That is, let A1

Then by addition, A1

= m times a
As = m times as = mas
A1 = n times a2 = Nɑ2,

A = n times a1 = na,

+ A,

= mag+na2 = (m+n) Ag = (m+nj times agı and As A = ma1 + na1 = · (m+n) a ̧ = (m+n) times a ̧.

Therefore A,+A, is the same multiple of a,, as 4, +4 is of a. That is, if the first magnitude be the same multiple of the second, as the third is of the fourth, &c.

COR. If there be any number of magnitudes A1, A2, A,, &c. multiples of another a, such that 4, ma, A2 = na, Ag=pa, &c.

=

And as many others B1, B, B, &c. the same multiples of another 6, such that B1 mb, B1 = nb, B1 = pb, &c.

=

Then by addition, A1+ A2 + A3 + &c. = ma + na + pa + &c.

=

(m +n+p+ &c.) a = (m +n+p+ &c.) times a:

and B1 + B2+ B2+ &c.

=

=

mb + nb + pb + &c. == (m+n+p+ &c.) times b:

(m + n + p + &c.) b

that is A+ A+ A+ &c. is the same multiple of a that

B1 + B+ B + &c. is of b.

Prop. 11. Algebraically.

Let A, the first magnitude, be the same multiple of a, the second, as A, the third, is of a, the fourth,

that is, let 1 = m times a,

=

and As = m times a1 = mas

If these equals be each taken » times,

then nA1 = mna2 = mn times a

or nA1, nA, each contain a,, a, respectively mn times.

Wherefore A,, nA, the equimultiples of the first and third, are respectively equimultiples of a, and a,, the second and fourth.

Prop. IV. Algebraically.

Let A1, a,, A3, a, be proportionals according to the Algebraica! definition: that is, let A, a, ¦¦ A‚: a

That is, if the first of four magnitudes has the same ratio to the second which the third has to the fourth; then any equimultiples whatever of the first and third shall have the same ratio to any equimultiples of the second and fourth.

The Corollary is contained in the proposition itself:

for if n be unity, then mA,: a2 :: mA2: as:
and if m be unity, also A1: na, :: As: na....
Algebraically.

Prop. v.
Let 4, be the same multiple of a.

that 4, a part of A1, is of a,, a part of a1. Then A, 4, is the same multiple of a1

-

a as 4 is of ai

For let A1 = m times a1 = ma1,

and Asm times as = mag' then A12 = ma ̧ — ma2 = m (a1 — a2) = m times (1 ɑgi). that is 41-42 is the the same multiple of (α, — a) as 1 is of . Prop. vi. Algebraically.

Let A, A, be equimultiples respectively of a1, a, two others,
=m times a1 = may,
A2 = m times a

that is, let A1

= mags

42 - B2 = ma2 — na2 that is, the remainders A, respectively. And if m n = 1, then A,

=

nay,

=⋅nag•

· (m − n) a1 = (m −n) times α, = (m — n) a2 = (m − n) times a2: B1, A, B2 are equimultiples of a1, 20

B1

=a1, and 42- B2 = α2: or the remainders are equal to a, a, respectively.

And since the fraction is equal to the following relations

only can subsist between A, and a; and between A, and α.

First, if A be greater than a,; then A, is also greater than a:
Secondly, if A, be equal to a,; then 4, is also equal to a:
Thirdly, if A, be less than a,; then As is also less than a:

Otherwise, the fraction could not be equal to the fraction

A3

and therefore a‚ : A1 :: α : 4 ̧.

Prop. c. "This is frequently made use of by geometers, and is necessary to the 5th and 6th Propositions of the 10th Book. Clavius, in his notes subjoined to the 8th def. of Book 5, demonstrates it only in numbers, by help of some of the propositions of the 7th Book; in order to demonstrate the property contained in the 5th definition of the 5th Book, when applied to numbers, from the property of proportionals contained in the 20th def. of the 7th Book: and most of the commentators judge it difficult to prove that four magnitudes which are proportionals according to the 20th def. of the 7th Book, are also proportionals according to the 5th def. of the 5th Book. But this is easily made out as follows:

First, if A, B, C, D, be four magnitudes, such that A is the same multiple, or the same part of B, which C is of D:

Then A, B, C, D, are proportionals: this is demonstrated in proposition (c).

Secondly, if AB contain the same parts of CD that EF does of GH; in this case likewise AB is to CD, as EF to GH.

Let CK be a part of CD, and GL the same part of GH; and let AB be the same multiple of CK, that EF is of GL: therefore, by Prop. c, of Book v, AB is to CK, as EF to GL: and CD, GH, are equimultiples of CK, GL, the second and fourth; wherefore, by Cor. Prop. 4, Book v, AB is to CD, as EF to GH. And if four magnitudes be proportionals according to the 5th def. of Book v, they are also proportionals according to the 20th def. of Book VII.

First, if A be to B, as C to D;

then if A be any multiple or part of B, C is the same multiple or part of D, by Prop. D, Book v.

Next, if AB be to CD, as EF to GH:

then if AB contain any part of CD, EF contains the same part of GH:

for let CK be a part of CD, and GL the same part of GII,
and let AB be a multiple of CK:

EF is the same multiple of GL:

take M the same multiple of GL that AB is of CK; therefore, by Prop. c, Book v, AB is to CK, as M to GL: and CD, GH, are equimultiples of CK, GL; wherefore, by Cor. Prop. 4, Book v, AB is to CD, as M to GH. And, by the hypothesis, AB is to CD, as EF to GH; therefore M is equal to EF by Prop. 9, Book v,

and consequently, EF is the same multiple of GL that AB is of CK.”

This is the method by which Simson shews that the Geometrical definition of proportion is a consequence of the Arithmetical definition, and conversely.

It may however be shewn by employing the equation

a с

= and taking b d'

ma, me any equimultiples of a and c the first and third, and nb, nd any equimultiples of b and d the second and fourth.

And conversely, it may be shewn ex absurdó, that if four quantities are proportionals according to the fifth definition of the fifth book of Euclid, they are also proportionals according to the Algebraical definition. The student must however bear in mind, that the Algebraical definition is not equally applicable to the Geometrical demonstrations contained in the sixth, eleventh, and twelfth Books of Euclid, where the Geometrical definition is employed. It has been before remarked, that Geometry is the science of magnitude and not of number; and though a sum and a difference of two magnitudes can be represented Geometrically, as well as a multiple of any given magnitude, there is no method in Geometry whereby the quotient of two magnitudes of the same kind can be expressed. The idea of a quotient is entirely foreign to the principles of the Fifth Book, as are also any distinctions of magnitudes as being commensurable or incommensurable. As Euclid in Books VII-X has treated of the properties of proportion according to the Arithmetical definition and of their application to Geometrical magnitudes; there can be no doubt that his intention was to exclude all reference to numerical measures and quotients in his treatment of the doctrine of proportion in the Fifth Book; and in his applications of that doctrine in the sixth, eleventh and twelfth books of the Elements.

and 43 = ma•

Therefore the third A, is the same multiple of a, the fourth.

wherefore, the third A, is the same part of the fourth a.

Prop. vII. is so obvious that it may be considered axiomatic. Also Prop. VIII. and Prop. Ix. are so simple and obvious, as not to require algebraical proof.

Prop. x. Algebraically.

Let A, have a greater ratio to a, than A, has to a.
Then A, > A,.

For the ratio of 4, to a is represented by 4,

and the ratio of A, to a is represented by

It follows that A1 > A,.

A

Secondly. Let a have to A, a greater ratio than a has to A1.