Then A1 - a1: A, — a1⁄2 :: α1 : ɑ„, is found proved in the preceding

-

process.

-

Prop. E. Algebraically.

Let A1: a,:: A3: α49

Then shall A,: A1 — a1⁄2:: A ̧: Â ̧

For since A, α2:: A2: α

Dividing the latter by the former of these equals,

Prop. xx. Algebraically.

Let A1, A2, A, be three magnitudes, and a1, a2, aз, other three, such that A: A2:: α1: ɑ2

and A2 A3 A2: Az

if A1> A, then shall a> az,

and if equal, equal; and if less, less.

and that A1 > Ag:

It follows that a1 is > a3.

In the same way it may be shewn

that if A1 = Ag, then a1 = a; and if A1 be < Ag, then a1 < az. Prop. xxi. Algebraically.

Let A1, A2, A3, be three magnitudes,

and a, a, as three others,

such that A: A2 :: ɑ2: αz,

and A2 A3 A1 A2.

If A1> Ag, then shall a1> ag, and if equal, equal; and if less, less.

Next if there be four magnitudes, and other four such, that

and similarly, if there were more than four magnitudes.

аг

a1

2

and similarly, if there be more than four magnitudes.

Prop. xxiv. Algebraically.

Let A1: a2: Ag: α4,

and A, a2 :: A。 : α,

Then shall A1 + A5 : a2 :: Ag + A ̧ : αs.

For since A, a2 :: A3 ; as,..

=

Α1 A3

and .. A1 + A5 : α2 :: A3 + A6 : as·

COR. 1. Similarly may be shewn, that

Let A1a: A3: α49

and let A, be the greatest, and consequently a the least.

Then shall A1 + α > α2 + A3.

Since A: a2 :: A3: αs9

but A > a, A, is the greatest of the four magnitudes,

"The whole of the process in the Fifth Book is purely logical, that is, the whole of the results are virtually contained in the definitions, in the manner and sense in which metaphysicians (certain of them) imagine all the results of mathematics to be contained in their definitions and hypotheses. No assumption is made to determine the truth of any consequence of this definition, which takes for granted more about number or magnitude than is necessary to understand the definition itself. The