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PROBLEM. To find the ratio of two similar rectilineal figures E, F similarly described upon straight lines AB, CD which have a givea ratio to one another. let G be a third proportional to AB, CD.

Take a straight line H given in migaitude ; 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, fo malie K to L; then the ratio of E to F is the same with the vario o il to L. for AB is to CD, as H to K, wherefore CD is to G, as K to L; 2od, ex aequali, as AB to G, fo is H to L. but the figure E is to b the figure F, as AB to G, that is as H to L.

b 2. Cor.

23. 6.

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PRO P. LV.

51. TF two straight lines have a given ratio to one ano

ther; the rectilineal figures given in fpecies described .. upon them, shall 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 species and doscribed upon them, have a given ratio to one another.

Upon the straight line AB describe the figure AG similar and similarly placed to the figure F; and because F is given in species, WAG is also given in species. there

fore fince the figures E, AG which
are given in species, are described A
upon the same straight line AB,
the ratio of E to AG is given'.

2. Dat. 53 and because the ratio of AB to - K CD is given, and upon them are described the similar and similarly placed rectilineal figures AG,F, the ratio of AG to F is given ". and the ratio of AG to E is given; B. 54. Dat. therefore the ratio of E to F is given C.

PROBLEM. To find the ratio of two rectilineal figures E, F given in fpecies, and described upon the straight lines AB, CD which have a giveu 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 line 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 furikot

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C 2. Dat.

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rectilineal figures AG, F are described upon the straight lines AT CD which have a given ratio, find their ratio by the 54. DE and make the ratio of K to L the fame.. the figure E has to F the fime ratio which H has to L. for, by the construction, as E is to AG, fo is H 10K; and as AG to F, so is K to L; therefore, ex aequali, as E to F, fo is H to L.

52.

PROP. LVI.
IF
F a rectilincal figure given in species be described

upon a straight line given in magnitude; the figure is given in magnitude.

Let the rectilineal figure ABCDE given in species be described upon the straight line AB given in magnitude; the figure ABCDE is given in magnitade.

Upon AB let the square AF be defçribed; therefore AF is go ven in fpecies and magnitude. and because the rectilineal figures ABCDE, AF given in species are described

C upon the same straight line AB, the ratio of 2. 53. Dat. ABCDE to AF is given'. but the square AF

B b. 2. Dat. is given in magnitude, therefore 6 also the

D
figure ABCDE is given in magnitude.

PRO B.
To find the magnitude of a rectilineal

E А.
figure given in species described upon a

I M straight line given in magnitude.

Take the straight line GH equal to the given straight line AB, and by the 53. Dat.

G 70 find the ratio which the square AF upon

K AB has to the figure ABCDE; and make the ratio of GH to HK the fame; and upon GH describe the square 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, tha is, as the figure GL to HM; and AF is equal to GL, therefore ABCDE is equal to HM.

C. 14. 5.

53

PRO P. LVII.
I F two rectilineal figures are given in fpecies, and if a

side of one of them has a given ratio to a side of the other; the ratios of the remaining fides to the remaining fides fhall be given.

1

jo} Let AC, DF be two retilincal figures given in fpecies, and let the ratio of the side AB to the file DE be given; the ratios of, the remaining sides to the remaining fides are allo given.

Because the ratio of AB to DE is given, is also a the ratios of 2. 3. Def. AB to BC, and of DE to LF; the ratio of BC to IT is girens. in b 10 Dat, the same manner, the ratios of the other sides to the other fides are

А given. The ratio which BC has to ET

B nay be found thus ; take a straight ine G given in magnitude, ao i beause the ratio of EC to BA is given, aake the ratio of G to H the labe; nd because the ratio of Ab to DE GI KL given, make the ratio of H to K le fame; and make the ratio of K to L the fame with the given tio of DE to EF. Gnce chercore as BC to BA, fo is G to H; and i BA to DE, fo is H to K; and as DE to EF, fo is K to L; ex quali, BC is to EF, as G to L. therefore the ratio of G to L is been found which is the lime with the ratio of BC to EF.

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PRO P. LVIII.
F two similar rectilineal figures have a given ratio se

to one another; their homologous fides have also a ven ratio to one another.

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Let the two funilar rectilineal figures A, B have a given ratio
one another; their homologous sides have also a givvn ratio.
Let the Gde CD be homologous to EF, and to CD, ET let the
aight line G be a third proportional. as therefore * CD to G,
is the figure A to B; and the ratio
A to B is given, therefore the ra-

of CD to G is given; anů C!), 2, G are proportionals, wherefere he ratio of CD to EF is giren.

DE TG The ratio of CD to EF may be ind thus; take a straight line H gi

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K i in magnitude; and because the ratio of the figure A to B is gi1, make the ratio of H to K tlfe fame with it. and, as the i 3. ects to be done, find a mean proportional L between "I and K; the

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ratio of CD to EF is the faine with that of H to L. iet Go third proportional to CD, EF; therefore as CD to C, fo is a B, and so is) H to K. and as CD to EF, fo is 1 to L, as is to

Dat.

in the 13.

Sec N.

54.

PRO P. LIX.
If two rectilineal figures given in species have a gis

ratio to one another; their fides fhall likewise is given ratios to one another.

Let the two rectilineal figures A, B given in species have a ratio to one another; their fides thall alfo have given ratios toc another.

If the figure A be similar to B, their homologous fides :have a given ratio to one another, by the preceding Propoie",

and because the figures are given in species, the sides of ead 2. 3. Def. them have given ratios ’to one another. therefore each side b. 9. Dat one of them has b to each side of the other a given ratio.

But if the figure A be not similar to B, let CD, EF be any ***
of their sides; and upon EF conceive the figure EG to be delo:
similar and similarly placed to
the figure A, so that CD, EF

Α.
be homologous files; therefore
EG is given in fpecies. and the C
figure B is given in species,

H
C. 53. Dat. wherefore the ratio of B to

K
EG is given ; and the ratio of

M
A to B is given, therefore b the

I ratio of the figure A to EG is 158. Dat. given. and A is similar to EG, therefore d the ratio of the side ()

to EF is given ; and confequently the ratios of the rema fides to the remaining fides are given.

The ratio of CD to EF may be found thus ; take a straighize H given in magnitude, and because the ratio of the figure A ABS given, make the ratio of H to K the same with it. and by the Dat. find the ratio of the figure B to EG, and make the ratio's to L the fame; between H and L find a mean proportional M;* ratio of CD to EF is the fime with the ratio of H to M. beco the figure A is to B, as H to K; and as B to EG, fo is Ktolis aequali, as A to EG, so is H to L. and the figures A, EG are

B

7., and M is a mean proportional between H and L; therefore, as Ĉis hewn in the preceding Proposition, CD is to EF, as H to M.

fecis

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PRO P. LX.

55 *F a rectilineal figure be given in species and magni

tude, the sides of it thall be given in magnitude.

Let the rectilineal figure A be given in species and magnitude ; Liboy fides are given in magnitude.

Take a straight line BC given in position and magnitude; and von PC de ribes the figure D fimilar, and similarly placed, to the 2. 19.6. gure A, and let EF be the

de of the figure A homo. ngous to BC the side of D; jerefore the figure Dis gi

A

L en in species. and because

F

B pon the given straight line C the figure D given in HI -hecies is defcribid, D is

M iven bin magnitu le. and

К.

b. 56. Dat. 2

ne figure A is given in magnitude, therefore the ratio of A to D ; given. and the figure A is fimilar to D; therefore the ratio of

he lide EF to the homologous fide BC is given and BC is gi- c. 58. Dat. en, wherefore EF is given. and the ratio of EF to EG is gi

d. 2. Dat. dienį, therefore EG is given. and, in the same manner, each of 6. 3. Det. - he other sides of the figure A can be sewn to be given.

PROBLEM. To describe a rectilimeal figure A finilar to a given figure D, and equal to another given figure H. It is Prop. 25. B. 6. Elen.

Because each of the figures D, H is given, their ratio is given, which may be found by making f upon the given straight line BC the f.Cor.45.1; parallelogram BK equal to D, and upon its fide CK making f parallelogram KL equal to H in the angle KCL equal to the angle MBC. therefore the ratio of D to H, that is of BK to KL is the fame with the ratio of BC to CL. and because the figures D, A are fimilar, and that the ratio of D to A, or H, is the fame with the ratio of BC to CL; by the 58. Dat. the ratio of the homologous sides BC, EF is the same with the ratio of BC to the mean proportional between BC and CL. find EF the mean proportion :); then EF is the side of the figure to be described, homologous to BC the

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