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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, so is H to L; therefore the ratio of AEB to AG is given. and the figure AEB is given in species, and to its side 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 similar to FD. for, by the construction, MN is similar to AG, and AG is similar to FD; therefore the parallelogram FD is similar to MN.

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IF
F the extremes of three proportional straight lines

have given ratios to the extremes of other three proportional itraight lines; the means shall 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 shall have to the other a given ratio.

Let A, B, C be three proportional straight 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 also given.

Because the ratio of A to D, as also 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 b to the rectangle A, C; and the square of E

to the rectangle b D, F. therefore the ratio of the square of B to C. 58. Dat. the square of E is given; wherefore also the ratio of

the straight line B to E is given.

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

АВС Because the ratio of B to E is given, the ratio of the D E F d. 54. Dat.square of B to the square of E is given d. therefore b the

ratio of the rectangle A, C to the rectangle D, F is given.

and the ratio of the fide A to the side D is given; theree.os. Dat.fore the ratio of the other side 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 shewn in the same manDer as in the foregoing Proposition.

PROP.

PROP

LXXII.

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F four straight lines be proportionals; as the first is to

the straight line to which the second has a given ratio; so is the third to the straight 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 straight line to which D has a given ratio.

Let E be the straight line to which B has a given ratio, and as B to E, so make D to F. the ratio of B to E is given', and therefore the ratio of D to F. and be

a. Hyr cause às A to B, so 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 Eis ABE the straight line to which B has a given ratio, and F that Ç DF to which D has a given ratio; therefore as A is to the straight line to which B has a given ratio, fo is C to that to which D has a given ratio.

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83.

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PRO P. LXXIII, I! F four straight lines be proportionals; as the first is to See N.

the straight line to which the second has a given ratio, so is the straight line to which the third has a given ratio to the fourth.

Let the Atraight line A be to B, as C to D; as A to the straight line to which B has a given ratio, so 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, so make F to C; because the ratio of B to E is given, the ratio of C to F is given. and be. A B E causé A is to B, as C to D; and as B to E, fo F to C; F C 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 given ratio, as F, to which C has a given ratio, is to D.

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64.

c. 1. 6,

E

PRO P. LXXIV.
TF

a triangle has a given obtuse angle; the excess of the

square of the fide which subtends the obtuse angle above the squares of the sides which contain it, shall have a given ratio to the triangle.

Let the triangle ABC have a given obtuse angle ABC; and produce the straight line CB, and from the point A draw AD perpen

dicular to BC. the excess of the square of AC above the squares 2. 12. 2. of AB, BC, that is a the double of the rectangle contained by DB,

BC, has a given ratio to the triangle ABC.

Because the angle ABC is given, the angle ABD is also given; b. 43. Dat. and the angle ADB is given, wherefore the triangle ABD is given 6

in species; and therefore the ratio of AD to DB is given. and as
AD to DB, so 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 also the ratio of twice the rectangle DB, BC to the
rectangle AD, BC. but the ratio of the rect-
angle AD, BC to the triangle ABC is given,

A
d. 41. s. because it is doubled of the triangle; there-
fore the ratio of twice the rectangle DB, BC

H €. 9. Dat. to the triangle ABC is given, and twice the

FG rectangle DB, BC is the excessa of the square

р В
of AC above the squares of AB, BC. there-
fore this excess has a given ratio to the triangle ABC.

And the ratio of this excess to the triangle ABC may be found thus; take a straight line EF given in position 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 EH perpendicular to FG. then the ratio of the excess of the square of AC above the squares of AB, BC to the triangle ABC is the same with the ratio of quadruple the straight line HF to HE.

Because the angle A B D 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 &. 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, so is the rectangle AD, BC to its half the triangle ABC; therefore,ex aequali, as twice

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, so is twice the rectangle DB, BC to the triangle ABC.

65.

PRO P. LXXV.
IF a triangle has a given acute angle; the space by which

the square of the side subtending the acute angle is less than the squares of the fides which contain it, shall have a given ratio to the triangle.

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 less than the squares of AB, BC, that is a the double of the rectangle a. 13. 27 contained by CB, BD, has a given ratio to the triangle ABC.

Because the angles ABD, ADB are each of them given, the triangle ABD is given in species; 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; there

А. fore the ratio of these rectangles is given, as also 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,

B DC therefore b the ratio of twice the rectangle CB,

b. 9. D: BD to the triangle ABC is given. and twice the rectangle CB, BD is a the space by which the square of AC is less than the squares of AB, BC; therefore the ratio of this space to the triangle ABC is given, and the ratio may be found as in the preceding Proposition.

L E M M A.
F from the vertex A of an Isosceles triangle ABC, any straight

line AD be drawn to the base BC; the square of the fide AB is
equal to the rectangle BD, DC of the segments of the base together
with the square of AD. but if AD be drawn to the base produred,
the square of AD is equal to the rectangle BD, DC together with
the square of AB.
CAS. 1. Bifect the base BC in E, and join

A AE which will be perpendicular * to BC; wherefore the square of AB is equal b to the

b. 41.16 squares of AE, EB. but the square of EB is equal to the rectangle BD, DC together D BDE C with the square of DE. therefore the square Dd 2

of

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of AB is equal to the squares of AE, ED, that is to b the square of
AD, together with the rectangle BD, DC. the other case is shewa
in the fame way by 6. 2. Elem.

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PRO P. LXXVI.
IF a triangle have a given angle, the excess of the square

of the straight line which is equal to the two sides that
contain the given angle, above the square of the third
fide, shall have a given ratio to the triangle.

Let the triangle ABC have the given angle BAC, the excess of
the straight line which is equal to BA, AC together above the
square of BC, shall 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 because AD is equal to
AC, BD is equal to BE; and BC is drawn from the vertex B of the
líosccles 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 square of BC; and therefore the square of BA,
AC together, that is of BD is greater
than the square of. BC by the rectangle

D
DC, CE; and this rectangle has a given
ratio to the triangle ABC. because the
angle BAC is given, the adjacent angle

B В

C
CAD is given; and each of the angles

H
ADC, DCA is given, for each of them is

E 2.5.& 32.1. the half of the given angle BAC; there

K K 6.43. Dat.fore the triangle ADC is given b in fpe

cies; and AF is drawn from its vertex to the base in a given angle, C. 50. Dat. wherefore the ratio of AF to the base CD is given and as CD d. 1. 6. to AF, so is the rectangle DC, CE to the rectangle AF, CE; and 6.41. 1. 'the ratio of the rectangle AF, CE to its half e the triangle ACE is

given; therefore the ratio of the rectangle DC, CE to the triangle € 37.1. ACE, that isf to the triangle ABC is givens, and the rectangle DC, S. 9. Dat. CE is the excess of the square of BA, AC together above the square

of BC; therefore the ratio of this excess to the triangle ABC is given.

The ratio which the rectangle DC, CE has to the triangle ABC is found thus. take the straight line GH given in position and magnitude, and at the point G in GHI make the angle HGK equal to

the

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