also C:D::G: H, by that same case, A: D::E:H. In the same manner is the demonstration extended to any number of magnitudes.

PROPOSITION XXIII. THEOREM.

If there be any number of magnitudes, and as many others, which, taken two and two in a cross order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last.

First, let there be three magnitudes, A, B, C, and other three, D, E, and F, which, taken two and two in a cross order, have the same ratio; namely, A: B:: E: F, and B:C:: D: E, then A: C:: D: F.

Take of A, B, and D, any equimultiples mA, mB, mD; and of C, E, F, any equimultiples nC, nE, nF.

Because A: B:: E: F, and because also A: B::MA:mB (V.15), and E:F::nE:nF; therefore, MA: MB::nE:nF (V. 11). Again, because B : C : : D: E, MB: nC ::mD:nE (V. 4); and it has been just shown that MA: MB::nE:nF; therefore, if m A nC, mDnF (V.21); if mA = nC, mD=nF; and if mAnC, mD∠nF. Now, mA and mD are any equimultiples of A and D, and nC, nF, any equimultiples of Cand F; therefore, A: C:: D: F (V. Def. 10).

A B, C, D, E, F. mA, MB, пC, mD, NE, nF.

Next, let there be four magnitudes A, B, C, and D, and other four, E, F, G, and H, which, taken two and two in a

cross order, have the same ratio,

namely, A: B::G:H, B:C::F:G,

and C: D:: E: F, then A:D :: E: H.

For, since A, B, C, are three magni

A, B, C, D.

E, F, G, H.

tudes, and F, G, H, other three, which, taken two and two in a cross order, have the same ratio, by the first case A:C::F:H. But C:D::E: F; therefore again, by the first case, A:D::E:H. In the same manner may the demonstration be extended to any number of magnitudes.

PROPOSITION XXIV. THEOREM.

If the first has to the second the same ratio which the third has to the fourth; and the fifth to the second, the same ratio which the sixth has to the fourth; the first and fifth together shall have to the second, the same ratio which the third and sixth together have to the fourth.

Let A: B::C: D, and also E: B:: F: D, then A+E:B::C+F: D.

Because E: B:: F: D, by inversion, B: E:: D: F. But by hypothesis, A: B:: C: D; therefore, by equality (V. 22), A:E:: C: F, and by composition (V.18), A+E: E::C+ F: F. Now, again, by hypothesis, E: B :: F: D; therefore, by equality, A +E:B::C+F: D.

PROPOSITION Ε. THEOREM.

Ratios which are compounded of the same ratios, are the same with one another.

Let the ratios of A to B, and of B to C, which compound the ratio of A to C, be equal, each to each, to the ratios of D to E, and E to F, which compound the ratio of E to F, A: C:: D: F.

For, first, if the ratio of A to B be equal to that of D to E, and the ratio of B to C equal to that of E to F, by equality (V.22), A:C::D:F.

A, B, C,

D, E, F.

And next, if the ratio of A to B be equal to that of E to F, and the ratio of B to C equal to that of D to E, by indirect equality (V. 23), A:C::D:F. In the same manner may the proposition be demonstrated whatever be the number of ratios.

PROPOSITION F.

The terms of an analogy are proportional by addition. If A: B:: C: D, then by addition A: A+B::C:C+D. For B: A:: D: C, by inversion (V. A); therefore A+B: A::C+D: C by composition (V. 18); hence, by inversion A: A+B=C:C+D (V. A).

PROPOSITION G.

The terms of an analogy are proportional by mixing.
If A: B:: C: D, then A + B : A-B::C+D:C-D.
For A + B : BC+D:D by composition (V. 18);

and by division A-B:B::C-D: D (V. 17), also by inversion (V. A)

B:AB::D:C-D;

but A+B: B::C+D:D;

therefore, by equality (V. 22), A+B:A-B::C+D: C-D.

If BA, it may be similarly proved that A+B: B-A::C+D:D-C.

PROPOSITION H. PROBLEM.

To find a common measure of two lines.

Let AB, CD, be the two lines.

Find the number of times that CD is contained in AB. If it be contained an exact number of times, then it is a

measure of it, and any part of it will also be a A

common measure. But if it be contained several

E B

times in AB, as 3 times, with a remainder EB; then if EB be a measure of CD, it will also be one of AE, which is a multiple of CD; and therefore it will be also a measure of AB; it would therefore be the common measure required. But if EB be not a measure of CD, let it be contained in it a certain number of times, as 2 times, with the remainder CF; then if CF be a measure of EB, it will also be a measure of DF, which is a multiple of EB; and therefore it will also be a measure of CD, and consequently of AE, and therefore also of AB. Let CF be contained 2 times in EB, then EB2CF; CD=2EB+CF=4CF+CF=5CF; and AB=3CD -+ EB = 15CF+2CF=17CF. And CF is therefore contained 5 times in CD, and 17 times in AB.

In the same manner, the common measure of any other two commensurable lines may be found.

COR. 1. If the process for finding a common measure of two lines never terminates, the lines are incommensurable.

COR. 2.-Any part of a common measure is also a common measure; and the measure found as above, is the greatest common measure.

COR. 3.-Any two commensurable lines are to one another, as the numbers denoting the number of times that they respectively contain their common measure.

PROPOSITION K. THEOREM.

The diagonal and side of a square are incommensurable. Let ABC be the half of a square, the side AB and diagonal AC are incommensurable.

Since the angle at B is a right angle, each of the equal angles at C and A is less than a right angle (I. 32); therefore ABAC. Again, AB+BC>

C

AC (I. 20), or 2AB-AC; hence

AC is greater than once AB, and less

than twice AB; and the same may be

proved of the diagonal and side of any square. Therefore, when the side of a square is taken once from its diagonal, there is always a remainder less than

D

its side.

From Cas a centre, with the radius CB, describe the arc DB; then CD=CB=AB, and AD∠AB. From D draw DE perpendicular to AC; then ED and EB being tangents (III. 16, Cor.), ED = EB (III. 37). But in the triangle ADE, the angle at D is a right angle, and that at A is half a right angle; therefore that at E is half a right angle, and therefore AD = DE = EB. When the first remainder AD therefore is taken from AB, the remaining part AE, from which AD is still to be taken, is the diagonal of a square, of which AD is the side. But this is the same as the former process; and when it has been performed in regard to AE and AD, the remaining lines to be compared will, therefore, again be the side and diagonal of a still smaller square; but since, when the side of a square is taken from its diagonal, there is a remainder, therefore in the above process there will always be found to be a remainder; the process therefore will never terminate, or no common measure can ever be found; therefore AC and AB are incommensurable.

EXERCISES.

1. If all the terms, or any two homologous terms, or the terms of either of the ratios, of an analogy, be multiplied or divided by the same number, the resulting magnitudes are still proportional.

2. If any number of magnitudes be in continued proportion, the difference between the first and second terms is to the first, as the difference between the first and last to the sum of all the terms except the last.

3. If the same magnitude be added to the terms of a ratio, it will be unchanged, increased, or diminished, according as it is a ratio of equality, minority, or majority.

4. The differences of the successive terms of continued proportionals are also in continued proportion.

5. The first term of an infinite decreasing series of quantities in continued proportion, is a mean proportional between its excess above the second, and the sum of the series.

SIXTH BOOK.

DEFINITIONS.

1. Straight lines are said to be similarly divided, when their corresponding segments are proportional.

2. A line is said to be cut in a given ratio, when its segments have that ratio.

3. Straight lines that meet in the same point, are called convergent or divergent lines, according as they are considered to verge towards or from the point of concourse.

4. A straight line is said to be cut in extreme and mean ratio, when the whole is to the greater segment, as the greater segment is to the less.

5. A straight line is said to be harmonically divided, when it is divided into three segments such, that the whole line is to one extreme segment, as the other extreme segment to the middle segment.

6. Three straight lines are said to be in harmonical pro