APPENDIX, 53
cube to the fifth, have their Logarithms in the ratio of the indices of their orders, that is of 3 to 5.
5.If a first sine be multiplied into a second producing a third, the Logarithm of the first added to the Logarithm of the second produces the Logarithm of the third. So in division, the Logarithm of the divisor subtracted from the Logarithm of the dividend leaves the Logarithm of the quotient.
6.And if any number of equals to a first sine be multiplied together producing a second, just so many equals to the Logarithm of the first added together produce the Logarithm of the second.
7.Any desired geometrical mean between two sines has for its Logarithm the corresponding arithmetical mean between the Logarithms of the sines.
[B]
8.If a first sine divide a third as many times successively as there are units in A; and if a second sine divides the same third as many times successively as there are units in B; also if the same first divide a fourth as many times successively as there are units in C; and of the same second divide the same fourth as many times successively as there are units in D: I say that the ratio of A to B is the same as that of C to D, and as that of the Logarithm of the second to the Logarithm of the first.
[C]
9.Hence it follows that the Logarithm of any given number ts the number of places or figures which are contained in the result obtained by raising the given number to the 10,000,000,000th power.
10.Also if the index of the power be the Logarithm of 10, the number of places, less one, in the power or multiple, will be the Logarithm of the root.
Suppose it is asked what number is the Logarithm of 2, I reply, the number of places in the result obtained by multiplying together 10,000,000,000 of the number 2.
But,
