CD and the parallelogram DT, the second and the fourth, any equimultiples whatever have been taken, viz., the base CL and the parallelogram CP; and if the base CH be equal to the base CL, the parallelogram CS is also equal to the parallelogram CP (P. 14) if less, less, and if greater, greater. Therefore as the base BC is to the base CD, so is the parallelogram BF to the parallelogrm DT (Def. 30). · Wherefore triangles and parallelograms, &c. Q. E. D.

PROP. XVI. COROLLARY. 1. (E. 5. 16).

If four magnitudes of the same kind are proportionals, they are also proportionals when taken alternately. That is, if A is to B as C is to D, then A is to C as B is to D.

A E

N

B C

ABC and ACD are two triangles, and CF, CE parallelograms. And ABC to ACD as ABF to ADE. Alternately ABC to ABF es ACD to ADE. Take of ABC and ACD, the first and the third, any equimultiples equal to CF and CE, and also of ABF and ADE, the second and the fourth, any equimultiples equal to FC and EC, and it is plain that if CF the multiple of ABC, the first, be less than FC, the multiple of ABF, the second, CE the multiple of ACD, the third is also less than EC the multiple of ADE the fourth, if equal, equal and greater, greater. PROP. XVI. COROLLARY. 2. (E. 5. 18).

If four magnitudes are proportionals, by composition, the first and second together are to the second; as the third and fourth together are to the fourth. That is, if A to B as C to D, then A+B is to B as C+D is to D.

In the above figure. ABC is to ABF as ACD to ADE. And also ABC+ABF is to ABF, as ACD + ADE is to ADE.

PROP. XVI. COROLLARY. 3.

If two magnitudes together are to one of the magnitudes as two other magnitudes together to one of them, the four magnitudes are proportionals.

ABC+ABF is to ABF as ACD+ADE is to ADE. But also ABC to ABF as ACD to ADE.

PROP. XVI. COROLLARY. 4. (E. 5. 11).

Ratios which are equal to the some ratio, are equal to one another. That is A to B as C to D; and C to D as E to F. Then

A to B as E to F.

ABC is to ACD as the base BC is to the base CD and the base BC is to CD as the parallelogram CF is to CE. It is plain that ABC is to ACD as CF is to CE..

PROP. XVI. COROLLARY. 5.

Magnitudes have the same ratio to one another, which their equimultiples have.

It is plain that ABC to ACD as parallelogram CF to CE and CF and CE are equimultiples of ABC and ACD.

PROP. XVI. COROLLARY. 6. (E. 5. 7, 9).

Equal magnitudes have the same ratio to a magnitude of the same kind. And conversely magnitudes which have the same ratio to a magnitude of the same kind, are equal.

It is plain from the 2nd figure of the proposition, that the parallelograms BT and BF which are equal to one another can not but have the same ratio to the parallelogram GS and conversely the parallelograms BF and BT which hay the same ratio to GS are equal.

PROP. XVI. COROLLARY. 7. (E. 5. 22).

If there be any number of magnitudes and as many others, which taken two and two in order, have the same ratio; the

first shall have to the last of the first magnitudes, the same ratio which the first has to the lust of the others.

A, B, C are three magnitudes, and as many others D, E, F; which taken two and two have the same ratio.

That is, if A to B as D to E

and B to C as E to F

then A to C as D to F.

It is plain from the figure first of the proposition, that
BC to CD as ABC to ACD

and CD to DL as ACD to ADL
But BC to DL as ABC to ADL.

PROP. XVII. THEOREM. (E. 6. 2).

If a straight line be drawn parallel to one of the sides of a triangle, it cuts the other two sides, or these sides produced proportionally, so that the segments between the base and the parallel are homologous, and if more straight lines be drawn parallel to the base, any segment is to any other segment in the one side, so are the corresponding segments in the other side to one another.

A

First, let DE be drawn parallel to BC, one of the sides of the triangle ABC. The sides AB and AC, or, AB, AC produced towards A, are out proportionally; that is, BD is to DA, as CE is to EA.

Join BE and CD. The triangle BDE is equal to the triangle CDE (P. 15). But ADE is another triangle; and equal magnitudes have the same ratio to the same magnitude (16. cor. 6). Therefore as the triangle BDE is to the triangle ADE, so is the triangle CDE to tho triangle ADE. But the triangle BDE is to the triangle

ADE, as BD is to DA (P. 16). For the same reason the triangle CDE is to the triangle ADE, as CE is to EA. Therefore as BD is to DA, so is CE to EA (16. cor. 4).

Next, let ADE be a triangle. Produce AD, AE towards D and E, and draw BC parallel to DE. Join CD, BE. It may be shown; as in the first case, that BD to DA a CE to EA.

Thirdly, let GE and FD be drawn paral

lel to BC, and through D, draw DK parallel to BF.

It may be shown that FD and GE are parallel to one another (8. cor.), and

that GF, DH, and GB, HK are equal to one another, being the opposite sides of the parallelograms FH and GK (P. 11).

According to the first case GF is to FA as ED is to DA. And KH is to HD as CE is to ED. But it has been shown that KH-BG and HD=GF. Therefore BG is to GF as CE is to ED.

as FA is to DA (16, cór. 7).

Therefore also BG is to CE

Cor. 1. Conversely, if the sides are cut proportionally so that the segments are homologous, then the straight lines between the points of section are parallel. (E. 6. 2).

Because if not, draw other straight lines as parallels. This may be proved as impossible.

Cor. 2. To divide a given straight line similarly to a given divided straight line, that is, into parts proportional to the parts of the given divided straight line. (E. 6. 10).

In the above figure, if AC be the straight line divided into parts at D and E, and AB to be divided similarly to it, place AB and AC so as to make any angle BAC. Draw

DF, EG, DK parallels, as in the figure. AB is divided into parts at F, G similarly to AC.

Cor. 3. To find a third proportional to two given straight lines. (E. 6. 11).

In the above figure. If AF and AD be the two straight lines and if FG be made equal to AD. Then AF is to FG (i.e., AD) as AD is to DE.

Cor. 4. To find a fourth proportional to three given straight lines. (E. 6. 12).

If AF, FG and AD be three straight lines, then AF is to FG as AD is to DE. Therefore DE is the fourth proportional found.

PROP. XVIII. THEOREM. (E. 6. 4).

The sides about the equal angles of equiangular triangles are proportionals and those which are opposite to the equal angles are homologous. Conversely, if the sides about each of their angles be proportionals, the triangles are equiangular, i.e., have these angles equal, each to each.

Let ABC and AED be two tri

angles, having the angle ABC equal to the angle AED, the angle ACB E to the angle ADE and the angle BAC to the angle EAD. AB is to С СЭ a AC as AE is to AD; AB is to BC as AE is to ED and AC is to BC as AD is to DE.

B

Place one of the triangles AED upon the other triangle ABC, so that the angle EAD and BAC may coincide, also the side AE be on the side AB, and consequently the side AD on the sile AC. If the triangles are equal they coincide and have their sides about each of their