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48.

I

PROP. LII.

F two triangles given in fpecies be described upon the fame ftraight line; they fhall have a given ratio

to one another.

Let the triangles ABC, ABD given in species be described upon the fame ftraight line AB; the ratio of the triangle ABC to the triangle ABD is given.

Thro' the point C draw CE parallel to AB, and let it meet DA produced in E, and join BE. because the triangle ABC is given in species, the angle BAC, that is the angle ACE, is given; and because the triangle ABD is given in species, the angle DAB, that is the angle AEC is given. therefore the triangle ACE is given in fpecies; wherefore the

ratio of EA to AC is

a. 3. Def. given a, and the ratio of CA to AB is given,

b. 9. Dat.

c. 37. I.

d. 1. 6.

as alfo the ratio of BA

C

E

H

A

BF

G

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to AD; therefore the ratio of EA to AD is given. and the triangle ACB is equal to the triangle AEB, and as the triangle AEB, or ACB, is to the triangle ADB, fo is the straight line EA to AD. but the ratio of EA to AD is given, therefore the ratio of the triangle ACB to the triangle ADB is given.

PROBLEM.

To find the ratio of two triangles ABC, ABD given in fpecies, and which are described upon the same straight line AB.

Take a straight line FG given in pofition and magnitude, and because the angles of the triangles ABC, ABD are given, at the . 23. I. points F, G of the straight line FG make the angles GFH, GFK

equal to the angles BAC, BAD; and the angles FGH, FGK equal to the angles ABC, ABD, each to each. therefore the triangles ABC, ABD are equiangular to the triangles FGH, FGK, each to each. through the point H draw HL parallel to FG meeting KF produced in L. and because the angles BAC, BAD are equal to the angles GFH, GFK, each to each; therefore the angles ACE, AEC are equal to FHL, FLH, each to each, and the triangle AEC equiangular to the triangle FLH. therefore as EA to AC,

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fo is LF to FH; and as CA to AB, fo HF to FG; and as BA to AD, so GF to FK; wherefore, ex aequali, as EA to AD, so is LF to FK. but, as was fhewn, the triangle ABC is to the triangle ABD, as the straight line EA to AD, that is as LF to FK. the ratio therefore of LF to FK has been found which is the fame with the ratio of the triangle ABC to the triangle ABD.

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F two rectilineal figures given in fpecies be defcribed See N. upon the fame ftraight line; they fhall have a given ratio to one another.

Let any two rectilineal figures ABCDE, ABFG which are given in fpecies, be described upon the fame ftraight line AB; the ratio of them to one another is given.

Join AC, AD, AF; each of the triangles AED, ADC, ACB, AGF, ABF is given a in species. and because the triangles ADE, a. 51. Dat.

a

E

G

A

D

b. 52. Dat.

c. 7. Dat.

B

F

KL MN

0 d. 9. Dat.

ADC given in species are described
upon the fame ftraight line AD, the
ratio of EAD to DAC is given b; and,
by compofition, the ratio of EACD to
DAC is given. and the ratio of DAC
to CAB is given, because they are de-
scribed upon the fame ftraight line
AC; therefore the ratio of EACD to
ACB is given "; and, by compofition, H
the ratio of ABCDE to ABC is given.
in the fame manner, the ratio of ABFG to ABF is given. but the
ratio of the triangle ABC to the triangle ABF is given b; where-
fore because the ratio of ABCDE to ABC is given, as also the
ratio of ABC to ABF, and the ratio of ABF to ABFG; the ratio
of the rectilineal ABCDE to the rectilineal ABFG is given a.

PROBLEM..

To find the ratio of two rectilineal figures given in fpecies, and defcribed upon the fame ftraight line.

Let ABCDE, ABFG be two rectilineal figures given in fpecies, and described upon the fame straight line AB, and join AC, AD, AF. take a straight line HK given in pofition and magnitude, and by the 52. Dat. find the ratio of the triangle ADE to the triangle ADC, and make the ratio of HK to KL the fame with it. find alfo

50.

the ratio of the triangle ACD to the triangle ACB, and make the
ratio of KL to LM the fame. alfo, find the ratio of the triangle
ABC to the triangle ABF, and make the ratio of LM to MN the
fame. and lastly, find the ratio of the triangle AFB to the triangle
AFG, and make the ratio of MN to
NO the fame. then the ratio of ABCDE

to ABFG is the fame with the ratio
of HM to MO.
Because the triangle EAD is to the
triangle DAC, as the ftraight line HK
to KL; and as triangle DAC to CAB,
fo is the ftraight line KL to LM;

D

E

C

A

B

៩.

F

therefore by using compofition as of- HKLMN

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ten as the number of triangles requires,
the rectilineal ABCDE is to the triangle ABC, as the straight line
HM to ML. in like manner, because triangle GAF is to FAB, as
ON to NM, by compofition, the rectilineal ABFG is to the tri-
angle ABF, as MO to MN; and, by inverfion, as ABF to ABFG,
fo is NM to MO. and the triangle ABC is to ABF, as LM to
MN. wherefore because as ABCDE to ABC, fo is HM to ML;
and as ABC to ABF, fo is LM to MN; and as ABF to ABFG,
fo is MN to MO; ex aequali, as the rectilineal ABCDE to ABFG,
fo is the straight line HM to MO.

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PROP. LIV.

two straight lines have a given ratio to one another; the fimilar rectilineal figures defcribed upon them fimilarly, fhall have a given ratio to one another.

Let the straight lines AB, CD have a given ratio to one another, and let the fimilar and fimilarly placed rectilineal figures E, F be defcribed upon them; the ratio of E to F is given.

To AB, CD let G be a third proportional; therefore as AB to CD, fo is CD to G. and the ratio of AB to CD is given, wherefore the ratio of CD to G is given; and confequently

a. 9. Dat. the ratio of AB to G is alfo given a. b. 2. Cor. but as AB to G, fo is the figure E to

20. 6.

AA

H

the figure F. therefore the ratio of E to F is given.

K L

PROBLEM.

To find the ratio of two fimilar rectilineal figures E, F fimilarly described upon ftraight lines AB, CD which have a given ratio to one another. let G be a third proportional to AB, CD.

Take a straight line H given in magnitude; and because the ratio of AB to CD is given, make the ratio of H to K the fame with it; and because H is given, K is given. as H is to K, so make K to L; then the ratio of E to F is the fame with the ratio of H to L. for AB is to CD, as H to K, wherefore CD is to G, as K to L; and, ex aequali, as AB to G, fo is II to L. but the figure E is to the figure F, as AB to G, that is as H to L.

I

PROP. LV.

F two ftraight lines have a given ratio to one another; the rectilineal figures given in fpecies defcribed upon them, fhall have to one another a given ratio.

Let AB, CD be two straight lines which have a given ratio to one another; the rectilineal figures E, F given in fpecies and described upon them, have a given ratio to one another.

Upon the straight line AB describe the figure AG fimilar and fimilarly placed to the figure F; and because F is given in species,

AG is also given in species. there

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the ratio of E to AG is given 2.

G

a. Dat. 53.

and because the ratio of AB to HK-L

CD is given, and upon them are

described the fimilar and fimilarly placed rectilineal figures AG, F,

the ratio of AG to F is given . and the ratio of AG to E is given; b. 54. Daṭ. therefore the ratio of E to F is given .

PROBLEM.

To find the ratio of two rectilineal figures E, F given in fpecies, and defcribed upon the ftraight lines AB, CD which have a given ratio to one another.

Take a straight line H given in magnitude; and because the rectilineal figures E, AG given in species are described upon the same straight line AB, find their ratio by the 53. Dat. and make the ratio of H to K the fame; K is therefore given. and because the fimilar

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c. 9. Dat.

52.

rectilineal figures AG, F are defcribed upon the straight lines AB, CD which have a given ratio, find their ratio by the 54. Dat. and make the ratio of K to L the fame. the figure E has to F the fame ratio which H has to L. for, by the conftruction, as E is to AG, fo is H to K; and as AG to F, fo is K to L; therefore, ex acquali, as E to F, fo is H to L.

IF

PROP. LVI.

Fa rectilineal figure given in fpecies be defcribed upon a ftraight line given in magnitude; the figure is given in magnitude.

Let the rectilineal figure ABCDE given in fpecies be defcribed upon the straight line AB given in magnitude; the figure ABCDE is given in magnitude.

Upon AB let the fquare AF be defcribed; therefore AF is given in fpecies and magnitude. and because the rectilineal figures ABCDE, AF given in fpecies are described

upon the fame ftraight line AB, the ratio of a. 53. Dat. ABCDE to AF is given. but the squate AF b. 2. Dat. is given in magnitude, therefore also the figure ABCDE is given in magnitude.

c. 14. 5.

53.

PROB..

To find the magnitude of a rectilineal figure given in fpecies defcribed upon a traight line given in magnitude.

B

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Take the straight line GH equal to the given ftraight line AB, and by the 53. Dat. G find the ratio which the fquare AF upon

H

K

AB has to the figure ABCDE; and make the ratio of GH to HK the fame; and upon GH defcribe the fquare GL, and complete the parallelogram LHKM; the figure ABCDE is equal to LHKM. because AF is to ABCDE, as the straight line GH to HK, that is as the figure GL to HM; and AF is equal to GL, therefore ABCDE is equal to HM c.

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PROP. LVII.

[F two rectilineal figures are given in fpecies, and if a fide of one of them has a given ratio to a fide of the other; the ratios of the remaining fides to the remaining fides fhall be given.

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