the respectively = s at the bases (placed in directum) of the given As, are proportional (alternando and ex æquali) the sides about the s subtended by those bases are proportional; and also the sides which are opposite to = Zs are homologous.

Schol. It is evident that the triangles are similar; and also that the sides, which are opposite to the angles are proportional.

Cor. 1. If a right line be drawn parallel to any side of a triangle, it shall cut off a triangle, similar to the whole.

For they will have a common and the other two s of the cut off ▲ are respectively to their internal opposite s, i, e. to the other s of the given 4 ; they are ... equiangular, and .. similar.

Cor. 2. If a right line be drawn parallel to any side of a triangle, the right line drawn from the opposite angle, will divide the parallels into proportional seg

ments.

For either segment of the line drawn parallel : the corresponding segment of the side to which it is parallel :: the segment of cutting line, taken from vertex : the whole cutting line; .. the segments are proportional to one another.

Note. This Prop. may be proved by assuming, from the vertex of any of the greater A on the legs about it, parts respectively to those sides of the lesser ▲ which are homologous to those on which the parts are assumed. For the right line connecting the extremities of those assumed parts, will be parallel to the third side of the given A ... &c.

Note 2. In the first Cor. the right line may be drawn to cut the sides produced; and in the second Cor. the right line may be drawn from the opposite to cut the parallels produced.

PROP. V. THEOR.

If two triangles have their sides proportional, they shall be equiangular, and the equal angles are subtended by homologous sides.

At the extremities of any side of either A construct

s to those at the extremities of the side homologous to this of the other A; the A thus formed is equiangular with that to which two of itss are made, and therefore similar to it (4. 6). But it is also equilateral with that ▲ on whose side it is constructed, for their common side bears the same ratio to conterminous sides of each, i. e. the ratio of two sides of the other given A; but since it is equilateral, it is also equiangular with the A on whose side it is constructed; and .. the given As are equiangular; and the equals are subtended by homologous sides.

Note This Prop. may be proved by a construction similar to that mentioned in the note of the preceding.

PROP. VI. THEOR.

If two triangles have one angle in each equal, and the sides about the equal angles proportional, the triangles are equiangular, and have those angles equal, which the homologous sides subtend.

same.

At the extremities of the proportional sides construct s, as in the preceding Prop.; and the proof will be the The only difference is, that in the former it, is of no consequence with which of the three sides you make the construction, in this it must be with one of the given proportional sides.

Note. This Prop. may also be proved by assuming on the proportional sides of the greater ▲ parts to the sides of the less, homologous to them.

PROP. VII. THEOR.

If two triangles have one angle in each equal, the sides about two other angles proportional, and each of the remaining angles either less, or not less than a right angle, the triangles shall be equiangular, and they will have those angles equal about which the sides are proportional.

First let each of the others be less than a right <, then the <s about which the sides are proportional shall be =. If they be not, cut off from the greater of

those <s a part to the less, with the side which is about the given = <. Then the line that cuts off this <, also cuts from the given A, a A which is equiangular with the other given one; and the remaining part of the cut A is isosceles, for two of its sides (i. e. the cutting line and remaining side about the cut <) bear the same ratio to the side with which the < is cut off, i. e. the ratio of those sides of the other given ▲ about the < to which the one cut off is.. that of the given A which is also an <at the base of the isosceles A is acute, and its external < (i. e. the contained by cutting line and part of cut side subtending cut off <) is obtuse, but this latter

is equal to one of the <s of the other A, which by hypothesis must be acute. Therefore those 4s about which the sides are proportional, are not unequal. In like manner it can be shewn, that if each of the other s be not less than a right, the 4s about which the sides are proportional

are &c.

Schol. The triangles are similar, if it be given only that one of the remaining angles is right. For if they are not similar construct as in the Prop.; then one of those As into which the given one is divided, is isosceles, and .. thes at its base are acute, but the external adjacent to one of these is right, .. that angle which is adjacent to it is right, but it is also acute, which is absurd.

PROP. VIII. THEOR.

If in a right angled triangle, a perpendicular be drawn from the right angle to the opposite side, it divides the triangle into parts similar to the whole, and to one another.

For each of the parts having a common with the whole, and each of them having a right <, they must be equiangular with the whole (Cor. 2. 32. 1.), they are.. similar with the whole, (4. 6), and .. equiangular and similar with one another.

Cor. Hence it is evident that in every right angled triangle the perpendicular is a mean proportional between the segments of the side on which it falls; also that the remaining sides are mean proportionals between their adjacent segments and the hypothenuse; and also, that the sides about the right angle and the perpendicular are proportional.

Note on Cor. The first analogy is derived from the similarity of the parts, for the sides about 4s are proportional, and the sides subtendings are homologous. And the second analogy is derived from the similarity of the parts with the whole A, &c. Observe always to make those sides homologous which subtend = s.

PROP. IX. PROB.

From a given right line to cut off any part required.

From either extremity of the given line draw a right line making any with it; assume on this drawn line from the point of concourse any portion, and also from the same point another portion, which shall be the same multiple of the first assumed part that the given line is of the part required to be cut off; connect the extremity of this multiple with the extremity of the given line, and draw through the extremity of the first assumed portion a line par. to this connecting line; this par. will cut from the given line the part required, adjacent to the point of

concourse.

For this part cut from given line given line the first assumed portion : to its multiple, (the second assumed portion) (Cor. 1. 4. 6); but the first assumed portion is a submultiple of the second (construc.), .. the part cut off by the par. line is an equisubmultiple of the given line, and is.. to the part required to be cut off.

PROP. X. PROB.

To divide a given right line similarly to a given divided right line.

Draw through either extremity of the undivided line a right line, making any < with it, and on this drawn line assume in continuum from the point of concourse, parts respectively to the several parts of the divided line; connect the extremity of the last assumed part, remote from the concourse with the extremity of the given undivided line, and draw through the extremities of the several assumed parts, lines par. to this connecting line; and those par. lines shall cut the given undivided line

D

similarly to the given divided line. The demonstration is the same as that of the foregoing proposition.

Note. By the assistance of this proposition, we may divide a given ▲ into any number of As which shall be to one another in a given ratio, by dividing the base of the given so that the parts will be to one another in that ratio; if then the extremities of those parts be connected with the opposite, the several As on those parts will be as the parts (Prop. 1. 6).

PROP. XI. PROB.

To find a third proportional to two given right lines,

Draw two right lines making any with one another; on those lines from the vertex of this 4, assume portions respectively to the given antecedent and consequent ; connect their extremities; from the connected extremity of the part = to the antecedent, assume in directum with it a part to the consequent; draw through the extremity of this a line par. to the connecting line, and it will cut from the other leg of the a segment (between the parallels) which shall be a third proportional to the given lines.

For the assumed antecedent: the consequent assumed in directum with it: the first assumed consequent is to this intercept. (7. 6). &c.

Note. If a ratio of lesser inequality is given, it may be continued till we come to a magnitude greater than any assigned.

It is thus demonstrated. Assume on an indefinite right line from one and the same point in it parts continually proportional, which shall be the successive terms of the series (suppose them to be to 1, 3, 9, 27, &c.); then since the first three of those are in continued proportion, (viz. 13:39) Convertendo, the first is to the dif. between the first and second, as the second is to the dif between the second and third, i. e. 1: 2 :: 3:6, permutando 1:3 :: 2 : 6 i. e. the first term of the series is to the second as the dif. between the first and second is to the difference between the second and third; and since the second term is greater than the first, it follows that the difference between the second and third is greater