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PROBLEM.

To find the ratio of two similar rectilineal figures, E, F, fimilarly described upon straight 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 same 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 same 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, so 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. 20.

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IF two straight lines have a given ratio to one another ;

the re&ilineal figures given in species descrited 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 described upon them, have a given ratio to one another.

Upon the straight line AB, describe the figure AG fimilar and similarly placed to the figure F; and because F is given in species, AG is also given in fpecies : Therefore, since the figures

C E, AG which are given in fpe- A B cies, are described upon the fame straight line AB, the ratio of

a 53. dat. E to AG is given ·, and because the ratio of AB to CD is given,

H

1. and upon them are described the fimilar and fimilarly placed rectilineal figures AG, F, the ratio of AG to F is given ; b 54. dat. and the ratio of AG to E is given ; therefore the ratio of E to F is given

PROBLEM.

c 9. dat.

To find the ratio of two rectilineal figures E, F given in fpecies, and described upon the straight 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, AĞ given in species are described upon the same straight line AB, find their ratio by the 530 dat. and make the ratio of H to K the fame; K is therefore given : And because the similar rectilineal figures AG, F are described

upon

upon the straight lines AB, CD, which have a given ratio, find their ratio by the 54th dat. and make the ratio of K to L the same : The figure e has to F the fame ratio which H has to L: For, by the construction, as E is to AG, fo is H to K; and as AG to F, fo is K to L; therefore, ex aequali, as E to F; so is H to L.

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IF a rectilineal 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 magnitude.

Upon A B let the square AP be described ; therefore AF is given in species and magnitude, and because the rectilineal fi. gures ABCDE, AF given in species are

described upon the same straight line AB, • 53. dat. the ratio of ABCDE. to AF is given a :

B
But the square AF is given in magnitude,

F 2. dat. therefore also the figure ABCDE is gi

DI
ven in magnitude.
PRO B.

E
To find the magnitude of a rectilineal L M
figure given in species described upon a
straight line given in magnitude.

Take the straight line GH equal to the given straight line AB, and by the 53d dat. find the ratio which the square

H K AF upon AB has to the figure ABCDE; and make the ratio of GH to HK the fame; and upon GH describe the square GL, and coniplete the parallelogram LHKM, the figure ABCDE is Cyual 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 e 14. 5.

equal to GL ; therefore ABCDE is equal to HM.

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IT

F two rectilineal figures are given in species, and if a

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

Let

Let AC, DF be two rectilineal figures given in species, and let the ratio of the side AB to the side DĘ be given, the ratios of the remaining fides to the remaining Gdes are also given.

Because the ratio of AB to DE is given, as also the ratios a 3. del. of AB to BC, and of DE to EF, the ratio of BC to EF is gi. ven). In the same manner, the ra

b 10. dat.

D tios of the other Gdes to the other fides are given.

А, The ratio which BC has to EF may be found thus : Take a straight B line G given in magnitude, and

E because the ratio of BC to BA is given, make the ratio of G to H the same ; and because the ratio of AB to DE is given, make the G H KL ratio of H to K the same; and make the ratio of K to L the same with the given ratio of DE to EF. Since therefore as BC to BA, so is G to H; and as BA to DE, so is H to K; and as DE to EF, so is K to L ; ex aequali, BC is to EF, as G to L; therefore the ratio of G to L has been found, wbich is the same with the ratio of BC to EF.

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IF

two similar rectilineal figures have a given ratio to See N.

one another, their homologous fides have also a given ratio to one another.

Let the two fimilar rectilineal figures A, B have a given ratio to one another, their homologous fides have also a given ratio.

Let the lide CD be homologous to EF, and to CD, EF let the straight line G be a third proportional. As therefore * CD a 2. Cor. to G, fo is the figure A to B ; and

20. 6. che ratio of A to B is given, therefore the ratio of CD to G is given ; А

B and CD, EF, G are proportionals; wherefore the ratio of CD to EF C D E F G

b 13. dat. is given.

The ratio of CD to EF may be found thus : Take a straight line H

H

IT given in magnitude ; and because the ratio of the figure A to B is given, make the ratio of H to K the same with it: And, as the 13th dat. directs to be done, find a mean proportional L

between

between H and K; the ratio of CD to EF is the fame with that of H to L. Let G be a third proportional to CD, EF; therefore as CD to G, fo is (A to B, and so is) H to K; and as CD to EF, so is H to L, as is shown in the 13th dat.

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See N.

F two re&ilineal figures given in species have a given

ratio to one another, their fides shall likewise have given ratios to one another.

I

a 3. def. b g. dat,

Let the two recļilineal figures A, B given in species, have a given ratio to one another, their fides fhall also have given ra. tios to one another.

If the figure A be fimilar to B, their homologous fides shall have a given ratio to one another, by the preceding propofition ; and because the figures are given in species, the fides of each of them have given ratios * to one another; there, fore each side of 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 two of their fides; and upon EF conceive the figure EG to be described Gimilar and similarly placed to the figure A, so that

G CD, EF be homologous fides ;

A therefore EG is given in fpe

с cies; and the figure B is given

D E B F 53. dat. in species; wherefore the ratio H of B to EG is given ; and the

K ratio of A to B is given, therefore the ratio of the fi

M gure A to EG is given ; and L đ 58. dat. A is fimilar to EG; therefore d the ratio of the fide CD to EF

is given; and consequently the ratios of the remaining fides to the remaining fides are given.

The ratio of CD to EF may be found thus : Take a straight line H given in magnitude, and because the ratio of the figure A to B is given, make the ratio of H to K the same with it. And by the 53d dat. find the ratio of the figure B to EG, and make the ratio of K to L the same : Between H and L find a mean proportional M, the ratio of CD to EF is the same with the ratio of H to M; because the figure A is to B, as H to K; and as B to EG, so is K to L ; ex acquali, as A

to

Α Τ

A T to EG, fo is H to L: And the figures A, EG are similar, and M is a mean proportional between H and L; therefore, as was fhewn in the preceding proposition, CD is to EF as H to M.

IF

PRO P. LX.

55. F a rectilineal figure be given in fpecies and magnitude,

the fides of it shall be given in magnitude.

Let the rectilineal figure A be given in species and magnitude, its fides are given in magnitude.

Take a straight line BC given in position and magnitude, and upon BC describe the figure D similar, and similarly a 18. 6. placed, to the figure A, and let EF be the side of the figure A homologous G to BC the side of D; A

D therefore the figure D is

B given in species. And be- E F

C cause : upon the given Atraight line BC the fi- H gure D given in Tpecies

M K is described, D is given b

,b 56. dat. in magnitude, and the figure A is given in magnitude, there. fore the ratio of A to D is given : And the figure A is fimilar to D; therefore the ratio of the side EF to the homologous Gide BC is given"; and BC is given, wherefore d EF is given; And C 58. dat. the ratio of EF to EG is given, therefore EG is given. “And, d 2. dal. in the same manner, each of the other fides of the figure A can

€ 3. def. be fhewn to be given.

PROBLEM.

1.

To describe a rectilineal figure A similar to a given figure D, and equal to another given figure H. It is prop. 25. b. 6. Elem.

Because each of the figures D, H is given, their ratio is gi. ven, which may be found by making upon the given straight f cor. 45. line BC the parallelogram BK equal to D, and upon its side CK making fthe 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 same with the ratio of BC to CL: And because the figures D, A are similar, and that the ratio of D to A, or H, is the same with the ratio of BC to CL ; by the 58th dat. the ratio of the homologous fides BC, EF is the Same with the ratio of BC to the mean proportional between BC and CL Find EF the mean proportional; then EF is the

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