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PROBLEM.

To find the ratio of two similar rectilineal figures, E, F, simi. larly 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 ČD is given, make the ratio of H to K the same with it; and because it is given, K is given. As H is to K, so make K to L; then the ratio of Eto 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 æquali, 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 Li b 2. Cor.

20. 6. PROP. LV.

51. IF two straight lines have a given ratio to one an. other; the rectilineal figures given in species 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 described 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, AG is also given in species: therefore, since the figures

E E, AG, which are given in spe A

B

с D cies, are described upon the same

F straight line AB, the ratio of E to AG is givena, and because the

G

a 53. dar, ratio of AB to CD is given, and HKL upon them are described the simi. lar and similarly placed rectilineal figures AG, F, the ratio of AG to F is given b: and the ratio of AG to E is given : there. b 54. dati fore the ratio of E to F is given

c 9. dat,

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PROBLEM. To find the ratio of two rectilineal figures E, F given in species, 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, AG given in species are described upon the same straight line AB, find their ratio by the 53d dat. and make the ratio of H to K the same: K is therefore given ;, and because the similar rectilineal figures AG, F are described

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upon the straight lines AB, CD, which have a given ratio, fi
their ratio by the 54th dat. and make the ratio of K to L the
same: the figure E has to F the same ratio which H has to L:
for, by the construction, as E is to AG, so is H to K; and as
AG, to F, so is K to L; therefore, ex æquali, as E to F, so is H
to L.

PROP. LVI.

52.

IF a rectilineal figure given in species be described upon a straight line given in magnitude, the figure is given in magnitude.

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Let the rectilineal figure ABCDE given in species be de-
scribed upon the straight line AB given in magnitude ; the
figure ABCDE is given in magnitude.

Upon AB let the square AF be described; therefore AF is
given in species and magnitude, and because the rectilineal
figures ABCDE, AF given in species are С
described upon the same straight line AB,

B a 53. dat. the ratio of ABCDE to AF is given a :

but the square AF is given in magnitude, D
b 2. dat. therefore balso the figure ABCDE is given
in magnitude.

PROB.
To find the magnitude of a rectilineal E
figure given in species described upon a

L

M straight line given in magnitude.

Take the straight line GH equal to the given straight line AB, and by the 530 dat. find the ratio which the square AF upon AB has to the figure ABCDE;

G H

K and make the ratio of GH to HK the same ; 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, that is, as the figure

GL to HM; and AF is equal to GL; therefore ABCDE is equal c 14. 5. to HM . 53.

PROP. LVII.

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IF two rectilineal figures are given in species; and if a side of one of them has a given ratio to a side of the other ; the ratios of the remaining sides to the remaining sides shall be given.

Let AC, DF be two rectilineal figures given in species, and Jet the ratio of the side AB to the side DE be given, the ratios of the remaining sides to the remaining sicles are also given,

Because the ratio of AB to DE is given, as also a the ratios of a 3. def. AB to BC, and of DE to EF, the ratio of BC to EF is given b. b 10. dat. In the same manner, the ratios of the other sides 10 the other sides are

D given.

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

B

F cause 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 ratio of H to K the same; and make the ratio of K to L

G H K 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 10 EF, so is K to Li er æquali, BC is to EF, as G to L; therefore the ratio of G to L has been found, which is the same with the ratio of BC to EF.

CE

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IF two similar rectilineal figures have a given ratio See N. to one another, their homologous sides have also a given ratio to one another.

tio

Let the two similar rectilineal figures, A, B have a given ra

to one another, their homologous sides have also a given ratio, Let the side DC be homologous to EF, and to CD, EF let the straight line G be a third proportional. As therefore a CD a 2. Cor. to G, so is the figure A to B ; and

20.6. the ratio of A to B is given, there. fore the ratio of CD to G is given ; and CD, EF, G are proportionals ; A wherefore by the ratio of CD to EF

B

b 13. dat. is given.

с D E F G The ratio of CD to EF may be found thus: take a straight line H

H

L 'K 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 ą inean proportional L between H and K; the ratio of CD to

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EF is the same with that of H to L. Let G be a third propor. tional to CD, EF ; therefore as CD to G, so 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.

IF two rectilineal figures given in species have a given ratio to one another, their sides shall likewise have given ratios to one another,

Let the two rectilineal figures A, B, given in species, have a given ratio to one another, their sides shall also have given ratios to one another.

If the figure A be similar to B, their homologous sides shall have a given ratio to one another, by the preceding proposition;

and because the figures are given in species, the sides of each of a 3. def.

them have given ratios a to one another; therefore each side of 9. dat. one of them has b to each side of the other a given ratio,

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

G CD, EF be homologous sides ; therefore EG is given in spe.

A

E cies; and the figure B is given

B. c53. dat. in species; wherefore the ra

с

D
tio of B to EG is given ; and
the ratio of A to B is given, H-
therefore the ratio of the figure K-

A to EG is given ; and A is si Ma 58. dat. milar to EG; therefore d the ra- L

tio of the side CD to EF is given ; and consequently the ratios of the remaining sides to the remaining sides 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 mean proportional M, the ratio of CD to EF is the sama

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

H

PROP. LX.

55. IF a rectilineal figure be given in species and magnitude, the sides of it shall be given in magnitude.

Let the rectilineat figure A be given in species and magni. tude, its sides 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 to BC the side of D; there.

D fore the figure D is given G A in species. And because

ΕΙ

L upon the given straight E F line BC the figure D given in species is described, D is given b in magnitude,

b 56. dat. M

K and the figure A is giv, en in magnitude, therefore, the ratio of A to D is given : and the figure A is similar to D; therefore the ratio of the side EF to the homologous side BC is given ; and BC is given, wherefore EF is given : 58.dai. and the ratio of. EF to EG is given , therefore EG is given. 2. dat. And, in the same manner, each of the other sides of the figure e 3. def. A can be shown to be given.

PROBLEM. 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 give en, which may be found by making f upon the given straight f Cor. 45. line BC the parallelogram BK equal to D, and upon its side 1. CK, making the parallelogram KL equal to H, and 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 sides BC, EF is the same with the rațio of BC to the mean proportional between BC and CL. Find EF the mean proportional; then EF is the

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