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then since A is the greatest term, it is greater than either B or C; but if A is greater than C, B is greater than D (V. 3, cor.), and by alternation A is to C as B is to D; therefore since A is greater than B, C is greater than D, so that D is the least term; then it is required to shew that A and D together are greater than B and C together.

For, by conversion, we have A: A-B::C:C-D,
and therefore alternately A: C:: A-B:C-D;
but AC,

therefore A-B> C-D.

To the terms of this inequality add B,

then AB + C-D.

Again, add D to the terms of this latter inequality,
then A+D B + C, therefore, etc.



(V. 10.) (V. 3.) (Hyp.)

(V. 2, cor.)

(I. Ax. 5.)

Ratios which are compounded of equal ratios are equal to one another.

Let there be given the ratios A to B as D to E, and B to C as E to F.

Then because there are three magnitudes A, B, C,

and three others D, E, F,

which, taken two and two in the same order,

are proportionals; therefore, by equality,

A is to C as D is to F.

But A to C is the ratio which is compounded of the ratios

of A to B and B to C,

and D to F that which is compounded of the ratios

of D to E and E to F.

Again, if we have the equal ratios A to B and E to F, and also the equal ratios B to C and D to E;



(V. Def. 10.)

A to B, E to F,
B to C, D to E

Then by indirect inequality we have A to C as D to F;
but the ratio of A to C is that which is compounded of
the ratios of A to B and B to C;
and the ratio of D to F that which is compounded of the ratios
of D to E and E to F; and so of any number of ratios.

(V. Def. 10.)

Q. E. D.



To find a line which shall be a common measure of two lines which are commensurable.

Let AB and CD be two lines which are commensurable;

it is required to find a line which shall measure both.

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From AB, the greater line, cut off as many parts as possible AO, OK, each equal to CD; if there is no remainder, CD is an exact measure of AB, and any part of it will be a common measure of the two lines. But if there be a remainder KB, which is necessarily less than CD, cut off parts equal to it from CD, namely, CL, LN; and if there now be no remainder, KB is a common measure; but if there be a remainder ND, on KB take parts equal to ND, and let these exhaust the line so that there is now no remainder; then is ND the common measure required.

For since ND measures KB,

it must also measure CN, which is a multiple of KB;

but it measures ND;

and measuring CN and ND, it must measure their sum CD.

Now AK is a multiple of CD,


therefore ND, measuring CD, must measure AK, a multiple of it; but ND measures KB,

therefore it measures the sum of AK and KB; that is, AB.

Now it has been shewn that it measures CD;

ND is therefore a common measure of the two given lines.

Again, any line which measures the two given lines

must be either ND, or a part of it;

for no line greater than ÑD can measure ND.

Let the line M be such a measure;

and if we suppose it to measure AB and CD,

it must measure AK, a multiple of CD,

and KB, the difference of AB and AK;

but CN is a multiple of KB, therefore M measures CN,
and therefore also ND the difference of CD and CN;
so that M is either equal to ND or a submultiple of it,
there cannot, therefore, be any measure of the two lines
greater than ND,

every other measure being either ND or a submultiple of it.
Wherefore ND is the greatest common measure. Q. E. F.


The diagonal and side of a square are incommensurable.

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From the diagonal DB cut off DO equal to DA, and join AO, then the angle DOA is equal to DAO;

but DAO is acute, being less than the right angle A, therefore DOA is acute and AOB obtuse,

and the angles OBA and OAB both acute. Hence the side BO is less than AB or DA,

so that AD is contained in BD with a remainder,

and is not therefore a measure of it.

From the point O draw OK perpendicular to BD; then the angles KOA and KAO,

being complements of the equal angles DAO and DOA, are equal to one another;

therefore AK and KO are equal.

Now, since BOK is a right angle, and OBK half of a right angle,

OKB is also half a right angle, and BO is equal to OK, and therefore to KA.

(I. 5.) (Const.)

(I. 14, conv.)

(I. 5, conv.)

(I. 23, cor. 5.)

(I. 5, conv.)

Again, BOK being a right angle, and BO and OK equal,
they are the sides of a square whose diagonal is KB;
from KB then cut off a part KL equal to KO, and join OL;
then it can be shewn in the same manner that BL,

the remaining part of the diagonal, is less than BO,

and hence the difference BO, between the side and the diagonal, is contained twice in the side AB with a remainder BL,

which is itself the difference between the diagonal and side of another square,

By repeating the process,

it would appear that BL would be contained twice in BO with a remainder,

which would be the difference between the side and diagonal of the square described on BL,

and the process might thus be continued indefinitely,

as a remainder would always appear.

Hence it follows that no line, however small, can be found which shall be contained without remainder in the side and diagonal, They are therefore incommensurable. Q. E. D.


A ratio which is compounded of other ratios is the same as that which is obtained by multiplying together the homologous terms.

Let the ratio of A to D be compounded of the ratios of A to B, B to C, and C to D; then the ratio of A to D is the same as that which is obtained by multiplying the antecedents together, and also the consequents.

For since A is to D as A is to D, if the terms of the second of these ratios be multiplied first by B and then by C, we shall have

A to D as Ax B x C is to BxCxD,

A to B

B to C

C to D

since magnitudes have the same ratio as their like multiples (V. 9). Now this result is the same as Ax B x C to BxCxD, that is, as A to D; so that A is to D as the product of the antecedents to the product of the consequents, the common factors disappearing.

Schol. 1.-If we take the ratios

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a: b

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That is, a has to the ratio compounded of the ratios of e to f, g to h, and k to l; so that compound ratio is the ratio of the products. It is manifest cd that if x, y, and z be supposed to be the measures y of the ratios of a to b, c to d, and e to f, then ef the product of the consequents will be contained 2 xyz times in ace, the product of the antecedents; that is, the measure of the compound ratio is the product of the measures of the simple ratios. This is no doubt the meaning of the definition of compound ratio, as given at the beginning of Book VI., as follows:

"Def. 5.-A ratio is said to be compounded of [other] ratios when it is produced by multiplying upon themselves the quantities [measures] of the other ratios."

14: 7




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Thus, in the case of the three ratios on the 2 margin, the measures 2, 3, and 4 multiplied upon 12: 4 themselves produce the measure, 24, of the com3 pound ratio 3360 to 140. The statement in the preceding Proposition, and that in Def. 11, are 3360: 140 plainly the same as this; for the common factors in the two ratios disappear when the terms are multiplied, and the ratio of 5 to 120 is the same as that of 4500 to 108,000, the measure of both being 3 3 × 4 × 2=24. Compound ratio is thus simply the 15: 60 ratio of the products [vide Book VI., Prop. 14, Note]; 4 and hence in regard to algebraic quantities and numbers, we may define compound ratio thus: In any number of ratios, if the product of the antecedents be taken, as also that of the consequents, the products are said to have to one another a ratio compounded of the other ratios.

5: 15

60: 120


*More literally thus: A ratio is said to be compounded of ratios when the quantities [numerical measures] of the ratios multiplied upon themselves make one [that is, a certain ratio]. The Greek word Taxorns, here rendered quantity, is the same as that which is employed in the definition of Ratio (V. def. 4). Wallis and Gregory, whom De Morgan follows, translate it by quantuplicity, but we consider quantity a more correct rendering; the reference being to the number of times one is contained in the other. This definition has, however, been regarded as spurious; such a definition can have in the ancient geometry no reference to lines, as in VI. 14 (Euc. VI., 23), unless their numerical representatives are meant (I. App., schol. 2).

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