Book V.

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
crofs order, have the fame ratio, viz.
A: B::G: H; B: C::F: G, and
C: D:: EF, then A: D:: E: H.
For, fince A, B, C are three magni-
tudes, and F, G, H other three, which, taken two and two, in
a cross order, have the fame ratio, by the firft cafe, A: C::
F: H. But C: D:: E: F, therefore again, by the first cafe,
A:D:: E: H. In the fame manner, may the demonstration
be extended to any number of magnitudes. Therefore, &c.
Q. E. D.

PROP. XXIV. THEOR.

F the firft has to the fecond the fame ratio which the third has to the fourth; and the fifth to the fecond, the fame ratio which the fixth has to the fourth; the first and fifth, together, shall have to the fecond, the fame ratio which the third and fixth_together, have to the fourth.

::

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

Because E: B:: F: D, by inverfion, B: E::D: F. a 22.5. But by hypothefis, A: B:: C: D, therefore, ex aequali*, b 18. 5. A: E::C: F, and by compofition, A+E : E::C+F: F. And again by hypothefis, E: B::F:D, therefore, ex quali, A+E : B:: C+F: D. Therefore, &c. Q. E. D.

ae

PROP. E. THEOR.

F four magnitudes be proportionals, the fum of the firft two is to their difference as the fum of the other two to their difference.

Let

L

Let A: B:: C: D; then if A > B,

A+B: A-B:: C+D: C-D; or if A <B

A+B: B-A:: C+D: D-C.

For, if A > B, then because A: B:: C: D, by divifion a,

A-BBC-D: D, and by. inverfion b,

B: A-B:: D: C-D. But, by compofitionc,
A+B B:: C+D: D, therefore, ex aequali d,
A+B: A-B:: C+D: C—D.

:

Therefore, &c.

In the fame manner, if B > A, it is proved, that
A+B: B-A:: C+D: D--C.
Q. E. D.

Book V.

a 17 5.

b A. S. c 18. 5.

d 22. 5.

PROP. F. THEOR.

ATIOS 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 aequali a, A: C :: D: F.

A, B, C,

D, E, F.

a 22. 5.

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 aequali inverfely b, AC :: D: F. In the fame manner may the b 23. 5. propofition be demonftrated, whatever be the number of ratips. Therefore, &c. Q. E. D.

ELE

Two fides of one figure are faid to be reciprocally proportional to two fides of another, when one of the fides of the first is to one of the fides of the other, as the remaining fide of the other, is to the remaining fide of the first.

III.

A ftraight line is faid to be cut in extreme and mean ratio, when the whole is to the greater fegment, as the greater fegment is to the lefs.

IV.

The altitude of any figure is the straight line drawn from its vertex perpendi cular to the bafe.

A

PROP.

Book VI.

Book VI.

PROP. I. THEOR.

RIANGLES and parallelograms, of the fame altitude, are one to another as their bases.

TRU

Let the triangles ABC, ACD, and the parallelograms EC, CF have the fame altitude, viz. the perpendicular drawn from the point A to BD: Then, as the base BC is to the base CD, fo is the triangle ABC to the triangle ACD, and the parallelogram EC to the parallelogram ČF.

Produce BD both ways to the points H, L, and take any number of ftraight lines BG, GH, each equal to the base BC; and DK, KL, any number of them, each equal to the base CD; and join AG, AH, AK, AL. Then, because CB, BG, GH are all equal, the triangles AHG, AGB, ABC are all a 38. 1. equal a: Therefore, whatever multiple the bafe HC is of the bafe BC, the fame multiple is the triangle AHC of the triangle ABC. For the fame reafon, whatever multiple the

F

H GBC

D

F

bafe LC is of the
bafe CD, the fame
multiple is the tri-
angle ALC of the
triangle ADC. But
if the base HC
be equal to the
bafe CL, the tri-
angle AHC is also
KOLL
equal to the tri-
angle ALC; and if the bafe HC be greater than the base
CL, likewife the triangle AHC is greater than the triangle
ALC; and if lefs, lefs. Therefore, fince there are four magni-
tudes, viz. the two bafes BC, CD, and the two triangles ABC,
ACD; and of the bafe BC and the triangle ABC, the firft
and third, any equimultiples whatever have been taken, viz.
the base HC, and the triangle AHC; and of the base CD and
triangle ACD, the fecond and fourth, have been taken any
equimultiples whatever, viz. the bafe CL and triangle ALC;
and fince it has been shown, that if the base HC be greater
than the bafe CL, the triangle AHC is greater than the tri-
angle