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PROP. 21.-THEOR. If from the ends of the side of a triangle there be drawn two st. lines to u point within the triangle, these lines shall be less than the other two sides of the triangle, but shall contain a greater angle. .

Con.—Pst. 2. Dem.-P. 20. Ax. 4. P. 16. E. 1, Hyp. Let the a be ABC, and from B

and C let the st. lines BD, CD,

meet within the A in the · D;
2 Conc. 1. then BD and CD shall be < BA

and CA;
., 2. but the L BDC shall be > than

ZBAC.
C. byPst. 2. Produce BD to meet CA in the

point E.
1 by P. 20. In A ABE the two sides AB and

AE are > BE;
2) Add. to each of these add the line EC ;

then the lines AB, AE, and EC, are > BE and EC. 4 P. 20. Again in A CED, the two sides CE and ED are > CD; 5. Add. to each of these add the line DB; 6 Ax. 4. then CE, ED, and DB together are > CD and DB together. 7 D. 3. But AB and AC are > BE and EC ; 8 à fort. much more are AB and AC > CD and DB.

9 P. 16. Again, in A CDE the ext. į BDC is > the int. _ CED, 10 P. 16. and in A ABE the ext. L CED is > the int. BAC; 11 à fort. I much more is 2 BDC > < BAC. 12' Rec. Therefore, if from the ends of a side, &c.

Q.E.D. APP.-In Optics this proposition is used to prove, that if from A we could see the line BC, and also from D a point nearer to the line, the base BC would appear less from A than from D: it does this on the principle that quantities seen under a greater angle appear greater. For this reason the apparent diameter of the sun measures more when the earth sin perihelion, than when it is in aphelion. Andthus,--according to Vitruvius,who composed his work on Architecture, about 15 B.C.--the tops of very high pillars should be made but little tapering, because they will, from the distance, of themselves seem less.

PROP. 22.—PROB. To make a triangle, of which the sides shall be equal to three given st. lines, but any two whatever of these must be greater than the third.

SOL.-P. 20, P. 3, Pst. 3, Pst. 1, Def. 21.-DEM.-Def. 15, Ax. 1. E. I Data. Given the three st. lines A, B, and C, any two of which

are greater than the third, | P. 20. A and B > C, A and C > B, and B and C >A; 2 Quæs. to make a triangle with sides = A, B, and C, each to each. 1 by Assum. Take a st. line DE unlimited towards E; 2 P. 3. make ĐF= A, FG = B, and GH = C; 3 Pst. 3. from centre F, with DF, describe the ODKL,

and from centre G, with GH, the O HLK;

join the points F,K, and G,K; · G Def. 21. then the fig. KFG is a A, having FK = A, FG = B, | Sol.

and GK = C.

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D. 1 byC.3,Def.15:: F is the centre of O DKL, FK = FD ;

2 Č. 2 & Ax. 1 but FD = the st. line A, .. FK also = A.
3 C.3, Def. 15 Again, :,: G is the centre of O HKL, .. GH = GK;
+ C. 2 & Ax.1but GH =C;... also GK=C;
5. C. 2. and FG is equal to B;
6 D. 2, 4. & 5.:, the three st. lines FK, FG, GK are respectively equal

I to the three st. lines A, B, and C. 7) Rec. And therefore the A FKG has its three sides, &c. Q.E.F. SCH.-1. In P. 22 it is assumed that the two circles will have at least one point of intersection.

2. If two of the given lines were together equal to the third, the circles would touch externally ; if the two were together less than the third, the circles would not touch at all: in either case no triangle could be drawn.

APP.-1. All rectil. figures being divisible into triangles, this Prop. is of very extensive use either for making one rectil. figure equal to another, or on the theory of Representative Values, making one figure like to another : in the first case the triangles into which the rectil. figure has been divided are repeated, side for side, in another rectil. figure of exactly the same linear dimensions: and the construction being completed, the two would correspond, angle to angle, line to line, and point to point : in the second case, the sides and angles of the first must be measured, and from a scale of equal parts, lines drawn in the second, representative of those in the first, and angles in the second equal to those in the first ;--for equality of angles, according to Def. 1, bk. vi., is essential to similarity of figure.

PROP. 23.-PROB. At a given point in a given line to make a rectilineal angle equal to a given rectilineal angle,

Sol.-Pst. 1, P. 22.- DEM.-P. 8. E. 1 Dat. Given the point A in

the st, line A B, and

the rectil. _ DCE; 2 Quæs. required at A in AB

to make an Z=DCE. C. i\by Assum In CD, CE take points Pst 1. D & E, & join D,E;

with AF=CD, AG=

CE, & FG = DE,
make the A AFG; D -

then the FAG shall be = L DCE. Dil by C. 2. | :::FA=DC, AG =CE,& DE=FG; (B

2 P. 8. l.. he LFAG= LDCE.
3 Rec. Therefore, at point Ā in AB &c

Q.E.F.

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Sch.-In Pr. 24 it is assumed that D and F will be on different sides of EG; or in other words that DH is less than DF or DG.

PROP. 25,-THEOR If two triangles have two sides of the one equal to the two sides of the other, each to each, but the base of one greater than the base of the other ; the angle contained by the sides of the one which has the greater base shall be greater than the angle contained by the sides equal to them of the other.

DEM.—P. 4, P. 24. E. 1]'Hyp. 1 In AS ABC, DEF,

let AB = DE, &

AC = DF, 24 2 but let BC be > EF, 3 Conc. then the L BAC is

> the Z EDF. D. 1 Sup. 1. If BAC beyn

EDF, it is either

equal or less. 2 Sup. 2. Suppose that _BAC

is = _ EDF: 3 Hyp. & P.4 then base BC = base EF : 4 Hyp. But BC is #EF; 5 Conc.

.. 7. BAC'is + LEDF. 6 Sup.

Again, suppose _ BAC < LEDF ; 7 H. & P. 24 then base BC is base EF :

| But BC is * EF; 9j Conc. 1.:. BAC is x L EDF. 10 D, 5 & 9. Now, _ BAC is E, nors, _ EDF; 11 Conc. 1. ĽBAC is > ZEDF. 12 Rec.

Therefore, if two triangles have two sides, &c. Q.E.F. SCH.-Prop. 24 and 25 have the same relation to each other as Props. 4 and 8, and the four may be combined thus:-Iftwo triangles have two sides of the one respectively equal to tro sides of the other, the rem. side of the one will be greater or less than, or equal to, the rem. side of the other, according as the angle opposed to it in the one is greater or less than, or equal to. the angle opposed to it in the other; or vice versa.-LARDNER'S Euclid, p. 56.

PROP. 26.-- THEOR.--(Important.) If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz., either the sides adjacent to the equal angles in each, or the sides opposite to them, then shall the other sides be equal, each to each, and also the third angle of the one to the third angle of the other.

Con.-P. 3, Pst. 1.- DEM.-P. 4, Ax 1, P. 16. E. 11 Hyp. I. In As ABC, DEF, let ABC = L DEF, and _ACB

= LDFE; 2 „ 2. also let one side = one side ; Conc. 1. then the other sides shall = the other sides;

, 2. and the third L of the one = the third L. of the other.

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