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Thé sines being given, the tangents and secants may be found from the following analogies (see Figure 3, for the definitions); because the triangles BDC, BAE, BHK, are equiangular, we have BD: DC::BA : AE; that is, Cos.: S.::R.: T. AE: BA::BH: HK; that is, T. : R.:: R.: Cot. BD : BC:: BA : BE; that is, Cos. : R.:: R. : Secant. CD: BC::BH: BK; that is, s. R.;: R.: Cosec.
1. The indices or exponents of a series of numbers in geometrical progression, proceeding from 1, are also called the logarithms of the numbers in that series *. Thus, if a denote any number, and the geometrical series, 1, a', a, a, a4, &c. be produced by actual mul. tiplication, then 1, 2, 3, 4, &c. are called the logarithms of the first, second, third, and fourth powers of a respectively. Consequently, if, in the above, a be equal to the number 2, then I is the logarithm of 2, 2 is the logarithm of 4, 3 is the logarithm of 8, 4 is the logarithm of 16, &c. But if a be equal to 10, then 1 is the
, logarithm of 10, 2 is the logarithm of 100, 3 is the logarithm of 1000, 4 is the logarithm of 10000, &c. The
1. series may be continued both ways from 1. Thus
1 1 1
1, a', a’, aș, a“, &c. constitute a series in geometrical progression, and, agreeable to the established notation in algebra, the indices, or logarithms,
- 4, -3, 2, — 1, 0, 1, 2, 3, 4, &c. If a be equal to the number 2, then -- 4 is the logarithm of
3 is the logarithm of 2 is the logarithm of 16
1 -:- 1 is the logarithm of-, o is the logarithm of 1, 1 is 4
2 the logarithm of 2, &c. If a be equal to 10, then
1 is the logarithm of
3 is the logarithm of
1 - 2 is the logarithm of i is the loga1000
100 1 rithm of O is the logarithm of 1, and 1 is the loga
10 rithm of 10, &c.
2. From the above it is evident that the logarithms * The reader ought to be acquainted with arithmetical and geometrical progression and the binomial theorem, before he enters on a perusal of any account of logarithms.
of a series of numbers in geometrical progression, constitute a series of numbers in arithmetical progression. Beginning with 1, and proceeding towards the right hand, the terms in the geometrical series are produced by multiplication, but their corresponding logarithms are produced by addition. On the contrary, beginning with 1, and proceeding towards the left hand, the terms in the geometrical progression are produced by division, but their corresponding logarithms are produced by subtraction.
3. The same observations apply to logarithms when they are fractions. Thus if a” denote any number,
2 1 1
a a”, &c. constitute a series
of numbers in geometrical progression, of which 之
no 0, n ñ ñ * &c. are the logarithms; and it is evident that the assertions in the last article hold true, both with respect to the numbers in geometrical progression and their corresponding logarithms. As a and n may be taken at pleasure, it follows that numbers in very different geometrical progressions may have the same logarithms; and that the same series of numbers in geometrical progression may have different series of logarithms corresponding to them.
4. If a be an indefinitely small decimal fraction, and successive powers of 1 +a be raised, then the excess of any power of 1 +a above that immediately preceding it will be indefinitely small. Thus let a = 00000000001, and then ] +a?=1•0000000000200000000001; and 1+al = 1.000000000030000000000300000000001; and proceeding by actual multiplication to obtain higher powers of 1.00000000001, it will be found that the difference between two successive powers is very small. If, instead of supposing, as above, that a='00000000001, we suppose it only one millionth part of this value, then the successive powers of 1+a will differ from one another by much smaller decimal fractions.
5. If, therefore, a be indefinitely small, and successive powers of 1+a be raised, a series of numbers in geometrical progression will be produced, of which the common numbers 2, 3, 4, 5, &c. will become terms. For on every multiplication by 1+a, an indefinitely
small addition is made to the power multiplied, and by this indefinitely small addition, the next higher power is produced. Some power of 1+a will, therefore, be
ta , equal to the number 2, or so nearly equal to it that they may be considered as equal. Continuing the advancement of the
powers of 1 + a, the numbers 3, 4, 5, &c., for the same reasons, will fall into the series.
6. The sum of the logarithms of any two numbers is equal to the logarithm of the product of the same two numbers. Thus if 1 +a raised to the nth power be equal to the number N, and if 1 +a raised to the mth power be equal to the number M, then, by the preceding articles, n is the logarithm of 1+al" or of its equal N, and for the same reason, m is the logarithm of M. Hence it follows that n+m= the logarithm of Nx M, for Nx M=l+a" x 1+am = 1+an + by the nature of indices. If the logarithm of N be subtracted from the logarithm of M, the difference is equal to the logarithm of the quotient which arises from the division of M by
M ita N. For
by the nature of indiN
1+al ces. The addition of logarithms, therefore, answers to the multiplication of the natural numbers to which they belong; and the subtraction of logarithms answers to the division by the natural numbers to which they belong.
7. If the logarithms of a series of natural numbers be ail multiplied by the same number, the several products will have the last-mentioned properties of logarithms. Thus, if the indices of all the powers of 1 +a be multiplied by l, then, using the notation stated in the last article, the logarithm of N is nl, and the logarithm of M is ml, and the logarithm of Nx M is ni+ml; for N> M=1+a" x 1+am = 1+and+mi, by the nature of
M indices. Also mlnl = the logarithm of
N N 1+a = 1+amni
. Hence the products arising 1+an?
+ from the multiplication of l into the indices of the powers of 1+a, are termed logarithms, as are also all numbers which have the properties stated at the end of article 6. It is on account of these properties that log
arithms are so very useful in calculations of the highest importance.
8. If the indices of the powers of 1+a, be multiplied by a, the products are called the hyperbolic logarithms of the numbers equal to the powers of 1+a. Thus, if the number N be equal to i tan, then na is the hyperbolic logarithm of N; and if the number M be equal to 1+alm, then ma is the hyperbolic logarithm of M. Hyperbolic logarithms are not those in common use, but they can be calculated with less labour than any other kind, and common logarithms are obtained from them.
9. If successive powers of a very small fraction be raised, they will successively be less and less in value. This truth appears most evident by putting the value
1 in the form of a vulgar fraction. 1
1 10000000000 100000 1000000000000000' &c.
10. Let it be required to determine the hyperbolic logarithm L, of any number N. Using the same notation as in the preceding articles, 1+an=N, and by extracting the nth root of each side of the equation, 1+a=N). Put m=;, and 1+x=N, and then N. 1
- 1 1+x)=(by the binomial theorem) 1 +mx +mx
m- -1 xx+mx
x2 + &c. = 1 +a. Now, as 2 3 a is indefinitely small, the power of 1+a, which is equal to the number N, must be indefinitely high; or, which is the same thing, n must be indefinitely great. Consequently m must be indefinitely small, and therefore may be rejected from the expressions m-1, m-2, m-3, &c. Hence 1 being taken from each side of the .
тх? mx3 above equation, we have a =mx
2 3 4 mxs -&c. Each side of this equation being divided by
23 x4 25
-&c. But m=n
-&c= + 2 3 x2
25 and therefore =an X
-&c. = L, 5
m, we have