81.

because therefore as AEB to FD, fo is H to K; and as FD to AG, fo is K to L; ex aequali, as AEB to AG, fo is H to L; therefore the ratio of AEB to AG is given. and the figure AEB is given in fpecies, and to its fide AB the parallelogram AG is applied in the given angle ABG, therefore by the 69. Dat. a parallelogram may be found similar to AG. let this be the parallelogram MN; MN alfo is fimilar to FD. for, by the conftruction, MN is fimilar to AG, and AG is fimilar to FD; therefore the parallelogram FD is fimilar to MN.

IF

PROP. LXXI.

F the extremes of three proportional ftraight lines have given ratios to the extremes of other three proportional ftraight lines; the means fhall also have a given ratio to one another. and if one extreme has a given ratio to one extreme, and the mean to the mean; likewise the other extreme fhall have to the other a given ratio.

Let A, B, C be three proportiona! ftraight lines, and D, E, F three other; and let the ratios of A to D, and of C to F be given. then the ratio of B to E is alfo given.

b

Because the ratio of A to D, as alfo of C to F is given, the ratio a. 67. Dat. of the rectangle A, C to the rectangle D, F is given. but the b. 17. 6. fquare of B is equal to the rectangle A, C; and the fquare of E to the rectangle D, F. therefore the ratio of the fquare of B to c. 58. Dat. the fquare of E is given; wherefore alfo the ratio of

the ftraight line B to E is given.

Next, let the ratio of A to D, and of B to E be given; then the ratio of C to F is also given.

АВ С

Because the ratio of B to E is given, the ratio of the DEF

d. 54. Dat. fquare of B to the fquare of E is given d. therefore the ratio of the rectangle A, C to the rectangle D, F is given. and the ratio of the fide A to the fide D is given; there

e. 65. Dat. fore the ratio of the other fide C to the other F is given.

COR. And if the extremes of four proportionals have to the extremes of four other proportionals given ratios, and one of the means a given ratio to one of the means; the other mean shall have a given ratio to the other mean. as may be fhewn in the fame manner as in the foregoing Propofition.

PROP.

PROP. LXXII.

F four straight lines be proportionals; as the first is to the straight line to which the fecond has a given ratio; fo is the third to the ftraight line to which the fourth has a given ratio.

Let A, B, C, D be four proportional straight lines, viz. as A to B, to C to D; as A is to the straight line to which B has a given ratio, fo is C to the ftraight line to which D has a given ratio.

Let E be the ftraight line to which B has a given ratio, and as B to E, fo make D to F. the ratio of B to E is given, and therefore the ratio of D to F. and because as A to B, fo is C to D; and as B to E, fo D to F; therefore, ex aequali, as A to E, fo is C to F. and E is ABE the straight line to which B has a given ratio, and F that CDF to which D has a given ratio; therefore as A is to the ftraight line to which B has a given ratio, fo is C to that to which D has a given ratio.

82

a. Hyp

PROP. LXXIII.

83.

I

F four straight lines be proportionals; as the firft is to See . the ftraight line to which the fecond has a given ratio, fo is the straight line to which the third has a given ratio to the fourth.

Let the ftraight line A be to B, as C to D; as A to the straight line to which B has a given ratio, fo is the straight line to which C has a given ratio to D.

Let E be the straight line to which B has a given ratio, and as B to E, fo make F to C; becaufe the ratio

of B to E is given, the ratio of C to F is given. and be- A BE
cause A is to B, as C to D; and as B to E, fo F to C; FC D
therefore, ex aequali in proportione perturbata, A is
to E, as F to D; that is A is to E to which B has a gi-
ven ratio, as F, to which C has a given ratio, is to D.

a 13.5.

64.

IF

PROP. LXXIV.

a triangle has a given obtufe angle; the excefs of the fquare of the fide which fubtends the obtuse angle above the fquares of the fides which contain it, shall have a given ratio to the triangle.

Let the triangle ABC have a given obtufe angle ABC; and produce the ftraight line CB, and from the point A draw AD perpendicular to BC. the excefs of the fquare of AC above the squares 2. 12. 2. of AB, BC, that is the double of the rectangle contained by DB, BC, has a given ratio to the triangle ABC.

c. I. 6.

a

Because the angle ABC is given, the angle ABD is alfo given; b. 43. Dat. and the angle ADB is given, wherefore the triangle ABD is given in fpecies; and therefore the ratio of AD to DB is given. and as AD to DB, fo is the rectangle AD, BC to the rectangle DB, BC; wherefore the ratio of the rectangle AD, BC to the rectangle DB, BC is given, as alfo the ratio of twice the rectangle DB, BC to the rectangle AD, BC. but the ratio of the rectangle AD, BC to the triangle ABC is given,

E

H

FG

D B

C

d. 41. 1. because it is double of the triangle, therefore the ratio of twice the rectangle DB, BC e. 9. Dat. to the triangle ABC is given. and twice the rectangle DB, BC is the excefs of the fquare of AC above the fquares of AB, BC. therefore this excess has a given ratio to the triangle ABC. And the ratio of this excefs to the triangle ABC may be found thus; take a ftraight line EF given in pofition and magnitude; and because the angle ABC is given, at the point F of the straight line EF make the angle EFG equal to the angle ABC; produce GF, and draw EI perpendicular to FG. then the ratio of the excess of the fquare of AC above the fquares of AB, BC to the triangle ABC is the fame with the ratio of quadruple the ftraight line HF to HE.

Because the angle ABD is equal to the angle EFH, and the angle ADB to EHF, each being a right angle; the triangle ADB is f. 4. 6. equiangular to EHF. therefore f as BD to DA, fo FH to HE; and g. Cor. 4. 5 as quadruple of BD to DA, fo is 8 quadruple of FH to HE. but as twice BD is to DA, fo is twice the rectangle DB, BC to the recth. C. s. angle AD, BC; and as DA to the half of it, fo ish the rectangle AD, BC to its half the triangle ABC; therefore, ex aequali, as twice

c

BD

BD is to the half of DA, that is, as quadruple of BD is to DA, that is, as quadruple of FH to HE, fo is twice the rectangle DB, BC to the triangle ABC.

PROP. LXXV.

IF a triangle has a given acute angle; the space by which

the fquare of the fide fubtending the acute angle is lefs than the fquares of the fides which contain it, fhall have a given ratio to the triangle.

6.5.

Let the triangle ABC have a given acute angle ABC, and draw AD perpendicular to BC; the space by which the square of AC is lefs than the fquares of AB, BC, that is a the double of the rectangle a. 13. 2 contained by CB, BD, has a given ratio to the triangle ABC.

A

Because the angles ABD, ADB are each of them given, the triangle ABD is given in fpecies; and therefore the ratio of BD to DA is given. and as BD to DA, fo is the rectangle CB, BD to the rectangle CB, AD; therefore the ratio of these rectangles is given, as alfo the ratio of twice the rectangle CB, BD to the rectangle CB, AD. but the rectangle CB, AD has a given ratio to its half the triangle ABC, therefore the ratio of twice the rectangle CB, BD to the triangle ABC is given. and twice the rectangle CB, BD is the space by which the fquare of AC is less than the fquares of AB, BC; therefore the ratio of this fpace to the triangle ABC is given, and the ratio may be found as in the preceding Propofition.

LE M M A.

B

DC

b. 9. Dat

IF from the vertex A of an Ifofceles triangle ABC, any traight

line AD be drawn to the base BC; the fquare of the fide AB is equal to the rectangle BD, DC of the fegments of the base together with the fquare of AD. but if AD be drawn to the base produced, the fquare of AD is equal to the rectangle BD, DC together with the fquare of AB.

A

b. 47. 1. of AB is equal to the fquares of AE, ED, that is to the square of AD, together with the rectangle BD, DC. the other cafe is fhewn in the fame way by 6. 2. Elem.

67.

IF

a triangle have a given angle, the excess of the fquare of the ftraight line which is equal to the two fides that contain the given angle, above the fquare of the third fide, fhall have a given ratio to the triangle.

Let the triangle ABC have the given angle BAC, the excess of the ftraight line which is equal to BA, AC together above the fquare of BC, fhall have a given ratio to the triangle ABC.

Produce BA, and take AD equal to AC, join DC and produce it to E, and thro' the point B draw BE parallel to AC; join AE, and draw AF perpendicular to DC. and becaufe AD is equal to AC, BD is equal to BE; and BC is drawn from the vertex B of the Hofccles triangle DBE, therefore, by the Lemma, the square of BD, that is of BA and AC together, is equal to the rectangle DC, CE together with the fquare of BC; and therefore the fquare of BA, AC together, that is of ED is greater than the fquare of BC by the rectangle DC, CE; and this rectangle has a given ratio to the triangle ABC. because the angle BAC is given, the adjacent angle CAD is given; and each of the angles ADC, DCA is given, for each of them is a.5.& 3.1. the half of the given angle BAC; thereb. 43. Dat.fore the triangle ADC is given bin fpe

d. 1. 6.

B

D

E

K

cies; and AF is drawn from its vertex to the bafe in a given angle,

c. 50. Dat. Wherefore the ratio of AF to the bafe CD is given. and as CD to AF, fo is the rectangle DC, CE to the rectangle AF, CE; and 8. 41. 1. the ratio of the rectangle AF, CE to its half the triangle ACE is given; therefore the ratio of the rectangle DC, CE to the triangle f 37. 1. ACE, that is f to the triangle ABC is given. and the rectangle DC, g. 9. Dat. CE is the excefs of the fquare of BA, AC together above the square of BC; therefore the ratio of this excefs to the triangle ABC is given. The ratio which the rectangle DC, CE has to the triangle ABC is found thus. take the ftraight line GH given in pofition and magnitude, and at the point G in GF make the angle HGK equal to

the