These results taken together suggest, but do not prove, the truth of the proposition that a relative multiple scale determines a ratio and therefore also the measure of the ratio.

Art. 64. The proposition just mentioned depends on the Fundamental Proposition in the Theory of Relative Multiple Scales, which is as follows:

If A and B be any two magnitudes of the same kind, and if D be any other magnitude; then there exists one and only one magnitude C of the same kind as D, such that the scale of C, D is the same as the scale of A, B*.

Assuming the truth of this proposition, let D be taken as the unit of number and represented by unity, then there exists a magnitude p of the same kind as the unit of number, such that the scale of p, 1 is the same as the scale of A, B†.

Since the magnitude p is of the same kind as the unit of number, it may properly be called a real number.

Since ρ is wholly determined by the scale of A, B it follows that any pair of magnitudes, which have the same scale as A, B, would also determine the same number

p.

Let therefore p be taken as the measure of the ratio of any pair of magnitudes having the same scale as A, B.

This implies that ρ is taken as the measure of the ratio of p to 1, because the scale of A, B is the same as the scale of P, 1.

Art. 65. NOTATION FOR RATIO.

If two magnitudes of the same kind be called A and B, then the ratio of A to B is written A: B.

A is called the antecedent or first term of the ratio, whilst B is called the consequent or second term of the ratio.

In this book, the fact that one ratio A: B is equal to another ratio C: D will be expressed thus:

A: B= C: D,

and not as it is written in most modern editions of Euclid :

which is read: the ratio of A to B is the same as the ratio of C to D, or more briefly, A is to B as C to D.

* Prop. 14 is a particular case of this.

In the case where A and B have a common measure the value of p has been actually determined, see Art. 50.

Art. 66. Def. 12. PROPORTION.

If there are four magnitudes such that the ratio of the first magnitude to the second is the same as that of the third magnitude to the fourth, then the four magnitudes are said to be proportionals, or in proportion.

If A, B, C, D are four magnitudes, such that

A: BC: D,

then A, B, C, D are proportionals.

A and D are called the extremes of the proportion.
B and C are called the means of the proportion.

D is called the fourth proportional to A, B and C.

The antecedents A and C of the two equal ratios are said to be corresponding* terms of the ratios; so also are the consequents B and D.

The case in which the means of the proportion are equal to one another requires special notice.

then the three magnitudes X, Y, Z are said to be in proportion; Y is said to be a mean proportional between X and Z, and Z is said to be a third proportional to X and Y.

Art. 67. Def. 13. EUCLID'S TEST FOR EQUAL RATIOS.

Euclid states this Test in the following manner :—

The first of four magnitudes is said to have the same ratio to the second, as the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth being taken; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth and, if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth: and, if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth.

The statement of the above Test in symbols has already been given in Art. 35.

Art. 68. A particular case of the conditions is often useful.
Those conditions must hold for all integral values of r and s.

Therefore they hold when r = s = 1.

* Euclid uses the term "homologous."

Art. 69. Def. 14. THE RATIO OF EQUALITY.

When the two terms of a ratio are equal, it is called a ratio of equality.

Art. 70. EXAMPLE 19.

Apply Euclid's Test for Equal Ratios to show that all ratios of equality are the same.

SECTION IV.

THE SIMPLER PROPOSITIONS IN THE THEORY OF RELATIVE MULTIPLE SCALES WITH GEOMETRICAL APPLICATIONS. SECOND SERIES. Nos. 17-24.

Art. 71. PROPOSITION XVII. (Euc. VI. 1.)

ENUNCIATION 1. The areas of parallelograms (or triangles) having the same altitude are proportional to the lengths of their bases.

ENUNCIATION 2. The scale of the areas of two parallelograms (or triangles) which have the same altitude is the same as that of the lengths of their bases. Any two parallelograms which have the same altitude may be placed so as to lie between the same parallels.

Let ABCD, EFGH, Fig. 51, be two parallelograms lying

between the same parallels.

It is required to prove that

[AB, EF]~[ABCD, EFGH].

C HG

A BEF

Fig. 51.

Take any integer r in the first column, and any integer s in the second column of the scale of AB, EF.

Then there are three alternatives shown by the figures

Then in order that the scale of AB, EF may be the same as that of ABCD, EFGH it is necessary to show that in these several cases

r (ABCD) = 8 (EFGH)

ABCDEFGH

Fig. 57.

r (ABCD) < s(EFGH).

r (ABCD) > s (EFGH)
On BA produced set off a length BK equal to r(BA).
Then draw KL parallel to BC cutting CD produced at L.
Then it is known by Art. 19 that BKLC=r(ABCD).
On EF produced set off a length EM equal to s(EF).
Then draw MN parallel to EH cutting HG produced at N.
Then it is known by Art. 19 that EMNH = s (EFGH).

Hence the scale of AB, EF is the same as that of ABCD, EFGH.

* This follows immediately from the proposition that parallelograms between the same parallels and on equal bases are equal in area.