The construction of the wonderful canon of logarithms

Appendix. 53
cube to the fifth, have their Logarithms in the ratio of the
indices of their orders, that is of 2, 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 secondproduces 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 desiredgeometrical mean between two sines has for
its Logarithm the corresponding arithmetical mean between
the Logarithms of the sines.
8. If a first sine divide a third as many times successively [B]
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 if 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 ^ is the same
as that of Q to D, and as that of the Logarithm of the
second to the Logarithm of the first.
9. Hence itfollows that the Logarithm of any given num- [C]
ber is the number ofplaces orfigures which are contained
in the result obtained by raising the given number to the
10,000,000,000"' power.
10. Also if the index of the power be the Logarithm of \o,
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.
G 3 But,