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Again M:1:::1.e : l'.e
But as e denotes Napier's base, l'.e=1. So that M=l.e, that is, the modulus of the common system, is equal to the common logarithm of Napier's base. Therefore, either of the expressions, l.e, or
l'.a used, to convert the logarithms of one of the systems into those of the other.
The ratio of the logarithms of two numbers to each other, is the same in one system as in another. If N and n be the two numbers; Then,
I.N :1.N::M: M'
l.n : 1'.n::M: M' Therefore, 1.N:l.n::l.N:l.n.
COMPUTATION OF LOGARITHMS.
68. The logarithms of most numbers can be calculated by approximation only, by finding the sum of a sufficient number of terms, in the series which expresses the value of the logarithms. According to art. 65. Log. N=MX( (N − 1) – }(N − 1)2 + (N-1)3, &c.)
Or, putting as before, n=N-1, Log. (1+n)=M(n-1n? +in-tìn5 – &c.) But this series will not converge, when n is a whole number, greater than unity. To convert it into another which will converge, let (1 — n) be expanded in the same manner as (1 +n), (Art. 65.) The formula will be the same, except that the odd powers of n will be negative instead of positive. We shall then have,
Log. (1+n)=M(n - n° +1n3 - 1n*+ins-&c.)
Log. (1-n)=M(-n-in-1n3-1n4 - 1N5 - &c.) Subtracting the one from the other the even powers of n disappear, and we have
2M (n+ n° +n +4n7+&c.)
But this, which is the difference of the logarithms of (1+n) and (1-n) is the logarithm of the quotient of the one divided by the other. (Art. 36.)
That is, Log.
1 Log.z= log. (2—2)+ 2M
This series may be applied to the computation of any number greater than 2.
To find the logarithm of 2, let z=4, Then (z – 1)=3, and the preceding series, after transposing log. (z—2) becomes
But as 4 is the square of 2; log. 4=2 log. 2. (Alg. 44.) So that log. 4- log. 2=log. 2. We have then
When the logarithms of the prime numbers are computed, the logarithms of all other numbers may be found, by simply adding the logarithms of the factors of which the numbers are composed. (Art. 36.)
69. In Napier's system, where M=1, the logarithms may be computed, as in the following table.
NAPIER'S OR HYPERBOLIC LOGARITHMS.
1 Log. 3=2
1 1 1 +
3.43 5.45 7.47 Log. 6=log. 3+log. 2.
1 1 1 1
=2.079441 =2.197224 =2.302585
70. To compute the logarithms of the common system, it will be necessary to find the value of the modulus. This is equal to 1 divided by Napier's logarithm of 10, (Art. 67.) that is,
This number substituted for M, or twice the number, viz. .36858896 substituted for 2 M, in the series in art. 68. will enable us to calculate the common logarithm of any number.
COMMON OR BRIGGS' LOGARITHMS.
1 1 1 Log. 2=.86858896
&c.) =0.301030 3.33 5.35 7.37 1
1 1 Log.3=.86858896
7.27 Log. 4=2 log. 2.
=0.602060 Log. 5=log. 10-log. 2=1-log. 2.
=0.698970 Log. 6=log. 3+log. 2.
=0.778151 1 1
+log. 5. =0.845098 Log. 8=3 log. 2.
=0.903090 Log. 9=2 log. 3.
=0.954243 Log. 10
SINES, TANGENTS, SECANTS, &c.
Art. 71. TRIGONOMETRY treats of the relations of the sides and angles of TRIANGLES. Its first object is, to determine the length of the sides, and the quantity of the angles. In addition to this, from its principles are derived many interesting methods of investigation in the higher branches of analysis, particularly in physical astronomy. Scarcely any department of mathematics is more important, or more extensive in its applications. By trigonometry, the mariner traces his path on the ocean; the geographer determines the latitude and longitude of places, the dinensions and positions of countries, the altitude of mountains, the courses of rivers, &c. and the astronomer calculates the distances and magnitudes of the heavenly bodies, predicts the eclipses of the sun and moon, and measures the
of light from the stars.
72. Trigonometry is either plane or spherical. The former treats of triangles bounded by right lines; the latter, of triangles bounded by arcs of circles.
Divisions of the Circle.
73. In a triangle there are two classes of quantities which are the subjects of inquiry, the sides and the angles. For the purpose of measuring the latter, a circle is introduced.
The periphery of every circle, whether great or small, is supposed to be divided into 360 equal parts called degrees, each degree into 60 minutes, cach minute into 60 seconds,