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c 1. dat. of LM to MN is therefore givenc: but as LM to MN, so is

GH to HK ; wherefore the ratio of GH to HK is given,

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See Note. IF each of the sides of a triangle be given in mag

nitude, the triangle is given in species.

Let each of the sides of the triangle ABC be given in mag

nitude, the triangle ABC is given in species. a 22. 1. Make a triangle a DEF, the sides of which are equal, each

to each, to the given straight lines AB, BC, CA, which can be done ; because any two of them must be greater than the third ; and let DE be e

A

D qual to AB, EF to BC, and FD to CA; and because the two sides ED, DF are equal to the two BA, AC, each to each, and the base EF equal to B

С E

F the base BC ; the angle b 8. 1.

EDF, is equalb to the angle BAC ; therefore, because the angle EDF, which is equal to the angle BAC, has been found, the angle BAC is givenc, in like manner the angles at B, C are

given. And because the sides AB, BC, CA are given, their d 1. def. ratios to one another are givend, therefore the triangle ABC is e 3. def. givene in species.

c 1. def.

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IF each of the angles of a triangle be given in mag. nitude, the triangle is given in species.

А

Let each of the angles of the triangle ABC be given in mag. nitude, the triangle ABC is given in species. Take a straight line DE given in

D position and magnitude, and at the * 23. 1. points D, E make a the angle EDF

equal to the angle PAC and the
angle DEF equal to ABC; there-

B

C E F fore the other angles EFD, BCA are equal, and each of the angles at the points A, B, C, is given,

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wherefore each of those at the points D, E, F is given : and because the straight line FD is drawn to the given point D in DE which is given in position, making the given angle ĘDF; therefore DF is given in position b. In like manner EF also is b 32. dat. given in position; wherefore the point F is given: and the points D, E are given; therefore each of the straight lines DE, EF, FD is givenc in magnitude ; wherefore the triangle DEF is

c 29. dat. given in speciesd: and it is similare to the triangle ABC : d 42. dat. which is therefore given in species.

1 def. 6.

S 4. 6.

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IF one of the angles of a triangle be given, and if the sides about it have a given ratio to one another; the triangle is given in species.

Let the triangle ABC have one of its angles BAC given, and let the sides BA, AC about it have a given ratio to one another; the triangle ABC is given in species.

Take a straight iine DE given in position and magnitude, and at the point D in the given straight line DE, make the angle EDF equal to the given angle BAC; wherefore the angle EDF is given; and because the straight line FD is drawn to the given point D in ED which is given in position, making the given angle EDF; therefore FD

А is given in position a. And because

a 32. dat.

D the ratio of BA to AC is given, make the ratio of ED to Df the same with it, and join EF; and because the ratio of ED to DF is given,

B с E F and ED is given, thereforeb DF is given in magnitude: and it b 2. dat. is given also in position, and the point D is given, wherefore the point F is givenc; and the points D, E are given, where-c 30. dat. fore DE, EF, FD are givend in magnitude : and the triangled 29. dat. DEF is therefore given in species; and because the triangles e 42. dat. ABC DEF have one angle BAC equal to one angle EDF, and the sides about these angles proportionals; the triangles are f similar ; but the triangle DEF is given in species, and there f 6. 6. fore also the triangle ABC.

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See Note.

IF the sides of a triangle have to one another given ratios; the triangle is given in species.

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Let the sides of the triangle ABC have given ratios to one another, the triangle ABC is given in species.

Take a straight line D given in magnitude ; and because the

ratio of AB to BC is given, make the ratio of D to E the same a 2. dat. with it; and D is given, therefore a E is given. And because

the ratio of BC to CA is given, to this make the ratio of E to F the same ; and E is given, and therefore a F; and because as AB to BC, so is D to E ; by composition AB and BC together are to BC, as D and E to F;

but as BC to CA, so is E to F; b 22. 5. therefore, ex æqualib, as AB

and BC are to CA, so are D

and E to F, and AB and BC e 20. 1.

B are greaterc than CA; there

D FF d A. 5. fore D and E are greaterd than

F. In the same manner any

two of the three D, E, F are e 22. 1. greater than the third. Makee

K the triangle GHK whose sides are equal to D, E, F, so that GH be equal to D, HK to E, and KG to F; and because D, E, F are each of them given, there

fore GH, HK, KG are each of them given in magnitude ; theref 42. dat. fore the triangle GHK is givenf in species : but as AB to BC,

so is (D to E, that is) GH to HK ; and as BC to CA, so is (E to F, that is,) HK to KG; therefore, ex æquali, as AB to AC, so is GH to GK. Wherefore & the triangle ABC is equiangular and similar to the triangle GHK; and the triangle GHK is given in species ; therefore also the triangle ABC is given in species.

Cor. If a triangle is required to be made, the sides of which shall have the same ratios which three given straight lines D. E, F have to one another; it is necessary that every two of them be greater than the third.

H

S 5. 6.

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IF the sides of a right-angled triangle about one of the acute angles have a given ratio to one another; the triangle is given in species.

Let the sides AB, BC about the acute angle ABC of the triangle ABC, which has a right angle at A, have a given ratio to one another; the triangle ABC is given in species.

Take a straight line DE given in position and magnitude ; and because the ratio of AB to BC is given, make as AB to BC, so DE to EF; and because DE has a given ratio to EF, and DE is given, therefore a EF is given ; and because as AB a 2 dat. to BC, so is DE to EF; and AB is less b than BC, therefore b 19. 1. DE is lessc than EF. From the point D draw DG at right an-c A. 5. gles to DE, and from the centre

A E at the distance EF, describe a circle which shall meet DG in

D

F two points ; let G be either of

с them, and join EG; therefore the circumference of the circle E

G is given d in position; and the

d 6. def. straight line DG is givene in position, because it is drawn to e 32. dat. the given point D in DE given in position, in a given angle ; thereforef the point G is given ; and the points D, E are given, f 28. dat. wherefore DE, EG, GD are giveng in magnitude, and the tri-g 29. dat. angle DEG in speciesh. And because the triangles ABC, DEG h 42. dat. have the angle BAC equal to the angle EDG, and the sides about the angles ABC, DEG proportionals, and each of the other angles BCA, EGD less than a right angle; the triangle ABC is equiangulari and similar to the triangle DEG : but i 7. 6. DEG is given in species; therefore the triangle ABC is given in species : and, in the same manner, the triangle made by drawing a straight line from E to the other point in which the circle meets DĞ is given in species,

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See Note.

IF a triangle has one of its angles which is not a right angle given, and if the sides about another angle have a given ratio to one another; the triangle is given

in species.

A

Let the triangle ABC have one of its angles ABC a gires, but not a right angle, and let the sides BA, AC about another angle BAC have a given ratio to one another; the triangle ABC is given in species.

First, let the given ratio be the ratio of equality, that is, let the sides BA, AC and consequently the angles ABC, ACB be e

qual; and because the angle ABC is given, a S2. 1.

the angle ACB, and also the remaining a

angle BAC is given ; therefore the trianh 43. dat. gle ABC is givenb in species : and it is B

evident that in this case the given angle ABC must be acute.

Next, let the given ratio be the ratio of a less to a greater, that is, let the side AB adjacent to the given angle be less than the side AC: take a straight line DE given in position and

magnitude, and make the angle DEF equal to the given angle c 32. dat. ABC; therefore EF is given c in position; and because the

ratio of B A to AC is given, as BA
to AC, so make ED to DG; and

A
because the ratio of ED to DG is

given, and ED is given, the el 2. dat. straight line DG is givend, and BA is less than AC, therefore ED

B

с e A. 5. is lesse than DG. From the centre D at the distance DG de.

D scribe the circle GF meeting EF

in F, and join DF; and because 6. def. the circle is givenf in position, as also the straight line EF, the point E

F g 28. dat. F is given; and the points D, E

are giren ; wherefore the straight 11 29. dat. lines DE, EF, FD are givenh in G i 49. dat. magnitude, and the triangle D.F k 18. 1. in speciesi, and because BA is less than AC, the angle ACB is I 1. 7. 1. lessk than the angle ABC, and therefore ACB is lessl than

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