5. If two sets of four magnitudes be proportionals, and if we divide corresponding terms, the quotients are proportionals.

6. If four magnitudes be proportionals, their squares, cubes, &c., are proportionals.

7. If two proportions have three terms of one respectively equal to three corresponding terms of the other, the remaining term of the first is equal to the remaining term of the second.

8. If three magnitudes be continual proportionals, the first is to the third as the square of the difference between the first and second is to the square of the difference between the second and third.

9. If a line AB cut harmonically in C and D be bisected in 0; prove OC, OB, OD are continual proportionals.

10. In the same case if O' be the middle point of CD; prove 00'2 = OB2 + O'D2.

BOOK VI.

APPLICATION OF THE THEORY OF PROPORTION.

DEFINITIONS.

1. Similar Rectilineal Figures are those whose several angles are equal, each to each, and whose sides about the equal angles are proportional.

Similar figures agree in shape; if they agree also in size, they are congruent.

1. When the shape of a figure is given, it is said to be given in species. Thus a triangle whose angles are given is given in species. Hence similar figures are of the same species.

2. When the size of a figure is given, it is said to be given in magnitude; for instance, a square whose side is of given length.

3. When the place which a figure occupies is known, it is said to be given in position.

II. A right line is said to be cut at a point in extreme and mean ratio when the whole line is to the greater segment as the greater segment is to the less.

III.

If three quantities of the same kind be in continued proportion, the middle term is called a mean proportional between the other two.

Magnitudes in continued proportion are also said to be in geometrical progression.

IV. If four quantities of the same kind be in continued proportion, the two middle terms are called two mean proportionals between the other two.

v. The altitude of any figure is the length of the perpendicular from its highest point to its base.

VI. Two corresponding angles of two figures have the sides about them reciprocally proportional, when a side of the first is to a side of the second as the remaining side of the second is to the remaining side of the first.

This is evidently equivalent to saying that a side of the first is to a side of the second in the reciprocal ratio of the remaining side of the first to the remaining side of the second.

PROP. I.-THEOREM.

Triangles (ABC, ACD) and parallelograms (EC, CF) which have the same altitude are to one another as their bases (BC, CD).

Dem.-Produce BD both ways, and cut off any num

ber of parts BG,

Now, since the several bases CB, BG, GH are all equal, the triangles ACB, ABG, AGH are also all equal [I. XXXVIII.]. Therefore the triangle ACH is the same multiple of ACB that the base CH is of the base CB. In like manner the triangle ACL is the same multiple of ACD that the base CL is of the base CD ; and it is evident that [I. XXXVIII.], if the base HC be greater than CL, the triangle HAC is greater than CAL; if equal, equal; and if less, less. Now we have four magnitudes: the base BC is the first, the base CD the second, the triangle ABC the third, and the triangle ACD the fourth. We have taken equimultiples of the first and third, namely, the base CH, and the triangle ACH; also equimultiples of the

second and fourth, namely, the base CL, and the triangle ACL; and we have proved that according as the multiple of the first is greater than, equal to, or less than the multiple of the second, the multiple of the third is greater than, equal to, or less than the multiple of the fourth. Hence [v. Def. v.] the base BC CD the triangle ABC: ACD.

2. The parallelogram EC is double of the triangle ABC [I. XXXIV.], and the parallelogram CF is double of the triangle ACD. Hence [v. xv.] EC: CF: the triangle ABC ACD; but ABC: ACD:: BC: CD (Part I.). Therefore [v. XI.] EC: CF:: the base BC: CD.

:

Or thus: Let A, A'denote the areas of the triangles ABC, ACD respectively, and P their common altitude; then [II. I., Cor. 1],

In extending this proof to parallelograms we have only to use P instead of P.

PROP. II.-THEOREM.

If a line (DE) be parallel to a side (BC) of a triangle (ABC) it divides the remaining sides, measured from the opposite angle (A), proportionally; and, conversely, If two sides of a triangle, measured from an angle, be cut proportionally, the line joining the points of section is parallel to the third side.

1. It is required to prove that AD: DB:: AE: EC.

Dem.-Join BE, DC. The triangles BDE, CED are on the same base DE, and between the same parallels BC, DE. Hence [1. XXXVII.] they are equal, and therefore [v. vII.] the B triangle ADE: BDE ::¡ADE : CDE ;

E

ADE: BDE :: AD: DB [1.],

but

and

Hence

ADE: CDE :: AE : EC [1.].

AD DB:: AE: EC.

2. If AD: DB:: AE: EC, it is required to prove that DE is parallel to BC.

Dem.-Let the same construction remain;

then

AD: DB:: the triangle ADE: BDE [1.], and AE EC the triangle ADE: CDE [1.]; but AD: DB:: AE: EC (hyp.).

Hence ADE: BDE :: ADE: CDE.

Therefore [v. Ix.] the triangle BDE is equal to CDE, and they are on the same base DE, and on the same side of it; hence they are between the same parallels [I. XXXIX.]. Therefore DE is parallel to BC.

Observation. The line DE may cut the sides AB, AC produced through B, C, or through the angle A; but evidently a separate figure for each of these cases is unnecessary.

Exercise.

If two lines be cut by three or more parallels, the intercepts on one are proportional to the corresponding intercepts on the other.

PROP. III.—THEOREM.

If a line (AD) bisect any angle (A) of a triangle (ABC), it divides the opposite side (BC) into segments proportional to the adjacent sides. Conversely, If the segments (BD, DC) into which a line (AD) drawn from any angle (A) of a triangle divides the opposite side be proportional to the adjacent sides, that line bisects the angle (A).

Dem.-1. Through C draw CE parallel to AD to meet BA produced in E. Because BA meets the