
find in the E. Column, of Half Dep., and it will give you
the meridional distance at the middle of the second line.
Thus 121.42 + 20.00 = ]41.'t2. Insert thi.j sum in the
column of Mer. Dist., immediately below the last entry, and
directly opposite to 2 in the first column. Proceed in this
manner until the column is filled up. The last sum must be
equal to the first, or the sum at the head of this column.
Next, multiply the meridional uistanct at the middle of
the first line, which in this example is 112.87, by the differ-
ence of latitude which will be found under « Diff. Lat." in
the column marked N., and which in this Example is 46.98;
and insert the product 5302.6326 in the column of North
Areas, because the Diff. Lat. ij N. The second meridional
distance in the middle of the line in this column of Mer.
Dist. is 141.42, but as the course is due East there is no dif-
lerence of Latitude, und conscn[UGntly no product to be in-
MBKSURATIOM OF LANDS,
U.i
Uff. Lat. and
•crted in either column of areas, The third meridional dis-
tance in the middle of the line is 170.14, and the different*^
of Latitude is 74.92 in the column marked S. The product
of these numbers is 8153.1088, and is to be placed in the
Column of South Areas, because the Difl*. Lat. is S. Pro-
ceed in this manner until these columns are completed.
^' B.— The meridional distance for the middle of the line
will always be found in the column of Mer. Dist., opposite
to, and m the same line with, the No. of the Station, the
Bcarmg, Distance, DifT. Lat. and Half. Dep. ♦
The sum of the products contained in the column of North
Areas is 10989.0006, and the sum of the products in the co-
lumn ot South Areas 15679.5043. The difference between
them 4690.5040 ch., or 469 ac, r. 8 ». is the area of the
r arm. "^
114
tMfD SURVETING.
o tf
ax
o
as
u
?5
W
u
Q
fa
ao
H
■<
O
1 —
®
CO
CO
iljjA
in
•^
Vi«
OS
S
<2
CO
IM
lO
iO
CO
i
CO
r>.
P
o
CO
8
o
o
CO
»-«
'I"
c?>
CO
o
!>•
»?» c» (n rf (r> o >* CO 00 CO o) vn e»
00 -^ -n- '^
»-< CO t-» »n
«0 »^ t^ r^ »f5 GO
G«
-H _■ r.
o CO «o Tf
•-< CO
»-^
(D CO »*
1—1
5» TT to 1^ t-
iOCO'<3'-^COaD<5iO
I-* »^
I-« 1— (
T-H »-<
1—*
""^
w
o
CO <
urn
o
»-4
s
o
n»
1
Gt
CO
o
I
1
•n
o
r«
c<
r-.
CO
if5
o
1^
»-«
»>•
^^
00
o
CD
t^
t^
-f
--
GO
T
O
ift
ao
If)
»n
o
CO
Ci
CO
QO
CO
CO
tn
o
o
o
d
in
o
Q
o
o
o —
o
o
in
o
c
in
in
o
o
d
CO
G»
CV
QO
&3
o
O
-3
o
o
C5<
o
O
OO
;?;
o
in
00
o
^.
-« G< CSO
in
CO
c «
~.J .
^ t o>
^ -^
• — n
Cw *rf *^
•sf i
a
'5
60
QJ < <
.S CO
s ■*
O .
C
.0
O)
«■<
o
3
«-*
ID
^^
d
o
v
.0
rt CO t:
el
oO =">
a o
§2;
§.2
O c3
£ =^ «
(■3 Z
KM
E-
£/
&
C
c
• — r;
g "" CO
Cw *rf *^
s< «
•sf i
^tiN a
«— . 60
QJ < <
.S 00
s ■*
o .
"13
o
s
o
o
«
.0
« 5t3
25
c "^
5 to «-■
o
>.
§2;
a o
§.2
0)
■^ CI
o u .2
si®
KM
MENSURATION OF L_JfDS.
Hi
X
IT.
§
00
H
<
'^»
00
U
u
^^
m
GO
CO
Oi
lO
QO
T
1^
irj
«>.
T
CO
r*
t"
CO
'X)
00 0;
— 'r
T
10
t^ t^ !05
•■s 'i"
■0
—
00
05
•* -T
e»
CO
•
"T
'.0
u"
>n .-.
-« Tj ^
'X)
,^
-
•^
ul
V
in
-T
•
'O
•-»
•>-
»
C(
^
'J ;o in ^ i-» O o O t~ rr lO
o
ICO
rr
CO
►^
u
c/j
12;
0.
u
Q
b.'
5:
-^«OCl»ftC»i-iO«05t>»
O O)
•-« CO
CO
o
—« CO •-«
CO
CO
'
1-1 rr
4- -f-
G<
CO
o
o
CO
o
CO
CO
CO
CO
00
B
O
(O
s
•a
a
e
e
o
o
GO
CO
CO
■^ t>.
o
CO
CO
00
r; 'o
w o
»
o
CD
o
in
o
O
c
o
CO
!>■ >
o
o
in
o
O
CO
o
vn
W w
o
o
<5«
o
>n
cr. ^ ^ ^ CD*
-« G«
iD
CO
6>
O
e
«
S
s
o
u
e
u
c
5
o
CO
2 fe
8 ^
^■Il^'''
•tr
IHf
i' ■'
ill
1
i
= ■
jf'Hi
■ HI
llHi
t
II
1
ll
■■5
m
LAIO SURVZTINO.
3 w
O «*
55 -«i
s
I-
.-J
§0 CO
in »fj
O "^ M •-<
• • ■ •
00 05 c» ■v
§<W O 00
cy5 c« -.
-< rn c«
-yj O Q
OJ Q O
t^ ift lf>
00 no oo'
rn Oi »-«
0» O CD
»^«« I'm
■^ 2 Q lO O CC <-( 05
-; CD Oi -f rr* t* erj lo
'■e ''j'ooiftooojor*
'3
♦«
a
o
s
en
-d
s
OS
»
0)
EiQ
B
in
«£> CJ ft*
>0 Cl i5«
G< *?< in
+4-4-
73
12;
oo o
— o
g'o*
CO rA
G«
•^ in o
o o o
in cc in*
C« G« -»
C C O Tf o o o
in in c as o o o
■n oj o" G« o in oo*
<30 5« (^ CO CO o« G«
«
■<
H
■I ^
\^,
^ ^ 00 O c» 00
. . . -x> (/}
-^ i^ ^ (ZJ i/5 !/J
>-i c» 9? t in -^ j>.
■5 «J.
o ^
c »^
o e
0) i-x
-C CO
fS CO
o '^
>^ 1-H
3 GO
* r«
X! "
~ 75
,« 75
' i
a
-S
a
o
a.
a
b
O
••«
«a
Q>
<t)
TJ
4>
.£3
fa ..
° s
'^ ^
rt 3
"" s
■s 2
ID U
«-i <?< G« CO rr -^
II II II II II II
t -T "rr 00 CO Q
in :o CO o r- ©
»n G« ft« o in o
^ GO »>• CO -n" CO
c» rr rr T cr CO
Xin CO 'o o J^
• • • • •
m G» G« o in
f^ .-^ CO t>» Oi -r
ll
£ &.
" a.
p . a.
5 u =
CD •• W
jl; • r
» u ^
(-.•'•
o '^ "
o, c •
Hi)'-
a. ^ >
fl - -
C4 *^
"1 r; (S (— ! tK (-» ■•iw
■S;o
§2
s
= -3
bo-3
C 3
~ O
7J
o
^rf
M^
o
«■«
o
«-J
so
m
_c
na
u.
IS
O
a.
u
Q}
ti
O O 'I" CI T
»-< O "V C? G»
• • • , •
!>• C t^ •^ f
G« G« G« G» C«
xxxxx
in o
in o
G« CO G»
t^ -< ,-4
00 o X -^i r-
an e
"S-S
cur
5 2
OS «
O a u
2 fi
M-i n -
- a -
Q « 3
^0,0
CO
CQ 2 "
. 5 3
<
'o
S $ cd
o ^ «>
DIVISION OF LANT>.
U7
(5* 00 -r --rj
II II II II
■^ 00 CO Q
«5 O r- O
'^i O irJ o
»>• crs Tf CO
•5 t
o w
E b
•' c
b> s
■ a.
E u 3
dS'S m
<D •• W
,1^ • r
(^ c u
o I. rt
° s a.
a, c at
WOT)
s « «
e! ^ >
fl - -
*" o "5
4)
a> c
1" T (T) CO
CO CO o J^
tI e< o »n
CO I— OJ 'T
+ -f-|-}-
O -^ CI T
o T CO <;;»
C t>-* Tj! TP
T3< rl Tf — <
5« G« &» C«
xxxx
D CI CO i5«
• • • •
s X c« r-
■^ *-• OS
r: « .
-O 60-
S S 3
OS §
C — la
O a u
=^ I °
2 "s
S «•-
« -
OS «
« a -
Q^ 3
O *»
. o u
"Jl
.s ^ S
•° .2 &
a .s =
.fc'u the Norf lungs a.ul Southing, into a w w,^^^^^^^^^^
"/'"'I. 'I'" « f the .olm , ;\ ■ Sou h te wiVr^'lr''? '"
DIVISION OF LAND.
PROBLEM J.
To divide a parallelogram in any propor-l, on hy a line run-
nmg parallel to a given side.
RUtE.
Since parallelograms of the same altitude are to one ano-
ther as their bases, (Euc. vi 1 > fir<l fir.=f t>,o ..
. I ^^"*- *'• ^>>' "'^<' nift the {>rea or quan-
tity of land contained in the whole figure. Then, as the
area of the whole parallelogram is to its base, ho is the area
or quantity of land to be laid off, to its base,
EXAMPLE.
i 20 1a' P'f '^'^«^!r .^ ^ ^ ^' <^^- ^4') the base A B
1.. 20 ch., and the side A D is 16 cA. It is required to de-
raised from these points, to the East or West acconlin. .0 .1
m,.p IS to be drawn on the East or West side 0^^^ ^ . ^ -^^''^
/'I,
I IB
LAND SUnVEYIKG.
terminc the point in the base A B, from which a right line
must coinnicnco, which, runnin^r i.arallel to b C until it
strikes the opposite side D C, shall form a rectangle E F B C,
containip-r lo ac. A D X A B = 16 X 20 = 320 cA. =
32 ac. = whole area. Then, as 32 ac. : 20 ch. : : 10 ac. :
G ch. 25 ;. = B F, the base of a rectangle E F B C, con-
taining 10 ac. Then from the point B lay off* 6 ch. 25 /. to-
wards A, and you will have the point required.
The preceding rule applies to all parallelograms, whether
Squares, Rectangles, Rhombuses, or Rhomboids,
PROBLEM ir.
From a ginen point in the boundaries of a Square or Rcc-
tan^le, to run a line which shall cut off a given quantity of
land in a giv."n direction.
EXAMPLE.
In the Rectangle A B C D, (Fig. 95,) containing 28 ac.
the line A B runs E. 14 ch., and the line AD runs N. 20 cA.
It is requirf'fl to lay off 15 ac, to the east by a line commcnc-
cing at E, 8 ch. from the / at B. The course and distance
of this division luie are also required.
RUIiE.
From the given point E run the line E F parallel to A D
or B C. Then determine the area of the rectangle A E F D;
thus, A D X A E = 20 X C == 120 c/t. =12 ac. = area of
A E F D. Subtract this area from the area of the whole
filture; thus, 28 — 12 = IG crc. = area of the remaining
rectangle E F C B. Find the difference between this area
and the area to be laid off; thus, 16 — 15 = 1 ac. Now it
will bo seen at once that this area is in the form of a riirht
angled triangle. Of this triangle there are known or deter-
mined the area which is 1 ac.,and the base line E F, which
is 20 ch. From these data, determine the length of :he per-
pendicular F S, by the following rule:
x)ivide the area by half the base, and the quotient will he
tho length of the perpendicular; thus, 1 ac. or 10 ch. -r
EF
DIVISION OF LAND.
UO
ch a right line
o h C until it
angle EFBC,
!0 = 320 ch. =r
ch. : : 10 ac. :
E F B C, oon-
' C ch. 25 /. to-
ll.
rams, whether
jidsj
quare or Rce-
fen quantily of
taining 28 ac.
run-s N. 20 ch.
line comnicnc-
e and distance
arallel to A D
nglcAEFD;
ac. = area of
of the whole
he remaining
eeii this area
1 ac. Now it
rni of a right
own or dcter-
e E F, which
th of
he per-
lotient will he
EF
or 10 ch = 1 ch., which is the length ol' the pcrj.endicular
i" b. As the area contained in the roctan^-le E 13 C F cut
of}- by the line E F, exceeds the <iuantitv of land (If, «o )
required to be laid off, it is evident that 'the 1 ac. mn.t be
taken tro,n that area; or, that the perpendicular F H must be
laul <,rt towards C, or the right. If, however, instea<i of the
Muant.ty to be laid oft" being less than the area or quantitv of
land contained m the remaining rectan-le E B C F it had
exceeded that quantity, the difference would have t'o be t',-
kcn from the area contained in the rectangle A D F F • or
m otJier words, the perpendicular F S would re.juire to be
laid ort towanls the West, or towards the left ht. .,!
The course is ascertained by Prob. V. Runnin^^ of Lines;
thus, as E F or 20 : F S or 1 : : 57.3^ ; / E F *< - ^^ r.o/
y.
Since^ie triangle^F S is right angled at F, ET^or
400 -I- F S= or 1 = S E"-or401 ch., the scjuare root of which
isj.0 ch. 3/. = E S. The course of E S therefore is N, 20-'
J~ E., and the distance 20 ch. '31.
PIIOBLEM Iir.
To divide a Trapezoid into two equal parts by a line rim-
nuiir perpc7idicularhj to the base or front.
EXAMPLE.
lu the Trapezoid A B C D, {Fig. 96,) containing 12 ac.
S r. Up., the parallel sides run due East 12 and 20 c// re-
spectively the si,Ie A D runs N. IQo E. 8 ch. 12 /,, and the
-!e B C N. 3^^ 34' W. 10 ch. 37 I. It is required to divide
the Irape/oul into two equal parts by a line running due
•>orth from tlie base or front A B.
From the angle at D let a perpendicular D S fall upon
ho base A B. By Rule I., Right-angled Trironon.etrv, we
;;;;•' that as is l ch. 39 /., and that D S is Hch. in length.
Iheu by Prob. II., Mensuration, we find thn nma of the
tnangie A D S to be 55000 /. Now the area of the whole
i rape.oul is 12 ac. 3 r. ^ p. or 1280000 /. Half the area of
no
LAND SURVEYING.
tho wliole Trapezoid is therefore 640000 /. But we have
already the ,area of the triangle A D S = 55600, which, sub-
tracted from 640000 leaves 584400 /. as the area remaining
to be laid off, which from an inspection of the figure, it in
evident must be laid off in the shape or figure of a rectangle.
In this rectangle we have the area 584400 /., and one of the
sides, D S, 8 ch. Hence,
584400
8.00
7 ch. 30 /. Then from
the point S run off 7 ch. SO I. due East to E,; and from E
run E F due North and the the trapezoid is divided by it
as wan required.
PROBLEM IV.
To divide a prn-altelogram into two equal parts by a line.
* riminng from a given point.
EXAMPLE.
Let A B C D, (J%. 97,) be a parallelogram. It is re-
quired to run a lino from the point F which shall divide the
parallelogram into two equal parts.
Draw the diagonal' D B, and bisect it. Draw a line from
the point F through the point of bisection of the diagonal,
and continue it till it strikes the opposite side D C iu the
point E; the line F E will divide the parallelogram A B C D
into two equal parts, as was required. (Euc. i. 34.)
PROBLEM V.
To divide a Trapezium..
EXAMPLE..
In the trapezium A B C D, (Fig. 98,) the two sides B C
and A D are parallel. It is required to cut off one third ol"
the whole area by a line running from the point A.
Produce the line B C to E, so that C E may be equal to
A D. From J hiy off B G equal one third of B E, and join
A G; the triaTigle A G B is the third part of the trapezium
A BCD. (Euc vi. 1.)
DIVISION OP LAND.
/. But we have
j600, which, sub-
3 area remainiinj
the figure, it is
re of a rectangle.
., and one of the
/. Then from
E,; and tVom E
is divided by it
ur
PROBLEM VI.
To divide a Trapezium into two equal parts, h, « line drawn
from one of its aiigles.
EXAMPLE.
Let A B C D, (Fig-. 99,) be the given trapezium, and A
the angle from which the dividing line is to be drawn. Draw
the diagonals A C and B D. Bisect D B in E. Throufrh
E draw G E F parallel to A C. Join A F and it will di-
vide the trapezium A B C D into two equal parts.
parts by a line
It is re-
shall divide the
gram
'raw a line from
)f the diagonal,
side D C iu the
logram A B C D
le. i. 34.)
; two sides B C
oft' one third of
oint A.
nay be equal to
if IJ E, and join
the trapeziuiiv
PROBLEM VIL
To divide a Triangle into any proposed number of equal
parts, by lines running- from a given angle.
EXAMPLE.
It is required to divide the triangle ABC, (Fig. iOO,) into
three equal parts, by lines running from the ano-le at C
According to Euc. i. 38, triangles upon equal bases,' and
between the same parallels, are equal to each other, there-
fore divide the line opposite the angle at C into three equal
parts, and from the points of division D, E, draw the lines
D C and E C. The areas of the triangles A D C, D E C, and
E B C, will be equal to each other.
PROBLEM Vm.
To divide a Triangle by lines running parallel to a given
side.
EXAMPLES.
1. In the triangle ABC, (Fig. 101,) the side A B mea-
sures 40 cA., the side A C 53 ch., and the side C B 56 c^i
It Ks requu-ed to divide the triangle ABC into two equal
parts, by a hue running parallel to A B
According to Euc. vi. 19, similar triangles are to each
other m the duphcate ratio of their homologous sides: there-
tore as the area of the whole triangle is to the square of its
mle, so ,s the urea to be cut oft' to the square of its side"
122
LAND SURVEYING.
The square root of this sum will he the side of rho trJanHe
to be cut ofl^ thus: the area of the triangle A B C is 106 of
niul consequently th(^-ea to be cut off is 53 ac. Then, us
AJB C or lOG ac. : A C^- or 3364 ch. : : C D E or 53 ai, :
C L)» or 1683 ch.', and V 1682 = 41 = C D. Then, draw
I) E from the point D, parallel to A B, and the trian-le is
divided into two equal parts, as was required.
2. Divide the above triangle into three equal parts, by
lines runninjr parallel to the base A B, (Fig.. 102.)
The whole area being 100 ac, one third of it is 35.3333 -f
nc; therefore as ABC oi^06 ac. : ATC^ ov 3364 ch. : :
C E U or 35.3333 -\- ac. : C E= or 1121.33 A-, and V J121
.33 + = 33 ch. 43 /. = C E.
Again as 106 : 3364 : : C G II or 70.6666 -f- • C~G- or
2242.65; then V 2242.65 = 47 ch. 35 /. = C G.
The:i through the points E and G draw ED and GH
parallel to A B, and the triangle is divided into three equal
jiarts, as was required.
^ By the same principle any specified amount of area may
oo laid off in a trapezium by a line running i)arallel to one
oi Its sides, for by producing some two of its sides until they
meet, a triungle Mill be formed.
EXAMPLE.
In the trapezium A B C D, (Fig. 103,) the side A D runs
S. 17- E. Ibch., A B runs E. ^20 ch., and B C runs 8. 20^
^'V . 1 7 ch. It is required to lay off 6 ac. towards the North
by a line running parallel to A B.
Produce the sides A D and B C until they meet in F
i'hcn ISO^ - (Z E A B -]- Z E B A) := 37^ = / A EB.
Bv Obli^ur Angi>ed Trigonometry.
To find the side \ E.
As Sine Z A E D 37°
Is to A B 20
Ho is Sine /ABE 70^
To A E 31.23
9.77046
1.30103
9.97299
1.4945G
DIVISION OF I. VND.
m3
To find the side B E.
As Sine Z A E B 37^
Is to A B 20
80 is Sine Z B A E 73°
9.77046
1.J0103
9.980G0
To BE 31.78 1.50217
Next, by Mensuration, Prob. II. Rule 2, the area of the
triangle A B E is found to be 29.86331 ac, from which take
the area to be cut off 6 ac, and the remainder is 23.8633 lac.
Then as 29.86334 : B E^ or 1009.9684 : : 23.88334 : ET-^
or 807.0507, and V 807.0507 = 23 ch. 40 I. Lay off B F =--
28 ch. 40 I, and from F run F G due West, and A B F (i
will contain 6 ac.
By the same rule we find that the triangle E C D contains
7 ac. 2 r. and 35 jo., and that the trapezium G F C D con-
tains 16 ac. and 23 p.
PROBLEM IX.
To divide land hy Calculation.
EXAMPLE.
It is required to divide the farm A B C D E F A, {Fig.
104,) into two equal parts by a line running from the point A,
Calculate first the area of the whole farm. Then draw a
line, or suppose a line to be drawn from the given point to
some other known station as at D, which will divide the
farm in the required proportion, as nearly as you can judge.
Then fill up the columns of a Calculation Table with the
courses and distances from A round to D. The difference be-
tween the sum of the North and of the South columns will
show the Difterence of Latitude, and the diflerencc between
the sum of the East and of the West columns will show the
Departure of D A. From these data find the area of the
part cut off A B C D A. Find the difference between this
area and the area of the half of the whole faru). That dif-
ftn-cncc will be a triangle. If the area of the piece cut off
exceed the half area of the whole figure, that triangle lies
within the figure cut off, but if it be less than half the area
^-24
LAND SURVEYING.
<)(■ the whole figure, the triangle lies on the opposite side of
the lin(^ A D. Suppose that tho quantity cut off exceed the
half of the whole area. Find the area of the triangle ADC.
Then as the area of the triangle A D C is to the square of
the side D C so is the area of the triangle A D H to the
square of the side D H. Then having the course and dis-
iuv ca of A D and D H, by a Traverse Table find the course
anl iistance of A H, the true dividing line. Set your com-
pass at A, and run the course and distance A H, and the
farm is divided as required.
Suppose the farm to contain 16 ac. 1 r. 21 p., and the field
notes of the survey to be as follows:
Commenced at station A, and run thence S. 80° E. 7 ch.
50 Z., thence S. 15° E. 7 ch. 90 /., thence S. 20° W. 10 ch.,
thence N. 75° W. 8 ch. 45 L, thence N. 8 ch., thence N. 15"
E. 8.50 /., to the place of beginning. Let the plan be laid
by a scale of 10 ch. to an inch.
DIVISION OP LAND,
125
pposite side of
off exceed the
riangle ADC.
I the square of
A D H to the
jurse and dis-
find the course
Set your com-
A H, and the
.J and the field
a s«
H <
S U
o a:
CC <J
s -;
H -!
Bh U
O Bj
iz; <
H
•r.
Q
•
O
•
u
<
s
Q
o
<
&■
—
b
o
»
HJ
■^
K
-3
25
O
Cfi
>5
h
o
6
1?;
en
o
4- -f
o
o
O
CD
!^»
o
GO
O
50
G^
CO
•^
GO
•
1— «
h-
Ci
CD
CO
eo
GO
GO
•
CO
CD
_,„.
T-t
i-H
o
o
Ci
o
o
o»
•
1^
o
05
»-i
•
o
^
1-^
I— (
ira
&
G«
CD
m
c«
»-<
GO •=?
CO 05
co'^
00
o
12;
o
Q
o
^
rr»o
C^l
G« '-'
(?»
OS
OS
CO
X>
>o
•^
G-«
SI
Oj
OS
1-H
1— (
T-^
X)
GO
CO
03
UO
c<
u
Q
•«1
triangle
E.
E.
ls^=
1-4
{j^ o o
O O CO
i]; <s< rt
O
CO
CO
<
a
o
03
o
<1
^
CO
o
03
o
fcO
s
CI
ci
3
o
in
>-i
u
>■
ii
i3
CO
Q.S
a
— <
e
S3
s
)i
S 2
^ 2
il § ^
' O ti
*>» M .
_, •-< >»
•• « 2
"^ tr. ^
CO aJ a>
— ;-* "
' Ph »
^ a>
« o
(=> c ca
".23 -w
'^ "^ TJ
c «
o
o
o
CO
ci
in
m
MM
3
o
C -J .23
i'-2G
T.AND StIRVKVING.
Cnfr.ula/ionor of Conslniction. "><'"ion of
liy Calcutation wo can ascortain flio true course- -in,! rJi«
PROBLEM X.
To divide a Triangular Lot of Land in certain proportions.
EXAMPLE.
Being employed to divide a ten acre lot of Marsh between
h CO clannants,-A, B and C; A clain.in, 4 acres, and B
and C 3 acres each. The lot is triangular. The base A B
( IS- 105,) measuring 20 ,/,., ,„j ^ perpendicular let fall
.hereon fr-om the opposite angle measuring 8 ch. 50 /. The
lot, therefore, it is evident, >vill not hold out its .neasure-
ment. Now snj)posing .;.. division lines to run from the
angle opposite to the base, how much land, and what nro^
portion of the base should each claimant receive
C D X A B = 20.00 X 8.50 = 170.0000 ^ 2 =, g^.oOOG
= 8 ac. 2 r. = area of the lot.
AslO:8^:.4:3ac. lr.24p. = A'sshare.
:3:2«c. 2n8;,. = B'sorC'sshare.
20 ch. : : 3 ac. I r. 24;,. ; 8 ch. = A's share
As 10 : 8^ :
As 8.5 ac. :
of base.
As 8.5 ac. :
share of base.
20 ch.
2 «c. 2 r. 8 ^. : 6 ch. = B's and C's
PROBLEM XL
To divide by Calculation a lot of land of a certain amount
of value <^nong different claimants, in proportion to the
amount of their claims and the estimated value of the land.
EXAMPLE.
A Testator leaves by ys\\\ a lot of land containing 500 ac.,
the value of which he estimates at £1470, to be divided
f»»,u
DIVISION OP LAND.
V27
fiinong h.s servants A B C D E F in the following propor-
uom, ucnonling to the value of the hind, viz: to A he bo-
T'lnn''' V'''"" "^""'^^ •^^^' t«B £20, to C £10, to D
^100, to E £400, and to F £1,000. Now the vah.e of the
land most convenient for A, B, and C, is estimated at 7s
per acre, vs le the land most convenient for D is worth 10s
per aero, for E 15.. per acre, and for F 12.. per acre [t is
roqu.re.l to determine the quantity of land which falls ta
the share of each.
nULE.
Divide the sum bequeathed to each by the value of the
and per acre which is to Le allotted to him, add the quo-
■ont, together, d.vule the whole given quantity of lani by
.ha sum; th.s quotient will be a common .nultiplier, by
« ch multiply each particular quotient and the jn-oduct
h/d viz, ' ''T"'"''r' "'""^^ '^''^ '^ '"^^^ .shareof each
ndmdual, or: Say as the sum of all the quotients is to the
Thus: A 7) 40( 5.714281
B 7) 20( 2.85714
7) 10( 1.42857 [
. 20( 2.85
f' 7) 10( 1.421
D JO) 100(10.
E 15) 400(26. GC666
* 12)1000(83.33333
Quotients.
Now,
Then,
Sum of Quotients, 129.99998 or ISO
500 -. 130 = 3.846153, the common multiplier.
5.71428X3.846153= 219778 a/. - \». i
2.85714 X 3.846153 = lo 9918 ac ~ R L'"''''
1.42857 X 3.846153 = .5 4941 ac - T' 'h""'^'
X 3.846153= 38:' 1 ^ : - g;; h'a::,^-
10.
26.66666 X 3.846153 = 10^ 5641 ar - F.'"r'""
83 S'n*?'^ V Q QAi-ir.o MZ ^^' J^^^s share*
^''^.dJJj., X 3.846153 = 320.5127 ac. = F's share.
Sum of the whole shares, = 500~o"o20 «c.
^in the same princinln if pny '^incrl- -hnv-. \ j .
land of different vah.^^. " ^^ ^ '^'""'^^ '''''^''
Again if there be different quantities of land as well as
<I'ffcrent values, find what each quantity is worth at it, "all"
LAND SURVEYING,
tion, and adtl their suhis toj^other: then, ns the sum of the
quantities is to this sum, so is one acre to its mean vahie.
LOCATION OF LANDS.
This section will treat of the method of laying oft' any
given quantity of land, in any specified form, from the len^^t
possible data.
As the quantity of land is generally given in acres, roods,
and perches, it is necessary to reduce them to square links,
which may be performed by the following
rules:
1. To the acres annex five cyphers on the right hand, and
the whole will be links.
2. Having annexed five cyphers to the right of the num-
ber of roods, divido the sum by 4, the quotient will be the
links.
3. To the right hand of the porches annex four cyphers,
and divide by 16, the quotient will be the links.
4. Add these sums together, and you have the square links
contained in the givei/quantity.
PROBLEM L
To lay out a given quantity of land, in the form of a Square.
RULE.
Extract the square root of the area, and you have the side
of the square required.
EXAMPLE.
It is required to lay out 200 acres of land in the form of
a squ" -1 the East side of a road running N. 10^ E.,
(FiV lo ) What is the length of the side of tho square?
Lay down also a plan from a scale of 25 chains to an inch.
200 ac. =_. 2000000 I., the square root of which is 4472 + /.
2= 44 ch. 72 I. = the side of the square; and since the course
of A C is N. lO*^ E„ the course of A B must be vS. 80^ E.
LOCATION OP LAWBS.
J2f»
ight hand, and
have the side
df'ROBLET^' If.
To lay out land in the form of jctangle, the length of one
side being given.
Divide the area by the given side, and the quotient ^viil
express the length of the other side.
EXAMPLE.
Being employed to lay out 80 ac. 2 r. 20 p. on the West
side of, and fronting on, a road running N. 3° i-. (Fi<r \m\
the lot to measure 12 chains in front? along Jaii rtl; ';'.
qmrcd the course and distances, with a plan of the same
80 ac 2 r. OQp. =8062500 /. -j- 12 cA. or 1200 /. = 6719 /
or 67 ch. 19 /. = length of side required.
The course of A B is S. 87^ E., and distance 67 ch. 19 /.,
and B C runs N. 3° E. 12 ch.
PROBLEM III.
To lay out land in the form of a Rhombus, one of the angles
being given.
RTILE.
Divide the area by the Nat. Sine of the given angle, and
the square root of the quotient will be the side required.
EXAMPLE.
Being employed to lay out 100«c. of land, in the fonn of
{ttg 108,) the course of the line from said road to be S. 80^
lot laid down by a scale o? 25 chain, to an inch.
Phe Nat. Sine of 80=, when radius is 1, is .98481, and
100«. ,0000000 /. Then 10000000-^ .98481 = lOlSl'"
PROBLEM IV.
To lay off land in the form of a Rhomboid, a side and an
angle being given.
RULE.
Divide the area by the product of the given side multi-
180
LAND SURVEVING,
plio.l into the Nat. Sine of the given an^Ie, the quotient will
be tlu, Dtiicr side.
EXAMPLE.
It is rcquirod to lay ofT 75«c. 23 p. in the form of :i rhom-
lK)i(l, on the Ncjith .side of, and fr tini,' on, n river, (Fifr.
109,) wiiich run.s duo East, the front to measure 15 chains
alonj,' said river, and the line from the river to tlie rear to
run N. l(P E. What mu.st be the length of the side line.'
Draw also a plan of the lot.
The Nat. Sine, as in the preeeding Example, is .98181 X
1500 /., the width of the front = 1477.21500. Then 75 ac.
Or. 23p. = 75M375-f- 1177.21500 = 5086 /. = 50 cA. 86/.,
the length of the .side lino A D.
I PUODLEM V.
To lay ojf land in a rectangular form, so that tJic length may
be a given multiple of the breadth,
RULE.
Divide the area by the given multiple and the .square
root of the quotient will be the width, and the width multi-
plied by the given multiple will be the length.
EXAMPLE.
«cing employed to lay off 78 ac. 2 r. 36 jt;. on the East
side of a line running N. 4° E. {Fig. 110,) in the form of
a reetangle, who.?o length .shall be three times that of its
breadth; required the courses, distances, and a plan of tho
lot.
78 ac. 2 r. 36 p. = 7872500 ?. -^ 3 = 2624166, the .square
root of which is 1620* very nearly. The breadth therefore
113 cK. 20 /., and the length == 16.20 X 3 = 48.60 or 48 ck
,60 /.
A D run.s N. 4^ E. 16 ch. 20 /.
'D C runs S. 86^ E. 48 ch. 60 I.
* By assuming 1G20 /. as tho vvicltii instead of 1619.9 -f- the
true widtii, tho above lot coiil;
una ; lOut 4 r. more liian tlic iriv
(Ml
quantity. Unless where land is exceedingly valuable, ji surveyor
would probably take 1C20 /. as the width, and lay off accordingly.
LOCATION OF LANDS.
191
ic quotient will
I'HOHLF.M VI.
To lay rut l„nd in a rectany:ular form, so that tl . lens-tk
may he Ic the hreaiUh in a certain proportion.
RULE.
Mujfij.ly tl.o area hy the loss and .livido i),„ ,,rn,I„rt I.y
thn grouter i.uml.or of tho proportion, mid tlu. sq.mre ro«'t
ot tl.o cp.otiont will 1,0 tho wi.lth: Ami tl.o >xi,ltl. .nultipliod
hy tho .qroator nn.l divi.lod l>y ;'.o loss nu„.!,or of the pro-
j)ortiou will be tl.o leiigih.
R^ 'I.E.
If it H ro(pnred to lay ou m ar. 17 p. in a rortnnjrnlnr
form, (Fig. Ill,) so that the l.roa.lth may l.o to tho lonuth
as 5 IS to8, uhat must tho ionjrth and tho hroadth l.rn-
speot.vcly? Draw u map of the lot hy a scale of .J5 ch. to
an inch.
I09«c.l7;>. = 109l0625/. X5=.54553l25^8=:f,H10Ma
the square root of uhioh is iitill = breadth of tho farm'
Ihon 2G11 X 8= 20888 -^ 5=.4179. Tho lon^th A |{
t .-^relore is 41 ch. 79 /., and the breadth A D 20 ch. 1 1 /.
PROBLEM VII.
To lay out land in the form of a Rectangle, so that the Un'-th
may exceed tkc breadth, by a certain given quantity "'
RULE.
Add the squai-e of one fourth of the given difloronoo to
the area, . .d f,-om tho square root of the sum .subtract half
theg.ven differonee for the less side. To the remainder
add tho nhole difference for tho greater side.
EXAMPLE.
It is required to layout 200 a., "n a rectangular form
{tig 112,) so that the length may exceed the breadth by
10 ch.
Y- = 250000 + 2000000 area, = 20250000, the
1000
.'•quart:
^■oot of >yhich i. 4500 - -^ or half the giyen difference =,
rS'i
LAND SURVEyiNG.
4000 /, = tho leas side, and 4000 -f- 1000 the whole difter-
ence = 5000 /., the greater side. The side A D therefore
must be 40 ch., and the side A B 50 ch.
PROBLEM VIII.
To lay out land in the form of a Rhomboid^ so that the
length may be a given multiple of the breadth.
RULE.
Divide the area by the product of the given multiple and
the natural sine of the given angle, and the quotient will he
tho breadth; and the breadth multiplied by the given multi-
jde '^'ill be ihe length.
EXAMPLE.
Required the sides of a Rhomboid containing 10 ac, {Fig.
113,) Avhose acute angle is 80°, and whose length is three
times greater than the breadth.
Nat. Sine 80^ to Rad. 1 is .98481 X 3 = 2.95443, and
10 ac. = 10.00000 /. -f- 2.95443 = 338474, the .square root
of which is 32 /. or 5 ch. 82 I. = width, and 5 ch. 82 /. X
3= 17 cA. ij /. = Icuffth.
PROBLEM IX.
To lay out land in the form of a Rhomboid, so that the length
may be to the breadth in any given proportion.
RULE.
Multiply the area by the less number in the given pro-
portion, nnd divide the product by the product of the Nat.
Sine of the given angle multiplied by the greater number of
the proportion: the sipiare root of the quotient will be the
•breadth; and the breadth multiplied by the greater and di-
vided by the less number in the given proportion gives the
length.
EXAMPLE.
l3(Mn2f employed to lay out 10 ac. in tho form oS a Rhoni-
btiid whose length shall be to its breadth as 2 to 5, and whose
DIVISION OF LAND,
ISS
\g 10 ac, (Fig.
\ of a Rhnm-
included angle shall be 80°; required the length and breadth.
(See the preceding Figure.)
Nat. Sine 80« is .98481 X 5 = 4.92405, and War =.
1000000 /. X ^ - 2000000 ~ 4.92405 = 4061(59, the square
root of which is C3S /. or G ch. 33 I., = the hrea.lth. and
038 X 5
—^-=1595 /. or 15 ch. 95 /., = the length.
PROBLEM X.
To lay out land in the form of a Trapezoid* Mho.e cenlral
length shall be any given multiple of the xmdlh.
RULE.
Divide the area by the given multiple, and the squnre
root ot the quotient will be the width; which multiplied by
the given multiple will give the central len-th
EXAMPLE.
It is desired to lay off 200 ac. on the East side of a roa<l
running N. 30^ W., (Fig. lU,) in the form of a trapezoid
^vho.se parallel sides shall run due East, and who.^e central
length sh.ll be double its breadth; required the courses and
distances, and a plan of the lot.
VlOff^^oriooOOOOOTr^ = 31G2 /. or 31 ch QP I =
the breadth, and 31 ch. 62 /. X 2 = 63 ch. 12 /. = cen'tr-il
length.
Then to find the length of the parallel sides, sinr-e in the
triangle A D N right angled at N, the side N D = B C is
known to be 31 ch. 62 L, the angle at D is al.o Known to
be 30^ and consequently the angle at A must be G0°; ..y
right angled Trigonometry, say ap Sine / A : D N : ;
Sine Z D : A N, from which we find A N to be 18 ch. 26 A
Again, since the central length E F = —±5.^ ^^„j ^^ j^^
7 ^^.~~ ^ *^'' '^ ^" ^'^" '-^'"t'vl length E F you add half
the ddlerence or half A iV, the sum will be the longest side
A B; and if from the cetUral leno-tl, R F y..)u .subtract half
* A Trapezoid is a rectilineal qiu.diilatcral li.r„re. oiilv ivva nf
whose opposuo sides are parallel. ° ' ^ ^""^ "*
M
"*«T« f>..Jt0::
tm
LAND SURVEYINO
the differenco or half A N, the remainder will be the length
of the shortest side DC.
6324 = 5411 or 54 ch. IW. = D C, the shortest side.
Wherefore commencing at A, A B runs East 72 cA. 37 l.
i^i"? "^^' ^^ '^^•' ^2 ^•' ^ ^ '•""^ W«st 54 ch. Ill
and D A runs N. SO'^ W., and the distance is ascertained by'
1 ngonometry to be 3(> ch. 51/.
PROBLEM xr.
To lay out land in the form of a Trapezium, having one
of its sides given.
, RULE.
Divide the given area into two parts, either equal or un-
equal, and then find the perpendicular that will lay out one
ot these parts in a right-angled triangle upon the given side
as a base. 1 his perpendicular will be a diagonal of the tra^
pczium, and a ba.e upon which theren,aining triangle must
be constructed. Then find the perpendicular, which, fall-
mg upon the-opposito side of this base, will lay out the other
part.
» ,f^;.5:^---'^''^f^, perpendiculars are found by dividing dou-
Ue the area ot the triangle by the given or known shfes
EXAMPLE.,
sure 8 c?' '^^' """" °^ ''^''''*' '''^'' ^ ^ '^^" '"«^-
Let the area be supposed to be divided into triangle., one
of which contams 5 ac. and the other 3.
5 ac. or 500000 I. X ^
S'ch'.'i^i^QCO ir'~ ^ ^-^^ '• = perpendicular. Then
from the poi.u U in tho given side B C, and perpendicular
to tt draw hA = laso L, and join A C; the triangle A B C
C0nt2,Hiri 5 uc.
Next --"''- °'' ^^^^^^' XJi
* I5r,0 = '^^O'- == perpendicular of
LOCATION OF LANDS.
ISS
remaining triangle. From A, and perpendicular to A'B,
draw A D 4S0 /. ; join D B, and tiic triangle A D B will
contain 3 ac.
The trapezium A C B D will also contain 8 ac, and the
side B C is 10 ch., as was required.
Pi..
>BLEM XII.
To lay out land in the form of a Triangle, of which one side
and the an^le at one of its erzlremities are given.
RULE,
Divide double the area by the product of the given side
multiplied into the Nat. Sine of the given angle, and tho
quotient will be the other side, including the given angle.
EXAMPLE.
From the Northern extrcTiiity of the line N C, (Fig-. 116,)
which runs due North 25 ch., it is required to run another
line C O, S. 34° 41' E., so that the triangle N O C may
contain 80 ac.
Nat. Sine of 34° 41' is .56C04 X 2500 or N C = 1422,
a«d 80 oc. X 2 = leOOOQOO I. -M422 = 112457. or 112 ch.
45 /., = side C Q.
PROBLEM XIII.
To lay off any quantity of land in a triangular form, be-
tween two lines forming an angle, one of the sides of the
triangle being given.
RULE.
Divide double the area to be laid off by the length of the
given side, and the quotient will be the length of a perpen-
dicular let f\ill from the opposite angle upon some part of
the base or line given. From the extremity of the given
line raise a perpendicular of the ascertained length. From
the end of that perpendicular run another line^ parallel to
the given line until it intersects the other line. Then a
line drawn from the point of intersection to tlie point frpm
136
I-\ND Sl'UVEYING.
which the porpendicular Avas raised will complete the tri-
angle containing the required number of acres.
EXAMPLE.
hi the corner or angle formed by the road A B (Fig. 117,)
and A C, I am required to lay off 4 ac. fronting on the road
A n 10 cL; rciiiiircd the termination of the lino A C.
Double of the area = 400000 X 2 = 800000 -f- 1000 /.,
(the length of A 13) = 800 /. or 8 ch. = length of perpen-
dicular. Then from B and perpendicular to the line A B
run a lino B S, 8 ch. in length. From the end of this per-
pendicular or from the point S run a line S C parallel to
B A, until it strikes or intersects the side line A C aforesaid
in C. Join C B. The triangle ABC contains 4 ac, as
was required.
N. B.— In' this case the quantity of the included ande
does not aitect the accuracy of the rule. It may be an an-
gle ol 80^ as C A B, or of 75° as C A B, or of 50^= as C" A B,
PROBLEM XIV.
To lay off land in the form of an Isosceles Triangle, the an-
gle contained between the equal sides and the area being
given.
RULE.
Divide doulle the area by the Nat. Sine of the given
angle, and the square root of the quotient will be the length
of one of the equal sides.
EXAMPLE.
It is rrq Mired to lay out 38 ac. 2 r. 18 p. in the form of an
Isosceles Triangle A B C, {Fig. 118,) the course of A B is
S. 25° W., and the coiuve of A C is S. 26° E.; What must
be the length of thf! sides end the course of B C.
38 ac. 2 r. IS p. = 33(11250/. X 2 = 772250, the double
nrea, ~ .77715, the Nat. Sine of / A, (25° + 20° = 51°)
= 9936940, the square root of which is 3151 = A C or A B.
Then 180^ - / A or 51- = (/ B -|- / O^or 129°. Now
since the / B and the / C are equal each of them is 64'^
SC. Whercforo, by Trigonometry:
LOCATION OF LA5DS,
137
plete the tri-
H (Fig. 117,)
J on the road
A C.
-f- 1000 /.,
h of perpcn-
he line A B
1 of this per-
^ parallel to
C aforesaid
ins 4 ac, as
;luded angle
ly be an an-
»°asC"AB,
ngle, the an-
'■ area being
f the given
e the length
! form of an
le of A B is
What must
/.
the double
2G° = 51°)
\CorAB.
29°. Now
them is 64''
As Sine of / C 64° 30' : B A 3151 : : Sine / A 51^ •
B C 2713.
The courhe of A B is S. 25° W., and the / B is IJI'- 30';
the course of B C is N. 89*-^ 30' E.
Hence the sides A B and AC are each 31 ch. 51 /., and
the side B C runs N. 89'^ SO' E. 27 ch. 13 /.
PROBLEM XV.
To locate land in the form of a Circle.
E,ULE.
Divide the area by .7854, and the square root of the quo-
tient will be the diameter.
EXAMPLE.
Required the diameter of a circle containing one acre.
{Fig. 119.)
V 1 oc. = 100000 /. -h 7854 = V 127323 . 65 =r 356 . 8 /.
or 3 c/^. iJG.S /,
PROBLEM XVI.
To lay out land m the form of an Ellipse.
CASE I.
When the Transverse Diameter exceeds the Conjugate by a
given quantity.
RULE.
Divide the area by .7854, to the quotient add the square
of half the difference between the diameters, from the square?
root of the sum subtract one half of the difference between
the diameters, and the remainder will give the Conjugate.
The difference added to the Conjugate will give the Tnuis-
verse.
EXAMPLE.
^ Required the Transverse and Conjugate diameters of an
^"llliprfo containing one acre, whose Transverse diameter
shall exceed thu Conjuo-ate by one chain. (Fig. 120.)
1'
138
LAffD SURVEYIlfO,
1 ac, = lOOOOO /. 4- . 7854 = 127323 -j- V — or 2500 «
2
129823, the square root of which is SCO — ~ or &0 ^
310 = Conjugate, and 310+ 100 = 410= Transverse.
The Transverse diameter therefore is 4 cA. 10 /., and the
Conjugate 3 ch. 10 /.
CASE II.
When the Transverse mid Conjugate Diameters are to each
other in a certain ratio.
RULE.
Multiply the area by the greater number in the propor^
tion, and divide that product by the product of the less num-
ber multiplied into .7854, and the square root of the quo-
tient will bo the Transverse diameter; then multiply the
Transverse by the less number in the proportion and di-
vide by the greater, and it will give the Conjugate. {See
the preceding Figure.)
EXAMPLE.
It is required to lay out one acre in the form of an Ellipse
whose Transverse diameter shall be to its Conjugate in the
ratio of 5 to 3,
100000 X
77854 X 3
» 276 = the Conjugate*
4C0 = the Transverse, and l^l^L?
APPENDIX.
It is presumed that in the preceding treatise nothing oe^
curs requiring a formal demonstration until the etudeoi ar.
rives at
RECTANGULAR TRIGONOMETRY.
THEOREM.
The sine versed-sine, tangent, and secant, of an arc
which IS the measure of any given angle, is to the «ine,
versed-sme, tangent, and secant, of any other arc which is
the measure of the same angle, as the radius of the fo-.st arc
18 to the radius of the second.
Let B D (Fiff. 121,) a„,, p H j^^. ^^^^ ^^^^ ^^.^^.^j^ ^^^_
sure the same an.^le BAG; and let A B be the radius of
the arc B D, and A If tlic radiu.. of the arc F H. Let D C
be the Sine, B C the tangent, and A C the secant, of the arc
l^aL'n/Tu ""V^ "'"^' " ^^ '''' ^''^"^^"^' -'* ^ F. the
parallel, accordmg to Euc. vi. 4, tang. B C : tang, il E
a'd V n''- ^ '^' ^"'^ ^^"" ^ ^ = «'"^ J^^ I ^^ rad.
; r A « „ ' '"'^ '^' '^'-''^- ^^ ^ •' '''' ^^ E : : rad. A B ,
Tdii. A H. Hence the trutli of the 'J^heorcm is obvious
Froni this Theorem if is evident, that, as the Trigonome^
trical lables exhibit in numbers, the sines, tangents, bo-
eants, &c., of certain angles to a given radius, they exhibit
THEOREMS.
i'.O
also the ratio of the sines, tangents, eecantfi, Stc, of the
same angles to any radius whatever.
Upon this principle the solutions of the difierent cases of
right-angled plane triangles depend; and from this Theoroin
the Rules for Ilcctangular Trigonometry are deduced.
HI
OBLIQUE-ANGLED TRIGONOMETRY.
THEOREM I.
The sides of a plane triangle are to each other as the sines
of the angles opposite to them.
Let ABC (Fig. 123,) be a triangle, and C D a perpen-
dicular let fall from the vertical angle at C upon the oppo-
site side A B; because the A C A D is right-angled at D,
C A : C D = R : Sine / A. For the same reason C B :
C D = R : Sine /.' B; and inversely, C D : C B = Sine B
: R; therefore, by indirect equality, C A : C B = Sine /
B : Sine / A. In the same v/ay it may be demonstrated
that C A : A B = Sine / B : Sine ^ C.
THEOREM II. .
If to half the sum of two quantities be added half their dif-
ference, that sum will be the greater quantity; and if from
half the sum of two quantities be subtracted half the differ-
C71CC, the remainder will be the less quantity.
Let the two quantities be represented by A E and E B,
( Fig. \2?,,) A E being the greater, and E B the less. Then
it is ovidint that A D is the sum, and C E the diftercnce of
the two quantities, and A D or D B their half sum, and
C D or D E their half difference. Now if to A D we add
D E Ave have A E, the greater quantity; and if from D B
we take D ]•: we have E B, the less quantity.
THEOREM in.
The sum of the two sides of a triangle is to their difference
as the langmt of half the sum of the angles at the base is
to the tangent of half the difference.
Let A B C {Fig. 121,) be any triangle. From A as a
140
Sec, of the
int cases of
•s Thporoiii
lui'ed.
'RV.
s the sines
> a perpen-
I the oppo-
jlcd at D,
ason C B :
i = Sine B
= Sine /
rnonstrated
' iheir dif-
nd if from
^ the differ-
\ and E B,
ss. Then
ftercncc of
sum, am!
D we add
from D B
difference
the base in
m A as a
141
APPENniX.
centre wilh the radius A 15 describe the semicircle D B E
Produce C A to D. Join D B and 13 E, and draAv E F pa-
rallel to B C. Then because the anj?le D A B is the exte-
rior angle of the triangle A B C, it is equal to the 6um of
the two interior and opposite angles ABC and A C B. But
the angle D E B is equal to half the angle D A B; therefore
the angle D E B is equal to half the sum of t'ho an-lcs
A B C and A C B. Now since A B is equal to A E, the
angle A B E is equal to the angle A E B. But the angle
A E B IS equal to the two angles E B C and B C E- where-
fore, also, the angle A B E is equal to the sum of the an-les
E B C and B C E. To each of these add the angle E B C-
then the whole angle A B C is equal to twice the angle
E B C together with the angle B C E. Whence it is plain
that the angle E B C or the alternate angle B E F is equal
to half the difFe.ence of the angles ABC and B C A Now
the angle D BE is a right angle. (Euc. iii. SI.) Therefore
to the same radius E B, D B will be the tangent of the an-
gle D E B and F B will be the tangent of B E F; so that
B D : B 1. : : tan. DEB:: tan. B E F : : tan. ^ (A B C
+ A C B) : tan. ^A B C - A C E). Also, A D and A E
are each equal to A B, it is evident that D C is the sum of
the sides A B and A C, and that C E is their differonc.
But because E F is parallel to B C, D C : C E : : D B •
B F; that is, the sum of the two sides of the trianHo ABC
IS to their difference, as the tangent of half the sum of tho
angles opposite to these sides is to the tangent of half their
uiflerence.
THEOREM IV.
Ina7iy right lined plane trianp;le the base h to the sum of
the other sides as the difference of these sides is to the
the difference of the segments of the base, made by a per^
pendieular let fall upon it from the angle opposite to it
In the ol)lique-anfflcd
triangle A B D, {Fig. 125,) pro-
duce B D until B G is equal to A B, the si
B as a centre, with the distance B G or B H describe
lortest side. On
a cir-
If'
149
MCN6CRATI0N OP StJPEnFICIEB.
cle A H G, cutting B D and A D in the points H and F..
Then D G is evidently equal to the sum of the sidca D B
and B A, and li D their difference. And since A E is equal
to E F, D F is the difference between D E and E A, the
segments into which the base is divided by the perpendicu-
lar let fall upon it from the opposite angle. Now (by Euc.
•iii. SC,) the rectangle contained by D G and D H is equal
to the rectangle contained by D A and D F; therefore, A D
.: G D : : H D : F D, i. c. the base is to the sum of the
other sides as the difference of these sides is to the differ-
ence of the segments of the base.
nOLE IT.
This is me'-'^ly another application of the same principle.
ROLE. III.
Let a = D A, 6 = D B, c r= A B, and a = D E, the great-
er segment; then a — a; = E A, the less segment.
Then, a : b ~\- c : : b — c:2x — a
and 2 ax — a^ = 6= — c'
Hence, 2 ax — o'-f6' — c'
X = a^-\-b'^~ c*
and
Hence the Rule.
1 -" a
1
Let i
perpen
1
C B E,
the lust
B : :C
2a
MENSURATION OF SUPERFICIES.
Ill
PROBLEM I.
nULE I.
The measuring unit of a superficies may be one inch, one
foot, one yard, one chain, or any determinate figure and mag-
nitude. Let A B C D {^Fig. 126,) be a rectangle, and M
the unit of measure. When M is contained a certain num-
l>er of times in A B and B C, it is only necessary to multi-
ply together the figures which exjjress the number of times
the linear unit M is contained in A B and B C. The pa-
rallelogram A B E F is equal to the rectangle ABC D-
<Euc. i. 86), Hence the reason of the rule is obvious.
AX
'■^.
■A
APPENDIX.
143
H and F.
idc8 D B
^ is equal
E A, the
•pcndicu-
(by Euc.
[ is equal
ore, A D
m of the
10 differ-
nULK II.
Let A B C D {Fig. 12G,) be a parallelogram, and C K, its
perpendicular altitude. Then in the right-angled triangle
C B E, Had. : Sine B : : C B : C E; then by nmltiplyi"ng
the last two terms of this proportion bv B A, as Rad • Sine
B : : C B X B A : C E X B A: but A B X C E =. area of
the parallelogram, and hence the rule.
nULE III.
The demonstration of this rule is ovidentlv comprised in
the preceding.
irinciplc.
he great-
PROBLEM II.
nui,K I.
The trutlrof this rule will be evident by comparing Fue
1. 41, with Rule I. Prob. II. ^ ^
RULE II.
This rule follows evidently and directly from Rule, II.
Prob
inch, on©
and mag-
and M
ain num-
tO JTJUlti-
of times
The pa-
. B C D.
ous.
PROBLEM IIL
Let the sides opposite to the angles A, B, and C, (Fift
122,) be represented by a, b, and ., respectively. ^'
Then6= = a= + c=~2cXDB,andDB = "!±£!z:*'.
2 c
Hence D C = V a'— -^liZzEi! '_
V
d -\- b -\- c
o
X V
a -f- b~\-
— «X V
^
a ~\-b -\~e
Ax V
a
-^ c. LetS^thesumofthesidesofiht
MKNSlfa.VTIOM OP SUPRRFICES.
144
y^j ^ii?_>li?-*?-= area of the trhuiglc. Thc-eforo tho
inula expressed in words, is tho rule.
his for-
PROIJLEM IV.
The area of the triangle A D D, = -^-— ^(Fj-. 128,)
DC DC
and the area of tho A B D C = -^ - X B F --p^X DK.
Then A B X D C X -;r^ = the sum, from which fornmla
tho rule is sufficiently obvious.
PROBLEM V.
In the Pentagon A B C D E, (Fig. 129,) let a perpcn-
dicular full from the centre II. Then A B X -^= area of
A A B R. Now tho area of the polygon is plainly equal to
the areas of as many As, each equal to the triangle A B R,
as the polygon contain.i sides. Hence the reason of the
Rule is manifest.
PROBLEMS VI & VH.
These two problems are simple applications of the rule
for determining the area of triangles.
The area A i —
m tXn s
PROBLEM VIII.
A DXmt
\-Ani, (Fig. 130;) the area
A D-f
A D -
Hence
Thes
land St
vestiga
This
ascertai
ms =
2
\- m n> &c. The area of the whole figure
If a SI
Northin;
sum of t
Let a I
first Stat
line. T
fi c, and
TJipn
■exactly f
sum of tl
Also, tl
n g -f k j
Ite added
to the lut
mm of tl:
In any
sides [yrv^j^
sum of tl]
they stand
APPENDIX.
^Jl± i!*J_fni -r ° >• -h £ g -;- B C
A D -;- B C
]45
6 X A B. Now ,
.i is an arithmetical mean between the two ends.
Hence the Rule.
PROBLEMS IX-XllI.
These Problems are of so little importunco in practical
land Hurvey.ng that I think it unnecessary to eivi the in-
vestJL'ations. •' ^ '^ '"
PROBLEM XIV.
This Problem is merely the application of the rule for
ascertainmg the area of u triangle.
PROBLEM XV.
THKOREM I.
If a survey has been accurately made the sum of tho
^orthmgs u.ll e.,ual the sum of the Southings, and the
sum 01 the Eastings will equal the sun, of the wLstin^s
Let a b c de, (Fig. m,) represent a f .Id. Let a be tho
^t stat.or, 6 the second, &c., and let N S be a meridian
hue. 1 hen a n b h, and c p, will be n.^ridians, and n i,
h ., and p d, Will be departure, or East and We.t lines
Ihon It IS evident that the Northings (//J- ,. ,vill* be
•exactly equal to the SotUhings, « n -^ /. A '- c ;r T th.
sum of the Northings is equa- .o the sum of 'the'souU;;,^;:;
Also the Departures, c h -' a o- = 6 w J- » ,/ • / ,. /„,
bo added to the first part of the preceding equation, and b n
to the latter, then c h -\- ag = n h -L pd^-fe i e tho
.^um of the Eastings is equal to the sum of ihe Westlnfeo.
THKOREM II.
1" any trapezium, as A B C D, {Fig. 1S,2,) having two
^i- perpendicular to a given side, the product of half the
side-
sum of the parallel sides multii
they stand will be th
e area of the I'ljure.
'I"'^ 'V^ the ba?c on which
n
146
MENSC7RATI0N OP SUPERFICIES,
1
ii^
Let Ds and m n be drawn parallel to A 13 , and R F pa-
rallel to B C or A D; Cs is the diiTerence between the sides
A D and B C, and C n = 7is = w D z::^ their half difference,
and the perpendicular E F will l)iscct Ds and in n. Now
a» the angles D F m and n F C are equul, Euc. 1. 2'2, and
the side F n :^ the side F in, and 7iC = mD, the ii iungles
are equal and similar. Now if A B be multiplied by
-= E F, the product will be the area of the
I
Trapezoid,
THEOREM HI.
In rectangular surveying (m which the work is always on
one side of the first Meridian,) if the departure of any sta-
tionary distance is East, and the work lios on the East side
of the North imd South line, the farther that that course is
run the greater will bo the departure or the distance from
the f'rst meridian. But the farther that a stationary dis-
tance having West departure extendi, the nearer will the
first meridian be a})proaclied.
Draw N S (^'Vg-. 133,) for a first meridian. Let a, b, c,.
and d, be stations on the East side of N S, Let also the
perpendiculars I a, h i, s; c, and e d, be raised upon the N.
and S. line. Draw a r, b t, and c p, parallel to N S. Now
the departin-e of the first stationary line rt 6, is 6 r, and lies
to the West si<le of its meridian a r. As the point h in the
line a b, is nearer the line N S than the point a, for a I —
i b — b i-, therefore b i is less than a I, x\wA consequently the
point b is nearer the first nitn-idian N S, thati the jioiiit a \.<.
In the next statiotui^y (listancc // r, the dcj);)rture lies to
the Eastward of its vueridian b I; but the ^ oiut c, or any
point in this distance line b c, is more remote t'roni the fir^t
meridian N S, than the point h is, for h i -\- c t = c g.
THEORRAI IV.
If the meridian distance in the middle of every stationary
line be mulr,i[)lied into the particular Northings or South-
ings of that line, .uul the dilfcrence between the sum of the
Norlh and llic su.-n of the Sout'i products be taken, that
daTerencc v.-ill ' i the area of the survey.
Let 1
whose a
plan for
from the
let meri(
F in, an(
Avill be t
The r
lines whi
Now I
^fXd,
and the s
or the wi
the midd
ward, ar(
By Th.
an .i y h\
Z AE^
Z A E D
the area 5
the remai
or the dil
*he sum o
Hence thi
The pr:
Land are 1
a formal d
In pcrus
ferred to tl
work, whi(
He will t
APPENDIX.
147
Lot A F B C D E (Fig. ,84,) be the plan of a «,r,ey
whose area ,s require,!. Dr.avv N S ,,„ the west .ide of ,he
plan tor a fir.st meridian. Let perpendiculars fall upon N S
rom the begmnins and middle of every stationary I „e and
ct ,ner,d.a„s be drawn through each station. The„ I"
'?',"";'";• ™^;'. ■>" ""^ Northings; and C I, Dp, and Ex,'
will be the Southmgs. - '
The meridian distances to •'.e middle of the stationary
« '''T^'r'^' "" Northward, are oa,dn,andgk
Now by Theor. II, Z H X » o = area Z A F H l-L
H/X,in = areaHFB/I,;„nd/«Xff*=area/BC6/
and the sum of the areas will bo the areaof Z A FBCJZ
or the whole North area. Again the meridian distance, ..l
warT ;"??" '''"T' """ '''•''' ■=— -» «»«".
wai(i, are (x R, e </, and u v.
By Theor. II, A j X G R = area of the trapezoid ;, D C 6v
Ta l\l ' tT "'•"' "f f J' *. and A Z X «» = Jl
Z A F DC A 7 T , '^'^ ""•"'^ "'" ""= '"0 urea of
/' A J!, U C * Z, or the whole of the South area Then if
the area Z A F b C 67 be taken from the area Z A E D cTz
tlie remainder will be the area of th , figure \ F B C D E A
rhelurn'o^H'^ "'l""^" "■" -^^ "^ "-^ N"'"' »«"» ««d
Hence the foundation of the rule is cvidenu
DIVISION OF LAND.
a form,. . ■"■" '" """'"" '■'■'"" '^"'^'''''^ Elements, that
a formal demonstratton of them is considered unnecessary
LOCATION.
fer'r"d''r^°V'"'' """ ■"■ "■" ^P'""'"'''' ""= Student is re-
-, I ... "^ '-"f"'ncu ill luui part 01 the nrecrdin<r
•vork, wh,eh treats of the Location of Lands. ^
He wdl take ,K,tiee, likewise, that in the following inve.-
148
LOCATION.
rigiitions the letter a, is employed to represent the area or
contents of a field, farm, &c.
PROBLEMS I & II.
The area of a rectangle is equal to the product of one
side X by the other. It is evident then that the area -f- by
one of the sides will give the other side. The area of a
square is the square of one of its sides. The square root
of the area therefore, must be equal to the length of one of
its sides. For a square, the formula stands thus: V « =
side of the square. For a rectangle it is thus expressed:
a
-f= the side required, in which expression b represents
the given side.
PROBLEM III.
Let S = nat. sine of the given angle, and jc = A B or B D.
a
a
Then S cb^ == a, and x~ = g ; therefore a: = V g"' which
affords the Rule.
PROBLEM IV.
Let S = nat. sine of the given angle, b — the side given,
and X = the side required. Then SX6Xa3orS6x=:o;
a
therefore x= r^ , which is the rule.
PROBLEM V.
Let m = the given multiple, and x = the breadth. Then
m K = the length, and m x X ^ or m .x' = a; hence x' =
— • Therefore, x = V — ' which gives the rule.
in w
= ICDj
5 a am
Rule.
Let
length
Xx =
6x4-^
x= V
Rule.
Let X
given m
XS=a
Let S:
2 : 5
breadth; i
and x2=s
ilule.
PROBLEM VI
Let X = width, then as 5 : 8
8 X . S X
X :— =— 3 wherefore ~v-
Lot T =
*or2x' =
APPEWDIX.
149
= length. Then
5 a and «2 = ^ '^
Rule.
8 X
8 ««
5 Xx = a,or-^:=^a;hence 8x^^
r- Therefore,a;=v~'
8~' ^vhich yields the
PROBLEM VII.
length rnlb";;;".!-! ^^^^^^-^nce between the
i= 1,2^ ""• Complete the square x'^ +
^*+T=«+4- Then,a;-f-4 = JTTT^ „. ,
- ^ ' ' 2 "^ « + — . Therefore
/.2 f - 4
4
Rule.
A — TT or a; = -— 1^
*— -2-— V a-j--^, ^vhich is the
PROBLEM VIII
Let X == breadth, S= „at. sbe of dven / on i
given multiple. Then «i r ,u , *= . ^' ^^d « := the
I nen w oj = the length. Then x^mx
XS=a,andwcc2S=a,anda:= = -:^. tk. r
a m S l^nereforc a; =^
"^ ~mH' '^^^'c^ is the Rule.
o;
PROBLEM IX.
Let S = nat. sine of / B A n «« i
5x '^ "^^> and X = width; then as
^ •' Y~ C B, and S 0. = D E, the perpendicular
breadth; hence ^ + Sx=anr^
and x'= -- therefore a; = v ~ - A n
n , ^^ — Vgg— AD, vv
hich affords t>ie
8 X
5
PROBLEM X.
Lot * — i»;^..i, iK -.
..... then 2 ^ = central long,!,. Then 2 :c X
*"-"=«''»''»=vf which is the R„,„.
!50
LOCATIOK.
PROBLEM XI.
The reason of this rule appears from the rule given at
Prob. II, because a triangle is just half a rectangle of the
same base and altitude.
PROBLEM XIL
Let b = given side, S = the sine of the given / , and « =»
bxS
the side required; then, — tt"— ^> nence we have fis x = aa
2«
and X =7—' from which the reason of the rule is obvious.
* PROBLEM XIIL
This Problem is merely a particular application of the
rule given at Prob. XI.
PROBLEM XIV.
Let S = the nat. sine of the given / , and x = one of tho
equal sides; then s x^ = 2 a, and x^ = — and x = V y»
which is the Rule.
PROBLEM XV.
According to Euc. xii. 2, circles are to each other as the
souares of their diameters. Now the area of a circle whose
diameter is 1, according to the calculation of the celebrated
Van Ceulen, is .78c>3i)8 -\-, but for practical purposes
. 7854 is sufliciently near the truth. When therefore x =
diameter, x' X -7854 = a. Thenx" =
a
.7854
> and X =
V -78(^4 ' which gives the Rule.
Ui
PROBLEM XVI.
Let 6 = .7854, and x = the Conjugate diameter, then
APPENDIX. 151
X -f- 100 /. = the Transverse diameter; then {x -\- 100) X
X X 6 = a, antl 6 a;^ -f- 100 6 i = a: and x* + 100 x = ?-
b
Complete the Square, and x' -|- 100 z + 2500 = ^^ -j- 2500.
Thenx-f-50 =V ^-1-2500 andx = v|-f 2500- 50,which
affords the Rule.
PROBLEM XVII.
Let X = the Transverse diameter, and 6 = , 785'!, Then
no 3x 5 b x^
as & : S : : I : --= the Conjugate; hence -^-p — = o, and
5a
5a
3 6 x' = 5 a and x'= "f^- Therefore x = V ||, which for-
mula expressed in words is the Rule,
FOR T
In the
B D = S
AP=i.
length of
and B to
that the t
Make t
tre Coft
and A B
A B F
gles is cq
must conl
n^ent A F
•'cnti-e of
(Euc. iii.
angle A
ABC nil
A COLLECTION OF
PROMISCUOUS PROBLEMS,
FOR THE FARTHER ILLUSTRATION OF THE
PRECEDING RULES.
PROBLEM I.
In the triangle A B D, (Fig. 135,) arc given A B = 5,
B D =: S, and A D = 4, and the line P D in position, viz:
A P _ 1. Required the construction of the figure, and the
length of the lines, A 0, and B, drawn from the angles A
and B to a vvnidmill at O, on such a point in the line P D
that the angle A O B shall contain 120°.
CONSTRUCTION.
Make the base, A B, the chord of 120^ and find the cen-
tre C ot the corresponding circle by making the angles BAG
and A B C each S0=. Because in every quadrilateral, a,
A O B F , inscribed in a circle, the; sum of the opposite an-
gles IS equal to 180-, (Euc. iii. 22;) and as the angle A O B
must contain 120° by the terms of the Problem, its supple-
n^ent A F B must contain 60^. And since the angle at the
ocmve of a circle is double the angle at the circumference,
(Euc. ni. 20,) the / A C B must be 120^ And as the tri-
angle A B C is an isosceles, each of the angles CAB and
ABC must contain 30^ The point in which the ciroumfe-
154
APPENniX,
rcnce of the circle cuts th«; line V D, given in position, will
be the situation of the windmill, and the angle A O IJ will
contain 120°, as was required.
Proportion hy which to find the segments A H and H B.
As A B (5) : A B + D B (7) : : A D — D B (!) : A H
— H B(1.4). Now A B = 5, and5+ 1 .4 = 6.4, the half
of which is S.2 = A H; and 5 — 1 .4 = 3.6, the half of
which is 1.8 = H B.
To find the perpendicular D H.
As the triangle A D B is right-angled at D, the perpen-
dicular D H is a mean proportional between the segmenta
A H and H B; therefore V 3.2 X 1.8 =/v/5.76 = 2.4 =
D H, and AH — PAz:=:2.2 = PH.
Proportion to find the / D P H.
As the side P H = 2 . 2 . 34242
Is to HD = 2. 4 0.38021
So is r ad. 90° 10.00000
To the tan. Z D P H = 47=^ 29' 10.03779
Hence 180° — 4^ 29' = 132° 31' = / D P A.
To find C B.
As the sine / B C A 120°
9.93753
Is to A B = 5
0.69897
So is sine / C A B 30°
9.69897
To the side C B = 2 . 88 . 46041
To find (Ae Z 5 B C P and C P B.
As B P + B C== 6.88 0.83759
IstoBP — BC = 1.12
So is tan. i sum = 75°
0.04922
10.57195
To tan. h diif. ^ 30° 17' 9 . 78358
Then 75° -\- 31° 17' = 106^ 17' = Z B C P.
And 75^^ — 31° 17' = 43° 43' == Z C P B.
PROMISCUOUS PKODLEMS.
To find C p.
As the sine / B C P = 10(J° 17' fl.OS'iiJiJ
Is to side B P = 4 O.OOiOtJ
So is sine ^ C B P = 30^ 9 69897
152^
To the side C P = 208
0.81881
Tojind ZsP O C andP C O,
As side C O =2.88
Is to sine / C P O 91" 12'
So is C P = 2 . 08
O.lfiOIl
9.99991
0.81881
To sine / P O C 4G'' 1 1' 9.85831
Then 130° — - 46° 11' — 91° 12' = 42"" .S'7' = / P C O..
To find the dist. P O.
As sine / C P O 91° 12' 9.99991
Is to the side C O = 2.88 0.4G041
So is sine / P C O 42" 37' 9.880H5
To the side P O = 1.95 0.29115
To find the Z s P O H and P B O.
As PB-fP = 5.95 077452
Isto P B— P O =2.05 0.31175
So is tan. ^ sum ^ s 6G° 15' 10.35654
To tun. -i difr. 38^4' 9.89;}77
Then 66° 15' -j- 38= 4' = 104° 9' = / P Q «.
And (j6'-' 15' -- 33^ 4' = 27'Mr = / P B O.
To find the side O B,
As sine / P O B 104° IG' 9.986SO
Is to side P B = 4 0.60206
J?o is sine / B P O 47" 29' 9.86752
To side O B = 3,04 0.48828
ii
!l
156 APPENDIX,
To find the side A 0.
As sine Z A O IJ li20'
h to side A B = 5
So is sine Z P B O 27^1'
9,93753
0.60897
9.65976
To side A O = 2.63 0.4'il'iO
yote.—'Vh'is Problem might have been solved more ex-
peditiously by Algebra.
ii i
PROBLEM II.
Near the middle of a certain farm or tract of land A B (',
(Fig-. 136.) whose form is that of an equilateral triangle,
and whoso side A C runs due North, is a spring of water at
0, so situated that the perpendiculars O T, O D, and S.
let fall from it upon the three sides of the triangle, are re-
spectively 18. '20, and 24 chains, viz: T r=: 18 c/i., O D :=r
20 ch., and OS — 24 eh. The owner of the farm has be-
queathed it to his three daughters, O, P, aiul Q; and to is
bequeathed the triangle A O C, to P the triangle A O B,
and to (i. the triangle B O C. Required the area of the
whole farm, the area of the portion bequeathed to each of
the daughters respectively, and the courses and distances of
the division lines.
It can be proved that the perpendicular C R is exactly
equal to the sum of the perpendiculars O D, O T, and O S,
in whatever part of the triangle the point O is situated.
The Problem is then solved by Trigonometry and Mensura-
tion of Superficies, as follows :
To find the side A C, (md consequently the other sides.
As the triangle A B C is equilateral, each of its Z s —
(iO'-; therefore •
As the .sine of the angle at A 60° 9.93753
U to C R = T + O D + O S = G2 1 .79239
Soisrad. 90= 30.00000
To the Bide A C r- 71.59
1.85486
PROMISCUOUS PROBLEMS.
<57
Then 71.59 c/t. (A B) X <i2 c^. (C R) = 1437 58 cA. ^
2 = 2iJ18 . 79 ch. ^r. 221 ac. 3 r. 20 p. = the arou of the tri-
angle A B C.
To find the share of each Daui^hter.
7l.59c/t. (AC) X24c/t. (Q S)
25 jo. = O's portion.
= 859.08 cA. = 85 ac. 3 r.
Ajraln
71-59 c A. (A B) X 20 cA. (O D)
71 ac. 2 r, 14 p. = P'.s portion.
^ //l.59cA. (BC)X18cA. (OS)
And — ^^
--= 715.00 rh.
464.31 ch..= 46 ac.
Ir. 28 j9. = Q's portion.
To find the Courses and Distances of the Division Lines.
Join S D, D T, and T S; then in the A D O S vvc have
given S O, O D, and / S O D, to find the side S D and
the Z S D O; then the / A D O = 90"^ — / S D O ==
/ADS. Having ascertained t.hisside and these angles, then
in the triangle A S D, all the /.s and the side 8 D will bo giv-
en to find A D and A S. Tlien D B and S C jnay be found
by Subtra(!tion, and B O and O C by the S(iuare Root.
Ne.xt, the /SOD -- 120^ subtracted from 180^ = 1)0°,
the half of which i.s SO'' — . half the sum of the opposite
andes.
Also, D O -•- O S = 20 :- 24 = 44 =
sum of the sides.
And O S — D O = 2 4 — 20 - 4 — difference of sidea.
To find the ZsSD O and D S 0.
As D o :- o s
44
Is to O S — D O = 4
So is i sum of /s 30~^
To tan. i diflr. 3°
To find the side S D.
As sine / D S O 27^
Is to zA^ L) O — 20
So is sine /SOD 120^
1 . G4345
0.t)020G
9.76144
8 . 72005
9.65705
1.30103
9.93753
To the side S D =38.15
1.58151
t&s
APPENDIX,
SS°
W:
Then 30« — 3° = 27^ « / D S 0, and 30^ -f S^
^ S D O, and 90^^ — 27' (Z D S O) =* CS"' « ^ A S D,
and 8C^ — 85° = 57« = A D S.
2'o/m(i A D.
As flinc Z D A S CO"*
Is to side D S = S8.15
So is sino Z A S D CS°
0.9S753
1.5S151
9.94G88
Tosido A D = 39,S9
1.59376-
To find A S.
As sino Z D A S 60°
U *oside D S=:38.1i»
So i.s sine / A D S 57°
9.93753
1.58151
9.92S59
To Bide A S = SG . 95 1 . 5G757
' To find the /, D A O,
As side A D = 39.29
Is to side D O = 20
So is rad. CO*^
1.59S7G'
1.30103
10.00000
To the tan. Z D A O '^7° 9 . V0727
Now the Z C A B = CO^ and Z D A O = 27°; their dif-
ference is 33?, and the course of A is N, 33° E.
Tofmd the Z ^^ O.
AsthcsMlc D B = 32.30
Is to the side D O = 20
So is rad. 90°
1 . 50920
1.30103
10.00000
To the tan. Z B B 31° 45' 9.79183
Since Z C A B = 60° and Z D B O = 31° 43', their sum
is 91° 43' ~ 90° = 1° 43' and 90^ — 1° 43' — 83° 17', Ilenco
the course of the line B O is S. 83° 17' E.
To find the length of A 0.
V(AD'-t-DO') = V1243.7041«44.08 = AO. The
length cf A O therefore is 44 ch. 8 /.
To find the length o/B 0.
V(DE» + DO') = v' 1443.2900=»S7,0D=*DO. Tbo
length of B O thcref jrc iy S7 ch. 9Q L
V(S
fore the
42 ch. 1
The I
lake v/h
area of
mencing
lake, I r
king the
course o
fore, tha
the disla
offsets a I
I
jNo.
o:
Tho
arc£
S3° «=
S D.
pnoMisctrou' problems. |^
To find tk» /, C S.
As dirt side C S = S4 . C'l 1 5S058
Is to the side S O =: ^4 r.asbci
^« '^ '•'^''- J <-" 10.00000
To tan. S C O 34° 4S' 9.840CS
To find the Unstk o/C 0. '
'^/(SC«-j-SO')=Vl77G.029fi = 42.14=cCO. There-
fore the course of C O is S. S4° 4S' E„ and the distance i.
42 cA. 1 4 /.
eirdif-
eirsum
Ilcnco
K The
L Tb«
PROBLEM III.
The Figure (1S7) is intende.l to represent that part of a
lake which 18 inchidcd within the boundaries of a farm, the
area of which part I am employed to determine. Com-
jnencng at that point C at which tl: /ear line intersects the
ake, I run S. 20c/,., thence Vi. 15c/i., thence N. i5c/l.,stri-
kmg the rear line at D. Setting my compass at D, I find the
course of 1) C, the rear line, to be E. I am satisfied there-
tore that the surv- r has been accuratoly made, and that
the distance between C and D is 15 ch. I then measure iii«
offsets and enter them into my field book as follows-
^l^^l^I^J'd^i^^di^fRP-ri^oucLEnvTirs:]
800/.
500 /.
SCO
200
500
SOO
(CO
200
4CO00O
COQQO
1500CO
120000
2)730000
S65000
5£l^!^/llL*li°i£LiS^JM0F Far. 81DESI BliiTA^ii:
J
4
7
8
^150 /.
COO
225
223
150/.
200
300
400
The area «f the whole Rectangle A B C D is
67500
120000
67500
£0000j
S4C000
li
]6<)
APPENi»;X.
B C X D C = 1500 / X aOOO I. = 30.00000/,
\nd the area of all the offsets is 3»35000-f- 345000= 7. 10000
Aroa o<'tho Lake 2iJ at. 3 r. 'Up.
22.9000
4
3 . 60000
40
24.00000
m
m
PROBLEM IV.
The multilincal Plot {Fig. 13S,) represents the bounda-
ries of an ungranied or reserved lot of land, of which the
bases and perpendiculars are given in the following field
book. Required the contents. It is required also to lay oflT
200 acres frotn the point B, towards A„
No. OF A.i
!\,SE.|
Perp.
Doubi-eAue.\s
1
1800
750
1350000
2
1800
850
1 530000
3
1270
350
444500
4
2100
GOO
1260000
5
1800
350
030000
1400
250
350000
7
3100
540
1674000
8
2500
1100
2750000
9
2S50
1280
3648000
10
2350
340
069000
11
3050
700
213.5000
12
^1740
1710
8247600
13
4200
2860
12012000
14
2200
1200
2640000
15
221.0
1(,00
3616000
16
3100
15!0
5134000
17
.3 /on
HOO
4440000
IS
lOSOO
4200
45360000
19
108U0
4400
47520000
2)145710100
7285505a
P20MISCUOUS PROBLEMS.
151
1.00000/.
.10000
[.9000
4
. 60000
40
4.00000
bounda-
vhich tho
i^ing field
to lay off*
Trapezoids,
rg =
IfJOO
of =
1000
ne =•
1 500
md =
1 060
<? c =
2050
S6 =
1900
ta =
2500
11610
Then the sum of these offset. = IIGIO -. 7, the number
■(.f them = 1658 the mean offset; and 1658 Xtr = 6-100 =
.<061 1200 ^ the area of the trapezoids, r f, o c, n d, mc,ei,
and a a, wh.ch added to the area of the trianghvs, (Nos
1--19,) give. 83166250 ^ 834 ac. 2 r. 26 p. = area of the
whole lot.
To lay off- i200 acres from the point B.
Let a conjectural lino bo drawn or run from B to M The
area of B M A D vv.ll be found to be 26288200, from which
subtract thc3 quantity to be laid ofl", 20000000, the remainder
.s 6288200, which X by 2 == 12576-100 == the double area of
tne A L A M, which divided by the conjectural line B M.
or 9150, gives 1374 for a perpendicular let lall froui M on
the hne A B. Then run B A and the operation i« completed.
PROBLEM V.
The following method of ascertaining the contents of a
hold, who.se ooundaries are ^curviliuear or irregular, is some-
urnes successfully a<l.,pted by skilful and experienced sur-
veyors.
Let the mixtilineal figure a c d ef g h C A, {Fk , 1.S9,)
represent the boundaries of a field or tract of land, the con-
tents ot which are required. Run the lines A B and B C,
so that the parts «, c, e and g, included between these lines,
^'Ktll be as nearly as can be estimated, equal to the parts b,
a and./, lyn,g l,eyoud fliem; then find the area of the iriun-
glo A B C.
In the same manner a curvilinear figure maybe reduced
' "''" ^"'"^ "* '' parallelogram, or any other rectilinear fi-
'Hi
6 til
It
is 5;
1C2
APPENDIX.
gure, and its area ascertained by the ordinary rules for de-
termining the area of such figures.
By this method a surveyor, of good judgment and exten-
sive experience, will come very near the true contents of a
field. A«, however, much will depend upon the formation
of a just estimate of the quantity of hind contained in de-
tached pieces, this method should only be adopted by skilful
practitioners, and where lands are not very valuable..
In!?!
PROBLEM VF.
The triangular lot of land, ABC, (Fig. MO,) lies on the
side of the road A B. It is required to divide it by a lino
running from the opposite angle C to the road A B, so that
the areas'ef the parts may be as 9 to 7, the side A B being
9i chains. What extent of front must bo assigned to each
parr?
As the two A s into which the field is tD be divided have
the same altitude, it is evident that they must be to each
other as their bases.
Therefore as IG : 9 : : 953 /. : 534 . 375 or 5 ch. 54 -f /. = A D.
And as. IG : 7 : : 950 /. : 41 5 . G'25 or 4 ch. li)-\-L=DB.
n 3:
PROBLEM Vn.
Divide the straight line A B, (Figs. 141 Si 142,) into thrco
parts, in the proportion of ?>, 5, 7.
From the i)oints A and B, draw the parallel lines A C and
B D, on opposite sides of the Koc A B. Prom a scale of
equal parts lay ofl' ?. iVom A towards E, and 7 from B t'l-
wards F. From the same scale lay off also 5 from E to-
wards C, and fr(,in F towan^s D. Tiien draw the iines E D
and C F, intersecting A B in H and R.
Then A H : H R : : 3 : 5, and 11 R : R B :
D
7.
In this %vay A B may be divided similarly to -any g' van di-
vided line,
JS'olc. — This method of dividing lines might be advantag'^-
ously employed lo find the point 1) in the preceding Prob.,
nni\ a-ho t^niiif P. in thif vvllu'Jl fnllllWS-
I for do-
id exten-
;nts of a
jnnation
ed in de~
)y skilful
es on tho
by a lino
5, so t!!at
I B being
d to each
ded have
) to each
L = A D.
?. = D B.
PKtMISCUODS PROBLEMS. gg
PROBLEM VIII.
The side A B of a triangular field containing 6 acre, is
4C0, and A C 420. It is required to lay ol 2 ac. Ty a
frS. '""'""^ ^"'" '^" ^'''''' ^' "'"^'^ '^ 22° *^-Lt
f E als between A and D, the point F will he in A C;but
^r not, tho^ponu F will be in B C. Now if wo join D C
and A 1< E would be parallel.
Therefore, As A D : A E : : A C : A F.
To find A E.
By Prob. VI., As G : 2 : ; A B (4GC) : A E (156).
To find A F,
As A D (230) : A E (155) : : A C (420) : A F (2S3)
Lay off 233 from A to F, in the direction of C, and ruu
the line l D, and the work is completed
If E ha<l fallen between D and B, the point F would have
fallen m ilie line C B.
into thrco
s A C and
1 .scale of
roni B t'T •
*om E to-
iines E D
PROBLEM IX.
A Gentleman after having taken the disncr ions of a
nqunro fiehl forgot all the dist mce.-:, and only rcccllected
that bavmg occasion to measure the diagonal hr oupd it to
exceed the .side by 10 ch. Required a ruie by which to find
tnc side.
^Lci X = one of the sides of the squa-r ... D C D, ( AV
MS,) and d = the difference bet^x c en tnc sido and the diagorVl
orlOcA. Then a,' -f-c;== the hypotenuse BD, Butaccor-
<ling to Euc. i. 47, lU;^ -j- D'^ = B D=^ or •:. B C= = B D »
Hence we have 2 x-^ =-. o;^ ^- ^2 d x + a^= or a;' - 2 rf x =J
C-. Comfjlete the Squerc and x- — ^dx -\- d- = rf- 4- d^
orj:= + 2 e^ x -{- d'~ = o d\ Then . - ./ = V 2 d\ and
a ~y 2 d"- -f a' or .T 4.14= sido of tho square, Hcnco
iosults the foiiowinj
•^.r
i
It
111
in
m
164
APPENDIX,
RULE!
To tho square root of twice the .sr(uare of the diflcrcncr,
add tho difference, and the sum will bo the side of tho
fiquarc,
PROBLEM X.
Required a rule by which to lay off a given quantity of
ijfind in the form of a triangle, two sides of which ahall be
equal, the included angle being given.
Let a represent the urea, S the nat. sine of the given /,
and a; one of the equal .sides. (i'Vg*. 144.) Then S x* — '2 a
'2 a
and x" = ~<i~' ' therefore a; == -v/
'J. a
Whence results
the rule. Divide double the area by the nat. sine of the in-
cluded angle, and the square root of the quotient will be the
length of one of the equal nidod.
PROBLEM XL
To re-es' ^ marks upon an old line, without running it.
IIULE.
Itun a lino at random in the direction of the old line, upon
which set up stak(>s equi-distant from each other. When
you arrive oi)posite to the end of the original line, th«;
marks upon which you wi^h to restore, remove your instru-
ment thereto, and let a perpendicular fall uj)on the random
line. Then as the length of that line is to the perpendicu-
lar let fidl upon it iV.j'u the extremity of tho olil line, so is
the distance to the first .stake to the distance which it must
be moved to the old line. 'J'hcjn if this distance be multi-
plied by 2, 3, 4, &,c.j as circumstances may require, tf* pr -
ducts will bo the several distances which the stakes must jo
moved, in order to st^nd on the old line.
Let C B, {Fig, 145/) be an old line, the mark.s upon which
it is rcquireil to re-establish, while at the same time it is in-
convenient to run the same. Run the random line C A 20 ch.
The length oi" the perpendicular A i^, or the distance of A
X:
liflcrencr,
itie of tho
[uantity of
h »hull be
PROMISCCOUS PROBLEMS.
165
from tho end of tho old line is found to be 5 ch. On this
lino, A C, set up stakes at a, c, and e, each 5 ch. distant from
each other.
Then, As C A (20) : A B (5)
As C A (20) : A B (5)
AsC A (20) : A B (5)
Or thus: Having found hy^tho first proposition that c fL
1.25, then 1 .25 X 2 = 2.50 = c d, and 1 .25 X 3 = 3.7D=:
a b.
Ce( 5) :(?/(!. 2.5.)
C c (10) : c rf(2.50.)
C a (15) : a h (3.75.)
I given Z,
S X- ^ 2 a
ICC results
; of the in-
will be the
)'unning it.
[ line, upon
5r. When
1 line, tho
our instru-
lie random
erpendicu-
[ line, so i.s
ich it must
; be multi-
re, th pr. •
tes must bo
ipon which
ime it is in-
5 C A 20 ch.
•tancc of A
PROBLEM XII.
A surveyor having run .30 ch. in a direction between South
and East, finds that .the sum of hi.s difference of latitude
and departure is 40 ch. Required the course, and the area
mcluded within the line run, the dilTerence of latitude, and
the departure.
INVESTIGATION.
Let a = the distance B C, (Fig-. UG,) and h = diff of
lat. + dep., or A B + A C; also let x = the dep. or A C.
then B — a- = diff. of lat.
Now, B C"- = B A= + A C"' (Euc. i. 47,) i. e. a^^b^^
2 6 a; -I- 2 x= or 2 .T^ - 2 6 a; = «= _ h\ Therefore, x^ -
— 2/,07 = A C the dep. Then -10 •— 27 07 = ]f> 93 =
A B the diff. lat.
Honcc we have the following rule to find tho remaining
f^idcs of a nght-angled triangle, when the hypotenusp and
the sum of the other two sides are given, viz:
Fioni twice the square of the hypotenuse, or distance run
--ubtract the square of the sum of the other sides, viz. the'
'i'«. lat. and dep.; divide the remainder by 4, and to tho
square root of the quotient add half the sum of the two
^-Jucs, and that sum will bo the departure.
m
Ik
If- ;. :
u
If
166 ArpsNDix.
To find the Course .
AsthedifT. of hit. 12.02
latothsdep. 27.07
So is rad. CO''
1.11125
1 . 4S248
10.00000
To the tan. of course G4' SO' 10.32122
7^0 find the Jlrea.
zIJU-^-J^-H^ = 17.\ ac. nearly. The course, therefore,
is S. C^° 30' E., and the area 17.i ac. nearly.
PROBLEM XIII.
To locate land in tbeform of a Bight-angled Triangle, the
Area and the Hypotenuse being given.
Let a = area, b — the hypotenuse A C, {Vig. 147,) and
"a
a; = base A B, tlien -- = perpendicular C B. Then, by
. 4 a'
Euc. i. 47,x--{---2-
= i'^; then x» -\~ 4 a- ~ b"^ x-, and x<
. i« x5 ^ _ 4 a'. Next, x^ — 6= x' -j- --- = -^ — 4 o»
S64
Then, z»~6= = V J^—^a^^ and x= = 6*-;-V
4 a'
---4a»J
' 4 >
which cx-
Whercforc x==vj6^iV[-^
prcssion afTtsrdo us the following
rule:
From the hiquadrate of the hypotenu.se divided by 4,"sub-
tracl four times the -^quarc of the area, and extract the
pquarc root of the remainder. To the root add, or from it
subtract (according as vou wish to ascertain the lon<,^c.st or
shortest side of the triangle) the square of the hypolnnu.so
divided by 2, the square root of the sum and of the differ-
««co Avillgive the length of the sides respectively.
•I
therefore,
'.angle t the
147,) and
Then, by
x^, and x<
4 o«
which cx-
64
4 "
:4
i by 4, "sub-
extract the
, or from it
5 lon<^c.st or
hypotenuse
' the diflcr-
y-
paoiiiscuous PROHLEMa. J€7
PROBLEM XIV.
Required the length and breadth of a rectangular meadow,
whose perimeter is GO chains, and area 20 acres.
Let a =. area, b = perimeter, and x = length, then --~
breadth, and 2x -;- ~ rrr b. Then 2 a-' -!- 2 « rrr 6 r. This
equation reduced tjives x
'4 "'-^ rro-^- i ^''c"
CO
a
c™20cA. r= length, and -- — 10 ch. — breadth. This fur-
mula, expressed in words, affords the following
rule;
Divide the square of the perimeter by IG, and from the
quotient subtract the area. To the square root of tha re-
mainder add one fourth of the perimeter, and the sum will
be the length, and the area divided by the length will b«tho
width.
PROBLEM XV.
An easy method of locating land in the form of a llhcmb'oid.
Let the figure A B C D, {Fig. 143,) represent a Rhom-
boid whose area and front A D arc ^iven.
By Trigonometry find the base and perpendicular A e and
c D, of the right-angled triangle A E D = C 6 B. Next,
find the areas of these triangles; then by Prob. II., Loc.,
by out the remainder in a rectangle as A 6 C e, and D e -f-
« C «= D C, a side of the Rhomboid.
PROBLEM XVL
The owner of a square field A B C D, {Fig. 149,) con-
taining 10 ac, wishes to lay off a walk half way around
which shall take up one acre. Required the width of the
walk.
Let a = one side of the square, and x = width of tho
H-alk; thon 2 a z — x' = 6, the area of tho walk. New this
t
I
u
168
APPENDIX.
equation rcrluccd gives a; = a + V (o'^ — ■ 6) = 52 /., the
width of tho vvulk. Henco tho
From the area of the square subtract tho area of the walk,
sulitra<;t tho square root of the remainder from the side of
the square, and this last remainder will be the width of thtj
walk.
JS^nte. — Perhaps it may not be alto<TPthpv superfluous to re-
mark that the square root of any (juaiitity may be either
positive or negative, i. <?„, it may have either the sif^n -4- or
~ before it, because — x X — x = x", as well as x X i".
Quadratic equations therefore admit of two solutions.
}l(!nce the reason of the use of the ambiguous expression T
placed; before the unknown quantity. The learner must
not, however, on that account, suppose that every Problem
leading to ;i Quadratic equation may have two answers.^
The Problem may be of such a nature as to reader one of
tho results wholly imulmissable, and which of them is to be
rejected will be easily determined by the conditions of the
Problem itself.
Wfi
PROBLEM XVII.
It is required to locate 2 ac. in the form of the figure
C Mine d, {Fig. 150,) so that the width C d shall be equal
to the width B n, the side C A being (i ch., and A B l ch.
What must be the width of C c? or B n?
I pj a __ m-ca, 6 = C A or 6 ch., c = A B or 4 ch., and
X = width, or C rf = Bn. Then b-\-c—x= length, and
/(, j^c — x)Xx—bx-\-cx — x^=ii substitute t for b -|- c,
then t X — X- = a, and this equation reduced gives the va-
lue of .t; thus, X =: -^- -^ V | ^ " " H '^^' ^- -= ^ ti or
B n, and from this expression we obtain the following
uule:
Add the distances A B and A C. Divide the square of
their sutn by 4. From the quotient subtract the area. Sub-
iruct the square root of the remainder from half the sum of
the two sides A B and A C : the remainder is the width.
PROMISCUOUS I'nOELEMfJ.
iod'
52 /., the
'the walk,
he side of
iltli of the
lions to rr-
be citlior
sign 4- or
as X X .r.
solutions.
n'ession T
rn<;r must
y Problem
auswera.
der one of
;m in to be
Dns of th«j
the figure
ill be equal
A B 4 ch.
4 ch., and
3ngth, and
t for b -|- c,
res the va-
.^C dot
swing
! square of
rca. Sub-
tile sum of
width.
PROHLF.M XVIII.
To Jindthc scale by which any plan has been drat^n ^nhen
the contents arc given, but the scale nmittcd.
nui.E.
Find the area oy any scale, and then institute the Adlow-
mg proportion:
As the content found
I^i to the s-iuare of the scale by which vou found it
oo IS the ;;'i\cn area ' '
To the square of the scale by wliich the plan was drawn
Ihe square rootof thi. nu.nbci^viJl be the scale required.
PROBLEM XIX.
To find a etruc area oj a survey, ihough it has been taken
by a chain xohich is dihcr too lo7ig or too short.
Lnl thcarcu l.o complotcl, „sif,hc ch.ni„ ha.l boon of the
proper length. U'he,, /bnn th„ fnllowh,, proportion •
As tlio square of the true chain
^s to the content found by the chain employed,
So IS tlie.square ol" the chain by which the survey was made
i o tiw true area.
PROBLEM XX.
To find the area of the inaccessible field A B C D E A
(Fig. J51.)
Jr H'T '"' ^^ ^ "' '""'" convenient distance H-om the
• ^Id. At t he cxtrennties of this line take the bearings of
Sysrr^ Tf ^ ^-^ ^--^^ ^^ ^^' ^^ ^--' -'i S A
and hi v"!; }^''''^'y^'^'-^''''^"S these several Tcourses
'"^^ huaiMg. thc.r points of intersection, as A B \ V <tp
.vou have the sides of th.e field ' '
Its shJes may also be A>und hy Trigonometry, thus- ..
^"i. z\ R fe A, you have all "
the /
s and the side R S to imd^
*70
APPZIODIS.
tho other side.?. Again in the A R S B, you have oil tho
/ s and i side II S to find the other sides. Having as-
certained ti.c sidc^i R A and R B, and knowing the / A R 13,
the side A B is easily determined. In the same way all tho
remaining sides of the field may ho ascertained.
PROBLEM XXr
Jlnoihcr method cf fmdins the area of ri^hUlined figures
by Calculalioii.
Find the DilTcrcnee of Latitude and tho whole Departure,
and fill up the Table as in f rob. XV. of Mensuration. Hav-
ing filled Ujithc column of Meridian Distances, add thcfirsi
M. D. to the soeond, for the first or upper number in the
Column of Sim.s; add the second M. D. to the third for the
second number in the colu:nn of Stms, &.C.; and add tho
lowest M. D. to tho uppci-mosl for the last number in the
column.
Then multiply each number in the column of sums by its
respective Northing or Southing. Insert tho product in the
column of North or South areas, and half the difercncc be-
tween these columns will be the area of the field.
Tho following example will .servo to illustrate tho Rule*
«
.M
a.
>n c
ai
00 1
o
CI .
^<.
•
M
^5 C
*^
C5 C
Q
CI c
^
u
at
K>
rt
O
M
O C
o
O
'^.>
_ ^ _^
<
o
<:
>^
ID
O
u
prtoMir;cuoc3 phselcmc.
IT
CO o o o
crj O O O
«r o ui cj
»^ 1^ m 00
1^ 00 O (XJ
00 rr> TT i-«
C» »^ CO ^.'
« «> C< V
O I-
I- ^
iO -JJ
■n CO
»>• —
o -.
c<
'u"» •:«
O V ;W
J l-i l^r
IO U:<
_»
t' OD
CO
\,'
QTt
-<•;
•-• f«
t~
•J*
r7;§
TP O
CTJ CJJ
30 O
TT re
1CC(
H
u
H
C/!
H
O
CO
o o
•-?
oco
r*
po
•-4
'; OT
■V
lO vn
C3
o>
m
TT
^
w4
OO~C0nr);2G0-;
»n -J c« c» CO m ffj
CO cr. G» — CO c»
■ ®3JI!***^oeoco
o o « — r^ — i ^
o —1 — < « i-» no
CO « « o c<
en
CO
c;
t^ 1" CO CO
r-
""a" »n oj «
^^
05 CTJ |>.
^4
c« c»
c;?
rf
CO lO
O
tj
^1^
O
in
o
CI
CO
;j
CI O C5 •-<
Tf O O G<
<o »r> ci irj
«—• lO »— 1
00
oo
>n o
CI •-<
rr CI 'CO
t^ G« CO
'-' OO •«,»
!OOOOOOOt>«
ocooocoo
Cr3O«if5OO<n0CI
C> GO 0-5 in CI cyj GO GO
o
- ., ^ ^ t> r* r.
o o
O O
• •
o
4-^
o o o
O C^ 00
•-» CD lO
o
^."^
C«
c/f (/: ^i
CO •-•
"• 7
O
V
c
171
*0 O M S; ^
;.^ ?:^^
•*• ** ~ «^
W « ^ S m
>-■ r rt 4*
t* > c fl
's «^-S— < «
"•5 -'«
,, re T3
- 5 -3 an ca
•- _n w _
dj •-• '^ tn
« nr "^
♦J t; -^3 '/! H
5 Oj J, O U
-;:; « r p
»- « 0) 2 2
c.^"S '- o
'' ^ i -9 -^
^'^S
'-I *3 ^ r-
o o es =: 3
— ^" O rt O
rt o jj w G,
o r; «5 c I;?
3
o J ^
J o o
o
rJ
C
o
P
o
J3 C S to
o
s
3^
- -r d a V. ^
■c ^•' :'' P =:
2 '■'■< o o •-<
c^ 2 - -^ &
X *-* o r— J^
I o s: « c>
« c^ I"
^ g S ^^ ">.
« e o o
*► H ■« >
^MAGE EVALUATION
TEST TARGET (MT-3)
/.
/
^
t-?/
&
f/x
1.0
I.I
1.25
IM 112.5
1144 _
j50
la IM 1112,2
II: 1^
2.0
1.4
1.8
1.6
V^
<^
/a
'a
I
c^.
C),
^m • j^^
.>'
A
>^
w/
°'W
Photographic
Sciences
Corporation
M
'h
-b
w
\\
^9)
V
.^^
^^o^
6^
^
%
'^
23 WEST MAIN STREET
WEBSTER, NY. 14580
(716) 872-4503
172
s^"
APpnrfDix.
:m
hnve not thou /> lit it advi.-vablr. to swell the sizo or increase the
price ot thi;* Treatise by their insertion. A Surveyor will
hnd his advantage in solectii-g- some particular system, in
rnakin- liirnsclf thorou,o-hly and faniiliarly acquainted with
It, and adhering to it with very few deviations.
Before concluding this branch of the subject, it may bo
serviceable to lay down a few Hxjlf.s for finding CGniained
anp^les.
1. If the first letters of the course nre unlike, and the last
likewise, subtract the lesi course from the greater;
Thus: S. 6'rW.
N. 15^ E.
47'= -. included /.
2. If the first letters arc unlike and the last alike, add the
bearings; *
Thus: S. SO'^W.
N. 40'-^ W.
70=" :.— included /.
3. If the first and l;ist letters are both alike, add the less
bearing to the supp]cn,ont of the greater.
4. If the first letters are alike and the last unlike, subtract
the sum of the bearings from 180°,
General Ra!e.~-\n oi,lcr to finu the quantity of an ans:le
suppose yourself stauiling at the angular point. Then re-
verso one or both courses, (as necessity may require,) and
the (pumtity of the angle will be easily ascertained.
LEVELLING.
Though it is not expected that every Land Surveyor will
possess all the qualifications, ;uid provide himself with all
the instruments necessary f >:• (;ivil engineering, yet he will
often find an acquaintance with the ger,crarprinciples of
Levelling of very great service. lie may bo employed to
8urvey a courr for condn.^tlng supplies of water, to deter-
mine the most suitable si;> f >r the erection of a mill, or the
most eligible route for the lunation of a road. If he make
hirwsolf completely master of the subject he is in this respect
LEVELLING.
I'r,
"•-heap and simnlv ■•„„ , , ^- ■'''"''° Circumfbre;,tor. a
'"eon.,-,,,,,,, o,...,,.:,', 'theta't':"' ■ '" ""'''•"" ""
b<! ompioyi-d. suivcyor caa c.vpct i„
WitI, regaril.t.) the rhnarv of T o,.„n-
«' nght angles ..vi.l, thi., v^'V^Lcl^Z ',' ' , r ■■""■"
'1.0 oartl?. .surf : ■;';";"' ,"'• "'"""'' "'"J '-^'■■"'T
=1.0 sun; CO „f ihll't] l."o, .o«evc.-,will s„„„ leave
•■'.0 -,i„, or.he ,,.',' :t::;,v: "^'""•' -^ ^'"''"'"'•' -^
"'• 'l.od«.a„ce to which hi r™''"'"'"" <"^ "«= square I
-ile i. i. rai. 8 /, v ,v ^ "" ^ '".° """ "^ "'^ --«' '
'ii.tanec is 2 ,n^les , V ^ T'"'" °'^ ""^ '^''^'""oc. The
carsh-a surface - -':;:;: ■ ■'" ";"° '•""■' ''' •■' 0...-V0 of the
--....oehav:;;;:t:r':::::ij:;-:— ^
f^uriacoequrJJy distant in ill it..; / "^ '^"'^^^ ^^ the oartn\s
^-pp..j.tho;:;;^;:;:;c;:::;r'-"™" ^■'-
viclcl into fee,, inches, a„a mZl^th "°' "'"
vanooi-aslkloatf-hpT. ■, . ' "* ''''^"i? n moveable
particular p ,, r' 1 ! " ' ""^ '"' '""''^ '"'''• '" ->'
">- -curo'thn a. ■ t„ eeof uT"""" """"''""^ "» -^"""^^
staves to nlac- thorn ,7,. '^■""'«'"'" '""" to carry the
an erect „,' e-nc ' ' !"'"\'="""'°"^' '" '«'op the,, i„
or ,n„,.e to ,0 Cc oi;:: ll!l"'"'.'7 '"^'-'-. -" ""^
iKuvc obstruction.^j from the
^vay.
ii; ij
174
APPENDIX.
Being thus furnished, the Surveyor directs the chain-
bearers to proceed, and measure or run tlie stationary dis-
tance. In consequence of the curvature of the earth\s sur-
flice the stationary distances shouM not exceed a few chains.
(The curvature of the earth in 10 ch. nmo«nts to an 'ghth
or . 1-25 of an inch.) He then plants his instrument in the
centre of t'lis distance, or midway between the two stations.
Then having carefnlly adjusted and levelled his instrument,
observing particularly that the air bubble is exactly in the
centre, he directs the men at the stations to raise or de-
press the vane until it is cut by the hair of the instrument,
and to note the height upon the staves. Suppose the height
of the vane ujjon one staflf is G feet from the surface, and
upon the other 3; then it is evident that the rise between the
places is exactly .'3 feet.
In a continuous ])roccss of levelling, or what is termed
compound levellir.g, it is not necessary to find by Subtrac-
tion the differences between every stationary distance. It
is sufficient to oi>ter each observation in its respective co-
lumn, under its title of f )re-sight or back-sight. Having
made the necessary entries, the back-stafl:" should be car-
ried f )rward and placed at a convenient distance before the
fore-stafl", which now becomes the back-stafi". Then place
the instrument in the centre, and proceed as before. In
this manner complete the survey.
Add up tho column of back-sights and the column of fore-
sights. If these sums are ef|ual, and the survey has been
correct, the first station and the last have the same elevation,
or they arc on the same level. If the sum of the back-
sights exceed the sum of the foresights, the terminating
point or last station is higher than the place of beginning
or first station, and vice versa, as will appear by the follow^
ing examples:
1
2
3
4
5
G
chain-
nary dis-
rtlrs sur-
IV chains,
m ghth
nt in tho
stations,
trument,
ly in the
10 or de-
truinent,
le height
ace, and
ween tho
3 termed
Suhtrac-
mcc. It
:tive co-
Having
he car-
!forc the
m placo
ore. In
of fore-
las been
evation,
le hack-
ninating
3ginning
! follow^
LCVELLIWa.
EXAMPLn I.
17a
No. OF
Stat.
COUR
SE.
Back-sights. || Fore-sights.
1
o
3
5
G
7
N. 10^ E.I
J>
N. 15° E.
Disi.
300
SCO
soo
soo
£00
SOO
300
£00
c
7
2
7
3
G
8
5
4
4
3
2
3
o
o
5
S
4
'y.00 1 1 42 I 4 f
37 3
8
1
?
3
cy
G
4
7
G
8
5
4
9
7
4
4
4 II 37 I 3 I 2
o
5
I
o
The terminating station is therefore 5/.'. l in. 2 fcn'Aa
higher than the first.
EXAMPLE ir.
No.o^
Stat.
1
2
3
4
5
G
Course.
S."45nv:
)>
s. ooHv.
DlST.
IN LIN.
Back -sights. || f ore-sights.
300
300
300
SOO
300
.FT.;lN.iTKNTHS||FT.jN.!'rENTH&
4
18C0 1 i^
1
G
2
5
1
o
4
5
1
5
1
3
4
7
o
o
•4#
ii i
S2| ,
25 8
8
7
Hence it appears ^hat the terminating point is ? feet low.
er than the first station.
The height of hills maybe ascertained likewise by Tri^o.
nometrical Calculations, by taking the angles of elevatTou
and depression, and measuring the slant sides.
M { S C E L L A N E IT S
• .V
Re-,-:stabi.i3:ime.nt of lost BouNnxRir-s,— Rem:.',?;'.
in attemp^s to .settle disputes about nkl lines, it often hap-
pens that tJio parties who were pre,ent at the ori-inal sur-
v-ey have forgotten tlie circuiT.stances or iuivo removed fron,
the country, or h:;ve thcniocivr ; l.eon removed by deatli
and consequently eufficlent cvidcneo cannot be procured to
restore the original boundaries. When a new line has been
ngrced to, in order to prevent liti-ation afterwards, it is ad-
visable that the parties formally release to each other alJ the
land quite to the newly established line.
The following i.s a copy of a Legal llclcasc for this pur-
pose : — ^
KNOW ALT MEN, by the.e presents, that I, Jl. «,, of
m the County of ,,ud Province of
tor ai". jn consideration of the sum of
good and lawi- money of tlic said Province of
r n r.r ■ ^'' ""^^ ^'' ^'^-''^h ^-'^'^^ ^ni] tru]v xmid by
in tlie County of and P^i-iucc of
at and l;eloro the ensealing and deliveiv of these
Zntn '^:^{y''lP\^l^-^^>^ { jo hereby acknowledge and
a^n,;^ -^li^ lf'^ 'r^^^^^^' i^^^VK remised, rdeLed,
am quit clain>, and by these presents DO remise, release
and quit clan,,, to the said C^l)..^ his heirs and ^.l^'Zl
vdntinf"-:^n-^'V'^^"' ''''T'l^ '^'^^"^" «»^1 ^'^'^''^"''' of
md tot .o'^.n' ^^-'."^^^-'^.^^•'■^'•' V, both -r in law orequiiv,of, in,
. n to tae .oilounig p.eco or parcel of Land, situate', Wing
'•'''' ^'''"- '''-' -'^n<i bounded as follows, naniclv
beginning at*
containing by
nUi'nnfu^J"' "''''"' '" "" '"'""''' "^ '^^''^ '=i"J ♦« •"'^'-''-t here iho
M:5C£LLANE0CS.
177
rrVliV,!?!. . r,,^ r~ ^"^ ^he ?amc, more o. !cs5.
ro HAVE and TO HOLD the above (In.crih.d premises,
to iiin the said C.l)., his lieirs, and assi-ns, to his and their
only j)roper boncht anrl behoof forever, together with ail
and suri^nlar, the buildiu-s, pnvilcives, and appurtenances
thereto bclongmnr, or in any Vri.se appertaining, and to every
part and pi-reel thereof. '
IN WITNESS whereof I have hereunto im- Hand and
T)Oi\\ .subscribed and set, thi:! day oC
in tlic year of our Lord One Thousand Eioht Hundred
and ^
Signed, Sealed, and Deli- '
vered in the presence of [ Jl, B. L, s.
Fj. F.
o. p.
Facts Respecting Magnetism,— VAniATion of run
Compass,
Iron, with its oxides, and alloys, is one of the .substances
most generally diffused through nature. It is not, however,
the only substance possessing the property of becoming
magnetical. The influence of Magnetism has been distinct-
ly observed in Nickel, Cobalt, and Titanium. It may be
detected in many clays, sands, stones, springs, and rivers,
in rain and snow, and even in many animal and vegetable
substances. According to M. Arago there is no substance
which, under favourable circumstances;, is incai)able of ex-
hibiting unequivocal evidences of magnetic virtue.
The opposite poles or ends of magnets attract each other,
i. e., the North pole of one magnet will attract the South
pole of another magnet.
Tho electric fluid, or lightning, generally destroys the
polarity of the needle.
Heat has a great influence on magnetism. A white heat
entirely destroys the magnetic virtue. According to the ex-
periment oi' Earlow on malleable iron, soft shear steel, and
hard shear steel, the magnetic power is about 4 times as
strong when the metal is at a red heat or at a blood red heat
as when it is cold.
Some substances will not exhibit any .symptoms of mag-
netic virtue until gently heated. Minerals which are not
II \t
! .! WZ
17a
APPENDIX,
V y.
W$
n<-
metallic arc almost all acted upon by the magnet after thej
liavc lieeji suojecrcd to the action of fire.
The effects of heat upon the magnetic virtue is very vari-
able. The principle upon which it operates is not under-
stood. No rule can therefore be given, by which its cfTccta
m \y be dercrmined.
Chemical action is said aKso to aficct the magnetic needle.
1 he statements ofcxperimentersui)onnagnetism arc often
very opposite and contradictory.
These consideration.,' would naturally lead us to expect
considerable disagreement between magnetic instruments,
and a want of uniformity of action in the magnetic needle.
This accordingly wo find to be the case. These f icts there-
fore afl'ord sufficient evidence of the inaccuracy of a method
Komctim^s employed by surveyors to ascertain the diflerenco
of variation. The following is the method to which I al-
lude : Knowing the original course and dale of an old line,
and having ascertained the prcc^ent course, they divide the
difference between the two courses by the nund^er of the years
which have intervened between the dates of the respective
surveys, and use the quotient as the mean annual difference
of variation, and allow it accordingly on all lines near that
place. The inaccuracy of this rule may be placed bevond
dispute, by the consideration of the following ficts:
The line of division between the Townships of Sackvillo
and Westmorland, in the Province of New Brunswick, runs
ucarJy four milco through a part of Sackvillc marsh', and
therefore affords peculiar facilities for observation.
In 17C2 its course was due North. In 171)1 it was traced,
and found to run by the compass N. 2° 4b' E., which gives
tjcarly 5' SO" as the yearly diffeience of variation. Accor-
dmg to a grant and plan made by George Sprowlc, then
Surveyor-General of New Brunswick, the course in 1813
was N. 2=^ SO' E, which exhibits a retrogade chan<.e of va-
nat.on of 10' in twelve years. In 1843 I found ii"to be N.
C° 15' E, or S° 45' in thirty years, giving an annual chann-o
of variation of 7' SO". From 17C2, when the line was first
run, until 1843, are cighty-oac yeans, and the change of va-
MISCELLANEOUS.
lio
nation during that time amounts to C !&', nliich gives 4' 37'
for the mean annual change of varit^tion. Again, by other
linos situated very near this lino, and run in 1823 the' dinbr-
cnce of variation in 184S was found to be 1« 20' or 4' every
year. ^
Whether we impute these discordant results to the inac-
curacy of the surveys, to the (ii.sagreenient of instruments or
to the n-regularitiesof the magnetic influence, or tj all these
causes combined, the inaccuracy of tho above rule is equally
proved. ^
Surveyors while using instruments governed by the mag-
not should take care that they have no substance about their
persons by which its actions may be affected. A delicate
needle n.ay be affected by a knife in the pocket, or buttons
composed of magnetic brass upon their clothes. Of this
fact any person may satisfy himself by placing a compass on
some solid object, and after the needle has settled, causing
some person having any ferruginous substance about him to
approach within two feet of the instrument. Tho move-
ment of the needle from its true position will indJrato tho
magnetic disturbance which his presence occasions. To
this cause may be referred many of the inaccuracies which'
arc so perplexing in old • irveys.
Surveyors would do well to devote some of their leisure
hours to the study of the geological structure of the earth.
Some acquaintance with this important and interesting sci-
cnce is so intimately connected with pr-ctical land su1-vcy-
ing, that it may legitimately be said to come within tho
range of his professional qualifications.
Si;CGESTiON AnouT Meridian Li.ves.
I would before dismissing this subject take the liberty of
suggesting the propriety and expediency of establishing ttuo
meridian lines in every County, or at convenient distances
from each other, by which every Surveyor might compare
his instrument as often as should be deemed necessary.
Those lines should be determined with great accuracy by
astronomical observation, and cstabli«hcd by permanent
I !
ill
nmM
ISO
APPEffOIX.
iiKirk.H not liable to an aJtcnitkm of position. Tht; courses
(rontiiiiu'd in all docunientt! conveyjn^r n titlo to hnulcd pro-
perty, aiui in all docuincntal rrconis tjhoulii bo taken from
this true ))iuri(lian, instead of from the magnetic. l)y this
mcuns many inac(Mu-;;cie.s avouM bo avoided, much nssist-
anco would be allbrdcHl in after times to Surveyors in tra-
v.'ini; ''"f-' ' v.hich wiil then !)o old, and bi etcne eventually the
.source of much valuable information upon the science of
magneiiim,— a .science of jrrcat importance in the airair.s of
mankind, a:id a science as yet but imperfectly understood.
My oI;jcct in the preceding treatise has been to select anti
include v/itJun as narrow limits as po.s^■iI)le the most Kimple
and mo.|t c:ctensively applIcaLle methods of Land Survey-
ing, and to adapt the .'jtiitcment and iilu.-^tration of them to
the most ordinary capacity.
The Surveyor must not expect to fmd in any treatise ex-
amples ut\i\cry case with which ho may meet in tl:(; course
of his practice. V/hcn we consider the numerous unac-
countable irrc:;;ularincs to v.hich tiie magnetic virtue is sub-
ject, the inumerous avA] unapj)reciable causes by Avhich it
may be distiirbed, the diUVrejices between i.M'trumcnts go-
ve.-ned ■ by the needle,, and the loose ujanner in which the
original surveys were frequently executed, lie need ntit be
astonished if after his utmost care and diligence:', he find it
almost Jmpossibla to restore lost boundaries with any dc"Teo.
of ccrtaiiiiy.
Most of the old grants in these Provinces contain more
land than they express. Sometime^, however, they contaiu
less. In eitlicr ease the recorded description differs from
their true dimensions and contents. Still the law ref]uire3
the Surveyor, in re-surveys, to follow as nearly as i)ossible
the original description.
Many errors have arisen from following courses as they
:ire literally e\-])resscd in the grant or reijordedj document,
without making the necessary ;dIowance for change of vari-
ation, &.C. Wlien this course has been pursued in subse-
MrSCELr-ANEOUS.
ISi
qucnt transfers of land, cxpei.^ive litijf^tion hu. soniefnio.
been the result.
Until meridian lines are estal.lisl.ed !.v oo.npot.-nt aiilh-,-
nty as suggested in u for.ner part of this work, in .Irauin-
deeds -.vlnch convey lands f^M'anfed or convey ed „iany verui
a^ro, u new survey should he nuuie, an.i the c<,urses and dis-
tance niserted accordingly. Tru., it i. that siuvrss an- e.s-
pensive, but lawsuits are ot'ten ruinous.
As in most cases the oldc.t liiie n.ust' be allowed, it oft,.,,
becomes a matter of n.uch importance to a.ceru.in the con.-
parat.vo ages of ditierent line.. An.ong the nunwrou- dif-
ficulties vy-h.ch the young Surveyor ha. to encounter, tins i.
about the greatest When several linos run frou. the .unl
mm forming with each other angles of considerable n,a..
the date ot then- favourite line, to determine which is th«
o dest IS often an exceedingly difficut task. lu a few in
stances the appearance of blazes on the troes-may 1 j
jome assistance. An appeal t^ the oldest inhabitants :' t^'
imu "",;"" P"""' 1 ^'" "••^^'"•^' ^"-•^•>' -• ^--- other
circumstances ..cquamted with the lines, is genoralk- the
most safe and satisfactory course to pursue. In d en
however, it is frequently difficult, and often utterlv i m ^ :
dete. """"''' '" '"^''^^'"» ""^^"-"^'^ ^"-" - -i-
Notwithstandiug the heavy responsibility which devolve*
on ho Surveyor, frequently he must depend, to n great ex
ent, upon his own judgment and discretion. I„ Tse cir
camstances an intimate and extensive knowledge of Mathe'
matical principles, and a general acquaintance tvith colla c
labyrinth, by suggesting to him numerous expedients whi<-h
-;^^ never occtir to the mind of . mere Lr;::^;-
1
4
' '' ,<•
ERRATA.
Page 7f», third line from tlie bottom— for '•' Sir William
MacLcan George Colebrooke, R. H. ^ 8lc., read Sii William
MacBeaii George Colcbiooke, K. H. &c.
Pag9 116, last line of tlic Calculation Ta!)!c, in the column
Half Departures — for "= 13.95 ' read — 13. OS,
Page 133, at the head— for <' Division of Land" read Lo-
cation of Land.
TRAVERSE TABLE,
CONTAINING TIIE
DIFFERENCE OF LATITUDE AND HALF DE-
PAIITUIJE,
TO EVEUY ^UAllTEU DECHliE OF TIIL COM VPS; THE DIS-
TANCE SUPPOSED 10 BE OWE CHAIN, OK FOUR RODS.
To find the Dijf. of La', and Dcp. by the following Table.
In tlic fi'-sit column tmder D. M. find the Degrees rncl
Minutes contained in the bearing; W) w line Avitli wliichj un-
der N. S. and Ft. W., you liavc the Diff. of Lut. and halt'
the D'^p. ior that hearing. ISIidtiply tliis Difl". of Lut. and
Dep. by the length of the Stationary Distance; and the pro-
duct will 1)0 the DiiT. of Lat. and half Dcp. for that line.
Thus, let the bearing be N. 10'^ 15' W., Distance 20 ch.
30 /. llcqnircd tlie Diif. of Lat. and Departure.
.9840 N.S. .0839 E.W.
20.30 20.30
295200
19G800
2(;G70
17780
1.9975200 Northin,?.
1.801(370 half Westing.
3.G09340 Westing.
N. 3. — In the following Table, if the Dep. be multiplied
by two, the Dei), and Diif. of Lat. will be the natural sine.-^
and co-.-^ines of the corresponding Courses, respectively; the
Radius being supposed to be one.
r
?MFFERENCn or LATITUDE AND DEPARTURE.
15
30
45
.8988
.8968
.8949
.892!)
i>j 90
,2210
,2230
,2250
5.
.(»
.99(32
.043(;
1(>.
.9()13
.1378
i27.
.8910
.2270
15
.99.58
.0457
15
.9{;00
.1399
15
.8890
.2239
,50
.9954
.0179
30
.9588
.1420
30
.8870
.2308
45
.09i;i .0501
45
.957«)
.1441,
45
.8850
.2323
C\
0;
15:
3(»i
M5|
7.1 01
i!n!
'30,
^15"
: ! 5;
'30
.!5!
1
9 JO!
ro,= 0,
.99451.0522
.99 10. or U
.993(>j.05{i6
.9931 L 0587
^91)25.01109
.9920!. On31
.99J4i.Oi352
.9'^0:!|.0ti74
'.1545'
29.
.1500
15
.15S'ji
30
.l()07i
45
j;^0
.9903]. oi;9(; !19.
.989!.;. 071 7
.9890|.0739
.98?'3 .0700
■38:7Tom2|:20,
.9870 .0801'!
.:'8!i3'.{;825
.9SJ5I.0847
.981S;.08i)8
.♦»-• IOJ.088:'
.'»^32j.OiJll
.9824'.0^;33
15
30
45
.8829
.8309
.8788
.8767
.8725
.8703
.8084
.2347
,23b"f)
,2335)
,24051
,2424
,-.;443
,2462
,2481
30.1
15
It'U
45
.8(;(io
.81)38
.St-IG
.8594
,2500
,2519
,2537
,2556
21.
1
15
.-JO
4 J
31.
.8571 .2575
15
.8549
.2594
30
.8527
.2612
451 .8503
,2031
32.
.8480
.2(i49
!5
.8457
.26(:8
30
.8431
. 'IbSO
^45
.8410'. 2705!
33.
■
15
30
'
45
34.
1 01
'15
30
i
45
35.
15
30
45
3(3.
15
30
45i
37.
15
30
1
4'5
38.
15
;}0 .
45 .
39.
.
15 .
30 .
i
-In
41.
15
30
45
^'
DIFFERENCF
• OP
LATITUDE AND
DEPARTt'RE,
ID. |.M.! iN.H. |lvVV.|iI).|M.| N.S. |E.W.||1). |M
• 1 N. «.
|E.VV.|
33.
(]
.8387
.2723
;44.
.7102
.3473
55.
f
..5731
.4095
■
10
.83()3
.2741
15
.7163
.3489
15
.5706
1.4108
3i
.833J)
.2759
30
.7132
.3.504
30
.5664
.4120
■ib
.8315
.8290
.'2718
'.2796
45
.7102
.3520
45
.5628
.4133
34.
45.
.7071
y r* '.> Tv
56.
.5592
.4155
If)
.8266
.2814
15
.7041
.3551
15
.5555
.4157
.id
.8241
.2832
30
.7009
.3.566
30
.5519
.4169
i
4b
.82K)
.2850
45
.6978
.3581
45
.5483
.4181
35.
.81SI1
.2667
46.
.694(i
.3596
'57.
.5446
.4193
lb
.816(i
.2885
15
.6915
.3612
15
.5408
.4205
30
.8141
.2903
1
30
.6883
.3627
1
30
.5373
.4216
45
.8116
.2921
1
1
45
.6852
.3642
45
.5336
.4228
3(3.
6
.80!.t0
.2939
147.
.6820
.3656
58.
.5299
.4240
15
.8064
.2956
15
.6788
.3671
15
.5269
.4251
30
.8058
.2974
30
.6756
.3686
30
.5225
.4263
45, .8012
.2991
45
.6724
.3701
45
.5188
.4274
37.
.71)8()
.3009
43.1
.6691
.3715 :59.
.5150
.4286
15
.7960
.3026
15
.6659
.3730
15
.5113
.4297
30
.7933
.3044
30
.6626
.3745
30
.5075
.4308
45
.7907
.3061]
45
.6593
.3759)
45
.5033
.4319
.4380
38.
.7880
.3078
49.
.6560
.3773
60.
.5000
15
.7853
.3095
15
.6527
.3788
15
.4962
.4341
30
.782(i
.3112
30
.6494
.3802
30
.4924
.4352
45
.7799
.3129
35
.6461
.3816
45
.4886
.4362
74373
'69.
.7771
.3146
50.
.6428
.3830
01.
.4848
lo
.7744
.3163
15
.6394
.3844
15
.4810
.4,383
30
.7716
.3180
30
.6361
..3858
30
.4771
.4394
1 1
i
45
.7(i88
.3197
45
.6327
.3872
45
.4733
.4405
■10.
.7660
.3214
51.
.6293
.3885
62.
.4695
.4415
lo
.7632
.3230
15
.6259
.3899
15
.4656
.4425
30
.7601
.3217
30
.6225
.3913
30
.4617
.4435
45
.7575 .3263}
45
.6191
.3926
45
.4579
.4445
■u.
.7547
.3280
52.
.6157
.3940
m.
.4540
.4455
lb
.7518
.S!i9,6
15
.6122
.3!>53
15
.4501
.4465
30
.7489
.3313
30
.6087
.396i]
30
.4462
.4475
4'2?
45
.7460
.3329
45
.6052
.3980
45
.4423
.4485
.7431
.3345
53.
.6018
.39931
64.
.4383
4494
15
.7402
.3361
15
.5983
.40o;>|
15
.4344
4503
'id
.7373
.3378
30
.5948
40191
^0
.4305.4513
15
.7313
S394I
15
.5912
4032,
i
65.
15
.4265 .4522
•13.1
.73 131.. 34 10
54.
.5878{
.4045
.4226
4532'
15
. 7232 j. 34 20
15
.58421
.40.58
15 .4186
1541
30
.72.54 ,3441|
30
.5807
4070
JO .4140
4550
45
.7224
.3457
15
.5771
4083;
1
15
.4177|.
4559
4" 'I
I-
■
^B
!
H^^^B
DJFFEUEKCR OF LATI TUDE A ND DEPARTURE.
1E.\V.||1)-|IV».| N. S. jE-VV-l
fD:j¥p. sT ii:.w.! |D.l->i.|N.s.
f)G.
15
30
45
.40G7
.,4027
.3087
.3947
G7.
or .3907
15
30
45
.38G7
.38i>7
.373G
.4507
.157G
.4550
.4594
.4011
.401!)
.4027
74.
75.
15
30
45
~0
15
30
45
,275(5
.2714
.27G2
.2G30
^2588
.2540
.2504
.2401
.4800
.4812
.4818
.4823
4829
4835
4840
4840
82.
83.
15
30
45
~d
15
30
45
.1392
.1348
.1305
.120 2
T2I8
.1175
.1132
.1088
.4951
.4954
.4957
.4900
.4902
.4905
.4908
.4970
08.
15
30
45
.3740
.3705
.3005
.3024
4030
4044
,4052
,4u0t»
70.
15
30
15
.2419
.2377
.2344
.2292
,4851
,4850
,4802
.4807
84.
15
30
45
.1045
.1002
.0958
.0915
.4972
.4974
.4977
.4979
09.
15
30
45
.3581
.3543
70.
71.
15
30
45
.4008
.407(i
.ooy^ .4( 8.3
_^GJU001_
.4!)98
.»
.3420
.3379
.3338
o
2207
15
30
.3255
211
f.i
.47(H)
.4713
.4720
.4727
.4734
77.1 0'
15
45
"0
15
3fi
73.
45
.3173J.4741
.3131 .474S
79.
15
45
.2249
.2207
.2104
.2122
T2079'
.203(;
.1993
.1951
TTlK>S'.4908
.18051.4912
.18221.4910
15
45
.3090
3007
,4755
,4'
,47;
H'^Z
.29C^5l.4775
50.
15
73.
15
30
45
29241.4781
,2882'. 4 /.'^S|
.2840|.47it4i
.2798'.4y^'0
15
30
.4872
.4870
.4881
.4880
^4890
.4895
.4899
.4904
85.
SO.
15
30
45
15
30
45
.0871
.0820
.078'!
.0741
.0097
.0054
.0010
.0500
.4981
.4983
.4884
.4980
4988
,4989
,4990
.4992
4920
.177!
.l73iJl.4!i24|
87.
.0.523
.4993
15
.0480
.4994
30
.0430
.4995
45
.0392 .4590
.1
.':oa
,4927
30 .1050.4931
45 .1007 .49.35
88.
15
30
45
.15(^4
.1521
.5 478
.14
o'-i
4938
,4941
,4945
.4948
89. 1
15
30
45
.0349
.0305
.0202
.0218
T0174
.0131
.0087
.0043
4997
49!I7
4998
4998,
4999
,4999
, 5000
.5000
90.1 0! .0000;. 5000
S. lE.VV.
32.
4951
48 .
4954
[15 .
4957
G2.
49G0
18
49G2
75
49G5
3:1
4908
8S
.4970
Tb
.4972
02
.4974
58
.4977
15
.4979
71
.4981
20
.4983
•m
.4984
r4i
.4980
J97
.4988
J51
.4989
ilO
.4990
3G()
.4992
323
.4993
ISO
.4994
43G
.4995
3i'2 .4596
34f
.4997
305
. 49!I7
■2{)-i
! .4998
21b
) .4998;
17^
I .4999
13]
.4999
08'
7 , 5000
104
} .5000
)000,.5000|
Fr;i L-j
T'l.
S /
\ ■'''(' iJt.
Tfi -• --ir —
f'M;
A-
E -i
L\,
/
V M;iMroe
n
■T.
N
/
.-^a?iw
f! I
111
I! :
)!)
,\
>A'
C^
11; 5^
/.,>V
'■ I q J.
xU
a*.
\':i ;
11
'.i I
.1"
* *
#
m
nk--
f
':'W^
i
I' f '• '
k
ill
HI V
A
m
•ill
'V
. > \
'V! 1
i .,i
/
iu/ 77
^^ *
3L.
li:^^ fji
ill
It I-
rv'
^f
'''/. •
-/.
::■/'
,/^
\
Jl.
i^.«.'
^i
rx:
)■(.
j;
/' - —
4^^.
T~
f--
'■<'l ,'Mi
-^
J'l. 1.
/'<) ,^»r,
^1'
~j—~
1 K' ' - a
\
)(<i ,'n
— i'- /
\
\
— ^H
T''<;.!),V.
'•'"; ,'/;>.
,/•'
>
'"',';■''
u'
■ V
y^
'J
JJ
/•M/'t'.>.
(r'
- -J
1-
1. 1:
K . — .
'•'mj. irv, ■'
It" — ^ „___,
W)
^A^.A
I
i
W '■
1
y 1
E—-
*^
iT'i
\ \
I i A
^ 1
1 ^'
I •-
D
L..
Si (ill' .'li'' (Ji.((i tin iiii'l
t'lj IIV ,
■ S' ruli- i'.'i rh tr n n m rh
— -'H
T Li,. ii>. lf)■<l-^ atutiT.iC
i
v..
'•-.'i
C
T. '
/<r
~- c
I'll- Hi.
I''ii //.;'
1,
if;
D^.-
!■ i>i- n-'^.
\.
.1* ir
\A-
n*
T'ni n i,.
F
A -I
'c /-
t'nj.U!
yri,
.1 (It:
A'
T^'.f/ l]<i.
1}
A
A
Fuj.ilS
i-Ki.iri.
B
r
i:
t7
.-_ii
rui,.i
U i
A I —
A-
<- F,
1^-
^■1
%
■!J
■f
■i
!'!
.H'
/
,1 -—
/•'''-;. //';,
FtijiV^i)
J I
/
//
\
r
X \
r
w
/ \
/
/
1 Fiy. i V.V
I \
'','/• ' '-\
t-
V'-
\\
\\
I
ll
\
I ff
6"
B
Ag
'■■'//■ ''.':),
f-^!) ivrj.
A
\S
— f — -»--
-3}
V
J
Vfl
7?-
^_--
-r--~>^
/)-=^-
/ 4
■ V. ,
C
P=!
/•Vf/.xpfi,
1 :-f
I . ■ '
i- I
-^L .._ .
' I !
t i
I I
/
— i~
:_..„_ J
M
K
II
c
w
I' "I- I \',S
\
7{
T.,h
' y ftlCW: .Vt'.iu
ly
/
I '
5V"
? ^~
irt^-
ni.
//I— =
V
^^*-li^^
vV
If,
.».-'.
y'i
/•
^
ft
I
11
ir I
I
PI
i^.;r
r
:,
If
i <3'
A<~-
VI. r.)
trN^
.^-
TO
li
I ;•
IT
rvh.Try Mev '' :At<iL' 0"
A^
Vii
i
1 1
i
I ;
•n^
^L
.4 _^i!ii(
/{-
Viij I',:'.
hnilli'i
l-'llh I '' I
A
l-iLj/li,'.!
i^lJ:50, ff
I'U h t.y JitfX if„IM>l
N.
1
o"
2
0.
3
0.
4
0.
_5
0.
6
0.
7
0.
8
0.
9
0.
10
1.
11
12
13
14
15
16
17
18
19
-
20
21
22
23
24
25
A TABLE
OP
LOGARITHMS OF NUMBERS
FROM 1 O 10,000.
N.
1
Lo?.
N.
26
Log.
N.
51
Loi?.
N.
76
liOg.
0.000000
1.414973
1.707570
1.880814
2
0.301030
27
1.431364
52
1.716003
77
1.886491
3
0.477121
28
1.447158
53
1.724276
78
1.892095
4
0.G02060
29
1.462398
54
1.732394
79
1.897627
5
0.698970
30
1.477121
55
1.740363
80
1.903090
6
0.778151
31
1.491362
56
1.748188
81
1.908485
7
0.845098
32
1.505150
57
1.755875
82
1.913814
8
0.903090
33
1.518514
58
1.763428
83
1.919078
9
0.954243
34
1.531479
59
1.770852
84
1 . 924279
U)
li
1.000000
1.041393
35
36
1.544068
1.556303
60
61
1.778151
1.785330
85
86
1.929419
1.934498
12
1.079181
37
1.568202
62
1.792392
87
1.939519
13
1.113943
38
1.579784
63
1.799341
88
1.944483
14
1.146128
39
1.591065
64
1.806180
89
1.949390
15
1.176091
40
1.602060
65
1.812913
90
1.954243
16
1.204120
41
1.612784
66
1.819544
91
1.959041
17
1.230449
42
1.623249
67
1.826075
92
1.963788
18
1.255273
43
1.633468
68
1.832509
93
1.968483
19
1.278754
44
1.643453
69
1.8.38849
94
1.973128
"■M
1.301030
45
1.653213
70
1.845098
95
1.977724
21
1.322219
46
1.662758
71
1.851258
96
1.982271
22
1.342423
47
1.672098
72
1.857333
97
1.986772
23
1.361728
48
1.681241
73
1.863323
98
1.991226
24
1.380211
49
1.690196
74
1.869232
99
1.995635
2b
1.397940
50
1.698970
75
1.875061
100
2.000000
N. B. In the following table, in the last nine columns of
each page, where th« first or leading figures change from Q's
to O's, points or dots are introduced instead of the O's through
the rest of the line, to catch the eye, and to indicate thai from
Ihence the annexed first two figures of the Logarithm in the
second column stand in the ne.\t lower line.
A TABLE OF LOGARITHMS FROM 1 TO 10,000.
'"'I
S
N.
1 |l|2|3i4i5i6|7|8|9|D. 1
100
000000
0434
08()8
1301
1734
2166
2.'i;'S. 3029
3461
3S91
432
101
4321
4751
5181
5609
6038
6406
6894
"321
7748
8174
428
103
8600
9026
9451
9876
.300
.724
1147
1570
1993
2415
424
103
012837
3259
3680
4100
4521
4940
5360
5779
0197
6616
419
104
7033
7451
7868
8284
8700
9116
9532
9947
.361
.775
416
105
021189
1603
2016
2428
2841
3252
3664
4075
44S6
4896
412
106
5306
5715
6125
6533
6942
7350
7757
8164
8571
8978
408
107
9384
9789
.195
.600
1004
1408
1812
2216
2619
3021
404
108
033424
3826
4227
4628
5029
5430
5830
6230
6629
7028
400
109
110
7426
7825
8223
2182
8620
2576
9017
291)9
9414
3302
9811
3755
.207
.602
4540
. 998
4932
390
3i)3
041393
1787
4148
HI
5323
5714
6105
6495
6885
7275
7664
8053
8442
8830
389
112
9218
9606
9993
.380
.766
1153
1538 1924
2309
2694
386
113
053078
3163
3816
4230
4613
4996
5378 5760
6142
6524
382
114
6905
7286
7666
8016
8426
8805
9185 9563
9942
. 320
379
11 -J
000698
1075
1452
1829
2206
2582
2958
3333
3709
4083
376
116
4458
4832
5206
5580
5953
6326
6699
7071
7443
7815
372
117
8186
8557
8928
9298
9668
..38
.407
.770
1 145
1514
369
118
071882
225(t
2617
2985
3352
3718
4085
4451
4816
5182
366
119
120
5547
5912
6276
9904
6640
.266
7004
.626
7368
,987
7731
1347
8994
1707
8457
8819
363
360
079181
9543
2067
2426
121
082785
3144
3503
3861
4219
1576
4934
.5291
5647
6004
357
122
6360
6716
7071
7126
778 1
8136 8490
8845
9198
9552
355
123
9905
.258
.611
.963
1315
1667 2018
2370
2721
3071
351
124 0931221
3772
4122
4471
4820
5169 5518
586()
6215
6562
349
125
6910
7257
7604
7951
8298
8644, 8990
9335
9681
..26
3''e
126
100371
0715
1059
1403
1747
2091
2434
2777
3119
3462
343
127
3 804
4146
4487
4828
5169
5510, 5851
6191
6531
6871
340
128
7210
7549
7888
8227
8565
8903' 9241
9579
9916
.253
338
129
130
110590
0926
1263
4611
1599
4i)14
1934
5278
2270
2605
5943
2940
6276
3275
(5608
3(509
6940
335
333
113943
4277
5611
131
72 n
7603
7934
821)5
,8595
8926
9256
958()
9915
.245
330
132
120574
09()3
1231
1560
.1888
2216
2544
2871
3198
3525
328
133
3852
4178
4504
4830
*5156
548 1
5806
6131
645(5
6781
325
134
7105
7429
7753
8076
8399
8722
9045
9368
9690
..12
323
135
130331
0655
0977
1298
1619
1939
2260
2580
2900
3219
321
136
3539
3858
4177
4496
4814
5133
545 1
5769
6086
6403
318
137
6721
7037
7354
7671
7987
8303' 8618
8934
9249
95(54
315
138
9879
.194
. 508
.8-.2
1136
1450; 1763
2076
2389
2702
314
139
140
143015
3327
6138
3639
3951
7058
4263
73(57
457114885
519(5
8291
5507
8603
5818
8911
311
309
146128
6748
7676
7985
141
9219
9527
9S35
. 142
.449
. 756
1063
1370
1676
1982
307
142
152288
2594
2900
3205
3510
3815
4120
4421
4728
5032
305
143
5336
5640
5943
6246
6549
6852
7154
7457
7759
806 1
303
144
8362
8664
8965
9266
9567
9868
.1(58
.469
.769
10(>8
301
145
161368
1667
1967
2266
2564
2863
3161
3460
3758
4055
299
146
4353
4650
4947
5244
5541
5838
6134
6430
6726
7022
297
147
73 J 7
7613
7908
8203
8497
8792
9086
9380
9674
9968
295
148
170262
0555
0848
1141
1431
1726
2019
2311
2603
2895
293
149
150
3186
3478
6381
3769
4060
6959
4351
7248
4641
4932
5222
5512
5802
8689
291
289
176091
6670
753(5
7825
8113
8401
151
8977
9264
9552
9839
.126
.413
.699
.985
1272
1.558
287
152
181844
2129
2415
2700
2985
3270
3555
3839
4123
4407
285
153
4691
4975
5259
5542
5825
6108
6391
6674
6956
7239
283
154
7521
7803
8084
8366
8647
8938
9'2(IM
9490
9771
..51
'281
155
190332
0612
0892
1171
1451
1730
2010
2289
2567
2846
279
156
3125
3403
368 1
3959
4237
4514
4792
5069
5346
5623
278
157
5899
6176
6453
6729
7005
7281
7556
7832
8107
8382
276
158
8657
8932
9206
9481
9755
..29
.303
.577
.850
1124
274
159
201397
1670
19'13
2216
2488
2761
3033
3305
3577
38481 272 1
.N. 1 1 1 1 2 1 3 1 4 i 5 1 6 i 7 1 8 1 9 1 D. 1
A TABLE OP LOGARITHMS FROM 1 TO 10,000.
9
D.
S[H
4:32
174
428
415
424
616
419
775
416
896
412
978
408
O'Jl
404
01^8
400
998
396
iVM
;393
8:30
389
(594
386
5-^4
382
•S-20
379
08:3
376
815
372
514
369
182
366
819
363
4:^6
360
004
357
552
355
071
351
562
349
.26
im
462
343
871
340
25:3
338
609
335
940
333
245
330
525
328
781
325
.12
323
219
321
40:3
318
564
315
702
314
818
311
911
30;)
982
:}()7
0:52
305
06 1
3U3
068
301
055
29;>
022
297
968
295
895
293
802
291
689
289
558
287
407
285
2:39
283
.51
'281
846
279
)623
278
!:382
276
1124
274
5848
272
9
D.
N. I 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 D. 1
160
204120
4391
466.1
1 4934
520']
t 54751 574f
) 60 le
. 628(
> 6.55e
> 271
161
6826
709(]
. 736!;
7634
7904
I 81 7.'^
t 8441
871f
> 897t
1 9247
' 269
162
951.^
978L
1 ..51
.319
.58f
» .85S
1 112!
138?
1 16.54
t 1921
267
163
212188
2454
c 272(J
2986
325S
! 351fe
378:^
I 404S
4314
4579
266
164
4844
510ii
5:37;^
5638
5902
, 6l6r
643f
1 66f>4
- 6957
' 7221
264
165
7484
7747
' 801(J
8273
853P
8798
906C
9323
958.1
984G
262
166
220108
037(J
0631
0892
11. 5L
1414
167.'5
1936
2I9f
2456
261
167
2716
29 7()
323fi
3496
3755
4015
4274
453.1
4792
.5051
259
168
5309
5568
582(j
6084
6342
660U
6858
7115
7372
7630
2.58
169
170
7887
8144
0704
84O0
0960
8657
1215
891:]
1470
9170
1724
9420
1979
9682
2234
993S
2488
.193
2742
2,56
2.54
230449
171
2996
3250
3504
3757
4011
4264
4517
4770
5023
5276
253
172
5528
6781
6033
6285
6537
6789
7041
7292
7544
7795
252
173
8046
8297
8548
8799
9049
9299
9550
9800
...50
.:300
2,50
174
240549
0799
1048
1297
1546
1795
2044
2293
2.541
2790
249
175
3038
3286
35:34
3782
4030
4277
4525
4772
50 1 9
5266
248
176
6513
5759 6006
6252
6499
6745
6991
7237
7482
7728
246
177
7973
8219 8464 8709
8<i54
9198
9443
9687
9932
.176
246
178
250420
066410908 1151
1395
1638
1881
2125
2368
2610
243
179
180
2853
3096
5Q14
33,381 3580
3822
6237
4064
6477
4306
6718
4548
69.58
4790
7198
5031
7439
242
241
255273
5755
5996
181
7679
7918
8158
8398
8637
8877
9116
9355
9594
98:33
239
182
260071
0310
0548
0787
1025
1263
1501
1739
1976
2214
238
183
2451
2688
2925
3162
3399
3636
3873
4109
4346
45 S 2
?-37
184
4818
5054
5290
5525
5761
5996
6232
6467
6702
6937
235
185
7172
7406
7641
7875
8110
8344
8578
8812
9046
9279
234
186
9513
9746
9980
.213
.446
.679
.912
1144
1377
1609
233
187
271842
2074
2306
2538
2770
3001
3233
3464
3696
3927
232
188
4158
4389
4620
4850
50 'J 1
5311
5542
5772
6002
6232
230
189
190
6462
6692
8982
6921
9211
7151
9439
7380
9667
7609
9895
7838
.123
8067
8296
.578
8525
.806
229
228
278754
.351
191
281033
1261
1488
1715
1942
2169
2396
2622
2849
3075
227
192
3301
3527
3753
3979
4205
4431
4656
4882
5107
5332
226
193
5557
5782
6007
6232
6456
6681
6905
7130
73.54
7578
225
194
7802
8026
8249
8473
8696
8920
9143
9366
9.589
9812
223
195
290035
0257
0480
0702
0925
1147
1369
1,591
1813
2034
222
196
2256
2478
2699
2920
3141
3363
3584
3804
4025
4246
221
197
4466
4687
490?
5127
5347
5567
5787
6007
6226
6446
220
198
6665
6884
7104
7323
7542
7761
7979
8198
8416
8635
219
199
200
8853
9071
1247
9289
1464
9507
1681
9725
1898
9943
2114
.161
2331
.378
2.547
.595
2764
.813
2980
218
217
3010:10
201
3196
;i4l2
3628
3844
4059
4275
449 1
4706
4921
5136
216
202
5:351
5566
5781
5996
6211
6425
6639
6854
7068
7282
215
203
7496
7710
7924
8137
8351
8564
8778
8991
9204
9417
213
204
9630
9843
..56
.268
.481
.693
.906
1118
1330
1,542
212
205
311754
1966
2177
2389
2600
2812
;3023
32.34
3445
3C56
211
206
:3867
4078
4289
4499
4710
4920
5 1:30
5340
.5551
5760
210
207
6970
6180
6390
6599
6809
7018
7227
7436
764.fi
7854
209
208
8063
8272
8481
8689
8898
9106
9314
9522
9730
9938
208
209
210
320146
0354
2426
0562
2633
0769
2839
0977
3046
1184
3252
1391
.3458
1598
3665
1805
3871
2012
4077
207
206
322219
211
4282
4488
4694
4899
5105
5310
5516
5721
5926
6131
205
212
6336
6541
6745
6950
7155
7359
7563
7767
7972
8176
204
213
8380
8583
8787
8991
9194
9398
9601
9805
...8
.211
203
214
3.30414
0617
0819
1022
1225
1427
1630
1832
2034
2236
202
215
2438
2640
2842
Ofj/tA \
3248
3447
3649
3850
4051
4253
202
216
4454
46551
4856
5057
5257
5458
5658
5859
6059
6260
201
217
6460
()660
6860
7060
7260
7459
7659
7858
8058
8257
200
218
8456
86.56
8855
9054
9253
9451
9650
9849
.,47
.246
199
219
340444 0642'
0841
1039 I337I
14:15 1632'
18:30 2028'
2225
198
^'- 1 <1 t Ul 2 1 3 4 1 5 1 6 7 1 8 1 9 D. 1
4
' I
■ft'
" it
8' .
11
r:l-
4
A TABLE OF LOGARITHMS FROM 1 TO 10,000.
N.
1 |l(2|3 4|5|6|7|8|9|D. 1
220
342423
2620
2817
3014
3212
3409
3606 3802
3999
41961 197 1
221
4392
4589
4785
4981
5178
5374
5670
6766
5962
6187
196
222
6353
6549
6744
6939
7135
7330
7525
7720
7915
8110
195
223
8305
8500
8694
8889
9083
9278
9472
9666
9860
...54
194
224
350248
0442
0636
0829
1023
1216
1410
1603
1796
1989
193
225
2183
2375
2568
2761
2954
3147
3339
3532
372^1
3916
193
226
4108
4301
4493
4685
4876
5068
6260
6452
6643
5834
192
227
6026
6217
6408
6599
6790
6981
7172
7363
7554
7744
191
228
7935
8125
8316
8.506
8696
8886
9076
9266
9456
9646
190
22U
23U
9835
..25
1917
.215
2105
.404
2294
.593
2482
.783
2671
.972
2859
1161
3048
1350
3236
1539
3424
189
188
361728
231
3612
3800
3988
4176
4363
4551
4739
4926
5113
.5301
188
232
6488
5675
5862
6049
6236
6423
6610
6796
6983
7169
187
233
7356
7542
7729
7915
8101
8287
8473
86.59
8845
9030
186
234
9216
9401
9587
9772
9958
.143
.328
.613
.698
.883
185
235
371068
1253
1437
1622
1806
1991
2175
2360
2544
2728
184
236
2912
3096
3280
3464
3047
3831
4015
4199
4383
4565
184
237
4748
4932
5115
5298
5481
6664
6846
6029
6212
6394
183
238
6577
6759
6942
7124
7306
7488
7670
7852
8034
8216
182
239
240
8398
8580
0392
8761
0573
8943
0754
9124
0934
9306
1116
9487
1296
9668
1476
9849
1656
..30
183V
181
181
380211
241
2017
2197
2377
2557
2737
2917
3097
327?
3466
3636
180
242
3815
3995
4174
4353
4533
4712
4891
6070
5249
6423
179
243
5606
6785
5964
6142
6321
6499
6677
6856
7034
7212
178
244
7390
7568
7746
7923
8101
8279
8466
8634
8811
8989
178
245
9166
9343
9520
9698
9875
..61
.228
.405
.582
.759
177
246
390935
1112
1288
1464
1641
1817
1993
2169
2345
2.521
176
247
2697
2873
3048
3224
3400
3575
3761
3926
4101
4277
176
243
4452
4627
4802
4977
5152
5326
6501
5676
58.50
6025
175
249
6199
6374
6548
6722
6896
7071
7245
7419
7592
7766
174
250
397940
8114
8287
8401
8634
8808
8981
9154
9328
9.501
173
251
9674
9847
..20
.192
.365
.538
.711
.883
1056
1228
173
252
401401
1573
1745
1917
2089
2261
2433
2605
2777
2949
172
253
3121
3292
3464
3635
3807
3978
4149
4320
4492
4663
171
254
4834
6005
5176
5346
.55-7
5688
5858
6029
6199
6370
171
255
6540
6710
6881
7051
7221
7391
7561
7731
7901
8070
170
258
8240
8410
8579
8749
8918
9087
9257
9426
9595
9764
169
257
9933
.102
.271
.440
.609
.777
.946
1114
1283
1451
169
258
411620
1788
1956
2124
2293
2461
2629
2796
2964
3132
168
259
260
3300
3467
5140
3635
5307
3803
5474
3970
6641
4 J 37
5808
4305
5974
4472
6141
4639
6308
4806
167
167
414973
6474
261
6641
6807
6973
7139
7306
7472
7638
7804
7970
8135
166
262
8301
8467
8633
8798
8964
9129
9295
9460
9625
9791
165
263
9956
.121
.286
.451
.616
.781
.945
lUO
1275
1439
165
264
421604
1788
1933
2097
2261
2426
2590
2754
2918
3082
164
265
3246
3410
3574
3737
3901
4065
4228
4392
4555
4718
164
266
4882
5045
5208
5371
5534
.5697
5860
6023
6186
6349
163
267
6511
6674
6S36
6999
7161
7324
7486
7643
7811
7973
162
268
8135
8297
8459
8621
8783
8944
9106
9268
9429
9591
162
269
270
9752
9914
1525
..75
1685
.236
1846
.398
2007
.559
2167
.720
2328
.881
2488
1042
2649
1203
2809
161
161
431364
2/1
2969
3130
3290
3450
3610
3770
39.30
4090
4249
4409
160
2V2
4569
4729
4888
5048
5207
.5367
5526
5685
5844
6004
159
273
6163
6322
6481
6640
6798
6957
7116
7276
7433
7592
159
274
7751
7909
8067
8226
8384
8542
8701
8859
9017
9175
158
275
9333
9491
9648
9806
9964
.122
.279
.437
.594
.7,52
1.58
276
440909
I0f)6
1224
1381
1538
1696
1852
2009
2166
oooo
157
277
2480
2637
2793
2950
3106
3263
3419
3576 i 3732
3889
157
278
4045
4201
4357
4513
4669
4825
49811513715293
5449
156
279
5604
5760
5915
60V\
62261 6382^ 6537 6692' 6848
7003
1.55
N.
1 |1|2|3|4|5|6|7|8|9|D. 1
N,
280
44
281
282
45
283
284
285
286
287
288
289
46
290
4(]
291
292
293
294
295
296
47
297
298
299
300
47
301
302
46
303
304
305
306
.307
308
309
310
4S
311
312
313
314
315
316
317
5C
318
319
320
6C
321
322
323
324
51
325
326
327
328
329
.330
61
331
332
52
333
334
335
336
337
338
339
53
N.
i
— -'.Z'S-.'Wfz
Id. I
6
197
7
196
195
4
194
9
193
6
193
4
192
4
191
6
190
9
189
4
188
1
188
9
187
186
3
185
8
184
5
184
4
183
6
182
181
V
181
6
180
8
179
2
178
9
178
9
177
,1
176
7
176
,5
175
ifi
174
11
173
,8
173
t9
172
.3
171
'0
171
'0
170
.4
169
• 1
169
!2
168
16
167
'4
167
(5
166
H
165
19
165
12
164
8
164
^9
163
'3
162
)1
162
)3
161
)9
161
)9
160
)4
159
)2
159
^5
158
)2
158
157
!9
157
[9
156
)3
155
ID. 1
A TAHLE OF LOGARITHMS FHOM 1
TO 10,000.
6
N,
|l|2|3i4l6|6|7!8|9|D. 1
280
447158
7;h3
7468
7623
7778
7933
8088
8242
8397
8552
1.55
281
8706
8861
9015
9170
9324
9478
9633
9787
9941
..95
154
282
450249
0403
0557
0711
0865
1018
1172
1326
1479
1633
1.54
283
1786
1940
2093
2247
2400
2553
2700
2859
3012
3165
153
284
3318
3471
3624
3777
3930
4082
4235
4387
4.540
4692
1.53
285
4845
4997
5150
5302
54.54
5606
5758
5910
6062
6214
1.52
286
6366
6518
6670
6821
6973
7125
7276
7428
7579
7731
1.52
287
7882
8033
8184
8336
8487
8638
8789
8940
9091
9242
151
288
9392
9543
9694
9845
9995
.146
.296
.447
.597
.748
151
289
460898
1048
1198
1348
1499
1649
1799
1948
2098
2248
1.50
290
462398
2548
2697
2847
2997
3146
3296
3445
3594
3744
1.50
291
8893
4042
4191
4340
4490
4639
4788
4936
5085
5234
149
292
6383
5532
5680
5829
5977
6126
6274
6423
6571
6719
149
293
6868
7016
7164
7312
7460
7608
7756
7904
8052
8200
148
294
8347
8495
8643
8790
8938
9085
9233
9380
9527
9675
148
295
9822
9969
.116
.263
.410
.557
.704
.851
.998
1145
147
296
471292
1438
1585
1732
1878
2025
2171
2318
2464
2610
146
297
2756
2903
3049
3195
3341
3487
3633
3779
3925
4071
146
298
4216
4362
4508
4653
4799
4944
5090
5235
5381
5526
146
299
300
5671
477121
5816
7266
5962
7411
6107
7555
6252
7700
6397
6542
7989
6687
8133
6832
8278
6976
8422
145
145
7844
301
8566
8711
8855
8999
9143
9287
9431
9575
9719
9863
144
302
480007
0151
0294
0438
0582
0725
0869
1012
1156
!299
144
303
1413
1586
1729
1872
2016
2159
2302
2445
2588
2731
143
304
2874
3016
3159
3302
3445
3587
3730
3872
4015
41.57
143
305
4300
4442
4585
4727
4869
5011
51.53
5295
.5437
5579
142
306
5721
5863
6005
6147
6289
6430
6572
6714
6855
6997
i42
31)7
7138
7280
7421
7563
7704
7845
7986
8127
8269
8410
141
308
8551
8692
8833
8974
9114
9255
9396
9537
9677
9818
141
309
310
9958
..99
1502
.239
1642
.380
.520
1922
.661
2062
.801
2201
.941
2341
1081
2481
1222
2621
140
140
491362
1782
311
'^-760
2900
3040
3179
3319
3458
.3597
3737
3876
4015
139
312
4155
4294
4433
4572
4711
4850
4989
5128
5267
5406
139
313
5544
5683
5822
5960
6099
6238
6376
6515
66.53
6791
1.39
314
6930
7068
7206
7344
7483
7621
7759
7897
80.35
8173
138
315
8311
8448
8586
8724
8862
8999
9137
9275
9412
9550
138
316
9687
9824
9962
..99
.236
.374
.511
.648
.785
. 922
137
317
501059
1196
1333
1470
1607
1744
1880
2017
21.54
2291
137
318
2427
2564
2700
2837
2973
3109
3246
3382
3518
3655
136
319
3791
3927
4063
4199
4335
4471
4607
4743
4878
5014
136
320
505150
5286
5421
5557
.5693
5828
5964
6099
6234
6370
1.36
321
6505
6640
6776
6911
7046
7181
7316
7451
7586
7721
135
322
7856
7991
8126
8260
8395
8530
8664
8799
8934
9068
135
323
9203
9337
9471
9606
9740
9874
...9
.143
.277
.411
134
324
510545
0679
0813
0947
1081
1215
1.349
1482
1616
17.50
1.34
325
1883
2017
2151
2284
2418
2551
2684
2818
2951
3084
133
326
3218
3351
3484
3617
3750
3883
4016
4149
4282
^'114
133
327
4548
4681
4813
4946
5079
.5211
5344
5476
5609
5741
133
328
5874
6006
6139
6271
64Q3
6535
6668
6800
6932
7064
1.32
329
330
7196
7328
8640
7460
8777
7592
8909
7724
9040
7855
9171
7987
9303
8119
9434
8251
9566
8382
9697
132
131
518514
331
9828
9959
..90
.221
.353
.484
.615
745
.876
1007
131
332
521138
1269
1400
1530
1661
1792
1922
2053
2183
2314
131
333
2444
2575
2705
2835
2966
3096
3226
3356
3486
36 1 6
130
334
3746
3876
4006
4136
4266
4396
4526
4656
4785
491.'-^
130
335
50 15
5174
5304
5l;54
5563
5693.
5H22
.5951
nop. 1
6VMI
]29
336
6339
6469
6598
6727
6856
6985
7114
7243
7372
75(4
1 129
337
7630
77.'i9
7888
8016
8145
8274
8402
8531
8660
87.SH' 129
338
8917
9045
9174
9302
9430
9559
96M7
9815
9943
..721 128
339
N.
530200 0328
0456
0.>S4i 0712' 0840i 0968' 1096
1223
13.'; I
' 128
1 1 1 2 1 3 i 4 1 5 1 r, 1 7 1 8 ! M
1 I).
13
il
' t
I '
c
A TABLE OP LCdARITHAlS FliOM 1 10 10,000.
N. I I 1 I 2 I 3 I 4 I 5 I 6 I 7 I 8 I 9 I U.
340
311
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
~n"."
531479
2754
4026
5294
6558
7819
9076
540329
1579
2825
544068
5307
6543
7775
9003
550228
1450
2668
3883
5094
556303
7507
8709
9907
561101
2293
3481
4666
5848
7026
568202
9374
570543
1709
2872
4031
5188
6341
7492
8639
579784
580925
2063
3199
4331
5461
6587
7711
8832
9950
591065
2177
3286
4393
5496
6597
7695
8791
9883
6()()!»73
1607
2882
4153
5421
6685
7945
9202
0455
1704
2950
4192
6431
6666
7898
9126
0351
1572
2790
4004
5215
6433
7627
8829
..26
1221
2412
3600
4784
5966
7144
8319
9491
0660
1825
2988
4147
5303
6457
7607
8754
9898
1039
2177
3312
4444
5574
6700
7823
8944
^._61
1176
2288
3397
4503
5606
670';
7805
8000
9992
1082
1734
3009
4280
5547
6011
8071
9327
0580
1829
3074
4316
5555
6789
8021
9249
0473
1694
2911
4126
6336
6544
7748
8948
.146
1340
2531
3718
4903
6084
7262
8436
9608
0776
1942
3104
4263
5419
6572
7722
8868
..12
1153
2-291
3426
4557
5686
6812
7935
9056
.173
1287
2399
3508
4614
5717
6ft 1 "
7914
9009
.101
1191
1862
3136
4407
5674
6937
8197
9452
0705
19,53
3199
4440
5678
6913
8144
9371
0595
1816
3033
4247
5457
6664
7868
9068
.265
1459
2650
3837
5021
6202
7379
85.54
9725
0893
2058
3220
4379
5534
6687
7836
8983
.126
1287
2404
3539
4670
5799
6925
8047
9167
.284
1399
2510
3018
4724
.5827
6927
8024
9119
.210
1299
1990
3264
4534
5800
7063
8322
9578
0830
2078
3323
4564
5802
7036
8267
9494
0717
1938
31.55
4368
5578
6785
7988
9188
.385
1578
2769
3955
5139
6320
7497
8671
9842
1010
2174
3336
4494
56,50
6802
7951
9097
.241
1381
2518
3652
4783
5912
7037
8160
9279
.396
1510
2621
3729
4834
.5937
7037
9228
.319
1408
2117
3391
4661
5927
7189
8448
9703
0955
2203
3447
4688
.5925
71.59
8389
9616
0840
2060
3276
4489
6699
6905
8108
9308
.604
1698
'3887
4074
.5257
6437
7614
8788
9959
1126
2291
3452
4610
.5765
6917
8006
9212
.355
1495
2631
3765
4896
6024
7149
8272
9391
..507
1621
2732
3840
4945
6047
7146
8243
9337
.428
1517
2245
3518
4787
6053
7315
8574
9829
1080
2327
3571
4812
6049
7282
8512
9739
0962
2181
3398
4610
5820
7026
8228
9428
.624
1817
3006
4192
5376
0555
7732
8905
..76
1243
2407
3568
4726
5880
7032
8181
9326
.469
1608
2745
3879
5009
6137
7262
8384
9503
1732
2843
3950
.5055
6157
7256
8353
9446
.537
2372
3645
4914
6180
7441
8699
9954
1205
2452
3696
4936
6172
7405
8635
9861
1084
2303
3519
4731
5940
7146
83^19
9.548
.743
1936
3125
4311
5494
6673
7849
9023
.193
1359
2.523
3684
4841
5996
7147
8295
9441
.583
1722
2858
3992
5122
6250
7374
8496
9615
.7,30
1843
2954
4061
5165
626/
7366
8462
9556
.646
2.500
3772
.5041
6306
7567
8825
..79
1330
2676
3820
.5060
6296
7529
8758
9984
1206
2425
3640
4852
6061
1 I 2
16251 1734
6 I 7
7267
8469
9667
.863
2055
3244
4429
5612
6791
7967
91 ■
.309
1476
2639
3800
4957
6111
7262
8410
9555
.697
1836
2972
4105
5235
6362
7486
8608
9726
.842
1955
3064
4171
5276
6377
7476
2627
3899
5167
6432
7693
8951
.204
14.54
2701
3944
5183
6419
7652
8881
.106
1328
2.547
3762
4973
6]82
7387
8589
9787
.982
2174
3362
4548
5730
6909
8084
9257
.426
1592
2755
3915
5072
6226
7377
8525
9669
.811
1950
3985
4218
5348
6475
7.599
8720
9838
.9.53
2066
3175
4282
5.386
6487
7586
8572' 8681
96651 9774
.7.55 .864
18431 19.'-.1
128
127
127
126
126
126
125
125
125
124
124
124
123
123
123
122
122
121
121
121
8
9
120
120
120
119
119
119
119
118
118
118
117
117
117
116
116
116
115
115
115
m
114
114
114
113
113
113
112
112
112
112
111
111
HI
110
110
110
110
109
109
109
D.
N. 1
400
60
401
402
403
404
405
406
407
408
61
409
410
61
411
412
413
414
415
416
417
62
418
419
420
62
421
422
423
424
425
1
426
(
427
63(
428
429
430
63;
431
/
432
,
433
<
434
435
i
436
(
437
64(
438
439
A
440
64r
441
A
442
f
443
f
444
•3
445
f
446
s
447
65(J
448
1
449
2
450
0.53
151
4
452
6
453
6
454
7
A r f
i.y.-j
8
456
8
457
9
458
660
45!)
1
N. I
(
A TABLE OF LOGARITHMS FROM I TO 10,000.
N.
1 |l|2|3|4 5|6|7!8|9|D
400
(j0206(]
2169
2277
238f
2494
2603
2711
2819
2928
3036
108
401
3144
3253
3361
3461J
3577
3686
3794
.•^902
4010
4118
108
4oy
4226
4334
4442
455C
4658
4766
4874
4982
5089
6197
108
4U:j
5305
6413
.5521
5628
5736
5844
.5951
6059
6166
6274
108
i
404
6381
6489
6,596
6704
6811
6919
7026
7133
7241
7348
107
«
405
7455
7562
7669
7777
7884
7991
8098
8205
8312
8419
107
406
8526
8633
8740
884?
89.5-3
906 1
9167
9274
9381
9488
107
407
9594
1 9701
9808
991^
..21
.128
.234
.341
.447
..5.54
107
40M
610660
0767
0873
0979
1086
1192
1298
1405
1611
1617
106
40'J
410
1723
1829
2890
1936
2996
2042
3102
2148
3207
2254
3313
2360
3419
2466
3525
2572
3630
2678
3736
106
106
612784
411
3842
3947
40.53
4159
4264
4370
4475
4.581
4686
4792
106
412
4897
5003
5108
6213
.5319
5424
5529
6634
6740
6845
105
413
5950
6055
6160
6265
6370
6476
6.581
6686
6790
6895
105
414
7000
7105
7210
7315
7420
7525
7629
7734
7839
7943
106
41,0
8048
8153
8257
8362
8466
8.571
8676
8780
8884
8989
106
416
9093
9198
9302
9406
9511
9615
9719
9824
9928
..32
104
417
620136
0240
0344
0448
0552
0656
0760
0864
0968
1072
104
418
1176
1280
1384
1488
1592
1695
1799
1903
2007
2110
104
419
420
2214
2318
3353
2421
.3456
2525
3559
2628
3663
2732
3766
2835
3869
2939
3973
3042
4076
3146
4179
104
103
623249
421
4282
4385
4488
4591
4095
4798
4901
5004
5107
.5210
103
422
6312
5415
5518
6621
6724
5827
5929
6032
6135
6238
103
423
6340
6443
6546
6648
6751
6853
6956
7058
7161
7263
103
424
7366
7468
7.57117673
7775
7878
7980
8082
8185
8287
102
425
8389
8491
8593, 8695
8797
8900
9002
9104
9206
9308
102
426
9410
9512
9613
9715
9817
9919
..21
.123
.224
.326
102
427
630428
0530
0631
0733
0835
0936
1038
1139
1241
i.342
102
■
428
1444
1545
1647
1748
1849
1951
2052
21.53
2255
2356
101
4ii9
430
2457
2559
3569
2660
3670
2761
3771
2862
2963
3973
3064
4074
3165
4176
3266
4276
3367
4376
101
100
633468
3872
431
4477
4578
4679
4779
4880
4931
.5081
5182
5283
5383
100
43^
5484
5584
5685
5785
.5886
5986
6087
6187
6287
6388
TOO
433
6488
6588
6688
6789
6889
6989
7089
7189
7290
7390
100
434
7490
7590
7690
7790
7890
7990
8090
8190
8290
8389
99
435
8489
8589
8689
8789
8888
8988
9088
9188
9287
9387
99
436
9486
9586
9686
9785
9885
9984
..84
.183
.283
.382
99
437
640481
0581
0680
0779
0879
0978
1077
1177
1276
1375
99
438
1474
1573
1672
1771
1871
1970
2069
2168
2267
2866
99
439
2465
2503
3551
2662
3650
2761
3749
2860
2959
3946
3058
4044
3156
4143
3255
4242
33.54
4340
99
98
440
643453
3847
441
4439
4537
4636
4734
4832
4931
5029
5127
5226
5324
98
442
5422
5521
5619
5717
5815
5913
6011
6110
6208
6306
98
443
6404
6502
6600
6698
679G
6894
6992
7089
7187
7285
98
444
7383
7481
7579
7676
7774
7872
7969
8067
8166
8262
98
445
8360
8458
8555
86.53
8750
8848
8945
9043
9140
9237
97
446
9335
94:J2
9.530
9627
9724
9821
9919
..16
.113
.210
97
447
650308
0405
0502
0599
0696
0793
0890
0987
1084
1181
97
44 S
1278
1375
1472
1.569
1666
1762
1859
19,56
2053
21.50
97
449
2246
2343
3309
2440
340.5
2.536
3502
2633
3598
2730
3695
2826
,379!
2923
3888
3019
3984
3116
4080
97
96
450
653213
45 1
4177
4273
4369
4465
4562
4658
4754
4850
4946
6042
96
452
5138
5235
5331
5427
5523
.5619
57 15
.5810
5906
6002
96
4o3
6098
6194
6290
6386
6482
6577
6673
6769
6864
6960
96
4o4
7056
7152
7247
7343
74381
7534
7629
7725
7820
7916
96
liJO
son
8107
8-^03
8298
8393
8488
8584
8679
8774
8870
95
456
8965
9060
9155
9250
9.346
9441
9536
9631
9726
9821
95
457
9916
..11
.106
.201
.296
..391
.486
..581
.676
.771
95
458
660865
0960
1055
11.50
1245
1339
1434
1529
1623
1718
96
45')
1813 1907' 20021
2096
2191
228(i
2380
24751
2569
2663
96
N. 1
<» 1 1 1 2 1 3 1 4 ! 5 1 6 1 7 1 8 1 9 1 D. 1
u '
■mffiSi
I
I
i
f
(
■ : IK r
m
'!!•'' i
8
N.
4fi0
461
462
463
464
465
466
467
468
469
476
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
612
513
614
5!5
516
517
618
519
N.
A TABLE OP LOGABITHMS FROM 1 TO 10,000.
T
1 I a I 3 I 4 I 5 I 6 i 7 I 8 I 9 | D.
662758
2852
2947
3041
3135
3230,
3701
3795
3889
3983
4078
4172
4642
4736
4830
4924
5018
5112
5581
5675
5769
5862 5956
6050
6518
6612
6705
6799' 6892
6986
7t63
7546
7640
7733
7826
7920
8386
8479
8572
8665
8759
8852
9317
9410
9503
9596
9689
9782
670246
0339
0431
0524
0617
0710
1173
1265
1358
1451
1543
1636
C72098
2190
2283
2375
2467
2560
3021
3113
3205
3297
3390
3482
3942
4034
4126
4218
4310
4402
4861
4953
6045
6137
5228
6320
5778
5870
5962
6053
6145
6236
6694
6785
6876
6968
7059
7151
7607
7698
7789
7881
7972
8063
8518
8609
8700
8791
8882
8973
9428
9519
9610
9700
9791
9882
680336
0426
0517
0607
0698
0789
681241
1332
1422
1513
1603
1693
2145
2235
2326
24 T
2506
2596
3047
3137
3227
3317
3407
3497
8947
4037
4127
4217
4307
4396
4845
4935
5025
5114
5204
5294
5742
5831
5921
6010
6100
6189
6036
6726
6815
6904
6994
7083
7529
7618
7707
7796
7886
7975
8420
8509
8598
8687
8776
8865
9309
9398
0285
9486
0373
9575
0462
9664
0550
9753
0639
690196
1081
1170
1258
1347
1435
1524
1965
2063
2142
2230
2318
2406
2847
2935
3023
3111
3199
3287
3727
3815
3903
3991
4078
4166
4605
4693
4781
4868
4956
5044
5482
5569
5657
5744
5832
5919
6356
6444
6531
6618
6706
6793
7229
7317
7404
7491
7678
7665
8101
8188
9057
8275
9144
8362
8449
9317
8535
9404
698970
9231
9838
9924
..11
..98
.184
.271
700704
0790
0877
0963
1050
1136
1568
1654
1741
1827
1913
1999
2431
2617
2603
2689
2775
2861
3291
3377
3463
3549
3635
3721
4151
4236
4322
4408
4494
4579
5008
5094
5179
5265
5350
5436
6864
5949
6035
6120
6206
6291
6718
6803
7655
6888
7740
6974
7059
7911
7144
7996
707570
7826
8421
8506
8591
8676
8761
8846
9270
9355
9440
9524
9609
9694
710117
0202
0287
0371
0456
0540
0963
1048
1132
1217
1301
1385
1807
1S92
1976
2060
2H4
2229
2650
2734
2818
2902
1 2986
3070
3491
3575
3G50
3742
! 3826
3910
4830
4414
4497
4581
1 4665
4749
5167
5251
5335
5418
! 5502
5586
3324 1
4266
5206
6143
7079
8013
8945
9875
0802
1728
2652
3574
4494
5412
6328
7242
8154
9064
9973
0879
1784
2686
3587
4486
5383
6279
7172
8064
8953
9841
0728
1612
2494
3375
4254
5131
6007
6880
7752
8622
9491
.358
1222
2086
2947
3807
4665
5522
6376
7229
8081
8931
9779
0625
1470
23 1 3
3154
3994
4833
5669
3418
436':
5299
6237
7173
8106
9038
9967
0895
1821
2744
3666
4586
5503
6419
7333
8245
9155
..63
0970
1874
2777
3677
4576
5473
6368
7261
8153
9042
9930
0816
1700
2583
3463
4342
5219
G994
6968
7839
8709
9578
.444
1309
2172
3033
3895
4751
5607
6462
7315
8160
9015
9863
0710
1554
2397
3238
4078
4916
5753
3512
4154
6393
6331
7266
8199
9131
..60
0988
1913
2836
3758
4677
5595
6511
7424
8336
9246
.154
1060
1964
2867
3767
4666
5563
6458
7351
8242
9131
^19
0905
1789
2671
3551
4430
5307
0182
7055
7926
8796
9664
.531
1J«5
2258
3119
3979
4837
5693
6547
740
8251
9100
9948
0794
1639
2481
3323
4162
5000
5836
3607,
4548
5487
6424
7360
8293
9224
.1.53
1080
2005
2929
3850
4769
5687
6602
7516
8427
9337
.245
n5l
2055
2957
3857
4756
5652
6547
7440
8331
9220
.107
0993
1877
2759
3639
4517
5394
6269
7142
8014
8883
9751
.617
1482
2344
3205
40G5
4922
5778
6632
7485
8336
9185
,.33
0879
1723
2566
3407
4246
50S4
5920
1 I 2 I 3
5 ! 6 I 7'T 8 ! 9
94
94
94
94
94
93
93
93
93
92
92
92
92
92
91
91
91
91
91
90
90
90
90
90
89
89
89
89
89
89
88
88
88
88
88
87
87
87
87
87
87
86
86
86
86
86
86
85
85
85
85
85
85
84
8j
84
84
84
84
"57
N. 1
620
71
621
522
623
624
625
72
626
627
528
529
630
72
631
i
532
i
533
(
534
•■
.535
«
536
(
637
J
538
73(
639
]
540
73i
541
542
r
543
<
544
£
645
e
546
7
547
7
648
8
549
9
550
74(J
561
1
562
1
653
2
554
a
555
4
566
5
.557
5
55»
6
559
7
560
748
561
8
662
9
563
750
564
1
565
2
566
2
567
3
568
4
,569
6
570
755
671
6
572
7
673
8
574
8
CVR
n
£7
576
760
577
1
578
1
579
2
N. 1
9 1
D.
fior
94
548
94
187
94
124
94
300
94
293
93
224
93
153
93
080
93
005
93
929
92
8n0
92
769
92
()87
92
002
92
516
91
427
91
337
91
245
91
151
91
055
90
957
90
857
90
756
90
652
90
547
89
440
89
331
89
220
89
107
89
993
89
877
88
759
88
639
88
517
88
394
88
269
87
142
87
014
87
883
87
751
87
617
87
482
86
344
86
205
86
065
86
922
86
778
86
632
85
485
85
336
85
185
85
.33
85
879
85
723
84
566
8j
407
84
-246
84
0S4
84
)920
84
9
1 D.
A TABLE OF LOGARITHMS FROM 1 TO 10,000.
N.
1 |l|2|3|4|5|fi 7|8|»|D. 1
620
71600C
\ 608 <
r 6170 6251
[ 6337| 6421
650.i
[ 6.58(1
6671
6754
83
621
683^
1 692 J
7004 708!:
1 7171
725^
733f-
1 7421
7504
758?
83
522
7671
7754
i 7837 792C
1 8005
1 8086
816L
1 825."]
8336
«4H
83
623
850S
! 858£
» 866^
1 8751
883^1
891?
' 900(J
9083
9165
9248
83
624
9331
9414
t 949?
' 958C
9663
974.':
982«
9911
9994
..77
83
625
720168
• 024'^
0325
040?
049U
0573
065.']
0738
0H2I
0903
83
626
0986
ima
1151
1233
131C
1398
1481
1563
1646
1728
82
627
1811
i89;j
197.'5
2058
2140
2222
2305
2.387
2469
2552
82
628
2634
2716
2798
2881
2963
3045
3127
3209
.329 1
3374
82
629
630
3456
3538
4358
3620
4440
3702
4522
3784
4604
3866
4685
3948
4767
4030
4849
4112
4931
4194
50 1 3
82
82
724276
631
5095
6176
5258
5340
5422
5503
6585
6667
6748
5830
82
532
6912
5993
6075
6156
6238
6320
6401
6483
6564
6646
82
633
6727
6809
6890
69Y2
7053
7134
7216
7297
7379
7460
81
634
7541
7623
7704
7785
7866
7948
8029
8110
8i91
8273
81
535
8354
8435
8516
8597
8678
8759
8841
8922
9003
9084
81
536
9165
9246
9327
9408
9489
9570
9651
9732
9813
9893
81
637
9974
..55
.136
.217
.298
.378
.459
•540
.621
. 702
81
538
730782
0863
0944
1024
1105
1186
1266
1.347
1428
1508
81
639
540
1589
1B69
2474
1750
2555
1830
2635
1911
2715
1991
2796
2072
2876
2152
2956
2233
3037
2313
3117
81
80
732394
541
3197
3278
3358
3438
3518
3598
3679
3759
3839
39 1 9
80
542
3999
4079
4160
4240
4320
4400
4480
4560
4640
47'>0
80
543
4800
4880
4960
5040
5120
5200
5279
5359
5439
5519
80
544
5599
5679
5759
6838
5918
5998
6078
6157
6237
6317
80
545
6397
6476
6556
6636
6715
6795
6874
69.54
7034
7113
80
546
7193
7272
7352
7431
7511
7590
7670
7749
7829
7908
79
547
7987
8067
8146
8225
8305
8384
8463
8543
8622
870 1
79
548
87>il
8860
8939
9018
9097
9177
9256
9335
9414
9493
79
549
9572
9651
9731
9810
9889
9968
..47
.126
.205
.284
79
550
740363
0442
0521
0600
0678
0757
0836
0915
0994
1073
79
551
1162
1230
1309
1388
1467
1546
1624
1703
1782
1860
79
552
193!
2018
2096
2175
2254
2332
2411
2489
2568
2646
79
553
2725
2804
2882
2961
3039
3118
3196
3275
3.353
3431
78
554
3510
3588
3667
3745
3823
3902
3980
4058
41.36
4215
78
555
4293
4371
4449
4528
4606
4684
4762
4840
4919
4997
78
656
5075
5153
5231
5309
6387
5465
5543
.5621
5699
5777
78
557
5855
5933
6011
6089
6167
6245
6323
6401
6479
6.556
78
55H
6G34
6712
6790
6868
6945
7023
7101
7179
7256
7334
78
559
560
7412
7489
8266
7567
8343
7645
8421
7722
8498
7800
8576
7878
8653
7955
8731
8033
8808
8110
8885
78
77
748188
561
8963
9040
9118
9195
9272
9350
9427
9504
9582
9659
77
562
9736
9814
9891
9968
..45
.123
.200
.277
.3.54
.431
77
563
750508
0586
0663
0740
0817
0894
0971
1048
1125
1202
77
564
1279
1356
1433
1510
1587
1664
1741
1818
1895
1972
77
565
2048
2125
2202
2279
2356
2433
2509
2586
2663
2740
77
566
2816
2893
2970
3047
3123
3200
3277
3353
3430;
3506
77
567
3583
3660
3736
3813
3889
3966
4042
4119
4! 95]
4272
77
568
4348
4425
4501
4578
4654
4730
4807
4883
4960
5036
76
569
570
5112
5189
6951
5265
6027
5341
6103
5417
6180
5494
6256
5570
6332
5646
6408
5722
6484
5799
6560
76
76
755875
671
6636
6712
6788
6864
6940
7016
7092
7168
7244
7320
76
572
7396
7472
7548
7624
7700
7775
7851
7927
8003
8079
76
573
8155
8230
8306
8382
8458
8533
8609
8685
8761
8836
76
574
8912
8988
9063
9139
9214
9290
9.366
9441
9517
9592
76
575
9688
V 1 'to
9819
9894
9970
. .45
.121
.196
. 272
.347
75
576
760422
0498
0573
0649
0724
0799
0875
0950
1025|
1101
75
577
1176
1251
1326
1402
1477
1552
1627
1702
1778
18.53
75
bV8
1928
2003
2078
2153
2228
2303
2378
2453
2529: 26041
75
579
2679
2754
2829
2904129781
3053 3128'
3203
3278: 3353'
75
N. 1
1 1 2 1 3 1 4 1
5 1 H 1 7 1 8 1 9 1 D. 1
I
in
A TABLE OF LOOARITIIMS PflOM 1 TO 10,000.
i
t
,
'*:- .!,
1.^
N.
1 |lf2|8|4|filfi 7|H|9|D. j
580
763428
3503
3578
3653
3727
3802 3877
3952
4027
4101
75
581
4176
4251
4326
4400
4475
4550 4624
4699
4774
4848
75
58'2
4923
4998
6072
5147
5221
5296 5370
5445
5520
5594
75
583
6669
5743
5818
5892
5966
0041 6115
6190
6264
6338
74
584
6413
6487
6562
6636
6710
6785' 6859
"933
7007
7082
74
585
7156
7230
730-1
7379
7453
7527
7601
7675
7749
7823
74
58«
7898
7972
8046
8120
8194
8'<»6.^
' 8342
8416
8490
8564
74
587
8638
8712
8786
8860
8934
90(1 n
9082
9156
9230
9303
74
588
9377
9451
9525
9599
9673
9746
9820
PS 94
9968
..42
74
58 a
770115
0189
0263
0336
0410
0484
0557
0631
0705
0778
74
590
770852
0926
0999
1073
1146
1220
1293
1367
1440
1514
74
591
1587
1661
1734
1808
1881
1955
2029
2102
2175
2248
73
592
2322
2395
2468
2542
2615
2688
2762
2835
2908
2981
73
593
3055
3128
3201
3274
3348
3421
3494
3567
3640
3713
73
59'1
3786
3860
3933
4006
4079
4152
4225
4298
4371
4444
73
595
4517
4590
4663
4736
4809
4882
4955
5028
5100
5173
73
596
6246
5319
5392
5465
5538
5610
5683
5756
5829
5902
73
597
5974
6047
6120
6193
6265
6338
6411
6483
6556
6629
73
598
6701
6774
6846
6919
6992
7064
7137
7209
7282
7354
73
599
7427
7499
7572
7644
7717
7789
7862
7934
80^6
8079
72
600
778151
8224
8296
8368
8441
8513
8585
8658
8730
8802
72
601
8874
8947
9019
9091
9163
9236
9308
9380
9452
9524
72
602
9596
9669
9741
9813
98S5
9957
..29
.101
.173
.24=i
72
603
780317
0389
0461
0533
0605
0677
0749
0821
0893
0905
72
604
r037
1109
1181
1253
1324
1396
1468
1540
1612
1684
72
605
1755
1827
1899
1971
2042
2114
2186
2258
2329
2401
72
606
2473
2544
2616
2G88
2759
2831
2902
2974
3046
3117
72
607
3189
3260
3332
3403
3475
3546
3618
3689
3761
3832
71
608
3904
39VO
4046
4118
4189
4261
4332
4403
4475
4546
71
609
610
4617
4689
5401
'.'760
5472
4S31
5543
4902
5615
4974
5686
5045
5757
5116
5828
5187
5899
5259
5970
71
71
785330
611
6041
6112
6183
6254
6325
6396
6467
6538
6609
6680
71
612
6751
6822
6893
6964
7035
7106
7177
7248
7319
7390
71
613
7460
7531
7602
7673
7744
7815
7885
7956
8027
8098
71
614
8168
8239
8310
8381
8451
8522
8593
8663
8734
8804
71
615
8875
8946
9016
9087
9157
9228
9299
9369
9440
9510
71
616
^581
9651
9722
9792
9863
9933
...4
..74
.144
.215
70
617
790285
0356
0426
0496
0567
0637
0707
0778
0848
0918
70
618
0988
1059
1129
1199
1269
1340
1410
1480
1550
1620
70
619
1691
1761
1831
1901
1971
2041
2111
2181
2252
2322
70
620
792392
2462
2532
2602
2672
2742
2812
2882
2952
3022
70
621
3092
3162
323 ij
3301
3371
3441
3511
3581
3651
3721
70
622
3790
3860
3930
4000
4070
4139
4209
4279
4349
4418
70
623
4488
4558
4627
4697
4767
4836
4906
4976
5045
5115
70
624
5185
5254
5324
5393
5463
5532
5602
5672
5741
5811
70
625
5880
5949
6019
608f
6158
6227
6297
6366
6436
6505
69
626
6574
6644
6713
6782
6852
6921
6990
7060
7129
7198
69
627
7268
7337
7406
7475
7545: 76141
7683i
7752
7821
7890
69
628
7960
8029
8098
8167
8238'
8305
8374 8443
8513
8582
69
629
8651
8720
8789
8858
8927
8996
9065 9134
9203
9272
69
630
799341!
9409
9478
9547
9616
9685
975^ 9823
9892
9961
69
631
800029 0098
0167
0236
0305!
0373
0442 0511
0580
0648
69
632
07n 0786
0854
0923
0992'
1061
1129 1198
1266
1335
69
633
1404
1472
1541
1609|
1678
1747
1S15 1884
1952
2021
69
634
2089
2158
2226
2295!
2363!
2432
2500 2568
2637
2705
69
635
2774
2842
2910
2979 1
3047: 31161
3184
3252
3321
3389
68
636
3457!
3525' 35941
36621
3730 3798!
3867!
3935
4003
4071
68
637
4139
420814276
43441
4412; 4480
4548
4616
4685
4753
68
638
4821
4889' 4957
5025
5093 5161
5229
5297
5365
5433
68
639
5501
5569' 5637
5705
5773' 5LU 590Si
5976
6044
6112
68
N. I
I 1 1 2 I G 1 4 i 5 1 6 1 7 8 I 9 1 D. i
F
1
640
806
641
6.
642
7.
643
8'
641
8t
645
9.
646
810;
647
01
648
].'
619
2:
6.'j0
812<
651
3f
652
45
653
4<
654
5f
655
65
656
6(
657
7J
658
85
659
8f
660
819f
661
8205
662
0*
663
1.'
664
2
665
2.''
666
3
667
4
668
4-
669
5-1
670
826(
671
6'
672
1{
073
8(
674
8(
675
91
676
9J
677
830.'
678
15
679
U
680
83??
681
3
682
3-
683
4-1
684
5{
685
66
686
6?
687
6r
688
7f
689
82
690
838S
691
94
692
8401
693
0?
694
i:^
r)H;i
H
696
2r
697
32
69,S
38
699
44
N. 1
1 D.j
01
Tf)
18
75
J'i
75
m
74
H'2
74
i'.i
74
R'l
74
[):j
74
12
74
78
74
14
74
18
73
il
73
13
73
14
73
ra
73
)2
73
i9
73
■A
73
m
72
)2
72
24
72
I'S
72
>5
72
M
72
)1
72
7
72
J2
71
6
71
)9
71
ro
71
)0
71
)0
71
»8
71
)4
71
71
5
70
8
70
,0
70
2
70
2
70
1
70
8
70
5
70
1
70
5
6g
8
69
69
2
69
2
69
i
69
8
69
5
69
1
69
5
69
9
68
1
68
3
68
3
68
2
68
lD.i
A TAHLE OF LO(;AniTUMS FiJOM f TO 10,000.
f!
64()
|0 1|2|3 4 5|6 71819 U. 1
806180
6248
6316 63-14
6451
65 19
6587
6655
6,23
6790
68
641
6858
6926
6994 7061
712J
7197
7264
7332
7400
7467
68
612
7535
7603
7670 773ft
7806
7873
7941
8008
8076
8143
68
643
8211
8279
8346
8414
8481
8549
8616
8684
8751
881S
67
644
8886
8953
9021
9088
9156
9223
9290
9358
9425
9492
67
645
9560
9627
9094
9762
9829
9896
9964
..31
..98
.165
67
046
810233
0300
0367
0434
0501
0569
0636
0703
0770
0837
67
647
0904
0971
1039
1106
1173
1240
1307
1374
1441
1508
67
648
1575
1642
1709
1776
1843
l.tlO
1977
2044
2111
2178
67
619
650
2245
2980
237C
3047
2445
3114
2512
3181
2579
3247
2646
3.il4
2713
3381
2780
3448
2847
3514
67
67
812913
651
3581
3648
3714
3781
3S48
3914
3981
4048
4114
4181
67
652
4248
4314
4381
4447
4514
4581
4647
4714
4780
4847
67
653
4913
4980
5046
5113
5179
5246
5312
5378
.5445
5511
66
654
5578
5644
5711
5777
5843
5910
5976
6042
6109
6175
86
655
6241
6308
6374
6440
6506
6573
6639
6705
6771
«)838
66
656
6904
6970
7036
7102
7169
7235
7301
7367
7433
7499
66
657
7565
7631
7698
7764
7830
7896
7962
8028
8094
8160
66
658
8226
8292
8358
8424
8490
8556
8622
8688
8754
8820
66
659
660
8885
8951
9610
9017
9676
9083
9741
9149
9807
9215
9873
9281
9939
9346
...4
9412
..70
9478
. 136
66
66
819544
661
82020!
0267
0333
0399
0464
0530
0595
0661
0727
0792
66
662
0858
0924
0989
1055
1120
1186
1251
1317
1382
1448
66
663
1514
1579
1645
1710
K75
1841
1906
1972
2037
2103
65
664
2168
2233
2299
236-1
2430
2495
2560
2626
2691
2756
65
665
2822
2887
2952
3018
3083
3148
3213
3279
3344
3409
65
666
3474
3539
3605
3670
3735
3800
3865
3930
3996
4061
65
667
4126
4191
4256
4321
4386
4451
4516
4581
4646
4711
65
668
4776
4841
4906
4971
5036
5101
5166
5231
5296
5361
65
669
670
5426
5491
6140
5556
6204
5621
6269
5686
6334
.^75'
639J
5815
6464
5880
6528
5945
6593
6010
6658
65
65
826075
671
6723
6787
6852
6917
6981
7046
7111
7175
7240
7-->()5
05
672
7369
7434
7499
7563
7628
7692
7757
7821
7886
7951
65
673
8015
8080
8144
8209
8273
8338
8402
8467
85:31
8595
64
674
8660
8724
8789
8853
8918
8982
9046
9111
9i75
9239
64
675
9304
9368
0432
9497
9561
9625
9690
9754
9818
9882
64
676
9947
..11
..75
.139
.204
.268
.3.32
•396
.400
.525
64
677
830589
0653
0717
0781
0845
0909
0973
1037
1102
1166
64
678
1230
1294
1358
1422
1486
1550
1014
1678
1742
1806
64
679
1870
1934
1998
2062
2126
2189
2253
2317
2381
2445
64
680
83?509
2573
2637
2700
2764
2828
2892
2956
3020
3083
64
681
3147
3211
3275
3338
3402
3466
3530
3593
3657
3721
64
682
3784
3848
3912
3975
4039
4103
•'^.66
4230
4294
4357
64
683
4421
4484
4548
4611
4675
4739
4805>
4866
4929
4993
64
684
5056
5120
5183
5247
5310
5373
5137
5500
5564
5627
63
685
5691
5754
5817
5881
5944
6007
6071
6134
6197
6261
63
686
6324
6387
6451
6514
6577
6641
6704
6767
6830
6894
63
687
6957
7020
7083
7146
7210
7273
7336
7399
746^
7.525
63
688
7588
7652
7715
7778
7841
7904
7967
8030
8093
SI 56
63
689
8219
8282
8345
8408
8471
8534
8597
8660
8723
8786
63
690
838849
8912
8975
9038
9101
9164
9227
9289
9352
9415
63
691
9478
9541
9604
9667
9729
9792
9855
9918
9981
..43
63
692
840106
0169
0232
0294
0357
0420
0482
0545
0608
0671
63
693
0733
0796
0859
0921
0984
1046
1109
1172
1234
1297
63
694
1359
1422
1485
1547
1610
1672
1735
1797
1860
1922
63
r)H;i
1 nsp.
2047
2110
2172
2235
2297
1.1 fiO
2422
2484
2547
62
696
2609
2672
2734
2796
2859
2921
2983
3046
3108
3170
62
697
3233
3295
3357
3420
3482
3544
3606
3669
3731
3793
62
rm
3855
3918
3980
4042
4104
4J66
4229
4291
4353
4415
62
699
4477
4539
4601
4664
4726
4788
4850
4912
4974
5036
62
N. t
1 1 1
2 3 i 4 1 5 1 6 '
8 1
"
D.
I
1 *<
! H
il
rii
•I
Hi
,f !i
12
A TABLE OF LOOAKITIIMS FROM 1 TO 10,000.
N.
1 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 'H 9 1 D. 1
700
845098
5160i 5222
5284
5346
5408
5470
5532
5594
5658
62
701
5718
5780
5842
5904
5966
6028
6090
6151
6213
6275
82
702
6337
6399
6461
6523
6585
6646
6708
6770
6832
8894
82
703
6955
7017
7079
7141
7202
7264
7326
7388
7449
7511
82
704
7573
7634
7696
7758
7819
7881
7943
8004
8066
8128
62
705
8189
8251
8312
8374
8435
8497
8559
8620
8682
8743
62
706
8805
8866
8928
8989
9051
9112
9174
9235
9297
9358
61
707
9419
9481
9542
9604
9665
9726
9788
9849
9911
9972
61
708
850033
0095
0156
0217
0279
0340
0401
0462
05':4
0.585
•:i
709
710
0646
0707
0769
1381
0830
1442
0891
0952
1014
1075
1138
1747
1197
1809
6~1
851258
1320
1503
1564
1625
1686
711
1870
1931
1992
2053
2114
2175
2236
2297
2358
2419
61
712
2480
2541
2602
2663
2724
2785
2846
2907
2968
3029
61
713
3090
3150
3211
3272
3333
3394
3455
.3516
3577
3637
61
714
3898
3759
3820
3881
3941
4002
4063
4124
4185
4245
61
715
4308
4367
4428
4488
4549
4610
4670
4731
4792
4852
81
716
4913
4974
5034
5095
5156
5216
5277
5337
5398
5459
81
717
5519
5580
5640
5701
5761
5822
5882
5943
6003
6084
61
718
6124
6185
6245
6306
6366
6427
6487
6548
6808
6888
80
719
720
6729
6789
6850
6910
7513
6970
7031
7634
V091
7894
7152
7755
7212
7272
7875
80
00
857332
7393
7453
75'-''4
7815
721
7935
7995
8056
8116
81'3
8236
8297
8357
8417
8477
60
722
8537
8597
8657
8718
8778
8833
8898
8958
9018
9078
80
723
9138
9198
9258
9318
9379
9439
9499
9559
9619
9879
80
724
9739
9799
9859
9918
99 /•'8
..38
..98
.158
.218
.278
80
725
860338
0398
0458
0518
0578
0637
0697
0757
0817
0877
60
726
0937
0996
1056
1116
1176
1236
1295
1355
1415
1475
80
727
1534
15')4
1654
1714
• 773
1833
1893
1952
2012
2072
60
728
2131
2191
2251
2310
2370
2430
2489
2549
2608
2688
80
729
730
2728
2787
2847
2906
2966
3025
3085
3680
3144
3739
320 i
371*9
3263
3858
60
59
863323
3382
3442
3501
3561
3620
731
3917
3977
4036
40O6
4155
4214
4274
4333
4392
4452
59
732
4511
4570
4630
4689
4748
4808
4867
4926
4985
5045
59
733
5104
5163
5222
5282
5341
5400
54.': 1
5519
f)578
5837
59
734
5696
5755
5814
5874
5933
5992
6051
6110
6169
(1228
59
735
6287
6346
6405
6465
6524
6583
6842
6701
6760
6819
59
73(»
6878
6937
6996
7055
7114
7173
7232
7291
7350
7409
59
737
7467
7526
7585
76'44
7703
7762
7821
7880
7939
7998
59
738
8056
8115
8174
8233
8292
8350
8409
8488
8527
8588
59
759
740
8644
8703
8762
.8521
8879
9466
8938
9525
8997
9584
9056
9642
9114
9173
9780
59
59
809232
9290
9349
9408
9701
741
9818
9877
9935
9994
..53
.111
.170
.228
.287
.345
59
742
870404
0462
0521
0579
0638
0896
0755
0813
0872
0930
58
743
^^989
1047
1106
1164
1223
1281
1339
1398
1456
1515
58
744
1573
1631
1690
1748
1808
1865
1923
1981
2040
2098
58
745
2156
2215
2273
2331
2389
2448
2506
2504
2622
2681
58
746
2739
2797
2855
2913
2972
3030
3088
3146
3204
i'^-'82
58
747
3321
3379
3437
3495
3553
3611
3669
3727
3785
3814
58
748
3902
3960
4018
4076
4134
4192
4250
4308
4366
4424
58
749
750
4482
4540
5119
4598
4658
4714
5293
4772
5351
4830
5409
4888
5466
4945
5003
5582
58
58
87506 1
5177
5235
5524
751
5640
5698
5756
5813
5871
592'^
5987
6045
6102
6160
58
752
6218
6276
6333
6391
6449
6507
6564
6822
8680
6737
58
753
6795
6853
6910
6968
7026
708;'
7141
7199
7256
7314
58
754
7371
7429
7487
7544
7602
7659
7717
7774
7832
7889
58
755
7947
8004
8082
8119
8177
8234
8292
8349
8407
8464
57
750
8522
8579
8637
8634
8752
SHOO
8866
S924
8981
9039
0/
757
9096
9153
9211
9268
9325
9383
9440
9497
9555
9812
57
758
9869
9726
9784
98J'
9898
9956
..13
..70
.127
.185
57
759
880243
0299
0356
J413
0471
052S
0585
0642
0899
07h8
57
N.
1 1 1 2 1 3 1 4 1 5 !() 1 7 1 8 1 9 1 D. 1
#
N.
i <
760
88u
761
1
762
1
763
2
764
3
765
3
766
4
767
4
768
5
769
5
770
886
771
7
772
7
773
8
774
8
775
9,
776
9i
777
890'
778
Oi
779
1.'
780
892(
781
2f
782
35
783
3<
784
4r
785
4S
786
5'j
787
5(
788
or
789
7C
790
897P
791
81
792
87
793
92
794
98
795
9003
790
09
797
14
798
20
799
25
800
9030
801
36
802
41
803
47
804
52
805
57
808
63
807
68
808
74
809
79
810
9084
811
90
812
95
813
9100
814
06
815
111
8!H
18'
817
22"
818
27,
819
32{
N. 1
U
i 1
D.
of)
62
75
62
91
62
11
62
28
62
4:3
02
58
61
72
61
85
•:i
97
f'l
09
61
19
61
29
61
37
61
45
61
52
61
59
61
04
61
(58
60
72
60
75
60
77
60
78
60
79
60
78
60
77
60
75
60
72
60
68
60
Si'A
60
i58
59
t52
59
)45
59
)37
59
!28
59
il9
59
t09
59
)98
59
)86
59
73
59
row
59
J45
59
)30
58
)15
58
)98
58
581
58
M52
58
HI
58
124
58
)03
58
i82
58
1(50
58
m
58
514
58
■IS9
58
164
57
)?.<J
512
57
185
57
/h(5
57
9
Id.
rrr
A TABLE OF LOGARITHMS ITROM 1 TO 10,000.
13
1
760
701
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
880814
1385
1955
2525
3093
3661
4229
4795
5361
6926
886491
7054
7617
8179
8741
9302
9862
890421
0980
1537
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
892095
2651
3207
3762
43i6
4870
5423
5975
6r,26
7077
897627
8176
8725
9273
9821
900367
0913
1458
2003
2547
0871
1442
2013
2581
3150
3718
4285
''852
6418
5983
6547
7111
7674
8236
8797
9358
9918
0477
1035
1693
0928
1499
2069
2638
3207
3776
4342
4909
6474
6039
2150
2707
3262
3817
4371
4925
5478
6030
6681
7132
6604
7167
7730
8292
8863
9414
9974
0533
1091
1649
0985
1556
2126
2695
3264
3832
4399
4965
5531
6096
4 I 6 I 6 I 7 1 8 I 9 I D.
903090
3633
4174
4716
6256
5796
6335
6874
7411
_- 7949
810 908485
811
812
813
814
815
«!6
817
818
SJ_9
7682
8231
8780
9328
9875
0422
0968
1513
2057
2601
9021
9556
910091
0624
1158
1690
2222
2753
3284
3144
3687
4229
4770
5310
5860
6389
6927
7466
8002
8539
9074
9610
0144
0678
1211
1743
2276
2806
3337
2206
2762
3318
3873
4^127
4980
5533
6085
6636
7187
7737
8286
8835
9383
9930
0476
1022
1567
2112
2655
6660
7223
7786
8348
8909
9470
..30
0589
1147
1705
1042
1613
2183
2752
3321
3888
4456
.5022
5587
6152
6716
7280
7842
8404
8966
9526
..86
0646
1203
1760
1099
1670
2240
2809
3377
3946
4512
6078
5644
6209
22fi'^
28i»
3373
3928
4482
5036
5588
6140
6692
7242
3199
3741
4283
4824
5364
5904
6443
6981
7519
8056
8593
9128
9663
0197
0731
1264
1797
2328
2859
3390
7792
8.341
8890
9437
9985
0.531
1077
1622
2166
2710
2317
2873
3429
3984
4538
5091
5644
6196
6747
7297
3263
3795
4337
4878
.5418
6958
6497
7035
7573
8110
7847
8396
8944
J492
..39
0686
1131
1676
2221
2764
6773
7336
7898
8460
9021
958-.
.J41
0700
1269
1816
1156
1727
2297
2866
3434
4002
4569
6135
6700
6265
2373
2929
3484
4039
4593
5146
5699
6251
6802
7352
6829
7392
7966
863 6
9077
'638
.197
0756
1314
1872
1213
1784
2354
2923
3491
4059
4625
6192
5757
6321
8646
9181
9716
0251
0784
13)7
1850
2381
2913
3443
3307
3849
4391
4932
6472
6012
6551
7089
7626
8163
7902
8461
8999
9547
..94
0640
1186
1731
2276
2818
8699
9236
9770
0304
0838
1371
1903
2436
2966
3496
3361
3904
4445
4986
6526
6066
6604
7143
7680
8217
2429
2985
3640
4094
i648
6201
5754
6306
6857
7407
6885
7449
8011
8673
9134
9694
.253
0812
1370
1928
7957
8506
9054
9602
.149
0695
1240
178d
232S
2873
8763
9289
9823
0358
0891
1424
i9&G
2488
3019
3549
3416
3968
4499
5040
5580
6119
6658
7196
7734
8270
2484
3040
3596
4150
4704
6257
5809
6361
6912
7462
1271
1841
2411
2980
3548
4 1 15
4682
5248
5813
6378
6942
7505
8067
8629
9190
9750
.309
0868
1426
1983
8012
8561
9109
9656
.203
0749
1295
1840
2384
2927
2640
3096
3651
4206
4759
.5312
5864
6416
6967
7617
8807
9342
9877
0411
0944
1477
2009
2.541
3072
3602
3470
4012
4653
5094
6634
6173
6712
7250
7787
8324
8860
9396
9930
0464
0998
1.530
2063
2594
3125
36551
8067
8616
9164
9711
.2.58
0804
1349
1894
2438
2981
1328
1898
2468
3037
3605
41V2
4739
6305
5870
6434
6998
7561
8123
8685
9246
9806
.365
0924
1482
2039
2595
3151
3706
4261
4814
6367
5920
6471
7022
7672
3524
r
«i'*8
5688
6227
6766
7304
7841
8378
8914
9449
9984
0518
1051
1584
2116
2647
3178
3708
8122
8670
9218
9766
.312
0859
1404
1948
2492
3036
3578
4120
4661
6202
5742
6281
6820
7358
7896
8431
8967
9603
..37
0671
1104
1637
2169
2700
3231
3761
67
57
57
67
57
67
57
57
57
66
56
56
56
56
56
66
66
56
56
66
56
66
56
55
55
65
55
66
65
55
66
55
55
55
55
55
55
64
54
64
64
54
54
64
54
54
54
54
54
64
54
54
63
53
53
53
53
53
53
53
fi I 7 I 8 I 9 I D.
S '"
f
ill n '
1.'
H I
;i
^. i \
lip'
II
la:*
14 A TABLE OP LOGARITHMS FROM I TO 10,000.
N. 1
ll!2|3|4|5l6|7|8|9lD. 1
820
913814 38671
3920
3973
4026
4079
4132
4184
4237
42901 53 1
821
4343
4396
4449
4502
4555
4608
4660
4713
4766
4819 ""
53
822
4872
4925
4977
5030
5083
5136
6189
5241
5294
5347
63
823
5400
5453
5505
5558
5611
5664
5716
6769
5822
58Vb
53
824
5927
5980
6033
6085
6138
6191
6243
6296
6349
6401
53
825
6454
6507
6559
6612
6664
6717
6770
6822
6875
6927
53
82G
6930
7033
7085
7138
7190
7243
7295
7348
7400
7453
53
827
7506
7558
7611
7663
7716
7768
7820
7873
7925
7978
52
828
8030
8083
8185
8188
8240
8293
8345
8397
8450
8502
52
829
830
8555
8607
9130
8659
9183
8712
9235
8764
9287
8816
9340
8869
8921
9444
89V3
9496
9026
9549
52
52
019078
9392
831
9601
9653
9706
9758
9810
9862
9914
996V
..19
..71
52
832
920123
0176
0228
0280
0332
0384
0436
0489
054 1
0593
52
833
0645
0697
0749
0801
0853
0906
0958
1010
1062
1114
52
834
1166J
1218
1270
1322
1374
1426
1478
1530
1582
1634
52
835
1686
1 738
1790
1842
1894
1946
1998
2050
2102
2154
52
836
2206
2258
2310
2362
2414
2466
2518
2570
2622
2674
52
837
2725
2777
2829
2881
2933
2985
3037
3089
3140
3192
52
838
839
840
3244
3296
3348
3399
3451
3503
3555
3607
3658
3V10
52
3762
3814
4331
3865
4383
3917
4434
3969
4486
4021
4538
4072
4124
4641
41V6
4693
4228
52
52
924279
4589
4744
841
4796
4848
4899
4951
5003
5054
5106
5157
5209
5261
52
842
5312
5364
5415
5467
5518
5570
5621
5673
5725
5VVb
52
843
5828
5879
5931
5982
60.34
6085
6137
6188
6240
6291
51
844
6342
6394
6445
6497
6548
6600
6651
6702
6754
6805
51
845
6857
6908
6959
7011
7062
7114
7165
7216
7268
7319
51
840
7370
7422
7473
7524
7576
7627
7678
V730
7781
7832
51
847
7883
7935
7986
8037
8088
8140
8191
8242
8293
8345
51
848
8396
8447
8498
8549
8601
86.52
8703
8754
8805
885V
51
849
850
8908
8959
9470
9010
9521
9061
9572
9112
9623
9163
9674
9215
9725
9266
9776
9317
9827
9368
9879
51
51
929419
851
9930
9981
..32
..83
.134
.185
.236
.287
.338
.389
51
852
930440
0491
0542
0592
0643
0694
0745
0796
084V
0898
51
853
0949
1000
1051
1102
1153
1204
1?54
1305
1356
1407
51
854
1458
1509
1,560
1610
1661
1712
: . ■ /<
1814
1865
1915
51
855
1966
2017
20b8
2118
2169
2220
2271
2322
23V2
2423
51
856
2474
2524
2575
2626
2677
2727
2778
2829
2879
2930
61
857
2981
3031
3082
3133
3183
3234
3285
333b
3386
3437
51
858
3487
3538
3589
36.39
3690
3740
3791
3841
3892
3943
51
359
860
3993
934498
4044
4549
4094
4599
4145
4650
4195
4700
4246
4751
4296
4801
4347
4852
4397
4448
49.53
51
50
4902
861
5003
5054
5104
5154
5205
5255
.5306
5356
5406
.5457
50
862
5507
55;i8
5608
5658
5709
5759
5809
5860
5910
5960
50
863
6011
6061
6111
6162
6212
6262' 6313
6363
6413
6463
50
864
6514
6564
6614
6665
6715
6765
6815
6865
6916
6966
50
86 1
7016
7066
7117
7167
7217
7267
7317
7367
7418
7468
50
866
7518
756ft
7618
7668
77i8
7769
7819
7869
7919
7969
50
867
8019
8069
8119
8169
8219
8269
8320
8370
8420
8470
50
868
8520
8570
8620
8670
8720
8770
8820
8870
8920
8970
50
869
9020
9070
9120
9170
9220
9270
9320
9369
9419
9469
50
870
939519
9569
9619
9669
9719
9769
9819
9869
9918
9968
50
871
940018
0068
0118
0168
0218
0267
0317
0367
0417
0467
50
872
0516
0566
0616
0666
0716
0765
0815
0865
091.5
0964
50
878
1014
1064
- 1114
1163
1213
1263
1313
1362
I4iy
146-^
60
874
1. 11
1561
1611
166(1
171C
1760
1809
1859
190y
19,5H
50
875
200H
2058
t 210?
' 2157
2?07
' 225G
2306
2355
2405
245.T
. 50
876
2504
- 2554
2602
1 265S
1 270S
! 2752
2801
2851
2901
295C
1 50
877
30011
304i:
1 309i
1 3l4(!
319i«
! 3247
' 3297
' 334t
3391
> 344 f
) 49
87
, 349£
> 3544
359r
( 364.^
3695
! 374S
, 3791
3841
3891
) 393i
) 49
879
' 398t
> 403t
) 4088
!Ul3?
' 4186
) 423f
) 428f
) 433f
) 438^
\ 4433' 49 1
N.
i 1 i 2 1 3 1 4 1 5 1 1 7 1 8 1 9 1 13. !
I l<
N.
880
94
881
882
883
884
885
886
887
888
889
890
94
891
892
95
893
894
895
896
897
898
899
900
95^
901
902
1903
1 904
905
906
907
908
8
909
8
910
959
911
9
912
9
913
960
914
915
]
916
1
917
2
918
2
919
3
920
963
921
4
922
4
923
6
924
6
925
6
926
6
927
7
928
7
929
8
930
968
931
8
932
9
933
9i
934
970.
935
0)
936
1'
ao-y
r
<j\j t j
938
2-
939
2(
N. 1
Id. 1
90
53
19
53
47
63
75
53
01
53
27
53
53
53
78
52
03
52
26
52
49
52
71
52
911
52
14
52
34
52
54
52
74
52
92
52
10
52
960
50
463
50
966
50
468
50
969
50
470
50
970
50
469
50
968
50
467
50
964
50
462
50
958
50
4iyO
50
950
50
445
49
939
49
433
49
9
1 ».
A TAI/;E OP LOOARITIIMS FEOM I TO 10,000.
15
N.
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
• 920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
:ju t
938
939
944483
4976
6469
5961
6452
6943
7434
7924
8413
8902
IM2|3|4|5|6|7|8|9;d.
949390
9878
950365
0851
1338
1823
2308
2792
3276
3760
453
5025
6518
6010
6501
6992
7483
7973
8462
895_l_
9439: 9488
9926 9975
4581
6074
5567
6059
6551
7041
7532
8022
8511
8999
0414
0900
1386
1872
2356
2841
3325
3808
954243
4725
5207
5688
6168
6649
7128
7607
8086
8564
959041
9518
9995
960471
0946
1421
1895
2369
2843
3316
4291
4773
5255
5736
6216
6697
7176
7655
8134
8612
963788
4260
4731
6202
5672
6142
6611
7080
7548
8016
968483
8950
9416
9882
97034r
0812
1276
2203
2666
9089
9566
..42
0518
0994
1489
1943
2417
2S90
3363
0462
0949
1435
1920
2405
2889
3373
3856
4631
5124
5616
6108
6600
7090
7581
8070
8560
9048
4339
4821
5303
5784
626r>6313
3835
4307
4778
5249
5719
6189
6658
7127
7595
8062
8530
8996
9463
9928
0393
0858
1322
17S6
2249
2712
6745
7224
7703
8181
8659
9137
9614
..90
0566
1041
1516
1990
2464
2937
3410
9536
..24
0511
0997
1483
1969
2453
2938
3421
3905
4680
5173
5665
6157
6649
7140
7630
8119
8609
9097
4387
4869
5351
5832
6793
7272
7751
8229
8707
3882
4354
4825
5296
5766
6236
6705
7173
7642
8109
8576
9043
9509
9975
0440
0904
1369
1832
2295
2758
9185
9661
.138
0613
1089
1563
2038
2511
2985
3457
9585
..73
0560
1046
1532
2017
2502
2986
3470
3953
4729
5222
5715
6207
6698
7189
7679
8168
8657
9146
4435
4918
5399
5880
6361
6840
7320
7799
8277
8755
9634
.121
0608
1095
1580
2066
2550
3034
3518
4001
4779
5272
5764
6256
6747
7238
7728
8217
8706
9195
3929
4401
4872
5343
5813
6283
6752
7220
7688
8156
9232
9709
.185
0661
1136
1611
2085
2559
3032
3504
3977
4448
4484
4966
5447
5928
6409
6888
7368
7847
8325
8803
9683
.170
0657
1143
1629
2114
2599
3083
3566
4049
4828
5321
5813
6306
6796
7287
7777
8266
8755
9244
4877
6370
5862
6354
6845
7336
7826
8315
8804
9292
9280
9757
.233
0709
1184
1658
2132
2606
3079
3552
4532
5014
5495
5976
6457
6936
7416
7894
8373
8850
9731
.219
0706
1192
1677
2163
2647
3131
3615
4098
8623
9090
9556
..21
0486
0951
3415
1879
2342
2804
4024
4495
4919 4966
5390 6437
5860
6329
6799
7267
7735
8203
8670
9136
9602
..68
0533
0997
1461
1925
2388
2851
5907
6376
6845
7314
7782
8249
9328
9804
.280
0756
1231
1706
2180
2653
3126
3599
0716
9183
9649
.114
0579
1044
1508
1971
2434
2897
4071
4542
5013
5484
5954
6423
6892
7361
7829
8296
4580
5062
5543
6024
6505
6984
7464
7942
8421
8898
9375
9852
.328
0804
1279
1753
2227
2701
3174
3646
9780
.267
0754
1240
1726
2211
2696
3180
3663
4146
4628
5110
5592
6072
6553
7032
7512
7990
8468
8946
4927
5419
5912
6403
6894
7385
7875
8364
8853
9341
9829
.316
0803
1289
1775
2260
2744
3229
3711
4194
9423
9900
.376
0851
K}26
180.1 1 1848
4677
5158
5640
6120
6601
7080
7559
8038
8516
8994
9471
9947
.423
0899
1374
8763
9229
9695
.161
0626
1090
1654
2018
2481
2943
4118
4590
6061
5531
6001
6470
6939
7408
7875
8343
8810
9276
9742
.207
0672
1137
1601
2064
2527
2989
2275
2748
3221
3693
4165
4637
5108
6578
6048
6517
6'>86
7454
7922
8390
8856
9323
9789
.264
0719
1183
1647
2110
2573
3035
2322
2795
3268
3741
2212
4684
5155
5625
6095
6564
7033
7501
7969
8436
8903
9369
9835
.300
0765
1229
1693
2167
2619
3082
49
49
49
49
49
49
49
49
49
J9
49
49
49
49
49
48
48
48
48
jlS
48
48
48
48
4 .-1
to
48
48
48
48
48
48
48
48
48
47
47
47
47
47
47
47
47
47
47
47
47
47
47
47
47
47
47
47
47
40
46
/L(i
46
46
46
tai
16
A TABIiE OP tOGABITHMS FROM 1 TO 10,000.
1* i '
t :
In-
i ^
f
•^ ! i
1 '
i
t 1
N. 1 |l 2|3|4|6|6|7|8l9|D. 1
940
973128
3174
3220
3266
33131
3359
.3405
3451
3497
3543
46
941
3590
3636
3682
3728
3774
3820
3866
.3913
3959
4005
46
942
4051
4097
4143
4189
4235
4281
4327
4374
4420
4466
46
943
4512
4558
4604
4650
4696
4742
4788
4834
4880
4926
46
944
4972
5018
5064
5110
5156
5202
5248
6294
5340
5386
46
943
5432
5478
5524
5570
6616
5662
5707
5753
5799
5845
46
946
5891
5937
5983
6029
6075
6121
6167
6212
6258
6304
46
947
6350
6396
6442
6488
6533
6579
6625
6671
6717
6763
46
948
6808
6854
6900
6946
6992
7037
7083
7129
7175
7220
46
949
950
7266
7312
7769
7358
7815
7403
7861
7449
7906
7495
7952
7541
7998
7586
8043
7632
8089
7678
8135
46
46
977724
951
8181
8226
8272
8317
8363
8409
8454
8500
8546
8591
46
952
8637
8683
8728
8774
8819
8865
8911
8956
9002
904V
46
953
9093
9138
9184
9230
9275
9321
9366
9412
9457
9o03
46
954
9548
9594
9639
9685
9730
9776
9821
9867
9912
99.18
46
955
9S0003
0049
0094
0140
0185
0231
0276
0322
0367
0412
46
95fi
0458
0503
0549
0594
0640
0685
0730
0776
0821
086Y
45
957
0912
0957
1003
1048 10931
1139
1184
1229
1275
1320
45
958
1366
1411
1456
1501
1547
1592
1637
1683
1728
17V3
45
959
960
1819
1864
2316
1909
2362
1954
2407
2000
2452
2045
2497
2090
2543
2135
2588
2181
2633
2226
2678
45
45
982271
961
2723
2769
2814
2859
2904
2949
2994
3040
3085
3130
45
962
3175
3220
:265
3310
3356
3401
3446
3491
3536
3581
45
963
3626
3671
3716
3762
3807
3852
3S97
3942
3987
4032
45
964
4077
4122
4167
4212
4257
4302
HP
47&7
4392
4437
4482
45
965
4527
4572
4617
4662
4707
4752
4842
4887
4932
45
966
4977
5022
5067
5112
5157
5202
5^47
5292
5337
5382
45
967
54'* 6
5471
5516
5561
5606
.5651
5696
5741
5786
5830
45
968
5875
5920
5965
6010
6055
6100
6144
6189
6234
6279
45
969
970
6324
6369
6817
6413
6861
6458
6906
6503
6951
6548
6996
6593
7040
6637
7085
6682
7130
bV2V
7175
45
45
986772
971
7219
7264
7309
7353
7398
7443
7488
7532
7577
V622
45
972
7666
7711
7756
7800
7845
7890
7934
7979
8024
8068
45
973
8113
8157
8202
8247
8291
8336
8381
8425
8470
8Di4
45
974
8559
8604
8648
8693
8737
8782
8826
8871
8916
8960
45
975
9005
9049
9094
9138
9183
9227
9272
9316
9361
9405
45
976
9450
9494
9539
9583
9628
9672
9717
9761
9806
9850
44
977
9895
9939
9983
..28
..72
.117
.161
.206
.250
.294
44
978
990339
0383
0428
0472
0516
0561
0605
0650
0694
0V38
44
979
980
0783
0827
1270
0871
1315
0916
1359
0960
1403
1004
1448
1049
1492
1093
1536
1137
1580
1182
1625
44
44
991226
981
1669
1713
1758
1802
1846
1890
1935
1979
2023
2067
44
982
2111
2156
2200
2244
2288
2333
2377
2421
2465
2!509
44
983
2554
2598
2642
2686
2730
2774
2819
2863
2907
2951
44
984
2995
3039
3083
3127
3172
3216
3260
3304
3348
3392
44
985
3436
3480
3524
3568
3613
3657
3701
3745
3789
3833
44
986
3877
3921
3965
4009
4053
4097
4141
4185
4229
4273
44
987
4317
4361
4405
4449
4493
4537
4.581
4625
4669
47 13
44
988
4757
4801
4845
4889
4933
4977
5021
5065
5108
51.W
44
989
990
6196
5240
5679
5284
5723
5328
5767
5372
.5811
6416
5854
5460
5898
5504
5942
5547
6986
5591
603C
44
44
995635
991
6074
6117
6161
6205
6249
629S
6337
6380
6424
6461^
44
992
6512
6555
6599
6643
G687
6731
6774
6818
6862
690t
44
993
6949
6993
703?
708C
712^
7l6g
7212
7255
7299
734t
1 44
994
7386
7430
7474
- 7517
7561
760.^
. 764S
7695i
7736
77VL
1 44
995
7823
7867
791C
1 7954
- 7998
8041
8085
8128
8172
82 U
) 44
996
fi9?10
sfiny
8347
' 8?iur
8434
L R4.7';
r Hii21
8564
. 8608
86.5i
J 44
997
8695
8731J
8785;
, 8826
886C
1 891?
\ 8956
» 900(
1 9043
gos-i
f 44
998
9131
9174
921.^
92-61
930f
) 934^
i 9395
! 9435
. 9479
952i
J 44
999
9565
960r
1 965i
t 969(
. 973'
) 978[
) 98261 9870
) 991.*3
995'/
f 43
N. |0|l|2|3l4l5 6 7|8 «! D.J
^
46
46
46
46
46
46
46
46
46
46
€
932
4b
45
382
45
830
45
279
45
727
45
'175
45
'622
45
!068
45
S514
45
^960
45
)405
45
)850
44
294
44
)738
44
1182
44
1625
44
2067
44
2509
44
2951
44
«92
44
3833
44
1273
44
4713
44
5152
44
5591
44
6030
44
6468
44
6906
44
7343
44
7779
44
8216
44
8652
44
9087
44
9522
44
9957
43
9
i P. 1
j
A TABLE
OP
LOGARITHMIC
SINES *AND TANGENTS
FOR ETEHT
DEGREE AND MINUTE
or THE QUADRANT.
N. B Tae minutes in the left-hand column of each page,
increasing downwards, belong to the degrees at the top ; and
those increasing upwards, in the right-hand column, belong to
the degrees below.
i t
Mi
•\\
I h
I! il'
18
(0 Degree.) a table of logarithmic
M.
Sine
D.
Cosine
Tang.
D.
CoianR.
1
2
3
4
5
6
7
8
9
10
11
12
\Z
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
GO
O.OOUUOOI
6.463726
764756
940847
7.065786
162696
241877
308824
366816
417968
463725
7.505118
542906
577668
609853
639816
667845
694173
718997
742477
764754
.'ioirn
293485
208231
161517
131968
111575
96653
85254
76263
68988
7.785943
806146
825451
843934
861662
878695
895085
910879
926119
940842
7.955082
968870
982233
995198
8.007787
020021
031919
043501
054781
065776
8.076500
086965
097183
107167
116926
126471
135810
144953
153907
16 2681
8.171280
179713
187985
196102
204070
211895
219581
227134
234557
241855
Co.'iine I
10.090000
000000
000000
000000
000000
000000
9.999999
999999
999999
999999
999998
62981
57936
53641
49938
46714
43881
41372
39135
37127
35315
33672
32175
30805
29547
28388
27317
26323
25399
24538
23733
9.999998
999997
999997
999995
999996
999995
999995
999994
999993
999993
22980
22273
21608
20981
203901
19831
19302
18801
18325
17872
9.999992
999991
999990
999989
999988
999988
999987
999986
999985
999983
9.999982
999981
999980
999979
999977
999976
999975
999973
999972
999971
00
00
00
00
00
01
01
01
01
Oj_
01
01
01
01
01
01
01
01
01
21
01
01
01
02
02
02
02
02
02
02
02
02
02
02
02
02
02
02
02
02
0.000000
6.403726
764756
940847
7.065786
162696
241878
308825
366817
417970
463727
501717
293483
208231
161617
131969
111578
99653
852.')4
76263
68988
7.. 505 120
542909
577672
609857
639820
667849
694179
719003
742484
764761
62981
57933
53642
49939
46715
4388
41373
39136
37128
35136
7.785951
806155
825460
843944
861674
878708
895099
910894
926134
940858
iiiiiiiiio.
13.536274
235244
059153
12.934214
837.304
758122
691175
633183
582030
536273
12
.494880
4.57091
422328
390143
360180
332151
305821
280997
257616
235239
17441
17031
16639
16265
15908
15566
15238
14924
14622
1433 3
14054
1.3786
13529
132.S0;
130411
12310
12587!
12372
12164
11963
9.999969
999968
999966
999964
999963
999961
999959
999958
999956
999954
9,'jy9952
999950
999948
999946
999944
999942
; '9940
9;i;>'138
i; 99936
*ii!'j'934
7.955100
968889
982253
995219
8.007809
020045
031945
043527
054809
065806
33673
32176
30806
29549
28390
27318
26325
25401
24540
23735
02
02
02
03
03
03
03
03
03 1
03
03
03
03
03
(^3
('4
04
04
04
04
8,076.531
086997
097217
107202
116963
126510
135851
144996
1.53952
162727
12.214049
193845
174540
156056
138326
121292
104901
089106
073866
059142
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
12
39
38
37
36
35
34
33
32
31
30
7
22981
22275
21610
20983
20392
19833
19305
18803
18327
1787 4
17444
17034
16642
16268
15910
15568
15241
14927
14627
14336
12.044900
031111
017747
004781
11.992191
9799.55
968055
956473
945191
934194
8.171328
179763
188036
196156
204126
2119,53
219641
227195
2.34621
241921
11.923469
913003
902783
892797
883037
873490
864149
855004
846048
837273
29
28
27
26
25
24
23
22
21
20
19'
18
17
16
15
14
13
12
11
10
14057
13790
13532
1.3284
13044
12814
12590
12376
12168
11967
11
828672
8202371
811964
8038441
795R74
788047
780359
772805
765379
758079
Sine
(J'jlang.
Taiic.
M.
8U Uogreoii.
M.
e
8.2
1
2
2
2
3
2
4
2
5
2
6
7
2
9
51
8.50
52
51
53
51
54
52(
n n
(J<i
0*
56
52J
57
53
58
53f
59
53(
60
54S
Cosi
;i- 1 1
u.
60
274
59
iU
58
15:5
57
214
56
^04
65
122
54
175
53
183
52
1)3()
51
273
50
S8()
49
091
48
328
47
143
46
ISO
45
151
44
821
43
997
42
616
41
239
4')
049
39
845
38
540
37
056
36
or
3844
1
8
5874
6
8047
4
0359
3
'2805
2
55379
1
)8079
SINES AND TA voENTs. (1 Degree.)
19
M.
Sine
1
2
3
4
5
6
7
8
9
10
11
12
13
i4
15
16
17
18
'0
21
23
24
25
20
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
4-8
49
50
51
52
53
54
55
56
57
58
59
60
8.241855
249033
256094
263042
269881
276014
283243
289773
296207
302546
308794
8,314954
321027
327016
:".2924
338753
344504
350181
355783
361315
366777
8.372171
3771i)9
382762
387962
393101
398179
403199
408161
413068
417919
8.422717
427462
432156
436''00
441394
445941
450440
454893
459301
463665
8
467985
472263
476498
480693
484848
488963
493040
497078
501080
505045
.508974
512867
516726
520551
521343
528102
531828
535523
539186
542819
D.
Cosine | P. | Tang. [ I) |
11963
11768
i 1.580
11398
11221
11050
10883
10721
10565
10413
10266
Coinng.
10122
9982
9847
9714
9586
9460
9338
9219
9103
8990
8880
8772
8667
8564
8464
8366
8271
8177
8086
7996
7909
7823
7740
7657
7577
7499
7422
7346
7273
7200
7129
7060
6991
6924
6859
6794
6731
6669
6608
6548
6489
6431
6375
6319
62G4
6211
6158
6106
6055
6004
9.999934
999932
999929
999927
999925
999922
999920
999918
999915
999913
999910
9.999907
999905
999902
999899
999897
999894
999891
999888
999885
999882
9.999879
999876
999873
999870
999867
999864
999861
999858
999854
999851
9.999848
999844
999841
999838
999834
999831
999827
999823
999820
999816
9.999812
999809
999805
999801
999797
999793
999790
999788
999782
999778
9.999774
999769
999765
999761
999757
999753
999748
999744
999740
999735
04
04
04
04
04
04
04
04
04
04
04
04
04
04
0")
05
05
05
05
05
05
05
05
05
05
05
05
05
05
05
06
06
06
06
06
06
06
06
06
06
06
06
06
06
06
07
07
07
07
07
07
07
07
07
07
07
07
07
07
07
07
8.241921
249102
25<" ' 65
26311.';
269956
276691
283323
289850
296292
302634
308884
11967
11772
11584
11402
11225
11054
10887
10726
10570
10418
10270
8.315046
321122
327114
333025J
338856
344610
350289
355895
361430
366895
8.372292
377622
382889
388092
393234
398315
403338
408304
413213
418068
10126
9987
9851
9719
9590
9465
9343
9224
9108
8995
8.422869
427618
432315
436962
441560
446110
8885
8777
8672
8570
8470
8371
8276
8182
8091
8002
7914
7830
7745
7663
7583
7505
450613
7428
455070
7352
459481
7279
463849
7206
8.468172
71.'^:>
472454
7066
476693
6998
480892
6931
485050
6865
489170
6801
493250
6738
497293
6676
501298
6615
505267
6555
8.509200
6496
513098
6439
516961
6382
520790
6326
524580
6272
628349
6218
532080
6165
535779
6113
539447
6062
543081
6012
Cosine
Sine
Cotaiig.
11
. 7.08079
750898
74383.'
736885
730044
723309
716677
710144
703708
697366
691116
11.684954
678878
672886
666975
661144
6.55390
619711
644105
638570
633105
11.627708
622378
617111
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
6119081,36
606766
601685
696662
591696
586787
581932
11.577131
6723821
567685
563038
558440
553890
649.387
644930
640519
536151
11
531828
527.546
623307
519108
614950
510830
506750
502707
498702
491733
11.490800
486902
483039
479210
47.5414
471651
467920
464221
460553
456916
35
34
33
32
31
30
29-
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
B
5
4
3
2
I
4
Tang.
Degrees.
M.
i :
i
1 '
1 "'*
!;;
20
M.
(2 Degrees.) a table op logarithmic
diue
!).
('osiiie
D.
Tftnp.
D.
("lotaiiR.
S
1
2
3
4
5
6
7
8
9
10
U
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
5^
56
67
58
59
60
.542819
646122
549'J95
553539
557054
560540
663999
567431
570836
574214
577566
6004
6955
5906
68)8
5811
5765
6719
6674
f630
/>587
5544
.5808!)2
584193
687469
6907il
693948
59715'i
600332
603489
606623
609734
.612823
615891
618937
621962
624965
627948
630911
633854
636776
639680
.642563
645428
648274
651102
653911
656702
659475
662230
664968
667689
.670393"
.73080
675751
678405
681043
683665
686272
688863
691438
693998
8.696543
699073
701589
704090
706577
709049
711507
713952
716383
718800
6502
5460
5419
5379
6339
6300
5261
6223
5186
5149
6112
6076
5041
6006
4972
4933
4904
4871
4839
4806
4775
4743
4712
4682
4652
4622
4592
4563
4535
450 6
4479'
4451
4424
4397
4370
4344
4318
4292
4267
4242
4217
4192
4168
4144
4121
4097
4074
4051
4029
4006
9.999735
999731
999726
999722
999717
999713
999708
999704
999699
999694
999689
9.999685
999680
999675
999670
' 999665
999660
999655
999650
999645
999640
.999035
999629
999624
999619
999614
999608
999603
999597
999592
999586
9.999581
999575
999570
999564
9995.1S
9995^3
999547
999541
999535
999529
9.99~9524
999518
999512
999506
999500
999493
999487
999481
999475
999469
.999463
999456
999450
999443
999437
999431
999424
999418
999411
999404
07
07
07
08
08
08
08
08
08
08
08
08
08
08
08
08
08
08
09
09
09
09
09
09
09
09
09
09
09
09
09
09
09
09
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
11
11
11
11
11
11
11
u
11
11
8.543084
646691
550268
553817
557336
560828
564291
667727
571137
574520
577877
8.581208
684514
587795
.591051
694283
597492
600677
603839
606978
610094
8,
613189
r 16262
619313
622343
625352
628340
631308
634256
637184
640093
.642982
645853
648704
651537
654352
657149
659928
662689
665433
668160
8.670870
673563
676239
678900
681544
684172
686784
689331
691963
694529
8.697081
699617
702139
704646
707140
709618
712083
714534
716972
719396
6012
6962
5914
5866
5819
6773
6727
6682
6638
5.595
6552
111,
5510
5468
6427
6387
5347
5308
5270
5232
5194
5158
5121
5085
5050
6015
4981
4947
4913
4880
4848
4816
4784
4753
4722
4691
4661
4631
4602
4573
4544
4526
4488
4461
4434
4417
4380
4354
4328
4303
4877
4262
4228
4203
4179
4155
413--4
4108
4085
4062
4040
4017
456916
453309
449732
446183
442664
439172
435709
432273
428863
425480
422123
11
.418792
415486
412205
408949
405717
402508
399323
396161
393022
389906
11.386811
383738
380687
377657
374648
371660
368692
365744
362816
359907
11.357018
354147
351296
348463
.345648
342851
340072
337311
334567
331840
11.329130
326437
323761
321100
318456
315828
313216
310619
308037
305471
11.302919
300383
297861
295354
*rr2S60
290382
287917
285465
283028
280304
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
Cosine
Sine
Cotanc.
Tang.
87 Degreefl.
M
1 Sii
d.7U
1
72
2
721
3
72{
4
72t
6
73(
6
73r
7
73£
8
737
9
73£
10
745
11
8.744
12
746
13
74fl
14
761
15
752
16
75S
17
757
18
760
19
762
20
21
764
8.766
22
768
23
770
24
773
25
775
26
777
27
779
28
781
29
783
30
31
786
8.787
32
789
33
791
34
793
35
795
36
797
37
799
38
801
39
803
40
41
805
8.807
42
809
43
811
44
813
45
815
46
817
47
819
48
821.
49
823'
50
825
51
8.827(
52
828J
53
830'
54
832(
55
834^
56
8365
57
838]
58
839f
59
8417
60
843£
R- 1 1
)916
60
1309
f)9
1732
58
)183
57
5664
56
)172
55
)709
54
5273
53
i863
52
vl80
51
5123
50
^792
49
i486
48
2205
47
^949
46
5717
45
i.008
44
J323
43
1161
42
3022
41
3906
40
7018
29
4147
28
1296
27
8463
26
5648
25
2851
24
0072
23
7311
22
4567
21
1840
20
9130
19
6437
18
3761
17
1100
16
8456
15
5828
14
3216
13
0619
12
8037
11
5471
10
2919
9
0383
8
7861
7
5354
6
2860
10382
4
7917
3
5465
2
3028
1
!0304
ig-
jmJ
SINES AND TANGENTS. (3 Degrees.;
fit
M.
Sine
D.
Cosine I).
'i:
.■•lie.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
8.718800
721204
723595
725972
728337
730688
733027
735354
737667
739969
742259
n.
8.744536
746802
749055
751297
753528
755747
757955
760151
762337
764511
8.766675
768828
770970
773101
775223
777333
779434
781524
783605
785675
8.787736
789787
791828
793859
795881
797894
799897
801892
803876
805852
8.807819
809777
811726
813667
815599
817522
819436
821343
823240
825130
8.827011
828884
830749
832607
834456
836297
838130
839956
841774
843585
4006
3984
3962
3941
3919
3898
3877
3857
3836
3816
3796
3776
3756
3737
3717
3698
3679
3661
3642
3624
3606
3588
3570
3553
3535
3518
3501
3484
3467
3451
3431
3418
3402
3386
3370
3354
3339
3323
3308
3293
3278
3263
3249
3234
3219
3205
3191
3177
3163
3149
3135
3122
3108
3095
3082
3069
3056
3043
3030
3017
3000
9.999404
999398
999391
999384
999378
999371
999364
999357
999350
999343
999336
9.999329
999322
999315
999308
999301
999294
999286
999270
999272
999265
9.999257
999250
999242
999235
999227
999220
999212
999205
999197
• 999189
1118.719396
9.999181
999174
999166
999158
999150
999142
999134
999126
999118
999110
9.999102
999094
999086
999077
999069
999061
999053
999044
999036
999027
9.999019
999010
999002
998993
998976
998967
998958
998950
998941
11
11
11
11
11
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
14
14
14
14
14
14
14
14
U
14
14
14
14
14
14
15
15
15
15
721806
724204
726588
728959
731317
733663
735996
738317
740626
742922
8.745207
747479
749740
751989
7.54227
756453
758668
760872
763065
765246
8.767417
769578
771727
773866
775995
778114
780222
782320
784408
786486
8.788554
790613
792662
794701
796731
798752
800763
802765
804758
806742
8.808717
810683
812641
814589
816529
818461
820384
822298
824205
826103
8.827992
829874
831748
833613
837321
839163
840998
842825
844644
("otanB.
4017
3995
3974
3952
3930
3909
3889
3868
3848
3827
3807
3787
3768
3749
3729
3710
3692
3673
36t)5
3636
3618
11.2806041 60
278194' 5!»
275796
273412
271041
268683
266337
264004
261683
259374
257078
11
3600
3583
3565
3548
3531
3514
3497
3480
3464
3447
254793
2.52521
250260
243011
245773
243547
241332
239128
236935
234754
58
57
56
55
54
53
52
51
50
11.232583
,230422
f'28273
r 26 134
224005
221886
219778
217680
215592
213514
3431
3414
3399
3383
3368
3352
3337
3322
3307
3292
3278
3262
3248
3233
3219
3205
3191
3177
3163
_31^50_
3136
3123
3110
3096
ouoo
3070
3067
3045
3032
3019
11.211446
209387
207338
205299
203269
201248
199237
197235
195242
1932.58
11.191283
1893J7
187359
185411
183471
181539
179616
177702
175795
173897
11.172008
170126
168252
166387
1 64529
162679
160837
159002
167175
155366
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
II
86 Degrees.
14
23
(4 Degrees.') a table op LooAniTiiMicj
M.
Siiiti
' I.
i
/ »
I-
1
2
;j
4
5
7
R
9
10
11
12
i;}
M
if)
Ifi
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
[34
35
36
37
38
39
40
S.HiarvM;')
Hir,3M7
847183
818971
8r)07r)l
852.W5
854291
8r)(;()19
85780 1
85951fi
_8(512t[3
8.S(';iOri
8(i4738
8(56455
808165
869868
871565
873255
874938
876615
878285
8.879949
881607
883258
884903
886542
888174
889801
891421
893035
894643
I).
'3005
2992
29H0
2967
2955
2943
293 1
2919
2907
2896
288 i_
',1873
2861
2850
2839
2828
2817
2806
2795
2786
2773
,896246
897842
899432
901017
902596
904169
905736
907297
908853
910404
41 8
42
43
44
45
46
47
48
49
50
61 8
52
53
54
ub
56
57
58
59
60
,911949
913488
915022
916550
918073
919591
921103
922610
924112
925609
,927100
928587
930068
931544
933015
934481
935942
937398
938850
94029 6
Cosine
27(53
2752
2742
2731
2721
2711
2700
2690
2ti80
2670
2660
2651
2641
2631
2622
2612
2603
2593
2584
2575
2566
2556
2547
2538
2529
2520
2512
2503
2494
2486
CoHliie i
9 . 99894 i
998932
9!)H923
998914
998905
99889(5
998887
998878
9988(59
9988(50
998851
I).
'I'aiiR.
9
,998841
998832
998823
998813
998801
998795
998785
998776
9987(56
99 8757
9.998747
998738
998728
998718
998708
998699
998689
998679
998669
998659
2477
2469
2460
2452
2'i'io
2435
2427
2419
2411
2403
9.998649
998639
998629
998619
998609
998599
998589
998578
998568
9 9 8558
9.998548
998537
998527
998516
998506
998495
998485
998474
998464
998453
9.998442
998431
998421
!5
15
15
15
15
15
15
15
15
15
]5
15
15
16
16
16
16
16
10
16
16
16
16
16
16
16
16
16
16
17
17
17
17
17
17
17
17
17
17
17
17
17
17
17
18
18
18
18
18
18
18
18
18
18
8.844644
81(5455
818260
850057
85 181 (5
853(528
855403
857 1 7 1
858932
860(58(5
862433
8.864173
86590(5
867(532
869351
8710(54
872770
874469
8761(52
877849
879529
8.881202
882869
884530
886185
887833
889476
891112
892742
894366
89 5984
8.897596
899203
900803
902398
903987
905570
907147
908719
910285
911846
"^3019
3007
2995
2982
2970
2958
2946
2935
2923
29 1 1
2900
8
998410 18
998388
998377
998366
998355
998344
18
18
18
18
18
.913401
914951
916495
918034
919568
921096
922819
924136
925649
927156
8.928658
930155
931647
933134
934616
936093
937565
939032
940494
941952
"117155356"'
153.545
151740
149943
1481.54
146372
144597
142829
1410t;8
2888
2877
2866
2854
2843
283?.
2821
2811
2800
2789
2779
2768
2758
2747
2737
2727
2717
2707
2097
2687
2677
2667
2658
2648
^638
2629
2620
2610
2601
2592
11.
139314
_nr/5(57
135827
134094
132368
130(549
128936
127230
12.5.531
123838
122151
120471
11.118798
117131
115470
113815
112167
110524
108888
1072.58
105634
101016
2583
2574
2565
25.56
2547
2538
2530
2521
2512
2503
2495
2486
2478
2470
2461
2453
2445
2437
2430
2421
11.102404
100797
099197
097602
096013
094430
092853
091281
089715
088154
GO
59
58
57
56
55
54
53
52
51
50
4!>
48
47
46
45
44
43
42
41
40
39
38
37
3(5
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
11.086599
085049
083505
081966
080432
078904
07738 1
075864
074351
072844
11.0 1342
009845
068353
066866
065384
063907
062435
060968
059506
058048
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
M
"IT
1
2
3
4
5
6
t
8
9
10
IJ
12
13
14
15
16
17 i
18
19
'^0
21
2i>
23
24
25
26
27
28
29
.30
31
32
33
34
35
36
37
38
39
40
8.9
9
9'
9
9'1
91
91
9:
9.^
9^
8.95
d5
95
9(i
9(i
9(5
I 96
9(5
96
JM5
8". 96
97
97
97;
97
97(
97'
97*
98(
98]
8.98i
98^
98f
98f
98fc
981]
990
991
993
991
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Sine
Colang.
Tang.
M.
8.995
997
91; s
099
9.000
002
003
OOti
OOfil
00 7(
9.008;
009.'
010'
0111
013]
OW,
01. 5f
oioe
018C
019S
Cosine
85 Degree*.
K' i
•AM]
f.O
Mf)
na
710
58
91 :j
r)7
IM
f>r)
M-Z
nr,
r)'j7
r)4
Hi>U
M
OfiH
r)a
:n4
r)()7
f)!
f)!)
BINES AND TANOKN'M. (5 DogroOS.)
23
M
Hino
I).
('osliio
I).
1
2
.1
4
6
fi
t
8
i)
H)
11
12
U)
11
IT)
Hi
17
I 18
lit
'ii)
21
2!)
21
25
20
27
28
2a
30
iii
.12
33
31
3;-)
30
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
TaiiK.
I>.
8.910296
91 1 738
913171
911606
916034
917456
948874
950287
95169(
953100
_9544J>9
8.955894
957284
958670
960052
961429
962801
964170
965531
966893
96'I249
8.969600
970947
972289
973628
974962
976293
977619
978941
980259
981573
8.982883
984189
98.5491
986789
988083
989374
990660
991943
9932221
994497
2103
2394
2387
2379
2371
2363
23.55
2348
2340
2332
2325
2317
2310
2302
2295
2288
2280
2273
2266
2259
2252
8
,99.5768
997'>36
91; ,i99
099560
000816
002069
003318
004563
005805
007044
2244
2238
223 1
2224
2217
2210
2203
2197
2190
2183
2177
2170
2163
2157
3150
2144
21.38
2131
2125
2119
.008278
009510
010737
011962
013182
014400
01,5613
016824
018031
019235
2112
2106
2100
2094
2087
2082
2076
2070
2064
20.58
2052
2046
2040
2034
2029
2023
2017
2012
2006
2000
9,998314
998333
998322
998311
998300
998289
998277
998266
998255
998243
99 8232
9.998220
998209
998197
998186
998174
998163
998151
998139
998128
998116
9.998104
998092
998080
998068
998056
998044
998032
998020
998008
997996
19
19
19
19
19
19
19
19
19
19
j9
19
19
19
19
19
19
19
20
20
20
9.997984
997972
997959
997947
997935
997922
997910
997897
997885
997872
.997860
997847
997835
997822
997809
997797
997784
997771
997758
997745
.997732
997719
997706
997693
997nrt0
997667
9976.54
997641
997628
997614122
20
20
20
20
20
20
20
20
20
20
20
20
20
20
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
22
22
22
22
22
22
8.9419.52
94310J
944852
. 946295
917734
949168
9.50597
952021
95.34 1 1
954856
956267
8.9.57674
959075
9()0473
961866
963255
964639
966019
967394
968766
970133
8.971496
972855
974209
9755C.0
970906
978248
979.586
980921
98225 1
983577
8.984899
986217
987532
9888421
990149
991451
992750
994045
995.337
996624
8.997908
999188
9.000465
001738
003007
004272
005534
006792
008047
009298
9.010.546
011790
01.3031
014268
ni?i5n9
016732
017959
019183
020403
021620
4'()tiiti|{.
2421
2113
2405
2.397
2390
2382
2374
2366
2360
2351
2344
2337
2329
2323
2314
2307
2300
2293
2286
2279
2271
2265
2257
2251
2244
2237
2230
2223
2217
2210
2204
2197
2191
2184
2178
2171
2165
21.58
2152
3146
2140
21.34
2127
2121
2115
2109
2103
2097
2091
2085
2080
2074
2068
2062
2056
3051
2045
2040
2033
2028
2023
Jopine
Sine
Cotang.
111.0.58048
I 056596
055148
053705
05226(i
0.50832
049403
047979
040559
045144
0^3733
11.042326
040<)25
039527
038134
036745
035361
03398 1
032606
031234
029807
11.028.501
027145
025791
024440
023094
0217.52
020414
019079
017749
016423
11.015101
013783
012468
0111.58
009851
008.549
007250
005955
004663
003376
10
11,002092
000812
, 999535
998262
996993
995728
994466
993208
9919.53
9 90702
10.9894.54
988210
986969
985732
98.3268
982041
98081/
979597
978380
60
59
58
57
56
55
54
63
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
.30
29
28
27
26
25
24
23
23
21
20
19
18
17
16
15
14
13
12
11
10
~9
8
7
6
u
4
3
2
1
X
64i)egrees,
I Tang. I M
24
(6 Degrees.) A TABLE OF LOGARmiMIC
iih
M
Hliie
D.
Comno 1 1).
Tang. 1
D.
Cutang. 1 1
9.019235
2000
9.997614
22
9.021620
2023
10.978380
'60
1
0204;ir)
1995
997601
22
022834
2017
977166
69
2
O'ZUy.VZ
1989
997588
22
024044
2011
975956
58
3
022825
1984
997574
22
025251
2006
974749
57
4
024016
1978
997561
22
026455
2000
973545
56
6
025203
:?73
997547
22
027655
1995
972345
55
6
0263H6
1967
997534
23
028852
1990
971148
54
7
0275f)7
1962
997520
23
030046
1985
969954
53
8
028744
1957
997507
23
031237
1979
968763
52
9
029918
1951
997493
23
032425
1974
967575
51
10
11
031089
9.0;J2257
1947
997480
9.997466
23
23
033609
1909
966391
10.965209
50
49
1941
9.034791
1964
12
033421
1936
997452
23
035969
1958
964031
48
13
034582
1930
997439
23
037 44
1953
96285^
47
14
035741
1925
997425
23
038316
1948
96168. ifil
15
036896
1920
997411
23
039485
1943
960515
4.1
16
038048
1915
997397
23
040651
1938
959349
44
17
039197
1910
997383
23
041813
1933
958187
43
18
040342
1905
997369
23
042973
1928
957027
42
19
041485
1899
9973{;^
23
044130
1923
955870
41
20
042625
1894
997341
23
24
045284
1918
954716
40
39
9.043762
1889
9.997327
9.046434
1913
lO.tJ.^3566
22
044895
1884
997313
24
047682
1908
952418
38
23
046026
1879
997299
24
048727
1903
951273
37
24
047154
1875
997285
24
049869
1898
950131
36
25
048279
1870
997271
24
051008
1893
948992
35
20
049400
1865
997257
24
052144
1889
947856
34
27
050519
1860
997242
24
053277
1884
946723
33
28
051635
1855
997228
24
054407
1879
945593
32
29
05274;;
1850
997214
24
055535
1874
944465
31
30
31
0538.')9
1845
997199
9.997185
24
24
056659
9.0,57781
1870
1865
943341
30
29
054966
1841
10.942219
32
056071
1836
997170
24
058900
1869
941100
28
33
057172
1831
997156
24
060016
1855
939984
27
34
058271
1827
997141
24
061130
1851
938870
26
35
059367
1822
997127
24
062240
1846
937760
25
36
060460
1817
997112
24
063348
1842
936652
24
37
061551
1813
997098
24
064453
1837
935547
23
38
002639
1803
997083
25
065556
1833
9344 '4
22
39
063724
1804
997068
25
066655
1828
933345
21
40
41
064806
9.065885
1799
997053
25
25
067752
1824
932248
20
19
1794
9.997039
9.068846
1819
10.9311.54
42
066962
1790
997024
25
069938
1815
930062
18
43
068036
1786
997009
25
071027
1810
928973
17
44
069107
1781
996994
25
072113
1806
927887
16
45
070176
1777
996979
25
073197
1802
926803
15
46
07 1 242
1772
996964
25
074278
1797
925722
14
47
072306
1768
996949
25
075356
1793
924644
13
48
073366
1763
996934
25
076432
1789
923568
12
49
074424
1759
996919
25
077505
1784
922495
11
50
51
075480
1755
996904
9.996889
25
25
078576
i 9.079644
1780
921424
10
9
9.076533
1750
1776
10.920356
62
077583
1746
996874
25
080710
1772
919290
8
53
078031
1742
996858
25
081773
1767
918227
7
64
079676
1738
996843
25
082833
1763
917167
6
55
080719
1733
996828
25
088891
1759
916109
5
56
081759
1729
996812
26
084947
17.55
Q15053
4
57
0a27fl7
1725
996797
26
OAROOO
1751
M/lfiOQ
s
58
083832
1721
996782
26
087050
1747
91295C
2
69
084864
1717
996766
26
088098
1743
911902
1
60
085894
1713
996751
26
089144
1738
910856
1 Cosine
Sine
1 (Jotanij.
I Tang. 1 M. 1
lif ;
83 Degrees.
M
1 8
U
y.ol
1
Oi
2
Of
3
1
4
1
5
r .J.
b
I ')'
7
8
(,"
9
\J.
10
rr
11
•K.
12
13
14
Ud
09
10
-5
10
16
10
17
10
18
10
19
10
20
10
21
9.10
22
10
23
10
24
10
25
11
26
11
27
11
28
11
29
11
30
11
31
9.11
32
11
33
11
34
11
35
12(
36
12
37
12
38
12;
39
12'
40
12i
41
9.12f
B-^^
12'
43
12'
44
12f
45
12f
46
13<
47
131
48
1.3S
49
l.^.*^
50
VM
51
9.13E
52
136
53
137
54
138
55
13t]
56
13fJ
r r/
kM
14u
58
141
59
142
60
143
1 1
so
60
w
60
)(\
58
1!>
57
If)
56
IS
55
18
54
54
53
^n
52
75
51
91
50
D'J
49
n
48
5r
47
?. i6|
15
4a
19
44
87
43
27
42
70
41
16
40
0(5
39
IH
38
73
37
31
36
92
35
50
34
23
33
93
32
65
31
41
30
19
29
00
28
84
27
70
26
60
25
52
24
47
23
'A
22
45
21
48
20
54
19
62
18
73
17
87
16
03
15
22
14
44
13
68
12
95
11
24
10
56
9
,90
8
!27
7
67
6
09
5
)53
4
>0Q
3
)5C
2
)02
1
i5G
|M.
♦aWEfl AND TANGENTS. (7 DcgrCf)3.)
2b
M.
Sine
I D. I r<w inB I I). I
TniiK.
I).
1
2
3
4
5
b
7
8
9
10
11
12
13
14
-5
16
17
18
19
20
21
22
23
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
-42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
o t
58
59
60
y. 086894
086922
087947
'. '-u,
08!;9frj
U0!.'.)8
0<>'^024
>!»;>, ^37
0<Mt i7
u,».//:56
_rnfi062
'>.»l9:;)65
U98066
099065
100062
101056
102048
103037
104025
105010
__105992
9.106973
107951
108927
109901
110873
111842
112809
113774
114737
115698
9.116656
117613
118567
119519
120469
121417
122362
123306
124248
125187
.126125
127060
127993
128925
129854
130781
131706
132630
133551
134470
9.135CS7
136303
137210
138128
139037
139944
1408.j0
1417.54
142656
143555
1713
1709
1704
1700
1696
1692
1688
1684
1680
1676
1673
1668
1665
1661
1657
1653
1649
1645
1641
1638
1634
1630
1627
1623
1619
1616
1612
1608
1605
1601
1597
1594
1590
1587
1683
1.580
1576
1.573
1569
1.566
1.562
1,559
1.5.56
1552
1549
1545
1.542
1539
1.535-
1532
1.529
1.525
1.522
1519
1516
1512
1.509
150G
1.503
1500
1496
9.996751
996735
996 720
996704
996688
99667 1
996657
99664 1
996625
996610
986594
9.996578
9965C2
996546
996530
996514
996498
996482
996465
996449
996433
9.996417
996400
996o84
996368
99635 1
996335
996318
996302
996285
996269
9.9962.52
996235
996219
996202
996185
996168
996151
996134
996117
996100
.996083
996066
996049
9960.32
J'^6015
995998
995980
995963
995946
995928
9.99.5911
995P94
99J876
995859
995841
995823
995806
995788
995771
995753
26
26
26
26
26
20
26
26
26
26
26
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
27
28
28
28
28
28
28
28
28
28
28
28
28
28
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
29
I <^>l«tlR, (
9.089144
090187
091228
092266
093302
094336
095367
096395
097422
098446
09946S
9.100487
101504
102519
103532
104.542
105550
106556
107559
108560
109559
9.
110.556
1U55I
112.543
1136.33
114.521
115.507
116491
117472
118452
1 19429
.120404
121377
122348
12.3317
124284
125249
126211
127172
128130
129087
9.130041
1.30994
131944
132893
133839
1.34784
135726
136667
137605
lo8542
9.1.39476
140409
141340
142269
143196
144121
145044
145966
146885
147803
1738
734
1730
1727
1722
1719
1715
1711
1707
1703
1699
1695
1691
1687
1684
1680
1676
1672
1669
1665
1661
1658
16.54
1650
1646
1643
1639
1633
16.32
1629
1625
1622
1613
1615
1311
1607
1604
1601
1.597
1594
1591
10.910856
909813
908772
907734
906698
905664
904633
903605
90 .;i78
9015.54
900532
10.899513
898496
897481
896468
895458
894450
893444
89244 1
891440
890441
1587
1.584
1581
1577
1574
1571
1.567
1564
1.561
1658
10.889444
888449
8874.57
886467
885479
884493
883509
882528
881.548
880571
10.879596
29
878623
28
8776.52
27
376683
26
87.5716
26
874751
24
873789
23
872828
22
871870
21
870913
20
60
59
58
57
56
55
64
53
62
51
50
''9
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
19
18
17
16
16
14
13
12
1.555
1551
1548
1545
1.542
1539
1535
1532
1.529
1.526
10.8;/>959
86'i, 06
86ft .06
867107
866161
865216
864274
863333
862395 11
861458 10
9
8
7
6
5
4.
3
2
1
10.860624
8.59591
858660
857731
856804
8.55879
854956
854034
863115
852197
Oosini!
Cotang.
Tung.
|M.
Wf
r- '-'
t
26
(«
Degrees. ) a table of LOGARixiimc
■
M.
Sine
D. 1
Cosine 1 D. |
Taii^'. 1 D. 1
Cotai^;. 1 i
prr
9.143555
1496
9 . 995753
30 •
9,147803 1526
10.8.52197
60
019
1
—
1
9
M4453
1493
995735
30
148718 1523
851282
59
1
145349
1490
995717
30
149632
1520
8.50368
58
r
3
140243
1437
995099
30
150541
1517
849456
57
.I
4
147136
1484
995681
30
151454
1514
818.540
56 1
4
M
5
148026
1481
995604
30
1.52303
1611
847637
55
5
C
148915
1478
995040
30
153269
1608
846731
54
6
t
7
149802
1476
995028
30
154174
1505
845820
53
7
8
150686
1472
995610
30
15.5077
1502
84^1923
52
8
9
151569
1469
995.591
30
155978
1499
844022
21
9
nM
* 1
10
11
152451
1466
1403
995573
9.9955,55
30
30
156877
9.157775
1496
843123
49
10
] 1
Ml
9 153330
1493
10.842225
9
f;"
12
154208
1460
995537
30
168671
1490
841329
48
12'
r .
lo
155083
1457
995519
30
159565
1487
840435
47
13
14
155957
1454
995501
31
160457
1484
839543
46
14
1
15
156830
1451
995482
31
161.347
1481
838653
45
15
16
157700
1448
995464
31
102236
1479
837764
44
16
1
17
158509
1^*45
995446
31
163123
1476
830877
43
17
1:^
18
150' 35
1442
995427
31
164008
1473
835992
42
18
1 '■
19
100301
1439
99.5400
31
104892
1470
835108
41
19
■
20
21
161164
1436
1433
995390
3'
31
165774
9.166054
1467
1404
834226
40
39
20
21
9.162025
9.995372
10.833340
9.
!;■■
\-
22
162885
1430
995353
31
16i'532
1401
832408
38
22
'^1
23
163743
1427
995334
31
168409
14.58
83159'
37
23
l!
24
164600
1424
99.5316
31
109284
1455
830716
36
24
25
165454
1422
995297
31
170157
1453
829843
35
25
26
106307
1-^19
995278
31
171029
14.50
828971
34
26
27
167159
1416
995260
31
171899
1447
828101
33
27
28
168008
1413
995241
32
172707
1444
827233
32
28
2j
168850
1410
995222
32
173634
1442
826366
31
29
30
31
169702
1407
1405
995203
32
32
174499
1439
1430
82.5.501
CO
29
80
31
9.170547
9.995184
9.175362
10.824038
9?
32
171389
1402
995165
32
1 70224
1433
823770
28
32
33
172230
1399
995140
32
177084
1431
822916
27
33
34
173070
1396
995127
32
177942
1428
822058
26
34
35
173908
1394
995108
ijf-*
178799
1425
821201
25
35
36
174744
1391
995089
32
1796.55
1423
820345
24
36
37
175578
1388
995070
32
180508
1420
819492
23
3/
38
176411
1386
99.5051
32
181360
1417
SI 8640
22 1 1
38
«
39
177242
1383
996032
32
182211
1415
817789
21
39
<
40
41
178072
9.1789«)0
1380
1377
99.5013
33
32
1830.'^,9
1412
816941
10.810093
20
19
40
41
(
9.994993
9.183907
1409
Q i
• ■ ;
42
179726
1.374
994974
32
184752
1407
81.5248
18
a: 1
42
i7 * A
s .
i
43
180551
1372
994955
32
185.597
1404
814403
17
43
A
44
181374
1369
994935
32
180439
1402
813501
16
44
45
182196
1366
994916
33
187280
139 J
812720
15
45
A
K
46
183016
1304
99-1896
33
188120
1390
811880
14
40
S
2
2
47
183834
1361
C94877
33
188958
1393
811042
13
47
48
184651
1359
994857
33
189794
' 1391
810200
12
48
49
185460
1350
994838
33
190629
1389
809371
11
''9
50
51
186280
] 353
1.351
994818
33
33
191462
9.192294
13S6
f384
808538
10
9
50
51
'I
9.187092
9.994798
10.H0770r
972
52
187903
1348
994779
33
193124
1381
800876
8
52
53
188712
1346
994759
33
193953
1379
806047
' 7
53
54
189519
1343
994739
33
194780
1376
80522(
I 6
54
2
2
2
55
190325
1341
994719
133
195600
1374
80439'
\ 5
55
56
191130
1338
99470(
33
- 190430
1371
803571
» 4
56
57
191933
1330
994G80
33
197253
1369
802747
^ 3
VI
1927-W
1333
994060
33
1 08074
1306
«0192f
) 2
* f
A,
69
193534
1330
994640
33
198891
1304
soiioe
) 1
60
194332
1328
994620
33
199711
! 1361
800287
1
(
io 2:
i
1 Cosine
1
1 ^'"^' i
1 (Jiitaiig.
1 'J'liiii.'. j M-
1 (Jot
' !l
tjl
Uesjf
ees.
^"P- 1 1
)2197
60
)1282
59
J0308
58
94r)(;
57
8540
56
.7637
55
16731
54
15826
53
t4923
52
14022
21
13123
5t>
12225
49
H329
48
10435
47
39543
46
38653
45
37764
44
36877
43
15992
42
35 108
41
34226
40
33346
39
32468
38
3159'
37
30716
36
29843
35
28971
34
28101
33
27233
32
26366
31
25501
CO
29
24638
23770
28
22916
27
22058
26
21201
25
20345
24
19492
23
18640
22
17789
21
16941
20
19
16093
515248
18
i 14403
17
! 13561
16
112720
15
! 11880
14
! 11042
13
n0206
12
W9371
11
^08538
10
<0770r
9
^06876
8
^06047
' 7
B0522(
I 6
30439'
t 5
303571
) 4
302741
f 3
■i0192f
) 2
BOiioe
) 1
300281
r
'iii.i.'. j M- 1
f
SINES AND TANGFNTS. (9 DcgrCCS
1
2
3
4
5
6
V
8
9
U)
II
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
80
31
32
33
34
35
36
3/
38
39
40
9.19433
195129
195925
196719
197511
198302
199091
199879
200666
201451
20223'i
41
42
43
14
45
40
47
48
19
50
51
52
53
54
55
56
57
9.203017
203797
204577
205354
2<)6131
206906
207679
208452
209222
209992
9.210760
211526
212291
21.3055
21.3818
214579
21.5.338
216097
216854
_2r7609
K 2 18363
219116
219868
220618
221.367
222115
222861
223606
224349
225092
9.22,58.33
226573
227.111
228048
228784
2295 1 8
230252
230984
231714
_232444
9.ii3.3172
233899
234625
23.5.349
236073
236795
237515
1328
1326
1323
1 32 1
1318
1316
1313
1311
1308
1.306
J^.304
1301"
1299
1296
1294
. 1292
1289
1287
1285
1282
1280
1278
1275
12V 3
1271
1268
1206
1264
1261
12,59
J257_
12,55
1253
1250
1248
1246
1244
J 242
1239
1237
1 235
12.33
1231
1228
1220
1224
1222
1220
1218
1216
_1^2M_
1212
1209
1207
1205
1203
1201
1199
1197
1 I or:
9
9
94600
9945H0
994560
994,540
994519
994499
994479
994459
994438
_994418
. 994397
994377
994357
994336
994316
994295
994274
9942.54
994233
994212
^994 191
994171
9941.50
994129
9941 OH
994087
994066
994045
994024
994003
.993981
993960
993939
993918
993896
993875
993854
993832
99.3811
_ ^9.3789
9.993768
993746
993725
9!)3703
9: 13681
993860
993638
n936l6
993594
__99;j572
9.993550
993528
993506
993484
99.3462
993440
993418
y93396
993374
993351
33
33
33
34
34
34
34
34
34
34
34
34
34
34
34
34
34
35
35
35
35
35
35
35
35
35
35
35
35
35
35
35
35
35
35
36
36
36
36
36
36
36
36
36
36
36
36
36
36
37
.37
■37
37
37
37
37
37'
37
37
37
37
201.345
202159
202971
203782
204592
205400
206207
207013
_207817
9.208619
200420
210220
211018
211815
212611
213405
214198
214989
215J780
9.216568
217356
218142
218926
219710
220492
221272
222052
222830
_223606
E». 224382
225156
225929
226700
227471
228239
229007
229773
23053'!
231,3(»2
1.3.54
1352
1.349
1.347
1345
1342
1340
J338_
1.3.35
1.333
1,331
1328
1326
1324
1321
1319
1317
1315
797841
797029
796218
70.5408
794600
793793
792987
792183
1312
1310
1308
1305
1303
1301
1299
1297
1294
1292
10.791381
790580
789780
788982
788185
7873S9
786595
78,5802
785011
784220
10
9.232065
232826
233586
234345
235103
235859
236614
237368
238120
238872
1290
1288
1286
1284
1281
1279
1277
1275
1273
1271
,783432
782644
7818.58
781074
780290
779508
778728
777948
777170
776394
60
69
5S
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
.30
9
1269
1267
1265
1262
1260
12.58
12,56
1254
1252
12,50
10.775618
29
774844
28
774071
27
773300
26
772529
25
771761
24
770993
23
770227
22
769461
21
.23'96;;i'
1248
240371
1246
',41118
1244
241865
1242
242610
1240
243354
1238
244097
1236
244839
12.34
245579
1232
2463 1 9
1230
768698 20
10.76793,5 19
767174 18
766414 17
765655 16
764897 15
764141 141
763386 13^
762632! 12
761880 11
761128 10
9
8
7
6
5
4
3
o
1
10.760378
759630
7588«2
7.58135
757390
7566-16
5003
Cotaiii;.
7550
755161
754421
7.5.368 1
IM.
fcO Degree's.
28
(10 Degrees.) a table op logarithmic
M.
smu
D. I Cosine
D.
Taiic.
D.
Coiaiig.
I
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
ao
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
14
45
46
47
48
49
50
9.239670
2403^"
241 iO
2418x4
242526
243237
243947
244656
245363
246069
246775
.247478
248181
248883
249583
250282
250980
251677
252373
253067
2.53761
1193
1191
1189
1187
1185
1183
1181
1179
1177
1175
1173
1171"
1169
1167
1165
1163
1161
1159
1158
1150
1154
9.2544.53
255144
255834
256523
2.57211
257898
258583
259268
259951
260633
9.261314
261994
262673
263351
264027
264703
265377
266051
266723
267395
9.268065
268734
269402
270069
270735
271400
272064
272726
273388
274049
51
52
53
54
55
56
58
59 I
60 I
9.274708
275367
276024
276681
277337
277991
278644
279297
279948
280599
11.52
1150
1148
1146
1144
1142
1141
1139
1137
J135
1133
1131
1130
1128
1126
1124
1122
1120
1119
1117
1115
1113
nil
1110
1108
1106
1105
1103
1101
1099
.993351
993329
993307
99C285
993262
993i:0
993217
993195
993172
993149
993127
9.993104
993081
993059
993036
993013
992990
9r,'>967
9.f2944
992921
992898
9.992875
992852
992829
992806
992783
992759
992730
992713
992690
992660
9.992643
992619
992596
992572
992549
992525
992501
992478
992454
992430
1098
1096
1094
1092
1091
1089
1087
1086
1084
1082
9.992406
992382
992359
992335
992311
992287
992263
992239
992214 40
992190 40
37
37
37
37
37
37
38
38
38
38
38
38
38
38
38
38
38
38
38
38
38
38
38
39
39
39
39
39
39
39
39
39
39
39
39
39
39
39
40
40
40
40
40
40
40
40
40
40
'0
9.246319
247057
247794
248530
249264
249998
2.50730
251461
252191
252920
253648
9.254374
2.55100
255824
256.547
257269
257990
2.58710
259429
260146
260863
.261.578
262292
263005
263717
264428
2651.38
26.5847
5^66555
267261
267967
1230
1228
1226
1224
1222
1220
1218
1217
1215
1213
1211
9.268671
269375
270077
270779
271479
272178
272876
273573
274269
274964
9
,992166
992142
992117
992093
992069
992044
992020
991996
991971
99194?
40
40
41
41
41
41
41
41
41
41
9.2756.58
276351
277043
277734
278424
279113
279801
280488
281174
281858
1209
1207
1205
1203
1201
1200
1198
1190
1194
1192
9.282542
283225
283907
284588
285268
285947
286024
2H?301
287977
288652
10.753681
752943
752206
751470
7.50736
750002
749270
748539
747809
747080
746352
10
1190
1189
1187
1185
1183
1181
1179
1178
1176
1174
1172'
1170
1169
1167
1165
1164
1162
1160
11.58
11.57
,745626
744900
744176
743453
742731
742010
741290
740571
7398.54
739137
10
,738422
737708
736995
736283
735572
734862
7341.53
733445
732739
732033
1155
1153
1151
1150
1148
1147
1145
1143
1141
1140
1138
1130
1135
1133
1131
1130
1128
il2G
1125
1123
10.731329
?30625
729923
729221
728.521
727822
727124
726427
72573 1
725036
10
,724342
723649
722957
7222( 6
721576
720887
720199
719512
718826
718142
10
,7174.58
716775
716093
71.5412
714732
7140,53
713376
7i2x';99
712023
711348
Cosine I
Bine
Cotani;.
Tanc
i
97 Degrees
M.
~0
1
2
3
4
5
6
7
8
9
10
U
12
13
14
15
16
17
18
119
120
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
/:j
'M
■t ■;■ I
■*bi
4V
48
49
50
IS
54
59
jOO^
IZJI
H
"^- 1 1
J681
60
IMS
59
5206
68
470
57
)736
56
)002
55
)270
54
^539
53
?809
52
ro80
51
3352
50
)626
49
1900
48
1176
47
3453
46
2731
4"=
2010
'i i
1290
43
0571
42
9854
41
9137
40
8422
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
W5
15
14
13
12
1'
JO
9
8
7
6
5
4
3
1
01
IM.
I
i
SIMES
AND TANGENTS, (ll Degrees.)
9.280599
281248
281897
282544
283190
283836
284480
285124
285766
286408
287048
9.287687
288326
288964
289600
290236
?90870
291504
292137
292768
293399
1082
1081
1079
1077
1076
1074
1072
1071
1069
1067
1066
. 294029
294658
295280
295913
296539
297164
297788
298412
299034
299655
1064
1063
1061
1059
1058
1056
1054
1053-
1051
1050
'M
.300276
300895
301514
302132
302748
303364.
303979
304593
305207
305819
1018
1046
1045
i043
1042
1040
1039
1037
1036
1034
9.991947
991922
991897
991873
991848
991823
991799
991774
991749
991724
991699
9.991674
991649
991624
991599
991574
991549
991524
991498
991473
991448
1.288652
289326
289999
290671
391342
292013
292682
293350
294017
294684
295349
o')7041
307f:50
308^59
■■U(88G7
:''^9474
3 1 0080
3106S5
311289
31 1893
9.312495
31^097
313698
314297
314897
3! 5'! 05!
316()<>'jj
316689
3 17284!
317879'
1032
1031
1029
1028
1026
1025
1023
1022
1020
lOiO
.991422
991397
991372
991346
991321
991295
991270
991244
991218
991193
1017
1016
1014
1013
1011
1010
1008
1007
1005
1004
1003
1001
1000
998
997
99n
994
993
991
990
.991167
991141
991115
991090
991064
991038
991012
990986
990960
990934
9.990908 44
990882! 44
990855! 44
990829; 44
990803 44
990777, 44
990750 44
990724 44
990697; 44
990671
9
44
.990644 44
990618 44
99«»59I 44
990565 44
9905-'8 44
90;..:; 1 45
.-^90485 4f
990458 45
990431 45
990404 45
1.296013
296677
297339
298001
298662
2i;9322
299980
300638
301295
301951
9.302607
303261
303914
304567
305218
305869
306519
307168
307815
30 8463
9.309109
309754
310398
811042
311685
312327
312967
313608
314247
__31.188oj
}.3155'?3
316 1. ')9!
3167951
317430
3180641
3186971
319329
3199611
320592;
_3'>!2J22!
.3i>l85I
322179
323106
323733
324358
124983
.::5607
3'362.'l
326853
•J27475i
1123
1122
1120
1118
1117
1115
1114
1112
1111
1109
1107
10.711348
710674
710001
7093J29
708658
707987
707318
706650
705983
705316
704651
1106
1104
1103
1101
1100
1098
1096
1095
1093
1092
1090"
1089
1087
1086
1084
1083
1081
1080
1078
1077
10.703987
703323
702661
701999
701338
700678
700020
699362
698705
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
1<^75
1074
1073
1071
1070
1068
1067
1005
1061
1062_
1061
1060
1058
J 057
055
1054
1053
1051
1050
1048
1047
1045
1044
1043
6980491 40
39
38
37
36
35
34
33
32
31
30
10.697393
696739
696086
695433
694782
694 1. J 1
693481
692832
692185
691537
10.690891 29
690246
689602
688958
688315
68 : ;373! 24
687033 23
G8639?' i2
•8
27
26
2''^
685753
685115
10.G8447'
683S41
68320
;^ .•<36
681303
680671
680039
?^7<?;'(i8
() 1^778
,".0
1041
iO-iO
1. ^9
1037
10' ;
103;.
678149
677521
676894
676267
675642
6750171
6743931
67.1,69'
673147
67252:
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
78 t) .^'■ivfi
i:'
Jh.
..''•.,^i- #■
Iff
u
t
30 "(12 Degrees.) a table of rooARiTHMic
M.
Sine
D.
Cimlne
I).
Tans.
I).
Cotaiiff-
1
2
3
4
5
6
7
8
9
!0
11
12
13
14
16
17
18
19
20
21
22
23
24
25
2()
27
28
29
30
31
32
33
31
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
60
51
52
53
54
55
56
u7
58
59
60
3.3178791
318473
319066
319658
320249
320840
321430
322019
322607
323194
323780
9.324366
324950
325534
326117
326700
327281
327862
328442
329021
329599
990
988
987
986
984
983
982
980
979
977
976
9
,330176
.380753
331329
331903
332478
333051
333624
334195
334766
335337
9.335906
336475
337043
337610
338176
338742
339306
339871
340434
340996
9.341558
342119
342679
343239
343797
344355
344912
345469
346024
9.347134
347687
348240
348792
349343
349893
350443
350992
351540
352088
975
973
972
970
969
968
966
965
964
962
961
960
958
957
956
954
953
952
950
949
948
946
945
944
943
941
940
939
937
936
935
934
932
931
930
929
927
926
925
924
[). 990404]
990378
990351 1
990324
990297
990270
990243
990215
990188
990161
990134
9.990107
990079
990052
990025
989997
989870
989942
9899 1 5
989887
989860
9.989832
989804
989777
989749
989721
989693
989665
989637
989609
__989582
9.989'^ 3
9
989413
989.384
989356
989328
989300
922
921
920
919
917
916
916
914
913
911
9.327474
328095
328715
329334
329953
330570
331187
331803
332418
333033
333646
1035
1033
1032
1030
1029
1028
1026
1025
1024
1023
1021
9.334259
334871
335482
336093
336702
337311
337919
338527
339133
339739
97340344
340948
341552
342155
342757
343358
343958
344558
345157
345755
9.346363
346949
347545
348141
348735
349329
349922
350514
351106
351697
9.989271 47
989243
989214
989186
983157
989128
989100
989071
989042
989014
9.988985
988956
988927
988898
988869
9S8840; 48
9S8Si ii 49
988782' 49
988753 49
9887241 49
9.3.52287
352876
353465
354053
? 54640
355227
i55813
356398
356982
357566
9.358149
358731
35931'.!
359893
360474
361053
3G1632
362210
362787
363364
10
1020
1019
1017
1016
1015
1013
1012
1011
1010
1008
1007'
1006
1004
1003
1002
1000
999
998
997
996
672526'
67l90i".
671285
670666
670047
669430
668813
668197
667582
666967
6663M
10.66.5741
665129
664518
663907
663298
662689
662081
661473
660867
660261
994
993
992
991
990
988
987
986
985
983
982
981
980
979
977
976
975
974
'.,73
971
10
,6.59656
659052
658448
657845
65/213
656642
656042
655442
C54843
654245
10.653647
6.53051
6,52455
651859
651265
650071
650078
649480
648894
_648303
10.647713
647124
646535
645947
645360
644773
644187
643602
643018
642434
970
969
968
967
966
965
962
961
960
10
.641851
641269
640687
640107
639.526
638947
•63Har.R
637 790
637213
636636
«T l>.«ree».
M.t !
9.:
1
2
3
A
; ■
9.3
52
3
53
3;
54
3!
55
3;
56
3J
57
3J
[iS
3^
59
3.'
60
3i!
SINES AND TANGENTS. (^13 Degrees.)
31
252"rv
190f.
1285
0606
0047
9430
i88l3
18197
i7582
16967
^354
1574 1
>5129
)4518
53907
53298
32689
i208l
51473
30867
B026 1
59656
59052
58448
57845
5/243
56642
560421
i55442|
;54843
15424 5
153647
i.53051
)i>2455
5518.59
551265
550671
550078
549-180
B48894
648303
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
261
251
24
23
22
21
20
(547713
647124
646535
645947
645360
644773
644187
643602
643018
642434
'641851]
641269
640687
640107
639.526
638947
•698368
63'^t790
637213
6 36636
Tuns I M.
19
18
17
16
15
14
13
12
9
8
7
6
5
4
3
2
1
M.
1 Sine
1 D.
1 Cosine 1 1).
1 'I'aiiK.
I 1^-
(!()iaii^. j
9.3.52(W8
911
9.988724
49
9.. 363364
960
I0.6.3663r
60
I
352635
910
988695
49
363940
9.59
636060
59
2
353181
909
988666
49
364515
958
6354815
58
3
3.53726
908
988630
49
365090
957
6.34910
57
4
,3.54271
907
988607
49
365664
955
634336
56
5
3.54815
905
988578
49
366237
9.54
633763
' 55
6
355358
904
988.548
49
366810
953
6.33190
54
7
355901
903
988519
49
367382
952
632618
53
8
356143
902
988489
49
367953
951
632047
52
9
356984
901
988460
49
368524
950
631476
5!
10
11
357524
9.358064
899
988430
49
49
369094
9.369663
949
630906
50
49
898
9.988401
948
10.630.337
12
358603
897
98B371
49
370232
946
629768
48
13
359141
896
988342
49
370799
945
629201
47
14
359678
895
988312
50
371367
944
628633
46
15
360215
893
988282
50
371933
943
628067
45
16
3607.52
892
98S252
50
372499
942
627.501
44
17
301287
891
988223
50
373064
941
626936
43
18
361822
890
988193
50
373629
940
626371
42
19
362356
889
988163
50
374193
939
62.5807
41
20
21
362S89
888
887
988133
9.988103
50
50'
374756
938
625244
40
r,9
9.363122
9.375319
937
10.624681
22
363954
885
988073
50
37.5881
935
624119
38
23
304485
884
988043
50
376442
934
623558
37
24
365016
883
988013
50
377003
933
622997
36
25
365.546
882
987983
50
377563
932
622437
35
26
366075
881
987953
50
378122
931
621878
34
27
366604
880
987922
50
378681
930
621319
33
28
367131
879
987892
50
379239
929
620761
32
29
367659
87r
987862
50
379797
928
620203
31
30
31
368185
9.368711
876
987832
9.987801
51
51
380354
927
619646
30
29
875
9.380910
926
10.619090
32
369236
874
987771
51
381466
925
618534
28
33
369761
873
987740
51
382020
924
617980
27
34
370285
872
987710
51
382575
923
617425
26
35
370808
871
987679
51
383129
i;22
616871
25
36
371330
870
987649
51
383682
921
616318
34
3/
371852
869
987618
51
384234
920
615766
23
38
372373
867
987588
51
384786
919
615214
22
39
372894
866
987557
51
385337
918
614663
31
40
41
37.3414
9.373933
865
987526
9.987496
51
.51
38,5888
917
614112
20
19
864
9.38643S
915
10.613562
42
374452
863
987465
51
386987
914
613013
18
43
374970
862
987434
51
387.5,36
913
612464
17
44
375487
861
987403
52
388084
913
611916
16
45
376003
860
987372
52
388631
911
611369
15
46
376519
859
987341
52
389178
910
610822
14
47
377035
8.58
997310
52
389724
909
610276
13
!^
377549
857
987279
52
390270
908
609730
12
49
378063
856
987248
52
.390815
907
609185
11
.h!.. .
378577
9.379089
854
9S7217
9.987186
52
52
391360
906
608640
10
9
853
9.391903
905
10.608097
i)2
379601
8.52
9871.55
52
392447
904
607553
8
i)3
380113
851
987124
52
392989
903
607011
7
54
380624
850
987092
52
393531
902
606469
6
55
381134
849
987061
52
394073
901
605927
5
56
57
381613
848
987030
52
394614
900
605386
4
382152
847
986998
52
395154
899
604 S4 6
3
58
3S2661
846
986967
52
395694
898
604306
2
59
383 1 68
845
986936
52
396233
897
603767| 1 1
bO 1
3S367;')|
844
986904
52
396771
896
603,:2;>'
.?;
■!■ -It
.1.
CiwitiH
hiiip
Cotiiiu;.
I am'.
M.
70 Dt'greea.
Ill
1
■ I
i
32 (14 Degrees.) a table of logarf' '"ic
M.|
SiriR I
I). 1
()
9.383075
844
1
384182
843
2
384687
842
3
385192
841
4
385697
840
fi
386201
«»39
6
396704
838
7
387207
837
8
387709
836
»
388210
835
10
388711
834
Cosine
D.
Tang.
I).
Cotung.
11
12
13
14
15
16
17
18
19
2a
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
9.389211
389711
390210
390708
391206
391703
392199
392695
393191
393685
9
833
832
831
830
828
827
826
825
824
823
y.9H6904
986873
986841
986809
986778
986746
986714
986683
986651
986619
986587
.394179
894673
395166
395658
396150
396641
397132
397621
398111
_398600
'.399088
399 )75
400062
400549
401035
401520
402005
402489
402972
403455
822
821
820
819
818
817
817
816
815
814
813
812
811
810
809
808
807
806
805
804
9.403938
404420
404901
405382
405862
406341
406S20
407299
407777
408254
iT 40,^7311
409i.H)7i
409682'
4101571
410632
4! 1106
411579,
4120521
4125241
412996'
52
53
53
53
63
53
53
53
53
53
^ 63
9.9865551 53
986.5231 53
986491,53
9864591 53
986427
986395
986363
986331
986299
986266
.396771
397309
397846
398383
398919
399455
399990
400.524
401058
401591
402124
803
802
801
800
799
798
797
796
795
794
794
793
792
791
790
7S9
V88
787
786
785
53
53
54
54
54
54
97986231 54
986202 .54
986169
986137
986104
986072
9S6039
986007
985974
985942
9.402656
403187
403718
404249
404778
405308
405836
406364
406892
407419
.985909
985876
985843
98.5811
985778
985745
985712
98567S
985646
98.5613
54
54
54
54
54
54
54
54
55
55
55
55
55
55
55
55
55
55
.407945
408471
408997
409.V21
410045
410.569
411092
411615
412137
412658
9.985.580
985547
985514
985480
985447
985414
9H5380
985347
985314
985280
9 . 985247
98.5213
985180
985146
985113
985079
985045
98.5011
984978
984944
9.413179
413699
414219
414738
41.5257
415775
416293
416810
41V326
417842
55
55
55
55
56
56
56
56
5_6
56
56
56
56
56
56
56
56
56
56
9
896
896
895
894
893
892
891
890
889
888
887
.418358
418873
419387
419901
420415
420927
421440
4219.52
422463
422974
.423484
423993
424503
49.5011
425519
426027
426531
427041
427547
42805
886
885
884
883
882
881
880
879
878
877
876
875
874
874
873
672
871
870
869
868
867
866
865
864
864
863
862
861
860
859
8.58
8.57
858
855
8.55
854
8.53
8.52
851
850
lU. 603229
602691
6021.54
601617
601081
600545
600010
599476
598942
598409
697876
10.597344
696813
696282
595751
695222
694692
594164
593636
593108
592.581
60
59
58
67
56
65
64
53
52
51
60
49
48
47
46
45
44
43
42
41
40
10.. 592055
591,529
691003
690479
689955
589431
.588908
688385
587863
587342
10.586821
586301
58578 1
685262
584743
584225
583707
583190
682674
.582158
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
10
849
848
848
847
846
845
844
843
843
842
,.581642
.581127
580613
680099
679585
579073
578560
578048
577537
577026
10
,576516
676007
575-^97
.574989
674481
573973
5734f)0
672959
{;72453
571948
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
75 Degrees.
M,l t
9.4
1
4
2
4
3
4
4
4
ii*^9
G91
154
617
081
545
010
476
942
409
'876
344
1813
1282
)751
)222
1692
U64
3636
J108
2581
60
59
58
67
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
2055
1529
1003
0479
9955
9431
8908
8385
7863
7342
6821
6301
1578 1
15262
!4743
!4225
13707
53190
J2674
J2158
^642
31127
30613
30099
79585
79073
78560
78048
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
77537
11
77026
10
9
76516
76007
8
75197
7
74989
6
74481
5
73973
4
734611
a
.72959
2
,72453
1
.71948
ang
EJ
SINES AND TA1SGENTS. (15 Dcgrees.)
MUJ aiiie I D. I Coaine | D. | Tang.
33
D.
C(.taiie.
1
2
3
4
6
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
49
50
51
52
53
54
55
56
5V
58
59
60
9.412996
413467
413938
414408
414878
415347
415815
416283
416751
417217
417684
9.418150
418615
419079
419544
420007
420470
420933
421395
4218.57
422318
9 422778
423238
423697
424156
424615
425073
425530
425987
426443
426899
9.427,354
427809
428263
428717
429170
429623
430075
430527
430978
431429
9.431879
432329
432778
433226
433675
434122
434569
43.5016
435462
785
784
783
783
782
781
780
779
778
777
770
775
774
773
773
772
771
770
769
768
767
9.984944)67
984910
984876
984842
984808
984774
984740
984706
984672
984637
984603
.984569
984535
984.500
98446s<
984432
984397
984363
984328
984294
984269
767
766
765
764
763
762
761
760
760
759
758
757
756
755
754
753
752
752
751
750
9.984224
984190
9841.55
984120
984085
984050
984015
983981
983946
983911
435908
742
9.436.3.')3
741
436798
740
437242
740
437686
739
438129
738
438572
737
/iQQm.i
•yon
439456
736
439897
735
440338
734 1
749
749
748
747
746
745
744
744
743
9.983875
983840
983805
983770
9837351 59
67
57
57
67
57
57
67
67
57
^
67
57
57
57
58
58
58
58
58
58
58
58
58
58
58
58
58
68
58
58
58
59
59
69
983700
983664
983629
983594
983558
9.983.523
9834S7
983452
98.3416
98.3381
983345
983309
983273 60
59
59
59
69
59
59
59
59
59
69
59
59
9.4280.52
428557
429062
429666
430070
430573
431075
431577
432079
432.580
4.33080
9.433580
434080
4,34579
435078
435676
436073
436570
437067
437663
43 8059
9.438.5.54
439048
439543
440036
440529
441022
441514
442006
442497
442988
9.443479
443968
444458
444947
445435
445923
446411
446898
447384
447870
983238
983202
60
60
9.983166 60
983130] 60
983094
983058
983022
982986
60
60
60
60
9.448366
448841
449326
449810
450294
450777
451260
451743
452225
452706
982914
982878
982842
60
60
60
9.453187
453668
4.54148
4i34628
4.55107
4.55586
456542
4.57019
457496
I C(wine |
Sine
842
841
840
839
838
838
837
836
835
834
883
832
832
831
830
829
828
828
827
826
825
824
823
823
822
821
820
819
819
818
817
816
816
815
814
813
812
812
811
810
809
809
808
807
806
806
805
804
803
802
802
801
800
799
799
798
797
795
796
795
794
10.571948
671443
570938
570434
669930
669427
668925
568423
567921
567420
56G920
10.666420
665920
56.5421
564922
564424
563927
563430
562933
562437
561941
10.561446
660962
560457
559964
659471
558978
558486
557994
557503
567012
10.6.56521
556032
655542
555053
554565
654077
553589
553102
552616
552130
10.551644
551159
560674
650190
549706
549323
548740
548257
547775
647294
iO.. 5468 13
546332
545852
545372
544893
514414
643930
54.3458
542981
542504 i
Tail?.
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
M.
I
li Degrees.
MV
'1 4f
34
(16 Degrees.) a tatile op logarithmic
M.
Sine
I).
Cosine
D-
TnuR. I D. I Cot:ii!(j (
1
2
3
4
6
6
7
8
9
10
II
ly
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
9.440338
440778
441218
441658
442096
442535
442973
443410
443847
444284
444720
9.445155
445590
446025
446459
446893
447326
447759
448191
448623
44 9054
9.449485
449915
450345
450775
451204
451632
452060
452488
452915
453342
.453768
454194
454619
455044
455469
455893
456316
456739
457162
457584
.458006
458427
45S848
459268
459688
460108
460527
460940
461364
461782
.462199
462616
463032
463448
463864
464279
4fi4r)04
465108
465522
465935
734
733
732
731
731
730
729
728
727
727
726
725
724
723
723
722
721
720
720
719
718
717
716
716
715
714
713
713
712
711
710
710
709
708
707
707
706
705
704
704
703
702
701
701
700
699
698
698
697
696
695
.982842
982805
982769
982733
982696
982660
982624
982587
982551
982514
982477
9.982441
982404
982367
982331
982294
982257
982220
982183
982146
982109
9.982072
982035
981998
981961
981924
981886
981849
981812
981774
981737
9.981699 6
60
60
61
61
61
61
61
61
61
61
61
61
61
61
61
61
61
62
62
62
62
62
62
62
62
62
62
62
62
02
62
981662
981625
981587
981549
981512
981474
981436
981399
981361
.981323
981285
981247
98 J 209
981171
981133
981095
981057
981019
980981
695
9.980942
694
980904
693
980866
693
980827
692
980789
691
980750
690
980712
690
980673
689
980635
688
980596
63
63
63
03
63
63
6o
63
63
63
63
63
63
63
64
64
64
64
64
64
64
04
64
64
64
64
64
64
9.457496
457973
458449
458925
459400
459875
460349
460823
461297
461770
462242
.462714
463180
463658
464129
464599
465069
465539
466008
466476
466945
.467413
467880
468347
468814
469280
469746
470211
470676
471141
471005
794
793
793
792
791
790
790
789
788
788
131.
786
785
785
784
783
783
782
781
7S0
780
9.472068
472532
472995
473457
47,3919
474381
474842
475303
475763
476223
9.476683
477142
477601
478059
779
778
778
777
776
775
775
774
773
7T3
772
771
771
770
769
769
768
767
767
766
765
765
764
763
478517
763
478975
762
479432
761
479889
761
480345
760
480801
759
9.481257
759
481712
758
482167
757
482621
757
483075
756
483529
755
483982
755
484435
754
484887
753
485339
753
10.642504
542027
541.551
541075
540600
540125
639651
539177
538703
538230
6 37758
10.537286
530814
536342
.53.5871
63.5401
.534931
534461
633992
533524
5330.55
10.532587
532120
.531653
531186
530720
530254
629789
529324
528859
528395
10.527932
527468
527005
526543
526081
525619
625158
624697
524237
623 777
10.. 5233 17
522858
522399
521941
621483
521025
520568
520111
519655
51919 9
"518743
518288
617833
617379
516925
516471
616018
515565
515113
5M66I
10.
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36 I
351
b^
33
32
31
30
29
'^8
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
I (Jo.~iue I
Sine
Cotang.
Tang.
M.
73 Degrees.
1
2
3
4
5
6
7
8
9
10
55
4«/
488
56
488
5'''
A DO
•iOO
.')8
489'
59
489.
60
489!
1
Cusin(
K«
(iO
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36 I
351
i>4
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
0111
i2
9655
11
9199
10
8743
9
8288
8
7833
7
7379
6
6925
5
L6471
4
6018
3
5565
2
15113
1
MfiBI
ang.
M.
SINES 7iND TANGfeNTs. (l7 Dcgrees.)
35
M.
>r<iiio
I)
Cosine
I).
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
3f
32
33
34
35
36
37
38
39
40
TaiiR.
D.
.465935
466348
466761
467173
467585
467996
468407
468817
469227
469637
4700.16
9.470455
470863
471271
471679
472086
472492
472898
473304
473710
474115
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
5?
58
59
60
9.474519
474923
475327
475730
476133
476536
476938
477340
477741
47814 2
9.478542
478942
479342
479741
480140
480539
480937
481334
481731
482128
9.482525
482921
48.3316
483712
484107
484501
484895
485289
485682
486075
9.486467
486860
487251
487643
488034
488424
A OODl \
-tooo I 'J
489204
489593
489982
Cosine
688
688
087
686
685
685
684
683
683
682
_68Jl_
68"0
680
679
678
678
677
676
676
675
674
674
673
672
672
671
670
669
669
668
667
667
666
665
665
664
663
663
662
661
661
600
659
659
658
657
657
656
665
655
654
) . 980596
980558
980519
980480
980442
980'J03
980364
980325
980286
980247
980208
T 980 1 69
980130
980091
9800.52
980012
979973
979934
979895
979855
979816
Cotang.
9.979776
979737
979697
979658
979618
979579
979539
979499
979459
979420
64
64
65
65
65
65
65
65
65
6.'^
65
65
65
65
65
65
65
66
66
66
66
.485339
485791
486242
486693
487143
487593
488043
488492
488941
489390
489838
755
752
761
751
750
749
740
748
747
747
746
9.979.380
979340
979300
979260
979220
979180
979140
979100
979059
979019
9.978979
978939
978898
978858
978817
978777
978736
978696
978655
978615
653
653
652
651
651
650
649
/.Jr.
648
9.978.574 68
97853
978493
978452
978411
978370
97S329
978288
978247
978206
66
66
66
66
66
66
66
66
66
66
06
66
67
67
67
67
67
67
67
67
67
67
67
67
67
67
67
68
68
68
9.490286
490733
491180
491627
492073
492519
492965
493410
4938.54
494299
9.494743
495186
495030
496073
496515
496957
497399
497841
498282
498722
746
745
744
744
743
743
742
741
740
740
9.499163
499603
500042
500481
500920
601359
601797
502235
602672
603109
68
68
68
68
68
08
68
68
68
Sine
■^
9.503546
503982
504418
504854
605289
505724
506169
506593
507027
507460
3.607893
508326
608759
509191
609622
610054
510485
610916
611346
51177 6
Cotaiig.
740
739
738
737
737
736
736
735
734
j;34
733~
733
732
731
731
730
730
729
728
728
727
727
726
725
725
724
724
723
722
722
721
721
720
719
719
718
718
717
716
716
10.514661 60
514209 S9
6137.58 68
513.307 67
612857 56
512407 65
611957 64
611508 .53
6110.59 62
610610 61
610162 50
10 .509714 49
509267 48
608820 47
608373 46
607927 45
507481 44
607035 43
5065S0 42
506146 41
505701 40
10.505257 39
604814 38
504370 37
503927 36
603485 36
603043 34
502601 33
502159 32
601718 31
501278 30
10.500837 29
600397 28
499958 27
499519 26
499080 25
498641 24
498203 2^3
497765 22
497328 I 21
496891 I 20
10.496454 19
496018
495582
495146
494711
494276
493841
493407
492973
492640
10.492107
491674
491241
490809
490378
489946
489515
489084
488654
488224
18
17
16
15
14
13
12
11
10
9
8
7
6
g
4
8
2
1
'i'ang. { M.
ir"
lit
fl^f
se
(18 Degrees.) a
TABLfl or LOOAKmtMIC
M.|
Sine 1
I). i
Cosine 1).
'I'miR. 1
D.
Ciitiine. 1
9.489982
648
9.978206
68
9.511776
716
10.488224
60
1
490371
648
978165
68
512206
716
487794
59
2
490759
647
978124
68
512635
715
487365
58
l)
491147
646
978083
69
513064
714
486936
57
4
491535
646
978 ' J
69
613493
714
486507
56
5
491922
645
978t'
69
513921
713
486079
55
6
492308
644
97795.)
69
514349
713
485651
54
7
492695
644
977918
61)
514777
712
485223
53
8
493081
643
977877
69
515204
713
484796
52
9
493466
642
977835
69
515631
711
484369
51
10
11
493851
642
977794
69
69
616057
9.616484
710
483943
10.483516
50
49
9.494236
641
9.977752
710
12
494621
641
977711
69
516910
709
483090
48
13
496005
640
977669
69
517335
709
4826()5
47
14
495388
639
977628
69
517761
708
4822;i9
46
15
495772
f.39
977.586
69
518185
708
481815
45
Ifi
496154
638
977544
70
^18610
707
481390
44
17
496537
637
977503
70
/) 19034
706
480966
43
IR
496919
637
977461
70
519458
706
480542
42
19
497301
636
977419
70
619882
705
480118
41
20
21
497682
636
977377
70
70
520305
705
479695
10.479272
40
39
9.498064
635
9.977335
9.520728
704
22
498444
634
977293
70
521151
703
478849
38
23
498825
634
977251
70
521573
703
478427
37
24
499204
633
977209
70
521995
703
478005
36
25
499584
632
977167
70
522417
702
477583
35
26
499963
632
977125
70
522838
702
477162
34
27
600342
631
977083
70
623259
701
476741
33
28
500721
631
977041
70
523680
701
476320
32
29
501099
630
976999
70
524100
700
475900
31
30
31
501476
629
976957
70
70
524520
699
47.5480
10.47.5061
30
29
9.501854
629
9.976914
9.524939
699
32
502231
628
976872
71
525359
698
474641
28
33
502607
628
976830
71
625778
698
474222
27
34
602984
627
976787
71
526197
697
473803
26
35
503360
026
976745
71
526615
697
473385
25
30
5(K1735
626
976702
71
527033
696
472967
24
37
504110
625
976660
71
527451
696
472549
23
38
504485
625
976617
71
527868
695
472132
22
39
504860
624
976574
71
528285
695
471715
21
40
41
505234
623
976532
71
71
528702
694
471298
0.470881
20
19
9.505608
623
9.976489
9.529119
693
42
505981
622
976446
71
529535
693
470465
18
43
506354
622
976404
71
529950
693
470050
17
44
506727
621
976361
71
530366
692
469634
16
45
507099
620
976318
71
530781
691
469219
15
46
507471
620
976275
71
531196
691
468804
14
47
507843
619
976232
72
531611
690
468389
13
48
508214
619
976189
72
532025
690
467975
12
49
508585
618
976146
72
532439
689
467561
11
50
51
508956
9.509326
618
976103
72
72
532853
689
467147
10
9
617
9.976060
9.533266
688
10.466734
52
509696
616
976017
72
533679
688
466321
8
53
510065
616
975974
72
534092
687
465908
7
54
510434
615
975930
72
634504
687
465496
6
55
510803
615
975887
72
534916
686
465084
6
56
511172
614
975844
72
535328
686
464672
4
57
511540
613
975800
72
635739
685
464261
8
58
511907
613
975707
72
53615(J
QHii
403S0U
2
59
612275
612
975714
72
636561
684
463439
1
60
512642
612
975670
72
536972
684
463028
L
; Cosine
1
1
1 Bine |
1 Cotaiig.
1
1 Tang. M. |
71
M.
9..'
1
2
3
4
5
6
7
,.. 1
i'ZU
60
?75M
59
^365
58
3936
57
3507
56
3079
55
i651
54
3223
53
1796
52
I3(i'j
51
3943
50
3516
49
3090
48
26t)5
47
i'ZMi
46
1815
45
1390
44
1)966
43
)542
42
[)118
41
1695
40
J272
39
H849
38
8427
37
8005
36
7583
35
7162
34
6741
33
6320
32
5900
31
5480
30
5061
29
4641
28
27
26
25
24
23
22
21
20
19
18
17
16
15
8804
14
8389
13
7975
12
7561
11
7147
10
6734
9
.6321
8
•5908
7
.5496
6
.5084
5
54672
4
54261
3
Job jU
X
53439
1
53028
iig.
M.
M.
Blue
1
2
3
4
5
6
7
8
9
10
ii
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
:!,>
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
f.O
9.512642
513009
5 1 3375
513711
514107
611472
6M837
515202
615566
515930
516294
SINES AND TANOENT8. (10 DeglCea.)
1>- I THny. I 1). I Ci„^i^
a?
Conine
9
516657
517020
517382
517745
518107
518468
518829
519190
619551
519911
520271
520631
520990
521349
521707
522066
522424
522781
623138
623495
.523852
524208
524564
624920
525275
625630
525984
526339
526693
527046
9.527400
527753
628105
528458
528810
629161
629513
529864
530215
530565
9.530915
531265
531614
631963
532312
632661
533009
533.?.57
533704
534052
Cosine I
612
611
611
610
609
609
608
608
607
607
606
605
605
604
604
603
a03
602
601
601
6oo
600
599
599
598
598
597
596
596
595
595
594
594
593
693
592
591
591
590
590
589
589
588
588
687
687
586
586
685
585
584
584
583
582
582
581
581
580
5m
579
578
.975670
975627
975583
975539
975496
975452
975408
975365
975321
975277
975233
9.975189
975145
975101
975057
375013
974969
974925
974880
974836
974792
9.974748
974703
974659
974614
974570
974525
974481
974436
974391
974347
9.974302
974257
974212
974167
974122
974077
974032
973987
973942
973897
9.973852
973807
973761
973716
973&n
973625
973580
973535
973489
_973444
9.973398
973352
973307
973261
973215
973169
973124
97307S
973032
972986
73
73
73
73
73
73
73
73
73
73
73
73
73
73
73
73
74
74
74
74
74
74
74
74
74
74
74
74
74
74
75
75
75
75
75
75
75
75
75
75
75
75
75
75
76
76
76
76
76
76
76
76
76
76
76
76
76
76
76
77
77
9.536972
637382
537792
538202
5.38611
539020
639429
639837
640245
540653
.54 1061
9.. 54 1468
541875
'4228 1
.)42688
543094
643499
643905
644310
644715
545119
9.54.5524
545928
546331
546735
547138
547540
647943
648345
548747
649149
9.649550
549951
550352
550752
5511.52
651.5.52
651952
652351
552750
563149
9.. 5,5,3548
553946
654344
554741
5,55139
555536
555933
556329
656725
557121
.5.57517
.5.57913
558308
658702
659097
5.59491
659885
5G0279
560673
561066
684
683
683
682
'82
680
680
679
679
10
I
Siiio
Until ngr.
678
678
677
677
676
676
675
675
674
674
673
673
672
672
671
671
670
670
669
669
.463028
462618
46220H
461798
461989
460980
460571
460163
459 A5b
459347
458939
668
668
667
667
666
666
665
665
665
664
664
663
663
662
662
661
661
660
660
659
659
6.59
658
658
657
657
6.56
656
655
655
10.4.58.532
4.58125
457719
457312
456906
4.56.501
456095
465690
4.55285
454881
10.4.54476
454072
453669
453265
452862
452460
452057
451655
461253
4.50861 30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
16
14
13
12
11
10
9
8
7
6
5
4
3
2
I
10.4504,50
450049
449648
449248
448848
448448
448048
447649
447250
446851
10.4464.52
446054
445656
445259
441861
444464
444067
443671
443275
442879
10.442483
442087
441692
441298
440903
440509
440115
439721
439327
438934
Tana.
M.
70 Dct'rees
u
^> # ^
IMAGE EVALUATION
TEST TARGET (MT-3)
1.0
I.I
1.25
• 56 I 3.2 |||ll /^ n
I! iSi^ ~
? ■- Ills
111=
U IIIIII.6
Photographic
Sciences
Corporation
m
iV
^^^
\\
^
<*
%
n> #1
9>'-
23 WEST MAIN STREET
WEBSTER, NY. 14580
(716) 872-4503
¥ m
>' i^.-
px
1
I
■P—HjHI— ■
38
(20 Degi
ees.) A
TABLE or LOGARITHMIC
M.
1 Bine
1 D.
! Cosine IX
1 Tang.
1 n.
CotiinR. 1 1
9.534052
578
19.972986
77
> 9.561066
6.55
10.438934
60
1
534339
577
973940
77
561459
654
438541
59
2
534745
577
972894
77
')61351
6.54
438149
58
a
£35092
577
972848
77
563244
653
437756
57
4
535438
676
972803
77
533636
653
437364
56
5
535783
576
973755
77
563028
653
436972
55
6
536129
575
972709
77
563419
652
436,581
54
7
536474
574
973663
77
563811
652
436189
53
8
536818
574
972617
77
564302
651
435798
.52
9
537103
573
973570
77
564592
651
435408
51
10
11
537507
573
973534
9.973478
77
77
564983
650
435017
50
49
9.537851
572
9..56537S
650
10.434027
12
538194
573
972431
78
565763
649
43423"
48!
13
538538
571
972385
78
5661,53
649
433847
47
14
538880
571
973338
78
566542
649
433458
46
15
539333
570
972351
78
56G932
648
433068
45
16
539565
570
973245
78
567320
648
432680
44
17
539907
569
972198
78
567709
647
432291
43
18
540249
669
973151
78
568098
647
431902
42
19
540590
568
973105
79
568486
646
431514
41
20
2T
540931
568
973058
78
78
568873
646
431127
10.430739
40
39
9.541272
567
9.973011
9.569261
645
22
541613
567
971964
78
569648
645
430353
38
23
541953
566
971917
78
570035
645
429965
37
24
542293
566
971.870
78
570433
644
429578
36
25
542632
505
971833
78
570809
644
429191
35
26
542971
565
971776
78
571195
643
428805
34
27
543310
564
971729
79
571.581
643
428419
33
28
543649
564
971682
79
.571967
642
4300J3
32
29
543987
563
971635
79
572353
642
437648
31
30
31
541325
9.544663
563
971588
79
79
573738
642
427262
30
29
562
9.971540
9.573123
641
10.426877
32
545000
562
971493
79
573.507
641
426493
28
33
545338
561
971446
79
573892
640
426108
27
34
545674
561
971398
79
574276
640
425734
26
35
646011
560
971351
79
674660
639
425340
25
36
546347
560
971303
79
575044
639
424950
24
37
546683
559
971256
79
575427
639
424573
23
38
547019
559
971308
79
575810
638
424190
22
39
547354
558
971161
79
576193
638
423807
21
40
41
547689
558
971113
9.971066
79
80
676576
637
423424
20
19
9.548024
557
9.576958
637
10.423041
42
548359
5.57
971018
80
577.341
636
422659
18
43
548693
556
970970
80
577723
636
422277
17
44
.549027
550
970933
80
578104
636
421896
16
45
549360
556
970874
80
578486
635
421514
15
46
549693
555
970837
80
578867
635
421133
14
47
550036
554
970779
80
579348
634
420752
13
48
550359
554
970731
80
579629
634
420371
12
49
550692
553
970683
80
580009
634
419991
11
50
51
551024
553
970635
80
80
580389
63,:
633
419611
10.419231
10
9
9.5513,56
552
9.970586
9.580769
52
551687
552
97'\538
80
581149
632
418851
8
53
552018
5.52
970*90
80
581528
632
418472
7
54
552349
551
970443
80
581907
632
418093
6
6b
552680
.551
970394
80
582286
631
417714
5
56
553010
550
970345
81
583665
631
417335
4
bV
553341
550
970397
81
583043
630
416957
3
58
553670
649
970349
81
583432
630
416578
2
59
554000
549
970200
81
583800
629
416200
1
ou
004339
r)4S
970152
81
584177
629
415N33
u
ma
(^'osiue 1
Sine I 1
Cotanp. 1
Tani'. M. 1
M.|
t
1 9
1
2
3
4
6
6
7
8
9
10
11
9
13
13
14
15
16
17
18
19
20
21
22
9.
|23
24
25
26
27
28
29
30
31
9.
32
33
34
35
36
37
(
38
(
39
t
40
t
41
9.f
42
r
43
f-
44
r-
45
f
46
R
t^
47
.'i
48
,'■
49
5
50
5
51
9.5
52
5
53
5
54
5
55
5
56
5
57
5
58
5
59
5
60
5
1
Co
09 Det^rtes.
inp. 1 1
38934
60
38541
59
38149
58
37756
o7
37364
56
36972
55
30581
54
36189
53
i5798
S2
35408
51
35017
50
49
34027
3423"
48 !j
33847
47
33458
46
J3008
45
J26S0
44
}229i
43
U902
42
U514
41
J1127
40
i0739
39
10352
38
•9965
37
59578
36
19191
35
J8805
34
!8419
33
S80J3
32
!7648
31
J7202
30
,6877
29
!6493
28
,6108
27
,5724
26
,5340
25
,4950
24
4573
23
4190
22
3807
21
3424
20
3041
19
2059
18
2277
17
1896
16
1514
15
1133
14
0752
13
0371
12
9991
11
9611
10
9231
9
8851
8
8472
7
8093
B
7714
5
7335
4
6957
3
6578
2
6200
1
5823
u
ns-
M.
SINES AND
M.
TANGENTS. (21 Degrees.;
39
Sine
D.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
50
57
58
59
60
9.554329
554658
554987
655315
555643
555971
556299
550626
556953
557280
557006
Conine | D. | Tnng. |
D.
9
557932
558258
558583
558909
559234
659558
559883
560207
660531
_560855
.561178
561501
561824
562146
562468
562790
.563112
563433
563755
56407.'
9 . 564396
.')64710
505036
66,5356
565676
565995
.560314
500032
500951
507209
9.. 50 758 7
567904
508222
508539
508850
509172
509488
509804
570120
57043S
9
,570751
571060
571380
571095
572003
572323
572036
5729.^0
573203
6735751
548
548
647
547
546
546
545
545
644
644
643
543
543
642
542
641
.541
540
540
539
539
9.9^0152
970103
970055
970006
969957
969909
969860
909811
969702
969714
969665
538
638
637
537
636
636
536
635
635
534
634
633
533
.532
532
531
531
531
530
_530_
529
529
528
628
528
527
627
526
626
525
81
81
81
81
81
81
81
81
81
81
81
82
82
82
82
82
82
82
82
82
82
82
82
82
82
83
83
83
83
83
83
9.968028 83
968578 83
968528 83
Cotang.
9.969616
969507
909518
969469
969420
969370
969321
969272
969223
969173
9.969124
969075
969025
968976
968920
908877
968827
968777
908728
908078
908479
908429
908379
908^29
908278
908228
968178
525
524
524
523
523
623
522
522
521
621
83
83
83
83
83
84
84
84
84
84
84
84
84
84
84
84
84
9.908128
968078
968027
967977
967927
967876
967826
967775
967725
96 7074
9.907024
967573
Gr,7.522
907471
907421
967370
967319
967268185
907217! 85
967166185
84
84
85
85
86
85
85
.684177
584555
684932
685309
685686
586062
686439
586815
687190
687566
587941
9.588316
688091
689066
689440
589814
690188
690502
590935
.591.308
.591681
9.592054
692426
592798
693170
693542
593914
594286
694656
59502V
695398
9.59.5768
696138
696508
696878
597247
597610
597985
598354
598722
699091
9
''199459
•99827
000194
600502
000929
601296
001002
002029
60239.-)
602761
9.603127
603493
003858
604223
604588
004953
605317
605682
000040
0004 lu
629
629
628
628
627
627
627
026
626
625
625
625
624
624
623
623
623
622
322
622
621
10.416823
415445
41.5068
414691
414314
413938
413661
4131S5
412810
412434
621
620
620
619
619
618
618
618
617
617
617
616
616
616
615
615
615
614
614
613
613
613
612
612
611
Gil
611
610
610
610
609
609
609
608
608
607
607
607
GOG
606
60
59
58
57
56
65
54
53
62
5
412059 I 60
49
48
47
46
45
44
43
42
41
40
39
38
37
36
36
34
33
32
31
30
29
28
27
26
26
24
23
22
21
20
19
18
17
16
lo
14
13
12
11
10
9
8
7
6
5
4
3
2
1
10.411684
411309
410934
410560
410186
409812
409438
409085
408692
4 0831 9
10.407946
407574
407202
406829
400458
406086
405715
405344
404973
404602
10.404232
403862
403492
403122
402753
402384
402015
401646
401278
400909
10.400541
400 17r
39980b
3994.38
399071
398704
398338
397971
397605
39723 9
10.396873
390507
390142
395777
395412
395047
394683
394318
393954
393590
m
40
(22 Degrees.; a
VABliE OP LOGARITHMIC
M.
1 Bine
1 D.
1 C'oaine j D.
Tang.
D.
Coiling.
•^
U
9. 573575
1 621
9.967166
86
9.606410
606
10.39.3590
60
1
673888
620
967115
85
606773
606
393227
59
2
674200
620
967064
85
607137
605
392863
58
3
574512
619
967013
85
607500
605
392500
57
4
674824
619
966961
85
607863
604
392137
56
5
575136
519
966910
85
608225
604
391775
55
6
675447
618
966859
85
608588
604
391412
54
7
575758
618
966808
85
608950
603
391050
53
8
576069
517
966756
86
609312
603
390688
52
9
576379
617
966705
86
C09674
603
390326
51
10
11
576689
616
966653
86
86
610036
9.610397
602
389964
10.389603
:,o
49
9.576999
616
9.966602
602
12
577309
516
966550
86
610759
602
389241
48
13
577618
615
966499
86
611120
601
388880
47
14
677927
615
966447
86
611480
601
388520
46
15
578236
514
966395
86
611841
00 r
388159
45
16
578545
514
966344
86
612201
600
387799
44
17
578853
513
966292
86
612561
600
387439
43
18
579162
613
966240
86
612921
600
387079
42
19
579470
513
966188
86
613281
699
386719
41
20
21
579777
9.. 580085
612
966136
86
87
613641
599
386359
10.386000
40
39
612
9 966085
9.614000
598
22
580392
511
966033
87
614359
698
385641
38
23
580699
511
965981
87
614718
598
38.5282
37
24
581005
611
965928
87
616077
697
384923
36
25
581312
510
965876
87
615435
697
384565
35
26
681618
510
965824
87
615793
597
384207
34
27
681924
509
965772
87
616151
596
383849
33
28
582229
509
965720
87
610509
596
383491
32
29
582535
509
165668
87
616867
596
383133
31
30
31
582840
508
965615
87
87
617224
595
382776
10.382418
30
29
9.583145
508
9.965563
9 617582
595
32
583449
507
965511
87
617939
595
382061
28
33
583754
507
965458
87
618295
594
381705
27
34
584058
606
96.5406
87
618652
594
381348
26
35
684361
606
965353
88
619008
694
380992
25
36
684665
"^06
965301
88
619364
593
380636
24
37
584968
605
965248
88
619721
593
380279
23
38
585272
505
965195
88
620076
593
379924
22
39
585574
604
965143
88
620432
592
379568
21
40
41
685877
604
965090
88
88
620787
592
379213
10.378858
20
19
9.. 586 179
603
9.965037
9.621142
592
42
586482
503
964984
88
621497
591
378503
18
43
686783
503
964931
88
621852
591
378148
17
44
587085
502
964879
88
622207
690
377793
16
45
687386
502
964826
88
622561
590
377439
15
46
687688
.501
964773
88
622915
590
377085
14
47
587989
501
964719
88
623269
589
376731
13
48
688289
501
964666
89
623623
589
376377
12
49
688590
500
964613
89
623976
689
376024
11
50
51
588890
9.589190
500
964560
9.964507
89
89
624330
9.624683
588
375670
10.375317
10
9
499
588
52
589489
499
964454
89
625036
588
374904
8
53
689789
499
964400
89
625388
687
374612
1-/
1
54
590088
498
964347
89
625741
687
374259
6
55
590387
498
964294
89
626093
587
373907
5
56
590686
497
964240
89
626445
586
37355a
4
57
590984
497
964187
89
626797
586
373203
3
58
591282
497
964133
89
627149
.586
372851
2
59
591580
498
964080
89
627501
585
372499
1
60
591878
496
9G4026'89
627852
.535
372 M-!
u
Co .le 1
Sine
C(it;iii),'. 1
Taiip. 1 M. [
M.
'~0
1
2
3
4
6
8
9
10
11
12
13
14
16
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
3:
34
35
36
37
38
39
40
44i
44
45
46
47
48
49
50
61
153
I 53
54
55
56
57
58
6y
60
(iT Dcuit'ci
irig.
359(r
60
3227
59
2863
58
2500
57
2137
56
1775
55
1412
54
1050
53
0688
52
0326
51
9964
:.o
9603
49
9241
48
8880
47
8520
46
8159
45
7799
44
7439
43
7079
42
3719
41
6359
40
GOOO
39
'i641
38
5282
37
4923
36
4565
35
42G7
34
3849
33
3491
32
3133
31
2776
30
3418
29
2061
28
1705
27
1348
26
0992
25
0636
24
0279
23
9924
22
9568
21
9213
20
8858
19
8503
18
8148
17
?793
16
7439
15
7085
14
6731
13
6377
12
6024
11
5670
10
6317
9
4904
8
4612
1-/
1
4259
6
3907
5
3555
4
3203
3
2851
2
2499
1
2M-i
iifi.
M.
M.
Sine
1
2
3
4
5
6
7
8
9
10
11 9.695T37
12^ 695432
595727
596021
596315
596609
596903
597196
597490
597783
9.591878
592176
692473
692770
693067
593363
693659
693955
594251
694547
594842
SINKS AM> ■^A^'(M;lv^s. (23 Degrees.;
_l !>■ I (.'osiiic I ]i, I Tarii.' | D.
41
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
3C
34
35
36
37
38
39
40
41
^2
43
44
45
46
47
48
49
50
496
495
495
495
494
494
493
493
493
492
492
9.964026189
C()taii)(.
9.598075
598368
598660
698952
r'9244
099536
699827
600118
600409
600700
9.600990
601280
601570
601860
602150
602439
602728
603017
603305
603594
491
491
491
490
490
489
489
489
488
488
963972
963019
903865
963811
963757
963704
963650
963596
963542
963488
9.963434
963379
963325
963271
963217
9631631 90
963108 91
89
89
90
90
90
90
90
90
90
90
90
90
90
90
90
9.627852
628203
628554
628905
629255
629606
629956
6303(]o
630656
631005
631,355
487
487
487
486
486
485
485
485
484
484
484
483
483
482
482
482
481
481
481
480
963054
962999
962945
9.962890
962836
962781
962727
962672
962617
962562
962508
962453
962398
9
61
53
53
54
55
56
57
58
6y
60
.603882
604170
604457
604745
605032
605319
605606
605892
606179
606465
9 606751
607036
607322
607607
607892
608177
608461
608745
609029
609313
480
479
479
479
478
478
478
477
477
476
9.962343
962288
962233
962178
962123
962067
962012
961957
961902
961846
9.961791
961735
961680
961624
961569
961513
961458
961402
961.346
961290
476
478
475
475
474
474
474
473
473
473
9.961235
961179
961123
961067
961011
9609.55
960899
960843
960786
960730
91
91
91
91
91
91
91
91
91
91
91
91
92
92
92
92
92
92
92
92
92
92
92
93
92
92
93
93
93
93
93
93
93
93
93
93
93
93
93
93
94
94
94
9.631704
6320.53
632401
632750
633098
633447
633795
634143
634490
634838
585
685
585
584
584
683
583
583
583
582
582
lO.. 372 148
371797
371446
371095
370745
370394
370044
369694
369344
368995
368645
9.636185
635532
635879
636226
636572
636919
637265
637611
637956
638302
9.6.38fi47
63K992
63i)337
639682
640027
640371
640716
641060
641404
641747
9.642091
642434
642777
643120
613463
643806
644148
644490
644833
645174
Cosine
.645516
64.5857
646199
646540
646881
647233
647563
647903
648243
648583
Sine
I
583
581
581
581
560
580
580
679
579
_57o
578'
578
678
577
577
577
577
676
576
576
10.368296
367947
367599
367250
366902
366553
366205
365857
365510
365162
10.364815
364468
364121
363774
363428
363081
362735
362389
362044
361698
575
676
676
674
674
574
673
673
673
672
10.. 36 135;,
361008
360863
360318
359973
359629
359284
60
59
58
57
66
56
64
53
62
61
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
36
34
33
33
31
30
29
38
27
26
26
34
23
358940 23
673
5^3
572
671
571
671
670
570
570
669
669
.569
569
568
568
568
667
5B'''
567
.566
368696
368353
10.357909
357566
357323
356880
356537
3.56194
3r.5852
35.5610
355168
354626
10.354484
354143
353801
353460
353119
352778
352438
21
201
19
18
17
16
16
14
13
13
11
10
9
8
7
G
5
4
3
Cdtang.
3517571 I
351417.
1 riim- fir
6& Degrees,
HP
42
(24 Degrees.") a table of logarithmic
M.
Sino
1 D.
1 Cobine 1 D.
Tang.
D.
Colarip. 1 1
U
9.609313
473
9.960730
94
9.648583
566
10.351417
60
1
609597
472
960674
94
648923
666
351077
59
2
609880
472
960618
94
649263
666
350737
58
3
610164
472
960561
94
649602
566
350398
57
4
610447
471
960505
94
649942
665
350058
56
6
610729
471
960448
94
6.50281
565
349719
55
6
611012
470
960392
94
650620
665
349380
54
7
611294
470
960335
94
650959
564
349041
53
8
611576
470
960279
94
661297
664
348703
52
611858
469
960222
94
661636
664
348364
51
10
11
612140
469
960165
94
95
651974
563
348026
10.347688
50
49
9.612421
469
9.960109
9.652312
663
12
612702
468
960052
95
652650
663
347350
48
13
612983
468
959995
95
652988
663
347012
47
14
613264
467
959938
95
653326
662
346674
46
15
613545
467
959882
95
653663
662
346337
45
1«
613825
467
959825
95
654000
662
346000
44
17
614105
466
959768
95
654337
661
345663
43
18
614385
466
959711
95
654674
561
345326
42
19
614665
466
959654
95
655011
661
344989
41
20
21
614944
465
959596
95
95
655348
661
344652
10.344316
40
39
9.615223
465
9.959539
9 . 655684
560
22
615502
465
959482
95
656020
560
343930
38
23
615781
464
959425
95
656356
560
343644
37
24
6 J 6060
464
959368
95
656692
659
343308
36
25
616338
464
959310
96
657028
559
342972
35
2(5
616616
463
959253
96
657364
559
342636
34
27
616894
463
959195
96
657699
559
342301
33
28
617172
462
959138
98
6580,?4
658
341966
32
29
617450
462
959081
96
658369
658
341631
31
30
31
617727
462
959023
96
96
658704
568
341296
10.340961
30
29
9.618004
461
9.958965
9.6.59039
658
32
618281
461
958908
96
659373
557
340627
28
33
618558
461
958850
96
659708
557
340292
27
34
618834
460
958792
96
660042
557
339958
26
35
619110
460
958734
96
660376
557
339624
25
36
619386
460
958677
96
660710
556
339290
24
37
619662
459
958619
96
661043
556
338957
23
38
619938
459
958561
96
661377
556
338623
22
39
620213
459
958503
97
661710
656
338290
21
40
41
62048P
458
958445
97
97
662043
655
337957
10.337624
20
19
9.620763
• 458
9.958387
9 662376
655
42
621038
457
958329
97
662709
554
337291
18
43
621313
457
958271
97
663042
554
336958
17
4-1
621587
457
958213
97
663375
654
336625
16
45
621861
456
958154
97
663707
554
336293
15
46
622135
456
958096
97
664039
663
335961
14
47
622409
456
958033
97
664371
553
335629
13
48
622682
455
957979
97
664703
553
335297
12
49
622956
455
95'!'921
97
665035
663
334965
11
60
51
623229
455
957863
97
9V
665366
9.665697
552
3346S4
10.334303
10
9
9.623502
454
9.957804
552
52
623774
454
957746
98
666029
552
333971
8
53
624047
454
957687
98
666360
551
333640
7
54
624319
453
957628
98
666691
551
.333309
6
55
624591
453
957570
98
667021
651
332979
5
56
624863
453
957511
98
667352
551
332648
4
67
625135
453
957452
98
6676S2
550
332318
3
68
625406
452
957393
98
668013
550
331987
2
59
625677
452
957335
98
668343
550
tin -I ffr »v
1
i
60
625918
451
957276
98
668672
550
33132S
1 Cosine
J Sine 1
Coiling.
1
1 Tang. 1 M. 1
M.
"o"
1
2
3
4
6
6
7
8
9
12
11
12
13
14
15
16
17
18
19
20
21
22
23
24
2b
26
27
28
29
30
31
32
33
34
35
36
S7
38
39
40-
41
42
43
44
45
46
47
48
49
50
53
54
55
66
57
58 I
60 I
65 Degrees.
'^. 1 1
417
60
077
59
737
58
398
57
058
56
719
55
380
54
041
53
703
52
364
51
026
50
S88
49
350
48
012
47
674
46
337
45
OOO
44
663
43
326
42
989
41
652
40
316
39
980
38
644
37
308
3G
972
35
636
34
301
33
966
32
631
31
296
30
961
29
627
28
292
27
958
26
624
25
290
24
957
23
623
22
290
21
957
20
624
19
291
18
958
17
625
16
293
15
961
14
629
13
297
12
965
11
684
10
303
9
971
8
640
7
309
6
979
5
648
4
318
3
987
2
f^rrt^f
1
i}o;
i
328
iP-
|M.
SINES AND TANGENTS. (25 Degrees.) 43
1
2
3
4
6
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
2b
26
27
28
29
30
31
32
33
34
35
36
S7
38
39
40
41
42
43
44
45
46
47
48
49
50
9.625948
626219
626490
626760
627030
627300
627570
627840
628109
628378
_(i28647
9.6289161
629185
629453
629721
629989
630257
630524
630792
631059
_ ^1^?^
9.631593
631859
632125
032392
632658
632923
633189
633454
633719
633984
451
451
451
450
460
450
449
449
449
448
448
.957276)
9572171
957158
957099
957040
956981
956921
956862
956803
956744
956684
447 9
447
447
446
446
446
446
445
445
445
9.634249
6.34514
634778
635042
635306
635570
635834
636097
636360
636623
444
444
444
443
443
443
442
442
442
441
. 956625
956566
956506
956447
956387
956327
956268
956208
956148
956089
51
52
53
54
55
56
57
58
ou
60
9.636886
637148
637411
637673
637935
638197
638458
638720
638981
639242
441
440
440
440
439
439
439
438
438
438
.956029
955969
955909
955849
955789
955729
955669
955609
955548
95.5488
9819.668673
981 669002
98 669332
98| 669661
669991
670320
670649
G70977
671.306
671634
671963
9.672291
672619
672947
673274
673602
673929
674257
674584
674910
675237
.955428
955368
955307
955247
955186
955126
955065
955005
954944
95488?
9.639503
639764
640024
640284
640544
640804
641064
641.324
G4 1 584
6418421
437
437
437
437
436
436
436
435
435
435
434
434
434
433
433
433
432
432
432
431
I 9.54823
954762
954701
954640
954579
954518
954457
954396
954335
954274
.954213
954152
954090
954029
953968
953906
953845
953783
953722
9536601
). 675564
675890
676216
676543
676869
677194
677520
677846
678171
67 8496
9.678821
679146
679471
679795
680120
680444
680768
681092
681416
681740
9.682063
682387
682710
683033
683356
683679
684001
684324
684646
68496 8
9.685290
685612
685934
686255
686577
686898
687219
687540
687861 1
688182
5.50
549
549
549
548
548
548
548
547
547
647
10
547
546
546
646
546
645
545
545
544
544
.331327
330998
330668
330339
330009
329680
329351
329023
328694
328366
10
544
644
543
543
643
643
642
542
542
642
60
59
58
67
66
55
64
63
52
61
. 328037 50
.327709
327381
327053
326726
326398
326071
325743
32.5416
325090
324763
10
641
641
641
641
540
540
640
640
639
639
,324436
324110
323784
323457
323131
322806
322480
322154
321829
321504
10
639
639
638
638
638
538
537
537
637
537
.321179
320854
320529
320205
319880
319,556
319232
318908
318584
318260
536
536
636
636
535
635
535
534
534
10
.317937
317613
317290
316967
316644
316321
31.5999
315676
31.5354
315032
10.314710
314388
314066
313745
313423
313102
312781
oi*4DU
312139
311818
49
48
47
46
4o
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
G
5
4
3
1
64 Degrees,
[IK •*'
44
(26 Degrees.) a
TABLE OF LOOABITIIMIC
M.
Bine
D.
t CoHinu 1).
■IV.tIL'. 1
I)
1 C.tttirig.
9.641842
431
U . 953660
103
9.688182
634
10.311818
60
1
642101
431
953599
103
088;j)2
534
311498
59
2
042360
431
9.13537
103
688823
634
311177
68
3
642618
430
953475
103
089143
533
310857
67
4
642877
430
953413
103
689463
633
310537
66
5
643135
430
953352
103
689783
533
310217
55
6
643893
430
953290
103
690103
533
309897
54
7
643650
429
953228
103
690423
533
309577
53
8
643908
429
953166
103
690742
532
309258
52
9
644165
429
953104
103
691062
632
308938
51
10
11
044423
9.644680
428
953042
103
104
691381
9. 69 1700
632
308619
10.308300
50
49
42S
9.952980
531
1*2
644936
428
952918
104
692019
631
307981
48
13
645193
427
952855
104
692338
631
307662
47
14
645450
427
952793
104
692656
531
307344
46
15
645706
427
952731
104
692975
631
307025
45
16
645962
426
952669
104
693293
530
306707
44
17
646218
426
952606
104
693612
530
306388
43
18
646474
426
952.544
104
693930
.530
306070
42
19
646729
425
952481
104
694248
630
305752
41
20
21
646984
425
952419
9.9.52356
104
104
694566
9.694883
,529
305434
10.305117
40
39
9 . 6-!
t7240
425
529
22
6^
7494
424
952294
104
69.5201
629
304799
38
23
6^
17749
424
952231
104
69.5518
529
304482
37
24
6'
8004
424
9.52168
105
695836
629
304164
36
25
6^
8258
424
952106
105
6961.53
528
303847
35
26
6^
18512
423
9.52043
105
696470
528
303530
34
27
6'
8766
423
951980
105
696787
528
303213
33
28
G4
9020
423
951917
105
697103
528
302897
32
29
6^
19274
422
9518.54
105
697420
627
302580
31
30
31
G-i
19527
422
951791
105
105
697736
9.6980.53
527
302264
10.301947
30
29
9.649781
422
9.951728
527
32
650034
422
951665
105
698369
627
301631
28
33
650287
421
951602
105
698685
.526
301315
27
34
650539
421
951.539
105
699001
626
300999
26
35
650792
421
951476
105
699316
526
300684
25
36
651044
420
951412
105
699632
626
30036b
24
37
651297
420
951349
106
699947
.526
300053
23
38
651549
420
951286
106
700263
625
299737
22
39
651800
419
951222
106
700578
525
299422
21
40
41
652052
9.652304
419
951159
106
106
700893
9.701208
625
299107
20
19
419
9.951096
524
10.298792
42
652555
418
951032
106
701.523
524
298477
18
43
652806
418
9.50968
106
7018.37
524
298163
17
44
653057
418
950905
106
702152
524
297848
16
45
653308
418
950841
106
702466
.524
297534
15
46
653558
417
9,50778
106
702780
52.
297220
14
47
653808
417
950714
106
70.3095
623
296905
13
48
G54059
417
950650
106
703409
523
296.591
12
49
654309
416
950586
106
703723
523
296277
11
50
51
654558
9.654808
416
950522
107
107
704036
9.7043,50
623
295964
10.29.5650
10
9
416
9.9.504,58
522
52
655058
416
950394
107
704663
522
295337
8
53
655307
415
9.50330
107
704977
522
295023
7
54
655556
415
950266
107
'^05290
522
294710
6
55
655805
415
950202
107
705603
521
294397
5
56
656054
414
9.501.38
107
705916
621
294084
4
57
656302
414
950074
107
706228
,521
293772
3
58
656551
414
950010
10'^
706541
521
293459
2
CO
658799
413
949945
10?
706854
S'-
1
60
657047
413
949881
107
707166
520
2928.34
_!
Cosine
Sine 1
Oota.ig.
1 I'ang. 1 M. J
63 Degrees.
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
191
20
21
22
23
24
25
26
27
28
29
30
31"
32
33
34
35
36
37
38
39
»40
41
42
43
44
45
46
47
48
49
60
51
9.
52
53
54
65
56
57
n»
t
59
(
60
(
)
C(
«■
318
60
li»8
59
177
58
if>7
57
537
56
il7
55
397
54
577
53
^58
52
^38
51
619
50
100
49
981
48
662
47
344
46
1)25
45
707
44
188
43
[)70
42
752
41
434
40
117
39
799
38
482
37
164
36
847
35
530
34
213
33
897
32
580
31
264
30
947
29
631
28
315
27
999
26
684
25
368
24
053
23
737
22
422
21
107
20
792
19
477
18
163
17
848
16
534
15
220
14
905
13
591
12
277
11
964
10
650
9
337
8
023
7
710
6
397
5
084
4
772
3
459
2
146
1
834
?• 1 M.
SINES AND TANOENTa. (27 Degrees.)
4.1
01 p. 657047
1 657295
657542
657790
f>^8037
658284
658531
658778
659025
659271
659517
9.659763
660009
6602.55
660501
660746
660991
661236
661481
661726
661970
.662214
662459
662703
662946
663190
663433
663677
663920
664163
664406
.664648
664891
665133
665375
^>65617
oG.5859
666100
666342
666583
666824
9.667065
667305
667546
667786
668027
668267
6(38506
668746
668986
669225
9.66'9464
669703
669942
670181
670419
670658
670896
413
413
412
4)2
412
412
411
411
411
410
410
410
409
409
409
409
408
408
408
407
407
9.949881
U49816
949752
949688
949623
949558
949494
949429
949364
949300
949235
407
407
406
406
406
405
405
405
405
404
404
404
403
403
403
402
402
402
<102
401
398
398
398
397
397
397
897
671372.1 396
60 1 671609 396
401
401
401
400
400
400
399
399
399
399
y. 949 170
949105
949040
948975
948910
948845
948780
948715
948650
94 8584
9.948519
948454
948388
948323
948257
948192
948126
948060
947995
947929
9.947863
947797
947731
947665
947600
947533
947467
947401
947335
947269
107)9.707166
1071 707478
707790
708102
708414
708726
709037
709349
709660
709971
710282
107
108
108
108
108
108
108
108
108
108
108
108
108
108
108
109
109
109
109
109
109
109
109
109
109
109
109
110
110
9.710593
710904
711216
711.'^25
711836
712146
712456
712766
713076
713386
9 947203
947136
947070
947004
946937
946871
946804
946738
946671
946604
110
110
110
110
110
110
110
110
110
110
9.713696
714005
714314
714624
714933
715242
715551
715860
716168
716477
9.946538
946471
946404
946337
946270
946203
946136
S46069
946002
945935
110
111
111
HI
111
111
111
111
Ul
m
111
111
111
111
112
112
112
112
112
112
9.716785
717093
717401
717709
718017
718325
718633
718940
719248
719555
9.719862
72C169
720476
720783
721089
721396
721702
722009,
722315
722621
I Cosine 1
9.722927
723232
723538
723844
724149
724454
724759
725066
725369
725674
520
520
520
520
619
519
519
619
519
518
518
518
618
618
547
517
617
517
616
516
516
10.2U28;M|f3u
292522 69
292210 ftfl
291898 57
291586 66
291274 65
290963 54
290651 53
290340 52
290029 61
289718 .50
516
516
615
4:5
6lN
615
614
614
514
514
614
613
613
613
613
613
612
612
512
612
612
511
511
511
511
511
510
510
610
510
510
509
509
509
509
509
608
808
508
608
10.289407
289096
288785
288475
288164
287854
287544
287234
286924
286614
10.286.304
285995
285686
285376
285067
284758
284449
284140
283832
2S3523
10.28.3215
282907
282^599
282291
281983
281670
281367
281080
280752
280446
10.280138
279831
279524
279217
278911
278604
278298
277991
277685
277379
10.277073
276768
276462
276156
27.5851
276646
275241
274835
: 7463 11
2743261
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
»5
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
I
5
4
3
2
I
\
62 Degrees.
\r^
s
46
(20 Degrees.; a TABLK of LOOAniTIIMTO
1
■11
M.
1 Kino
1 '»■
I UoHhU! 1 U.
1 Taiiu.
1 l>.
("..taiii;. 1 1
9.671009
396
"97945935
112
9.725674
608
10.274323
I5(r
1
671847
395
945868
112
725979
508
274021
59
2
672084
395
945800
112
7262H4
507
273716
5«
n
672321
395
945733
112
726588
507
273412
57
4
672558
395
945666
112
726892
507
273108
56
5
672795
394
945598
112
727197
507
2?2803
55
f)
673032
394
945531
112
727501
507
272499
54
7
673268
304
945464
113
727805
506
272195
53
8
673505
394
945396
113
728109
506
271891
52
9
673741
393
945328
113
728412
506
2V1588
51
10
ifi
673977
393
945261
9.945193
113
113
728716
506
271284
50
49
9.674213
393
9.729020
506
10.270980
12
674448
392
945125
113
729323
505
270677
48
i;{
674ti«'i
392
945058
113
729626
505
270374
47
14
674919
392
944990
113
729929
505
270071
46
15
675155
392
944922
113
730233
505
269767
45
If)
675390
391
944854
113
730.')35
505
269465
44
17
675624
391
944786
113
730838
504
269162
43
18
675859
391
944718
113
731141
504
268859
42
19
676094
391
944650
113
731444
504
268556
41
20
21
676328
390
944582
114
114
731746
504
268254
40
39
9.6765(52
390
9.944514
9.732048
504
10.267952
22
676796
390
944446
114
732351
503
267649
38
23
6T7030
390
944377
114
732653
503
267347
37
24
677264
389
944309
114
732955
503
267045
36
25
677498
389
944241
114
733257
503
266743
35
26
677731
389
944172
114
733558
503
266442
34
27
677964
388
944104
114
733860
502
266140
33
28
678197
388
944036
114
734162
502
265838
32
29
678430
388
9:3967
114
734463
502
265537
31
30
31
678683
388
943899
114
114
734764
502
265236
10.264934
30
29
9 678895
387
9.943830
9.735066
502
32
679128
387
943761
114
735367
502
264833
28
33
679360
387
943693
115
735668
501
264332
27
34
679592
387
943624
115
735969
501
264031
26
35
679824
386
943555
115
736269
501
263731
25
36
680056
386
943486
115
736570
501
263430
24
37
680288
386
943417
115
736871
501
263129
23
38
680519
385
943348
115
737171
500
262829
22
39
6S0750
385
943279
115
737471
500
262529
21
40
41
680982
385
943210
115
115
737771
500
262229
20
19
9.681213
385
9.943141
9.738071
500
10.261929
42
681443
384
943072
115
738371
500
261629
18
43
681674
384
943003
115
738671
499
261329
17
44
681905
384
942S34
115
738971
499
261029
16
4ft
682135
384
942864
115
739271
499
260729
15
46
682365
383
942795
116
739570
499
260430
14
47
682595
383
942726
116
739870
499
260130
13
48
682825
383
942656
116
740169
499
159831
12
49
683055
383
942587
116
740468
498
259532
11
50
51
683284
382
942517
116
116
740767
498
259233
10
9
9.683514
382
9.942448
9.741066
498
10.258934
02
683743
382
942378
116
741365
498
258635
8
53
683972
382
942308
116
741664
498
258336
7
54
684201
381
942239
116
741962
497
258038
6
55
684430
381
942169
116
742261
497
257739
5
56
684658
381
942099
116
742559
497
257441
4
57
684887
380
942029
116
742858
497
2.57142
3
58
685115
380
941959
116
743156
497
256844
2
09
GS5343
38U
941889 117
743454
49/
496
256540
1
60
685571
380
.. 941819 117
743752
256248
<l
"T Cosine I
[0
11
12
13
14
15
16
17
18
19
20
21
122
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
43
44
45
46
47
48
49
.50
51
52
53
54
55
56
57
58
59
60 I
Sine
Cotang.
Tang.
M.
81 Degrees.
•If-
1
i3^
-G^
K)21
59
r/i«
68
J412
57
1108
56
!80:)
55
,499
54
,195
53
891
52
ms
51
284
50
980
49
677
48
;j74
47
071
46
7G7
45
465
44
162
43
859
42
556
41
254
40
952
39
649
38
347
37
045
36
743
35
442
34
140
33
838
32
537
31
236
30
934
29
633
28
332
27
031
26
731
25
430
24
129
23
829
22
529
21
229
20
929
19
629
18
329
17
029
16
1729
15
1430
14
H30
13
1831
12
t532
11
)233
10
UJ34
9
?635
8
<336
7
^038
6
-739
5
?'441
4
n42
3
)S44
2
■» ^ a />
1
30i!J
3248
g-
SINES AND TANOENTS.
(20 Degrees.)
47
.685571
685799
68602V
08C254
686482
686709
686936
687163
6S7389
687616
687843
. 688069
688295
68*<5«1
688747
688972
689198
689423
689648
689873
690098
.690323
690548
690772
690996
691220
691444
691668
691892
692115
692339
.692.562
692785
693008
693231
693453
693676
693898
694120
694342
694564
380
379
379
379
37«
378
378
378
378
377
_?77_
3/7
377
376
376
376
376
375
375
375
_375
374
374
374
374
373
373
373
373
372
372
9.W18I9
■Ml 74 9
941679
941609
941539
941409
941398
941328
9412.581
941187
941 U7
9.941046
940975
940905
940834
940763
9406931
940622
940.551
940480
940409
9
J. 694786
6950071
6952291
695450
695671
69.5892
696113
696334
6965.54
696775
.696995
697215
697435
6976.54
697874
69S094
698313
698532
6987.51
698970!
372
371
371
371
371
370
370
370
370
369
.940.338
940267
940196
940125
940054
939982
939911
939840
939768
939697
9.713752
744050
741348
744645
744943
745240
745538
745835
746132
746429
746726
9.747023
747319
747616-
747913
748209
748505
748801
749097
749393
749689
496
496
496
496
496
496
495
495
495
495
495
10.256248160
255950 59 ,
25.5652 58
255:i55 57
255057
254760
2.54402
£.54165
25.3868
253.571
253274
56
65
Ui
.531
52 i
51
6«i
369
369
369
368
368
368
368
367
367
367
9.939625
939554
939482
939410
939339
939267
939185
939123
939052
938980
9.749985
750281
750576
750872
751167
751462
751757
752052
752347
752642
367
366
366
366
366
365
365
op-
365
364
9.938008
938836
9.38763
938691
938619
938547
938475
938402
9.38330
__938258
9.9.38185
9.38113
938040
937967
937895
937822
937749
ucs/fivOi
937604
9375311
9.752937
753231
753520
753820
7.54115
754409
754703
7.54997
7.55291
755585
9.7,55878
756172
766465
756759
757052
757345
757638
7579:^ 1
758224
758517
493
492
492
492
492
492
■1
4l>.
490
490
490
490
490
490
489
10.2.50015
249719
249424
249128
218P33
248538
^ .243
^7948
"653
'58
39
38
37
36
36
34
33
32
31
30
•3 29
9.758810
759102
759395
759687
759979
760272
760564
760856
761148
761439
489
489
489
489
489
488
488
488
488
488
.1
1
2'l^ucJ5
24.5591
246297
245003
244709
244415
10
488
487
487
487
487
487
487
486
486
486
244122
243828
243535
243241
242948
242655
24236S
242069
241776
241483
10
.241190
240898
240605
240313
r'<r0021
2.J9728
239436
2.19144
238852
238561
28
27
26
26
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
2
I
60 Degrees
i^^TI
' «
%>i
48
(30 Degrees.) a table op logahithmic
M. I Hiim I I).
3
4
5
6
V
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
38
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
159
[60
(.'odillR
TaiiR.
f).
9.698970
099 1 89
699107
699626
699844
700062
700280
700498
700716
700933
701161
9.701368
701585
701802
702019
7022:16
7024.'i2
702669
702885
703101
_7033_17
9.703533
703749
TK)3y64
704 179
7C4395
704610
704825
705040
705254
_ 705469
9 705683
705898
706112
706326
706539
706753
706967
707180
707393
707606
9.707819
708032
708245
708458
708670
708882
709094
709306
709518
709730
9 7099-il
710153
710364
710575
710786
710997
711208
711419
*y I 1 con
s 1 i \T ,g J
711839
364
364
364
364
363
363
363
363
363
362
36S
363
362
36]
361
,%1
361
360
360
360
360
359
359
359
359
359
358
358
858
358
357
357
357
357
356
356
356
356
355
355
355
355
354
354
354
354
353
353
353
353
353
352
352
352
351
351
351
351
350
9.937531
937458
937385
937312
937238
937135
937092
937019
936946
936872
936799
9,936725
936652
936578
936505
936431
936357
936284
936210
936136
__r "6062
9.93598b
9359 1'i
935840
935766
935692
935618
935543
935469
935395
93 5320
9.935246
935171
935097
935022
934 48
934873
934798
934723
9.'«4649
934574
9.934499
934424
934349
934274
934199
934123
934048
933973
933898
933822
9.933 '7
933/1
933596
933520
933445
933369
933293
933217
nnn ^ A t
933066
21
22
22
22
22
22
22
22
22
22
22
i,2
23
23
23
23
23
23
23
23
23
23
23
23
24
24
24
24
24
24
24
24
24
24
24
24
24
25
25
25
25
25
25
25
25
25
25
25
25
26
26
26
26
26
26
26
26
26
26
26
(-((laiit'
r
9.761439
486
761731
486
762023
480
762314
486
762606
485
762897
485
763188
485
763479
485
763770
485
764061
485
764352
484
9.764643
484
764933
484
765224
484
765514
484
76580r^
484
766095
484
766335
483
766675
483
766965
483
767255
483
9.767545
483
767834
483
768124
482
763413
482
768703
482
768992
482
769281
482
769570
482
769860
481
770143
481
9.770437
481
770726
481
771015
481
771303
481
771592
481
771880
480
772168
480
772457
480
772745
480
773033
480
9.773321
480
773608
479
773896
479
774184
479
774471
479
774759
479
775046
479
775333
479
775621
478
775908
478
9.776V05
478
776482
478
776769
478
777055
478
777342
478
777628
477
777915
477
778201
477
1 1 cid 1
'I i 1
778774
477
10
238.561160
238269 59
237977
237686
237394
237103
236812
23652!
236230
235939
235648
10.235357
235067
234776
234486
234195
233905
233615
233325
233035
232745
10.232455
232166
231876
231587
231297
231008
230719
23043 J
230140
229852
10.229563
229274
228985
228697
228408
228120
227832
227543
227255
226967
10.226679
226392
226104
225816
225529
225241
224954
224667
224379
224092
10.223805
223518
223231
222945
222658
222372
222085
221799
221226
Cosine
j Sine I I Cotang. I
58
57
56
55
54
63
52
51
50
49
48
47
4b
45
44
43
42
41
40
39
38
37
38
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
i
I Thiin- I M.
uQ £)'^(eui
1
2
3
4
5
8
7
8
9
]0
11
12
13
14
15
16
17
18
|l9
I 20
^21
122
iV3
24
TiO
26
27
28
129
30
'i^
32
33
34
35
36
37
38
139
40
41
9.'
42
43
44
45
7
40
7
47
7
48
7
49
7
50
7
152
'53
5.;
55
.'J6
57
58
59 I
601
lit'
n
S5«nj60|
S2()9
59
7977
58
768fi
57
■/•39-1
56
7103
55
1812
54
j52!
53
?230
52
VJ39
51
J<i48
50
)357
49
5007
48
177G
47
H86
46
1195
45
3905
44
<615
43
)325
42
J035
41
J745
40
i455
39
2166
38
1876
37
1587
38
1297
35
[008
34
)719
33
)43J
32
)140
31
)852
30
)5f)3
29
)274
28
^985
27
^697
26
^408
25
n20
24
r832
23
r543
22
r255
21
)967
20
5679
19
)392
18
)104
17
J81G
16
)529
15
>241
14
1954
13
1667
12
t379
11
1092
10
3805
9
3518
8
3231
7
2945
6
2658
6
2372
4
2085
3
1799
2
ibl)i
I
1226
M.
Hire
9 7^18391
7120501
71226C*
712469
712679]
712889
713090
71330]
713517
713726
__713935
9.714144|
7143521
714561
714769
714978
715186
715394
715602
715809
716017
). 71 6224
716432
716639
716646
717053
717259
717466
717673
71 7879
718085
1^1—L. !i!i:i!i« I «>. I Tmll'. I
348
347
347
347
347
347
346
346
346
346
'9.9330661 126
93J!')90| 127
9329141 127
932838! 127
33i:7f)'> 127
93268.)
93*i609
932533
932457
932380
932304
''.718291
7i0497
718703
718909
719114
719.320
719525
719730
719935
720140
345
345
345
345
345
344
344
344
344
343
9.9.32228
9,32151
y32075
931998
931921
931845
931768
931691
931614
931.537
9.J3146
931.383
931306
931229
9311.52
931075
93099S
930921
930843
930766
127
127
128
128
128
128
128
138
128
128
9.720345
720.549
720754
7; 9958
721162
721366
721570
721774
721978
72!3181
9.930688
930611
930533
930466
9.S0378
930300
930223
930145
930067
929989
x28
!28
128
129
129
129
129
129
129
129
.778774
779060
779346
779632
779910
7R020?
7804891
7N0775
781060'
7813461
__78163l|
9.781916
78',520l
782486
782771
783056
783341
783626
783910
784.95
784479
9.72238.'>l
722Ji88|
722791
722994
723197
723400
723603
723805
7240071
724210 1
Cosine I
9.929911
929833
929755
929677
929599
929521
929442
929364
929286
929207
9.929129
929050
928972
928893
928815
928736
928657
928578
t\i>OAf\n
928420
9.784764
7850481
785332
78.5016
785900
786184
786463
786752
7870361
78 73191
9.78760'
787886
788170
788453
788736
789019
789302
7895851
789868
_790151
1.790433
790716
790999
7912811
791563
791846
792128
792410
792692
792974
10. 221 22BnKr
220940| 59
8206.'i4 58
!i203f)8 57
82008r> 66
2197971 f,o
219511 -
219225
218940
2186,54
2]83^
10.218084
217799
217514
217229
216944
216659
216374
210090
21. -1805
215.521 40
1.215236
214952
214668
214384
214100
21.3816
213.')32
213248
212'J64
212681
9.793256
793538|
793819
794101
794383'
794664!
794945I
795227]
795,008!
7957891
.212397
212114
211830
211,547
211264
210981
210698
210415
210132
209849
10.209567
209284
209001
208719
298437
208154
207872 .,
207590 12
207308 II
207026 1 10
10.206744
206462
206181
205899
205617
205336
205055
2O4773I
2044921
204311
UJ^
no
(32 I)c<;frces.) a tablk of LofiAHiTHMic
M.
Sii:e
I).
('osiiiH
I I). I Tiing. [ P.
CoiniiR.
1
2
3
4
5
6
7
8
1)
10
11
12
1:3
14
15
16
17
\H
19
20
21
22
2:3
24
25
2fi
27
28
20
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
4f]
47
48
49
50
51
52
53
54
55
56
57
58
59
60
.724210
724412
724014
""■24810
725017
725219
725420
725622
725823
726024
726225
,726426
72»)626
726827
727027
727228
727428
727628
727828
728027
_728227
.728427
728626
7^28825
729024
729223
729422
72962
729820
730018
730216
.73041.5
730613
7308 1 1
731009
731206
731404
731602
731799
731996
732193
9.732390
732587
732784
732980
733177
733373
733569
733765
733961
7341.57
9.7.?4353
734549
734744
734939
735135
735330
73552
735719
735914
736109
:«37
337
336
336
336
336
335
335
335
335
334
334
334
334
334
333
333
833
333
333
332
332
332
332
331
331
331
331
330
330
330
330
330
329
329
329
329
329
328
328
328
328
328
327
327
327
327
327
326
326
9 . 928120
928342
928263
928183
928104
928025
927946
927867
927787
927708
92762t>
9.927549
927470
927390
927310
92723 1
92TI51
927071
926991
926911
92683 1
9.92'6751
926671
926591
926511
926431
926351
926270
926190
926110
926029
1321
132
132
132
132
132
132
132
132
132
132
132
133
133
133
133
133
133
133
133
1.33
326
326
325
325
325
325
r ■.. C
324
324
324
9.925949
925868
925788
925707
92.5626
925545
925465
925384
925303
92522 2
9. 925141
925060
924979
924897
924816
924735
924654
924572
924491
9244(t9
133
133
133
134
134
134
134
134
134
134
134
134
134
134
134
135
135
135
135
135
1.35
135
135
135
135
136
136
136
136
136
.795789
796070
796351
796632
796913
797194
797475
797755
798036
798316
798596
9.798877
799157
799437
799717
799997
800277
800557
800836
801116
801396
.801675
801955
802234
802513
802792
803072
803351
803630
803908
804187
468
468
468
468
468
468
468
468
4f\'T
467
j467
467"
467
467
467
460
466
466
466
466
466
9.804466
804745
80.5023
805302
805580
805859
806137
806415
8066J3
806971
9
9.924328
924246
924104
924083
924001
923919
92375.5
923673
923591
1.36
136
136
136
136
I3r.
I'M.
137
137
137
,807249
807.527
807805
808083
808361
808638
808916
809193
809471
809748
9.810025
810302
810.580
810857
811134
811410
811687
81196.
812241
812517
466
466
465
465
465
465
465
465
465
465
464
464
464
464
464
464
464
463
463
463
463
463
463
463
463
402
462
462
462
462
462
462
462
462
461
461
461
461
461
461
10.204211
203930
203649
203368
203087
202806
202525
202245
201964
201684
201404
10.201123
200843
200563
200283
200003
195)723
199443
199164
198884
198604
10.198325
198045
197766
197487
197208
196928
196649
196370
196092
195813
10.195534
19.5255
194977
194698
194420
194141
193863
193.585
193307
193029
10.192751
192473
192195
191917
191639
191362
191084
190807
190529
190252
10.189975
189698
189420
189143
188866
188590
188313
; 88036
187759
187483
60
59
58
57
56
55
54
.53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1
2
.3
4
fi
7
8
9
10 I
u
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
4f
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
nS
59
60
Cosine
Sine
t
Cotansf.
'J'ang.
M.
S7 Decrees.
HiR. 1
1211 60
M)30
59
3649
58
13fiH
57
5087
56
i80«)
55
252.')
54
2245
53
PJfil
52
l()8'l
51
1404
50
1123
49
)843
48
)5f.3
47
^283
46
1)003
45
.(723
44
9443
43
9104
42
8884
41
8604
40
8325
39
8045
38
7760
37
7487
36
7208
35
6928
34
6649
33
6370
32
6092
31
5813
30
5534
29
5255
28
4977
27
4698
28
4420
25
4141
24
3863
23
3585
22
13307
21
>3029
20
12751
19
12473
18
)2195
17
)1917
16
)1639
15
)1362
14
H084
13
)0807
12
)0529
11
)0252
10
•!9975
9
^9698
8
39420
7
39143
6
38860
5
88590
4
88313
3
S803fc
) 2
87755:
1 1
874 8r
1
ang. 1 M.
m
SIXrs AND TA.VOKNT8. (33 DegrPGS.)
fil
9.736109
736303
736498
736692
736886
737080
737274
737467
737661
737855
_72H0J8
'738241
738434
738627
738820
739013
739206
7393;J8
739590
739783
_739975
.740167
740359
740550
740742
740934
741125
741316
741508
741699
741889
9
. 742080
742271
742462
742652
742842
743033
743223
743413
743602
743792
9.743982
744171
7443G1
744550
744739
744928
745117
745306
745494
745683
9
.745871
746059
746248
746430
746624
746812
746999
747187
747374I
7475621
324
324
324
323
323
323
323
323
322
322
322
322
322
321
321
321
321
321
.320
320
320
320
320
319
319
319
319
319
318
318
318
318
318
317
317
317
317
317
316
316
316
316
316
31.5
315
315
315
315
314
314
314
.']14
314
313
313
313
313
313
312
313
312
y. 923591
923509
923427
923345
92.'}26;i
923181
923098
923016
922933
922851
_9227J58
9.922686
922603
922.'>20
922438
922355
922272
922189
922106
922023
921940
9.921857
921774
921691
921607
921.524
921441
921357
921274
921190
9 21107
9.921023
920939
920856
920772
920688
920604
920520
920436
920352
920268
9.920184
920099
920015
919931
919846
919762
919677
919.593
919.508
919424
9.9L9339
919254
919169
919085
919000
918915
918830
918745]
918659
918574
.812517
812794
y 13070
813347
813623
813899
814175
8144.52
814728
81. -3 004
_ 815279
9.81.5555
81.5831
8i6107
816.382
8166.58
816933
817209
817484
817759
8J8035
9.818310
818.585
818860
81913;
819410
819684
819959
820234
820.508
820783
9
821057
821332
821606
821880
8221.54
822429
82270
822977
823250
_823524
.823798
824072
824345
824619
824893
825166
825439
825713
825986
826259
lAO
9.826532
826805
827078
827351
827624
827897
828170
828442
828715
828987
10.187482
187206
186930
186653
186.377
186101
185825
18.5.548
185272
184996
184721
10
,184445
184169
183893
183618
183342
183067
182791
182516
182241
181965
i0.18Ki90
181415
181140
180865
180590
180316
180041
179766
179492
179217
457 10
457
178943
178668
178394
178120
177846
177571
177297
177023
176750
176476
. 176202
175928
17.5655
17.5381
175107
174834
174.561
174287
174014
17.3741
.173468
173195
172922
172649
172376
172103
171330
1715.58 2
171285 1
1710131
60
59
58
57
56
55
64
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15 »
14
13
12
11
iO
9
8
7
6
Til I 111.
M.
56 D'!gre«;s.
52
(34 Degrees.) a
TABLE OP LOGARITHMIC
"mT
Sine
D.
1 Cosine | D.
1 Tang.
1 D
1 Cotaiig. 1 1
9.747562
312
9.918574
142
9.828987
454
10.171013
60
1
747749
312
918489
142
829260
454
170740
59
2
747936
312
918404
142
829532
454
170468
58
3
748123
311
918318
142
829805
454
170195
57
4
748310
311
9182.33
142
830077
464
169^^23
66
5
748497
311
918147
142
830349
453
169(351
56
6
748683
311
918062
142
830621
453
169379
54
7
748870
311
917976
143
830893
453
169107
53
8
749056
310
917891
143
831165
453
168835
52
9
749243
310
917805
143
831437
453
168563
51
10
11
749429
310
917719
143
143
831709
9.631981
453
453
168291
50
49
9.749615
310
9.917634
10.168019
12
749801
310
917548
143
832253
453
167747
48
13
749987
309
917462
143
832525
453
167475
47
14
750172
309
917376
143
832796
453
167204
46
15
750358
309
917290
143
833068
452
166932
45
16
750543
309
CI 7204
143
833339
452
16G661
44
17
750729
309
917118
144
833611
452
166389
43
18
750914
308
917032
144
833882
452
166118
42
19
751099
308
916946
144
834164
452
165846
^.1
20
21
751284
308
308 ■
916859
144
144
834425
462
165575
10.165304
40
39
9.751469
9.916773
9.834696
452
22
751654
308
916687
144
834967
452
165033
38
23
751839
308
916600
144
835238
452
164762
37
24
7^2023
307
916514
144
835509
452
164491
36
25
752208
307
916427
144
835780
451
164220
35
26
752392
307
916341
144
836051
451
163949
34
27
752576
307
916254
144
836322
451
163678
33
28
752760
307
916167
145
836593
451
163407
32
29
752944
306
916081
145
836864
451
163136
31
30
31
7.53128
9 753312
306
915994
145
145
837134
451
162866
30
29
306
9.915907
9.837405
451
10.162595
32
753495
306
915820
145
837675
451
162325
28
33
753679
306
915733
145
837946
451
162054
27
34
753862
305
915646
145
838216
451
161784
26
35
754046
305
915559
145
838487
450
161513
25
36
754229
305
915472
145
838757
450
161243
24
37
754412
305
915385
145
839027
450
160973
23
38
754595
305
9152P7
145
839297
450
160703
22
39
754778
304
915210
145
839568
450
160432
21
40
41
754960
304
915123
146
146
839838
9.840108
450
160162
20
19
9.755143
304
9.915035
450
10.159892
42
755326
304
914948
146
840378
450
159622
18
43
755508
304
914860
146
840647
450
159353
17
44
755690
304
914773
146
840917
449
159083
16
45
755872
303
914685
146
841187
449
158813
15
46
756054
303
914698
146
841467
449
158543
14
47
756236
303
914510
146
841728
449
158274
13
48
756418
303
914422
146
841996
449
158004
12
49
766600
303
914334
146
842266
449
157734
11
50
51
756782
302
914246
147
147
842535
449
157465
10
9
9.756963
302
9.914158
9.842805
449
10.1,57195
62
757144
302
914070
147
843074
440
156926
8
53
757326
302
913982
147
843343
449
156657
7
54
757507
302
913894
147
843612
449
156388
6
55
757688
301
913806
147
843882
448
1.56118
6
56
757869
301
913718
147
844151
448
15.5849
4
57
758050
301
913630
147
844420
448
155580
3
KO
•vcQoon
om
147
aAAdan
AAQ
'559,1 I
o
rf' 7
? ?,-\.' vjrf
«-?VF 1
•w'-r-i'-f* ■•^^
TT-J?
59
758411
301
913453
147
844958
448
155042
1
60
758591
301
913365
147
845227
448
154773
CoKirie 1
1
Sine 1 1
Coiang.
Tang. M j
11
12
13
14
15
16 1
17
18
19
20
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
142
43
44
45
46
47
48
49
50
51
52
53
54
55
57
58
59
60
55 Degrees.
■Jllg.
1
013
60
)740
59
»468
58
(195
57
K>23
56
(iol
55
1379
54
1107
53
1835
52
663
51
1291
50
iOl9
49
'747
48
'475
47
'204
46
.932
45
.661
44
.389
43
(lis
42
.846
'^.1
.575
40
.304
39
.033
38
1762
37
t491
36
1220
35
1949
34
1678
33
1407
32
tl36
31
!866
30
,595
29
!325
28
!054
27
784
26
513
25
243
24
»973
23
»703
22
»432
21
1162
20
1892
19
1622
18
1353
17
1083
16
!813
15
>543
14
!274
13
!004
12
'734
11
r465
10
'195
9
1926
8
.657
7
.388
6
ill8
6
.849
4
.580
3
.Ol 1
o
.042
1
t773
8-
M
SINES AND TANGEWT9. (35 Degrees.)
9.768591
758772
758962
759132
759312
759492
759672
759852
760031
760211
__760390
9.760569
760748
760927
761106
761286
761464
17 761642
i« 761821
761999
_ 762177
9.762356
762534
762712
•^62889
763067
763245
763422
763000
763777
763954
TaiiE.
.913365
913276
913187
913099
913010
912922
912833
912744
912655
912566
912477
D,
9.912388
912299
912210
912121
912031
911942
911853
911763
911674
911584
9.845227
845496
845764
846033
846302
846570
846839
847107
847376
847644
847913
Cotaiig.
9.764131
764308
764485
764662
764838
765015
76519J
765367
766644
785720
>. 766896"
766072
766247
766423
766698
766774
766949
767124
767300
767475
9.911496
911405
911316
911226
911136
911046
910956
910866
910776
91068 6
9.910596
910506
910415
910325
910235
910144
910054
909963
909873
909782
9.848181
848449
848717
848986
849264
849622
849790
860058
850325
860593
448
448
448
448
448
447
447
447
447
447
447
9.860861
861129
851396
851664
861931
852199
862466
852733
853001
853268
447
447
447
447
447
447
446
446
446
446
10. J 54773
154604
154236
163967
1636S8
153430
153161
162893
152624
152366
152087
1.767649
767824
767999,
768173
768348
7686971
768871
769045
9.909691
909601
909510
909419
909328
909237
9091^^6
909056
908964
908873
9.863635
863802
864069
854336
854603
854870
855137
865404
866671
855938
446
446
446
446
446
446
446
445
445
445
9.908781
908690
908699
908507
908416
908324
908233
908141
908049
907958
151 9.866204
8.56471
856737
857004
857270
857637
867803
1S2| 858069
152 858336
152| 86 8602
162
162
152
152
153
163
445
445
445
445
445
445
445
445
444
444
9.858868
8.59134
859400
859666
85993'^-
---, 86019^
163 86046 1
153| 860730
860995
861261
444
444
444
444
444
444
444
444
444
443
60
59
58
57
66
65
54
63
62
51
^ 60
10.151819 49
151551 48
151283 47
151014 46
150746
150478
160210
149942
149675
149407
10.149139
148871
148604
148336
148069
147801
147534
147267
146999
146732
10.146465
146198
145931
145664
145397
146130
144863
144596
144329
144062
10
443
443
443
443
4/ia
443
44?
443
443
443
143796
143529
143263
142996
142730
142463
142197
141931
141664
141398
10.141132
140866
140600
140334
139802
139536
139270
139005
138739
1!»
I
54
(3G Degrees.) a
TIBLE OF L06AU1T1IMIC
M.l
Sine
D.
fosine 1 D. |
'Pnnff. 1
D. 1
Ci'ianp. 1 1
9.769219
290
9.9079.581 153
9.861261
443
10.1.38739
60
1
769393
289
907866 153
861527
443
138473
59
2
' 769566
S189
907774
153
861702
442
138208
58
3
769740
289
907682
153
862058
442
137942
J>7
4
769913
289
907590
153
862323
442
137677
56
5
770087
289
907498
153
862589
442
137411
55
6
770260
288
907406
153
862854
442
137146
54
7
770433
288
9073 M
154
863119
442
136881
53
R
770606
288
907222
154
863385
442
1,36615
52
9
770779
288
907129
154
863650
442
136.3.50
51
10
1 1
770952
288
288
907037
154
154
863915
442
136085
50
49
9.771125
9 906945
9.864180
442
10.135820
!2
771298
287
906852
154
864445
442
1355.55
48
13
771470
287
906760
154
864710
442
13.5290
4'/
14
771643
287
906667
154
864975
441
13.5025
46
1ft
771815
287
906575
154
865240
441
134760
4.'d
16
771987
287
906482
154
865505
441
134495
44
17
77'>.159
287
906389
155
865770
441
134230
43
18
772331
286
906296
155
866035
441
133965
42
19
772503
286
906204
155
866300
441
133700
41
20
21
772675
286
906111
1.55
155
866564
9.866829
441
133436
40
39
9.772847
286
9.906018
441
10.133171
22
773018
286
905925
155
867094
441
132900
38
23
773190
286
90.5832
155
867358
441
132642
3V
24
773361
285
305739
155
867623
441
132377
3b
25
773533
285
905645
155
867887
441
132113
3b
26
773704
285
905552
1.55
868152
440
131848
34
27
773875
285
905459
1,55
868416
440
131584
33
2S
774046
285
905366
156
868680
440
131320
32
29
774217
285
905272
156
868945
440
131055
31
30
31
774388
284
905179
156
1.56
869209
440
130791
30
29
9.774558
284
9.905085
9.869473
440
10.130.527
32
774729
284
904992
1.56
869737
440
130263
28
33
774899
284
904898
156
870001
440
129999
2/
34
775070
284
904804
1.56
870265
440
129735
2b
35
775240
284
904711
1.56
870529
440
129471
2b
36
775410
283
90*617
156
870793
440
129207
24
37
3S
775580
775750
283
283
904523
904429
1.56
157
871057
871321
440
440
128943
128679
23
22
39
775920
283
904335
157
C71.585
440
128415
21
40
41
776090
283
904241
157
1.57
871849
439
439
128151
20
19
9.776259
283
9.904147
9.872112
10.127888
42
776429
282
904053
1.57
872376
439
127624
18
43
776598
282
903959
1.57
872640
439
127360
17
44
776768
282
903864
1.57
872903
439
127097
lb
45
776937
282
903770
1.57
873167
439
126833
lb
46
777106
282
903676
1.57
873430
439
126570
14
47
777275
281
903581
1.57
873694
439
126306
13
48
777444
281
903487
1.57
873957
439
126043
12
49
777613
281
903392
158
87422C
439
12578C
11
50
51
777781
281
1 281
90329*=
1.58
1 158
874484
9.874747
439
125516
10
1 9
9.77795C
9.90320?
' 439
10.1252.5S
52
778 11 f
» 281
903106
t 1.5f!
8750 K
t 439
12499(
) 8
53
778281
r 280
9030 H
[ 15S
87527J
i 438
12472T
7
54
77845{
) 280
9029H
i 1.5S
! 87553(
) 438
12446'
1 6
55
77862'i
1 280
90282^
1 1.5(-
) 87580(
) 438
12420(
) h
i)(i
77879'.
i 280
902721
) 1.5^
I 438
12393'
1 4
57
778961
) 280
90263-^
1 1.55
I 87632(
5 438
12367^
1 3
58
779 1 2J
i 280
902535
) 15i
) 87658!
) 438
12341
1 2
5M
77929.
i 279
90244'
t 1.55
) 87685
1 438
12314<
:> 1
60
77946.
5 279
90234!
) 1.5(
) 87711-
1 438
1 12288
b
1 Cosine
1
t Sine 1
1 Cotiini;.
1
j Tauf!. 1 M.
i
•Jl)e«
rc'CS.
.\I
J__
9.
1
2
3
4
5
6
7
8
9
10
11
9.
12
13
14
15
16
17
18
19
20
21
9.
22
23
24
25
26
27
»
28
29
r
30
r
31
9.'
32
33
c
34
35
36
.•
(
37
"i
38
7
39
7
40
7
4i
9.7
42;
7
43
7
44
7
45
7
46
7
47
7
48
7
49
7
50
7
51
9.7
52
7
53
7
54
7
55
7
'jfi
7
57
7
is
7
i9
7
(
:;o
7!
1
(Jo.
^
3
39 60 1
73
59
08
58
42
57
.77
56
m
55
46
54
!81
53
515
52
150
51
)85
50
i-ZO
49
355
48
J90
47
)25
46
reo
45
195
44
J30
43
)65
42
700
41
im
40
171
39
300
38
B42
37
377
36
113
35
348
34
584
33
320
32
055
31
791
30
527
29
263
28
999
27
735
26
471
25
207
24
943
23
679
22
415
21
151
20
888
19
624
18
360
17
097
16
.833
15
>570
14
1306
13
5043
12
)780
11
J516
10
1253
9
199(1
8
1727
' 7
146']
[ 6
420C
) 5
^93'
r A
367^
1 3
341
2
3141
) 1
288(
i
11^. 1 M.
SINES AND TANGENTS. \^S7 Dcgraes.)
65
M.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
2G
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
5S
59
60
Sine
9.779463
779631
779798
779966
780133
780300
780467
780634
780S01
780968
781134
.781301
781468
781634
781800
781966
782132
782298
782464
782630
_782796
.782961
783127
783292
783458
783623
783788
783953
784118
784282
784447
.784612
784776
784941
785105
785269
785433
785597
785761
785925
786089
. 786252
78G416
786579
786742
786906
787069
787232
787395
787557
7S7720
9.787883
788045
783208
78S370
788532
7.'^^694
788856
789018,
789180
7893421
T>.
Cosine
D.
'r-int;.
n.
279
279
279
279
279
278
278
27S
278
278
278
277
277
277
i;r7
277
277
276
276
276
276
27'4
276
275
275
275
275
275
275
274
274
274
274
274
274
273
273
273
273
273
273
272
272
273
272
272
272
271
271
271
_271_
271
271
271
270
270
270
270
270
270
269
9.902349
902253
902158
902063
901967
901872
901776
901681
901585
901490
901394
Cotiiiin.
.901298
901202
901106
901010
900914
900818
900722
900fS26
900529
900433
.900337
900240
900144
900047
899951
899854
899757
899660
899564
899467
9.899370
899273
899176
899078
898981
898884
898787
898689
898592
898494
9.898397
898299
898202
898104
898006
897908
897810
897712
897614
897516
O.S97"418
897:i20
897222
897123
897025
896926
896828
896729
896631
896532
.877114
877377
877640
877903
878165
878428
878691
878953
879216
879478
879741
1.880003
880265
880528
880790
881052
881314
881576
881839
882101
882363
.882625
882887
883148
883410
883672
f 83934
^'84196
8.H457
8S4719
88'i.<i80
438
438
438
438
438
438
438
437
437
437
437
10,
437
437
437
437
437
437
437
437
437
436
VZxHSO
122623
122360
122097
1218.35
121572
121309
121047
120784
120522
120259
1(53
.885242
885503
885765
886026
886288
886549
886810
887072
887333
887594
.887855
888116
888377
888639
888900
889160
889421
889682
889943
890204
436
436
436
436
436
436
436
436
436
436
436
436
436
436
435
435
435
435
435
10.119997
119735
119472
119210
118948
118686
118424
118161
117899
117637
10.117375
117113
116852
116590
116328
116066
11,5804
115543
11.5281
11.5020
10,
z
Sine
9.890465
890725
890986
891247
891507
8;) 1 768
892028
892289
892549
892SI0
Colati''.
435
435
435
435
435
435
435
435
435
434
114758
114497
114235
I i. 39 74
113712
113451
113190
112928
112667
112406
10.
434
434
434
434
434
4;>4
434
434
434
434
112145
111884
111623
111361
II 1100
110840
110579
110318
1 10057
109796
10.109535
109275
109014
108753
108493
108232
107972
107711
107151
107190
Toil''.
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
30
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
I
52 Degieus
5«
(3fi Degrees.) a table or logarithmic
^^B
M.l
Bine 1
0. 1
(.'osine 1 D. Tang. |
D.
Cotanp.
M 1
iB
9.789342
269
9.896532
164
9.892810
434
10. 1071901
80
1
^^^■t
1
789504
269
896433
165
893070
434
106930
69
9
V,
789665
269
896335
165
893331
434
106669
68
1
I^K'
^
789827
269
896236
105
893591
434
106409
67
2
3
4
6
6
7
8
9
10
1 |«'i
4
789988
269
896137
165
893851
434
106149
56
5
790149
269
896038
165
894111
434
105889
55
6
790310
268
895939
165
894371
434
105629
54
7
790471
268
895840
165
894632
433
10.5368
53
8
790632
268
895741
165
894892
433
105108
62
■^Hi
q
790793
268
895641
165
895152
433
104848
61
H
•
10
790954
268
895542
9.895443
165
166
895412
433
104588
50
49
48
^B
IT
9.791115
268
9.895672
433
10 104328
11 9.)
12 (
13 f
14 ^
^H
12
791275
267
895343
166
895932
433
104068
'■
791436
267
895244
166
896192
433
103808
47
^^H
14
791596
267
895145
166
896452
433
103.548
4b
^H
15
791757
267
895045
166
896712
433
103288
4b
J
I
IB
791917
267
894945
166
896971
433
103029
44
lu c
16 6
17 g
18 8
17
792077
267
894846
166
897231
433
102769
43
18
792237
266
894746
166
897491
433
102509
42
19
792397
266
894646
166
897751
4.33
102249
41
20
?,1
792557
266
894546
9.894446
166
167
898010
433
101990
10,101730
40
39
I
2
n
a
8
9.792716
266
9.898270
433
1 r\ t>
9,^
792876
266
894346
167
898530
433
101470
38
'il J>.C5
22 8
23 8
?,3
79Q035
266
894246
167
898789
433
101211
3/
24
793195
265
894146
167
899049
432
100951
36
?,5
793354
265
894046
167
899308
432
100692
36
O
c 1 <-i
86
793514
265
893946
167
899568
4a2
100432
34
26 8
27 8
28 8
on o
27
793673
265
893846
167
899827
432
100173
33
28
793832
265
893745
167
000086
432
099914
32
ifl^^i
29
793991
265
893645
167
900346
432
099654
31
30
81
794150
264
893544
167
168
900605
432
099395
30
29
3
3
3;
3;
3^
3f
3(
3^
3f
3£
4C
41
42
43
44
4*1
D 8
jSH
9.794308
264
9.893444
9.900864
432
10.099136
1 9.8(
32
794467
264
893343
168
901124
432
098876
28
33
794626
264
893243
168
901383
432
098617
2/
^ ai
34
794784
264
893142
168
901642
432
098358
2b
3.'i
794942
264
893041
168
901901
432
098099
2o
1 o/
36
795101
264
892940
168
902160
432
097840
24
37
795259
263
892839
168
902419
432
097581
23
1 or
38
795417
263
892739
168
902679
432
097.321
22
1 fit
39
79557C
263
892638
168
902938
432
097062
21
1 Of\
40
41
795733
263
892536
168
169
903197
9.903455
431
431
096803
10.096545
20
19
' oil
80
9.795891
263
9.892435
9.80
an
42
796049
263
892334
169
903714
431
096286
18
43
796206
263
892233
169
903973
431
096027
17
fi/t
44
790364
262
892132
169
904232
431
095768
16
fin
^^Hl
45
796521
262
892030
169
904491
431
095509
16
fin
^^Ht
^^
796679
262
891929
169
904750
431
095250
14
46
fin
^^^^H i
47
796836
262
891827
169
905008
431
094992
13
47
fin
48
796993
262
891726
169
905267
431
094733
12
48
49
50
51
fin
\ i.
49
797150
261
891624
169
905526
431
094474
11
fin
■ .1^
50
51
797307
9.797464
261
891523
9.891421
170
170
905784
9.906043
431
094216
10
9
80
£ .^?
261
431
10.093957
9.80
fin
-, *
52
797621
261
891319
170
906302
431
09369S
8
52
53
797777
261
891217
170
906560
431
093440
V
53
80
An
aH
54
79793'!:
261
891115
170
906819
431
093181
6
54
JHj
55
798091
261
891013
170
907077
431
092923
5
55
80
60
80'
Hn'
SB
261
890911
170
90733P
431
092664
4
;5fi
j^i
57
79840:i
260
890803
170
907594
431
092406
3
57
^H '
58
798560
260
89070?
170
907852
431
092148
2
58
^H
59
798716
260
89060f
170
908111
430
091889
1
69
act'
|H
60
798872
260
89050:3
170
908369
430
091631
60
80^
|l.
1 Cosine
1
1 Sine 1 1 Cotang.
1
j 'I'ang. M.
.
Cosi
51 Dcgrsea.
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
74
11
16
10
57
9
9S
8
40
7
81
6
23
5
64
4.
t06
3
48
8
89
1
31
|M.
n
SINES AND TANGENTS. (39 Degrees,)
57
Sine.
1
2
3
4
6
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
9
.798872
7IJ9028
799184
799339
799495
799651
799806
799962
800117
8002721
800427
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
9.800582
800737
800892
801047
801201
801366
801511
801665
801819
801973
3.802128
802282
802436
802589
802743
802897
803050
803204
803357
803511
9.803664
803817
803970
804123
804276
804428
804581
804734
804886
80 5039
9.805191
805343
805495
805647
805799
805951
806103
806254
806406
806557
9.806709
806860
807011
807163
807314
607465
807615
807766
807917
808067
260
260
260
259
259
259
259
259
259
258
258
9.8905031 170
890400 171
258
258
258
258
258
257
257
257
257
267^
267
266
256
256
256
256
256
266
256
255
690298
890195
890093
889990
889888
889785
889682
889579
889477
9.889374
889271
889168
889064
888961
888858
888755
888651
888648
888444
171
171
171
171
171
171
171
171
171
255
265
255
256
254
264
254
254
254
254
9.888341
888237
888134
888030
887926
887822
887718
887614
887510
887406
172
172
172
172
172
172
172
172
173
9.9083691
908628
908886
909144
909402
909660
909918
910177
910435
910693
910961
172 9.911209
9.887302
887198
887093
886989
886885
886780
886676
886571
886466
886362
173
173
173
173
173
173
173
173
174
911467
911724
911982
912240
912498
912766
913014
913271
913529
173 9.913787
254
253
253
253
253
253
253
253
262
25a
9.886257
886152
886047
886942
885837
885732
885627
885522
885416
885311
174
174
174
174
174
174
174
174
174
175
914044
914302
914560
914817
916076
91.5332
915590
915847
916104
Cosine
9.885205
885100
884994
884889
884783
884677
884572
884466
884360
884254
sine i
176
175
175
175
175
175
176
176
176
176
176
176
1761
176
17R
176
176
176
176
177
9.916362
916619
916877
917134
917.391
917648
917905
918163
918420
918677
9
,918934
919191
919448
919706
919962
920219
920476
920733
920990
921247
1.921503
921760
922017
922274
922530
922787
923044
923300
923557
923813
Cotang.
430
430
430
430
430
430
430
430
430
430
430
10.01,16311 60,
0913721 59
091114
090856
090598
090340
090082
089823
089665
089307
089049
430
430
430
430
430
430
430
429
429
429
429
429
429
429
429
429
429
429
429
429
429
429
429
429
429
429
429
428
428
428
428
428
428
428
428
428
428
428
428
428
428
428
428
428
A no
428
428
428
427
427
10.088791
088533
088276
088018
087760
087502
087244
086986
086729
086471
10.086213
085956
085698
086440
085183
084925
084668
084410
084153
083896
10.083638
083381
083123
082866
082609
082352
082096
081837
081680
081.323
10.081066
080809
080552
080295
080038
079781
079524
079267
079010
078753
10.078497
078240
077983
077726
077470
077213
076956
076700
076443
076187
58
57
56
56
54
53
512
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
Tang.
IM.
59 D«gree«i
68
(40 Degrees.') a table op logarithmic
M.|
B'.ne
I).
Costne 1 I) 1
Tung. 1
I>. 1
Coliintt. 1 j
9.808067
251
9.884254] 177|
9.923813
427
10.076187
()0
1
808218
251
884148
177
924070
427
073930
39
3
808368
251
884042
177
924327
427
073673
38
3
808319
2.30
883936
177
924383
427
073417
37
4
808669
230
883829
177
924840
427
073160
36
5
808819
230
883723
177
92.'J096
427
074901
33
6
808969
250
883617
177
923352
427
074648
34
7
809119
250
883510
177
923609
4'57
074391
53
8
809269
230
883404
177
923865
427
074133
32
9
809419
249
883297
178
926122
427
073878
31
10
11
809569
249
883191
9.883084
178
178
926378
9.926634
427
073622
10.073366
30
'To
9.809718
249
427
12
809868
249
882977
178
926890
427
073110
48
13
810017
249
882871
178
927147
427
072833
47
14
810167
249
882764
178
927403
427
072397
46
15
810316
248
882657
178
927659
427
07234 1
43
16
810463
248
882330
178
927915
427
072083
44
17
810614
fJ48
882443
178
928171
427
071829
43
18
81076'j
248
882336
179
928427
427
071373
42
19
810912
248
882229
179
928683
427
071317
41
20
21
811061
248
882121
9.882014
179
179
928940
9.929196
427
071060
10.070801
40
39
9.811210
248
427
22
811338
247
881907
179
929432
427
070348
38
23
811507
247
881799
179
929708
427
070292
37
24
811633
247
881692
179
929964
426
070036
.36
25
811804
247
881384
179
930220
426
069780
33
26
811932
247
881477
179
930473
426
069323
34
27
812100
247
881369
179
930731
426
069269
33
28
812248
247
881261
180
930987
426
069013
32
29
812396
246
881153
180
931243
426
068V37
31
30
31
812544
246
881046
180
180
931499
9.931735
426
426
068301
10.068215
30
29
9.812692
246
9.880938
32
812840
240
880830
180
93'^,010
426
06/990
28
33
812988
246
8807'^r2
180
932266
426
067734
27
34
813135
246
880613
180
932522
428
067478
26
35
813283
246
880305
180
932778
426
067222
23
36
813430
245
880397
180
933033
426
066967
24
37
813578
245
880289
181
933289
420
066711
23
38
813725
245
880180
181
933545
426
066433
22
39
813872
245
880072
181
933800
426
066200
21
40
41
814019
9.814166
245
879963
9.879853
181
181
934056
9.9.M311
426
426
063944
20
19
1 245
10.0636H'J
42
814313
245
879746
181
934367
426
063433
18
43
814460
i 244
879637
181
934823
426
063177
17
44
814607
1 244
879329
181
93.3078
426
064922
16
45
814753
244
879420
181
933333
426
064667
15
46
814900
' 244
879311
181
933389
426
064411
14
47
815046
: 244
879202
182
935844
426
004136
13
48
815193
; 244
879093
182
936100
426
063900
12
49
815339
, 244
878984
182
936353
426
063643
11
30
51
81 5485
9.815631
243
878873
182
182
936610
9.936866
426
063390
10.0631.34
10
9
i 243
9.878766
423
52
813778
243
878656
; 182
937121
425
062879
8
53
813924
243
878547
182
937376
425
062624
7
54
816069
243
878438
182
937632
425
06236H
(]
53
81621fi
243
878328
; 182
937887
425
062113
5
50
HlGoOi
1 243
878219
; 183
938142
423
061H5H
4
37
81630?
i 242
878109
: 183
938398
423
061602
3
58
816632
, 242
877999 183
938633
423
061347
2
59
81679S
1 242
877890: 183
93H908
42,5
06I0!»2
1
60
81694?
t! 242
877780 183
939163
1 425
0608:57
j Cohirie
I
1 Sine 1
1 Coiaiig.
1
1 Tani,'. 1 M.
49 Uegreva
M.
1
2
8
4
6
6
7
8
U
U)
II
12
13
14
15
16
17
!8
19
2()
21
22
23
24
2!)
26
27
28
29
30
31
32
33
34
33
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
f)\
32
33
34
53
56
82
57
82
58
82
59
82
00
82
1 1
i7
HO
<(l
f)5)
r.)
M
17
ru
)()
M
>.i
f):)
tH
M
H
M
Jf)
r)2
78
51
f)0
')(')
'10
10
4H
')H
47
)7
'!<)
11
'If)
htf)
41
!i«»
4:j
73
4 a
17
41
60
40
01
•A'J
AH
:w
!»-2
:r<'
'ir.
34
CO
33
i:{
32
r)7
31
01
30
If)
20
00
28
;m
27
78
20
TZ
25
(57
24
11
23
r)5
22
00
21
44
20
8'J
10
:j:)
18
77
17
aa
1(5
i\7
15
11
14
5(5
13
00
12
45
11
00
10
34
U
70
8
24
7
f)H
(5
13
5
5H
4
02
3
47
2
!»2
1
37
|MT
SlWrS AND TANOKNTf. ^41 DogrOCS.)
BO
HIlKt
0.81(5043
81708N
HI 7233
HI 7370
HI 7524
HI 7(5(58
817813
817058
818103
818247
^818302
0.818530
818(581
818H25
HI 80(51)
H10II3
8 1 0257
8 1 0101
HI 054 5
810(580
HI 0832
„i__'L„J <'"Hiim i l». I TaiiH. ( p |
0.81007(5
820120
H202(53
820400
820550
820(503
H2083(5
820070
821122
_82I2(55
31 9.827407
8215.50
821(503
821835
821077
822120
8222(52
822404
822.54(5
822088
.822830'
822072
823114
8232.55
823307
823530
823(580
82.3821
8230(53
824104
0.8 24 24 .5 1
8243801
8245271
824(5(58
824808'
8240401
825000
8252301
82.5371 j
8255111
a4a
242
242
242
241
241
241
241
241
241
241
240
240
2-10
210
240
240
240
230
230
_ ?''^'^
'2.30
230
230
230
238
238
238
238
238
__238
2.38^
2.38
237
2.37
237
237
237
237
237
23(5
230
23(5
23(5
230
2.3(5
23r»
235
235
235
235
23.5'
235
235
234
0.87? 780
877(570
8775(50
H77450
8773'|(t
H7 7230
877120
877010
87(58!M>
87«78<)
87(5(578
0.87(55(58
87(5457
870347
87023(5
87(5125
87(50 M
875004
87570.3
875082
875571
0.87.5'150
87534 H
875237
875120
87.50 M
874003
87470!
H74080
874508
__874450
0.874344
874232
874121
874000
873800
873784
873(172
873500
87.3448
87.3335
183
183
183
183
183
IH'l
184
184
184
184
IS4
]84
184
184
185
185
185
185
185
185
185
l85
185
185
180
180
180
180
180
180
180
18(5
187
187
187
187
187
187
187
187
187
'J
87.3223
187
873110
188
872008
188
872885
188
872772
188
872050
188
872547
188
872434
188
872321
188
872208
188
234
234
234
234
234
0.872005
87 1 08 1
87I8()H
871755
87!fM!
871528
871414
871.301
871187
871073
180
180
180
180
IHO
189
180
180
180
190
0.0301(53
030418
030(573
030028
040183
040438
040(504
W40040
04 1 201
0414.58
041714
0.041008
042223
042478
042733
042088
04324.3
043408
043752
044007
0442(52
li. 0145 1 7
044771
iM5020
045281
045535
04!, "0
04(5045
04(5200
040554
040808
». 04 7003
047318
047572
047820
048081
04833(1
948500
048844
040090
040353
.04'0«07
0108(12
9.50 no
9.50370
950025
950870
9511.33
95 1 388
951042
_951890
9.0.52150
952405
952059
ii529l3J
953 If;?;
9,5342)1
953075!
9.530291
9.54183
9.544371
425
425
425
425
425
425
425
425
425
425
425
425
425
425
425
425
425
425
425
425
425
425
424
424
424
424
424
424
424
424
424
424
424
424
424
424
424
424
424
424
_424
424
424
424
424
424
424
424
424
424
424
424
424
424
424
423
423
423
423
423
423
• '(ll(IM((.
'0.000837
0(50582
000327
0(1007
0508 1 7
05il5(l2
050300
05905 1
05879(1
05H542
05828(1
10.0.'i8(»32
057777
057522
0572(1/
«)570I2
05(5757
05(5502
050248
055993
055738
i 0.055483
055229
051074
051710
0.544(15
054210
053955
0.53701
05341(1
053 1 92
10.052937
052(582
052428
052174
051910
0510041 24
(10
59
58
57
50
55
54
53
52
51
50
40
48
47
40
45
44
43
42
41
40
39
.38
37
3(5
35
34
33
32
31
.30
29
28
27
20
25
051410
051150
0.5(»90 1
050047
T0.05?)i«9.3
05(»i.38
019884
049030
049375
049121
0488(57
048012
048358
_ 048104
l(). 04 7850
047595
047.341
047(;87
040833
040579
010325
04007)
0458 1 7
0455031
t'tiiiji. I
23
22
21
20
)9
)8
17
10
15
14
13
12
II
10
9
8
7
6
4
3
2
1
hi.
4« U<-greeN.
eo
(42 Degrees.) a table of LOGARirHMio
M.
1 8iMe
I).
1 (,'o»in(i 1 I).
TniiL'.
1 D.
1 Colnnc. 1 1
9.825511
234
9.871073
190
9.9.54437
423
10.045563
60
1
825051
233
870960
190
9.54691
423
045309
59
2
82.5791
233
870846
190
9.54945
423
045055
58
3
82.5931
233
870732
J 90
955200
423
044800
57
4
826071
233
870618
190
95.54.54
423
044546
56
5
826211
233
870504
190
955707
423
044293
55
S
826351
233
870390
190
955961
423
044039
54
7
F26491
233
870276
190
9.56215
423
043785
53
8
826631
233
870161
190
956469
423
043.531
5?,
9
826770
232
870047
191
956723
423
043277
51
10
11
826910
232
869933
191
191
956977
423
043023
60
49
9.827049
232
9.809818
9.957231
423
10.042769
12
827189
232
869704
191
957485
423
042515
48
13
82732ft
23i
869.589
191
957739
423
042261
47
U
827467
232
869474
191
957993
423
042007
46
15
827606
232
869360
191
958246
423
0417.54
45
16
827745
232
869245
191
958.500
423
041.500
44
17
827884
231
8691.30
191
958754
423
041246
43
18
828023
231
869015
192
959008
423
040992
42
19
8'>8162
231
868900
192
959262
423
040738
41
20
21
828301
9.828439
231
868785
192
192
9.59516
423
040484
40
39
231
9.868670
9.9.59769
423
10.040231
22
928578
231
868555
192
960023
423
039977
38
23
828716
231
868440
192
960277
423
039723
37
24
8288.55
230
868324
192
960531
423
039469
36
2f)
828993
230
868209
192
960784
423
039216
35
26
829131
230
868093
192
961038
428
038962
34
27
829269
230
867978
193
961291
423
038709
33
28
829407
230
867862
193
961545
423
038455
32
29
829.545
230
867747
193
961799
423
038201
31
30
31
829683
230
867631
193
193
962052
423
037948
10.037694
30
29
9.829821
229
9.867515
9.962306
423
32
829959
229
867399
193
963560
423
037440
28
33
830097
229
867283
193
962813
423
0.37187
27
34
830234
229
867167
193
963067
423
036933
26
35
830372
229
867051
193
963320
423
036680
25
36
830.509
229
866935
194
963574
423
036426
24
37
830646
229
866819
194
963827
423
036173
23
38
830784
229
866703
194
964081
423
035919
22
39
830921
228
866586
194
9643.35
423
035665
21
40
41
831058
228
866470
194
194
964.588
422
035412
10.0351.58
20
19
9.831195
228
9.866353
9.964842
422
42
831332
228
866237
194
965095
422
0.34905
18
43
831469
228
866120
194
965349
422
034651
17
44
831606
228
866004
195
965602
422
U34398
16
45
831742
228
865887
195
965855
422
034145
15
46
831879
228
86o770
195
966109
422
0.33891
14
47
832015
227
865653
195
966362
422
033638
13
48
832152
227
865536
195
966616
422
033384
12
49
832288
227
865419
195
966869
422
033131
11
50
51
832425
227
865302
195
195
967123
422
032877
10
9
9.832561
227
9.865185
9.967376
422
JO. 032624
52
832697
227
86.5068
195
967629
422
032371
8
53
832833
227
864950
195
967883
422
032117
7
54
832969
226
864833
196
968136
422
031864
6
55
833105
226
^64716
196
968.389
422
031611
5
833241
226
864598
196
968643
422
4
57
8.33377
226
864481' "6j
068896
422
031104
3
68
833512
226
86436; ,
969149
422
030861
2
59
833648
226
864245
969403
422
030597
1
60
833783
226
864127' X
969656
422
030344
_J
Cosine 1
Sine 1 1
Cotang. 1
Taiig. 1 M. 1
M.
47 Dsgrees.
2
3
4
6
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
40
47
48
84
84
49
84
50
84
61
9.84
62
84
53
84
54
84
55
84
56
84
57
84
58
84
59
84
60
84
1 Cosi
■ 1 1
'i63
60
109
59
055
68
'^00
57
51(5
56
J!)3
55
):}9
54
785
53
331
52
277
51
)23
60
rog
49
515
48
261
47
J()7
46
('54
45
JOO
44
M6
43
)93
42
r.is
41
184
40
J3l
39
)77
38
r23
37
K)9
36
>16
35
J62
34
mo
33
r55
32
401
31
M8
30
594
29
140
28
87
27
)33
26
580
25
126
24
73
23
)19
22
565
21
U2
20
58
19
)05
18
551
17
J98
16
45
15
<91
14
538
13
584
12
31
11
577
10
524
9
571
8
117
7
mi
6
511
5
«<-~
j;ji
'1
04
3
551
2
597
1
544
1
M.
"NESANDTANOENTS. (43 DcjrreCS.)
61
1
2
3
4
6
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
9.833783
833919
834054
834189
834325
834460
834595
834730
834865
834999
835134
9.835269
835403
835538
835672
835807
83594 1
836075
836209
836343
836477
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
40
47
48
49
50
51
52
53
54
55
56
57
58
59
60
9.836611
836745
836878
837012
837146
837279
837412
837546
837679
837812
9.837945
838078
838211
838344
838477
838610
838742
838875
839007
839140
9.839272
839404
839536
839668
839800
839932
840064
840196
840328
840459
9.840591
840722
84085*
840985
841116
841247
841378
841509
841640
841771
226
225
226
225
225
225
225
225
225
224
224
224
224
224
224
224
224
223
223
223
223
9.864127
864010
863892
863774
863650
863538
863419
863301
863183
863064
862946
19619.969656
223
223
223
222
223
222
223
223
223
222
9.862827
862709
862590
862471
862353
862234
862115
861996
861877
861758
9.861638
861619
^61400
861280
861161
861041
860922
860802
860682
860562
196
197
197
197
197
197
197
197
197
198
198
198
198
198
198
198
198
198
198
199
969909
970162
970416
970669
970922
971175
971429
971682
971935
972188
222
221
221
221
231
221
221
221
221
220
220
220
220
220
220
220
219
219
219
219
9.860442
860322
860202
860082
859962
859842
859721
859601
859480
859360
219
219
219
219
318
218
318
218
218
218
9.859239
859119
858998
858877
858756
858635
858514
858393
858272
858151
.858029
857908
857786
857665
857543
857422
857300
857178
857056
856934
199
199
199
199
199
199
199
199
200
200
200
200
200
200
200
200
201
201
201
201
201
201
201
201
202
202
202
202
202
202
9.972441
972694
972948
973201
973454
973707
973960
974213
974466
974719
9.974973
975226
975479
975732
975983
976238
976491
976744
976997
977250
9.977503
977756
978009
978262
978515
978768
979021
979274
979527
979780
202
202
202
203
203
203
203
203
203
203
9.980033
980286
980538
980791
981044
981297
981550
981803
982056
982309
9.982662
982814
983067
983320
983573
984079
984331
984584
984837
423
422
423
433
433
433
422
422
423
423
422
423
422
422
422
422
423
423
433
423
422
422
422
422
422
422
422
422
423
432
422
423
422
422
422
422
422
422
422
423
423
433
433
423
431
431
431
431
431
431
431
421
421
421
421
421
421
421
421
421
10.030344
U30091
029838
029584
03933 1
029078
02HH25
028571
028318
028065
027812
10.027559
027.306
027052
026799
026546
026293
026040
025787
025534
025281
10.025027
024774
024621
024268
024015
023762
023609
023256
023003
022750
10.022497
022244
021991
021738
021485
021232
020979
020726
020473
020220
10.019967
019714
019462
019209
018966
018703
018450
018197
017944
017691
10.017438
017186
016933
016680
016427
016174
015S21
016669
0164161 I
0161631
60
69
58
67
56
56
54
63
63
61
60
49
48
47
46
45
44
43
42
41
40
39
38
37
36
36
34
33
33
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
IG
15
14
13
12
11
10
46 Degrees.
i
6S
hi,
i« Si.
(44 Degrees.) a table of LooARtTiirnc
M.|
Hiiifi 1
"• 1
Ccmine i>. j
T;mi?. i n. 1
('(llMIIS 1
9.841771
218
9.8.56934
203
9.98 '.837
421
10.015163
6U
I
841902
218
856812
•..M)3
985090
421
014910
59
2
842033
218
850690
204
985343
421
0146.57
58
3
842163
217
856568
204
985696
421
014404
57
4
842294
217
856446
204
985848
421
0141.52
56
»
842424
217
856323
204
986101
421
013899
55
6
842555
217
8.56201
204
986354
421
013646
54
7
842685
217
856078
204
986607
421
013393
53
8
842S15
217
855956
204
986860
421
013140
52
9
842!)4<)
217
855H33
204
987112
421
012888
51
10
ll
843076
9.vS43206
217
216
855711
205
205
987305
421
012635
50
49
9.855.''j88
9.987618
421
10.012382
12
843336
216
85.5465
205
987871
421
012129
48
13
843466
216
855342
205
988123
421
011877
47
14
843595
216
855219
205
988376
421
011624
46
1ft
843725
216
855096
205
988629
421
011371
45
in
843855
216
854973; 205 1
988882
421
011118
44
17
843984
216
854850
205
989134
421
010866
43
18
844114
215
8.54727
206
989387
4.1
010613
42
19
844243
216
854603
206
989640
421
010360
41
20
21
844372
9.844502
215
854480
9.854356
206
206
989893
421
010107
40
39
215
9.990145
421
10.009855
22
844631
215
154233
206
990398
421
009602
38
23
S44760
215
854109
206
990651
421
009349
37
21
844889
215
8i;3980
206
990903
421
009097
36
2r)
845018
215
853862
206
991156
421
008844
35
26
845147
215
853738
206
991409
421
008591
34
27
845276
214
85.3614
207
991662
421
008338
33
28
845405
214
853490
207
991914
421
008086
32
20
845533
214
853366
207
992167
421
007833
31
30
31
845662
214
853242
207
207
992420
421
007580
30
29
9.845790
214
9.853118
9.992672
421
10 0G7328
32
845919
214
8.52994
207
992925
421
007075
28
33
846047
214
8.52869
207
993178
421
006822
27
34
846175
214
852745
207
993430
421
006570
26
35
8*6304
214
852620
207
993683
421
006317
25
3fi
846432
213
852496
208
993936
421
006064
24
37
846560
213
8.52371
208
994189
421
005811
23
38
846688
213
852247
208
994441
421
005559
22
39
84C316
213
852122
208
994694
421
005306
21
40
41
846944
213
851997
9.851872
208
208
9S»4947
421
005053
20
19
9.847071
213
9. 95199
421
10.004801
42
84T"09
1 213
6 ^747
208
995452
421
004548
18
43
8't,,: '
?13
851622 208
995705
421
004295
17
44
847iti'.
■ 2.2
851' f»7 209
995957
421
004043
16
4f>
P,47;;82
1 2i2
-^513/2
209
996210
421
003790
15
4R
84770;*
212
851246
209
996463
421
003537
14
47
847836
212
851121
209
996715
421
003285
13
48
847964
212
8.50996
209
996968
421
003032
12
49
818091
212
850870
209
997221
n
002779
11
50
51
848218
212
212
850745
209
209
997473
1
421
002527
10
9
9.848345
9.850619
9.997726
10.002274
52
848472
211
850493
210
997979
421
002021
8
53
848599
211
650368
210
998231
421
00176S
7
54
848726
211
860242
210
998484
421
0015U
1 6
55
848852
211
8.50116
210
998737
421
00126S
1 5
56
84S„79
211
849990
210
99^:89
421
001011
4
57
849106
211
849864
2i0
999242
421
000758
i 3
58
849232
211
849738
210
999495
421
00050?
) '^
59
849359
211
849611
210
999748
421
0002.5C
) 1
leo
849485
211
849485
210
10.000000
421
00000(
)
Iz
Cosine
1
1 Sine 1
1 Cotaiig. 1
1 Tang. M.
4
i5 D^
;ree8.
"*"
>
M
8
00
1
00
2
00
3 00
f
00
6
7
S
9
10
11
12
13
14
16
17
18
19
20
21
22
23
24 ,
I
29
30
38
39
40
46
47
48
49
50
51
54
55
56
57
58 I
59
M
" 1 i
)ifi;j
«U
910
59
M7
58
1404
57
H5'2
56
J899
55
Jt)4fi
54
33!):)
53
}14()
52
WHH
51
HV.i^
50
i-.iS'Z
49
il29
48
1877
47
1624
46
1371
45
1118
44
0866
43
0613
42
0300
41
0107
40
9855
39
9602
38
9349
37
9097
36
8844
35
8591
34
8338
33
8086
32
7833
31
7580
30
29
7328
7075
28
6822
27
6570
26
6^17
25
6064
24
15811
23
15559
22
)5306
21
)5053
20
19
)4801
)4548
18
)4295
17
)4043
16
)3790
15
)3537
14
)3285
13
J3032
12
)2779
11
32527
10
9
32274
32021
8
31769
r
315H]
6
01263
5
01011
4
00756
3
0050f
» 2
00255
\ 1
OOOOC
)
ng. ■ ■ W
A TABLE OF JVATUUAL SIIVES.
M
1
2
3
4
5
6
7
S
9
10
11
12
13
14
Dt•^r
"NatT j pT (;o-
00000
00029
ooor>8
00087
00116
00145
001751
0020 1
00233
0026-^
0029 1
0032')
00349
00378
00407
_[5 00436
16
17
18
19
20
21
22
23
24
Unit.
00000
00000
00000
'00000
00000
00000
iooooo
I'OOOOO
Iooooo
00000
99999
99999
99999
99999
39999
1 Deg.
00465
00495
00524
00553
00582
006 1 1
00640
00669
00698
25 00727
26 00756
27 00785
28 00814
29 00844
00873
Nut.
Sine
0l7'15
01774
101803
()1832
!01H62
01891
019201
1019491
101978
02007
02036
02065
02094
02123
02152
02181
N. (;o.
nine
99985
99984
99984
99983
99!)83
999'>t2
[99982
99981
99980
99980i
99979
99979
99978
99977
99977
999/6
,> Dog.
30
31
32
33
34
00902
00931
00960
00989
35'0I018
36|01047
37 01076;
38
39
40
41
01105
01134
01164
01193
42101222
43' 01 251
44 01280
45 01309
46
47
48
49
50
51
53
54
55
56
57
58
59
01338
01367
01396
01425|99990
01454 99989
99996
99996
99995
99995
99995
99995
,99994
99994
99994
99993
99993
99993
99992
99992
9999!
02211
02240
02269
0:.^298
02327
02356
02385
02414
02443
02472
02501
02530
02560
02589
02618
99991
99991
99990
01483
01513
01542
01571
01600
01629
01 658
99989
99989
99988
99988
99987
99987
99986
02647
02676
02705
02734
i»2763
02792
02821
02850
02879
<(2908
02938
02967
02996
03025
03'»54
99976
99975
99974|
999741
99973
99972
99972
99971
99970|
99969!
99969
99968|
99S67
999661
9 9966 ,
99965 i
999641
99963!
99963
99962
9'J961
99960
99959
99959
99958
99957
99956
999551
999541
99953;
iVut.
Hiiio
o:ri90
035 1 9
03548
03577
03606
03635
(3664
03693
03723
0375t;
0378 1
03810
03839
03868
03897
03926
03955
03984
04013
04042
04071
04100
04129
04159
04188
04217
04246
04275
04304
04333
04362
9993U
99938
39937
99936
99935
99934
99933
99932
99931
99930
99929
39927
99926
99925
99924
99923
04391
04420
04449
04478
04507
04536
04565
04594
04623
04653
04682
[04711
04740
04769
04798
99922
9992 1
99919
99918
99917
99916
99915
99913
99912
99911
99910
99909
99907
99906
99905
3jl)uir.
"Sat. iNrTJo^
J^ine I Hiiie
05234
052(53
0.5292
05321
05350
05379
05408
05437
k)546r
0549
05524
05553
05582
05611
05640
05669
05698
05727
05756
05785
05814
05844
05873
05902
05931
05960
05989
06018
06047
06076
06105
99863
9986 1
99860
9985S)
99857]
99855
99854!
9985v;'
99851
99849
99847
99846
99844
99842
99341
99839
4 Ueir.
01687' 99986
01716 99985
89 Deff.
03083
03112
03141
03170
03199
03228
03257
03286
03316
03345
03374
03403
03432
03461
99952
99952;
99951;!
99950 i
99949 i
99948!
99947
99946,
99945
9 9 004
99902
99901
99900
99898
99897
99896
99894
99893
99892
99890
99889
99888
99886
99885
04827
04856
04885
04914
04943
04972
05001
05030
99883
99882
99881
99879
99878
99876
99875
99873
06134
06163
06192
06221
06250
06279
06308
06337
06366
06395
0(5424
06453
06482
06511
06540
99SiS
9!i-..";
99
99833
99831
99829
99827
99826
99824
99822
9982J
99819
99817
99815
99813!
Or.OPiQ Q0fl70
N. Co-
Sine
99944;;05088'99870
05117 99869
99943
99942
99941
99940
Nat.
Sine
88 Deg.
05146
05175
05205
N. Ci)-
Sine
99867
99866
99864
Nat.
Sine
87 Deff.
06569
06598
06627
06656
06685
06714
06743
06773
99812
99810
99808
99806
99804
99803
99801
99799
99797
99795
99793
99792
99790
99788
99786
99784
99782
99780
99778
99776
99774
99772
199770
Nat.
Hine
06976
07005
07034
07063
07092
107121
07150
07179
07208
07237
072(.''
0729.j|
073i54
07353
0/382
(37411
07440
07469
1)7498
07527
07556
07585
07614
07643
07672
07701
07730
07759
07788
07817
07846
99756
3754
99752
!"»750
99748
99746
99744
99742
99740
99738
99736
99734
99731
99729
99727
99725
M
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
06803 193788
0683' '99766
06860
06889
06918
07875
07904
07933
07962
07991
08020
08049
08078
08107
08136
08165
08194
08223
08252
08281
99723
99721
99719
99716
99714
99712
99710
99708
99705
99703
99701
99699
99690
99694
99_692
99689
99687
99685
99683
99680
99678
99676
99673
99671
99668
99666
99664
99661
99659
99657
06947 99758
N. Co-
Sine
99764
99762
99760
Nat.
Sine
86 Peg. 85 Deg,
08310
08339
08368
08397
08426
08455
08484
08513
A Q »^ Ac\
08571
08600
08629
08658
08687
\. Co-
Sine
99654
99652
99649
99647
[99644
99642
99639
99637
99632
99630
99627
99625
996?2
Nat.
Sine
44
43
42
41
40
39
38
37
36
35
34
3?
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
1^
14
1?.
12
11
10
9
8
7
G
5
4
3
2
_l
'n
i
64
A TABLE or NATtTRAX SINES.
-
o Deg.
! 6 L>eg.
7 Dog.
i « l^eg. 1
9 D
t'g.
M
60
>I
N. S.
08716
N. CS.
A.S.
10453
N. CS.
iN. S.
N. CS.
99255
j N.S.
13917
1
N.CS.'
99027
N.S.
15643
N. CS.
99619
99452
12187
98769
1
08745
99617
10482
99449
12216
9925:.
13946
99023!
15672
98764
59
'Z
08774
99614
10511
99446
12245
99248
113975
990191
15701
98760
58
8
08803
99612
10540
99443
12274
99244
114004
990151
15730
98755
57
4
08831
99609
10569
99440
12302
99240
,14033
9901 Ij
15758
98751
56
b
08860
99607
10597
99437
12331
99237
114061
99006!
15787
98746
55
b
08889
99604
10626
99434
12360
99233
il4090
990021
15816
98741
54
V
08918
99602
10655
99431
12389
99230
J14119
98998
15845
98737
53
8
08947
99599
10684
99428
12418
99226
114148
98994
15873
98732
52
y
08976
99596
10713
99424
12447
99222
114177
98990
15902
98728
51
10
09005
99594
10742
99421
12476
99219
14205
98986i
15931
98723
50
ii
09034
99591
10771
99418
12504
99215
14234
98982 j
15959
98718
49
12
09063
99588
10800
99415
12533
99211
14263
989781
15988
98714
48
la
09092
99586
10829
99412
12562
99208
14292
989731
16017
98709
47
14
09121
99583
10858
99409
12591
99204
114320
98969J
16046
98704
46
lb
16
0Q150
99580
99578
10887
10916
99406
126:20
12649
99200
99197
14349
14378
98965
98961
10074
98700
98695
45
44
09179
99402
16103
IV
09208
99575
109*5
99399
12678
99193
114407
98957
16132
98690
43
18
09237
99572
10973
99396
12706
99189
14436
98953
16160
98686
42
19
09266
99570
11002
99393
12735
99186
14464
98948
16189
98681
41
2U
09295
99567
11031
99390
12764
99182
14493
98944
16218
98676
40
5il
09324
99564
11060
99386
12793
99178
14522
98940
16246
98671
39
"Z'Z
09353
99562
: 11089
99383
12822
99175
14551
98936
16275
98667
38
23
09382
99559
11118
99380
12851
99171
14580
98931
16304
98662
37
24
09411
99556
11147
99377
12880
99167
14608
98927
16333
98657
36
2b
09440
99553
11176
99374
12908
99163
14637
98923
16361
98652
35
26
09469
99551
11205
99370
12937
99160
14666
98919'
16390
98G48
o4
2V
09498
99548
11234
99367
12966
99156
14695
98914:
16419
98643
33
28
09527
99545
U263
99364
12995
99152
14723
98910
16447
98638
32
2'J
09656
99542'
j 1129 1
99360
13024
99148
14752
98906
16476
98633
31
31
09535
09614
99540
'11320
11349
9935?
99354
13053
13081
99144
14781
98902,
98897
16505
16533
98629
30
29
99537
99141
14810
98624
32
09642
99534
! 11378 99351
13110
99137
114838
98893
16562
98619
28
33
09671
99531
11407
99347
13139
99133
il4867
98889
16591
98614
27
34
09700
99528
11436
99344
13168
99129
14896
98884
16620
98609
26
3b
09729
99526
11465
99341
13197
99125
i 14925
98880
16648
98604
25
36
09758
99523
11494
99337
13226
99122
i 14954
98876
16677
98600
24
3V
09787
99520
11523
99334
13254
99118
114982
98871
16706
98595
23
38
09816
99517
11552
99331
13283
99114
15011
98867
16734
98590
22
39
09845 99514
11580
99327
13312
99110
15040
98863
16763
98585
21
40
09874
99511
11609
99324
13341
99106
15069
98858
16792
98580
20
41
09903
99508
] 1638
99320
13370
99102
15097
98854
16820
P«575
19
42
09932
99506
11667
99317
13399
•9098
i 15126
98849
16849
98570
18
43
09961
99503
11896
99314
13427
99094
; 151 55
98845
16878
98565
17
44
09990
99500
11725
99310
i 3450
99091
15184
98841
16906
98561
16
45
46
10019
99497
99494
11754
11783
99307
13485
13511
99087
15212
98836
98832
16935
16964
98556
15
14
10048
99303
99083
15211
98551
47
10077
99491
11812
99300
13543
99079
,15270
98827
16992
98546
13
48
10106
99488
11840
99t:'37
13572
99075
115292
98823
17021
98541
12
49
10135
99485
11869
99293
13600
99071
15327
98818
17050
98536
11
50
10164
99482
11898
99290
13629
99007
15356
98814
17078
98531
10
51
10192
99479
11927
99286
13658
99063
; 15385
98809
17107
98526
9
52
10221
99476
11956
99283
113687
99059
115414
98805
17136
98521
8
53
10250
99473
11985
99279
^13716
99055
115442
98800
17164
98516
7
54
10279
99470
12014
99276
113744
99051
15471
98796
17193
98511
6
55
10308
99467
12043
99278
113773
99047
45500
98791
1722298506
5
56
10337
99464
12071
99269:^13802
99043
15529
98787
17250
! 98501
4
57
10366
99461
12100
99205 j 13831
99039
15557
98782
17279
198496
3
58
10395
99458
12129
99262 13S60
99035
115586
98778
17308
198491
2
59
M
10424
99455
12 ■58
N. CS.
99258 13889
99031
N.S.
15615
N. CS.
98773
N.S.
17336
98486
1
M
N. CS. N. S.
N.S.
1 N.CS.
N. CS.
N.a
H4 Desr.
U-i Dog.
1 Ut 1
Jeg.
81 Deg.
80 Ueg.
41
185
42
185
43
185
44
186
45
186.
46
1'86(
47
187
48
187;
49
187(
50
1871
51
1885
62
188f
53
188^
54
1891
55
189r
56
189f
0/
iSHi
58
190S
59
1905
M
N.Ci
79
,
cs.
M
769
60
764
59
760
58
755
57
751
56
746
55
741
54
737
53
732
52
728
51
723
50
718
49
714
48
709
47
704
46
700
45
695
44
690
43
686
42
681
41
676
40
671
39
667
38
662
37
657
36
652
35
G48
34
643
33
638
32
633
31
629
30
624
29
619
28
614
27
609
26
604
25
600
24
!595
23
!590
22
!585
21
1580
20
i575
19
!570
18
!56o
17
!561
16
i556
15
S551
14
i546
13
!541
12
!536
11
5531
10
*526
9
i521
8
J516
7
^511
6
^506
5
^501
4
^496
3
^491
2
=1486
1
^. a
M
S-
A TABLE OP NATUHAL SINKS.
65
U
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
10 Ueg.
N. 8.
17365
17393
17422
17451
17479
17508
1 75371
17565
17594
17623
17651
17680
17708
17737
17766
17794
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
62
53
54
55
56
Hi
58
69
M
17823
17852
17880
17909
17937
17966
17995
18023
18052
18081
18109
18138
18166
18195
18224
N.CS.
98481
98476
98471
98466
98461
98455
98450
98445
98440
98435
98430
98425
98420
98414
98409
98404
11
N.8.
De^.
18252
18281
18309
18338
18367
18395
18424
18452
18481
18509
18538
18567
18595
18624
18652
98399
983S4
98389
98383
98378
98373
98368
98362
98357
98352
98347
98341
983361
98331
98325
19081
19109
19138
19167
19195
19224
19252
19281
19309
1^338
19366
19395
19423
19452
19481
19509
1'8681
18710
18738
18767
18795
18824
18852
18881
18910
18938
18967
19024
19052
98320
98315
98310
98304
98299
98294
98288
98283
98277
98272
98267
98261
98256
98250
98245
N.CS.
98240
98234
98229
98223
98218
98212
98207
98201
98196
98190
98185
98179
98174
98168
N. 8.
19538
19566
119595
119623
19652
19680
19709
19737
19766
19794
19823
19851
19880
19908
19937
19965
19994
20022
20051
20079
20108
20136
20165
20193
20222
20250
20279
20307
20336
20364
N.CS.
98163
98157
98152
98146
98140
98135
98129
98124
98118
98112
98107
98101
9S096
98090
98084
98079
12 \)eg.
N.S. N.CS.
98073
98067
98061
98056
98050
98044
98039
98033
38027
98021
98016
98010
98004
97998
97992
20791
20820
20848
20877
20905
20933
^20962
20990
21019
21047
21076
21104
21132
21161
21189
21218
97987
97981
9797[
97969
97963
97958
97952
97946
97940
97934
97928
97922
97916
97910
97905
20393
20421
20450
20478
20507
20535
20563
20592
20620
20649
20677
20706
20734
21246
21275
21303
21331
21360
21388
21417
21445
21474
21502
21530
21559
21587
21616
21644
97815
97809
97803
97797
97791
97784
97778
97772
97766
97760J
97754
97748
97742
97735
97729
97723
97899
97893
97887
97881
97875
97869
97863
97857
97851
97845
97839
97833
97827
21672
21701
21729
21758
21786
21814
21843
21871
21899
21928
21956
21985
22013
22041
22070
97717
97711
97705
97698
97692
97686
97680
97673
97667
97661
97655
97648
97642
97636
97630
N.CS.
13 P eg
N.S.
22495
22523
22552
22580
|22608
'22637
22665
22693
22722
22760
22778
22807
22835
22863
22892
22920
22098
22126
22155
22183
22212
22240
22268
22297
,22325
J22353
22382
97623
97617
97611
97604
97598
97592
97585
97579
97573
97566
97560
97553
97547
97541
97534
22948
22977
23005
23033
23062
23090
23118
23146
23175
23203
23231
23260
23288
23316
, 23345
123373
23401
97437
97430
97424
97417
97411
97404
97398
97391
97384
97378
97371
97365
97358
97351
97345
97338
97331
97325
97318
97311
97304
97298
97291
97284
97278
97271
97264
97257
97251
97244
97237
U Dbg.
N.S.
24192
24220
24249
24277
24305
24333
24362
24390
24418
24446
24474
24503
24531
24559
24587
24615
97230
97223
23429197217
2076397821
N.CS. N.S".
22438
22467
N. CS.
97528
97521
97515
97508
97502
97496
97489
97483
97476
97470
97463
22410 97457
97450
97444
N.S.
79 Peg. 78 Peg. [| 77 Peg. || 76 Peg. {"irP^.
23458
23486
23514
23542
123571
23599
23627
23656
23684
23712
23740
23769
23797 97T27
23825 97120
23853 97113
23882 97106
23910 97100
97210
97203
97196
97189
97182
97176
97169
97162
97155
97148
97141
97134
24644
24672
24700
24728
24756
24784
24813
24841
24869
24897
24925
24953
24982
25010
25038
N. CS .
97030
97023
97015
97008
97001
96994
96987
96980
90973
96966
96959
96952
96945
96937
96930
96923
AI
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
23938
23966
23995
24023
24051
24079
24136
24164
97093
97086
97079
97072
97065
97n.'iS
25066
25094
25122
25151
25179
25207
25235
25263
25291
25320
25348
25376
25404
25432
25460
96916
96909
96902
96894
96887
96880
96873
96866
96858
96851
96844
96837
96829
96822
96815
96807
96800
96793
96786
96778
96771
96764
96756
96749
96742
96734
^6727
96719
96712
96705
24108 97051
97044
97037
25488
25516
25546
25573
25601
25629
25657
25685
25713
25741
257fia
25798
25826
25854
N.CS. N.S. I N.CS. N.S
96697
96690
96682
96675
96667
360
o53
96645
96638
96630
96623
96615
96608
96600
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
6
4
3
2
J.
M
!
(
66
A TABLE OF NATURAL SINES.
15 De^.
M
1
2
3
4
5
6
7'
9'
10
11
12
13
14
15
K)
17
18
19
20
21
22
23
24
25
26
27
28
29
30
N.8.
■)882
259 10
25938
259r)G
25994
20022
2(5050
2(5079
2(5107
20135
20163
26191
26219
26247
26275
26303
26331
26359
26387
26415
26443
26471
26500
26528
26556
26584
26612
26640
26668
26696
26724
16 Ueg.
31
32
33
N. CS.
96593
96585
96578
96;" 70
96.. .2
96555
965471
96540 I
96532';
96524
96517
9(5509
96502
96494
96486
96479
96471
96463
96456
96448
96440
96433
96425
9(5417
96410
96402
16394
96386
96379
96371
96363
27564
27592
27620
27(548
27676
7704
27731
;J7759
znsi
27815
27843
27871
27899
27927
27955
27983
N. 8. N. CS .
96126
96118
96110
96102
96094
96086
9(5078
96070!
96062
96054
96046
96037
96029
96021
96013
96005
17 L>eg.
N.S. IN.C'S.
28011
28039
28067
28095
28123
28150
28178
28206
28234
28262
28290
28318
28346
28374
128402
26752
26780
,26808
34126836
35126864
36 26892
37 269;>0
38 26948
26976
27004
27032
27060
2708M
271 i(
39
40
41
42
43
44
95997
95989
95981
95972
95964
95956
95948
95940
9593 1
95923
95915
95907
95898
95890
95882
45127144
96355
96347
96340
96332
96324
96316
96308
96301
96293
96285
96277
96269
96261
9(5253
96246
29237
29265
29293
2932 1
29348
2937(5
29404
29432
29460
29487
29515
29543
29571
29599
29626
29654
29682
29710
29737
29765
29793
29821
29849
29876
29904
29932
29960
29987
30015
30043
95630
95622
95613
95605
95596
95588
95579
95571
95562
95554
95545
95536
95528
95519
95511
95502
_m Peg.
NTJT^N. C8.
309(V2
30929
30957
30985
31012
31040
31068
31095
3 1 1 2:»
31151
31178
31206
31233
31261
31289
31316
2H429
28457
28485
28513
28541
28569
28597
28625
28652
28680
28708
28736
28764
28792
28820
46 1 27 172
47127200
48127228
49 1 -^7256
50127284
51127312
52;27340
53 127368
54:27396
55127424
56 27452
57j274S0
58127508
59 27536
96238
96230
96222
96214
96206
96198
96190
96182
96174
96166
96158
96150
96142
96134
M N.CS
95874
95865
95857
95849
95841
95832
95824
95816
95807
95799
95791
95782
95774
95766
95757
30071
95493
95485
95476
95467
95459
95450
95441
95433
95424
95415
95407
95398
95389
95380
95372
95106
95097
95088
95079
95070
95061
95052
95043
95033
95024
95015
9500(5
949i)7
94988
94979
94970
30098
30126
30154
30182
30209
30237
30265
30292
30320
30348
3037(5
30403
30431
30459
30486
28847
2^875
28903
28931
28959
28987
29015
29042
29070
9098
29126
29154
29182
29209
N. 8.
74 Dejr.
95363
i)5354
1)5345
95337
9532S
95319
95310
95301
95293
952H4
95275
95266
95257
95248
95240
31344
31372
31399
31427
31454
31482
31510
31537
31565
31593
31620
31(548
31675
31703
31730
9496 1
94952
94943
94933
94924
94915
9490(5
94897
94888
94878
9486i)
94860
194851
94842
94832
19 Peg.
N. 8. "IsTCiB
94552
94542
91533
94523
94514
94504
94495
94485
94476
94466
94457
<)4447
94438
94428
94418
94409
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
95749
95740
95732
95724
95715
95707
95698
95690
95681
95673
95664
95647
95639
nTcsT N_8^
73 PefT.
31758
31786
51813
31841
31868
31896
31923
31951
31979
3200(i
32034
32061
32089
32110
32144
;52997
33024
3305 1
33079
33106
33134
33161
33189
33216
33244
3327 1
33298
3332(5
33353
3338 1
30514
30542
30570
30597
30625
30653
30680
30708
30736
30763
30791
50819
30846
30874
95231
95222
95213
95204
95195
i)5186
95177
95168
95159
95150
95142
95133
95124
i95115
94823
94814
94805
94795
94786
94777
94768
94758
94749
94740
94730
94721
94712
94702
94(593
94399
94390
94380
94370
94361
94351
94342
94332
94322
94313
94303
9429;5
94284
94274
94264
3340S
3343(5
334(53
33490
33518
33545
33573
33(500
33627
33655
33682
;53710
33737
33764
33792
32171
32199
32227
32254
32282
32309
32337
32364
32392
32419
32447
32474
32502
32529
N.CS. I N.8.
72 Peg.
94684
94674
94665
i)465(5
91646
94637
94627
94618
94609
94599
94590
945«0
,94571
94561
94254
94245
94235
94225
94215
94206
i)4196
941 ^'6
94176
94167
94157
94147
94137
94127
)4118
14
43
42
41
40
39
38
37
3<i
&»
34
33
32
31
30|
29
;8
27
26
25
24
23
22
21
20
19
18
17
16
15
N.CH. I N.S .
71 Peg.
94108
94098
94088
94078
94068
94058
94049
94039
94029
i 940 19
,94009
93999
1 93989
,93979
,53819
;5384(5
33874
33901
33929
3395(5
33983
540 1 1
;54()38
34065
34093
34 ! 20
.34147
3417.'=
N. C8. I N. 8^
70 Peg.
14
13
12
11
10
9
8
7
6
5
4
3
2
_l
JVl
1
2
3
4
r>
O'lO^H
9
94049
8
94039
7
94029
«
94019
5
94009
4
93999
'A
93989
'i
93979
I
N.S.
AI
)epr.
A TAHLE OF NATURAL SINES.
JVl
1
2
3
4
f)
<;
7
H
9
10
11
12
13
14
ll>
l(i
17
18
19
r>7
iJ02 y39("iy
_N
34
34229i93!)r)9
34 2r) 7 93949
34284 93939,1 3r)9 18
31311 !>;i()i;<jl;}rj94r)
34339^93919 3r)973
343(;<i 93909 3()()0()
i N._H.
3r)837
3.')N(i4
3r>891
31393
34421
34448
l^38H9
93879
34 '17.01 93809
34.')03 938r)!>
31 r)30 93849
34 r)r)7 93839
34r>84|93829
34012 93819
22
24
20
27
28
29
30
31
32
33
34
3.'>
30
34039
34000
3'i094
34721
20134748
21 3477.0
34833
.34830
34857
34884
34912
34939
J4900
34993
35021
3004 8 i
93H()9
93799
93789
93779
!i3709
9.3759
93748
93738
93728
93718
93708
93098
93088
93077
03(i07
93(J57
I 30027
3i;0.04
30081
30108
30135
30102
30190
30217
^30244
30271
30298
.30325
3(i352
.30379
:3(i400
,30434
;3040l
30488
30515
3(i542
30509
30590
3fi()23
30050
:KWi77
39 35200
40 35293
41 35320
4213.5347
43 '3.5375
4413.5402
N. VH.
93'35N
93348
93337
93327
13310
93300
!».3295
93285
93274
93204
93253
93243
93232
93222
»32ll
93201
!i3l90
93180
93109
J3159
93148
93137
J3127
93110
93100
93095
930H4
93074
9300.3
93052
93042
9303 1
93020
93010
92999
J2988
92978
92907
92950
J.5075 9.30471 30704
}5102|93037i 30731
35130193020; 307.58
35157;930ini3078.'y
35183.93000! 30812
37 35211|93590|30839
38 i 35239 93.585,30807
93575 30894
93505130921
9.3.555| 30948
93514:30.975
93534' 37002 92902
93524 37029 92892
I 45|;i5429i935^[_4;,3705<; !i^H81
40i35450l935():j''37O83 <)2870
47 35484i93'193 37110
48 3551 119.3483' 37137
49{3.5538 93472*37104
.50 35505 93402 37191
51i35592 9.3452 37218
52j35019i93441!:37245
.53j35047 93431, 37272
.54135074; 93420 37299
55135701 9341 0i!37320
50 1 35728 ' 93400 37353
57! 35755 ' 93389 ! 373.MO
58,35782'9.3379
59 1. 3 58K) 193308
Miiv.cH. iior
!1294.0
1)2935
:i2924
92913
12859
92849
92838
92827
928 1
112805
!i2794i
92784 i
92773;
92702
j^r.ii
N. H.
37401
37488
375 1 5
37512
37509
37595
37022
3701!)
3707(
37703
37730
37757
37784
17811
17838
.37805
37892
37919
37940
.37973
37999
mvM
38053
38080
38107
38134
3810!
3818H
18215
23 Dtjr.
3824 I
38208
:J8295
138322
1.38349
38370
38403
38430
38450
18483
38510
38.537
38504
138591
380 1 7
38044
38071
38098
38725
38752
38778
38805
N.Cis.
92718
92707
92097
!)20N0
92<)75
920041
920531
!)2042
92031
92020
92009
925!)H
92.5N7
92570
925()5
92554
92.543
92532
9252 1
925 1
92499
92488
92477
92400
924.55
92444
92432
92421
92410
I N.H.
'39073
1^39100
139127
39153
'39180
39207
.39234
39200
J39287
39314
39341
39307
39394
39421
39418
39474
19.501
39528
39555
3958 1
39008
N.rrt.'
92(')50
92039
92028
'M Dog.
N.N.
40074
10700
40727
91833
ill 822
39035 91810
1)2399
!)2388
92377
92300
92.355
92343
92332
92321
92310
92299
922H7
92270
.3900 1
39088
.39715
3974 1
39708
39795
39822
39848
92OI0ti40753
40780
4080(1
40833
40800
108N0
40913
40939
4090<i
40992
41019
41045
4^072
41098
41125
41151
4 11 78
4 1 204
41231
91994
91982
91971
9 I 959
91948
!)I930
!) I !)25
91914
9 1 902
1)1891
91879
91808
91850
91845
N.CH.
9135.5
M
00
59
58
57
:1
91799
91787
91775
91704
9 1 752
!) 1 74 I
9 1 729
91718
39875191700
39902 91094
39-.;28j9l083
199559 107 1
91000
91048
91030
'39982
'40008
'40035
:40002J9I025
:40088!91013
|40 115 91001
40141 91.590
92205 '40108 91578
92254i40195'91500
92243,40221 19 1 5-05
922.3 1
92220
92209
92 1 98
37407"!iii740
92729
09 l3
etr.
40248,91.543
! 40275 ^91531
40301 19 1519
,:40328;91508
92180 40355191490
92175 40381 J9 1484
92104140408,91472
38832 92152140434 91401
388.59 92141 40401191449
38880 92130'40488'91437
38912 92119,10514
38939 92 1 07 1 4054 1
38900 920901,40507
3H993 92085 : 40594
39020 92073:40021
39^40 192002, 40047
4 1 257
4 1 284
41310
41337
41303
41390
41410
141443
4M09
14T490
4 1 522
41.549
•11575
4 1 002
41028
410.55
41081
4 1 707
41734
913 13
91331
91319
9 1 307
91295155
)I2H;»|.54
9 1 272
91200
91248
91230
9 1 224
91212
9 1 200
9 1 1 88
91170
91104
9 1 1 52
91140
91128
9 1 1 Hi
91104
91092
91080
91008
91050
91044
111032
9 1 020
91008
90990
90984
90972
909f;0
90948
909.30
90924
90911
90899
90887
90875
N.t'H.'ljV^K
07 JJop.
N. f:H.
91425
91414
91402
91390
9 1 378
iiUiOO
N.H.
4 1700,' 90803
4I787|90851
'41813|90839
4l840j9O820
4\Hm 908 1 4
41892 90802
|419l9l90790
419451907781
4 1972 1 90700
41998 90753
42024 90741
42051 1 90729
42 1 04 190704
4213090092
421.50! 90080
42 183 '90008
42209190055
42235 '90643
N.f:8. ! nVh.
05 Dejf.
.53
52
51
50
49
48
47
40
45
44
43
42
41
40
39
38
37
30
35
34
33
32
31
110
29
28
27
20
25
24
23
22
21
20
19
18
17
10
K5
14
13
12
U
10
9
8
7
5
4
3
2
1
M
J
-yir
68
A TABLE OP NATURAL SINES.
\m
M
1)
1
2
3
4
h
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
%b De^r.
4226*^
42288
42315
42341
423(57
423'J4
42420
42446
42473
42499
42525
42552
42578
42604
42631
42657
'N. ca.
90631
90618
90606
90594
90582
90569
90557
90545
90532
90520
90507
90495
90483
90470
90458
90446
42683
42709
42736
42762
42788
42815
42841
4286?
42894
42920
42946
42972
42999
43025
43051
31
32
33
34
35
36
37
38
39
40
41
42
43
90433
90421
90408
90396
90383
90371
90358
90346
90334
90321
90309
90296
90284
90271
90259
t^ Deg.
N.H.
N. CS.
43837
43863
43889
43916
43942
43968
43994
44020
44046
44072
44098
44124
44151
44177
44203
44229
44255
44281
44307
4433;)
44359
44385
44411
44437
44464
44490
44516
44542
44568
44594
44620
89879
89867
89854
89841
89828
89816
89803
89790
89777
89764
89752
89739
89726
89713
89700
89687
43077
43104
43130
43156
43182
43209
43235
43261
43287
43313
43340
43366
,43392
44 43418
45 43445
46
47
48
49
50
51
-11 Dt-y.
89674
89662
89649
89636
89623
89610
89597
89584
89571
89558
89545
89532
89519
89506
89493
>J^8^ N.(J8.
4M99
45425
45451
45477
45503
45529
45554
45580
45606
45632
45658
45684
45710
45736
45762
45787
43471
43497
43523
43549
43575
43602
43628
53J43654
64143680
55143706
56143733
67 43759
43785
43811
58
59
M
90246
90233
90221
90208
90196
90183
90171
90158
90146
90133
90120
90108
90095
90082
90070
90057
90045
90032
90019
90007
89994
89981
89968
89956
89943
89930
89918
89905
89892
N. C8. N. 8,
64 Deg.
44646
44672
44698
44724
44750
44776
44802
44828
44854
44880
44906
44932
44958
44984
45010
45813
45839
i5865
45891
45917
45942
45968
45994
46020
46046]
460721
46097
46123
46149
46175
89480
89467
89454
99441
89428
89415
89402
89389
89376
89363
893ii0
89337
89324
89311
89298
89101
89087
89074
8906 1
89048
89035
8902 1
89008
88995
88981
88968
88955
88942
88928
88915
88902
88888
88875
88862
88848
88835
88822
88808
88795
88782
188768
88755
88741
88728
88715
88701
46947
46973
46999
47024
47050
47076
47101
47127
47153
47178
47204
47229
47255
47281
47306
47332
45036
45062
45088
45114
45140
45166
45192
45218
45243
45269
45295
45321
45347
45373
N. CS. N. S,
46201
46226
46252
46278
46304
46330
46355
46381
46407
46433
46458
46484
46510
46536
46561
89285
89272
89259
89245
89232
89219
892061
891931
89180
89167
89153
89140
89127
89114
46587
46613
46639
46664
46690
46716
46742
46767
46793
46819
46844
46870
46896
46921
N. C8.
88688
88674
88661
88647
88634
88620
88607
88593
88580
88566
88553
88539
88526
88512
88499
88485
88295
88281
88267
88254
88240
88226'
88213
88199
88185
88172
88158
88144
88130
88117
88103
88089
±'^ Ueg.
N. 8. IN.CS.
47358
47383
47409
47434
47460
47486
47511
47537
47562
47588
47614
47639
41665
47690
47716
88075
88062
88048
88034
88020
88006
87993
87979
87965
87951
87937
87923
87909
87896
87882
48481
48506
48532
48557
48583
48608
48634
48659
48684
48710
48735
48761
48786
48811
48837
48862
47741
47767
47793
47818
47844
47869
47895
47920
47946
47971
47997
48022
48048
48073
48099
63 Deg.
88472
88458
88445
88431
88417
88404
88390
88377
88363
88349
88336
88322
88308
N. 8.
87868
87854
87840
87826
87812
87798
87784
87770
87756
87743
87729
87715
87701
87687
87673
M
87462 00
8 7448; 59
87434158
87420'57
87406 56
87391
87377
87363
87349
87335
87321
87306
87292
87278
87264
87250
55
54
53
53
51
50
49
48
47
46
45
48888
48913
48938
48964
48989
49014
49040
49065
49090
49116
49141
49166
49192
49217
49242
87235
87221
87207
87193
87178
87164
87150
87136
87121
87107
87093
87079
87064
87050
87036
44
43
42
41
40
39
38
37
36
35 i
34
33
32
31
30
49268
49293
49318
19344
49369
49394
49419
49445
49470
49495
49521
49546
49571
49596
49622
48124
48150
48175
48201
48226
48252
48277
148303'
48328
48354
48379
148405
48430
4845 6
N. csT
87659
87645
87631
87617
87603
87589
87575
87561
87546
87532
87518
87504
87490
87476
N.S.
62 Deg. 1) 61 Deg
87021
87007
86993
86978
86964
86949
86935
86921
86906
86892
86878
86863
86849
86834
86820
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
49647
49672
49697
49723
49748
49773
49798
49824
49849
49874
49899
49924
49950
49975
86805
86791
86777
86762
86748
86733
86719
86704
86690
86675
'86661
86646
86632
86617
N.CS.
N.S.
14
13
12
11
10
9
8
7
6
6
4
3
2
_1
M
60 Deg.
48
512
49
512
50
512
51
512
52
513
63
513
54
513
55
613
66
514
57
614
58
514
69
614
M
N.C
5t
A TABLE OP NATURAL SINES.
fi9
21
1
2
3
4
5
6
7
30 Deg.
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
I 47
48
49
50
51
52
53
54
55
66
57
58
59
N. S, I N. C8.
8^603
86588
86573
86559
86544
86530
86515
86501
86486
86471
86457
86442
86427
86413
86398
86384
50000
50025
50050
50076
50J01
50126
50151
50176
8 50201
50227
50252
50277
50302
50327
50352
50377
50403
50428
50'153
50478
50503
50528
50553
50578
50603
50628
50654
50679
50704
50729
50754
86369
86354
86340
86325
86310
86295
86281
86266
86251
86237
86222
86207
86192
86178
86163
51504
51529
51554
51579
51604
51628
51653
5 J 678
51703
51728
51753
51778
51803
,51828
51852
151877
31 Peg.
N. 8. [ n.cb;
85717
85702
85687
85672
85657
85642
85627
85612
85597
85582
85567
85551
85536
85521
85506
85491
50779
50804
50829
50854
50879
50904
50929
50954
50979
51004
51029
51054
51079
51104
51129
51154
51179
51204
51229
51254
51279
51301
51329
51354
51379
51404
51429
51454
51479
86148
86133
86119
86104
86089
86074
86059
86045
86030
86015
86000
85985
85970
85956
85941
51902
51927
'51952
51977
52002
52020
52051
52076
52101
52126
52151
52175
52200
52225
52250
85926
85911
85896
85881
85866
85851
85836
85821
85806
85792
85777
85762
85747
85732
52275
152299
52324
52349
52374
52399
52423
52448
52473
52498
52522
52547
52572
52597
52621
85476
85461
85446
85431
85416
85401
85385
85370
85355
85340
85325
85310
85294
85279
85264
85249
85234
85218
85203
85188
85173
85157
85142
85127
85112
85096
85081
85066
85051
85035
52992
53017
53041
53066
53091
53115
53140
53164
53189
53214
53238
53263
53288
53312
53337
53361
53386
53411
53435
53460
84805
84789
84774
84759
84743
847281
84712
84697
84681
84666
84650
84635
84619
84604
84588
84573
84557
84542
84526
84511
53484184495
5350984480
53534 84464
54464
,54488
(54513
154537
54561
54586
54610
54635
154059
54683
54708
54732
54756
54781
54805
54829
83867
83851
83835
83819;
S3804
83788;
83772'
837561
83740,1
83724
83708
83692
83676
83660
83645
83629
"sTn
eg.
53558
53583
53607
53632
53656
53681
53705
53730
N. CS. I N. 8.
59 Deg.
52646
52671
52696
52720
52745
52770
52794
52819
52844
52869
52893
529 IS
52943
52967
N. CS.
85020
85005
84982
84974
84959
84943
84928
84913
84897
84882
84866
84851
84836
84820
N.S.'I
53754
53779
53804
53828
53853
53877
53902
53926
53951
53976
54000
54024
54049
54073
54097
84448
84433
84417
84402
84386
84370
84355
84339
84324
84308
84292
84277
84261
84245
84230
84214
84198
84182
84167
84151
84135
84120
84104
54854
54878
54902
54927
54951
4975
j4999
55024
55048
55072
55097
55121
55145
55169
55194
54122
54140
54171
54195
54220
54244
54269
54293
54317
54342
54366
54391
54415
54440
58 Peg.
84088
84072
84057
84041
84025
84009
83994
83978
83962
83946
83930
83915
183899
83883
N. <J8. I N.8 .
57 Peff.
55218
55242
55266
55291
55315
55339
55363
65388
55412
55436
55460
56484
55509
55533
55557
83613
83597
83581
83665
83649
83533
83517
83501
83485
83469
83453
83437
83421
83405
83389
N.8.
55919
55943
55968
55992
56016
56040
66064
56088
50112
56136
56160
66184
56208
56232
56256
56280
55581
55606
55630
55664
55678
55702
j'55726
56760
55775
55799
55823
55847
55871
55895
N.CS.
83373
83356
83340
83324
83308
83292
83276
83260
83244
83228
83212
83195
83179
83163
83147
56305
56329
56353
56377
|5640l
56425
56449
56473
66497
56521
56545
56669
56593
56617
56641
N.CS.
82904
82887
82871
82856
82839
82822
82806
82790
82773
82757
82741
82724
82708
82692
82676
82659
82643
82626
82610
82593
82577
82561
82544
82528
82511
82495
82478
82462
82446
82429
82413
83131
83115
83098
83082
83066
83050
83034
83017
83001
82985
82953
82936
82920
N.8.
Peg. I 56 Peg.
56665
56689
56713
56736
56760
56784
56808
56832
66856
56880
56904
56928
56952
56976
57000
57024
57047
57071
57095
57119
57143
57167
57191
57215
57238
57286
57310,
57334
182396
i82380
82363
82347
82330
82314
82297
82281
82264
82248
82231
82214
82198
82181
82165
S2148
82132
82115
82098
82082
82065
82048
82032
J82015
81999
I o 1 r\ .^ »-*
181965
81949
81932
N.C8. 1 X. 8 .
55 Peg."
60
59
58
57
56
55
54
53
52
51
60
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
26
24
23
22
21
20
19
18
17
16
15
14
13
12
II
10
9
8
7
6
3
2
J
M
70
A TABLE OP NATURAL SINES.
M
1
2
4
5
6
7
9
10
n
12
13
14
35 Deg.
N. a, I N. c;h.
57381
57405
57429
57453
57477
5750 1
57524
57548
57572
57590
57619
57643
57667
57691
15 57715
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
57738
57762
57786
57810
57833
57857
57881
57904
57928
57952
57976
57999
58023
58047
58070
58094
58118
58141
81915
81899
81882
81865
81848
81832
81815
81798
81782
81765
81748
81731
81714
81698
81681
81661
81647
81631
81614
81597
81580
81563
Si 546
81530
81513
81496
81479
81462
81445
81428
81412
34158165
35
36
37
38
39
40
41
58189
58212
58236
58260
58283
58307
58330
12158354
43J58378
44 58401
45 58425
46
47
48
49
50
51
52
53
54
55
56
57
58
59
58449
58472
58496
58519
58543
58567
58590
58614
58637
58661
58684
58708
58731
58755
81395
81378
81361
81344
81327
81310
81293
81276
81259
81242
81225
81208
81191
81174
81157
81140
81123
81106
81089
81072
81055
81038
81021
81004
80987
8O970
80953
80936
80919
36 Dofj.
5S779
58802
58826
58849
58873
58H96
58920
58943
58967
58990
59014
59037
59061
59084
59108
59131
59154
59178
59201
59225
59248
59272
59295
59318
59342
59365
59389
59412
59436
59459
59482
59506
59529
59552
59576
59599
59622
59646
59669
59693
59710
59739
59763
59786
59809
59832
80902
80885
80867
80350
80833
80816
M0799
80782
80765
80748
80730
80713
80696
80679
80662
80644
80627
80610
80593
80576
80558
80541
80524
80507
80489
80472
80455
80438
80420
80403
80386
80368
80351
8U334
80316
80299
80282
80264
80247
80230
80212
80195
80178
80160
80143
80125
M N. eg. N. S_.
54 Deg.
59856
59879
59902
59926
59949
59972
59995
60019
60042
60065
60089
60112
60135
60158
37 Ot'g^
NTsr"NVm_
79864
79846
79829
70811
79793
79776
79758
79741
79723
79706
79688
79671
79653
79635
79618
79600
60182
60205
60228
60251
60274
60298
60321
60344
(U)367
60390
60414
60437
60460
60483
60506
60529
60553
60576
60599
60622
60645
60668
6069 J
60714
60738
60761
60784
60807
60830
60853
60876
60899
60922
60945
60968
60991
61015
61038
61001
61084
61107
61130
61153
61176
61199
61222
79583
79565
79547
79530
79512
79494
79477
79459
79441
79424
79406
79388
79371
79353
79335
79318
79300
79382
79264
79247
79229
79211
79193
79176
79158
79140
79122
79105
79087
79069
80108
80091
80073
80056
80038
80021
80003
79986
79968
79951
79934
79916
79899
79881
N. C8. N. S
53 Deg.
61245
61268
61291
61314
61337
61360
61383
61406
61429
61451
61474
161497
61520
61543
61566
61589
61612
61635
61658
61681
61704
61726
61749
61772
61795
61818
61«U
61 64
61887
61909
3U Dtig.
N. fl. N. Crt.
61932
61955
61978
62001
62024
62046
62069
62092
62115
62138
62160
62183
62206
62229
62251
62274
62297
62320
62342
62365
62388
62411
62433
62456
62479
62502
62524
62547
62570
62592
78801
78783
78765
78747
78729
78711
78694
78676
78658
78640
78622
78604
78586
78568
78550
78532
78514
78496
78478
78460
78442
78424
78405
78387
78369
78351
78333
78315
78297
78279
78261
39 Deg.
79051
79033
79015
78098
78980
78962
78944
78926
78908
78891
78S73
788o5
178837
78819
78243
78225
78206
78188
78170
78152
78134
78116
78098
78079
78061
78043
78025
78007
77988
N.S.
62932
62955
62977
63000
63022
63045
63068
630;.0
63113
63135
63158
63180
63203
63225
63248
03;Tn
63293
63316
63338
63361
63383
63406
63428
63451
03473
63490
63518
63540
63563
63585
63608
N.CS.
77715
77696
77678
77660
77641
77623
77605
77586
77568
77550
77531
77513
77494
77476
77458
77439
77421
77402
77384
77366
77347
77329
77310
77292
77273
77255
77236
77218
77199
77181
77162
62615
62638
62660
62683
62706
62728
627511
62774'
62796
62819
62842
62804
62887
62909
63630
63653
63675
63698
637^0
63742
63765
63787
63810
63832
63854
63877
63899
63922
63944
N.CS.I N.S
62 Deg.
77970
77952
77934
77916
77897
177879
77861
177843
77824
77806
77788
777uU
77751
77733
77144
77125
77107
77088
77070
77051
77033
77014
76996
76977
76959
76940
76921
76903
76884
M
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
N.CS.I N.S.
r 51 bog.
63966
63989
64011
64033
04056
64078
64100]
641231
64145
64167
64190
04212
64234
64256
76866
76847
76828
76810
76791
76772
76754
76735
76717
76698
76679
176642
176623
N.CS .I N.B.
50 Deg.
14
13
12
11
10
9
8
7
6
5
4
o
2
_l
M
39
40
41
42
43
44
45
46
47
48 6J
49|6i
50 1 6;
51 6;
52 6.'
53 6^
64 6£
60
M
60
59
58
57
56
55
51
5:)
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
_l
M
I
41
N. s:
65606
n6628
65650
75471 66913 74311 1 fi8^
??h^^««935 74295 |:6822l
75433
,MJ5672 75414
fi5694 75395
65716 7537^
llo-n:?! 753371
■ 75318
752991
73135
73116
73096
73076
1^^,^5^74276 68242
66978 7-J 256 68264 rM)7n
fi6999 74237 68285 7'1nJ?
67043 74198 68327 7S0R
67064 74178 68349 Sir
67086 74159 68370 S?fi
67107 74139 6839? S?7
671^0 74120 68412 72S
74100 68433 72917
740S0 68455 72897
^ 1,... .^-, 740G1 68476 TQavv
75203 67215,74041 68497 7^1?
75W 67237 74O226I5I8 S
75280 67129
75261 67151
75341 67172
75222 67194
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
cmnn i..^?" V'''87 73885 68666 72697
69466
69487
69508
69529
'695491
,'69570
'69591
169612
69633
69654
69675
69696
69717
69737
69758
69779
73865
73846
73826
73806
64967
649891
650111
65O33I
65055
65077
65099
65122
65144
65166
65188
65210
65232
65254
65276
46 651 J8
47 65320
76022
76003
j 75984
75965
75946
75927
75908
75889
75870
75851
75832
758131
75794
74915
74896
74876
. 74857
6632774838
66349 74818
75770
75756
48
49
50
51
52
53
64
65342
65364
65386
65408
65430!
65452
654741
654961
655 I 8| 75547
56
•5^165562
75738
75719
75699
75680
75661
75642
75623
75604
75585
75566
66371
|66393
|66414
,66436
66458
66480
66501
IC6523
66545
166566
J66588
166610
66632
66653
66675
66697
,66718
166740
166762
74799
,6868872677
6870972657
6873072637
6875172617
'6877272597
67516 73767 68793 72577
67538 73747l!68814 72557
67559 73728
67580 73708
6760273688
'67623 73669
67645 '
67666
74780 67688
74760
74741
74722
74703
74683
74664
74644
74625
74606
74586
74567
74548
74528
74509
74489
74470
74451
67709
67730
6883572537
68857725T7
6887872497
6889972477
68920 72457
69800
69821
69842
69862
69883
69904
69925
69946
69966
169987
70008
70029
70049
70070 I
70091
71610
71690
71669
71549
71529
71508 ;
71488 L.
71468 37 i
7144736
70112
70132
70153
71427
71407
171386
71366
71345
71325
71305 29
71284
71264
73610
73590
73570
67752|Y3551
6777373531
67795 73511
65540
75509
65584175490
65606 75471
n-csT! n. s.
,'66783 74431
66805 74412
[66827
[66848
'66870
66891
66913
N.CS
_49Deg. ii~48"D;^
74392
74373
74353
74334
74314
N.8.
67816
67837
67859
67880
67901
,67923
67944
67965
67987
68008
68029
68051
68072
68093
681 15
68136
68157
68179
73491
73472
73452
73432
73412
73393
73373
73353
73333
73314
73294
73274
73254
73234
73215
73195
73175
73155
68200:73135
68962 72417
72397
72377
72357
72337
72317
72297
68983
69004
63025
69046
69067
69088
69109 72277
69130172257
69151 172236
|69172 722T6
6919372196
6921472176
69235
69256
69277
69298
69319
69340
72156
72136
72116
72095
72075
72055
N.CS.I nX
47 Peg.
69361 72035
69382172015
69403
69424
69445
69466
N.CS.
71995
71974
71954
71934
N.S.
'7017471243
7019571223
70215 71203
70236 71182
70257 71162
,70277 71141
70298 71121
,70319 71100
,70339 71080
70360 71059
70381 71039 .„
70401 71019 15
70422 70998 14
70443 70978 13
70463170957 12
70484170937 11
70505170916 10
70525 1 70896 9
7054670875 8
70567, '70855 7
70587170834 6
(70608 1 708 13 fi
/-0rj28|70793 4
70649 70772 3
7067070762 2
70690170731 1
7071170711
46 Peg.
N.C8.I N .S M
45 IJog,~'