The construction of the wonderful canon of logarithms

Remarks on Appendix.
6i
the quoiients and the Logarithms of the divisors will be
reciprocally proportional.
Otherwise the ratio will not be exactly the same on both
sides ; nevertheless, if the divisors be very small, and the
dividends sufficiently large, so that the quotients are very
m.any, the defect from- proportionality will scarcely, or not
even scarcely, be perceived.
Hence it follows that the logarithm )
[C]
Let two numbers be taken, lo and 2, or any others you
please. Let the Logarithm of the first, namely 100, be
given ; it is required to find the Logarithm of the second.
In the first place, let the second, 2. multiply itself continuously
until the number ofthe products is exceeded, by unity
Then let the last
only, by the given Logarithm of the first.
product be divided as often as possible by thefirst number,
10. and again in like m-anner by the second number, 2.
The number ofquotients in the latter case will be 100, ^for
the product is its hundredthpower ; and if a number be multiplied
by itself a given number of times forming a certain
product, then it will divide the product as m,any times and
once more ; for exam,ple, if 2)
be multiplied by itself four
times it makes 243, and the same 3 divides 243 five times,
the quotients being Bi, 27, 9, 3, i.) In the former case,
where the product is continually divided by 10, it is manifest
that the number of quotients falls short of the number
of places in the dividend by one only. Therefore (by the
preceding proposition) since the same product is divided by
two given numbers as often aspossible, the numbers of the
quotients and the Logarithms of the divisors will be reciprocally
proportional. But, the number of quotients by the
second being equal to the Logarithm of the first, the num-
H 3 ber