Cor. If A, A,
Then A, -a,

process.

Prop. E. Algebraically.

or

a2

and multiplying these equals by

A2

or

a2

A1 A2

=

A, -a a1
A2 -a2

A1 ar

but

A2

a1

--

=

a2

A, a

but

=

=

a1 an

A2- a2 :: α1: α, is found proved in the preceding

-

2

a2

A2

=

1

Let A, a,:: A3: α,
Then shall A1: A1
For since A,
A1

a2 A3 A3
α2:: As: α

Ag
3

..

a2

a4

subtracting 1 from each of these equals,

A1

Ag

1,

a

As

1=

=

=

=

=

- 1,

a1

A3

as

A3

a2
A1

სი

as

Dividing the latter by the former of these equals,

A3 A3-as

A1 A1- an
=

=

a2

a2

as

as

as

or

Prop. xx. Algebraically.

Let A1, A2, A, be three magnitudes, and a1, a2, ag, other three,
such that A: A2:: α1: α2r
and A2 A3: A2: Az:
if A1> Ag, then shall a1 > ag,

and if equal, equal; and if less, less.
A1 a1

Since A: A2 :: α1: ɑ2, ..

A1 ar

A3

and since the fraction

=-9

A1

A3

and that A1> Ag:

It follows that a1 is > a3.

or

For since A: A2:: α2: α3,

and since A2 A3 :: α1: α2,

=

A1
As

and since the fraction

=

In the same way it may be shewn

=

that if A1 A3, then a1 = a; and if A1 be < Ag, then а1 < αz. Prop. xxi. Algebraically.

Let A1, A2, A3, be three magnitudes,

and a1, a2, as three others,

such that A: A2 :: α2: α3,

..

is equal to

and A2 A3 A1 A2.

If A1> A3, then shall a1 > ag, and if equal, equal; and if less, less.

Multiplying these equals,
A2 a2 αι

A1

X 9 az

a2

..
A2

a1

=

..

A3

and that A1 > A3.

is equal to

A2 α-3

A2

a1

A3

a1

It follows that also a > Az. Similarly, it may be shewn, that if A1 A3, then a and if A1 < A3, also α1 < ɑ3.

=

Prop. XXII. Algebraically.

Let A1, A2, A3 be three magnitudes,
and a1, a2, as other three,
such that A: A2 :: α1: αg,

and A2 A3 Az: Az• Then shall A1: A3 :: α1: αz.

For since A: Ag :: α1 : αg,

and since A2: A3 :: Ag: as,

or

Multiply these equals,

A2 α1 Ay

= X -,

=- 9

A3 Az

A1 a1
A3 Az

and A1: As :: A1: Az.

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

..

=

..

..

Multiplying these equals,

A1 A2 A3
X
A3 A1

A

and since A: Ag :: α1: A2,

..

A2

Ag

A3

A3

A1

=

A2

=

and A1: A:: a: as

and similarly, if there were more than four magnitudes.

Prop. xx. Algebraically.

Let A1, A2, A3 be three magnitudes,

and a1, a2, as other three, such that A1: A2 ¦¦ αz: Az, and A2 A3 :: A1: Ag. Then shall A1 : Ág :: α¡ : αz. For since A: A2:: αz: α3,

A1

..

=

a1

az

as

az

Multiplying these equals,

A1 A2 a2 a1

X

X

A3 a3

A1

a1

A3 az

..

and A1: A,:: A1: Az.

If there were four magnitudes, and other four,
such that A1: A2 :: α3: α1,

A2 A3

α2: Az,

α1: a2.

A1

A2

Ag: A

Then shall also А1

:

For since A: A2 :: αz: αs,

A2:A3: a2 A3,

A3

..

A3 A4 α1: Az A4

Multiplying these equals,
A2 A3 Az C-2
123 x 1
A3 A

X

X

X

as

az

A1 a1

A1

.. A1: Д :: α1: α,

and similarly, if there be more than four magnitudes. Prop. xxiv. Algebraically.

or

or

=

or

=

=

and since As: α2 :: A: α, ..

=

:: α : a.

=

A3

A

Let A1: a2 :: Ag: α49 and А : α2 :: Á ̧ : α,

Then shall A1 + A5 : a2 :: Ag + A。 : α4.

A1

A3

For since A1: 42 :: Ag : ɑ4,..

a4

=

=

=

a2 as

Divide the former by the latter of these equals,

A1

A

As A

as

A1

As

adding 1 to each of these equals,

A3
+ 1 = + 1,
A

A1 + As - A3 + A6

=

A,

A

=

=

or

=

=

Prop. xxv. Algebraically.

Let A1a: A3: α49
and let Д be the greatest, and consequently a the least.
Then shall A1 + α4 > α2 + A3.
Since A 2 :: A3: αs

A1 A3

=

Աշ as

Multiply these equals by

=

1 =

A1 a2
.. =
A3

"

as
subtract 1 from each of these equals,

A1

a2

1,

A3

as

A1- A3

A3

a4

Multiplying these equals by

A1

A3

A3

a2

A1

but

a2

A1 A3 A1

Α

=

a2

a4 a2

but A1 > a,: A, is the greatest of the four magnitudes,

.. also A1 A3 > α2 — ɑ4,
add A3+ as to each of these equals,
.. A1 + αs > az + Az.

=

a

=

=

A6

Աշ

A3'

"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