Next, let mn, or n 7 m, then m(A+B) may be greater than nB, or may be equal to it, or may be less; first, let m(A+B) be greater than nB; then also, mA+mB7nB; take mB, which is less than nB, from both, and mA 7nB-mB, or mA7(n-m)B (6. 5.). But if mA7(n-m)B, mC7(n-m) D, because A: B:: C: D. Now, (n−m)D=nD-mD (6. 5.), therefore mC7nD-mD, and adding mD to both, mC+mD7nD, that is (1. 5.), m(C+D)7nD. If, therefore, m(A+B)7nB, m(C+D)7nD.

In the same manner it will be proved, that if m(A+B)=nB, m(C+D) =nD; and if_m(A+B)≤nB, m(C+D)≤nD; therefore (def. 5. 5.), A+ B:B::C+D:D

PROP. XIX. THEOR.

If a whole magnitude be to a whole, as a magnitude taken from the first is to a magnitude taken from the other; the remainder will be to the remainder as the whole to the whole.

If A: B:: C: D, and if C be less than A, A-C: B-D :: A : B. Because A: B:: C: D, alternately (16. 5.), A: C :: B: D; and therefore by division (17. 5.) A-C: C:: B-D: D. Wherefore, again alternately, A-C: B-D::C: D; but A: B:: C: D, therefore (11.5.) A -C: B-D :: A: D.

If four magnitudes be proportionals, they are also proportionals by conversion, that is, the first is to its excess above the second, as the third to its excess above the fourth.

If A B C D, by conversion, A: A-B :: C: C-D.

:

:

For, since A: B:: Č: D, by division (17. 5.), A-B: B:: C-D: D, and inversely (A. 5.) B: A-B:: D: C-D; therefore, by composition (18.5.), A: A-B:: C: C-D.

COR. In the same way, it may be proved that A: A+B :: C:C+D.

PROP. XX. THEOR.

If there be three magnitudes, and other three, which taken two and two, have the same ratio; if the first be greater than the third, the fourth is greater than the sixth; if equal, equal; and if less, less.

If there be three magnitudes, A, B, and C, and other three D, E, and F'; and if A : B :: D: E; and also B : C :: E : F, then if A7C, D7F; if A=C, D = F; and if A/C, D ZF.

A, B, C,
D, E, F.

First, let A7C; then A: B7C: B (8. 5.). But A: B:: D: E, therefore also D: E7C: E(13.5.). Now B: C:: E: F, and inversely (A.

5.), CB F: E; and it has been shewn that D: E7C: B, therefore D: E7F: E (13. 5.), and consequently D7F (10. 5.).

Next, let A=C; then A: B:: C: B (7. 5.), but A: B:: D: E; therefore, CBD E, but C: B:: F: E, therefore, D: E:: F: E (... 5.), and D F (9. 5.). Lastly, let A/C. Then C7A, and because, as was already shewn, C: B:: F: E, and B: A:: E: D; therefore, by the first case, if C7A, F7D, that is, if A/C, D/F.

PROP. XXI. THEOR.

If there be three magnitudes, and other three, which have the same ratio taken two and two, but in a cross order; if the first magnitude be greater than the third, the fourth is greater than the sixth; if equal, equal; and if less, less.

If there be three magnitudes, A, B, C, and other three, D, E, and F, such that A: B:: E: F, and B: C:: D: E; if A7C, D7F; if A=C, D=F; and if A/C, D/F.

First, let A7C. Then A: B7C: B (8. 5.), but A: BE: F, therefore E: F7C: B (13. 5.). Now, B: C:: D: E, and inversely, C: B::E:D; therefore, E: F7E: D (13. 5.), wherefore, D7F (10. 5.).

A, B, C,
D, E, F.

Next, let A=C. Then (7. 5.) A: B::C:B; but A: B:: E : F, therefore, C: B:: E: F (11. 5.); but B: C:: D: E, and inversely, C: BED, therefore (11. 5.), E: F:: E: D, and, consequently, D=F (9. 5.).

Lastly, let A/C. Then C7A, and, as was already proved, C: B:. E: D; and B: A :: F: E, therefore, by this first case, since C7A, F7 D, that is, DF.

If there be any number of magnitudes, and as many others, which, taken two ana two in order, have the same ratio; the first will have to the last of the first magnitudes, the same ratio which the first of the other has to the last.*

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

A,

B, C,

D, E, F,

Take of A and D any equimultiples whatever, mA, mD; and of B and D any whatever, nB, nF: and of C and F any whatever, qC, qF. Because A: B:: D: E, mA : nB :: mD: nE (4. 5.); and for the same reason, nB: qC:: nE: qF. Therefore (20. 5.) according as mA is greater than qC, equal to it, or less, mD is greater than qF, equal to it, or less; but mA, mD are any equimultiples of A and D ; and qC, qF are any equimultiples of C and F; therefore :: D: F. Again, let there be four magnitudes, and other four which, taken two

mA,

nB, qC,

mD,

nE, qF.

(def. 5. 5.), A : C

*N. B. This proposition is usually cited by the words "ex æquali," or "ex æquo."

and two in order, have the same ratio, viz. A: B:: E: F; B: C:: F G; C: D:: G: H, then A: D:: E: H.

A, B, C, D,

E, F, G, H.

For, since A, B, C are three magnitudes, and E, F, G other three, which, taken two and two, have the same ratio, by the foregoing case, A: C:: E: G. And because also C: D:: G: H, by that same case, A : D :: E: H. _n the same manner is the demonstration extended to any number of magnitudes.

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 will 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, viz. 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.

C,

A,

mA, mB, nC,

D, E, F,

Because A: B:: E: F, and because also A : B :: mA : mB (15. 5.), and E F nE: nF; therefore, mA: mB:: nE: nF (11. 5.). Again, because B: C:: D: E, mB: nC:: mD: nE (4. 5.); and it has been just shewn that mA : mB:: nE: nF; therefore, if mA 7 nC, mD 7 nF (21.5.); if mA=nC, mD=nF; and if mA/nC, mDnF. Now, mA and mD are any equimultiples of A and D, and nC, nF any equimultiples of C and F; therefore, A: C:: D: F (def. 5. 5.).

mD, nB, nF.

A, B, C, D,
E, F, G, H.

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, viz. 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 magnitudes, 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. 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

PROP. XXIV. THEOR.

But C: D::E:

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.

:

::

N. B. This proposition is usually cited by the words "ex æquali in proportione pertur bata:" or, 66 ex æquo inversely."

Because E: B: F: D, by inversion, B: E:: D: F. But by hypothesis, ABC: D, therefore, ex æquali (22. 5.), A: E:: C: F; and by composition (18. 5.), A+E : E::C+F: F. And again by hypothesis, EB:: F: D, therefore, ex æquali (22. 5.), A+E: B:: C+F: D.

PROP. E. THEOR.

If four magnitudes be proportionals, the sum of the first two is to their diffe rence as the sum of the other two to their difference.

A+BA-B:: C+D: C-D; or if A/B

A+B: B-A :: C+D: D−C.

For, if A7B, then because A: B :: C: D, by division (17. 5.),
A-B: B: C-D: D, and by inversion (A. 5.),

B: A-B:: D: C-D. But, by composition (18. 5.),
A+B B:: C+D: D, therefore, ex æquali (22. 5.),
A+B:A−B::C+D:C−D.

:

In the same manner, if B 7 A, it is proved, that
A+B B-A :: C+D: D-C.

PROP. F. THEOR.

Ratios which are compounded of equal ratios, are equal to 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 D 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, ex æquali (22. 5.), 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, ex æquali inversely (23. 5.), A : C:: D F. In the same manner may the proposition be demonstrated, whatever be the number of ratios.

PROP. G. THEOR.

If a magnitude measure each of two others, it will also measure their sum and

difference.

Let C measure A, or be contained in it a certain number of times; 9 times for instance: let C be also contained in B, suppose 5 times. Then A=9C, and B=5C; consequently A and B together must be equal to 14 times C, so that C measures the sum of A and B ; likewise, since the difference of A and B is equal to 4 times C, C also measures this difference. And had any other numbers been chosen, it is plain that the results would have been similar. For, let A=mC, and B=nC; A+B=(m+n)C, and A—B= (m-n)C.

COR. If C measure B, and also A−B, or A+B, it must measure A, for the sum of B and A-B is A, and the difference of B and A+B is also A

ELEMENTS

OF

GEOMETRY.

BOOK VI.

DEFINITIONS

1. SIMILAR rectilineal figures are those which have their several angles equal, each to each, and the sides about the equal angles proportionals.

In two similar figures, the sides which lie adjacent to equal angles, are called homologous sides. Those angles themselves are called homologous angles. In different circles, similar arcs, sectors, and segments, are those of which the arcs subtend equal angles at the centre.

Two

equal figures are always similar; but two similar figures may be very unequal.

2. Two sides of one figure are said to be reciprocally proportional to two sides of another, when one of the sides of the first is to one of the sides of the second, as the remaining side of the second is to the remaining side of the first.

3. 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.

4. The altitude of a triangle is the straight line drawn from its vertex perpendicular to the base. The altitude of a parallelogram is the perpendicular which measures the distance of two opposite sides, taken as bases. And the altitude of a trapezoid is the perpendicular drawn between its two parallel sides.

PROP. I. THEOR.

Triangles and parallelograms, of the same altitude, are one to another as their bases.

Let the triangles ABC, ACD, and the parallelograms EC, CF have the same altitude, viz. the perpendicular drawn from the point A to BD: Then,