For let AM be an arc = m. AB, then the angle or sector between OA and OM (reckoned correspondingly to the arc AM) will be m times the angle or sector AOB.

And let A'N be an arc=n. A'B', and A'O'N an angle or sector n times the angle or sector A'O'B':

and according as AM is >, =, or < A'N,

so is the angle or sector AOM>,=, or < the angle or sector A'O'N;

and therefore as AB : A'B' :: angle or sector AOB: angle or sector A'O'B'.

BOOK V.

PROPORTION.

INTRODUCTION.

[For the use of those for whom it may be thought well to defer the study of the complete, but more difficult, mode of treatment of Proportion in Book IV., the following Definitions and Propositions referred to in that Book are here collected, with an indication of the principles of an incomplete mode of treatment by which they may be established for commensurable magnitudes.]

Def. 1. One magnitude is said to be a multiple of another magnitude when the former contains the latter an exact number of times. According as the number of times is 1, 2, 3...m, so is the multiple said to be the 1st, 2nd, 3rd...mth.

Def. 2. One magnitude is said to be a measure or part of another magnitude when the former is contained an exact number of times in the latter.

Def. 3. If a magnitude can be found which is a measure of two or more magnitudes, these magnitudes are said to be commensurable, and the first magnitude is said to be a common measure of the others.

It is easy to prove that commensurable magnitudes have also a common multiple, and conversely that magnitudes which have a common multiple are commensurable.

1

A measure of a line is any line which is contained in it an exact number of times. Thus an inch is a measure of a foot; and a yard is a measure of a mile. So too the measure of an area is any area which is contained an exact number of times in it. A square inch is thus a measure of a square yard. A measure is therefore an aliquot part of any magnitude which it measures. The length of a line, the extent of an area, or any other magnitude, is completely known when we know a measure of it, and how many times it contains that measure.

In measuring any magnitude we take some standard to measure by. Thus in measuring length we take a yard, or a foot, or an inch. In measuring solids we take a cubic inch, a cubic foot, or the like. The standard so taken is called the unit. It may be a precise measure of the magnitude measured, or it may not. The number, whether whole or fractional, which expresses how many times a magnitude contains a certain unit is called the numerical value of that magnitude in terms of that unit. Thus in speaking of a line as 7 yards long, a yard is the unit of length, and the numerical value of the line in terms of that unit is 7.

Two lines or magnitudes of the same kind are said to have a common measure when there exists a unit of which they can both be expressed as multiples. Thus 15 inches and I foot have a common measure, for with the unit 3 inches, their numerical values would be 5 and 4; and with the unit I inch their numerical values would be 15 and 12. All whole numbers have unity as a common measure.

The following problem gives a method of finding the greatest common measure of two magnitudes, if any common measure exists, and illustrates the familiar Arithmetical method.

PROBLEM.

To find the greatest common measure of two magnitudes, if they have a

common measure.

Let AB and CD be the two magnitudes. From AB the greater cut

off parts, AE, EF... each equal to CD the less, leaving a remainder FB which is less than CD.

From CD cut off parts, CG..., equal to FB, leaving a remainder GD less than FB.

From FB cut off parts FH, HB... equal to GD: and continue this process until a remainder GD is found which is contained an exact number of times in the previous remainder, so that no further remainder is left. The last remainder is then the greatest common measure.

For, firstly, since GD measures FB, it also measures CG; and therefore measures CD. But CD=AE and EF; and therefore GD measures AE, EF and FB; that is it measures AB. Hence GD is a common measure of AB and CD.

And again, since every measure of CD and AB must measure AF, it must measure FB or CG, and therefore also GD: hence the common measure cannot be greater than GD; that is GD is the greatest common

measure.

So also, in the figure adjoining, the first remainder is GB; the

second HD; the third IB; the fourth KD, which is contained exactly twice in IB. Hence KD is the greatest common measure, and it will be seen to be contained twice in IB, and therefore five times in HD, seven times in GB, 12 times in CD, and 43 times in AB.

Hence AB and CD have as their numerical values 43 and 12 in terms of the unit KD.

COR. Every measure of KD is a common measure of AB and CD.

When magnitudes have a common measure they are called commensurable. But it is very frequently the case in Geometrical figures, that lines and other magnitudes have no common measure; the process above given continuing indefinitely; the remainder becoming smaller at each step of the process but never actually disappearing. In this case the lines are said to be incommensurable.

Def. 4. The ratio of one magnitude to another of the same kind is the relation of the former to the latter in respect of quantuplicity.

The ratio of A to B is denoted thus, A: B, and A is called the antecedent, B the consequent.

The complete examination of the nature of the comparison of two magnitudes according to quantuplicity is contained in Book IV. For numbers, and for magnitudes generally, so far as they are commensurable (and it is to be noted that this is not the normal, but the exceptional, case), the comparison may be made in a more simple manner either

(1) (As is usual in Arithmetic) by considering what multiple, part, or multiple of a part one magnitude is of the other;

or (2) by considering what multiples of the two magnitudes are equal to one another.]

Def. 5. When the ratio A: B is equal to the ratio PQ, i.e. either

(1) When A is the same multiple, part, or multiple of a part of B as P is of Q; or,

(2) When like multiples of A and P are equal respectively to like multiples of B and Q;

the four magnitudes are said to be proportionals, or to form proportion.

The equality of the ratios is denoted by the symbol :: ; and the proportion thus, A: B :: 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 to one another, and so also are the consequents.