The treatises on book-keeping have described this difference in as peculiar a manner. They call these accounts the fictitious accounts. Now they represent the merchant himself; their credits are gain to the business, their debits losses or liabilities. If the terms real and fictitious are to be used at all, they are the real accounts, end all the others are as fictitious as the clerks whom we have supposed to keep them.

This theorem shews that what is called reducing a fraction to its lowest terms (namely, dividing numerator and denominator by their greatest common measure), is correctly so called.

For that which measures a measure is itself a measure; so that if a measure of a could have a measure in common with b, a itself would have a common measure with b.

A prime number is one which is prime to all numbers except its own multiples, or has no divisors except 1 and itself.

Expand (a-1)ᵇ by the binomial theorem; shew that when b is a prime number every coefficient which is not unity is divisible by b; and the proposition follows.

The principle of this mode of demonstration of Horner’s method was stated in Young’s Algebra (1823), being the earliest elementary work in which that method was given.