to be 5, 1, and so on, and the line X would be said to be the unit of length.

III. We have seen (Note to Def. 55) that the terms multiple, measure, equimultiple, &c., are applied to the particular kinds of ratios which can be denoted by single numbers, and it is important to remark that these terms are extended to all kinds of ratios without restriction.

Thus, is said to be a multiple of, because it is equal to 2 x (Bk. V. Prop. 3), and and are said to be equi

multiples of and respectively.

So, also, the ratio of two commensurable ratios is defined to be the answer to the question how many times they respectively contain the same common measure.

Thus the ratio of and 35 is, because is equal to 3×, and 35 is equal to 5 × 7.

It is easily seen that the ratio of two ratios must be the same as the quotient of the antecedent divided by the consequent.

For if a and b be the numerator and denominator of the antecedent, and c and d those of the consequent, we

da

know that the ratios are identical with the ratios and db

bc

bd

respectively (Bk. V. Prop. 5);

I

therefore they contain the ratio

ad and be times

therefore their ratio is

respectively,

ad a

bd

с

or = (Bk. V. Prop. 8). bc b d

The student may omit Section II. of this Book, and pass

on immediately to Book VI.

SECTION II. ON THE COMMENSURABILITY AND INCOMMENSURABILITY OF MAGNITUDES OF LIKE KINDS.

PROPOSITION 13.

If a magnitude be a measure of each of two other magnitudes, it shall be a measure both of the sum and difference of any multiples of these other magnitudes.

Let A, B, and C be any three magnitudes of like kind, as, for instance, three straight lines, and let C be a common measure of A and B, then C shall be also a measure of the sum or difference of any multiples of A and B.

Let EF be any multiple of A, and FG any multiple of B, and let EF and FG be placed in the same straight line with a common extremity at F; and first let FG be placed in the prolongation of EF, then the length of EG will be the sum of some multiple of A together with some multiple of B.

Let EF be divided into portions EH, HK, &c., each equal to A, in the points H, K, &c., then the last of these points will coincide with F, because EF is a multiple of A.

Similarly, let FG be divided into portions FL, LM, &c., each equal to B in the points L, M, &c., then the last of these points will coincide with G, because FG is a multiple of B.

If now portions be marked off upon EG equal to C, the points H, K, &c., will coincide with points of division, because EH, HK, &c., being equal to A are multiples of C, and therefore one point of division will coincide with F, and similarly the last point of division will coincide with G.

Therefore EG is a multiple of C, or C is a measure of

EG.

Next, let FG lie upon EF, then EG will be the difference of some multiple of A and of some multiple of B.

If portions be marked off upon EG equal to C beginning with E, it follows from reasoning similar to the above that one point of division will fall upon E and another upon G;

therefore EG is a multiple of C, or C is a measure of EG; therefore C is a measure both of the sum and difference of any multiples of A and B.

Similar reasoning applies to other like magnitudes besides the lengths of straight lines.

Note.-The following shorter proof may also be given of this proposition.

Let A be equal to C multiplied by the number represented by m, or let A be equal to mC, and let B be equal to nC.

Let p and q be any other numbers, and therefore p. A and q. B any multiples of A and B ;

therefore . A is equal to p×m times C,

and q. B is equal to qxn times C.

But px mC+qxnC is equal to C multiplied by the sum of the numbers pm and qn ;

therefore pA+qB is some multiple of C. Similarly, also, pA~qB is some multiple of C.

DEFINITION.

64.—When as many parts as possible, each equal to some smaller magnitude, are taken from a larger magnitude, the larger magnitude is said to be divided by the smaller, the number of parts taken from the larger is called the quotient, and the part of the larger which remains after the subtraction is called the remainder of the division of the larger by the smaller.

Also the smaller magnitude is called the divisor, and the larger the dividend.

Thus, if parts AE, EF, &c., each equal to the smaller straight line CD, be marked off upon the larger straight line AB until a portion GB remains less than CD, then AB is said to be divided by CD, the number of parts AE, EF, &c., equal to CD, is called the quotient of the division, and the portion GB is called the remainder.

When GB is equal to zero, or the last point of division coincides with B, AB is a multiple of CD.

PROPOSITION 14

If the larger of two like magnitudes be divided by the smaller, and then the divisor be divided by the remainder, and the process be continually repeated, the divisor and remainder at any step being the dividend and divisor of the next succeeding step, every common measure of the first dividend and first divisor shall be a common measure of each succeeding dividend and divisor, and conversely every common measure of any dividend and divisor shall be a common measure of the first dividend and first divisor.

For distinctness sake, let the two magnitudes be represented by two straight lines as AB and CD.

Let AB be divided by CD, and let the remainder be GB. Let CD be divided by GB, and let the remainder be MD, and so on.

Then AG is a multiple of CD.

Because AG is a multiple of CD, therefore every common measure of AB and CD is a measure of AB~AG, that is, of GB, and it is also a measure of CD;

therefore every common measure of AB and CD is a

common measure of GB and CD.

Also every common measure of GB and CD is a common measure of the sum of GB and any multiple of CD.

But AG is a multiple of CD;

therefore every common measure of GB and CD is a common measure of AG+GB, that is, of AB, and it is also a measure of CD ;

therefore every common measure of GB and CD is a
common measure of AB and CD.

Similarly, it may be proved that every common measure of GB and CD, i.e. of AB and CD, is a common measure of MD and GB, and that every common measure of MD and GB is a common measure of GB and CD, i.e. of AB and CD.

And so on for any number of steps in the process.

PROPOSITION 15.

If at any step of the process described in the last Proposition the remainder becomes zero, the corresponding divisor shall be the greatest common measure of the two given magnitudes;