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1883.
P]
r
Pre^
ELEMENTARY
4
Practical Mathematics
FOR
HIGH SCHOOLS,
INCLUDING
E RUDIMENTS OF TRIGONOMETRY, NAVIGATION,
MENSURATION, AND DYNAMICS,
7ITH THE USE OF LOGARITHMS,
BY
FRANK H. EATON, A.B. (Harv.),
'^CHBK OF MATHEMATICS IN THE NOVA SCOTIA NORMAL SCHOOU
Prescnbed by the Council of Public Instruction for use in the
Public Schools in Nova Scotia.
TRURO, N. S. !
D. H. SMITH&CO
1883.
<^^^
y
c
I ^
Eiitt red according to Act of rurlinnipnt of Canada, in the year 1883, by
D. H. SMITH & CO.
Ill the Office of the Minister of Agr.'oulture at Ottawa.
6'.^>
PREFACE.
^
In ^he compilation of this manual, the author has had in
view the requirements of students who have been well
taught in the rudiments of Algebra and Geometry, and
has aimed to give it such a character that its use as a
text-book will conduce to the development of mathe-
matical power, as well as to the acquisition of valuable
practical i:n<>wledgj.
It i hoped that, while pains has been taken to guard
aginst the enervatirg efieocs of wire-drawn explanations
:nd awev'ib..^uH jUustratioin, the text has been made
suiiicieotl}' precis, and ^nieili(.ible,and the questions asui
exercises there^u bafficiont^y typical and suggestive to
render a complete ccmprehtnsion of the principles and
methods discussed, e -fily attainable by all for whom :t
is designed.
The purely geometrical demonstrations of Chapters
IX. and X. are rendered necessary by the fact, that
editions of Euclid still in use do not contain an adequate
treatment of the geometry of solids.
W lile most of the exercises are original, not a few
have been selected from various works without specific
acknowledgment.
The author is under obligation to several friends for
material aid in preparing the tables and in correcting the
proof, but more especially to Alexander Mackay, Esq.,
of the Halifax High School, for a critical and suggestive
revision of the manuscript.
HRIDEI.EKRO, Ortohtr, 1S8S.
INTKOUUCTION.
Pure Mathematics inv(!Htij:jatos tht» relations of numbers
ftnd of 8pac(; magnitudes.
Applied Mathematics shows how the principles of
Pure Mathematics may be utilized for practical purposes.
The ultimate aim of this book is to render some of the
elementary principles of Pure Mathematics available in the
computation of dist>\nces, directions, areas, volumes, and
capacities, and of the theoretical relations and effects of
forces in their simpler combinations. But, since the
method of Logarithms and the principles of Trigono-
metry are of special utility in many of such computa-
tions, the earlier chapters will be devoted to such a con-
sideration of these subjects as is absolutely essential to
the general aim.
Furthermore, since an accurate representation of dis-
tances, directions, and outlines by diagrams is frequently
necessary, it will be appropriate to describe briefly, in a
preliminary chapter, the construction and use of simple
implements by means of which lines and angles can be
plotted and measured.
)
CONTENTS.
V.
CHAP. I.-^IMPLE MATHEMATICAL INSTRUMENTS.
Diagonal Scale, ..... |
Protractor, ••-... j
Scale of Chorda, - • • . . 3
EXBROiaiM, • • • . .
9
10
10
10
CHAP II-TRKiOyOMETRICAL DEFINITIONS.
Trigonometry.
Rectilineal Figures Composed of Right Triwiglea,
Similar Kight Triangles,
Trigonometrical Ratios,
Sine and Cosine,
Tangent and Cotangent,
Secant and Cosecant, -
Illustration,
Functions of Obtuse Angles,
EXKKCtSlkS,
4
6
6
7
9
10
11
12
12
12
12
12
18
18
13
II
14
14
CHAK III.-ABSOLUTE VaLUES OP TRIGONOMETRICAL
FIATIOS.
fiimits of Trigonometrical Ratios, - . . 13
Functional Values for Angles of 46', 30\ 60", and 18°, 14
Tables of Natural Functions, • • • - 1ft
How to Use the Tables, • • • • 16
EXBRCISSS, • • • . .
16
16
17
16
18
CHAP. IV. -SOLUTION OF PLANE TRIANGLEa
Definition,
Four General Cases, -
Solution of Right Triangles,
Law of Sines, -
Ambiguous Case,
Two Sides aad Contained Angle,
Another Formula,
When the Three Sides are Known,
KXERCISKS,
17
18
19
20
21
22
23
24
20
20
20
21
21
22
23
23
2ft
CONTENTS,
CHAP. V.-.LO<»ARITHMS.
I>«f)nitioii, .....
Uiffuront. Ix>{{arithini of th« SAnte Nrmh^r. •
Common or Dvimry Synti-ni.
Pro|M)rtioR of Coiumoii Lo^ttritlims, •
f^ )garithm> from the Tahlua, •
Algebraio Priituiiiles, ....
\A)iitkrit\iin» of I'rottuotb ainl Quotionti,
Tables of fiOgarithniH of Augiilnr FuDotioiu,
KXKIU'IHRM, ....
AST.
»Aua
85
27
n
87
27
88
88
8tt
80
89
ao
89
31
30
88
30
31
cuAiv VI. rKKuns and distancrs computkd by
THK AID OH" TttlGONOMETKY.
|)«'tiuitionit, .....
Vertical DiatAitccH, ....
Horizontal DiatanceM, ....
Aiipliohtioti of the (ieomotry of Cin-Ics,
Ratio of Diamuter am! Circumference,
KXKRCMKM, ....
CHAI'. VlI.-i>'AVIOATION.
Klciiieiitary Conoeptioua,
Auxiliary CoiiceptionM,
Kepresentationa by Mercator'c Chart,
KxpressioQ for Diifeionce of Longitut'e,
Expression for Meridional Parts,
Trigonometrical Relations of Departure, Difference
of Latitude, Diatance, and Course,
Trigonometrical Relations Demonstrated,
Trigonometrical Relations of Difference of Longitude,
Nioridional Difference of Latitude, and CourHe,
RoMolution of a Traverse,
Variation of the Compass,
Deviation,
lipdway,
CJorrectiou of the Log,
KXBKCIMKS,
CHAP. VIIL-COMPUTATION OF PLANE AREAS.
Area of Parallelogram derived from Base and Altitude, 51 50
Area of Triangle derived from Base and Altitude, • 62 50
Area of Triangle derived frum its Three Sides, - 63 51
33
33
34
38
35
34
.W
35
37
35
37
38
39
•
39
.19
m
40
41
m
41
41
•
42
42
m
4^
42
•
44
4»
'♦
45
44
46
45
47
45
48
45
49
40
50
46
47
OONTKMTH.
▼H
Atm fA Triangle derircd from Sitiss aad I^iaa of
Iiuorib«(l ('iroU, - • ■ • .
Arsftof OiroU denved from Uadtoa ar Ctmamfsi^noo,
Aw* of CirouUr Sector derivud from it« Arc,
AmA of Qu»4lril«tei«l darirod from DiaguoAli and
iiiei? Triclin»ticD, . • ■
Area of QuAdriUtond derived from its Hide* and
Inciinatiuu of Diagonal*, • . .
Area of QtiadrilaterKi Imicriiitible in a Cii-olt,
Aroa« of SimM.ir Kiguros, ....
EXKKCMNM, • .
*ai
S4
SA
M
»7
68
fi9
60
rAoa
fit
n
fit
R9t
U
54
S5
16
CHAP. IX.-COMI»aTATlON OK POLYUKDhAL AND CUllVBD
AKKAS.
I'clyhodron, ....
PrUm, .....
liateral Surface of Prinni,
(cylindrical Surface. C'ylind r,
liateral Surface of Cylinder,
Pyramid, - , . ,
Frustum of Pyramid, •
Lateral Surface jf Pyramid an.l i^Vutttum,
Cone, .....
Lateral Surface of Right Cone,
Conic Frustum, and Surface ui Frustum of Right
Cone, • • • .
Spherical Surface,
EXSRCISEH, ..... 05
81
<i
6i
69
63
60
61
eo
66
60
66
61
67
62
68,
6t
69
62
70
6.1
71
64
72
64
CHAP X.- VOLUMES AND CAPACITIES.
Equivalent Prisms,
Volume of Prism and Cylinder,
Volume of Triangular Pyramid,
Volume ot any Py ramid or Cone,
Volume of Frustum of Cone or Pyrauj
Volume of Sphere,
Volume of Spherical Sector, •
Volumes of Similar Pyramids,
Volumes of Similar Solidn in (ioneral,
EXRRCIHRM,
78
74
76
76
77
78
79
80
81
67
67
68
69
69
TO
70
70
71
71
viii
CONTENTa
CHAP. XL -DYNAMICS.
Haws of Motion,
Definitions,
Graphic Representation of Forces,
Resultant of Forces Acting in same T
Parallelogram of Forces,
Triangle of Forces,
Polygon of Forces,
Moment of a Force,
Equal Opposite Moments,
Parallel Forces,
Centre of Gravity,
CMitre of Gravity of Systems
Elementary Machines,
The Inclined Plane,
Tke Wedge and the Screw,
The Lever,
Weighing,
The Pulley, •
Velocity Varying Uniformly,
Relation Ci Velocity to Pressure and
Momentum, ...
Motion Varying in Direction,
EXERCI»E!},
EXEKCISES FOR RbVIBW,
Answeks,
Mass,
ART.
PAOK
82
72
83
73
84
73
85
73
86
73
87
74
88
74
89
75
90
75
91
76
92
76
93
77
94
77
95
78
96
79
97
80
98
81
99
81
100
83
101
84
102
84
103
84
85
90
96
TABLES.
L — Natural Functions, . . - - . 102
II. — Logarithms, ...... lOg
III. — Logarithmic Functions, - - - - - 114
iV.— Meridional Parts, - - - - - 122
v.— Traverse Table, ...... l'J5
APPENDIX.
A. — The Compass Card, ----- 127
B. — A General Method Appliv;able to the Computation of
Areas, used in the Surveyixig of Land, - - 128
PRACTICAL MATHEMATICS.
CHAPTER I.
SIMPLE MATHEMATICAL INSTRUMENTS.
1. Diagonal Scale— Con8t7'uction. — Lei AK* (fig. 1) be
a liae of convenient length, divided into ten equal parts,
AB, BG, etc. Draw Aa perpendicular to AB, and equal
to it; and through B, G, D, etc., lines parallel to Aa;
likewise draw ak parallel to AK. Divide Aa and AB
into ten equal parts each; through the division points of
the former draw horizontal parallels, and oblique parallels
through those of the latter, as in the figur.^.
^
li
Fig. 1.
A scale so constructed is a decimal one, containing one
thousand parts ; en it hundreds are read off to the right
of the point marked 0, tens to the left, and units down-
wards on the vertical line OB.
Use Illustrated. — ^A distance of 478 feet may be repre-
Only part of the scale can be shown on the page.
10
PRACTICAL MATHEMATICS.
I
I
sented by the lino Im, and a distance of 295 miles by
the line tm.
Note.—M a distance of more than a '^housantl unita is to be plotted,
it can he done by drawing a line equal in length to the scale, and then
extending it to represent the excess.
2. Protractor. — This instrument is used for plotting and
measuring angles when great accuracy is not required. It
is simply a semicircle of paper, brass, tin, or other con-
venient material, with its circular edge graduated for
180 degrees.
In measuring an angle, the centre is placed on the
vertex, and one radius on one of the side? of the angle ;
the graduation of the arc will then indicate the number
of degrees in the angle.
3. Scale of Chords.— The use of this instrument is
similar to that of the protractor. In constructing such
a scale, chords are drawn from one extremity of a
quadrantal arc to successive degree points, and their
lengths then laid off on a horizontal line, measuring
always from one extremity.
An angle may be plotted with the scale of chords by
describing an arc with the chord of 60° as radius, since
the chord of 60° is equal to radiuG, and placing in it,
from the scale, the chord corresponding to the given
angular magnitude. Lines drawn from the centre to the
extremities of the chord will contain the required angle.
EXERCISES.
1. Construct a decimal scale containing 1000 parts.
2. Plot a triangular field whose sides are .35, 40 2, and 53 6 rods, re-
spectively, in length.
3. The base of a triangle is 4*01, and the perpendicular from its middle
point is 3*29; measure the other sides.
i
SIMPLE MATHEMATI(;\L INSTRUMENTS.
11
4 Meware the sides of a parallelogiam whose diagonals are at nght
anglfw to one another, and whose lengths are represented by the num-
ber* 352 and 728.
5. Construct, of paper, cardboard, or tin, a aemicircular protractor,
gradnated at intervals of 5°.
6. Make a triangle and measure its angles.
7. Make a triangle, two of whose angles shall be respectively 3fi°
and 65°.
8. Construct a scale of chords for angles at intervals of 10 .
9. Make an angle of 30* by means of the protractor, and measure it
with the scale of chords.
10 The sides AB, BC, CD of a pentagon are 362, 468, and 504, and
the angles A, B, C, and D 100% 105°, 90% and 125% respectively; con-
struct the pentagon, and vletermine its remaining parts by mstniiaents.
CHAPTER II.
TKiGONOMETRICAL DEFINITIONS.
4. Trig-onometry. — The investigation of the mutnal
relations of the sides and angles of a triangle is called
Trigonometry.
5. Rectilinear Figures composed of Right Triangles.
— The right triangle is the most elementary of all recti-
linear figures; for an oblique triangle may be considered
as the sum or the difference of two right triangles; and
all polygons are divisible into triangles, either right or
oblique.
Note. —The discussions in this and the following chapter appertain,
primarily, to right triangles ; their application to oblique triangles will
appear in Chapter IV.
6. Similar Right Triangles.— It is a well-established
geometrical principle that the sides about the equal
angles of mutually equiangular triangles are proportional
(Euc. Bk. VI., Prop. IV.). Hence, if two right triangles
have an acute angle in one, equal to an acute angle in
the other, the ratio of any two sides of the one is equal
to that of the two corresponding sides of the other. In
other words:— r^ ratio of any two aides of a right
triangle, one of whose acute angles is of a given magni-
tude, is the same, whatever the lengths of those sides.
7. Trigonometrical Ratios. — For every value, there-
fore, of the acute angle, which is called the angle of
reference, there are six of these fixed ratios, three of
which are the reciprocals of the others. These ratios
f
TRIGNONOMBTRICAL DEFINITIONS.
18
-i; ■■■
sure called Trigonometrical Functions of the angle of
reference.
8. Sine and Cosine. — The ratio of the side opposite
the angle of reference to the hypotenuse is called Sine,
and that of the adjacent perpendicular side to the hypo-
tenuse is called Cosine. ^L: i '
Note.-
-The difference butween unity and ine cosine is called Versed-
9. Tangent and Cotangent- — ^The ratio of the side
opposite the angle of reference to the adjacent pei*pen-
dicular side is called Tangent, and its reciprocal, the
ratio of the adjacent perpendicular side to the opposite
side, ic called Cotangent. ' ■'^ \.
10. Secant and Cosecant. — The ratio of the hypo-
tenuse to the perpendicular side adjacent to the angle
of reference is called Secant, and that of the hypotenuse
to the opposite side is called Cosecant. ^ '
11. Illustration. — In the triangle
ABC (fig. 2) denote the sides opposite
the angles A, B, C, by a, b, c, respec-
tively; then, from preceding articles —
A
/
/
sine of .4 =^ ; cosine of -4 = r ;
tangent of A = -; cotangent of .4 = - ;
c a
secant of ^ = - ; cosecant of ^ = - .
/
^
c
B
Fig. 2.
Note 1. — These functions are written sin A, cos A, tan .4, cot A^
sec A, cosec A ; and their squares, sin ^A, cos "-4, &c.
Note 2.— Since the angles A and Care complementary, an application
of the definitions for functions of C will show that the sine, tangent, and
tiecant of an angle are the cosine, cotangent, and cosecant, respectively, of
Us complement. Hence the names co-sine, co-tangent, co-secant.
Note 3. — The functions of an angle are frequently regardec! as the
fu«ctioua of the arc which measures tbat angle. Kencu the luiiuo CV-
ctUar Functions sometimes given to them.
I '-i
I
14
PBACnCAl< MATHKMATICB.
12. FunctioM of Obtoia Angles. — In the irmnirle
il5C(fig.3)- ^. *^
wax A=.
BO
whatever the positions of
ilCand^C. Now.ifilC
were to revolve about the
point A to the position
AC, the angle A would
become BAO\ BG would //'
take the position B'C
and consequently the sine of BAC would be —
AG
Let AC be drawn equal to AG, and making z BfAO^^ ^ BAC
.-. iBAC^X^^-A, B'C'^-BC, and^=:^ that ia.
AC AC ^
8in-4=8in(180''-i4).
Similarly—
COB BAC=i
AB'
AC
But linoe AR and AB are measured in opposite directions, one must
be considered positive and the other negative.*^ Hence, if ii S' = W^ J
■AH AB J •
2^--^^,»nd_
cos>4= -cos (ISC-il).
From the above considerations it appears that tlie sine
of an angle ia equal to the sine of its mpplement, while
the cosim of an angle is equal to the cosine of its mppU-
ment with a contrary sign*
EXERCISES.
1. Verify the following identities (Art. 11);
sin A
cos A
coiA:^
1
=tan A ;
1
sin A
tau^;
— cosec A.
cos A
sin A
1
cos A
~ cot A ;
- sec -4 ;
tb! /l^tf.^^^V^u'JSht best not toj|ter fully into t'hg discussion of
#r:«„;r::::?"*";^"L:"!:"!!-!*'if'"^' ^:!r^***^** eutireiy that relating to
functions of angles greater than 180°.
0,r^^^
c
TRIGONOMBTRICAL DEFINITIONS.
15
le
r
n
2. ExpreM in words the roUtions indicftUd in Ex. 1.
3. <tiven the nine and cosine of an angle, oxpreM all the other
functions in terms of these twa
4. Show that the cosecant, secant, and cotangent of an angle are the
reciprocals nf its sine, cosine, and tangent, respectively.
^4r\i>'n. Express all the other functions of an angle in terms of its sine.
Apply Euc. I. 47.
K. Establish the following identities : —
tan A
n/6«
sin »il = 1 - cos 'il ; sec *il = 1 + tan *A.
i
7. Construct the two angles whose common sine is { ; the angle whose
cosine is 2 ; the angle whose tangent is |.
8. Find tan A and sec A when sin ^ =f.
9. Given coseo .4=1, find cot A and cos i4.
^^0. When cim A = \, what are sin A and cosec A T
^^^*^^11. Express all the functions of an angle in terms of the sine of its
complement.
12. With what limitations is the following statement true?— Tho
functions of au angle are equal to those, respectively, of its supplement.
CHAPTER III.
ABSOLUTS VALUES OF TRIGONOMETRICAL RATIOS.
8KCTI0N I.
Numerical Valtiea 0/ Functioru.
ABC ^l^ of f ri^ooometrical Batios.-I„ the triangle
.hn.?/5®\?' ^ ^^^ '"PP"'^'' <="?'''''« of revolution
about A; the nearer it approaches coincidence with AB
«. the .mailer the angle A. the les., CB becomes, and the
nearer AB approaches to equality with AG; while the
tZTn?*^^>T»'^'""'"l°'^^^^''^<=°"'^«' the nearer
the ength of AB approaches zero, and that of GB to
equality with the length of AG It follows then that, a,
e^e angle A varies from 0' to 90'. its sine increases from
" to i, and its cosine decreases from 1 to
Similar processes of reasoning will demonstrate the
hmtting values of the other functions (see Ex 1-6)
lA-*' ^^?*'T' ^*'"''*" ^°' *°«^'^= °^«°. 30°, 60°, and
"T r processes by which functional value.
may be determined for most angles between 0° and 9o'»
cannot well be explained in this book, yet these values
for a few angles are readily obtainable, thus —
ABSOLUTE VALUES OF TRKiONOMSTRICAL RATIOS. 17
AngUmf m*.— In the triMigle ABC {Bjt. 4), if ArzW
AV\ and AC^BVsf^, But tm il = ^^ ' -'- '" *
2 BO* -
= 4\/2, and
Angles of 30' and fiO".— If ^(7Z> (fig. 5) be equiUt«ra], uid C^ per
pendiottl«r to the boie, then AGB^SO', AB~^^, and BC* = ^ AC*
.'. sin 30* = 4, and ain 60'=^.
AngU$ of 18* '.-Let ABE (fig. S) be iioaceles, and t ABC- ^i? ^ 18*.
IfBZ)« = i5i4i4Z)then
BD*-BA {BA-BD) = BA^-BDBA
. BD'BD ,
. BD _ -l+^/8 .. ... . ,
' ^ST — 9 * positive root only
being taken.
Butein ^-BC=^^= _^(Euc. IV. 10)
A Jo "AH
.'. tin 18'=^^^.
iVo^e.-- Having found values for sin 45°, sin
30°, sin IV, and sin 18°, the values of the other >
functions may be deduced from them (Chap.
II., Ex. 6) ^
SECTION J I.
Trigonometrical Tables.
15. Tables of Natural Functions.— The functional
values for consecutive angles from 0" to 90° may be com-
puted and tabulated for reference.
It is, however, unnecessary to extend the Tables be-
yond 45°, since the sine, tangent, secant, cosine, cotangent,
and cosecant of angles from 0° to 45" are respectively
the cosine, cotangent, cosecant, sine, tangent, and secant
of angles from 45° to 90° (Art. 11, Note 2).
Note 1. —Since the functions of supplementary angles are numericatlv
equal (Chap. II., Ex. 12), the same Table will furnish functional values
for angles from 90° to 180°, due ragard being had to the appropriate
algebraical signs. '^
* This demonstration involves the solution of a quadratic equation,
and may be omitted, if necessary.
n
18
PRACTICAL MATHEMATICS.
JS^oit 2.—ln the Tablet arrtnf^d for this book (Tabb I.) the valaee
nvtn are for angles at intervala of 10', and are, in general, earned ta
four place* of decimala only; the nae of nuch Tables giving reaultfl of
luthoittnt accuracy fur Otdinary computations. In Tables to be used in
very precise calculations, however, the iutcrvak must be smaller, and
tlie number of decimal places greater.
Notti 3.— Secants and cosecants are to be derived, when required, from
the tabulated cosines ard smes (Chap. II.).
16. How to Use the Tables— Proportional Parts.— The
method of finding from the I'ables the function of an
angle lying between two succensive tabulated angular
▼alues, called the Method of Proportional Parts, is based
upon the assumption that through the given tabular iii-
terval the angle and its function vary proportionally.
Illvstration. — Required from the Tables sin 43° 3'.
Sin 43'i: -6820; sin 43* 10'= 6841 ;
Difference of sine for this intervals '0021
.', sin 43* 3'= sin {43" + ^ tabular interval) = *in 43*+^, funotionAl
difference = 6820 + A x 002 1 = •682a
Again. — Required ccs 25° 37'.
Cos 26' 37' = cos (26' 30' + /« tabular interval)
= cos 25» 30' - ,V (cos 25° 30' - cos 25' 40*)
= •9026 -1^x0013 =-9017.
Note 1. — The error incurred by the method of proportional parti di<
minishes, of course, with the interval for which the functions are
tabulated.
Note 2. — The process of obtaining the angle corresponding to a given
functional value which lies between two tabulated functions demands
no special elucidation, being simpl}' the reverse of that just illustrated.
EXERCISES.
1. What is meant by saying that cos 0*= 1?
2. Why are the sine and the cosine invariably proper fractions?
3. Give the limiting values of each of the functions of angle* from
0" to 90°.
4. Of what angles is 1 the sine, the tangent, the cosine, the secant,
respectively?
5. What angle has i tor ^ its sine? 2 for its secant! V3 for its
tang.^nt?
(J/tf--
AHSOr.UTK VALUB8 OK TKiaUNOIlKTHK AL KATI08. 19
«. Oive th« limits of variation of the tangent and of the M«uat of
anglua in the aecond quadrant.
(JlAkJL'^' **»"** ^ ^^'o oot ^ = VJ ^hen aeo A^ 1-
2 VI. •
8. Find all the functional valaaa of 46* without oiing the Table*.
aX9. Show that ooi 60* = 4, and deduce the remaining functional valoea.
to. Oirea tan -^ =^ , find A, and de<luce valaea for the renuuning
functions.
I 11. Given sin W-"^ — , find values for the remaining functions.
12. <Jiven that sin il and cos A are each equal to -6446, find A and
B from the Tables.
13. Find from the Tables the sine and the coaine of .17' 20*, deduce
therefrom the remaining funotioual values, and compare rtaults witii
the tabulate<l values.
14. What are the sine and cosine respectively of 32" 25' and 17* 12-6'?
'8. Find tan A aud secant A, whoa A =57" 22' 30".
16. Given sec ^4 = IM, find A from the Table of sines.
17. Show that Tables constructed for angles from 0* to 45* mav serve
for angles from 0" to 180*. ^
18. Find the angles whose sines are -6678, -3584, -8397.
SufjgeaUon —There will be two angles in each instance (Art. 12).
19. Find the angles whose cosines are -6439, -8390, 7805.
20. Find the sine of 78" 36' 6 ", and the cotangent of 10* 23' 30".
CHAPl'ER IV.
SOLUTION or PLANI TPTAIfOfJ
RECTION I.
l.xlrodnction. Right TriangUi.
17 Definition —Of the six parts of a fcrian^lo, any
three being given except the three angU'H, the rcHt may
U computed. Thi« computation is called Soliring the
triangle.
yoff.—pie methods of hoI^jiik 4II tri»nfl[lei, right And obIiqn«, are in
tho main bMed on the r«l»tion« <liiicuc^e«i in Uh»pt«r II,
18. Four Oenc sJ Oases.— It will be necessary to dis-
cuss four distinct cases, since the data may involve (a)
two angles and o.ie side; (b) two sides and an angle
opposite one of .hem; (c) two sides and the included
angle; (d) three sides.
iVbte.— In cue (»), four parte are known itince two angle* immediately
determine the third. Hence, when two ^naileii and a aide conatitute
th« daU, it la immaterial whether the side is adjacent to both, or to
only one of the angle*. _
10. Solution of Right Triangles.
— If the triangle is right, all the
cases of Art. 18 may be solved by
an immediate application oi i! e
formula,
Sin A (or C, fig. 7)=a(or c)-h6 (Art II),
except that in case (c) if the given
angle is B, the right angle, it will be
A^^
necessary to apply first, either
6= Ja*+c* ; or tan il =*
e
Nntf 1. — Th** f'^rmnhL ni%> A Inf />! — « /..,_ ^i_^ i. :ii
... - ■ •. •--*• •-• yji '^i .-V, vein
that given.
.ib^vw iae vrau «•
tOJ^UTON or PLANS THUNGLKS.
SI
\ V 2. -The matbod of io\ving right triiu<gi«a Ui baMd imnMdklaly
oit t.ti« il«rtnitu;na of Arte. 8, 9, anil 10 ; hut tpccial dainoiiaintioM
wiU b« iiMMMM-y for th« tolutiun uf th« Mveral omm when tha imn^
H oblique.
<S^.-~Bolvt Kxaroiaes 110 aI the end of thia chapter.
MOTION II.
SolutUm of Oblique TriangUt.
20. L&w of Bines— From the vertex H of \Se obliqii*
triangle ABC (Sg. 8),
«lraw th« p'Tpencl'^iijlar
fW\ then by articles 8
and 11. —
17)
Aa
ainy4.
•iu C
BC
AH-
BD
no'
Whence the important
theorem, that tfie aides A
of a triangle a "c pro-
portioned to the sines of
Fig. 8.
the angles which they s^cbtend. By means of this theorem
cases (a) and (6) may be solved
il?M|/.— Solve Exercues 10 and 11.
21. The Axnbiguoas Case. — In .solving case (J), in
which an angle must be determined from its sine, the
re.sult would appear to be ambiguous, since the same
value of the sine belongs to two supplementary an'^les;
but this ambiguity occurs only when the side opposite
the kmnun angle is less than the other known hide.
Thus, in fig. 8 the known parts being the sidep AB .md
BG and the angle A, if BG he less than AB, it is plain
that two constructions are possible, one making the
'^"o*'' ■"-■^-jjyVTSit.v, .ixsj usjj/usu, wic; oilier iiiaiviii^ i\i acube and
supplementary to the former. Otherwise only one con-
22
f'HACTlCAli MATHEMATI{\S.
struction is pospibl^s and of the two values obtainable
from the Table for the unknown angle, the smaller must
necessarily be taken.
<8'Mi7.— Solve Exercises 11-17.
22. Two Sides and Contained Angle— For the solu-
tion of oblique triangles when the data consist of two
sides and the contained angle— case (c)— the follow-
ing theorem has been demonstrated:— 2%e ratio of the
mm and difference of any two aides of a triangle is equal
to the ratio of the tangents of half the sum and half
the difference of the two angles adjacent to the third side.
if
Fig. 9.
Thus in the triangle ABC (fig. 9), the angle B being
included by the sides c and a —
c-a Hnl(G-A)'
Demonstration.— With radius BC, the shorter of the
two sides, describe a circle; produce AB to F; join FG,
GE ; and draw ED parallel to FG. Then—
BCE=Bi:G=i FBC=i(G+A) (Euc. I. 32),
^ i?Ci> (or BOA -B0£J) = G-1i (C+^) = i (G-A);
^^^ Ev^=tan FEG (E\ic. III. .•:i. anil Art, 9):-t£" i tn^ a\ -„''—
DF
^^=tan EGD (Euc. I. 29)=tan i {G-A),
SOLUTION OF PLANE TRIANOLEa
23
•■•^I'^t"^ M§r7)' *''*^' ^^ "=^»"*y of triangles (Euc. VL 4).
FC _AF_c + a
DE AE c-a
• c + ct_ tan ^ (G +A)
' ' c-a tan ^ {C-A)'
Sote 1.— Observe that C+ ^ = 180" - B; hence three of the quantities
in the equation just proven are known, if c, a, and B constitute the
data; the fourth, tan i (C-A), and hence i (0-A), can be found.
Xote2.-1iiC+A) + UC-A) = G; and ii (G+A)-^ (G-A)zzA.
Note 3. — To fiad the remaining side, apply the principle of Art. 20.
tVMgr.— Solve Exercises 18-20.
23. Another Formula. — A formula which is sometimes
preferable to that of Art. 22 in solving case (c) is thus
demonstrated: — In the triangle ABC (fig. 10), either acute
or obtuse, draw AL perpendicular to 3G.
A A
First. -AB^ = BG^ + CA^-2BGGD (Euc. II. 13); but C2)=i4C.co3
(Art. 8),
.*. c'=a''-i ')«-2a5.co8 G.
Second.- AB^=BC^ + CA^ + 2BG-GD (Euc. 11. 12); but C'Z)=^C.eo8
ACD= -AG.cosACB (Art. 12),
.*. c» = a'» + 6'-2a6.cos G.
Note 1. — By the aid of this formula, c can be found if a, b, and G
are known ; the remaining angles may be found by Art. 20.
Note 2, — It is obvious that the formula of this article is equally ser-
viceable wherx the three sides are given, but in the next article a formula
more commonly used for this case will be investigated.
iiy^flr.— Solve Exercises 21 and 22.
24. wnen the Three Sides are Knowii. — Describe a
circle (fig. 11) with AG, any part of a side of the angle
i
u
PRACTICAL MATHEMATICS.
A, as radius; draw CB perpendicular to the other side,
which must be produced to meet the circuinference in E;
join EC.
Fig. 11.
Tan ^ =tu> ^( Euc. III. 20) = ?,9 (Art. 9) = ^^ = ^^
2 ' EB^ '"^ EA+AB CA+AB
B\\tBC-AC%\nA,a.ndiABz=AC.coiA,
A AG. %in A sin A
tan .-=.-— ^
2 AC+AC.cobA 1 + cos^'
sin' A
1 - cos« A
tan'»"-=
2 (1+C08J)« (1 + cos^)
From Art. 23, cos ^ = *' "^ *^' " **'
-i^,(Chap.II.,Ex.6) = i^
1- coa A
oobA'
tan«4=
2
1-
2bc
26c '
26c-6«-c« + a»
I . ft' + c'-g" 26c+6» + c'»-a" ,
2fc
Factoring (Todhunter's Algebra, Art. 87),
tan* 4 ~ {a + c-b){a + b-c)
2 (a + 6 + c) {b + c-a)' .
Let 2a=a + * + c; then a + c-6 = 2(«-6); o-r 6-c=2(« - c)j and
6+c-o=2 (8-o).
.•.taa«^=*J-«zl)iiZ^
.-. t»n:l-
2
^ /(^-6)>-(
\/ «(«-«)
A^ote. -In applying this formula in case {d), it is advisable generally
of sines.
11
(ifrou.
SOLUTION OF PLANE TRIANGLES.
25
il.
EXERCISES.
1. What is mnant by solving a triangle? Explain the exception
in Art. 17.
2. Why is there not an additional case (Art. 18) when the data are
two angles and the side adjacent to both?
J 3. Find ell the unknown parts of the isosceles right triangle whose
(^^^^^^ hypotenuse is 6 s/2.
Sug. —By Geom. A - 45*, and o = c = 6.
. 4. The perpendicular sides of a right triangle are respectively 6 and
OUn^^jQjg . what are the other parts?
.Va^.— a = Vi08 = 6 V3, . •. C- 30°, etc.
Afru^ 5. In aright triangle a =400, -4 = 53°; find the other parts without
^^^ the Tables; sin 63°= 8.
(ijff^^JL 6- Given sin ^ = cos A, a=14, and A + 0=8; find all the unknown
'''^^iparts.
J , , , ^ 7. If sec A = — ^ find the ratios a : 6 : c, when B= 90".
Cl' ^ cos C^
8. Solve triangles, in each of which ^=90°, from the following addi-
tional data, using Tables when necessary:—
a=30, 6=40. JAy^-^"*—
^=40°, a=62. d<yviJ-
C7=52° 30', a=525. dU^'^^
a=625, 6=800. (ht^^UL-
a
CL&^y^.J^■
"=1,6=12.
c
??= -5,0=100. dj^^^JL
h
£_^ a=50.
a '
cify
Sec il = 1-41, and 6=6338.
.^yj^ 9. Solve the triangle in which sec A= ^ and 5=c=6.
-■V
10 Apply Art. 20 in solving the triangle of which the hypotenuse is
680, and one angle 65°; also that in which 5=90°, 6 = 700, and a =545.
<Sm^.— Sin90°=l.
A 11. Solve the triangle in which a = 95, c = SO, and A = 55° 30'.
^^^■*' 5u^._FromArt.20, «!"^=H, .'. sin (7=8in ^ «-=sin55°30'x|%tc.
^' sin -4 a a w<J
I 12. Find &Jl the other parts of triangles in which the data are respec-
^^^^^-^ively:—
6 = .300, C = 48° 20', and 5 = 85°;
a=240, (7-^36° 27', and 5=45° 35'.
13. When will case (6) afford an ambiguous result? Illustrate and
demonstrate geometrically.
14. How many of the unknown parts in the ambiguous case will have
double values? What is the maximum value of the given augle m tue
ambiguous case?
26
PRACTICAL MATHEMATICS.
15. fjin/l the double values when the data are:-^ =45- a-fiO &-flft
J7.SoWe the uoscele-tnangla whose vertical angles Ig-'^and whose
^^^'^^^n"":^^^^^ 22. when
aide m each case be found ? ' ^"""^ "^'^ *^® remaining
20 Ifj^'-^V'T"^^' ^" ^^^°^ ''=^8. c=46. and 5 60' 35'
.ngle hi.li'^'r'a^al^^^^^^ ^« perpendicular, the
ments of B and , ma/e ^^6^^ rLpect'^ly^''^'^ ''°°^ ^'' ^'^ *^« "««•
to;uitt^JU: *5:4tetf^^^^^^^^ ^T^» °f Art. 23. modiBed
22. How may the formurof A^ o Jl " . J T? '>''^' diff«'-«nces.
data are a, b, and c. ^^- ^^ ^® modified to find ^ when the
2a Write the equivalents of tan ^ and tan ^ r«,n«nf;- i • x
2 o» respectively, m terms
of o, b, and c. " £
24. Apply both methods 'in finding the angles when-
(1) a=5, 6=6, c = 7.
(2) a=525, 6=450, c = 300.
(3) o=6=45, c=10.
(4) a=3, =4, c=5.
ii
CHAPTER V.
LOGARITHMS.
Section 1.— Nature and Properties of Logarithms.
26. Definition.— Assuming a'^b,x is the exponent of
that power of a which equals b, and as such receives the
name of the logarithm ofh to the base a; more briefly
»=loga 6; or loga b=:x.
Note.— A series of values for x corresponding to consecutive values
of 6 constitutes a system of logarithms, or table of logarithms.
26. Different Logarithms of the same Number-
Since a may have any value whatever in the expression
log. b = x,it would appear that any number may form
the base of a system of logarithms (see Note 5); and that
the logarithm oi- a given number will vary with the base.
Illustration.— log^ 64 = 6 ;
log4 64 = 3;
log8 64 = 2;
log4=-6;
log4=-3;
logs 6^ =-2.
Note 1. — The logarithm of a number which is an exact power of the
base, is a positive integer, while that of its reciprocal is an equal nega-
tive integer.
Note 2. — If the number lies between two exact powers of the base or
their reciprocals, the logarithm lies between two consecutive integers,
either positive or negative.
Illustration.-S* 7 68 7 8» .*. log- 68 lies between 2 and 3:
1111
also — 7 A 7 A- . •. logg
8« 68 8»
58 ^ --S between - 2 and - 3
68
1
that is, logg 68 =2 -fa fraction, and log^ - = -3 + a fraction.
Note 3.— The integral part of a logarithm is called the characteristic,
and the fractional part, which can be only approximately found, and
lience is conveniently expressed as an interminate decimal, is called the
viantisBU. *
Algeljra.
For the method of finding approximate mantissas, see Advanced
28
PRACTICAL MATHEMATICa
over
*»^e; ^ogf,^ = - 8 + a fraction might have been ezprcMed log, J- = - 2 -
a fraction, but the former is the mode adopted. ^
logS;^^* " *''^'"' ***** * ^°°* '«'•'" *^« base of a .y.tem of
SKUTION II.
Common Logarithms.
27. Common or Denary System— The base of ihv
coiumon system of logarithms is 10, which is usually
omitted m expressing the logarithms of numbers.
Thus by tog 100 = 2 is meant logu> 100 = 2.
18 2^m2q'lftls^'!fn^*' • *^^ ^'»Pi?"an System, so called from its inventor
la ^ 718^81828, and is generally denoted bv e Thin «v.flL Ik i!
.iniK)rUnt theoretically.^^ foJno furthVcUide^rtSoi^uSs^b^^^^
28. Properties of Common Logarithjis.— The follow-
ing properties of logarithms in the common system are
due to the decimal nature of the Arabic notation of
numbers : —
(1.) Every integral or mixed number whose integral
part consists of n digits, lies between lO""' and
10", and hence the characteristic of its loga-
rithm is w - 1. ^
(2.) Every remove of the decimal point to the right
or to the left, increases or diminishes the log-
arithmic characteristic of the number by a
unit, since such r move either multiplies or
divides the number by 10.
(3.) In a decimal fraction, the position of the ifirst
significant digit, in reference to the decimal
point, determines the characteristic of its
logarithm.
iZ^us^ra^io/i.— If the mantissa of log 4356 = -639 then
log 4356 = 3-639 , . - ' '
log 435-6 = 2-639
log 43-56 = 1-639
log 4-356 =
lo^ '4356 = r+ -639 = 1-639
log -04356 = 2+ -639 = 2~639, etc.
LOGARITHMS.
S9
29. Logarithma from the Tables.— r/t/Tt! DigltH.-Axi
a table of logarithms constructed tor numbers from 1 to
1000 (Table II.), the mantissa for any number consisting
of three digits may be found in the column numbered
above by its right hand digit, and ttorizontally oppotdte
the first two figures of the number as found in the column
marked iV above.
Illustration.— l^og 467 = 2-6698.
Four or more Digits. — When the antilogarithm, that
is, the number whose logarithm is sought, consists of more
than three digits (e.g., 4676), its logarithm is found
from the table by the method of proportional parts
illustrated in Art. 16.
Illustration. — Required log 4676.
Log 46-70 = log 46*7 = 1 6693 ; similarly
log 46«C = log 46-3 = 1-6702.
Hence the logarithmic difference corresponding to the
difference of yu ( = tiftt) between these two numbers is
•0009. Assuming then, that the difference betv/een the
logarithms of two numbers, differing from each oth jr by
only a small fraction of either, is proportional to the dif-
erence between the numbers : —
Log 46-76 = log 46-70+T% X XKWO = 1 6693+ 0005 = 16698.
Note. — The method of finding the antilogarithm, being tlie reverse of
that just illustrated . needs no explanation.
a-
SECTION III,
Utilitif of Logarithms.
30. Algebraic Principles- — Assume a*
' = c; from the laws of exponents, —
log„ be =-x + p (Art. 25)
h, and
he
.«+»
a, . .
c
h ■= a
«
loga 6"
fix
v^
= a"
X
'. log„7f- = -.
'.rf
SO
PRACTICilL MATHEMATICS.
I
31. tofirarithms of Products and Quotients—The re-
siUts ot Art. liO are embodied in the following atatements:-
(1.) 7//« logarithm of a product is the mm of die
logarithms of its factors; and as a result, the.
logarithm of a power of a number is the
logarithm of the number multivlied by the
exponent of the power.
(2.) The logarithm of a quotient is the difference
between the logarithm of the dividend and thit
of the divisor; hence, also, the logarithm of a
root of a number is the hgaHthm of the
number divided by the index of the root
JVb;« l.—Thus, by the appropriate use of locarithms thb reanltn nf
.£^^^ 2.— Instead o( subtracting a logarithm, its co-loffarithm. i f th.
of «?f °?!?K»"'!"n,'m>y be reaJily written down by Bubtraotinir osoli
-SuflT.— Solve Exercises 1'12.
SECTION IV.
Logarithmic Functicma.
32. Tables of Logarithms of Angular Functions—
bmce the values of circular functions usually involve
four or more places of decimals, it is best in operations
involving these values as factors (Chapter IV.) to employ
logarithms. The logarithms of the natural functions of
angles from 0' to 90° have been tabulated, therefore
constituting a^afe^e of Logarithmic Functions (Table III )'
''^^ntZt'^^r-'' ^ *'^ -^ ^' '^^ tabAe^tX^o^^
Note 2. — Thn iiifit.hnd nt fii.^'^/* *i._ i -a.^. • .
and that of l^nain, ^^^^o^^^^SuSn^^i^^SS ^^.^S^i
LOGAUlTHMa
81
being limiUr to the methods already doiioribod for ocing the other
tables, needi no farther explanation.
Note 3, — Instead of Bubtraotins the lop;arithm of a fanotion we may
add the logarithm of its ruciprocid ( ^ rt. 31 (2), and Chap. IL Ex. 1).
£X KRC«.S]SS«
1 . Define logarithm.
2. What is the logarithm of »oe base in any system t
3. Of what number is the logarithm the rame in all systems ? What
is its logarithm ?
4. Of what numbers U 6 the logarithm to the bases 2, 3, 4, 6 re«neo<
tirely ?
Sug. 2» = 32.-. logj32=6.
6. What are the respective bases of systemji in which log 2*25 = 2,
log 64 = -1, log ^ = 2, logi = -2, log 6 = 2, log 2 = 2. log 2 =3?
1\-1.
64
log^64 = -l.
6. Name three numburs whose logarithms to the base a are integral
numbers ; three numbers whose logarithms are mixed numbers greater
than 4; three numbers whose logarithms are proper fractions.
7. Distinguish between manttsaa and characteristic. Is it necessary,
^ or only convenient, to consider the mantissa as always positive ? Why ?
*^™ Illustrate.
S. What are the logarithms of 16 to the bases 5, j, ^ . respectively ?
Generalize. 2 4 10
Sug. (ly^= 16, etc.
9. Show that in the common system the number of units in the ohar-
acteristio of the logarithm of a number is always one leas than the
number of digits in the number. Illustrate.
10. What are the logarithms of '1, "Ol, "OOl respectively, to the base
10?
11. Show that in the common system the characteristic of the loga-
rithm of a decimal fraction % U be - {n+ 1) if, in the fraction, n ciphers
precede the first significant digit.
12. Given log 2 = -30103, and log 3 = '47712, find without tables the
logarithms of the following numbers :— 8, \'Q, '27, .5^, -081, 5, 6, U
310
•06. ^ 36 -064. (i;:)*.
*f
PRACTICAL MATHKMATICa.
Sug,. rx)g i^ = log J2 . log (I6x i) = log 2* +log * = WIOS x 4 ^
(-1)= -20412.
Log 6 = log ^^, etc.
18. What obvious advantages has the common system OTor all othera f
14. Find from Table 11. the logarithms of 7606, 35-49, •000«542:
also the antdoganthms of 3-4567, 1-0999. '
^^mx'i^m^fJei^'^ *"**'*'* (3+ -4667) = mntUog 3 x aatUog -4567
15. Perform by means of logarithms the operations indicated i—
4569 X Wir,, 00.36 x 689-6 :
76-08-7-69, -9897 -^-9011 J
V6869*, 864* -706*.
oJ QA'^'"'\ ^^u".J*^^". "^- **»« logarithmic sines of angles denoted by
-» .W, and 78 25 ; also the logarithmic uines of thoir complements
and their supplements.
17. Find log tan 13° 10', and log cot 48 50'.
18. Find log sec 65°, log cosec 67^ 40', log tan 25° 38' 30 '.
>Sug. Sec 66°
1
cos 55^ • ■ ^^^ '^'^ ^'^" ^ ^°« ^ "^^^ "^^ ''^•'''°' •^^ ^® *"
result (Art. 32, Note I).
19. Test the correctness of the logarithmic functions of the animlar
magnitudes 36°, 65°, 75°, as given in Table III, by the aid of Tables I.
and 11.
• "^n ^tl,"^® nurnKv-s in Table III. are but the logarithms of values given
in Table I. with 10 added. *
20. Of what angles ia 1-9235 the actual logarithmic sine, cosine,
tangent, and cotangent respectively ?
21. Why, in the tables of logarithmic functions, is a single difference
column sufficient for tangent and cotangent, while sines and cosines
require one for each ?
22. Why is it impossible to obtain from the Tables accurately the
natural or logarithmic sines of very small angles ? Is there the same
diHiculty m regard to the cosines of small angles?
23. Solve by the aid of logarithms the triangles in which—
(1) 5 = 90°, i4 = 47° 30', and a = 5689.
(2) B = 90% a = 466, and 6 = 784 -5.
(3) a --- 100, 6 =600, c =781.
(4) a =. 2-56, 6 = 74 78, ^ = 35° 30'.
(6) A = 5^° 25' 30 ", B = 27° 28' 17", c
(6) a = 630-6, b = 4283, c = 396-5.
236.
CHAPTER VI.
HEionra and distances computed by the aid op
TKIOONOMETRY.
33. Dehaitions— Tho application of the principles
explained in the preceding chapters in the computa-
tion of distances, heights, and directions, presupposes
the ntcossary data to be furnished by actual measurc-
niunt.
The distance between two remote objects is said to
mibtend an angle at the observer's eye. This angle is a
vertical or a horizontal angle, according as it is subtended
by the vertical or by the horizontal distance between the
objects.
The acute angle included between the oblique line
connecting two positions of ^ ^
different altitudes, and the
horizontal through either of
them, is called the angle of
elevation of the higher, and
the angle of depression of the
lower, thus : —
DAB (fig. 12) is the angle of elevation of ^ aa seen from A; and
CBA (= DAB) is the angle of depression of .^ as seen from B.
34. Vertical Distances- — If the elevation of a distant
point above an observer's level is required, any horizontal
base line may be measured; li'^ewise, the angled made
with the base by lines drawn from its extremities to
the foot of the vertical through the distant point ; and
lastly, the angle of elevation of that point froir one
extremity of the base.
12.
I
f
34
PRACnOAL MATUEMATICB.
one
TbMr^If Z) (Bg. IS) is the dcTaUd point, and O the obMrrer"!
/) poflition, the neceM'^ry <UtA Mr*
the bM« HC, and th« Mtaltt
ABO,AaB,mAABD.
In the horizontnl trtAngle A BC,
AB ctMlte found (Art. '20); theu
in the vortical triaugle A BD, A D
ia ubtainablo (Art. 10).
Note 1.— If the iMwa line cm
be meaaured in the aame plane
with tho vertical, that ia, directly
towarda or away from its foo^
the date neoMaary are the base
-, ,- - l*n« •nd the angles of elevation
*^fi' »'• from ita extremities.
If the foot of tho vertic*' is aocessible, the problem is a still simpler
n»^?l*?;~'^^**®'«^^ ?'•,*"*• fl*«^ff. •*«•. m*y ^ found by com-
CI with that of an upright atick whose length ia
35. Horizontal Distances. - In compufcing the hori-
zontal distance between two inaccessible points, the
necessary data are,~a measured base, and the two
angles formed at each extremity of the base by lines
drawn from them to both the inaccessible points.
Fig. 14.
Thus, let P, <;> (fig. 14) be the two points, and XY the measured
tnanglesPAr and OXY. resDeciivtlv. f^an ha co^^"*^-* -_3^r^.,„
PQ in the triangle fXQ. ' '" ' ~" '"'*'"•"-'' ="-» "ajwi/,
BEIOHTH ANli DI8TA3fCS&
85
Kote. Th« problan ia dmpU if on« of the pointt, P, b tcoowibU
from X, •ud the other, Q, viaibU from both I* md X (fig, U).
86. Application of the Geometry of Oirclei— There
are many trigonometrical probloma whose solution in-
volvijM a recognition of the properties of circlea
Thus, it is frequently necessary to find the distance of
a station of observation from three points whose dis-
tances from each other, and the horizontal angles at the
observer's position, are known ; as, for instance, refiuired
the distance of a ship at sea from
each of three heatilands whose
distances from each other are
laid down upon a chart, and
whose directions from the ship
are observed.
In fig. 15, AB, BC, CA denote the
known di«tancea; the anfflea CAD and
DC A are constructed eciual, respectively,
to the angles under which the distances
d.'tioted by iiC and AB i^re observed, a
circle is 'escribed about ADO, BD is
produced to meet the circumference in
A', and A E and CE are joined.
E denotes «he o'tserver's position,
iiince the angles A 4'Z) and ACD are F'«- l^*
ciiual, as also &rifCED muI CAD (Euc. III. 21)
Then AD ot the triangle ADC n
BA C of the triangle A BC ; whence
in the triangle
for^j; BE, a
triangle A CE.
triangle A
in the triangle BAD ; the next step . „ .^
for AE, BE, and the angle BAE ; lastly, Ci- i» obtainea by solving the
*/»
37. Ratio of rotameter and Circnmference of a
Circle.— Many compui^auions of lengths, both those which
involve the trigonometrical methods and those which do
not, require that either the circumference or the diameter
of a circle shall be expressed in terms of the other, ^o
do this exactly can be shown to be impossible, that is, the
two quantities are incommensurable, but in various ways
n-c xai^iu uui»wc\/il ineili may Do luUnCi 10 WUfilil U'fiy
required degree of approximation.
96
PKACTICAL MATHEMATICS.
One of the siinple.st of these methods is *o compute by
trigonometry, if necessary, the perimeter of a regular
inscribed polygon in terms of the radius ; then, consider-
ing this polygon as the tirso of a series of regular inscribed
polygons, each of which has twice as many sides as that
which precedes it in the series, to compute the values of
the successive perimeters by the data furnished at each
step. And since the circumference is the limit of such
inscribed perimeters, it is evident that the successive
values, thus found, more and more nearly express the
ratio of the circumference to the radius.
In this way the following results have been obtained,
regarding the radius as 1 : —
No. of Sides in In-
scribed Polygon.
Perimeter.
6
6-
12
6-211658
24
6-2()o257
48
6 -278700
98
6-282066
192
6-282905
384
6-28.3115
768
6-283163
I sue
6-283181
3072
6 283183
A comparison of the last two results shows that the
ratio sought has been obtained true to five decimal places;
that is, —
Circum. = diam. x 3-14159+ {see Ex. 20.)
This rallo is generally denoted by ir ; hence the formula
Circum. =27rr.
Note 1.— From this formula it is easy to find the length of a circular
arc approximately, in terms of the radius, having given either the
number of degrees m the arc, or the length of the chord subtending it.
Note 2.— For many purposes it is sufficient to regard ir as equal
to ot.
HEIGHTS AND DISTAKCES.
37
EXERCISES.
1. A horizontal angle is measured by an arc of the horizor., while a
vertical angle is measured by an arc of a celestial meridian. Explain
this statement
Sug.— An observer's celestial meridian is in the same plane as his
terrestrial meridian.
2. Distinguish between antfle of elevation and angle oj depression.
3. Devise some simple means of measurine horizontal and vertical
angles.
4. The jhadow cast by a tree is 40 feet long, and that of an upright
stiiik 3 feet long is 2 feet 6 inches; how tall is the tree? what is the
altitude of the sun? and what will be the length of the shadow when
tlie altitude of the sun is only 10" 40' ?
^ 5. From the foot of a tree standing on a horizontal plane, a straight
line is measured to a distance of 155 feet, and the angle of elevation of
its toj) found to be 34°, Required the height of the tree.
6. The angle of elevation of the top of a hill from the foot of its slope
is 45" 50'; while, at a point 500 feet horizontally distant from the foot,
the angle of elevation is 24° 30'. What is the vertical height of the hill?
7. A tower stands on a hill, the angle of elevation of which from one
point of observation is 40° 30', that of the tower being 59° 50' ; from a
second point of observation 275 feet horizontally more remote, the
angle of elevation of the top of the tower is 26° 25'. Required the
height of the tower.
8. Fifty feet from the foot of a tower, situated on the summit of an
incline, the angle subtended by its height was 52° ; seventy-five feet
further down the incline the angle subtended was 31° 40'. Required
the height of the tower.
6'ug.—(l) Sin 20° 20' : sin 31° 40' : : 75 : ?; (2) apply Art. 22.
9. From the foot of a mountain the ground slopes away at an angle
of I(F with the horizon. At a certain position on the slope the vertical
angle subtended by the mountain is 50° 30'; 200 yard? rurther up the
(slope the subtended angle is 55°. What is the vertica I distance of the
mountain top above the first station ?
10. From the north end of a chuich 60 feet long and 40 feet high, a
tower rises to the height of 55 feet from the ground. How tar must an
observer, whose eye is 5 feet 6 inches from the ground, stand from
the basement on the south end, in order that he may just see the top of
the tower ?
11. A lighthouse is standing on the summit of a precipitous cliff; the
keeper sees a distant ship in line with a buoy which is moored i of a mile
from the foot of the cliff, the angle of depression of the ship and of the
buoy being 20° 30', and 50° 30', respectively. How far away is the ship?
12. From a point beyond the end of a mountain spur, both extremities
of a proposed tunnel through it are visible, subtending an angle of 69° .W;
the distance to one point of emerireiice is l.\ miles, to the othf^r- l n-Ue,'
How long will the tunnel hi ? " '
88
PRACTICAL MATHEMATICS.
13. An observer on the hank of a river wishinff to juinonf.;« uu
distance from an objec# on the opposite hZ] sfght tL^bS^d also
«^oJl kT ^"^ ^r "'^" H^ ^^^ "^«'- 75 yards further dowCfindSitte
2?aii K.* *'"',9^ "•^e of *he nver he obser^-es, simUarly. that the
5Mflr.— Compare Art. 35.
««?«;„ ^^®?°''® ^^^^ ? circular race-course is 1 ^ milo* lone. Three
''^y^^f *^ ** """^ selected, such that the angle under which the aeronS
Tr^t'^nAfXtrt ?r '^' ^."^ ^' ''^' 37'. iSid that 8u£tend^' bythe
toJces of th«i5nw '"'°"^' " ^*f 30'. Required the shortest dis
XZ porttsTthe'feTce?'^ '^"°*'"' '"^ *^^ '^^^^^ '' *^« -*«-
«t,fej:^Vov^'''!^^"^ proportional to the circumferential angles they
Sug.— Find the angles ^, B, and C
17. Lines joining three objects are respectively 125-6. 130-4 and 112
Sl?ro'f/r'''i' '""^ ^l^ ^:S"*'°^ °"*"^« ^f th« triangle ?hefi^? 1*1
tte last of these lines subtend angles which are respectively 48» 58' ^d
objecte. "^""'""^ *^' ^'--"^'^ ^'^ *^^ «*"*^^^ from each ^of the th^e
iS'Mgf.— Apply Art. 36.
« i^ J« ^®'f^* "^i.^^^/v '5.^^ ^"°^^«' »^^ *^e chord of half the arc
w 4 feet 6 mches. Fmd the diameter of the circle.
^ugr. —Construct diagram. From similar triangles —
Diameter = (4-6)2 ~ 1-25. ' ^
19. 'The chord of an arc is 20 feet, and its height, 4 feet. Find the
cir^cumference of the circle and the length of the arc subtendeHy tue
Au^^'~^^^ piameter=4+I0»^4 by similar triangles: (2) arc • t x
diam. ; : angle at centre : 360'. * , \^/ am . r x
20. Find for circle whose radius is 8, the perimeters of a series of in
s;,el;et"''^'"''^ "^'" ^'''''"^ '' 6,T2rrd"i? sidS;
CHAPTER VII.
NAVIGATION.
38. Elementary Oonceptiona. — In addition to the
principles of trigonoiiietry, the computation of distances
and directions at sea involves the application of mathe-
matical principles based on the fact of the earth's
virtual sphericity.
The fundamental elements of less difficult problems in
Navigation are known technically as : —
Distance — Expressed in nauticd miles, i.e. miles of
60V 6 ^eet each; the track of the ship being called a
Rhumb Line.
Course— The angle made by the rhumb line with a
meridian, as indicated by the compass.
Difference of Latitude and Difference of Longitude.—
Respectively the differences between
the latitude and longitude left and
the latitude and longitude arrived
at.
Any two of these elements may be
the data from which the remaining
two are to be found.
Let PE and PQ (fig. 16) denote meridians
which intercept EQ, a A, Bh, and AB, area re-
spectively of the equator, two parallels, and a
rnumb line; then EQ denotes difference of
l.i^gitude, aB or Ah difference of latitude,
and the angle hAB the course. pj»^ jg^
39. Auxiliajpy Conceptions. — To facilitate computa-
tion, certain subsidiary elements are employed : —
Meridian Distance. — The number of nautical miles
40
PRACTICAL MATHEMATICS.
between two given meridians at a given latitude.
It is greatest at the equator, diminishing gradu-
ally as the poles are approached (ficr, 16).
Departure. — The number ot* miles easting or westing
actually made in sailing from one meridian to
another. Since the meridian distances at the
initial and final latitudes are unequal when the
course is obliqr.e, the departure in that case is
equal to the mean meridian distance, which is not
found, however, on the parallel of the mean lati-
tude, but a little nearer the poles. Thus, departui-e
may be denoted by dp (fig. 10), more than half-
way from ^a to bB.
Meridional Difference of Latitude.— A minute of lati-
tude, which is everywhere the same, and a minute of
longitude at the equator are equal, each
being a nautical mile in length. Hence,
if two meridians are a mile apart at the
equator, because of their convergence
the meridian distance diifers more and
more from a minute of latitude as the
distance from the equator increases. In
other words — relatively to the corre-
sponding meridian distance, a minute
of latitude becomes greater and greater
the higher its latitude. From this it
follows that, if. to represent two con-
secutive meridians, two vertical straight
lines be drawn (fig. 17), thus making the
meridian distances at all latitudes the
same, then the distances between the
^'
2'
J'
\ ^
I ^
B
i ^
1 ^
C
\ b
B
[..-(?.
A
i
xi.\ji.i.£t\jixua,i.
iinca
Q
Fig. 17.
^ - intersecnng
them, designed to represent consecutive
minutes of latitude, must be made sue-
NAVIGATION.
41
cessively greater antl greater, in order that, at any given
latitude, the <rue proportion between a minute of latitude
and the corresponc^ing meridian distance may be preserved.
Thus— Of the successive increments, Aa, xb, yc, zd, etc. (fig. 17), each
after the first is greater than the preceding one ; Qa, Qb, Qc, Qd, etc.,
are called meridional parts for the latitudes 1', 2', .3', etc.; and ae, ad,
bd, etc., are meridional differences of latitude corresponding to the true
differences of latitude, AG, AD, BD, etc.
Note. — Maps and cha^-ts of portions of the earth's surface executed
upon t'lis principle are said to be drawn on Mercator's Projection, so
called frorh its inventor.
40. Representations by Mercator's Chart — It is
evident that if a rhumb line be projected upon a Merca-
tor's chart it will be longer than if drawn to the same
scale on a sphere, in the ratio of the meridional difference
of latitude to the true difference of latitude, and each of
the meridian distances in the former case, being equal to
the difference of longitude, will exceed in the same ratio the
mean meridian distance — that is, the departure — in the
latter.
It follows, then, that the lines in such a chart denoting
respectively meridional difference of latitude, difference
of longitude, and rhumb, are to one another as true
difference of latitude, departure, and true distance.
41. Expression for Difference of Longitude. — Let P
(fig. 18) represent a pole of the earth,
and PA and PB meridians intercepting
AB and ab, arcs of the equator and the
parallel of latitude x° respectively. Draw
the radii OA, OB, OP; draw also aD
and hD, making the sector aDb similar
to the sector AOB.
Then «-^- = «^ = "^
AB AO aO
= cos DaO = cos aOA ;
But aOA -
dist. at lat. cc°
a;", AB = diff. Ion, and ah — mer.
diff. Ion. =
mer. dist. at lat. x"
cos lat. x"
Fig. 18.
= mer. dist. at lat. x° x 8ed> lat. x\
42
PRACTICAL MATHEMATICS.
Hi
42. E^pre.*^8ion for Meridional Parts— Since the dif-
ference of longitude exceeds the meridian distance at
any latitude m tne ratio of the secant of that latitude
(Art. 41). It follows that. if. as in fi;;. 17. the meridian dis-
tances are all made equal to the difference of londtude
ea.h exceeds the distance it represents in the ratio of the
secant of its latitude, which is therefore the ratio in
which eaxjh corresponding minute division of the meridian
must be mcrea^sed to preserve the proportion (Art 39)
In other words, since the true meridian distance at lat V
ha^ been multiplied by sec. 1'. arid the meridian distance
at lat. 2 ha^ been multiplied by sec. 2'. etc., hence in
representing latitude, Qa - 1' x sec. r, a6 - 1 x sec 2'
6c = 1 X sec. 3', etc. and consequently the distance of
the parallel of latitude a:" from the equator is equal to 1' x
(sec 1 + sec. 2' + etc. ... + sec. x'), that is --
Mer part, for lat. x" = sec. V + sec. 2' + etc. . . .'+ sec. x\
4d£n~af ^aft^^rX^^^^ tables of
43. Tngonometrical Relations of Departure, Difference
^ of Latitude, Distance, and Course.
—•The lines on a Mercator's Chart
which denote respectively a rhumb
line and the corresponding meridio-
nal difference of latitude and differ-
ence of longitude form a right tri-
angle, in which the course is denoted
by the acute angle adjacent to the
meridian line; so, also, may the cor-
^.nof ?• \ responding parts of a similar triangle
denote respectively, actual distance, difference of lati-
tude, departure, and course (Art. 40).
Let ACB (%. 19) be the triangle from a Mercator's
s
NAVIGATION.
48
Chart of which AC and CB and the angle A denote
njspectively me,, diff. lat. diff. Ion., and c^mvse; then if
AD denotes true aifference of latitude, DE and AE will
be the departure and distance respectively; and the
solution of the triangle ADE, will give any two of the
four elements under consideration, that is, diff. lat, den
diM and course, the other two being furnished as data! '
44. Trigronometrical Relations Demonstrated.— That
a plane nght triangle may represent the mutual relations
of course, departure, difference of latitude, and distance
can be directly proven : —
Let AB (fig. 20) be a rhumb line cut by meridians
into portions, so small that each may be
considered as a straight line; correspond-
ing to these minute rhumbs Ac, cd, etc.,
hB, let hc,ed, etc., jB, at right angles to the'
consecutive meridians, denote the minute
departures, the aggregate of which con-
stitutes the total departure; and Ab, ce,
etc., hj, the minute differences of latitude,'
whose aggregate makes up the whole
difference of latitude; then since the ^-e- ^».
triangles are all similar, one angle in each being equal
to the course, —
Ab:bc:Ac::ce:ed:cd:: etc. : : hj :jB : hB,
.'.Ab:hc '.Ac:'.Ab^ce-\-ctc. +hj: ',j + ed+etc. +JB :Ac + cd + etc.+hB.
Hence AbibcAc:: diff. lat. : dep. : dist.
^ But the elementary triangle Abe is similar to any plane
nght triangle, one of whose acute angles equals the course;
hence the relations affirmed in Art. 43 are established.
/f«S \Z^}^ T*^°1 °^ computation, thus suggested, is called Plane
baiUng, because it involves the properties of a pl^e triangle.
rdSL7tL^k^!.±^l7?i!?.^^^^^^^ K^^Jo detennine the
Note a.-A table, in which are recorded the departures and differences
Fig. 20.
44
PRACl'ICAL MATHEMATICa
of Utitude, computed for coniocutive connea And diiUnoes. iu c»Iled »
Traver$e Tablt (««« TabU V.).
Note 4.— It ii uieful to know the departure, chiefly as an aid in det. r-
mining the difference of longitude, as ahown in the next article,
^u^.— Solve Exerciaea 1-23.
45. Trigonometrical Relations of Difference of Longi-
tude, Meridional Difference of Latitude, and Course.-
At aoiiie latitude intermediate between the initial and
^ final latitudes the meridian distance
must equal the departure (Art 39);
denote this latitude by z, then, —
diff. Ion. =dep. x aec. z (Art. 41)
If, then, the parts of the triangle
ADE (fig. 21) denote departure, etc.
W in Art. 43, and AD be produced till
AG =^ AD ^ sec. z, the side (75 will de-
note difference of longitude, and At\
meridional difference of latitude, as
Fig. 21.
in Art. 43. Hence from the similar triangles —
Diff. Ion. _ mer. diff. lat.
dep. diff, lat. >
Diff. Ion. = mer. diff. lat. x
dep.
diff; lat.
= mer. diff. lat. x tan. course.
This method of findinc the difference of longitude is called
because the relations involved are those of lines on
Notel
Mercator'a SaiUmj,
a Mercator'a chart.
Note 2.— If the course is on a parallel, the difference of latitude is
nothing, and the meridian distance is equal to the distance sailed
Hence (Art. 41) diff. Ion. rrdist. x sec. lat.
In such cases the solution is said to be by Parallel Sailing.
Note 3.— The methods thus far discussed are sufficient for the solution
of all problems relating to simple courses.
Note 4.— In low latitudes, and when the course is greater than 45°
the intermediate latitude at which the meridian distance is ecual to the
departure (Art. 39) may be assumed, without serious error, 'to be the
mean of the initial and final latitudes. On this assumption—
Diff. Ion. = dep. x sec. mid. lat.
Longitude found by this method is said to be found by Middle Latitude
oailtng.
Sug. —Hoive Exercises 24-37.
NAVIOATION.
i6
46. Resolution of a Traverse— In all but exceptional
voyages a ship's course is a compound one, of which the
various simple courses and distances are recorded in the
log book.
When such a compound course or traverse is to be
resolved, it is necessary to ccipute, or to find directly
from the Traverse Table, the departures and differences
of lacitude for the several courses. The algebraic sums
of these will be, respectively, the net departure and differ-
ence of latitude for the traverse.
The distance in direct course, and the difference of
longitude, can then be found as if the course were a
simple one (Arts. 43 and 45).
N^ote 1. — The reaolution of a traverse is called Traverse Sailing.
Note 2. — The first course on a traverse is usually called Departure
Course. It is simply the bearing and distance of ;?me headland, light-
house, etc., from the ship, observed just before the land is lost sight of.
Tliis bearing ust be reversed, of course, in the computation.
47. Variation of the Compass.— As the magnetic and
geographical poles are not coincident, it is only in certain
longitudes that the needle points directly towards the
North. The readings ^^^ the compass in all other longi-
tudes must therefore be corrected for vacation, which is
either easterly or westerly according as the north end of
the needle is deflected towards the east or towards the
west.
Note. — The amount of variation in any place may be known by con-
sulting a chart, or by comparing the direction of the needle with that
of the sun at noon.
4S, Deviation. — The influence of the beams and other
iron in a ship upon the needle occasions an additional
error in the compass readings. The amount of this error,
which is called Deviation, varies with the direction in
which the ship is heading, being least, generally, when
„ ._„,,,.._ I. j._,j TTXVil fiiU XtlCSlTiiCUiV; l.ll'Ct.l.'Liiaill. A
record of the errors for all directions of a ship's head,
f
46
PRACTICAL MATHEMATICS.
Obtained by sxHnging the ship, as it is called, constitutes
a demahon table for that ship. "•-"uwa
JVo<<!--The drcuiTnUnce* of each cms will #l«fAm,;„» i *i- ..
49. Leeway.-Un.ler the infiuence of the wind the
8h.p H course usually lies either nearer to the mcridi^, or
farther from it, than the compos indicates, and allowance
tor Leeway must be made accordingly.
hJ^: °''"*f."''° °V'V' I-osr-Generally, the navigator
bases his estimate of the distance run by his ship upon
cated by the log glass, and the number of feet in a knot
ot the log hne, are supposed to be in the ratio of ^ m
that the number of knots reeled off while the sS' i,
running out may indicate directly the rate of the sh"p i
miles per hour. But. owing to an error in the glass or
the knot, or both, the ratio between them is someT^L
m excess, and sometimes in defect of this; con :equently
the estimated rate and with it the estimated distance
Z'»r ''^ P^PO't'on^tely in excess ordefectof the true
distance The necessary correction, however, muv be
St " PllVoni.n,-.aetual ratio of '.laJa^l
hnot ^to^^as the estimated distance is to the true
diManee.
JVb^e—Sometimes, for the ratio?*'?? ia Bn)>rf;h,f<.^ S . j
jg(l0 """(joys' "'""'"""'«<' g. and sometimes,
«08d'
NAVIQATION.
EXERCISES.
1. Dofino lihutnb Line, Covr»e, Dintanrf, and Dead Xeehnhff,
2. A rhumb-line it a vpinl curve. Explain.
8. What is the differeuce jf latitude in milet
(1) When lat A = 46" 20' N., and lat B = l?" 30' V.?
(2) When lat A =: 4fi' 20' 8., and lat fl = 17* 30" N.t
4. What ie the difference of longitude in miles—
(1) When Ion A ._ 73* 17' 30" E., and Ion fl = 62* 18' E.?
(2) When Ion A = 28' 18' 30" E., and Ion // = li;' W.?
(3) When Ion A = 120* E., and Ion ^ ^ 96* W. ?
5. Why ii it necessary to specify the latitude vrhen allusion ia made
to meridian diat*aoe ?
6. What is the difference betwen meridian distance at the equator
and difference of longitude ?
7. Under what circumstances are distance, departure, and differenoe
of longitude the same ?
8. Under what circumstances are meridian distance and deoarture
the same? *
^•. )^^y " n-'fc the riean meridian distance, i.e., the departure, the
meridian distance as measured on the parallel of mean latitude ? Would
it be so if the earth's surface were a plane ?
10. Why does the length of a degree of longitude vary in different
latitudes, while that of a degroe of latitude is practically constant?
11. Is the degree of longitude at 'latitude 45° more or less than half
as long as a degree of longitude at the e(juator ? Why ?
12. In the construction of a Meroator's chart the eartl is conceived
of as being a cylinder. Explain.
13. Explain why, in such a chart, consecvtive latitude lines in higher
latitudes muf t be farther apart than in lower latitudes.
14. Describe in detail Mercator's projection.
15. What obvious objection is the:e to the use of school maps drawn
on Mercator's projection ?
Sug. — Exaggeration of high l^^itude areas.
16. Define Meridional Parts, and Meridional Diferenee of Latitude.
17. Derive the formula dif. lot , = mer. dist. x sec. lat.
18. Explain the method of constructing and using a table of meri-
dional parts.
19. Find from Table IV. the meridional differences of latitude corre-
sponding to the true differences of latitude in Ex. 3.
20. Given the distance 105 miles, the course N.E. by N., and the
latitude left 50°, required the latitude arrived at, and the departure.
Sug.— Solve the triangle of Article 43.
21. Difference of latitude =114-4, d. tance = 150 miles. Required the
course and departure.
r,ii,j. — Express the course in points and quarter points as nearly as
possible. ^ ^
I
48
PIIACTICAL MATUKMATKU
dep«rtur« 80 milot, wlut
22. Coiiri* S.R.r:., Ifttitu.le left 44' 30' N.
U the (liataiico run 4ti<l latitude in ?
23. KxpUin tilt Mturo o! • Travem Table, aod the method of
uitng It.
M. Diatinguiah between Plam Sailing and -/erra/or'« SaUing, lo called
fi^-i^. -By th^ former, latitude may be found; by the latter, longitude.
25. Derive the formula, dif. Ion. = nur. dif. lot. x tan eourte.
^nu* /^ i' '* ''«/"''*' *«'^'7t/ ^ l^n«l«r what oircumeUnoe. u the for-
mnla d^f. Ion. = </«<. « ^c. /a<. available ?
8 W . 1024 mile.. Utquired the latitude and longitude in.
AA^}j *^»''«**.;rom latitude 48' 28' N., longitude 6" 3' VV.,to latitude 37*
44' N., longitude 2.r 4^ W. Required tlie coume and dieUnoe
cn!!!./lt?'t"*t K^ '^''J^' ^- ^"8it«de left 15- W., latitu.b, in 4G- '25'K,
couree N.L by L. Required the distance .ai!«d. an«l the longitude in!
lon^tudeTr / """'^ * ■*''^ '*" *^"" ""' '° '""""^* ^^ *° *'^'°«« *»•'
«i *!i' ^". ''^•^ '*.***,"'*® j" !* *^»* «^««"y *>«»• • -Wp goes 8 mUes she
cimngeh her longitude 9-5 milea T *^ "
• ^'i' i.^y^** '"-.**** ^^^^^^^ **' * <lpgree of longitude in each of the follow-
ing lat.tudee, 0% 20', .Sir, 4fl», 60^ 60', 76' ?
33. What ia the mid Uaitudf for the foUowinc limita :—
( 1) 36» 6' 20 " N . , and 34' 22' S. ? **
(2) 1' 16' N., and 7' 20' N.?
34. What ia meant bv Middle Latitude Sailing f With what limita-
tions must the method be applied ? Why ?
Svg.—ln low latitudes and when tho angle made by the rhumb with
Ihe' dlpairelVt^^'' ^'^''''''* "^ *'*^*"^' '' ""*" *^"'"i'*^"^ ^i'»»
35. Derive the formulae—
DiJjT. Ion. = dep. x a^c. mid.-lat.; and
i)//?'. /on. = dist. X sin course x sec. mid-lat.
^?}H *!i®,"l^*^°'^ °^ "^''^^^^ latitude sailing to ihe following :—
(1) Left latitudo 62° 6' N., longitude 35^6' W., course N.W
by W., distance 229 miles. Required latitude and longi-
tude m. ^
(2) Required course and distance from ^ to J5. when lat
.iriK,- w""-- ^ = "• '"■ '"- "^ ^ = «• ^' ^^
meSod^^** "* ™*^* ^^ -ffeso/yinfl- a 2'rarcr«e ? Explain in fuU the
39. Define Traverse Sailing and DejtaHure Course.
40. Resolve the following compound course for latitude and longitude
mix m^^^
..if ?.>.*: 25; 34; S.^ Ion. 92° 18' R; saUed W. bv N. 34 miW
«.i^. u. 00 miiea, E.S.ii;. 46 miles, a. 38 miles, S.W. f W. 75 milea.
36.
NAVrOATIOJf.
49
41. Enumerate the e&rretthna ftpplicfthle to th« rMuling of th« iimcU*.
Diatinguuh olearly between Variation and Deviation.
42. Suppose the compau (ournt, that ta, the oourae m in.Iicated by
the compiua, ia W.N.W., and the variation is 2 pointii easterly; what
is tht5 trufl course T
iSug. Since the north end of the needle points too far towards the east,
the tnio course must be farther east than is indicated.
43. Wba* I the true ooors* when the eompass ooane is S.S.E.
variation 27' .«' E.1
5«.7. -The north end points V 30* too far towards the east, honct
the louth end poiuts 27* .'W toe far towards the wust, hence the trot
oourae lies 27" 30' farther towards the west than is indicated.
44. Compft«* coarse 8.R. fy" . 'ariation 22' 30' W., deviation 4' 15'
E., leeway li |)oints, wind w*?.: Hequired true course.
8ug. — Variation and deviation are in opposite directions, hence apply
difference 18* 16' W.j wind drives the ship IJ points fnrther east than
the compasu reading correcte ' ' t variation and deviation,
45. A ship in latitude 13' 5' N.. longitude 123* 23' E., sails 225 miles
a W. 4-W by compass; deviatiou 6° 35' W. ; variation I Jpts. E. Required
latitude ak X longitude in.
Sug, — First find true course.
46. What is the distance and the eoiirso as indicated by a compass
with 17' easterly variation and J point easterly deviation, from Cape
Men.locino, Itvt. 40' 29' N., Ion. 124" 29' W., to the Solomon Isles, lat.
7M5'8., Ion. 157' 4: E.?
<S'«.7.— Find the true course and apply oorretidonB inversely for compost
course.
47. Leaving a place in lat. 37' S., Ion. 151* E., I sail E.N.E. by com-
nasa 39 miles, with wind S.E., making 1^ points lee-way, variation 9"
29' E., deviation 6' W. The wind now shifts to the east, and I change
my apparent courts to S.E. by S., making 2 points lee-way ; deviation
for this position of the ship's head ia 3" E , variation unchanged. I
sail in this course for 6 hours at the rate of 5 miles. Find latitude and
longitude in.
48. On June 4, at noon, I sighted a rock in lat. 39' 40* S., Ion. 87'
15' E., bearing N.N.E., distant 15 miles. Afterwards, during the next
24 hours, I sailed as follows :-37 miles E. by S., 47 miles E.N.P]., 51
miles N.^-VV., and 29 miles E.S.E. Required the course, distance, and
latitude and longitude in, on June 5 at noor.
Su(j. — Departure course to be inverted.
49. What are potsible sources of error in taking the logf What
should be the length of a knot on the log line, when the sand runs out
in 28 seconds ?
Sug. Suppose a nautical mile = 6080 fent.
60. What are the true distances in th^ following cases ? —
(1) Estimated distance 78 miles, knot 47 feet, glass 27 seconds.
(2) Estimated distance 415 miles, knot 48 feet, glass 28 seconds.
/0\ 1?-i.: i._J J^_i Ao :i-_ 1 i. t;i\ *_-j. _i oo j_
CHAPTER VIII
h\
I I
COMPUTATION OF PLANE AREAS.
61. Area of Parallelogram derived from Base and
Altitude. — A square, whose side is a linear unit, consti-
tutes a superficial unit or area unit of like denomination
with the lineai' unit.
In any rectangle the number of area units is equal to
the number of linear units
of the same name in the
base multiplied by the
number of like units in
the altitude (fig. 22) : and
since all parallelograms,
with bases and altitudes
^^S- 22* equal respectively to those
of the rectangle, are equivalent to it, the area of any
parallelogram is denoted by the product of the number
of units in the base and the number of units in the
altitude.
Note 1.— Whenever area is to be denoted by the product of numbers
representing the lengths of certain lines, it is conventionally said, to be
equal to the product, of those lines.
Note 2.— If two adjacent sides of a parallelogram and their included
angle are given, the altitude may be obtained by trigonometry in terms
of" the sine of that angle.
52. Area of Triangle derived from Base and Al-
titude. — From Article 51, and well-known geometrical
relations, the truth of the following proposition i,s ap-
parent : —
COMPUTATION OF PLANE AKEAS. 51
The area of a triangle ia equal to half the product of
the base and altitude.
Note 1.— It may often be necessary first to obtf.In from the data, bv
Ih^T^T^^^l *^^ h^^' ""'. *t^ ^*^*"^«' ^'^ ^^^' »«' for instance, when
the data are two szdes and the included angle, or three sides. For the
^«;t«^''T' ^«^7«r' a formula, whose application aflFords a simpler
method of computing the area, is developed in the next article.
J\ro<e2.— Thearea of any rectilinear figure maybe computed from
data which are sufficient for finding the bases end altitudes of t'-e severS
triangles into which it may be divided by diagonal lines. (A'ee Art. 57.)
53. Area of Triangle derived fSrom its Three Sides.—
In the triangle ABC (fig. 23)—
Fig. 23.
a^ = b^ + c^-2bj(EMc. II. 13)
'''J =
ft' + c'-ga
26
.•.. = yo»-i^l±^^(Euc.I.47,,
,-. area = * A = ^ / JfeV^ -(6«h-c' -g')'
2 2\/ 4P
= /(26C + 6'' + c* - a''')(2ic -b^- c^' + an
V 16
V ^
2 2 ^~~
~ \/s(s-a)(s-6)(s-c). (Compare Art. 24.)
52
PBACTICAL MATHEMATICS.
64. Area of Triangle derived from Sides and Radius
of Inscribed Circle— In the triangle ABC (tig. 24), let
OD be the radius of the inscribed circle. Join OA, OB,
OG, then-
Area 50(7=?'*;
2 '
area AOC-^^4'^ area AOB = ^-^.
2 2
Hence area AB0=^~tA±^xr=8r.
Note 1. — This method involves simply a conception of the given
triangle as composed of triangles whose altitudes are equal, and
whose bases are the sides of the given trians'^*.
Note 2. — An extension of the foregoing principle to circumscribed
polygons determines that the area of any polygon is equal to half the
product of the perimeter and the radius of the inscribed circle.
Note 3.— If tht polygon is regular, its area is evidently equal to half
the continued product of one side, the number of sides, and the radius
of the inscribed circle or apothem.
55. Area of Circle derived from Radius or Circum-
ference. — Since the circle is the limit of inscribed
and circumscribed polygons (Art. 37) its area is equal to
half the product of its radius and circumference, and is
indicated by the formula.
Area = 7rr« = J irD^.
56. Area of Circular Sector derived from its Arc.
— If radii be drawn to various points on the circum-
;%.
COMHl'TATION OP PLANE AREAS.
53
ference of a circle, the areas of the sectors into which
the circle is thus divided are evidently proportional to
the arcs which form their respective bases ; hence the area
of a sector is equal to half the product of its arc and
radius.
Xote 1---It is often necessary, in the solution of problems, first to
hnd the length of the arc (Art. 37, Note 1). . ««
^/ote 2.— The chord subtondinrr an arc of a sector forms a triangle with
the radu d.awn to its extremities, and it h apparent that the area of
this triangle added to the area of the sector greater than a semicircle,
and subtracted from the area of the sector leas than a semicircle, will
give the area of the corresponding circular segment.
Xoie 3.— The area of a lune is simply the difference between the areas
ot two segments having the same chord and unequal arcs.
57. Area of Quadrilateral derived from Diagonals and
their Inclination.— In the quadrilateral ABGB (%. 25).
the diagonals d and d' and their inclination ^, are supposed
to be known : —
Fig. 25.
Area A nG= is AG xBP = ^ AG X BR X sin ^ (Art. U).
f^raa ADO =i^AGxDQ = ^ AG xDBx Bin ^
,: area ABCD=l AGxBDxsin^ = dd' x ?i?-^
•>
S'^'^oteJ^"^ ^^ ^^™^^^ *" illustration of the method suggested in Art.
58. Area of Quadrilateral derived from its Sides and
Inclination of Diae-onals — Dpnnfp An Tif' nn ha
(tig. 25). by «, 6, c, d, respectively, also AE, RB, CR,
54
M
PRACTICAL MATHEMATICS.
fnd'iafJ^' ""' ^' ^' ""^'P^'^^^^y 5 ^^'^^ (Euc. II 12
«' = »»' + n« + 2mn cos ».
i» = n' + 2^9 - o„p C03 ^
c» = ;j» + ,ya + 2pq coa ^.
tZ» = 7«» + 2" - 2m7 cos a.
.-. a* +c« -6« -'d- = {mn+pq+m2 + np)2coB^
= {m+p){n + q) 2 coa ^
= 2(/(i' cos S
= 2c/(i' ?yL^.
tanS^
= 4 area ^/^Ci) -^ tan S (Art. 57)
.*. area = (a' + c^ - i^" - d')
tanS
59. Area of Quadrilateral Inscriptible in a Circle -
If, ma quadrilateral, the opposite angles are supplement-
ary, it can be demonstrated geometrically that a circle may
be circumscribed about it. Let ABGD (fig. 26) be such
a quadrilateral, and denote the angle contained by any
two of the sides, as a and d,hyx:—
n
^\^. 26.
( 1 ) // the sides and x are known—
AteaASOD = area ABD +araa BCD= (ad+hc) 5™?, rfnoc
{.'i) J/ only the sides are known—
BS= a. in..: AS = a-^mi^. ■. SD = a +aVrriiHv
COMPUTATION OF PLANE AREAS.
M
I 12
le.—
lent-
may
such
any
e
Q X.
But BS^ + SD"" = BD* = TB^ +^3'
.'. d« + 2orfN/l - sin'x + a« = c» - 26cVl - niu'i + &•
2(ad + 6c)
.'. sir -= V4(adj^6c)^_--J6 9 + c« - a« - rf»)«
2(arf + 6c)
.'. 2Bmx{ad+bc)
= V 2^^rfT2&c + 6'' H-c«-a«-(/a)c.»arf4.26c-6'-c' + a''+d«)
= V{(6 + c)8-(a-"£i)«}|(a + d)M6^"7
= V(6 + c+a-rf)(6T^:r^c/)(a + rf+6-c)(a + d-5+c)
. . area= ?!£5(aii + 6c)
^2
= V(6 + c+a-rf) 6 + c-o + d
X ^^ X
o + rf4-6-c a + rf - 6 + c
V2 ^ V2
= \/(«-a)(«-6)(a-c)(8-d}
'>r2
~^2
.„5f ^ ^TJ^'^ demonstration includes, of course, the cases of th > rect-
angle and the square.
Note 2.— If one side of the quadrilateral as d is 0, then the figure be-
comes a triangle, and its area = \/s(8 - a)(a - b){s -c), as in Art. 53.
Xote 3.— If the sides of the quadrilateral are in arithmetical progres-
sion, then ?L±A±^Z_^ ^ „ a + b-c + d _, a-b + c+d ^ .
2 2 "■ ' 2 ~ ' ^
b + c + d-a , , ,
- - — 2" =di hence area = ^yalJcd.
60. Areas of Similar Pigu -.—The areas of tri-
angles are as the compound ra^xos of their bases and
altitudes (Art. 52). In similar triangles this compound
ratio is the duplicate ratio of their bases and of all other
homologous lines; hence the areas of similar triangles
are as the squares of homologous lines ; and in general,
the areas of similar plane figures are proportional to tlte
squares of homologous liiies.
n'x
u^x
56
PRACTICAL MATHEMATICS.
^^
rV-
If
ilij
EXERCISES.
*ni" 4«f ^'^*' Vlt relationship between units of surface ami linear units,
and mterpret the expression -arm e^iuals the product of t}^ Until
, 2. What geometrical prin ^les are involved in the motho»ls of obtain-
f5«?A ^^^** ^- *^® ^^°«*^' *? *^® '*«*'■«"* foot, of the side of a square
dfigonX-i^itr ' '""^ "'^^ ^ *'« '^''^'' ^' ^ ^-- --"-"J
itstfaTtMntrr^flts^Uga^^ "'°^^ ^^^«*^ ^« *^^-,- «-** «
5 How many area units are there in a rectangle whose diagonal
contammg x linear units, is twice aa long as the shortest side ? ^
'V^altUude"^ *12! *''*'* ""^ * »-^«™bus, one of whose angles is 120», and whose
\r.'!i rT^}"^} *' *^*' T'' °^ a parallelogram whose adjacent sides are 5
and 6, and one of whose angles is 135"?
whose'^aUituiLiser" °' "" '^""''^'"^ *"^"8^^' "^°»« "^« ^ 10?
9. Find the altitude of an equilateral triangle whose area is 60
;fa*?"i^''"'T^''''^^t^^ ^^^ ^-P"" ^"*^'"« *^« area of a triangle in terms of
Its sides; and hnd the area of a triangle whose sides are in the ratio of
5, 12, and 13, the perimeter being 50 yards.
11. The side of a square is 100 feet ; a point is taken inside the sauare "
which IS distant GO feet an. 80 feet respectively from the two enTof a
Ji ?he f^ourVo^rneTo? I'^iqu^. '"''^^^" ^^^"^' '^ ^°^^^"« *^« P-*
follows')-2! 3. t ri f3r23.l3!'4Vf""''" "'°^^ *'"^ ^'^'^ ^^« ^« -
., 13. Jind the altitude of an equilateral triangle whose area is equal to
the difiFereuce between the areas of two triangles, in both of which two
adjacent sides are 235 and 640 respectively, the included ancle of the
one being 50°, and that of the other 40". ^
14. WhaJ are the sides, respectively, of two equal rhombuses, the
diagona 8 of one of which are 20 a..' 30 respectively, while one of the
diagonals of the other is 50 ?
o«*-^' J^® ""^i^^V^ i!"^ ''''■''l^ circumscribing an equilateral triangle is
25 inches. What is the area of the triangle ? ^
^. ^^;.i^°^ *]*** *^^ area of a trapezoid is equal to the product of its
breadth and its mean length. ^ "^^uui, ui its
*i.^'^' ^H? ,*^*^^ °^ .* ^^'^P'-^zoid is 8 acres 2 roods 17 poles : the sum of
beL^etn themt' '" ''^ '^"^'- ''^'^'''''^' perpe?idicular dist^e
18 Demonstrate a rule for finding the area of a triangle in terms of
Its sides and the radius of the inscribed circle, ^pplv f he rvrJt,.;^!^ 4.-
iHuitiiateral figures. ^^■•' -« P*^— V*- tk.
rOMPUTATlON OF PLANK AREAS.
57
19. What is the area of a heptagon of which the radius of the in-
scribed circle is 7 ?
20. Find the radius of the circle that can he inscribed in a trinnj^ular
field containing an acre, the perimeter of which is the least possible.
21. Explain fully the steps by which, from an expression for the area
of a triangle, an expression for the area of a circle may be arrived at.
Sug.—HeQ Arts. 37, 52, and 65.
22. What is the difference between the area of an equilateral triangle
and that of the inscribed circle, the radius of the circumscribintr circle
being 10. *
'23. A horse is hitched by a long rope to one corner of an equilaterall)
triangular field. How long must the rope be that the horse may feed off
a quarter of an acre ?
24. An equilateral triangle and a regular hexagon have the same
perimeter. Show that the areas of their iascribed circles are as 4 to 9.
25. Show that the area of a triangle is equal to one-fourth the pro-
duct of its three sides divided by the radius of the circumscribing circle.
Sug. — From the vertex of one angle draw a perpendicular upon the
opposite side, also a diameter, then from similar triangles, etc. (Euc.
V A. V//.
26. The three sides of a right triangle are respectively 6, 8, and 10.
Find the areas of the two lunes formed by describing upon the sides
semicircles whose convexities lie towards the same direction. By how
much does their sum differ from the area of the triangle ?
27. The area of a sector is 150 square feet ; its angle is 50'. Find the
perimeter of the sector.
i'S. The chord of a sector is 6 inches ; the radius is 9 inches. What
IS the area of the sector. /e&t
29. The radius of a circle is 10 iaebes, two parallel chords are drawn
on the same side of the centre at distances from it of 4 feet and 6 feet
respectively. Find the area of the zone between the chords^
30. What are the areas of segments whose dimensions are given as
follows : —
Chord 17-32; heights.
Chord 10; height 1-339.
31. Find the arei of a quadrilateral whose diagonals are 66 and 64
respectively, their inclination being 48°. '
32. W^hat is the area of a quadrilateral, two of whose oppos'te sides
are 300 and 250 respectively, the other two, 450 and 275 respectively,
and the inclinatioi: of their diagonals S5°? r j.
_33. What is the area of the quadrilateral whose sides are respectively
15-6, 13-2, 10, and 26, and whose opposite angles are supplementary?
.34. Given two sides of a triangle, which are 20 and 40 poles respec-
tively, how long must the third side be that the triangle may contain
just an acre ? a j
35. If from a triangle, whose sides are 13, 14, and 15, there i?. i-.wt
off, by a Ime parallel to the longest side, a triangular area denoted by
J4, what are the lengths of the sides of the part cut off?
5 H
I
l\
d8
PRACTICAL MATHEMATICS.
thiril in tho aorioi b«ing 20 teat? '"'•*• "' '»• ">• lli«ui«t«r of the
•econd l,4,g 17 chaiM ? ^ ' ""°« ""■ «'«'"!«« of the lirst ud
h.t?-th^e S:-,;: sr " sr;t"fh'eS^ji "" » "«""- ""•^o-
LT T."" '"" °' "■"" '•-"^'1 CVtt oa:s " °^"" *"
cefe73^TheTJl,"/.')TriAt"»t^ JZff ^' «?™'« "«""■ ••«
u the same. Why ? * triangle the relation of their are.w
CHAPTER IX.
COMPUTATION OF POLYHEDRAL AND CURVED AREAa
61. Polyhedron.— A polyhedron is a solid whose faces
are plane surfaces, and whose edges are consequently
straight lines.
Nofe.— The names tetrahedron, hexahedron, octahedron, dodecahedron
and icoaahedron are given to polyhedrons of four, six, eight, twelve and
twenty faces respectively.
62. Prism.— If a polygon be sup-
posed to move along a line not in
its plane, remaining always parallel
to its first position (fig. 27), a prism
is said to be generated.
A prism, therefore, may be de-
fined as a polyhedron, two of whose
faces, called bases, are equal and
parallel polygons, and whose other
faces, called collectively its lateral
Hurface, are all parallelograms.
yote 1. — A prism is HgfU or oblique accord-
ing as its lateral edges are perpendicular or
<»l»lique to its basal edges; that is, as tlie
lateral edges are equal or unequal to the
altitude.
Note 2.— A regular prism is a right prism, ^8- 27.
whose bases are regular polygons ; in other words, whose lateral facea
are equal rectangles.
Note 3. — A right section of a prism is a section perpendicular to its
lateral edges.
Note 4» — The n"*"b«»«* «' »'-
prism as triangular, quadrangular, pentagonal, etc
60
PRACTICAL MATnEMATlCS.
hiii
*
I
63. Lateral Surface of PriaiiL—Tho lateral aroa of
any prism is donoted by the product of its altitude
and the perimeter of its base, or by the product of a
lateral edge, and the perimeter of a right section (Arts.
61 and 62).
^ot€.-n IB obvious that the Uteral wrface of a right prism is enaal
to the product of a lateral e<lge and the baHftl perimetor.
64. Cylindrical Surface. Cylinder.— If there be two
^ ' parallel circles (fig. 28), and
if a common tangent be sup-
posed to revolve so as always
to be parallel to the lino
joining their centres, the re-
volving tangent is said to
generate a circular cylhul-
rical sui'face.
A solid whose lateral
surface is cylindrical, and
whose two bases are equal
parallel circles, is a cir-
.. ,. cular cylinder. Thegene-
ratmg Ime is called an element of the cylinder.
A cylinder may likewise be conceived of as generated
by the motion of a circle along a line not in its own plane
keeping always parallel to its first position.
«„^'?'' ^'T'^^^ line joining the centres of the bases is called the axia
and according as it is perpendicular, or oblique to the bie and hence
equal to or greater than the altitude, the cyliider is rUjTov oufqu!
^^ir^t^^T^ '■'^'f .q/Wftrfer, called also a cylinder ofrevolutUm, mav be
generated by revolving a rectangle about one side as an axis. ^
65. Lateral Surface of Cylinder.-If a rcgukr prism
of any number of sides be inscribed in a ^^^\^g 29)
each lateral edge will lie in the cylindrical surface^ and
be equal m length to an element of the cylinder and if
the number of faces in the prism be supposed to in-
)
Fig. 28.
POLYHEDRAL AND CURVED AREAS.
61
crcaae without limit, the perimeter of the base of the
prism will approach the circumference of the base of the
cylinder a.s a limit, and the surface of the prism that of
.the cylinder as a limit. Then, because the lateral surface
of the pr'^'m is always equal to its altitude multiplied
by its perimeter (Art 63), " owever great the number of
■ides, —
The lateral eurface of the cylinder ia equal to the
product of its altitude and the circumference of its
base.
Fig. 29. Fig. .^0.
Note. — Lateral surface of a cylinder =2irr^.
66. P3a*amid.— If from a point in a polygon a line be
drflwn not in its plane (fig. 30), and if the polygon be
supposed to move abng this line, remaining constantly
parallel to its first position, and diminishing uniformly
in size without alteration of shape until it is reduced to
a point, the solid generated is called a pyramid: which
may therefore be defined as a polyhedron boiinded by a
es
PRACTICAL MATUE1UTIC8.
Fig. 31.
polyp^on called its baae, and three or mo*e triangles which,
together, constitute its lateral Hurface.
JS^^Tf^ T-^^" pyramid t« • pyramid who«« hue ii • nffnkr
pdygw and who«« v.rt«x that i.. the common rert«x of the triiSSSr
»o«i» la in the p«rp«ndiculAr from tb« middle point of its bsM. ^*
*v.^^/f?r'^!j® '^I** *«<?*< of A wguUr pyramid k
the altitude of any lateral fao«.
67. Frustum of Pyramid. —A fruFtum
of a pyramid ia that portion of it in-
eluded between the base and a section
parallel to the base (fig. 31). The lateral
faces of a frustum are therefore all trape-
zoids; and in the frustum of a regular
pyramid they are all ecjual, the altitude
of any one being the slant height of the
frustum.
68. Lateral Surface of Pyramid and Frustum— r^i^
lateral area of a regular pyramid is equal to one half the
product of its slant height and basal perimeter; while
that of the frustum of a regular pyramid ia equal to the
product of its slant height and mean basal perimeter
(Arts. 52, 66, and 67).
.^?T ^--^f^o^ing the lower and upper perimeters of a frustum by P
and p respectively, and the slant height by S,-^ » u«i uy /-
Lot. surf. z=Sx^-tP
2
Sd*ite7ru™t^' ^~^' "^ *^** *^® **'"® formula serves for the pyramid
Note 2.— Since neither the oblique pyramid nor its frustum has a
uniform slant height, the lateral surface of either can be obtained o" J
by finding separately the areas of the lateral faces. ^
69. Cone.— If a line be supposed to revolve, so that in
any position it shall pass through a fixed point and be
tangent to a circle, it is said to generate a conical surface
POLYUEDBiX ANb CUBVID auvai^
<(S
(fig. I. , of which the fixed point is the apex, and the
generati;.^ line an element
A solid, whose lateral surface is conical, and whoM
base is a circle, ii\ a cone,
A cone may also be conceived of as generated by the
motion of a circle along a line not in it« plane, rema'
mg always parallel to that plane, but diminishing uni-
formly m size until it is reduced to a point.
axi'T?? fehrP" ^^«>""»°K }^^ "P^x With the centre .,f the bate i. th«
£-.:L}tore ;5«itr;:;:^:jr '' " ^•'^-^'^-^^ - °»>"^'- '« ^*
iVo/< 2.— A rij7A< co«<!, or con« qf revolution, may evidently be oeneratMl
Fig. 32. Fig. 33,
70. Lateral Surface of Right Cone.— A cone is evi-
dently the limit of inscribed pyramids whose bases are
regular polygons; and a right cone is the limit of inscribed
regtdar pyramids. In the latte.. case the element of tne
cone is the limit of the slant heights of the inscribed
pyramids, and the circumference of the conic base the
imiit of their ba&al perimeters. Hence,—
64
PRACTICAL MATHEM:A.TICS.
The Icteral surface of a. right cone is equal to half the
'product of its element and basal circumference (Art. 68).
71. Conic Frastum, and Surface of Frustum of Right
Cone. — A frustum of a cone is that portion of it included
between the base and a section parallel to the base. The
frustum of a right cone is the limit of inscribed frustums
of rep^ular pyramids (fig. 34); hence, —
The lateral surface of a right conic frustum is equal
to the product of its element and its n ean basal circum-
ference (Arts. 68 and 70).
Kote 1.— If a trapezoid revolve about a side which is perpendicular
to the two parallel sides, it will generate a right conic frustum, of which
the lateral surface is equal to the product of the remaining side of the
trapezoid and the circumference generated by its middle point.
Note 2.— Denoting the slant height of the cone or frustum by .S", the
radius of its lower base by It, and that of the upper base— which in the
case of the cone is 0— by r, —
Lat. mirf. o/conr.orfrustnm=Tr(R + r)S'.
Fig. 34. Fig. 35.
72. Spherical Surface. — If a regular semi-perimeter
inscribed in a circle be supposed to revolve about the
diameter, each side will generate a conic suriaee, tue area
POLYHEDRAL AND CURVED AREAS.
65
of which will be denoted by the product of the revolving
side and the mean circumference (Art. 71, Notes 1 and 2).
Thus (fig. 35),
Surf. AB = AB> drc. GX = HZiie. BY) x circ. CZ,
wince the triangles ^J5Fand ZCX are similar.
In like manner the surfaces generated by the other
sides arc equal to the circumference whose radius is the
apotherr- tha^" is, circ. CZ — multiplied by the altitudes
of the c< jsponding trapezoids (or triangles); hence, —
Surf: GBAFK - circ. CZ{GH + HZ+ZL + LK)
= circ. CZ X diaTtieter.
Since the semi-circumference is the lixnit of inscribed
regular semi -perimeters, the sphere generated by the re-
volution of the former is the limit of the inscribed cones
and frustums generated by the latter, the radius being
likewise the limit of apothems; hence, —
Th^ surface of a sphere is equal to the product of the
circumference of a great circle and its diameter.
Noie. — A spherical surface = 4^^* (Aj efcr^afl^a
spherical zone = 2irr x height of iione.
fl*p' and that of any
EXERCISES.
\. IVstinguish between a geometrical solid and a material solid.
2. Describe each of the polyhedrons »lacussed in tlils chapter, dis-
tinguishing clearly ^etween tL^se which are right, those which are
regular, and those which are oblupie.
3. State explicitly the relationship between cylinders and prisms ;
between cones and pyramids.
4. Explain how a cylinder vnay be generated by the motion of a circle.
Make a similar explanation in regard to cones.
5. Interpret the formula S=PxH as an expression for the lateral
surface of a prism ; and adapt it to the case of a cylinder.
P + P'
6. Show how the formula S= — ,2~ x H', may be applied as an expres-
frusta. Adapt the same formula to the case of cylinders and prisms.
60
PRACTICAL MATHEMATIC&
7. Find the total area of each of the regular prisms whoso bases are
nespectiyely tnangular. pentagonal, and hexagonal, the height of «SS
being 6 mclies. ai>a one side of the base in each, 6 inches.
Sug. —Total area = lateral area + terminal p reas.
8. Construct a series of exercises sim.lar to those of Ex, 7. substitutiiiC
pyramids for prisms, and solve each. suDsncuiUig
9. How many square nches are there in the interior surface of a
&e"f ^""^ "''^ * ^^'^ '°^''"' '^ ^*" ^^*"^«*«' «d height^; each
♦• ^?' YiJ'x* " ?*® ^**®''*^ ^nxifice of the largest hexaeonal stick of
squaJe fSV "" '''* "^ ^'^^^^ each^Lection at 5 cents per
12. A tent is to be made in the form of a frustum of a right circular
cone, surmounted by a cone, the dimensions to bo as follow7:-herKh[
o* anex of cone frnm flio «T.m,«/1 lo *«„4. . _i-_x i. • , , , . ueignt
o:
, '' , — ^ "» ""^ uiuiciioiuuB Lu DO as loixows : — neiirht
apex of cone from the ground. 12 feet ; slant height of frustum. 8
i^lL^T'^"''' °^ ^i'^'V'"'' '-^^ ^^^ ^S f««* respectively. r£d the
number of square yards of canvas necessary.
• l^; ^^'^^ ? description of the solid generated by the revolution of a
right triangle about its hypotenuse, find the total area of such a soUd
If the sides of the revolvmg trianglo are 3. 4, and 6 respectively.
14. Demonstrate a formula for the surface of a sphere.
Ji'l^^ l^® difference between the internal and external surfaces
of a shell whose thickness is 1 mch, and whose internal diameter is 4
16. Assuming the earth to be a perfect sphere, whose diameter is 8000
miles; what is the area, m square luiles, of each of the climatic zones?
CHAPTER X.
VOLUMES AND CAPACITIES.
73. Equivalent Prisms.— If equivalent polygons be
supposed to move simultaneously along two equal lines, to
which their planes remain in all positions respectively
perpendicular (fig. 36), two equivalent prisms will be
generated (Art. 62); that is,—
Prisma of equivalent bases and equal altitudes are
equal in volume.
««S*;~" the bases of a prism are parallelograms, it is caUed a
paraUeloptped, either right, or, oblique, according to the inclination
of the edges to the bases. If the bases of a right parallelopiped are
rectangles, it is called a rectangular parallelopiped.
Fig. 36.
74. Volume of Prism and Cylinder.— A cube whose
edge is a linear unit constitutes a cubical unit, or unit
volume, of like denomination with the linear unit.
In any rectangular parallelopiped the number of
volume units is AviHftTsflv annol fY% iVic^ ««»v,l.^^ ^C —^-.i-i,
-1 — J — _. — J ,_,,^.,ij._i ^..^. t.ii-^. iiuiii^ijci. v/i super-
■
68
PRACTICAL MATHEMATICS.
i|«
I
}
fjcial units in the base multiplied by the number of like
linear units in the altitude (fig. 37).
Hence (Arts. 65 and 73) the number of volume units
of given denomination in a prism or cylinder is found by
nmltiplying the number of like area units in the ba.se by
the number of like linear units in the altitude; more
briefly, —
The volume of a pHsm or cylinder is equal to the pro^
diict of its base and altitude.
of Xe bis;:Ti';;«'ir.'"' '^ ' '^""^''■' ''p^*"'''* ^^ ^'^^ °^ *»»« ^^^^^
\. ^°^^?T}^ is evident that the volume of a rectangular paraUelopiued
13 denoted by the product of its three dimeuaions. P"^aueiopipea
Fig. 37. Pig 38_
75. Volume of Triangular Pyramid.— If in the trian-
gular prism ABG-DFE (fig. 38) a plane be passed through
the points CD.E, it will cut off the triangular pyramid
i^-VJiJ^- and if another plane be passed through the
P"?!!"*^. ^'/'^' ""^ ^^^ remaining quadrangular ^ rramid, it
will divide it into two triangular pyramids G-ABE and
C-ADE, which are equal, for their bases and altitudes are
equal (compare Arts. 66 and 73); but G-ABE, that is,
E-ACB,B>Iidi G-DFE havinnro/^nol Uno^.. «^J „1J.-J.._.i
VOLUMES AND CAPACITIES.
69
B
equal. Hence, any triangular prism may be divided into
three equal triangular pyramids, two of which have the
same base and altitude aa the prism. In other words,—
The yohmve of a triangular pyramid is denoted hy
one-third the product of its hose ar}d altitude.
76. Volume of any Pyramid or Cone.— A prism and a
pyramid, whose altitudes and polygonal bases are equal,
may be divided, the one into triangular prisms, the other
into triangular pyramids, so that the bases of the former
series will be respectively equal to those of the latter
(fig. 39), the altitudes of all being equal; and since each
triangular pyramid is one-third the volume of the corre-
sponding prism (Art. 75), the aggregate volume of the
pyramids is one-third che aggregate volume of the
prisms; that is, —
The volume of any pyramid or cone is equal to one-
third the product of its base and altitude.
J\ro<e.— The volume of a cone expressed in terms of the radius of the
base 18 ^ vr^If.
xr:
Fig. 39.
77. Volume of Frustum of Cone or Pyramid.— The
volume of a pyramidal or conic frustum may most readily
be found by calculating the altitude of the missing seg-
V*
70
PRACTICAL MATH EM ATIC8.
ment (figs. 81 and 34), and also that of the whole pyra-
mid or cone thus restored ; the ditference between their
volumes will evidently be the volume of the frustum.
78. Volume of Sphere. — Any polyhedron, whose faces
are equal regular polygons, may be considered as com-
posed of pyramids whose common apex is the centre of
the circumscribing sphere, whose bases are the equal
faces of the polyhedron, and whose altitudes are conse-
({uently equal to the apothem of the polyhedron.
Hence the volume of the polyhedron will l>e denoted by
one-third the product of its apothem and superficial area;
and since the sphere may be regarded as the limit of such
polyhedrons inscribed, the spherical surface and the
radius being respoctively the lim'ts of the surface and
apothem of the polyhedron, it follows that, —
The volume of the sphere ia denoted by one-third the
product of its surface and radius; or,
vol. sphere = } surf, x rad. = |irr' = ^tD*.
79. Volume of Spherical Sector. — A spherical sector
is the solid generated by the revolution of a circular
sector about the radius drawn to the middle point of its
arc; hence the ratio of the volume of a sphere to that of
an equi-radial spherical sector is the ratio of the circum-
ference of a great circle of the sphere to the arc of the
generating circular sector, — that is, of the spherical sur-
face to the base of the spherical sector. Hence, —
The volume of a spherical sector ia equal to oiu-third
the product of its ha^e and radius.
Note. — A spherical sector is the sum or difference of a cone of re-
volution and a ^herical segment, according as it is less or greater than
a hoDiisphere. Hence the methc^ o^ finding the volume of the segment.
80. Volumes of Similar Pyramids. — The volumes of
two nvramids are in the 2*^ of the Droducts of their
- - jr„- - - - - - ^
bases and altitudes (Art. 76). ' f they are similar, that is,
VOLUMES AND CAPAC1TII<:S.
71
if corresponding dimensions ai\ proportional, their bases
are in the diq licate ratio of corresponding banal edges
(Art. 60), and their altitudes are in the ratio of those
edges, hence, —
The volumes of similar pyramids are in the triplicate
raiio of their edges, or of any homologous lines.
81. Volumes of Similar Solids in General. — Since the
pyramid may be regarded as the elemental solid, —
Any two similar solids are as the cubes of homologous
lines.
(1)
(2)
(3)
(4)
(5)
EXERCISES.
1. The following formulae for volumes are appMoable respectively to
what varieties of sulida?—
V = AxB
V=i AxB
V = AxirR«
V = JAxirR«
V = $irR»=iTD».
2. Demonstrate each of the furniulae in Ex. 1.
3. How may the volume of a spherical segment be obtained, its height
and the radius of the sphere beiiie known?
4. The angle made by each of the faces of a regular quadran^lar
pyramid with its base is 40°, the altitude being 3 mches ; what u its
volume ?
5. The opposite faces of a regular hexagonal pyramid are inclined to
one another at an angle of 60°, the slant height of the pyramid being 12
inches ; what is the volume of a similar pyramid of double the height?
6. The length of a box which holds a bushel is 2 feet ; what is the
corresponding dimension of a similar box of 8 times the capacity?
7. The upper and lower diameters of a milk pan are 15 mches and 12
inches respectively, the width of the lateral surface being 5 inches ; how
many quarts does the pan hold?
Sug. — An imperial gallon contains 277.274 cubic inches.
8. What is the diameter of a hemispherical bowl that holds a gallon?
a litre?
9. The depth of a cubical box, the depth and diameter of a cylindrical
tub, and the diameter of a hemispherical bowl are all the same ; com-
pare their capacities.
10. The internal diameters of two water pipes are, respectively, 3 and
4 inches ; compare their carrying capacities.
11. Two parallel planes divide a ball, whose diameter is 12 inches, into
three segments of equal thickness ; how many cubic inches are there in
each seijinent?
12. Of all solids of given volume, which has the lea^t surface?
CHAPTER XI.
DYNAMICS.
Section l.-^ Immediate Aiyplication of Fnrce.
82. Laws ofMotion. -Experience justifies the followin.r
propositions : — ''
(a.) That only by the application of external force
can an inanimate body at rest be made to
move, or a moving body be made to chancre the
rate or direction of its motion.
{h.) TImt the tendency of force instantaneously
npphed is to produce uniform motion in a
straight line ; while a constant pressure tends
to produce uniformly accelerated motion,
(c.) That the resultant effect of two or more forces
acting simultaneously upon a body is the same
as it they acted consecutively.
{d.) That, when a force acts, two portions of matter
are affected equally, but in opposite directions
^ Newton embodied these generalizations upon experience
m his celebrated laws of Motion ;—
(1.) Every body perseveres in its state of rest or
moving uniformly in a straight line, unless
compelled to change this state by external
forces.
(2.; Change of motion is proportional to the
impressed force, and takes place in the direction
m which the force acts.
(3.) Reaction is always equal and opposite to action;
that is to Hay, tli;; actions of two bodies on each
DYNAMICS.
78
other are always equal and in opposite
directions.
Note I.— -Under ordinary circumatanceB, all motion meeta with a
re«i8tance due to friction and the difficulty of displacing the medium
through which the motion is directed.
Note 2.— Since all bmlies are constantly acted upon by the force of
gravity, the ellect of other forces is always subject to modification from
this cause.
«3. Definitions.— Any number of forces may act siiirul-
taneously upon a body, at the same point or at different
points, in the same direction or in different directions.
The single force that would be equivalent in effect to
several forces is called their Resultant
A force equal to the resultant, but opposite to it in
direction, is called an Equilibrant.
Note I.— Problems in Dynamics deal with the mutual relations of
component loTc^fi and their resultant or equilibrant. If the problem
relates to forces m equilibrium.-that is to say, any one of which may be
considered m the equilibrant of all the others, -it is a problem in Statics:
otherwise, it is a problem in Kinetics.
Note 2.— Composition of forces is the process of finding the resultant
when the components are known; while the reverse process is called
resolution ofjorccs. ^
84. Graphic Representation of Forces.— The points of
application, the intensities, and the directions of forces,
maybe represented by straight lines. The geometrical
relations of such representative lines would therefore
indicate the relations of the forces.
85. Resultant of Forces Acting in the same Line.—
Forces acting in one direction may be considered positive,
and those in an opposite direction, negative ; hence, the
resultant of forces acting in the same line is their
algebraic sum ; while that sum with its sign changed is
their equilibrant.
86. Parallelogram of Forces.— If two forces act upon
the same point, but not in thpi suttia lino fVio^r -„.;n
>- • " • ; ■-■""-J' TTiix
partially neutralize each other; so that their resultant
I ji
74
PIUCTICAI. MATHK!VfATIC«.
11'
I
r
I
I
will
"••)
iOfe than their sum, and its direction will lie
'*e directions of the forces.
If the adjacent sides,
A U and AG, of the paral-
lelogram -4 J5Ci> (fig. 40)
represent the intensities
and inclination of two
forces acting .simultane-
ously upon the same
point, their resultant will
Fig. 40.
be represented in intensity and direction by the diagonal
AD (Arts. 82, 3); and their equilibrant, by DA.
87. Triangle of Forces.-The line CD (fig. 40), equal
and parallel to AB, may also denote the force repre-
sented by AB; hence the triangle ACD may represent two
forces and their resultant, or three forces in equilibrium.
tioat^tJtwo fo±/^^ " °' '°""' Hupplementary to the inclina:
♦
88. Polygon of Forces.—
Let AB, AX, A Y, and AZ
(fig. 41), represent four forces
acting at a point A ; complete
the successive parall jlograms
ABGX,AGDY,ADEZ. Then
-4 C will represent the resultant
of the first two forces, and AD
the resultant of that resultant
and the third force— that is,
of the first three forces;
similarly . E will represent
the resultant of the four
forces.
_ Fig. 41.
Tiie principle of "the x^olygon of forces may be stated
DYNAMICS.
75
tluia : — Tli^ resultant of aeimral forces ckctirig upmi the
same point in different directions rnay he represented
hy the line joinimf the extremities of a crooked line, whose
consecutive portions are rettpectively proportional to the
forces and parallel to their dii'ections.
Note.— It ia apparent that such forces are in equililiriuni when the
oro<»ked line forms a olosed polygon.
80. Moment of a Force— If to one end of a rod
capable of rotation about a fixed point in its length, a
force be applied, not in the direction of the rod, its
effectiveness to produce rotation will depend partly upon
the intensity of the force, and partly upon the perpen-
dicular distance from the centre of rotation to the line
in which the force acts. Hence the product of this
Fig. 42.
distance and the force may be taken as a measure of
the tendency to produce rotation, that is, of the moment
of the force.
Thus, if F (fig. 42) denotes a force applied to the
rod AD, tending to make it rotate with jB as a
centre, the measure of the moment of F about B is
—FxBG.
r^j\ ^n .
if\j, ftuUciii i/ppuol(»6 xTxiliiiwiii/S* — SltV lH'CliiiiY vWU pcir<ili6i
forces F and F (fig. 43) can be applied, either on the same
76
I'HACTICAL MATHEMATirs.
11
t ♦
side or on oppcsito aides of B, in such a wiy that their
moment« will be tqual and in
opposite directions. In «uch
a case— •
"Y?/* ,1 — From this equation of
equilibrium may be obtained the
proportion,—
fif'.'.DB'.AB.
Notf. 2. —The forces are evidently
most effective when their directioun
are perpon<licular to the lino joininit
their point* of applica ion.
Fig. 43.
91. ParaUel Porces.-Tf a number of p- rallol forces
a^t upon a rod supported at a single point, each produces
the same effect at the point of support as if it were
apphed directly at that point.
Hence, if a number of rods, united at a common centre
ot support, are acted upon by parallel forces, the resultant
pressure upon the support is the algebraic sum of tue
forces; and, if the point of support be such that a
moment in any one direction is balanced by an equal .ind
opposite moment, the forces will be equilibrated b/the
resistance of the support. '
92. Centre of Gravity.-^If, for the system of ro!
unioed at the point of support (Art. 91), there be
substituted a rigid body supported, similarly, ai. a sin<Tle
point and acted upon by parallel forces, the condiilons^'of
equilibrium will be precisel> the same.
Again, jf thfe forces acting upon the body are simply
the weights of component particles of the body, the point
ot application of their resultant—that is, the point which
must be supported in order that the body may rest ir
differently m any position— is called the Centre of Gravity.
Noit\.—lt is ofteu convenient to consider the wnicrhf nf o \.^a. j
discussion as concentrated at the centre of gravity ^ ^^^ ''''^^^
OYNAMICm
77
i
''" ^.^'^y ■opported at a point, not th« centre of gravity, {■ in
■" u If the Doint of aupport ia vertically above, or vortically
« «•»! litre of gravity. In the former lue ther« ia $tahU, and
ter, umtnblf eiiuUtbrium.
^V -' ~T^« ."""''■f °' irravity of a bwly aunpended iacce«»iively from
1W0 r. T^rent pomU, it evidently at the point of intenectiou of the two
'' "V ! linen drawn through those points, when the position of aUble
«4auiDrium ba^ btiun issuuiod.
93. Centre of Qravity of Systems.— The followir<?
propo.sitions relating to the centre of gravity of syHteins
of bodies ntod no elucidation : —
(1.) The centre of gravity of two bodioH divi(le.s
the line joining tlieir individual centres of
gravity into segments, which are inversely as
the weights of the bodies (Art. 00, Note 1.)
(2.) The centre of gravity of three equal bodie.^,
situated at the angular points of a triangle, is
one-third the distance from the middle of
either side of the triangle to the angle opposite.
(3.) If lines, joining the centres of gravity of four
equal bodies, form a parallelogram, the point
of intersection of the diagonals is the centre
of gravity of the system.
£^ii<7.— Solve Exercises 1-31.
IP
SECTION II.
Forces Mediately Applied.
94. Elementary Machines.— The utility of machinery
is due to the fact that in accomplishing work, i.e. in over-
coming resistance, an indirect application of force may
be more effective than a direct applies tion.
Every machine consists of some modification or compli-
cation of three elementary machines — The Inclined
Plane, The Lever, and The Pulley.
78
U
■
I
is
ill
PRACTKJAL MATHEMATICS.
95. The Inclined Plane. (1) Direction of Force
parallel to Slope.— U a body be supported on an inclined
plane by a force whose direction is parallel to the length
of the plane, the ratio of the power to the weight —
friction being neglected — is the ratio of the height of the
plane to its length, that is, the sine of the inclination.
For, since the weight is supported, in part, by the
resistance of the plane, let
the vertical line DF{Rg. 44)
denote the weight W, then
BE perpendicular to the
surface and EF parallel to
the direction of the force
will denote the components
of the weight which are supported by the resistance li,
and the power P respectively.
Then, from th-^ similar triancrles. —
P
W
BO
AB
shi A.
(2.) Direction of Force parallel to Base.—li a weight
be supported on a plane by a power acting horizontally,
the ratio of the power to the weight is the ratio of the
height to its base, that is, the tangent of the inclination.
For, if DF perpendicular
to the plane (fig. 45) and EF
parallel to its base denote, re-
spectively, the resistance R,
and the power P, then the
vertical line DE, which repre-
sents their resultant, may
Fig. 46. obviously represent the
weight F; and ^ince DEF and ABC are similar,—
P
BC
AC
~ fan A.
DYNAMICS.
79
Nott 1. In the first case,
In the second case,
R
W
cos
p = see A.
Note 2.— That R is greater than W in the second case, is due to the
fact that it is resolvable into the two components P and W, acting at
right angles.
96, The Wedge and the Screw.— Two modifications of
the inclined plane, namely, the Wedge and the Screw, are
commonly enumerated among the elementary machines.
The wedge usually consists of two inclined planes
placed base to base, though it is sometimes a single plane.
The screw may be conceived of as a vertical section of
an inclined plane, wrapped about a cylinder in such a
way that its base forms the circumference of a circle
perpendicular to the axis of the cylinder.
f!C>
fi^
^
c ^
Fig. 46.
As the power is usually applied perpendicularly to the
axis of the screw, the law of equilibrium would be anal-
ogous to that of case second, Art. 95, giving the formula, —
P _ d
W c
in which d denotes the distance between two threads,
and c the circumference of the cylinder.
Note 1.— Because of the difficulty of estimating the force of the blow
by which the Vrcdge is driven, and because friction forms so large a pare
of the resistance to be overcome, it ia imposi/ible to give a useful law
for determining the efficacy of this machine.
Note 2. — For the combination of the lever and the screw, see Art. 97,
\i
I
'■ '
m
PIIACTICAL MATHEMATICS.
fj'
'zs:
DW
97. The Lever.— In the varied use of the lever, the law
of equilibration of parallel forces is constantly illustrated.
For, whether the fulcrum , that is, the point of support^
be at one end, or be-
tween the ends (fig.
47), the power and
weight— in case of
equilibrium and neg-
lecting the weight
of the bar — are in
the inverse ratio of
the arms on which
they act.
Denoting the
power-arm and
weight-arm, respec-
tively, by A and
(Z)
K
wb
f5)
^
uw
Fig. 47.
P ^ A'
W -'A'
V . .'" tu'^\''^ '^'^ ^^^"'' ^^ *^^^ ^«^«r figured above.
miT hp ;7 •5''' 1*^^ "^^'S^*' ^' °^ *^« ^eam must be recognized it
may be considered as concentrated at the centre of DrLv^r If
distance, D, from the fulcrum ; then,- gravity, at a
PxA+QxD=]VxA'- or.
P>iA^WxA' + QxD,
L^s^^;y^lt^tr^eVl*'^ ^^^^^ ^^ *^^ ^^^--^ °^ *•- -^^^^t is.
thilt artle l"n1'%'' J^ ^ '^.PPJi^d at the circumference of a screw -
ference of the screw by c, and by ^ that de^cdbed by ^;^ '''' '''''''^-
But(Artyi3),—
Hence, —
F
P
r
c
C'
w'
F
d
d
W ~ c"
DYNAMICS.
81
of
Note 3.— In the use of the Wheel and Axle, P is applied at the cir-
c imference of a large wheel and W at the circumference of a smaller
wheel rigidly connected with it ; the radii of t^ wheels constitute
the arms of a lever of the first order ; hence, —
P _ r_c
98. Weighing.— Various adaptations of the lever are
used in weighing, the simplest being the steelyard and
the balance.
The steelyard is a lever of the first order, the centre of
gravity of which is situated in the shorter arm. The
graduation of the longer arm is effected by the aid of the
principle in Art. 97, Note 1.
The balance is essentially a lever with equal arms, it.«
sensitiveness being increased by placing the fulcrum
above the centre of gravity.
If the arms of a balance are unequal, the weight of a
body a; can be obtained by weighing it in both - an^;
for, if If and W denote the apparent weights, and A and
A', the arms, —
jcx^ :: jrx.4'; and—
X X A'= W X A; whence—
a? =VlFx /r.
99. The Pulley —It is evident that, of the t\\ o ends of
a cord passed freely round a weight, if one be attached
to a support, while to "<^^r;5A i=i-^u-d-
the other a power is ap
\}^ d, the portions of the
rope being parallel to each
other, the power and the
support will each sustain
one half the weight. The
pulley is simply a grooved
wheel, by whose aid the
advantage which this de-
vice gives, may be more coirveniently scoured.
Fig. 48.
P
mm-
'}
82
PRACTICAL MATHEMATICa
From figs. 48 and 49, it appears, —
(1.) That the fixed pulley confers no mechanical advao -
tage beyond a change of direction.
(2.) That, in a system of n movable pulleys with »
single cord, two portions of which support each pulley, —
P _ 1
w zJT
(3.) That when there are as many separate cords as
there are movable pulleys, —
Ir'
1
Fig. 49.
^ote 1. In a similar way the ratio of power and weight in any system
can be ascertained. Practically, howevor, allowance must be made for
mction and for the weigb'^ of the system itself.
^ote2.— The six mac.mes,— f«c«?ie(i Plane. Wedge, Screw, Lever,
Wheel and Axle, and Pulley, are called the Mechanical Powers.
Sug.~^o\vQ Exercises 32-51.
DYNAMICS.
83
sEcrrioN III.
Variable Motion..
100. Velocity varying uniformly.-Motion Tesultin-
from a uniformly persistent pressure or attraction h
uniformly acceler Jed motion (Art. 82). Such is the
motion of a falling body, gravity being the constant
force ; and, if a body be projected upwards, its motion
will be uniformly retarded.
The space passed over in a given time by a body
moving with a variable velocity is denoted by the
product of the time and the mean velocity,— which in
the case of velocity vai'ying uniformly, is the mean of
the initial and final velocities.
Hence, denoting by S the distance parsed over during
the time t, the initial velocity by 7, and the final velo-
city by V, —
Again, denoting the gain or loss of velocity occasioned
by gravity in one second, which may be taken as 32-2
feet per second, by g, —
v= V:i:gt; hence, —
S=Ft-h^gt\
Note 1.— From the equations,—
V = V dt^ gt, and
S^ Vt±iigt\~
the following equation may be derived by eliminating t,—
V' = V d^ 2gS.
Note 2. — If, in the equation v* = F' - 9 « .c « — a ^i.- u
when^a_body projected Vertically upwards at'tfini' l^^ SgL^^UnTol
and if, in the equation v»= V^ +2 q S, F=0 which is. fmo ^f +k
body at the initil point of its descent.- ^^ *^^ ^^^
hence the velocity of projection and the final velocity of descent are tl.«
84
PRACTICAL MATHEMATICS.
101. Relation of Velocity to Pressure and Mass. -
Since an acceleration of 82*2 feet per second is given to
a mans of x lbs. by gravity— *.«. by a force of x lbs.—
every second the incr ment given by a force of 1 lb.
woultl be —
1
X
X 32-2 feet
and that given by a force of y lbs. would be—
I X 32-2 feet; that is—
The acceleration due to constant pt^essure varies directly
as the intensity of the pressure, and inversely as the
whole mass moved.
Note.— By the aid of a device known as Attwood's Machine, this law
can be experimentally establiahed.
102. Momentum. — The power of overcoming resist-
ance possessed by a moving body is called its momentum,
the measure of winch is the product of the mass and
velocity of the body.
The changes of motion produced by the impact of a
moving inelastic body on another already in motion, or
free to move, are only illustrations of the operation of the
third law of motion, that action and reaction are equal.
The momentum gained by one from the impact is lost by
the other.
103. Motion varying in Direction.— Pa^^ of a Projec-
^ B tile. — The combined ac-
tion of two forces not in
the same line, one, at
least, of which is con-
stant, results in curvi-
linear motion. Such is
the motion of a projectile
^B thrown in any other
Fig. 50. than a vertical directioa
DYNAMICa
85
Thus a body projected homontally from A (fig 60)
with a velocity that would carry it to i? in the time
m which it would fall to G if dropped, describes the curve
AD; and EAD, which, if it were not for the resistance
ot the air, would be a parabola, is the path of a body
shot obliquely upwards with an energy whose horizontal
and vertical components may be represented by EC and
CA respectively.
Circular Motion.— U, in obedience to one of two equal
and constant forces acting at right angles upon a body it
continually tends towards a fixed point, the resulti'n^
motion will be circular.
This ip Ulustrated in whirling a Imll at the end of a
string, the tension of which fui- . hes the centripetal
force. If some centre-ward attraction could be substi-
tuted for the tension of Lhe string the conditions would
be analogous to those which determine the substantially
circular motions of some of the heavenly bodies.
EXERCISES.
^1. define -.-Dynamics, Statics, Kinetics, Component, Resultant,
tqmhhrant.
2. Show by iUustrations that the operation of Newton's three laws
of motion IS of necessity imperfect in all but exceptional cases.
3 What is meant by Oraphic Repreaentatim of Forces? CompoHition
of Iforces ? Resolution of Forces ?
4. What are the Parallelogram of Forces, and Polynan of Forces
respectively ? ^./ ^ »
5. The sides of a triangle taken in order may represent three forces in
eqmhbnum, but when three forces act along the sides of a trianglu
they cannot be in equilibrium. Explain.
„„?! Y".*^?^^**^*^*^^**^*^**'^ !L*^® resultant of forces acting upon the
sauie point at a laaximum ? When is the resultant^f two such forces ?
86
PRACTICAL MATHEMATICS.
f ?
7. If two forcea, P and Q, »ci at the Mine point, their reraltMit beiBg
R, and their iuchuation wi, ihow that the following propoaitiuna are
If m equalc,—
(1). 90*.
It*
(2). m\
/?•
(.3). 120%
R*
(4). 4*i",
R*
(5). 135*,
R*
(6). 30-,
R*
(7). 150*,
R*
Bug.— Apply Euc. II. 12, l.^
P* + r^« - PQ ;
■ P* + Q^ + s/2. PQi
P' + Q-'-s/'2. PQ;
P« + (;>» + V3. f'Q;
. ?'u ^ "^JP "o^^" 'orward 24 feet while a ball is falling frtm the most
to the deck, a distance of 64 feet ; how far did the ball move ?
9. A boat is moored in a stream by two ropes, one from each bank,
and inclined to the direction of the current, at angles of 3(f and 4fi» :
what IS the ratio of the tensions on the ropes ?
,*^: .^Z'': a vertical force of 6 lbs. two forces are to be substituted, one
of which IS horizontal, and the other inclined to the horizon at an auirle
of 46"; what are those forces ?
11. What is the resultant of two forces of 5 and 11 lbs. acting at an
angle of 60'? 90'? 120'? 135"? 160'?
12. A horizontal force of 12 lbs. is resolvt^d into two components, one
of which IS a vertical force of 25 lbs. ; what is the magnitude and
direction of the other component ?
Svff. — Find direction of obliqu3 force by trigonometry.
13. Under what circumstances will three equal forces acting at the
same point be in equilibrium?
14. Compare the resultant of two forces of 6 and 12, acting at an
*?§!fo *** *-^'*' ^^*^ ^^^^ °^ *^° forces of 6 and 6, acting at an ancle
of 60 . «» o
^•^l5"'^«*®"' ™*^'"ff «" a"g^e of 60% support a chandelier whose
weight is 90 lbs.; what is the pressure along each, rafter?
1 1 oe J-^® resultant of two equal forces acting upon a point at an angle
of 135 18 10 lbs.; find the valur of each component.
17. The larger of two forces which act at right angles is 129-5 lbs
and the sum of the resultant and the smaller force is 136-9 lbs. ; find the
resultant and the smaller force.
18. Three pegs are fixed in a wall at the corners of an upright equi-
lateral triangle ; a cord, vv'hose length is four times that of a side of the
tnangle, is hung over the pegs, its ends are tied, and a weight of 5 lbs.
IS attached below ; what is the tension on each of the pegs ?
IS- The successive angles, at which four equal forces act, are ,30°,
60, and 90"; what is the direction and magnitude of their eauili-
oraut? ^
r
DTMAMIC&
a7
so. Explain the principle of moments.
21. Show by the parallelogram of forcea, that, if two forces act at
the same point, their moments about a point in ths line of their
resultant are equal.
122. What are the conditions of equilibrium of three forces acting
upon a body.
Sug. : — They must act at the same point, eto. Apply the principle
of moments.
23. A weight of 152 lbs. carried on a pole by two men, is placed
three timeR as far from une end as from the other; what weight is
supported reapectiveiy by the men, each of whom grasps an end of the
pole ?
24. Of two parallel forces acting in opposite directions, the greater is
10 lbs. and acts at a diBtauce of 8 inches from tho resultant, which is
lbs. ; find the distance between the forces.
25. How may a couple be equilibrated f
Sug : — A couple consists of two equal parallel forces acting in
opposite directions.
26. A bent lever, considered without weight, has equal arms making
an angle ^f 150°; what is the ratio of the weights attached to the extre-
mities of the arms, if, in tho position of equilibrium, one arm is horizontal?
27. Weights of 1, 2, .3, 4, and 5 lbs., respectivelv, are hung at equal
distances along a rod wliose length is 20 inches, and which is suspended
at a single point so as to remain horizontal ; locate the point of
suspension.
28. Define — Centre of Gravity, Stable and Unstable Equilibrium.
What sort of a body resting on a horizontal plane is in Neutral
equilibrium ?
29. A uniform rod, weighing 5 lbs., is 6 feet long ; at the ends are
placed weights of 6 and 8 lbs. respectively ; where is the centre of
gravity of the whole ?
30. Show that, if equal weights be placed at the aneular points of a
triangle, the centre of gravity of the system will be one-tnird the distance
from the middle ot either side, to the vertex of the opposite angle.
31. Where would be the centre of gravity of weights of 3, 6, 2, and 6
lbs., placed consecutively at the corners of a square whose side is 12
inches ?
32. Describe each of the Mechanical Powers, and state the law of
equilibrium for each.
33. Show that when motion is produced by the aid of a naachine the
power and weight are in the inverse ratio of the distances described by
them, and therefore, of their velocities a^ weU. Explain the statement
— the work done by the power is equal to that done Dy the weight.
Sug.:— In the use of the inclined plane, the distance described by the
weight is the vertical height to which it rises. When the power is
o'"l" 'HSLrt; of iii is fififsctiT©*
«\cis*a1lAl ^.r% 4rVt
naock
1
t
ill
ii
PRACTICAL MATHEMATICS.
84. Tho linno of an inclinrd plane ia 10 fnet, and the height 3 feet ;
what furuu acting p^ralloi to tho base Hill balance a weight uf 2 tonii 7
35. A safe weighing I50<) ll)a. ii to be raised 5 feot by an inclined
plane ; the grentcat power that can be applied is 250 Iba. ; what it the
ahortest plane that can be uaed ?
36. A force, applied parallel to the bafie of a plane whose inclination
ia 20°, aaatuins a curtain weight ; a aecond force a|)pliud along the length
of anotht-r plane whoso inclination is 40' sustains an equal weight ;
what is the ratio of the forces ?
37. Mention a variety of common implements each of which is n
lever, diatingui8hin'» whothtr it is of the first, aecond, or third order.
To which oruer does an oar belong?
38. Four feet from the fulcrum of a lever of tho first clasR, wlioBe
leiiKth is 10 feet, is attached a weight of 50 lbs. ; 3 feet from the fulcrum,
and on the same side of it, is another weight of 30 lbs. ; what weight at
the other end will balance them both T
39. If a uniform beam, which is 12 feet long and weighs 40 lbs., is
used as a lever of the second order, what is the least power that can
raise a weight of 500 lbs. attached 3 feet from the fulcrum ?
iSug.:—G. G. of beam is in the middle.
40. A beam 18 feet long is supported at both ends ; a weight of 1
ton is suspended 3 feet from one end, and a weight of 14 cwt., 8 feet
from the other end ; what is the pressure on each point of support ?
41. A power of 50 lbs. acts upon the long arm of a lever of the first
class ; the arms of this lever are 5 and 40 inches respectively. The
other end acts upon the long arm of a lever of the second class ; the
arms of tnis lever are 6 and 33 inches respectively, vi'ind the weight
that may be thus supported.
42. The Wheel and Axle is an endless lever ; explain.
43. The pilot-wheel of a boat ia 3 feet in diameter; the axle, 6
inches; the resistance of the rudder, 180 lbs.; what power applied to
the wheel will move the rudder ?
44. Two men capable of exerting forces of 260 and 300 lbs.,
respectively, work the handle of a winch and axle ; the radius of the
axle is 6 inches ; what must be the length of tho arm of the winch that
the men may be just able to raise a weight of 4480 Ibe.?
45. Explain clearly why a pulley must be movable in order to give
mechanical advantage. In constructing a pulley it is desirable to have
the radius of the wheel as long as possible. Why ?
46. A man whose weight is 150 lbs. supports a weight equal to his
own, by means of a sy. tern of 3 pulleys with a single cord ; what is his
pressure upon the floor ?
47. Which is the more advantageous, a movable block with 3 sheaves,
or a system of 3 movable pulleys with separate ropes attached to a
beam above ?
48. Figure and describe a system of pulleys with separate cords
attached to the weight. If, in such a system, there are three such
DYNAMIC'S.
89
cunis. Will it bo more or less advAntftgcoQi th»n if th«rA w«i« thr«« ooi<l«
atUoh«<i to the beam.
49. If, in th« second ay item of palleyi (Art. Od) thei-e are 4 movablu
who«e weights, b«'ginninj^ with the highest, are 2, ~
palleyi
.< lbs
3, 4, and
respectively, what weight can b« sustained Ly a power of
12 lbs.?
60. A screw with threads I| inches apart is driven by a lever 4^ feet
long ; what in the ratio of power to weight ?
61. An endless screw which is turned by a wheel 10 feet in circum-
ference, acts upon a wheel having 81 teeth ; this wheel has an axle 18
incheH in circumference ; the power ia 76 lbs. ; what weight can be
supported from the axle 7
62. Which of Newton's laws ia illustrated by a body falling from
rest?
6.3. Show that the spaces dcHoribed in flucc«8sive seconds by a body
falling from ruttt are as the odd numbers 1, 3, 6, 7, &, etc.
54. What is the mean velocity of a body projected vertically down-
wards with a velocity of 30 feet per second, aunng the first 6 seconds
of descent?
55. A body is projected vertically upwards with a velocity of 180 feet
per second ; bow far will it ascend in 6 seconds ? How long before it
will return to the ground ?
56. A body is thrown upwards with a velocity of 100 feet per second;
at the same instant another is dropped from a point 20U feet high ;
where will tiiey meet? and with what velocities will they be iaoving
at that time ?
57. Two stones are dropped from different heights and reach the
ground at the same time, the first from a height of 81 feet, and the
second from a height of 49 feet ; find the interval between their
starting.
68. A rifle is pointed horizontally over the rail of a vessel which is
9 feet above the water ; with what velocity must the ball bo discharged
that it may strike the water (JOG feet off?
Sug.: — Consider flr = .32 ft.
69. What will be the effect of the mutual impact of two inelastic
bodies of equal weights, whose velocities in opposite directions are as
1: 2?
60. Three inelastic balls whose weights are respectively 6, 7, and 8
lbs., lie in the same straight line ; the first is made to impinge on the
second with a velocity of 60 ft. per second ; the first and second to-
gether impinge in the srune way upon the third ; find the final velocity,
Sug.: — No change of momentum is produced by the successive
impacts.
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IMAGE EVALUATION
TEST TARGET (MT-3)
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23 WEST MAIN STREET
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EXERCISES FOR REVIEW.
NAVIGATION.
1. Define the termi— Distance, Rhuiab, Course, Difference of Latitude,
Vtfftrence qf Lon<jUude,—w used in problema in Navication. Illustrate
by a diagram.
2. What is meant by Dead Reckoning f
3. Define Meridian Distance tad Departure,
4. \yhat is the length of a degree of longitude at the equator? At
the poles i o ^ «.«
1 ^L.u^*l^ ^ "^* * degree of longitude in latitude 45' just half the
length of a degree at the equator ? Is it more or less than half*
6. Would meridian distance vary in diflFerent latitudes if the earth
were a plane ? A cylinder ?
al/i tT^d T *^®^'"®®* °^ latitude of substantially the same length in
.nS' ^^*^ " ^^^ departure of a ship which sails 100 miles due east?
100 miles due north ?
5. Why is not the departure equal to the arithmetical mean of the
initial and final latitudes ?
10. Explain the principle on which a Mercator's map is constructed.
y*, ,^^^y ",* "^*P ^^ *^^ world, drawn on Mercator's projection, un-
suitable for school-room purposes ? ir * -.
12. Explain what is meant by Meridional Difference of Latitude.
13. Give a formal definition of Meridional Parts.
^J^' ^^***°» ^'"0™ *^'® ^al'le. the meridional parte for latitudes 23"' 45'
42 , and 66° 30', respectively. '
15. Find by the aid of the table the meridional diflferences of latitude
corresponding to the following terminal latitudeo :—
46»N., .36° N.;
27° N., 48° 30* N,j
10° S., 13° N.;
25° N., 7° 30' S.
16. Demonstrate the formula —
Diff. Ion. =mer. dist. xsec. lat.
17. Show that the formula of Ex. 16 may be used to find the length
of a degree of longitude in a given latitude.
18. What is the length of a degree of longitude lu each of the follow-
ing latitudes— 10°, 23° 30', 45°, 66° 30' ?
19. In what latitudes is the lencth of a degree of longitude 2C miles ^
30 miles ?
20. What is the velocity of the earth's rotation at the parallel of 60°?
EXERCISES FOR REVIEW.
91
21. Show that the mutu&l relations of course, distance, departure,
And difference of latitude are those of tlie three sides and an acute
angle of a plane right triangle.
22. Deiine Plane Sailing, and establish the following equation* : —
Sin course = dep. -;- dist.
Tan course = dep. -f- diff. lat.
Dist. « = dep. ' + diff. lat.».
23. Find the unknown elements in each of the following cases : —
No.
1
Courte.
Diatanoe.
Deimtiira.
Latitude left.
I^ttitii 1e ill.
N.E.
'•() niilos.
?
20° N.
?
S.S.E.
100 „
•
?
6°N.
3
?
3:]i> „
2G° .30' S.
C5-S.
4
?
400 „
250 iniies.
0°
f
5
?
?
20° K
CO°N.
Sufj. — Distance, departure, and difference of latitude are expressed in
nautical miles.
24. Explain the construction and use of a Traverse Tahle.
25. Is the method of Fiane Sailing competent to find the course and
distance when the initial and final latitudes and longitudes are given ?
26. What is meant by Mercaior's Sailing ? In what respect does it
supplement the method of Plane Sailing ?
27. Demonstrate a formula to be used in finding the difference of
longitude when the initial and final latitudes and the course are given.
SJ8. What are the latitude and longitude of a ship after sailing N.N. W,
4(10 miles from lat. 37° 10' N., Ion. 49° 25' W.?
A'uj/.— Find— (1) diff. lat.; (2) merid. diff. lat.; (3) diff. Ion.
29. What course must a vessel take in sailing from Cape Sable, lat.
43° 26' N., lou. 65° 38' W.— (a) to Bermuda, lat. 32° 15' N., Ion. 64° 40'
W. ? (6) to Hatteras, lat. 35' 14' N., Ion. 76° 30' W.? What distance
is sailed in each case?
30. Find the unknown elements in the following problems : —
No.
Lat. left.
Lat. in.
Lon. hft.
148''40'W.
5° 10' W.
?
Lon. in.
Course.
Distiince.
1
2
8
18°30'N.
50° N.
?
20° S.
37° N.
48° 30' S.
154° 20' W.
?
158° 42' E.
•
S. 49° W.
W.byN.
?
?
420 miles.
92
PRACTICAL MATHEMATICH.
sit**'™' "" "°"""'*'' "''"■^'"^ '- "" '""owi-'S problem, m P.rrild
No.
1
Lat.
Dlir. I^n.
Oistanos.
23° 30'
?
100 miles.
2
45°
2° 20'
?
3
?
9 '5 miles.
8
— ^ ,
A tofi!- ^^ "^'^^^^ ^^'^''^' «*"i"« '^^ «o°"e and distence from
Lat. A 33° 10' S., Ion. A 52° E.
.. n- ,. . I^**B42°S., Ion. B 145° E.
dS. Given lat. A, 15" 20' S.. Ion A 5° F Inn « i<»oA/^»r ,. ,
|™^ A to B. «» .Ue,; and the-;:o^,e^ fL'X-to^ .^a Zl tS
36. DefiM r™.»r«e, 7V«w« &iK„y, and Apart»„ Coar*^
SK?w^'iT •?® ^oUowing courses and distances: S.W.fW 62 mile«
S Jf-^'^I!-!^^'' ^.-^^J ^ ^"^«' SW.iW. 29 miles. S b^K 30mlS'
2° 48^-47"! P°^"*=^^^ 1^'' 2 P*- = 8' 26' 15"; i pt.=5°37' 30"; i pt.=
46 miles, S.S.E. 30 ^ile* khyW^%\,ilefi^F WmTa ''i^'^'^^^;
the latitude and lon^tude inf Id fhetScf cfursf an^d^d^t^ ^"^'^^
co^cidlSlT ^ "^^"'''^ '°"''' ^^ * ^^^P ^^ i<« ^^«^ <'ourse seldom
KXERCISES FOR REVIEW.
m
42. Di8ting;ui8h clearly between Variation and Deination.
43. Find the true or apparent course, as may be necessary, in each of
the following problems: —
1
Couiso.
True Course.
VuriKtion.
Deviation.
Leeway.
Wind.
W.byN.
?
2pta. K
ir vv.
2 pts.
N.byW.
2
?
S. 63M8'45"W.
lipts.W.
6' 30' W.
lipts.
E.byS.
3
N.
?
li pts. W.
2» 40' E.
Ipt.
EbyS.
4
?
S.
1 pt. E.
i pt. W.
ipt.
N.W.
Suff. — Draw a diagram in each case.
44. A ship is sailing N.E.^E., and the direction of the head is changed
by 118° through the north; what is the new course of the ship in points
to the nearest i .4arter?
45. A ship is in lat. 38° 44' N., Ion. 18° 33' W., and sails by the
compass E.N.E. 70 miles; required the latitude and longitude in, if the
variation is f pts. W., deviation 8° E., leeway 1 pt., direction of the
wind E.S.E.
46. How is a ship's rate of speed observed? Explain the method of
correcting the log.
47. What length of knot should be used with a 27-second glass? A
35-second glass ?
48. Find the true or apparent distance run by a ship in each of the
following cases : —
No.
1
2
3
Glaes.
Knot.
Apparent Dist.
True Dist.
32 sec.
27 „
28 „
47 ft.
47 „
53 „
315 miles.
?
37 miles.
?
80 miles
V
il
w.
PL/.:NE AREAS.
1. Explain the relation between unit area and linear unit.
2. What is the convention under which the area of a parallelogram is
said to be equal to the product of its base and altitude ?
94
PRACTICAL MATHJCMATICa
JpJStl :f^*::':SJ.^t^^e.V'' •"• °' * P»rallalogra« equal to
p.yeb7ram'' ''*^'' ''^J'^"' "^* "^^ ^ '^^'^^ '^ '^^ »>a«e of the
tnr'ti^^^^ geometrical priaciples are involved in obtaining an expreasion
for the area of a tnangft in terms of .ts base and altitude? ^''P'*"'*'"
loJ; Vu ■ i" ***°, "i^^ °' ^^'^ ^"'^ °f » barn whose rafters are 20 feet
bng.^the ndge-pole being 30 feet from the ground, and the pJsU 18 iZl
%/.— Find the altitude.
6Mr7.— See Art. 53. • . .
♦1,1?' -^^1 ?^'"' °^ *^® *'^''°^® inscribed in a trianc^le io 2 the sidex of
the tnangle being respectively 7, 15. 20 ; what is theTrea of tt tr^lel
thetrillgfes ot*Ex.'^'^f °' *'' '*"'"« "' ^^° «-«l« --"'>«^ - «-h of
5uflr.— Apply Art. 54.
5«Jt: tWrSTSor' diag„nal/fro„ the o,^^Z^,
fee"" *'°"'P"*<' "" »"» °' » '««>■■»>• pentagon, one of whose sides is 30
18. ?'"«J tlio ares of each of the following regular nolvirona thoi^i,,.
rXs^J-^l^Sh i'"^ d1<J,nXs" ^°''°'""' "^'''"^ " • -«•=. «»
are^s^eS^r2rS^;th^e7Sdra:g°e^:!'■f^^
3ofinl'^riSffilg«f:l^-'jL°rfei^^t'r^^^^
u.«l.
EXERCISES FOR REVIEW.
9d
22. Find the »re» of a rhombua whoio porimeter is 100, one of the
extenor angles being 60°.
^iaP*"*?"*® *'*® *'"®* **' * trapezoidal lold whose parallel boundaries
are 100 fcid 120 rods respectively, the length of a fence on one end of
the held perpendicular to the sides being 40 rods.
24. Show that the area of a circle is equal— (1) to .^-UISS times the
square of its radius ; (2) to '7854 times the square of the diameter.
2r>. How many square acres are there within a circumference whose
radius is half a mile ?
26. In a circ'e whose radius is 10 feet a regular pentagon is inscribed;
m another whose radius is 15 feet is inscribed an equilateral triangle
How much more space is included between the rectilinear and circular
boundaries '.n the one case than in the other?
^?* Ji?*^ *^® *''®* °' * circuUr cector whose radius is 10 and whose
arc IS 30 . ""«•«
28. There is a field in the form of a regular hexagon; the rope by
Which a horse is tethered is fastened in one angle ; how long must the
rope be that the horse's feeding-ground may be i of an acre ?
29. What is the area of a circular segm'^nt, the arc of which is that
of a quadrant, the diameter being 12 feet T
30. The arc of a segment of a circle is 65«, and its radius 25 jet ;
what IS the area of the segment T
31. Find the area of a circular sector, of which the arc is subtended
I by a chord 8 feet long, and half the arc by a chord 6 feet long.
32. The height of the arc of a sector is 2-6, and the chord of half the
arc IS o ; what is the area of the sector ?
Sug. —Diameter = (chord of i arc) » -j- height of arc ; why ?
33. Explain how to find the area of a circular zone ; of the Tina
between the circumferences of two concentric circles.
34. How many square yards are there in a quadrangular plot of
^^V^J'J!"® <l»ago^al8 of which are 18 and 21 feet, and their contained
angle 30 ?
35. Demonstrate a formula for finding the area of an irregular four-
sideu hgure. two of whose opposite angles are supplementary, its sides
^^taJ^^^^^^A*"^^ °^ * quadrilateral field, two of whose opposite sides
are 500 and 400 links, and the other two 450 and 350 links reapectivelv
the diRgonals being inclined at an angle of 80°. '
37. A triangular board is cut in a line which bisects two adjacent
sides ; what is the ratio of the areas of the 8ogments?
Sug.— The triangular segment and the whole board are similar.
38. A circular coin is four times as large as another ; compare their
39. The sides of a triangle are to one another as 3, 4, and 5 ; compare
the areas of the semicircles described on these sides as diameters.
40. If the semicircle on the longest side in Ex. 39 includes the
triangle, compare the sum of the areas of the lunes formed outside of it
with the area of the triangle.
ANSWERS.,,
CHAPTER I. Page 10.
». 8-85. /
4. 404. /
10. 120^ 101, 625.
CHAPTER II. Page 14
•• IVfi; WE.
10. I ; |.
CHAPTER III, Page 18.
8. Sine 0, 1 ; Cosine 1, ; Tangent 0, oo ; Cotangent «, : Secant
1, oo ; Cosecant oo, 1.
4. 90° ; 45> ; 0° ; 0°. | b. 30' ; 60° ; 60'.
6. 00, 0; 00,-1. j 7. 450. 3^0
8. Pine = Cosine = iV2; Tangent = Cotangent=l ; Secant = Co-
secant =^2.
9. Sine = i^; Tangent=V3; Cotangent 4 V^ ; Secant=2; Co-
secant =1^3.
^°* ^^'6ose?ant-2^°"'^^'^^'^^' ^°**''««"* = V3 ; Secant = §V3;
11. Cosine, -9511 ; Tangent, '3249 ; Cotangent, 3 078 ; Secant, J 0409:
Cosecant, 3-236.
12. 33" ; 67". I 13. -8148 ; -5398.
14. -6361; -8442; -2959; 9553.
15. 1-562; 1-855. ( le. 57» 17'
18. 34»36'; 21°; 57' 6' 15"; or, 145' 24'; 159'; 122' 63' 45".
19. 57° 3' ; 32° 58' ; 38' 41' 40". | 20. 9802 ; 5 454.
CHAPTER IV. Page 25.
6. C = 37'; 6 = 500; c = 300.
6. ^ = C=45°; a = c = 14; 6 = 19-8.
& A =48' 35' ; 1^41' 25' ; c = 26-4.
C = 50'; 6 = 80-89; c=61-98.
I 7. 1 : V2 : 1.
^ =37' 30' ; 6=862-4 ; c=684-l.
^=51' 23'; C=38'37'_; c= 499-4.
^ = C=45'; a = c = 6V2.
^=30°; C=60'; a=57 -7+ ; 6=115-5.
.4=30°; ^=60°; 6 = 100; c = 50V3.
^=44' 50' ; C=45° 10' ; c = 4493 6 ; a = 4466-88.
14
ra
ANSWERS
t.
A.BziO.
10.
C = 2fl^ a = 6lfl-28; r=S87-87.
A =61"%' I C=38'52'; «=439a
u.
= 4^67; 5=80^33'; 6=118-7.
19.
il^iOMC; a = 219+; c=224 0.
il=97"58'; 6=1731; c = 143-9.
16.
B=63^ 168' ; C=8r 44-2' ; c = 83 9.
W7
Secant
= Co-
:; Co-
•0409;
1
Of-
.8=126' 44 2* ; C=r IS'S ; c=12'2.
IX CZ>=48-08; AD = 48-0S; BD :^ 35 8%
IT. B = C=8r; 6 = c=4a». \*3 iJ5.
19. i4 = 70' 52-9' ; (7 = 48' 32 ■ r ; /* = ft:) 48.
90. (1)43 MJ; 44 3; 43.
(•:) S/?^>neDta o: B 19' 71', 41 27 9' ; M^ 8-65', 26^ 26-45'.
iefc.m .i.ta of 6 1600, 3842 ; :i366. -JUSS,
V. r») A - 4-y 25' ; ^--f>7° 7 6' ; C=68° 27-5'.
(2) A -86* 24-8' B=z6k' 48' • C=34' 33-2'.
(H) .! - ^ = {J3' 37 ' ; C- 12' 46'.
si) A - 36' 6^ ; ^=63' t' ; C^90.
OHi^PTER V. Page 31.
ft. 32 ; 243 ; 1024 ; 0126. A'
8. 1-6 ; 015625; -C 2; -s/fll; 's/2 ; -^2.
19. -90309 ; 20412 ; 1-43136 ; 3 09162 ; 2!>0848 ; -69897 ; 1*82391 j
i -77816; 2-82391 ; 2-9*2082; 1-55630; 280618; '37482.
14. 3-8754; 15501; 4 8157; 2862; 12-58 + .
16. Results obtaiT^ed by logarithms differ more or less from those other*
wise obtai..ed..t^\\ •"'^ ^i?^^^ •
16. 1-6340; I <t»44 ; 1 9555 ; T-3027 ; 1-6340; 1-9911.
^0. 56' 59'; 123' 1'; 33" 1'; 39' 69'; 60' 1'.
3. (1) (7=42' 30'; 6 = 7716; c = 6213.
(2) A = 36' 32' ; C= 54' 28'; c = 638 -4.
(3) ^ = 39° 60'; 5=50' 12'; (7=89' 58'.
(4) 5 = ^6' 46' or 143' 14'; C=107' 44' or P 16'; c = 119 or -276.
(5) C7=99'6' 13"; 6 = 1103; flC^192.
(6) ^ = 79' 59-2' ; 5=5-2' 17'; C:=47' 43-8'.
CHAPTER VI
4. 48 ft.; 50° 12'; 264 8 ft.
6 104-55 ft.
6. 408-8 ft.
7. 96-72 ft.
8. 91 -45 ft.
16. 882 ft., 916-2 ft., 1553'6 ft.; '923 ft., 9626 ft., 3724 -4 ft.
16. 6-9, 5 02, 4-87 miles.
a
Page 37.
9. i^Vda.
10. 138-4 ft.
11. 713-9 yds.
12. 2610 yds. , .
13. 79'*7d8.7^-b^*'*^AA
98 PRACIICAL MATHEMATICS.
IT. 105 4. I23-2, 234-3 furlong*.
18. 16 2 ft. I -tt. 9\'\ ; 22-07 ft.
20. Jiisori»»o<l— 48; 49GS1>6; 50-l!20. ^
CircuiMcrib«d- S.'i 4;M)4 ; 014368; 50-5728.
CHAPTKR VII. Page 47.
3. (1) 1070 mil«i ; (2) a770mil«M.
4. (1) 6o.)A miles ; (2) 87 18^ miltti ; 8640 milei.
19. (1) 19U1; (-2)4123. | 20. 51' 27-.r ; 3S-3 mile* ^ ,
21. 31 pointB ; 1)7 + inilea. | 22. *209 miles ; 47- 43* N. -*' '^21
27. Lftt. in 37° 0-5' N ; Un. in 8"'t«-»' B. JLJL" -tf 6'</'<rW
28. 8. 64* 23" W. ; 1106 mile*. | 29. 2666 ; ^8e» 46^8. ^^ir^i "zy. ^^
80. 90 miles. ( 81. 32^ 38' 8".
32. 60; 66-38; 61-06; 42-42; .38-57 L^j 15-53.
83. (1) 0" 62' 10" N.; (2) 4» 18' N. ^
88. (1) Ut. in 64" 13-2' N.; I^. in 40' 23' 6" W.; (2) S. 74' W.,
471-7 miles.
40. Lat in 20" 16' S.^ Lon, in 90^ 68-7' E.
42. N.VV. I 43. S. 5» W.
44. S 85° 45' E.
45. (1) Lat. in 11" 15-6' N.; (2) Lon. 120' 2' E.
46. S. 30^ 59^' W.; 5180 miles.
47. Lat. in 37" 7 7' S.; Ixm. in 15r 37 7' E.
43. Lat. in 39* 3*3' S.; Lon. in 89' 23 5' E.
49. 4^28 ft.
80. (1) SOlt miles ; (2) 421, V» mile* ; (3) 43Vo'(i miles.
Page i30.
, 144 22 ft.
. 13-05 nearly, 10 03 + ; 0.
. 53-20 ft.
. 27 -52 sq. in.
. 34 5-2.
. 61-42; 9 05.
, 133163.
. 67366-4.
. 215 04.
. 23*1 poles nearly.
, 6-947; 7-483; 8017.
. 10; 10 V^; 20; 20V2 ; 40.
. 2-071 +.
. 2-C38 + .
. 14 ac. 1 rood.
L.1.V3.N/3
4t' 16' 36 ' -J4'
CHAPTER VIIL
3.
208 7 + ft; 293 H-ft.
23
4.
ld\
26
aj« ,-
27
6.
~4 'V ^•
28
6.
166 27.
29
7.
21-21+.
30
8.
(1)43-3 + . (2)20-78.
81
9.
1019.
32
:o.
83J yds.
33
11.
'2400, 2600, 3200, 1800.
34
12.
2 89; 24-24; 379 47.
35
13.
126 67.
36
14.
18 0-2; 25-7.
37
15.
-801-873. 'tSII-^yi'-
33.
16.
280-5 yds.
39
19.
lfia-1888.
20.
91-05 ft.
40.
22.
61 •'^^'
AH&yNI^Ji.
H.^i '721
-^/^^^ay.^"-^
74' W.J
CHAFIKR IX. Page 65.
T. 139 176 iq. in.; .303-84 sq. in.; 403 056 sq. ia
$. 71-793 3q. in.} i7i-16i<|. in.; 236*39 iq. in.
9. 1690 m. in. t U9'^ M
11. $3-25. ^
18. 621%.
10. 72 iq. ft
12. 71 6sq. y(!0. ,
16.6l2-83aq. in. ai'^^Mj -.j
16. Torrid, 80173632 sq. m. e»ob t«mi)«rat«, 52100294 m. m.; eaoh
frigid, 63440'J0 sq. m.
II
CHAITER X. Page 71.
«.
8.
6.
7.
51 14 eu. in.
I'M uu. in. ; 3450 ou. in.
4 ft.
9'Sy (luartii.
. . , fda
8. -8-03 in.; 1 Utr*. 1 0«/ ♦«.•,'/.(,
10.
1 : Z.
H
w
•- L
CHAPTER XI. Pago 85.
;"0.
8. 68.
9. 2 ; ^/2.
10. 5 Ibe ; 6V2 Ibfc
11. (1) 14 1 + ; (2) 12+; (3)
9 5-i-;(4)8-2+;(6)7'l + .
12. 27-7; 64' 20'.
14. 10 3 + .
18. ,30^3.
16. 13 06591bi.
17. 7 2; 129-7.
«i
19. Vs/A ; 120° with first force.
23. 38 lbs.; 114 lbs.
24. 28jor 12-
26. V3 : 2.
27. 13i in. from 1 lb.
23. ,31>iu. f6^--)
31. ^ in. from centre.
34. 1200 Ibp.
3l. 30 ft + .
36. Tan 20": Sin 40*.
38. 48i lbs.
39. 14.) lbs. +
40. 2288} Ibi.; lUlUbs.
41. 2200 Ibi.
43. SOlb'S. +.
44. 3 ft. 4ill.+.
46. 125 lbs.
49. 151 lbs.
80. 1 : 271-4 + .
51 40500 lbs.
84, 1 10^ ft.
58. 497ift.; lli»,V«econdi.
86. 135-6 ft. from ground ; .35-6
ft. ; 64-4 ft per aecoud.
87. i second.
68. SOO ft. per sec.
69. Final velocity of the two
denoted by J.'
60. 15 ft
'2 ; 40.
*ar
II
TABLES.
102
lU
i
PRACTICAL MATHEMATICS.
Table I.
NATUI-AL FUNCTIONS.
X
P
0'
10'
20'
30'
40'
60' jp
X
0-
Sin.
Cos.
Tan.
Cot.
1.0000
.0029
1.0000
.0029
.0058
1.0(K)0
.0068
.0087
1.0000
.0087
.0116
.9999
.0116
.0145
.9999
.0146
Cos.
Sin.
Cot.
Tan.
Cos.
Sin.
Cot.
Tan.
Cos^
Sin.
Cot.
Tan.
Cos.
Sin.
Cot.
Tan.
89°
87°
86"
85°
34°
V
4°
'Sin.
C08.
Tan.
Cot.
Sin.
Cos.
Tan.
Cot.
Sin.
Cos.
Tan,
Cot.
Sin.
Cos.
Tan.
Cot.
.0175
.9998
.0175
57.29
.0204
.9998
.0204
49.10
.0233
.9997
.0233
42.96
.0262
.9997
.0262
38.19
.0291
.9996
.0291
34.37
.0320
.9995
.0320
31.24
.0494
.9988
.C495
20.21
.0669
.9978
.0670
14.92
.0349
.9994
.0349
28.64
.0378
.9993
.0378
26.43
.0552
.9985
.0553
18.08
.0407
.9992
.0407
24.54
.0581
.9983
.0582
17.17
.0436
.9990
.0437
22.90
.0611
.9981
.0612
16.35
.0465
.9989
.0466
21.47
.0640
.9980
.0641
15.60
.0814
.9967
.0816
12.25
.0523
.9986
.0524
19.08
.0698
.9976
.0699
14.30
.0727
.9974
.0729
13.73
.0756
.9971
.0758
13.20
.0785
.9969
.0787
12.71
.0843
.9964
.0846
11.83
Cos.
Sin.
Cot.
Tan.
Cos.
Sin.
Cot.
Tan.
&"
Sin.
Cos.
Tan.
Cot.
.0872
.9962
.0875
11.43
.0901
.9959
.0904
11.06
.0930
.9957
.0934
10.71
.0959
.9954
.0963
10.39
.0987
.9951
.0992
10.08
.1016
.9948
.1022
9.788
6'
7"
Sin.
Cos.
Tan.
Cot.
Sin.
Cos.
Tan.
Cot.
.1045
.9945
.1051
9.514
.1074
.9942
.1081
9.255
.1103
.9939
.1110
9.010
.1132
,9936
.1139
8.777
.1161
.9932
.1169
8.556
.13.34
.9911
.1346
7.429
.1190
.9929
.1198
8.345
.1363
.9907
.1376
7.269
Cos.
Sin.
Cot.
Tan.
83°
.1219
.9925
,1228
8.144
.1248
.9922
.1257
7.953
.1276
.91)18
.1287
7.770
.1305
.9914
.1317
7.596
Cos.
Sin.
Cot.
Tan.
82°
X
F
60'
50'
40'
30'
20'
10'
pL.
Sec x~
Sin. «
NATURAL FUNt'TIONS.
Table I.
NATURAL FUNCTIONS.
103
«l F
0'
10'
20'
30'
40'
50 ! F
M
Sin.
.1392
.1421
.1449
.1478
.1507
.1536
Cos. 1
u (Jos.
^ Tan.
.9903
.9899
.9894
.5 890
.9806
.9881
Sin.
81'
.1405
.1435
.1465
.1495
.1524
.1554
Cot.
Cot.
7.115
6.968
.1593
6.827
6.691
.1650
6.661
.1679
6.435
ran.
(^08.
Sin.
.1664
.1622
.1708
9*
Cos.
.9877
.9872
.9868
.9863
.9858
.9853
Sin.
80°
Tan.
,1684
.1614
.1644
.1673
.1703
.1733
Cot.
Cot.
6.314
6.197
6.084
.1794
6.976
.1822
5.871
5.769
Tan.
Cos.
— .
Sin.
.1736
.1765
.1851
.1880
Cos.
.9848
.9843
.9838
.9833
.9827
.9822
Sin.
79°
^^°T8.i.
.1763
.1793
.1823 .1853
.1883
.1914
Cot.
Cot.
6.671
5.576
.1937
5.485
5.396
.1994
5.309
5.226
.2051
Tan.
Cos.
Sin.
.1908
.1965
.2022
ir
Cos.
.9816
.9811
.9805
.9799
.9793
.9787
Sin.
78°
Tan.
.1944
.1974
.2004
.2035
.2065
.2095
(Jot.
Cot.
5.145
.2079
5.066
4.989
4-915
4.843
4.773
Tan.
Cos.
Sin.
.2108
.2136
.2164
.5>193
.2221
12'
Cos ' .9781
.9775
.9769
.9763
.9757
.9750
Sin.
77°
Tan.
.2126
.2156
.2186
.2217
.2247
.2278
Cot.
—
Cot.
4.705
4.638
4.574
.2300
^.511
4.449
4.390
.2391
Tan.
Cos.
Sin,
.2250
.2278
.2334
.2363
13»
Cos.
.9744
.9737
.9730
.9724
.9717
.9710
Sin.
76°
Tan.
.2309
.2339
.2370
.2401
.2432
.2462
Cot.
Cot.
^in
4.331
4.275
4.219
.2476
4.165
.2504
4.113
4.061
Tan.
Cos.
.2419
.2447
.2532
.2560
Cof^
.9703
.9696
.9689
.9681
.9674
.9667
Sin.
75°
14"
Tnn
.2493
.2524
.2555
.2rSG
.2617
.2648
Cot.
Cot.
4.011
3.9G2
.2616
3.914
3.867
3.821
3.776
Tan.
Cos.
Sill
.2588
.2644
.2672
.2700
.2728
15°
Co.^
.9659
.9652
.9644
.9636
.9628
.9621
Sni.
74°
Tan
.2679
.2711
.2742
.2773
.2805
.2836
Cot.
Cot
3.732
3.689
3.647
3.606
3.566
3.526
Tan.
X
F
60'
50'
40'
30'
20'
10'
F
.
Sec. »=7^
Cosec. »=£,-
— X
OS.
1
— X
in.
Limits 1 rr„J „ A —
^ J. an. X u, w«
; Cos. X, 1, 0:
; Cot. x^, G.
104
PRACTICAL MATHEMATICS.
J I
16
IT
Sia
. Cos.
Tan
Cot.
Shi!
, Coa
Tan.
Cot.
Table I.
NATURAL FUNCTIONS.
0'
.2766
.9613
.2867
3.487
10'
18'
19
UO
Sin.
Cos.
Tan.
Cot.
SiU;
Cos.
Tan.
Cot
Sin!
Cos.
Tan.
Cot
.2924
.9563
.3056
3.271
.3090
.9611
.3249
3.078
.2784
.9605
.2899
3.450
.2952
.9665
.3089
3.237
20 30'
.3256
.9466
.3443
2.904
21'
22
23
Sin.
Cos.
Tau.
Cot.
Si^
Cos.
Tan.
Cot.
Sin.
Cos.
Tan.
Cot.
.3420
.9397
.3640
2.747
.3118
.9502
.3281
3.047
.2812
.9596
.2931
3.412
.2979
.9546
.3121
3.204
.3283
.9446
.3476
2.877
.3584
.9336
.3839
2.606
.3448
.9387
.3673
2.723
.3611
.9395
.3872
2.583
.3145
.9492
.3314
3.018
.3311
.9436
.3508
2.850
.2840
.9588
.2962
3.376
40'
.300?
.9537
.3153
3.172
.3746
.9272
.4040
2.475
.3907
.9205
.4246
2.356
.3773
.9261
.4074
2.466
.3934
.9194
.4279
2.337
.3476
.9377
.3706
2.699
.3638
.9315
.3906
2.560
.3800
.9260
.4108
2.434
.3961
.9182
.4314
2.318
.3173
.9483
.3346
2.989
.3338
.9426
.3541
2.824
.3502
.9367
.3739
2.676
.2868
.9580
.2994
3.340
.3035
.9628
.3185
3.140
.3201
.9474
.3378
2.960
.3665
.9304
.3939
2.639
.3366
.9417
.3574
2.798
.3529
.9356
.3772
2.661
X
60' 50' I 40'
Sec. x=
Cosec. x;=
Cos.
1
Sin.
.3827
.9239
.4142
2.414
.3987
.9171
,4348
2.300
30'
.3692
.9293
.3973
2.517
.3854
.9228
.4176
2.394
.4014
.9159
.4.383
2.282
"20'
60'
.2896 iCr^
.9672 I Sia
.3026
3.306
X
.3062
.9620
.3217
3.108
.3228
.9465
.3411
2.932
.3393
.9407
.3607
2.773
Cot. 73"
Tan. j
Cos.'
Tan.
Cos.
Sin.
Cot,
Tan,
.3567
.9346
.3805
2.628
Cos.
Sin.
Cot.
Tan.
W
.3719
.9283
.4006
2.496
.3881
.9216
.4210
2.375
.4041
.9147
.4417
2.264
Cos.
Sin.
Cot.
Tan.
Cm.
Sin. i
Cot.
Tan.
Cos.
Sin.
Cot.
Tan.
70"
69°
68*
67°
Cos.
Sin. ofsoi
Tan.'
10'
P
X
Limits -SS,^"- ^J. 1; Cos. a: 1.0:
(Tan. x 0, 00 1 Cot. x or , 0.
p
1
X
'/6.
a.
)t.
73*
71
70
69'
68*
NATURAL FUNCnONa
Tablb I.
NATURAL FUNCTIONS.
105
X
r
0'
10'
20' 30'
40'
50'
r
X
24*
26'
26°
27°
28°
29°
30°
31'
X
Sin.
Cos.
Tan.
Cot
.4067
.9135
.4452
2.246
.4094
.9124
.4487
2.229
.4120
.9112
.4622
2.211
.4147
.!)i00
.4667
2.194
.4305
.9026
.4770
2.097
.4462
.8949
.4986
2.006
.173
.9088
.4692
2.177
.4200
.9075
.4628
2.161
Cos.
Sin.
Cot.
Tan.
Cos.
Sin.
Cot.
Tan.
Cos.
Sin.
Cot.
Tan.
66'
64°
63°
Sin.
Cos.
Tan.
Cot.
Sin.
Cos.
Tan.
Cot.
.4226
.9063
.4663
2.146
.4253
.9051
.4699
2.128
.4279
.9038
.4734
2.112
.4331
.9013
.4806
2.081
.4358
.9001
.4841
2.0«6
.4384
.8988
.4877
2.050
.4410
.8975
.4913
2.035
.4436
.8962
.4950
2.020
.4488
.8936
.5022
1.991
.4514
.8923
.6059
1.977
Sin.
Cos.
Tan.
Cot.
Sin.
Cos.
Tan.
Cot.
.4540
.8910
.5095
1.963
.4566
.8807
.5132
1.949
.4592
.8884
.6169
1.935
.4617
.8870
.5206
1.921
.4643
.8857
.5213
1.907
.4669
.8843
.5280
1894
Cos.
Sin.
Cot.
Tan.
Cos.
Sin.
Cot.
Tan.
Cos.
Sin.
Cot.
Tan.
Cos.
Sin.
Cot.
Tan.
Cos.
Sin.
Cot.
Tan.
62°
61°
60°
59°
58°
.4695
.8829
.5317
1.881
.4720
.8816
.5354
1.868
.4746
.8802
.5392
1.855
.4772
.8788
.5430
1.842
.4797
.8774
.5467
1.829
.4823
.8760
.5505
1.816
Sin.
Cos.
Tan.
Cot.
Sin.
Cos.
Tan.
Cot.
.4848
.8746
.6543
1.804
.4874
.8732
.5681
1.792
.4899
.8718
.5619
1.780
.5050
.8631
.5851
1.709
.5200
.8542
.6088
1.643
.4924
.8704
.5658
1.767
.4950
.8689
.5696
1.756
.4975
.8675
.6735
1.744
.6000
.8660
.5774
1.732
.5025
.8646
.6812
1.720
.5175
.8557
.6048
1.653
.5075
.8616
.5890
1.698
.6100
.8601
.5930
1.686
.5125
.8587
.5969
1.676
Sin.
Cos.
Tan.
Cot.
.6160
.8672
.6009
1.664
.6225
.8526
.6128
1.632
.5250
.8511
.6168
1.621
.5275
.8496
.6208
1.611
F
60'
50'
40'
30
20'
10'
F
X
Sec. X-
Cos.x Tiniita ^ ^"^ « 0, 1; Cos. z 1, 0:
Cosec. x = -L ( ran. a: 0, 00 ; Cot x « , 0.
Sin. X
106
ff'
PRACTICAL MATHEMATICS.
Table I.
NATURAL FUNCTIONS.
X
F
10'
20'
30'
40'
50
F x\
Sin
.6299
.5321
.5.348
.5373
.6398
.5422
<\)S
■ ^amm
82'
• Cos
.8480
.8465
.8460
.8434
.8418
.8403
Sin
Tun.
.e5>49
.6289
.6330
.6.371
.6412
.6453
Cot
6T
Cot.
Sin.
1.6(K)
1.690
1.680
.6495
1.670
1.660
.6544
1.650
Tan
M\{\\ .5471
.6519
.6568
Cosl 1
goo Cos.
.8:}H7
.8371
.8355
.8339
i .8323
.8307
Sin
lan.
Cot.
.6494
1.540
.6536
1.530
.6577
1.620
.5640
.6619
1.511
.6664
.6661
1.501
.6703
1.492
Cot.
Tan.
Cos.
56"
Sin.
.5592
.5616
.5688
.6712
34°
Cos.
1...
.8290
.8274
.8258
.8241
.8225
.8208
Sin
jian.
.6745
.6787
.68.30
.6873
.6916
.6959
Cot
bh'
Cot.
1.483
.5736
1.473
.5760
1.464
' .5783
1.455
.5807
1.446
1.437
Tan.
Sin.
.5831
.5854
Cos
35°
Cos.
.8192
.8175
.8158
.8141
.8124
.8107
Sin
Tan.
.7002
.7046
.70fiO
.7133
.7177
.7321
Cot.
64°
Cot.
1.428
1.419
.590J
1.411
J. 402
1.393
1.385
Tan.
Sin.
.5878
.5925
.5948
.5972
.5995
Co.s
36°
Cos.
.8090
.8073
.8056
.8039
.8021
.8004
Sin
Tan.
.7265
.7310
.7355
.7400
.7445
.7490
(^ot
53"
Cot.
S7n.
1.376
1.368
1.360
1.351
1.343
1.335
Tan.
.6018
.6041
.6065
.0088
.6111
.6134
Cos
37°
Cos.
.7986
.7969
.7951
.7934
.7916
.7898
Sin.
Tan.
.7536
.7581
.7627
.7673
.7720
.7706
Cot. ^'^"I
Cot.
Sin.
1.327
1.319
1.311
1.303
.6225
1.295
1.288
Tan.
.6157
.6180
.6202
.6248
.6271
Cos
«jco Cos.
.7880
.7862
.7844
.7826
.7808
.7790
Sin
I'an.
.7813
.7860
.7907
.7954
.8002
.8050
Cot
61'
Cot.
1.280
1.272
1.265
1.257
1.250
1.242
Tan.
(7os.
Sin.
.6293
.6316
.6338
.6361
.6383
.6406
39°
Cos.
.< / < 1
.7753
.7735
.7716
.7698
.7679
Sin,
Tan.
.8098 ; .8146
.8195
.8243
.8292
.8342
Cot.
50"
X
C6t.
1.2.35
1.228
1.220
1.213
1.206
1.199 '
ran.
P
60'
5G^
40'
30
20'
10'
p A
Sec. a;-;.- ^.
Cos. X
1
Lii
-*«{lis
. a; 0,1;
Cos. tB 1, 0:
Cose
e, T- "■
1. X 0, CO
; Cot. a; 00 . 0.
Sill X
J
NATURAL FUNCTIONa.
107
Tablk I.
NATURAL FUNCTIONS.
X
X
40"
41'
42°
43°
44°
F
Sin.
Cos.
Tan.
Cot.
Sin.
Cos.
Tan.
Cot.
Sin.
Cos.
Tan.
Cot.
Sin.
Cos.
Tan.
Cot.
Sin.
Cos.
Tan.
Cot.
0'
10' 20 ! 30'
40'
50
F
X
.6428
.7660
.8391
1.132
.6450
.7642
.8441
1.185
.6472
.7623
.8491
1.178
.6494! .6517
.7004 . .7585
.8511 .8591
1.171 t 1.164
.6026 .6648
.7490 1 .7470
.8847 .8899
1.130 1.124
.6r).39
.75(50
.8942
1.157
Sin.
'^ot.
Tan.
Cos.
Sin.
Cot.
Tan.
Cos.
Sin.
Cot.
Tun.
Cos.
Sin.
Cot.
Tan.
49-
48'
47°
46°
.6561
.7547
.8693
1.150
.6583
.7528
.8744
1.144
.6713
.7412
.9057
1.104
.6841
.7294
.9380
1.066
.6967
.7173
.9713
1.030
.6604
.7509
.8796
1.137
.6734
.7392
.9110
1.098
.6862
.7274
.9435
1.060
.0070
.7451
.8952
1.117
.6799
.73.33
.9271
1.079
.6926
.7214
.9601
1.042
.6691
.7431
.9004
1.111
.0766
.7373
.9163
1.091
.6884
.7254
.9490
1.054
.6777
.7353
.9217
1.085
.6905
.7234
.9545
1.048
.7030
.7112
.9884
1.012
.6820
.7314
.9325
1.072
.6947
.7193
.9657
1.036
.6988
7153
.9770
1.024
.7009
.7133
.9827
1.018
.7050
.7092
.9942
1.000
C€«.
Sin.
Cot.
Tan.
45°
X
F 60'
60'
40'
30' 20'
10'
F
X
Sin. 45''=co3. 45''=\/2-T-2=-7071
Tan. 45' = cot. 45"= 1.
Cos. X Limits F*°- ^ <^' 1; Cos. X 1, 0:
^ 1 *^™'** ITan. X 0, CO ; Cot. a; oo , 0.
Cosec. X— . - . » f
bin. X
JOS
PRACTICAL MATHI^MATICa
Tablk II.
LOGARITHMS.
7' »
, i
i J
in an
N 12
3
. 4
1 6 6
7
8
9
10 0000
0043
0086
0128
0170
0212
0253
0294
0334
0374
11 0414
0453
0492
0531
0569
0607
0645
0682
0719
0756
12
0792
0828
0864
0899
0934
0969
1004
1038
1072
HOC
13
1139
1173
1206
1239
1271
1303
1336
1367
n99
iA30
14
1461
1492
1790
1523
1818
1663
1847
1684
1876
1614
1903
1644
1673
1703
1V32
15
1761
1931
1969
1987
2014
16
2041
2068
2095
2122
2148
2175
2201
2227
2253
2279
17
2304
2330
2355
2380
2405
2430
2455
2480
2604
2529
18
2653
2577
2601
2626
2648
2672
2695
2718
2742
2766
19
20
2788
2810
3032
2833
2866
2878
3096
2900
3118
2923
2946
2967
2989
3010
3064
3075
3139
3160
3181
3201
21
3222
3243
3263
3284
3304
3324
3345
3365
3385
3404
22
3424
3444
3464
3483
3502
3522
3641
3560
3579
3598
23
3617
3636
3655
3674
3692
3711
3729
3747
3766
3784
24
3802
3820
3838
3856
4031
3874
3892
3909
3927
4099
3946
4116
3962
4133
26
3979
3997
4014
4048
4065
4082
26
4150
4166 4183
4200
4216
4232
4249
4265
4281
4298
27
4314
4330 4346
4362
4378
4393
4409
4425
4440
4456
28
4472
4487 4502
4518
4533
4548
4564
4579
4594
4600
29
4624
4771
4639
4654
4669
4814
4683
4698
4713
4728
4742
4767
30
4786
4800
4829
4843
4857
4871
4886
4900
31
4914
4928
4942
4955
4969
4983
4997
5011
5024
5038
32
5051
5065"
5079
5092
5105
5119
5132
5145
5159
6172
33
5185
5198
5211
5224
5237
5250
5263
5276
5289
5302
34
5315
5441
5328
5453
5340
5465
5353
5366
5490
5378
5391
5403
5416
5428
35
6478
6502
6614
5527
6639
6661
36
5563
5575
5587
5599
5611
5623
5635
5647
5658
5670
37
5682
5694
5705
5717
5729
6740
5752
5763
5776
5786
38
5798
5809
5821
5832
5843
5855
5866
5877
5888
5899
39
40
5911
6021
5922
5933
6042
5944
5955
5966
5977
6085
5988
5999
6010
6031
6053
6064
6075
6096
6107
6117
41 6128
6138 6149
6160
6170
6180
6191
6201
6212
6222
42 6232
6243 6253
6263
6274
6284
6294
6304
6314
6?26
4.'i fi.'^Sfi
6345 : 6355
6365
6375
6385
6395
6405
6415
6425
44 6435 6444 | 6454
6464
6474
648-1
64U3 6503 1
6513
6522
I
LiXJAKITHMH.
Tablk II.
IX)GAilITHMS.
109
N
46
1
2 :
3 , 4
5 6
7
8
9
6532
6542
6551
6561
6571
6580 ' 6590
6699
6609
6618
46
6628
6637
6646
6656
6665
6675
6684 1
6693
6702
6712
47
6721
6730
6739
6749
6758
6767
6776
6785
6794
6803
48
6812
6821
6830
6839
6848
6857
6866
6875
6884
6893
49
50
6902
6911
6998
6920
7007
6928
7t)16
6937
6946
7033!
6955
7042
6964
6972
6981
7067
6990
7024
7050
7059
01
7076
7084
7093
7101
7110
7118
7126
7135
7143
7152
52
7160
7168
7177
7185
7193
7202
7210
7218
7226
7235
53
7243
7251
7259
7267
7275
7284
7292
7300
7308
7316
54
7324
7404
T332
7340
7419
7348
7427
7356
7435
7364
7443
7372
7451
7380
7388
7306
55
7412
7459
7466
7474
56
7482
7490
7497
7505
7513
7520
7528
7536
7543
7551
57
7659
7566
7574
7582
7589
7597
7604
7612
7619
7627
58
7634
7642
7649
7657
7664
7672
7679
7686
7694
7701
59
60
7709
7716
7789
V723
7731
773S
7810
7745
7818
7752
7760
7832
7767
7774
7782
7796
7803
7825
7839
7846
61
785?
7860
7868
7875
7882
7889
7896
7903
7910
7917
62
7924
7931
7938
7945
7952
7959
7966
7973
7980
7987
63
7993
8000
8007
8014
8021
8028
8035
8041
8048
8055
64
65
8062
8129
8069
8075
8082
8089
8156
8096
8102
8169
8109
8176
8116
8182
8122
8189
8136
8142
8149
8162
66
8195
8202
8209
8215
8222
8228
8235
8241
8248
8254
67
8261
826;
8274
8280
0287
8293
8299
8306
8312
8319
68
8325
8331
8338
8344
8351
8357
8363
8370
8376
8382
69
8388
8395
8401
8407
8470
8414
8420
8426
8488
8432
8439
8445
3506
70
8451
8457
8463
8476
8482
8494
8500
71
8513
8519
8525
8531
8537
8543
8549
8555
8561
8567
72
8573
8579
8585
8591
8597
8603
8609
8615
8621
8627
73
8633
8639
8645
8651
8657
8663
8669
8675
8681
8686
74
8692
8698
8704
18762
8710
8768
8716
8722
8727
8785
8733
8739
8^45
8802
75 8751
8756
'8774
8779
8/91
8797
76
8808
8814
8820
8825 1 8831
8837
8842 8848
8854
8859
77
8865
8871
i 8876
8882 ' 8887
8893
8899 8904
8910
8915
1 ^_ —
ni\Af\ onr, 4
1 Qnan
anar.
«Q71
78
89211
hM'Z 1
»9.ji:
j \jij\j-j
'.fx^'j"
79
8976
8982 8987
8993 i 8998
9004 9W9 9015
9020
9025
110
PllACTICAL MATHEMATIC&
1
Iaule II.
tOGAIUTHMS.
M
2 ! 3
4
5
6
7 8,9
80
9031
9036
9042 1 9047
9053
1 9058
9063
9069 9074 1 9079
81
9086
9090
9096
9101
9106 1 91 12
9117
9122 1 9128
9133
82
9138
9143
9149
9154
9159
9166
91V0
9176
9180
9186
83
9191
«J196
9201
9206
9212
9217
9222
9227
9232
9238
84
86
9243
9294
9248
9299
9253
9268
9309
9263
9315
9269
9320
9274
9325
9279
9330
9284
9289
9310
9304
9336
86
9345
9350
9356
9360
9365
9370
9376
9380
9386
9390
87
9?&6
9400
■ 9406
9410
9416
C42()
9426
9430
9436
9440
88
9445
9460 ; 9456
9460
9465
9469
9474
9479
9484
9489
89
90
9494
9542
9499 9')04 9509 9513
9518 ' 9523
9566 ' 9671
9628
9576
9633
9638
9586
!)547
9552
i 9557 > 9562
9681
91
9590
9595
'9600
9606 9609
9614
i 9619
9624
9628
9633
92
9638
9643 9647
9652 , 9657
9661
9666
9671
9676
9680
93 ' 9685
9689 Qh' 4
9699 . 9703
9708
9713
9717
9722
9727
94 9731
9736 9741
9782 1 9786
9746
9791
9750
9795
9754
9800
9759
9763
9768
9773
9818
95
9777
9806
9809
9814
9(5
9823
9827 1 9832
9836
9841
9845
9860
9864
9859
9863
9'/
9868
9872 9877
9881
9886
9890
9894
9899
9903
9908
98
9912
9917 9921
5^926
9930
9934
9939
9943
9948
9952
99
100
9956
0000
9961 9966
9969
0013
9974
0017
9978
0022
9983
9987
9991
9996
0039
0004
0009
0026
0030
0035 1
101
0043
0048
0052
0056
0060
0065
0069
0073
0077
0082
102
0086
0090
0095
0099
0103
0107
0111
0116
0120
0124
103
0128
0133
0137
0141
0145
0149
0154
0158
0162
0166
104
105
0170
0212
0175
0179
0183
0224
0187
0228
0191
0233
0195
0237
0199
0241
0204
0208
0216
0220
0245
0249
106
0253
0257
0261
0265
0269
0273
0278
0282
0286
0290
107
0294
0298
0302
0306
0310
0314
0318
0322
0326
0330
108
0334
0338
0342
0346
0350
0354
0358
0362
0366
0370
109
0374
0378
0382
0422
0386
0390
0430
0394
0434
0398
0438
0402
0441
0406
0410
110
0414
0418
0426
0445
0449
111
0453
0457
0461
0465
0469
0473
0477
0481
0484
0488
112
0492
0496
0500
0504
0508
0512 1 0515
0519
0523
0527
!ll3
0531
0535
0538
0542 1
0546
0550 0j54
0558
0561
OAfin
|li4
0569
0573
0577
0580 !
0584
0588 0592
0596 05i>9 , 0603 1
LOOAHlTUMa
111
Tabli IL
LOGARITHMS.
N
116
1
2
3 i 4
6
6
7
8
9
0607
0611
0616
0618 0622
0626 :
0630
0633
0637
0641
116
0645
0648
0652
0656
0660
0663
0667
0671
0674
0678
117
06S2
0686
0689
0693
0697
0700
0704
0708 0711 1
0716
118
0719
0722
0726
0730
0734
0737
0741
0745
0748
0762
110
120
0756
0769
0796
0763
0799
0766
0770
0774
0777
0813
0781
0786
0788
0792
0803
0806
0810
0817
0821
082
121
0828
0831
0335
0839
0842
0846
0849
0853
0856
0360
122
0864
0867
0871
0874
0878
0881
0885
0888
0892
0896
123
0899
0903
0906
0910
0913
0917
0920
0924
0927
0931
124
0934
0969
0938
0941
0946
0948
0983
0952
0986
0955
0990
0959
0993
0962
0997
0966
1000
125
0973
0976
0980
126
1004
1007
1011
1014
1017
1021
1024
1028
1031
1036
127
1038
1041
1045
1048
1052
1055
1069
1062
1065
1069
128
1072
1075
1079
1082
1086
1089
1092
1096
1099
1103
129
130
1106
1139
1109
1143
1113
1116
1149
1119
1153
1123
1156
1126
1159
1129
1163
1133
1136
1146
1166
1169
131
:173
1176
1179
11 83
1186
1189
1193
1196
1199
1202
132
1206
1209
.212
1216
1219
1222
1228
1229
1232
1235
133
1239
1242
1245
1248
1252
1255
1258
1261
1265
1268
134
1271
1303
1274
1307
1278
1231
1313
1284
1316
1287
1319
1290
1323
1294
1297
1329
1300
1332
135
1310
1326
13(i
1335
1339
1342
1345
1348
1351
1355
1358
1361
1364
137
1367
1370
1374
1377
1380
1383
1386
1389 ; 1392
1396
138
1399
1402
1405
1408
1411
1414
1418
1421
1424
1427
139
14!)
1430
1461
1433
1436
1467
1440
1471
1443
1474
1446
1477
1449
1480
1452
1455
1468
1464
1483
1486
1489
141
1492
1495
1498
1501
1504
1508
1511
1514
1517
1520
142
1523
1526
1529
1532
1535
1538
1541
1544
1547
1560
143
1553
1556
1559
1562
1565
1569
1572
1575
1578
1581
144
1584
1587
1590
1593
1623
1596
1500
1629
1602
1605
^308
1611
1641
145
1614
1617
1620
1626
1632
1635
1638
146
1644
1647
1649
165".
1655
1658
1661
1664
1667
1670
147
1673
1676
1679
1682
1685
1688
1691
1694
1697
1700
148
1703
1706
1708
1711
1714
1717 1 1720 , 1723
1726
1729
149
1732 1735
1738
i 1741 il744
1748 i 1749 j 1752
i 1765
1
1758
lis
I'UACTICAL MATUKMATICS.
Tabl« II.
LOGARITHMS.
N
1
2
3
4
6 6*7
8 I 9
1
160
151
152
153
154
1761
1790
1818
' 1847
1876
1764
1793
1821
1860
1878
1767
1796
1824
1853
1881
1770
1798
1827
1855
1884
1772
1801
1830
1858
1886
1776
1804
'833
1861
1839
1917
1946
1973
2000
2028
1778
1807
1836
1864
1892
1781
1810
1838
1867
1895
1784
1813
1841
1870
1898
1787
1816
1844
1872
1901
1928
1956
1984
2011
2038
20fi6
2092
2119
2146
2172
2198
2225
2251
2276
2302
2327
2353
2378
2403
2428
2453
2477
2502
2526
2550
166
156
157
158
159
1903
1931
1959
1987
2014
1906
1934
1962
1989
2017
1909
1937
1965
1992
2019
1912
1940
1967
1995
2022
1915
1942
1970
1998
2026
1920
1948
1976
2003
2030
2057
2084
2111
2138
2164
1923
1961
1978
2006
2033
1926
1963
1981
?009
2036
160
161
162
163
164
204:
2068
2095
2122
2148
2044
2071
2098
2126
2151
2177
2204
2230
2256
2281
2307
2333
2358
2383
2408
2433
2458
2482
2507
2531
2555
2579
2603
2627
2651
2047
2074
2101
2127
2164
2040
2076
2103
2130
2156
2052
2079
2106
2133
2169
2055
2082
2109
2135
2162
2060
2087
2114
2140
2167
2193
2219
2245
2271
2297
2063
2090
2117
2143
2170
2196
2222
2248
2274
2299
2325
2350
2375
2400
2425
2450
2475
2499
2524
2548
165
166
167
168
169
2175
2201
2227
2253
2279
2304
2330
2355
2380
2405
2430
2455
2480
2504
2529
2180
2206
2232
2258
2284
2310
2335
2360
2385
2410
2435
2460
2485
2509
2533
2558
2582
2605
2629
2653
2183
2209
2235
2261
2287
2312
2338
2363
2388
2413
2438
2463
2487
2512
2536
2185
2212
2238
2263
2289
2188
2214
2240
2266
2292
2191
2217
2243
2269
2294
170
171
172
173
174
175
176
177
178
179
2315
2340
2365
2390
24tl5
2440
2465
2490
2514
2538
2317
2343
2368
2393
2418
2443
2467
2492
2516
2541
2565
2589
2613
2636
2660
2320
2345
2370
2395
2420
2445
2470
2494
2519
2543
2322
2348
2373
2398
2423
2448
2472
2497
2521
2545
2570
2594
2617
2641
2665
180
181
182
183
184
2553
2577
2601
2625
2648
2560
2584
2608
2632
2655
2562
2586
2610
2634
2658
2567
2591
2615
2639
2662
2572
2596
2620
2643
2667
2574
2598
2622
2640
2669
LOaARITHM&
lis
\
Tabu II
LOGARmrMS.
&
ir
1R5
1
2
3
4
6
6
7
8
H
2fl73 1 2674
S676
2679
2681
2683
2686
2688 2«J)0 2H93
18f>
209f. 2697
2700
2702
27i>4
2707
2709
2''ll 2714 2716
187
2718 , 2721
2723
2726
27':rt
2730
273i
2736 i 2737 2739
188
2742
2744
2746
2749
2761
2763
2756
2768 27(50 ; 2762
189
2766
2767
2769
2772
2774
2797
2776
2799
2778
8801
2781
2804
27a3 2786
2806 ! 2808
190
.2788
2790
S798
8794
191
2810 2813
2815
2817
2819
2822
2824
2826
2828
2831
192
2833
2836
2838
2840
2842
2844
2847
2849
2851
2863
193
2856
2858 ' 2860
2862
2866
2867
2869
2871
2874
2876
194
2878
2880 1 2882
J
2885
2887
2909
2889
-
2911
2891
2894
8896
2898
196
2900
2903 2905
2907
2914 2916
2918 1 2920
196
2923
2925 2927
2929
2931
2934
2936
2938
2940 , 2«>42
197
2945
2947 ' 2949
2951
2953
2956
2958
2960
2962 2964
198
2967
2969 2971
2973
2975
2978
2980
2982
2984
2986
199
2989 ' 2991 2993
2995
3997
2999
3002 3004
3006
3008
H
114
nUCTlCAL MA'**HKMATIC&
Tablk [II.
LOGARITHMS OF TRIGONOMETRICAL RATIO&
■JO
I Hill.
» 10
1 Un.
f 10 1 cot f 10
1 COR. f 10
a
0* &
1
10.0000
90" 0'
0* 10'
7.4037
3011
7.4637
3011 18.6363
(!
10.0000
89' 60'
0' 20'
7.7648
1760
7.7648
1761 1 12.2368
lO.OCKX)
89' 40'
0* 30'
7.9408
1250
7.9409
1249 12.0891
10.(KK)0
89 .30'
{)' 40'
8.0658
969
8.0658
969
11.9342
10.00(K)
89 MY
o'^y
8 1687
792
8.1627_
792
11.8373
<»
lO.O'KX)
89" 10'
V c
n.2419
669
8.2419
670
11.7681
9.9999
89 0*
V 10'
H.30H8
680
8.3089
680
11.6911
9.999f)
88 60'
r 20'
8.3668
611
8.3G69
612
11.6.331
9.9999
88" 40'
1' 30'
8.4179
468
8.'1181
467
11.6819
9.99!)9
88 30'
1*40'
8.4637
413
8.46.38
416
11.5362
1
9.9998
88^ 20*
1' fto'
8.5050
378
8^5053
378
11.4947
9.9998
88" 10'
2" 0'
8.6428
348
8.5431
348
11.4669
1
9.9997
88' 0*
2' 10'
8,5776
321
8.6779
322
11.4221
9.9997
87" 6(V
2° 20'
8.6097
3k .0
8.6101
3(K»
11.3899
1
9.9996
87° 40'
2" 60'
8.6397
280
8.6401
281
11.3599
9.9996
87' 30'
2' 40'
8.6677
^33
t^.6682
263
11.3318
1
9.9996
87" 20'
2' f)0'
8.6940
248
8.6946
249
11.3056
9.9995
87° 10'
3' 0'
8.7188
235
8.7194
235
11.2806
1
9.9994
87' 0'
3° 10'
8.7423
222
8.7429
223
11.2571
1
9.9993
86' 50'
3" 20'
8.7645
212
8.7652
213
11.2348
9.9993
86° 40'
3° 30'
8.7857
202
8.7865
202
11.2136
1
9.Ci)92
86' 30'
3° 40'
8.8059
192
8.8067
1P4
11.1933
1
9.9991
86 20'
3" 50'
r 0'
8.8251
8.8436
185
177
8.8261
8.8446
186
11.1739
11.1554
1
1
9.999^)
9.9989
83' 1(1'
89^ 0'
178
r 10'
8.8613
170
8.8624
171
11.1376
9.9989
85° 50'
4° 20'
H.8783
163
8.8795
luo
i;.i2r6
1
9.9988
86' 40'
4" 30'
8.8946
158
8.8060
ms
n.Mv.'D
1
9')87
85° 30'
4' 40'
8.9104
152
8.9118
i 1.0lo2
1
9.9986
85° 20'
4' 50'
5 0'
8.9256
8.9403
147
142
8.9272
U8
143
11.0728
11.0580
1
2
9.9985
9.9983
85° 10'
85^ 0*
8.9420
a' 10'
8.9545
137
8.9563
138
11.0437
1
9.9982
84" 50'
b" 20'
8.9682
134
8.9701
135
11.0299
1
9.9981
84° 40'
5° 30'
8.0816
129
8.9836
130
11.0164
1
9.9980
84° 30'
5° 40'
8.9945
125
8.9966
127
11.0034
1
9.9979
84" 20'
5° 50'
9.0070
122
9.0093
123
10.9907
2
9.9977
84° lv>'
6" C
9-0192
119
90216
l-2() 10-9784
I
9-9976
84 0'
X
1 COS.
+ 10
1 cot. -
h 10 ' tan. + 10
1
Is
(ill. + 10
X
L«Ha» HUMM OK THUJO.NnMKTIlHAI. KAVIOK
115
Tablk hi.
U)(JAIUT».M8 OF TlUCJONOMilTKlCAL IL\TI08.
m
6' 0'
1 sin.
+ 10
t Ua.
» 10 Icut. tlO
I
con. + 10
X
9.0192
119
9.02 H$
120 10.97H4
I
9.9976
84 Of
6 •0'
9.U3I 1
li:.
9.03:'.6
117 I0.9<im
1
9.997t
H3 50'
6 20*
9.0426
113
9.0453
114
10.9647
2
1 9.9973
8.V40'
6 30'
9.0.')39
109
9.05^7
111
10.9433
I
!»9972
83* 30'
«" 10'
9.064H
107
9,0<178
lOH
10.9322
1
. MU71
83' ii)'
6' flO'
9.0766
104
9.0786
I or.
10.9214
>>}
Ad
9.99><9
83* lo'
r 0'
9.0869
102
9.0891
104
10.9109
1
9.99<i8
8:'° 0-
7M0'
9.0961
99
9.0996
lOl
10.}KK)6
2
9.9966
H2' 5i>'
7' 20'
9.1060
97
9.109({
98 ! 10.H!)()4
2
9.9964
H-r to'
7* 30'
9.1167
96
9.1194
97
10.8806
1
9.9963
b- "'
7" 40'
9.1262
93
9.1291
94
10.8709
2
9.9961
82^ 2* '
r 60'
9.1346
91
9.1385
9.') 10.8616
2
9.9959
82° 10'
8' (y
9.1436
8f^
9.1478
91
10.8622
I
9.9958
82' C
8 10'
9.1626
87
9.1669
89
10.8431
2
9.9956
81" 60'
8' iO'
9.1612
85
!).1668
87
10.8342
2
9,9954
8r40'
8" ?,0'
9.1697
84
9.1746
8(J
10.8265
2
9.9952
81° 30'
8° -10'
9.1781
82
!).1831
B-i
10.8169
2
9.99.50
81° 2(''
_8'' .'iO'
9.18(}3
80
9.1918
_82
10.8086
2
_9.9948
81° 10'
9^ 0'
9.1943
7'»
9.1997
81
10.80O3
2
9.9946
81* Of
9" 10'
9.2022
78
9.2078
80
10.7922
2
9.9944
80° 5(»'
9°1'0'
9.2100
7(5
9.2158
78
10.7H42
2
9.9942
80° 40'
9" 30'
9.2176
75
9.2236
77
10.7764
2
9.9940
80° 30'
9° 40'
9.2261
73
9.2313
7(5
10.7687
2
9.9933
80° 20'
9"^ CO'
9.2324
73
9.2389
74
/ 0.76 11
2
9.9926
80° 10'
10' c
9.2397
71
9.2463
73 10.7537
2
9.9934
80° a
10° 10'
9.2468
70
9.2636
73 1 10.7464
3
9.9931
79' 50'
10" 20'
9.2538
68
9.2609
71
10.7391
.)
9.9929
79° 40'
10' 30'
9.2606
68
9.2680
70
10.7320
o
9.9927
79' .30'
10° 40'
9.2674 '
66
9.2760
69
10.7250
3
9.9924
79 20'
10' 50'
11° 0'
9.2740
9.2806
66
64
9.2819
9.2887
68
66
10.7.81
2
3
9.9922
9.9019
79" TO'
79° 0'
10.7113
11° 10'
9.2870
64
9.2953
67
10.7047
2
9.9*- 17
78° 60'
11° 20'
9.2934
63
9.3020
65
10.6980
3
9.9914
78° 40'
ir3o'
9.2997
61
9.3085
64
10.6915
2
9.9912
78' 30'
11° 40'
9.3058
61
9.3149
63
10.6851
3
9.9909
78° 20'
11° 50'
9.3119 '
60
9.3212
63
10.6788
2
9.9907
78' io'
12° 0'
9.3179
69
9.3275
61 10.6725 1
3 '
9.9904
78° 0'
X 1 1 (;08. -
f 10
1 cot -
MO Itan. + 10| 1 t
»in. + 10
X
lie
PRACTICAL MATHEMATICS.
If
Table III.
LOGARITHMS OF TRIGONOMETRICAL RATIOS.
X
1 sin. + 10
I tan. + 10
1 cot. + 10
1 COS. + 10
X
12' C
9.3179
59
9.3275
61
10.6725
3
9.9904
IQ' C
12° 10'
9.3238
58
9.3336
61
10.6664
3
9.9901
77° 50'
12° 20'
9.3296
57
9.3397
61
10.6603
2
9.9899
77° 40'
12° 30'
9.3353
57
9.3458
59
10.6542
3
9.9896
77° 30'
12° 40'
9.3i:o
56
9.3517
69
10.6483
3
9.9893
77° 20'
12° 50'
9.3466
9.3521
55
64
9.3576
9.3634
58
10.6424
10.6366
3
3
9.9890
9.9887
77° 10'
77" C
13° C
67
13° 10'
9 3575
54
9.3691
57
10.6309
3
9.9884
76" 50'
13° 20'
9.3629
53
9.3740
56
10.6252
3
9.9881
76° 40'
13° 30'
9.3682
52
9.3804
55
10.6196
3
9.9878
76° 30'
13' 40'
a3734
52
9.3859
55
10.6141
3
9.9875
76° 20'
13° 50'
14° C
9.3786
61
60
9.3914
64
53
106086
106032
3
3
9.9872
9.9869
76° 10'
76° C
9.3837
9.3968
14^ 10'
9.3887
50
9.4021
63
106979
3
9.9866
75° 50'
14° 20'
9.3937
49
9.4074
53
106926
3
9.9863
75° 40'
14° 30'
9.3986
49
9.4127
62
106873
4
9.9859
75° 30'
14° 40'
9.4035
48
9.4178
51
105822
3
9.9856
75° 20'
14° 50'
9.4083
47
9.4230
51
50
105770
105719
3
4
9.9853
9.9849
75° 10'
75° C
16° C
9.4130
47
9.4281
15° 10'
9.4177
46
J.4331
50
105669
3
9.9846
74° 50'
15° 20'
9.4223
46
9.4381
49
10.5619
3
9.9843
74° 40'
15° 30'
9.4269
45
9.4430
49
106570
4
9.9839
74° 30'
15° 40'
9.4314
45
9.4479
48
105521
3
9.9836
74° 20'
15° 50'
16° C
9.4359
44
9.4527
48
105473 S 4
8.9832
74° 10'
9.4403
44
9.4575
47
105425
4
9.9828
74° C
16° 10'
9.4447
44
9.4622
47
105378
3
9.9825
73° 50'
16' 20'
9.4491
43
9.4669
47
105331
4
9.9821
73° 40'
16° 30'
9.4533
42
9.4716
46
105284
4
9.9817
73° 30'
16° 40'
9.4576
42
9.4762
46
10.5238
3
9.9814
73° 20'
16° 50'
9.4618
41
9.4808
9.4853
45^
45
105192
10.5147
4
9.9810
73° 10'
17° 0*
9.4659
41
4
9.9806
/2° 0'
17° 10'
9.4700
41
9.4898
45
105102
4
9.9802
72° 50'
17° 20'
9.4741
40
9.4943
44
10.5057
4
9.9798
72" 40'
17° 30'
9.4781
40
9.4987
44
105013
4
9.9794
72° 30'
17° 40'
9.4821
40
9.5031
44
10.4969
4
9.9790
72°2t>'
17° 50'
9.4861
39
9.5075
43
104925
4
9.9786
72° 10*
18^ Of
9.4900
39
9.5118
43
104882
|T
9.9782
72° C
X
1 COS.
4-10
1 cot. + 10
1 tan. + 10
It
jin. + 10
X
*
LOGARITHMS OF TRIGONOMETRICAL RATIOS. 117
Tablb III.
LOGARITHMS OF TRIGONOMETRICAL RATIOS.
X
1 sin.
+ 10
i tan. + 10 1 cot. + 10
I
cos. + 10
X
18° 0'
9.4900
39
9.5118
43
10.4882
4
9.9782
72^ (y
18° IC
9.4939
38
9.5161
42
10.4839
4
9.9778
71° 60'
18" 20'
9.4977
38
9.5203
42
10.4797
■ 4
9.9774
71° 40'
18° 30'
9.6015
38
9.5246
42
10.4765
4
9.9770
71° 30'
18° 40'
9.5052
38
9.5287
42
10.4713
6
9.9766
71° 20'
18° 50'
19° C
9.5090
9.6126
36
9.5329
9.5370
_41
41
10.4671
10.4630
4
4
9.9761
9.9757
71° 10'
71° C
37
19° 10*
9.5163
36
9.5411
40
10.4589
5
9.9752
70° 50'
19° 20'
9.6199
36
9.6451
40
10.4549
4
9.9748
70° 40'
19° 30'
9.6235
35
9.6491
40
10.4509
6
9.9743
70° 30'
19° 40'
9.6270
35
9.6531
40
10.4469
4
9.9739
70° 20'
19° 50'^
20° 0'
9.5306
9.6341
35
9.6671
40
10.4429
5
5
9.9734
9.9730
70° 10'
70° 0'
34
9.6811
39
10.4389
20° 10'
9.5376
34
9.5650
39
10.4350
6
9.9726
69° 50'
20° 20'
9.6409
34
9.5689
39
10.4311
4
9.9721
69° 40'
20° 30'
9.6443
34
9.6727
39
10.4273
5
9.9716
69° 30'
20° 40'
9.5477
33
9.6766
38
10.4234
5
9.9711
69° 20'
20° 50'
21° C
9.6510
33
33
9.5804
_38
38
10.4196
10.4158
5^
4
9.9706
9.9702
69° 10'
69^ C
9.5643
9.5842
21° 10'
9.557H
33
9.6879
38
10.4121
5
9.9697
68° 50'
21° 20'
9.6609
32
9.5917
37
10.4083
5
9.9692
68° 40'
21° 30'
9.5641
32
9.5954
37
10.4046
5
9.9687
68° 30'
21° 40'
9.5673
32
9.5991
37
10.4009
6
9.9682
68° 20'
21° 50'
22° 0'
9.5704
9.5V36
32
31
9.6028
37
36
10.3972
5
5
9.9677
9.9672
68° 10'
68° 0'
9.6064
10.3936
22° 10'
9.5767
31
9.6100
36
10.3900
5
9.9667
67° 50'
22° 20'
9.5798
30
9.6136
36
10.3864
6
9.9661
67° 40'
22^ 30'
9.5828
30
9.6172
36
10.3828
5
9.9656
67° 30'
22° 40'
9.5859
30
9.6208
36
10.3792
6
9.9651
67° 20'
22° 50'
23° 0'
9.5889
9.5919
30
30
9.6243
9.6279
35
36
10.3757
10.3721
5
6
9.9646
9.9640
67° 10'
67° 0'
23° 10'
9.5948
30
9.6314
35
10.3686
5
9.9635
66° 50'
23° 20'
9.5978
29
9.6348
35
10.3652
6
9.9629
66° 40'
23° 30'
9.6007
29
9.6383
35
10.3617
5
9.9624
66° 30'
23° 40'
9.6036
29
9.6417
34
10.3583
6
9.9618
66° 20'
23° 60'
9.6065
28
9.6452
34
10.3548
5
9.9613
66° 10'
24° C
9,6093
28
9,6486
34
10.3514
6
Q QKf\*7
66° 0'
X
1 COS. -
HO
1 cot. + 10
Itan. + 10
1 i
3in. + 10
X
118
PRAfTTK^AI. MATHEMATKJS.
il
5'
1
Table III.
LOCJARITHMS C7 TRirONOMETHICAL RATIOS.
X
1 sin.
+ 10
1 tan.
+ 10
loot. + 10| 1
COS. + 10
1 ^
24r 0'
9.6093
1 2:s
9.6486
34
10.3514
6
9,9607
66 0'
24^ 10'
!)6121
28
9.6520
34
10.3480
5
9.9602
65° 50'
24° 20'
9.6149
28.
9.6553
34
10.3447
6
9.9596
65° 40'
24° 30'
9.6177
28
9.6587
33
10.3413
6
1 9.9590
65° 30'
24° 40'
9.6205
27
9.6620
33
10.3380
6
9.9584
65° 20'
24° 50'
9.6232
9.6259
27
27
9.6654
9.6687
33
33
10.3346
10.3313
5
6
9.9579
65' 10'
26° 0'
9.9573
66 C
25° 10'
9.6286
27
9.6720
33
10.3280
6
9.9567
64° 50'
25° 20'
9.6313
27
9.6752
33
10.3248
6
9.9561
64 40'
25° 30'
9.6340
i ii6
9.6785
33
10.3215
6
9.9555
64° 30'
25" 40'
9.6366
26
9.6817
32
10.3183
6
9.9549
64 20'
25^ 50'
26' 0'
9.6392_
9.0418
_26
26
9.6850
9.6882
32
32
10.3150
10.3118
6
6
9.9543
9.9537
64° 10'
64° 0'
26° 10'
9.6444
26
9.6914
32
10.3086
7
9.9530
63° 50'
26° 20'
9.6470
25
9.6946
32
10.3054
6
9.9524
63° 40'
26° 30'
9.6495
25
9.6977
32
10.3023
6
9.9518
63° 30'
2()° 40'
9.6521
25
9.7009
31
10.2991
6
9.9512
63° 20'
2(r 50'
27 0'
9.6546
9.6570
25
25
9.7040
31
31
10.2960
t
9.9505
9.9499
63° 10'
9.7072
10.2928 6
10.2897 7
63° C
27° 10'
9.6595
25
9.7103
31
9.9492
62° 50'
27° 20'
9.6620
24
9.7134
31
10.2866
7
9.9486
62° 40'
27° 30'
9.6644
24
9.7165
31
10.2835
7
9.9479
62° 30'
27° 40'
9.6668
24
9.7196
31
10.2804
6
9.94/3
62° 20'
27° 50'
28° 0'
0.0692
9.6716
24
24
.i.7226
31
10.2774
10.2743
7
9.9466
62° 10'
9.7257
30
7
9.9459
62^ C
28° 10'
9.6740
24
9.7287
30
10.2713
6
9.9453
61° 50'
28° 20'
9.6763
24
9.7317
30
10.2683
7
9.9146
61° 40'
28° 30'
9.6787
23
9.7348 !
30
10.2652
7
9t>439
or 30'
28° 40'
9.6810
23
9.7378 :
30
10.2622
7
9.9432
61° 20'
28^ 50'
29 0'
9.6833
9.6856
23
23
9.7408
30
10.2592
7
7
9.9425
9.9418
61° 10'
61° 0'
9.7438
30
10.2562
29' 10'
9.6878
23
9.7467
30
10.2533
7
9.9411
60° 50'
29' 20'
9.6901
23
9.7497
30
10.2503
7
9.9404
60° 40'
29° 30'
9.6923
23
9.7526
29
10.2474
7
9-9397
60° 30'
29° 40'
9:6946
22
9.7556
29
10.2444
7
9.9390 60° 20'
29° 50'
9.6968
22
9.7585
29 i
1
10.2415
7 '
9.9383 60° 10'
30° 0'
9.6990
22
9=7614
i
29
10=2386
8
9.9375
ao^ 0'
.' 1 COS. -f
•10
1 cot. +
10 ]
tan. + 10
Isi
n. + 10
X
X
6
0'
5"
60'
5"
40'
5°
30'
5°
20'
5^
10'
5''
C
A °
60'
40'
^o
30'
20'
i°
10'
lO
0'
>0
50'
}0
.in'
A'
\
L00AEITHM8 OF TRIGONO>IKTRICAL RAl.OS. 119
Table III.
LOGARITHMS OF TlUdONOMETRICAL RATIOS.
X
30°
30°
30
30^^
0'
10'
20'
30'
30 40'
30^^ 50;^
31 C
3r 10'
31 20'
3r 30'
3r 40'
;ir50'
32° 0'
32° 10'
32° 20'
32° 30'
32° 40'
32' 60'
33 0'
33° 10'
33° 20'
33° 30'
33° 40'
33° 50'
lain. + 10 I 1 tan. + 10 I cot. + 10
34
34°
34°
34°
34°
34°
0'
10'
20'
30'
40'
50'
36° 0'
35° 10'
35° 20'
35° 30'
35° 40'
35° 50'
9.69:)0
9.7012
9.7033
9.7055
9.7076
9.7097^
9.7118
9.7139
9.7160
9.7181
9.7201
9^7222
9.7242
9.7262
9.7282
9.7.302
9.7322
9J342
9.7361
9.7380
9.7400
9,7419
9.7438
9^7457
9.7476
9.7494
9.7513
9.7531
9.7550
9.7568
9.7586
9.7004
9.7622
9.7640
9.7657
9.7675
22
21
22
21
21
21
21
21
21
20
21
20
20
20
20
20
20
19
Sft°
X
19
20
19
19
19
29
18
19
18
19
18
18
18
18
18
17
18
17
Q 7f5C»-2 17
1 COS. + 10
9.7614 I
9.7644 i
9.7673 i
9.7701 I
9.7730 ;
9.7759
9.7788 I
9.7816
9.7845
9.7873
9.7902
9.7930
9.7958
9.7986
9.8014
9.8042
9.8070
9^8097_
9.8125
9.8153
9.8180
9 1208
9.8235
9.8263
9.8290
9.8317
9.8344
9.8371
9.8398
9.8425
9.8452
9.8779
9.8506
9.8533
9.8559
9.8586
29
29
29
29
29
29
29
29
28
28
28
28_
28
28
28
28
28
_28
28
28
28
27
27
27
27
27
27
27
27
_27
27
27
27
27
27
27
9.8613
10.2386
10.2.356
10.2.327
10.2299
10.2270
10.2241
10.2212
10.2184
10.2155
10.2127
10.2098
10.2070
1 COS. + 10
10.2042
10.2014
10.1986
10.1958
10.1930
10.1903
10.1875
10.1847
10.1820
10.1792
10.1765
10.1737
27
1 cot. + 10
10.1710
10.1683
10.1656
10.1629
10.1602
iai575
10.1548
10.1521
10.1494
10.1467
10.1441
10.1414
8
7
7
8
7
8
7
8
8
7
8
8
10.1387
Itan. + 10
8
8
8
8
8
8
8
8
9
8
8
A
8
9
8
9
9
9
8
9
9
9
9
9
9.9375
9.9.368
9.9361
9.9353
9.9.346
9^9338
9.9331
9.9323
9.9315
9.9308
9.9300
9.9J92
9.9284
9.9276
9.9268
9.9260
9.9252
^^9244
9.9236
9.9228
9.9219
9.9211
9.9203
9.9194
9.9186
9.9177
9.9169
9.9160
9.9151
9.9142
9.9134
9.9125
9.9116
9.9107
9.9098
9.9089
9.9080
1 sin. + 10
60° 0'
59° 50'
59° 40'
59° 30'
59° 20'
59° 10'
59^ 0'
58' 60'
68° 40'
58° .30'
68° 20'
68° 10'
68° 0'
57° 50'
57° 40'
57° 30'
57° 20'
57° 10'
67° 0'
56° 50'
66° 40'
56° 30'
56° 20'
56° 10'
66° 0'
55° 50'
55° 40'
55° 30'
55° 20'
SSJIO'
65° 0'
54° 50'
54° 40'
54° 30'
54° 20'
54° 10'
54° 0'
X
Ijn
PRACTICAL MATHEMATICa
i-
If. .'S
%
Table III.
LOGAIUTHMS OF TRIGONOMETRICAL RATIOS.
X
1 Bin. + 10
1 tan. + 10
Icot. + 10| 1 COS. + 10
X
36° 0'
9.7692
17
9.8613
27
10.1387
9
9.9080
54° (y
36° 10'
9.7710
17
9.8639
27
10.1361
10
9.9070
53° 50'
36° 20'
9.7727
17
9.8666
26
10.1334
9
9.9061
53° 40'
36° 30*
9.7744
17
9.8692
26
10.1308
9
9.9052
53° 30'
36° 40'
9.7761
17
9.8718
26
10.1282
10
9.9042
53° 20'
36° 50'
37° C
9.7778
9.7795
17
16
9.8745
9.8771
26
26
10.1255
9
10
9.9033
9.9023
53° 10'
53° 0'
10.1229
37° 10'
9.7811
17
9.8797
26
10.1203
9
9.9014
52" 50'
37° 20'
9.7828
13
9.8824
26
10.1176
10
9.9004
52° 40'
ii7° 30'
9.7844
17
9.8850
26
10.1150
10
9.8995
52° 30'
37° 40'
9.7861
16
9.8876
26
10.1124
10
9.8985
52'^ 20'
37° 50'
38° 0'
9.7877
9.7893
16
9.8902
26
10.1098
10.1072
19
10
9.8975
52° 10'
52" (y
17
9.^928
26
9.8965
38° 10'
9.7910
16
9.8954
26
10.1046
10
9.8955
51° 50'
38° 20'
9.7926
15
9.8980
26
10.1020
10
9.8945
51° 40'
38° 30'
9.7941
16
9.9006
26
10.0994
10
9.8935
51° 30'
38^ 40'
9.7957
16
9.90."^
26
10.0968
10
9.8925
51° 20'
38° 50'
39° O'
39° 10'
39° 20'
9.7973
16
15
9.90;
9.908
26
26
10.0942
10
9.8915
51° 10'
51° G'
9.7989
10.0916
10
9.8905
9.8004
16
9.91 lu
26
10.0890 10
9.8895
50° 5(V
9.8020
15
9.9135
26
10.0865
11
9.8884
50° 40'
39° 30'
9.8035
15
9.9161
26
10.0839
10
9.8874
50° 30'
39° 40'
9.8050
16
9.9187
26
10.0813
10
9.8864
60° 20'
39° 50'
9.8066
15
15
9.9212
9.9238
26
26
10.0788
11
10
9.8853
9.8843
50° 10'
50° C
40° 0' 9.8081
10.0762
40° 10' 9.8096
16
9.9264
26
10.0736
11
9.8832
49° 50'
40° 20' 9.8111
14
9.9289
28
10.0711
11
9.8821
49° 40'
40° 30'
9.8125
15
9.9315
26
10.0685
11
9.8810
49° 30'
40° 40'
9.8140
15
9.9341
26
10.(;659
10
9.8800
49° 20'
40° 50'
9.8155
9.8169
14
9.9366
26
10.0634
10.0608
11
11
9.8789
49° 10'
41° 0'
15
9.9392
26
9.8778
49° 0'
41 10'
9.8184
14
9.9417
25
10.0583
11
9.8767
48° 60'
41° 20'
9.8198
15
9.9443
25
10.0557
11
9.8756
48° 40'
41' 30'
9.8213
14
9.9468
25
10.05.''^
•1
9.8745
48° 30'
41° 40'
9.8227
14
9.9494
25
10.050. 1-2 1
9.8733
48° 20'
41° 50'
42° "¥
9.8241
''
9.9619
25
10.0481
11
9.8722
48° 10'
9.82n5 1 Li.
9.9544
25
10.0456
11
9.8711
48° 0'
X
1 008. + 10 1 cot. + 10 1
tan. + 10
1 . li... + 10 1 X
i
LOGARITHMS OF TRIGONOMETIUCAL RATIOa
121
!
Table IIL
T-^GARITHMS OF TRIGONOMETRICAL RATIOS.
.: 1 sin.
+ 10
1 tan. + 10
I cot + 10
1
COS. + 10
X
42° C 9.8265
14
9.9544
25
10.0456
11
9.8711
40° C
42° i(y
9.8269
14
9.9570
25
10.0430
12
9.8699
47° 50'
42° 20'
9.8283
14
9.9595
26
10.0405
11
9.8688
47° 40'
42° 30'
9.8297
14
9.9621
25
10.0379
12
9.8676
47^ 30'
42° 40'
9.8311
13
9.9046
25
10.0354
11
9.8665
47° 20'
42° 50'
43° 0'
9.8324
9.8338
14
13
9.9671
9.9697
_25
26
10.0325
10.0303
12
12
9.8663
47°J0^
47^ C
9.8641
43° 10'
9.8351
14
9.9722
25
10.0278
12
9.8629
46° 50'
43° 20'
9.8365
13
9.9747
25
10.0263
11
9.8618
46° 40'
43'^ 30'
9.8378
13
9.9772
25
10.0228
12
9.8606
46° 30'
43° 40'
9.8391
14
9.9798
25
10.0202
12
9.8594
46° 20'
43° 50'
9.8405
13
9.9823
25
10.0177
12
9.8582
46° 10'
44' 0'
9.8418
13
9.9848
§5
10.0152
13
9,8569
46° C
44° 10'
9.8431
13
9.9874
25
10.0126
12
9.8557
45° 50'
44° 20'
9.8444
13
9.9899
:>o
10.0101
12
9.8545
45° 40'
44° 30'
9.8457
12
9.9924
25
10.0076
12
9.8532
45° 30'
44° 40'
9.8469
13
9.9949
25
10.0051
12
9.8520
45° 20'
44° 50'
9.8482
13
9.9975
25
10.0025
13
9.8507
45° 10'
46° 0'
9.8495
13
0.0000
25
10.0000
72
9.8495
45° C
X
1 COS. -
f 10
1 cot. + 10
Itan. + 10
1 sin. + 10 1 X 1
0'
122
PRACTIf'A!, MATHEMATICS.
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124
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APPENDIX A-
THE COMPASS CARD.
One point
ir 15'
Three quarters of a point = 8" 26' 15"
One hJf point = 5° 37' 30"
One qtuuter point «= 2° 48' 45"
128
APPENDIJL
APPENDIX B.
A. GENERAL METHOD APPLICABLE TO THE COMPtrTATION
Of AREAS, USED IN THE HirilVEYINO OF LAND.
I^t ABCDEFO be any irregular jwlygon whose area it
required.
Draw any indofinite line NS, let fall perpendiculara AA'
BB', CC, DD; KE', FF'. GG' uiK)n it from each of the anglei
of the polygon, and measure or compute their lengths. Obtain
also the lengths of the several parts of NS intercepted between
successive perpendiculars, •.#., A'B' B'C, G'D\ D'E' E'F' F'O'
GA'. ' ' •
The area of the given polygon is evidently a^ual to the
difference between the sum of the trapezoids A'ABB', B'BCC,
C'CDD', D'DEE', and the sum of the tiupozoids A'A^G''
G'GFF', F'FEE'; that is.—
Area of polygon = ^{(A'A + B'B)A'B' + (B'B + C'C)B'C' +
(CO + D'D)C'p' + (D'D + E'F)D'E' - (E'E + F'F)E'F' - (F'F +
G'G)F'G - (G'G + A A)G'A;
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