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COURSE OF
MATHEMATICS:
COMTAIHINa
THE PRINCIPLES OP
PLANE TRieONOMETRY,
MENSURATION,
NAVIGATION, AND SURVEYINQ.
ADAPTED TO THE METHOD OF INSTRUCTION IN THE
AMERICAN COLLEGES.
BY JEREMIAH DAY, D.D.LL.D.
PRESIDENT OF YALE COLLEGE.
NEW HAVEN:
PUBLISHED BY DURRIE AND PECK.
NEW YORK — COLUNS, KEESE, AND CO.
354 Pearl Street.
1838.
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TREATISE
OF
PLANE TRIGONOMETRY.
TO WHICH IS PREFIXED
A SUMMARY VIEW OF THE NATURE AND USE OP
LOGARITHMS;
BEZNO
THE SECOND PART
OP
A COURSE OF MATHEMATICS.
ADAPTED TO THE METHOD OF INSTBUCTION IN THE
AMERICAN COLLEGES.
BY JEREMIAH DAY, D.D. LL.D.
PREfllDENT OP TALE COLLEGE. ^ ,^ ' ^7
;Xv,..
NEW HAVEN;
PUBLISHED BT DURRIE AND PECK.
HEW YORK— COLUNS, KEESE, AMD CO.
354 Pearl Street.
1838.
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XFnsi
/^HARVARD \
university)
I LIBRARY
^ Entbbxd,
â–²eeordinc to Act of Congress, in the year 1831, bj
^ JEREMIAH DAY,
In the Clerk's Office of the District of
OONmCTZGDIL
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The plan upon which this work was originally commen-
ced, is continued in this second part of the course. As the
single object is to provide for « doss in college^ such matter
as is not embraced by this design is excluded. The mode
of treating the subjects, for the reasons mentioned in the pre-
face to Algebra, is, in a considerable degree, diffuse. It was
thought better to err on this extreme, than on the otheri
especially in the early part of the course.
The section on right angled triangles will probably be
considered as needlessly nunute. The solutions might, in
all cases, be effected by the ^leorems which are given for
oblique angled triangles. But the applications of rectangu-
lar trigonometry are so numerous, in nav%ation, survejring,
astronomy, &c., that it was deemed important, to render
familiar the various methods of stating the relations of the
sides and angles ; and especially to bring distinctly into view
the principle on which most trigonometrical calculations are
founded, the proportion between the parts of the given tri-
angle, and a similar one formed from the sines, tang^its, Ac^
in the tables.
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CONTENTS,
LOOA&ITHMS*
Section I. Nature of Logarithms * * ••••.. • 1
It. Directions for taking Logarithms and their
Numbers from the Tables 10
tIL Methods of calculating by Logarithms.
Multiplication 17
Division 21
Involution 22
Evolution 25
Proportion 27
Arithmetical Complement 28
Compound Proportion 80
~ Compound Interest .32
Increase of Population 35
Exponential Equations 39
IV. Different Systems of Logarithms 42
Computation of Logarithms 45
TRIGONOMETRY.
Section I. Sines, Tangents, Secants, &c 49
II. Explanation of the Trigonometrical Tables . 58
III. Solutions of Right angled Triangles .... 66
IV. Solutions of Oblique angled Triangles ... 80
V. Geometrical Construction of Triangles ... 91
VI. Description and use of Gunter's Scale ... 97
VII. Trigonometrical Analysis 105
VIII. Computation of the Canon 123
IX. Particular Solutions of Triangles ... * . 127
Notes . . i . 137
Table of Natural Sines and Tangents • . . 147
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LOGARITHMS.
SECTION I.
l^ATURE 0Â¥* liOOARIf HMs/
Art, 1. The operations of Multiplicatiou and Ettvisioil,
when they are to be often repeated, become so laborious^
that it is an object of importance to substitute, in their stead-
m^ore simple methods of calculation, such as Addition and
Subtraction. If these can be made to perform^ in an expe^
ditious manner, the oflice of multiplication and division, a
great portion of the time and labor which the latter processes
require, may be saved^
Now it has been shown, (Algebra, 233, 237^) that powers
may be multiplied, by adding their ea;p<ments, and divided,
by subtracting their exponents. In the same manner, roots
may be multiplied and divided, by adding and subtracting
their fractional exponents. TAlg. 280, 286.) When these ex-
ponents are arranged in taoles^ and applied to the general
purposes of calculation, they are called Logarithms.
2. LOGARITHMS, then, are the EXPONENTS oP
A SERIES OP POWERS AND ROOTS.f
In fcrming a system of Ic^arithms, some particular num*
ber is fixed upon, as the bcise, radio:, or first power, whose
logarithm is always 1* Prom this, a series df powew is rais*
ed, and the exponents of these are arranged in tables for Use#
To explain this, let the mmiber which is chosefi foi' the first
* Mask^yne'a Prefoce to TaVtor^s Logaritlltn^ Ininxiuetiotf to Hutton'*
Tables. Keil on iK^arithms. llatfares Scriptoria LogarithmicL Bri^^ Log'
arithms. Bodaoa's Anti-logarithmic Canon. Euler'a Algebnu
t See note A<
1
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2 NATURE OF LOGARITHMS.
power, be represented by a. Then taking a series of pow-
ers, both direct and reciprocal, as in Alg. 207 ;
a*, a^, a^, a', a«, a~^, a~^, a-^, a~*, <fcc.
The logarithm of a^ is 3^ and the logarithm of a^^ is — 1,
of a» is 1, of a-3 is — 2,
of a « is 0, of a""3 is — 3, <fcc.
Universally, the logarithm of o* is x.
3. In the system of logarithms in common use, called
Briggs* logarithms, the number which is taken for the radix
or base is 10. The above series then, by substituting 10 for
a, becomes
lOS 103, 102^ 101, 100, 10-1, 10-2, 10-3, &c.
Or 10000, 1000, 100, 10, 1, ^V, tU, tttVit, <fcc.
Whose logarithms are
4, 3, 2, 1, 0, —1, —2, —3, <kc.
4. The fractional exponents of roots, and of powers of
roots, are converted into decimals, before they are inserted
in the logarithmic tables. See Alg. 255.
The logarithm of a^, or o®- ^ ^ 3 3^ ig 0.3333,
of a% or a«-« « « «, is 0.6666,
of a^, or a«-4 a 8 S is 0.4285,
of a'8^,ora3-««««, is 3.6666, <fcc.
These decimals are carried to a greater or less number of
places, according to the degree of accuracy required.
6. In forming a system of logarithms, it is necessary to
obtain the logarithm of each of the numbers in the natural
series 1, 2, 3, 4, 5, (fee; so that the logarithm of any number
may be found in the tables. For this purpose, the radix of
the system must first be determined upon ; and then every
other number may be considered as some power or root of
this. If the radix is 10, as in the common system, every
other number is to be considered as some power of 10.
That a power or root of 10 may be found, which shall be
equal to any other number whatever, or, at least, a very near
approximation to it, is efvident from this, that the exponent
may be endlessly varied ; and if this be increased or dimin-
ished, the power will be increased or diminished.
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NATURE OF LOGARITHMS. 3
If the exponent is a fraction, and the nwmerator be increas-
ed, the power will be increased ; but if the denonUncttor be
increased, the power will be diminished.
6. To obtain then the logarithm of any number, according
to Brigffs' system, we have to find a power or root of 10
which shall be equal to the proposed nuipber. The eTponent
of that power or root is the logarithm required. Thus
7^109.9451
30 = 10>*^^»
400=10«-««*®
therefore the
logarithm
of 7 is 0.8451
of 20 is 1.3010
of 30 is 1.4771
of 400 is 2.6020, &c.
7. A logarithm generally consists of two parts, an integer
and a dedmcd. Thus, the logarithm 2.60206, or, as it is
sometimes written, 2+.60206, consists of the integer 2, and
the decimal .60206. The integral part is called me charac-
teristic or index* of the logarithm ; and is frequently omitted,
in the common tables, because it can be easily supplied, when-
ever the l(^[arithm is to be used in calculation.
By art. 3d, the logarithms of ^
10000, 1000, 100, 10, 1, .1, .01, .001, &c.
are 4, 3, 2, 1, 0, —1, —2, —3, <fcc.
As the logarithms of 1 and of 10 are and 1, it is evident,
that, if any given number be between 1 and 10, its logarithm
will be between and 1, that is, it will be greater than 0, but
less than 1. It will therefore have for its index, with a
decimal annexed.
Thus, the logarithm of 5 is 0.69897.
For the same reason, if the given number be between
10 and 100, ) the log. C 1 and 2, i. e. 1-fthe dec. part.
100 and 1000, ) will be ^ 2 and 3, 2+the dec. part.
1000 and 10000, ) between ( 3 and 4, 3+the dec. part.
We have, therefore, when the logarithm of an integer or
mixed number is to be found, this general rule:
* The term index, as it is used here^ may possibly lead to some confusion in
the mind of the learner. For the logantbm itself is the index or exponent of a
power. The characteristic, therefore, is the index of an index.
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£ ir ATURB OF I.06ARITHMS.
8. 7%^ indez of the logarithm is always one less, than the
number of integral figures, in the natural number whose logor-
rithm is sought : or, the index shows how far the first figure
of the natural number is removed firom the place of units*
Thus, the logarithm of 37 is 1.56820.
Here, the number of ^ures being two, the index of the
logarithm is X.
The logarithm of 253 is 2,40312,
Here, the propcfeed number 253 consists of three figures,
the first of which is in the second place from the unit figure.
The index of the logarithm is therefore 2.
The logarithm of 62.8 is 1.79796,
Here it is evident that the mixed number 62,8 is between
10 and 100. The index of its logarithm must, therefore,
be 1.
9. As the logarithm of 1 is 0, the logarithm of a number
less than 1. that is, of any jproi^T fraction, must be negative.
Thus, by art. 3d,
The logarithm of yV <^r .1 is — 1,
of tItt or .01 ig— 2,
of y^^T or .001 is — 3, &c.
10. If the proposed number is between tttt *^^ ttViti
its logarithm must be between — 2 and — 3. To obtain the
logarithm, therefore, we must either subtract a certain firac-
tional part from — 2, or add a fractional part to — 3 ; that is,
we must either annex a negative decimal to — 2, or a posi-
tive one to — 3,
Thus, the logarithm
of .008 is either — 2 — .09691, or — 3+.90309.*
The latter is generally most convenient in practice, and is
more commonly written 3.90309. The line over the index
• ' '■' ' ' ' <: ' • ' •
* That these two expressions are of the same yalne wiU be evident, if we sab-
tract the same quantity, -|-.90309 from each. The remainders will be equal, and
therefore the quantities from which the 8ubtractio^ is made must be equal. S^
note B.
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If ATURE OF LOGARITHMS. e
denotes, that thai is negative, whiie the decimal part of the
ic^^arithm is positive.
c of 0.3, isT47712,
The logarithm 5 of 0.06, is^77815,
( of 0.009, is"3:95424,
And universally,
11. The negative index of a logarithm shows how far the
first significant figure of the natural number j is removed
from the place of units, on the right; in the same manner as
a positive index shows how far the first figure of the natural
number is removed from the place of units, on the left. (Art.
8.) Thus, in the examples in the last article,
The decimal 3 is in the first place from that of units,
6 is in the second place,
9 is in the third place ;
And the indices of the logarithms are 1, 2, and 3.
12. It is often more convenient, however, to make the in-
dex of the logarithm positive, as well as the decimal part.
This is done by adding 10 to the index.
Thus, for — 1, 9 is written ; for — 2, 8, &c.
Because — 1+10== 9, — 2+10= 8, <fcc.
With this alteration,
CT90309 ) C 9.90309,
The logarithm I 2^90309 ( becomes I 8.90309,
' 5:90309 J ^ 7.90309, &c.
This is making the index of the logarithm 10 too great.
But with proper caution, it will lead to no error in practice.
13. The sum of the logarithms of two numbers, is the log-
arithm of the product of those numbers ; and the difference
of the logarithms of two numbers, is the logarithm of the
r)tient of one of the numbers divided by the other. (Art. 2.)
Briggs' system, the logarithm of 10 is 1. (Art. 3.) If there-
fore any number be multiplied or divided by 10, its logarithm
will be increased or diminished by 1 : and as this is an in-
teger, it will only change the index of the logarithm, without
acting the decimal part
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NATURE OP LCN3ARITHMd.
Thus, the logarithm of 4730 is 3.67486
And the logarithm of 10 is 1.
The logarithm of the product 47300 is 4.67486
And the logarithm of the quotient 473 is 2.67486
Here the index only is altered, while the decimal part re-
mains the same. We have then this important property,
14. The DECIMAL PART of the logarithm of any number
is the same^ as that of the number multiplied or divided by
10, 100, 1000, <fcc.
Thus the log. of 45670, is
4567,
456.7,
45.67,
4.567,
! 4.65963,
3.65963,
2.65963,
1.65963,
0.65963,
.4567,
1.65963, or 9.65963,
.04567,
2.65963, 8.65963,
.004567,
3.65%3, 7.65963.
This property, which is peculiar to Briggs' system, is of
great use in abridging the logarithmic tables. For when we
have the logarithm of any number, we have only to change
the index, to obtain the logarithm of every other number,
whether integral, fractional, or mixed, consisting of the same
significant figures. The decimal part of the logarithm of a
fraction found in this way, is always positive. For it is the
same as the decimal part of the logarithm of a whole num-
ber.
16. In a series of fractions continually decreasing, the
negative indices of the logarithms continually increase.
Thus,
In the series 1, .1, .01, .001, .0001, .00001, <fcc.
The logarithms are 0, — 1, — 2, — 3, — 4, — 6, &c.
If the progression be continued, till the fraction is reduced
to 0, the negative logarithm will become greater than any as-
signable quantity. The logarithm of 0, therefore, is infinite
and negative. (Alg. 447.)
16. It is evident also, that all negative logarithms belong
to fractions which are between 1 and ; while positive loga-
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NATURE OF LOGARTTHBIS. J
rithms belong to natural numbers which are greater than 1.
As the whole range of numbers, both positive and negative,
is thus exhausted in supplying the logarithms of integral and
fractional positive quantities ; there can be no other numbers
to fiirnish logarithms for negative quantities. On this ac-
count- the logarithm of a negative quantity is, by some wri-
ters, considered as impossible. But as there is no diflference
in the multiplication, division, involution, <fcc. of positive and
negative quantities, except in applying the signs ; they may
be considered as all positive, while these operations are per-
forming by means of logarithms ; and the proper signs may
be afterwards aflix:ed.
17. If a series of numbers be in geometrical progression,
their logarithms will be in arithmet ic al progression. For,
in a geometrical series ascending, the quantities increase by
a common multiplier ; (Alg. 436.) that is, each succeeding
term is the product of the preceding term into the ratio.
But the logarithm of this product is the sum of the logarithms
of the preceding term and the ratio ; that is, the logarithms
increase by a common addition, and are, therefore, in arith-
metical progression. (Alg. 422.) In a geometrical progression
descending, the terms decrease by a common divisor, and
their logarithms, by a common difference.*
Thus, the numbers 1, 10, 100, 1000, 10000, &c. are in
geometrical progression.
And their logarithms 0, 1, 2, 3, 4, &c. are in ariflmaeti-
cal progression.
Universally, if in any geometrical series,
a = the least term, r = the ratio,
Z = its logarithm, / = its logarithm ;
Then the logarithm of ar is L+l, (Art. 1.)
of ar^ is L+21,
of ar^ is L+31, &c.
Here, the quantities a, ar', ar^, ar*, <fcc., are in geomet-
rical progression. (Alg. 436.)
And their logarithms L, L+l, L+21, L+31, &c., are in
arithmetical progression. (Alg. 423.)
* See note C.
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S tOGARlTHBOO CURVE^
THE LOGARITHMIC CURVE.
19. The relations of logarithms, and their corresponding
numbers, may be represented by the abscissas and ordinates
of a curve. Let the line AC (Fig. 1.) be taken for unity.
Let AF be divided into portions, each equal to A C, by the
points 1, 2, 3, &c. Let the line a represent the radix of a
given system of logarithms, suppose it to be 1.3 ; and let a»,
a^j <fcc. correspond, in length, with the different powers of a-
Then the distances from A to 1, 2, 3, <fcc., will represent the
logarithms of a, a^, a^, <fec. (Art. 2.) The line CH is called
the logarithmic curve, because its abscissas are proportioned
to the logarithms of numbers represented by its ordinates.
(Alg. 627.)
20. As the abscissas are the distances from AC, on the line
AF, it is evident, that the abscissa of the point C is 0, which
is the logarithm of 1 = AC. (Art. 2.) -The distance from A to
1 is the logarithm of the oroinate a, which is the radix of
the system. For Briggs' logarithms, this ought to be ten
times AC. The distances from A to 2 is the logarithm of the
ordinate a^ ; from A to 3 is the logarithm of a^, <fcc.
21. Thd logarithms of numbers less than a unit are nega-
tive, (Art. 9.) These may be represented by portions of the
line AN, on the opposite side of AC. (Alg. 507.) The ordi-
nates a~*, a~2, a-~^, &c., are less than AC, which is taken
for unity ; and the abscissas, which are the distances from A
to — 1, — 2, — 3, &c., are negative.
22. If the curve be continued ever so far, it will never
meet the axis AN. For, as the ordinates aire in geometrical
progression decreasing, each is a certain portion of the pre-
ceding one. They will be diminished more and more, the
farther they are carried, but can never be reduced absolutely
to nothii^. The axis AN is, therefore, an asymptote of the
curve. (Alg. 545.) As the ordinate decreases, the abscissa
increases j so that, when one becomes infinitely small, the
other becomes infinitely great. This corresponds with what
has been stated, (Art. 15.) that the logariUim of is infinite
and negative.
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LOGABimmC CURVE. g
.23. To find the equation of this curve,
Let a=the radio; of the system,
a: = any one of the abscissas,
y=the corresponding ordinate.
Then, by the nature of the curve, (Art. 19.) the ordinate
to any point, is that power of a whose exponent is equal to
the abscissa of the same point ; that is, (Alg. 528.)
* For other propertiei of the logarithinio curvei see FlunoBo.
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10 THE LOOARITHiaC TABLED
SECTION II.
DIRECTIONS FOR TAKINQ LOGARITHMS AND THEIR NUM-
BERS FROM THE TABLES.*
Art. 24. The purpose which logarithms are intended to
answer, is to enable us to perform arithmetical operations
with greater expedition, than by the common methods. Be-
fore any one can avail himself of this advantage, he must
become so familiar with the tables, that he can readily find
the logarithm of any number ; and, on the other hand, the
number to which any logarithm belongs.
In the common tables, the indices to the logarithms of the
first 100 numbers, are inserted. But, for all other numbers,
the decimal part only of the logarithm is given ;,' while the
index is left to be supplied, according to the principles in
arts. 8 and 11.
25. To find the logarithm of any number between 1 and
100:
Look for the proposed number, on the left ; and against
it, in the next column, will be the logarithm, with its index.
Thus,
The log. of 18 is 1.25527. The log. of 73 is 1.86332.
26. To find the logarithm of any number between 100 and
1000 ; or of any number consisting of not more than three
significant figures, with ciphers annexed.
In the smSler tables, the three first figures of each num-
ber, are generally placed in the left hand column ; and the
fourth figure is placed at the head of the other columns.
Any number, therefore, between 100 and 1000, may be
found on the left hand ; and directly opposite, in the next
column, is the decimal part of its logarithm. To this the
index must be prefixed, according to the rule in art. 8.
• The best English Tables are Hutton's in Svo. and Taylor's in 4to. In theses
the logarithms are carried to seven places of decimals, ana proportioaal parts are
placedin the margin. The smaller tables are numerous ; and» when accurately
printed, are sufficient for common calculations.
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THE LOQAHrrHMIC TAK^BB.
11
The log. of 468 is 2.66087, The log. of 935 is 2.97081,
of 796 2.90091, of 386 2.68659.
If there are ciphers annexed to the significant figures, the
logarithm may be found in a similar manner. For, by art.
14, the decimal part of the logarithm of any number is the
' same, as that of the number multiplied into 10, 100, <fcc. All
the difference will be in the index ; and this may be suppUed
by the same general rule.
The log. of 4580 is 3.66087, The log. of 326000 is 5.51322,
of 79600 4.90091, of 8010000 6.90363.
27. To find the logarithm of any number consisting of
wovR figures^ either with, or without, cipheis annexed.
Look for the three first figures, on the left hand, and for