Then there is a certain interdistribution of the pillars among the palings, or in other words of the multiples of the distance between the pillars among the multiples of the distance between the palings. The 2nd pillar lies between the 2nd and 3rd palings, the 5th pillar between the 6th and 7th palings, and so on.

Now if the distance between the palings or between the pillars were altered by any quantity however small, then the distribution would be changed if the series were continued without limit. For if the distance between the palings were changed by a distance equal to say the nth part of the distance between the pillars, then the nth paling would be changed by the whole distance between two pillars, and therefore its position among the pillars would be changed.

Def. 4. The ratio of two magnitudes is said to be equal to that of two other magnitudes (whether of the same or of a different kind from the former), when any equimultiples whatever of the antecedents of the ratios being taken and likewise any equimultiples whatever of the consequents, the multiple of one antecedent is greater than, equal to, or less than that of its consequent, according as that of the other antecedent is greater than, equal to, or less than that of its consequent.

Or in other words:

The ratio of A to B is equal to that of P to Q, when mA is greater than, equal to, or less than nB, according as mP is greater than, equal to, or less than nQ, whatever whole numbers m and n may be.

It is an immediate consequence that:

The ratio of A to B is equal to that of P to Q; when, m being any number whatever, and n another number determined so that either mA is between B and (n+1) B or equal to nB, according as mA is between ”B and (n + 1) B or is equal to nB, so is mP between nQ and (n+1) Q or equal to nQ.

The definition may also be expressed thus:

The ratio of A to B is equal to that of P to Q when the multiples of A are distributed among those of B in the same manner as the multiples of P are among those of Q.

That is, if a model were constructed of the pillars and palings, it would be correct, or the ratio of the distances of pillars and palings in the street is the same as the ratio of the distance of pillars and palings in the model, if every pillar in the model fell between the same palings in the model, as the corresponding pillar in the street did among the palings in the street, the street being supposed to be of indefinite length.

It will be observed that this is a method of ascertaining whether four magnitudes are in proportion which is wholly independent of any arithmetical representation of the numbers.

Def. 5. The ratio of two magnitudes is greater than that of two other magnitudes, when equimultiples of the antecedents and equimultiples of the consequents can be found such that, while the multiple of the antecedent of the first is greater than or equal to that of its consequent, the multiple of the antecedent of the other is not greater or is less than that of its consequent.

Or in other words:

The ratio of A to B is greater than that of P to Q, when whole numbers m and n can be found, such that, while mA is greater than nB, mP is not greater than nQ, or while mA=nB, mP is less than nQ.

Def. 6. When the ratio of A to B is equal to that of P to Q, the four magnitudes are said to be proportionals or to form a proportion. The proportion is denoted thus:

AB: P:Q,

which is read, "A is to B as P is to Q." A and Q are called the extremes, B and P the means, and Q is said to be the fourth proportional to A, B and P.

The antecedents A, P are said to be homologous*, and so are the consequents, B, Q.

* That is, occupy the same position in the ratio.

Def. 7. Three magnitudes (A, B, C) of the same kind are said to be proportionals, when the ratio of the first to the second is equal to that of the second to the third: that is when A: B :: B: C.

In this case C is said to be the third proportional to A and B, and B the mean proportional between A and C.

Def. 8. The ratio of any magnitude to an equal magnitude is said to be a ratio of equality. If A be greater than B, the ratio A: B is said to be a ratio of greater inequality, and the ratio B: A a ratio of less inequality. Also the ratios A: B and B A are said to be reciprocal to one another.

THEOREM I.

Ratios that are equal to the same ratio are equal to one another.

Proof. Let A: B: P: Q, and also A: B:: X: Y, then shall P Q :: X : Y.

and

For since mA >=<nB according as mP>=<nQ

(Def. 4.)

mA>=<nB according as mX>=<nY,

therefore

mP>=<nQ according as mX>=<nY,

and therefore (Def. 4) P: Q :: X : Y.

THEOREM 2.

If two ratios are equal, as the antecedent of the first is greater than, equal to, or less than its consequent, so is the antecedent of the second greater than, equal to, or less than its consequent.

Proof. Let A : B :: P: Q, then as A>=<B so is P>=<Q.

For by Def. 4, as mA>=<nB so mP>=< <nQ,

whatever integers m and n are.

Let m and n each equal r;

A><B so P>=<Q.

then as

THEOREM 3.

If two ratios are equal, their reciprocal ratios are equal.

Proof. Let A: B: P: Q, then B: A :: Q: P.

For, since the multiples of A are distributed among those of B as the multiples of P among those of Q, the multiples of B are distributed among those of A as the multiples of Q among those of P; and therefore

BAQ P.

: (Def. 4.)

THEOREM 4.

If the ratios of each of two magnitudes to a third magnitude be taken, the first ratio will be greater than, equal to, or less than the other as the first magnitude is greater than, equal to, or less than the other: and if the ratios of one magnitude to each of two others be taken, the first ratio will be greater than, equal to, or less than the other as the first of the two magnitudes is less than, equal to, or greater than the other.

Proof. Let A, B, C be three magnitudes of the same kind, then

and

A: C> or <B: C, as A>= or <B,

C: A> or <C: B, as A <= or> B.

If A = B, it follows directly from Def. 4 that A: C :: B: C and CA :: B: A.

If A > B, m can be found such that mB is less than mA by a greater magnitude than C.

Hence if mA be between nC and (m+ 1)C, or if mA=nC, mB will be less than nC, whence (Def. 6) A: C>B:C;

Also, since nC>mB while nC is not > mA (Def. 6) C: B>CA or C: A < C : B.

If A < B, then B>A and therefore B: C>A : C, that is A: C<B: C, and so also C: A > C : B.

COR. The converses of both parts of the proposition are true, since the "Rule of Conversion" is applicable.

THEOREM 5:

The ratio of equimultiples of two magnitudes is equal to that of the magnitudes themselves.

Proof. Let A, B be two magnitudes, then mA : mB :: A: B.

For as A> or <qB, so is m.pA> or <m. qB; but m.pA=p.mA and m.qB=q.mB, therefore as pA >= or <qB, so is p. mA >= or <q. mB, whatever be the values of and 4, and hence mA : mB :: A: B.

THEOREM 6.

If two magnitudes (A, B) have the same ratio as two whole numbers (m, n), then nAmB: and conversely if nA = mB, A has to B the same ratio as m to n.

Proof. Of A and m take the equimultiples nA and n. m, and of B and ʼn take the equimultiples mB and m. n, then since A Bm : n, therefore as nA is >=< mB, so is nm >=<m.n; but since n. m = m. n, it follows (Def. 4) that nA = mB.

Again since mB : nB :: m : n we have, if nA = mB, nA : NB :: mn; whence it follows (Theor. 5) that A : B :: m : n.