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whilst the line u, is projective of line u and perspective of

line u

2

COR. II. If the homologous rays of two pencils O and O, meet upon the same line u,; or, in other words, if the two pencils O and O, be perspective of each other; and if, upon a second line u intersecting u1, we project the points a,b,c,d, of u1, first from the centre O in abcd, and then from O, in a'b'c'd'; we shall obtain two series of homologous points, situate upon the line u and having the points of intersection of u with u, and OO, in common. These two points coincide when the intersection of u with u, falls upon the line of centres OO2. With the centres O and O, in the relative positions given in Fig. 6, the homologous systems abcd and a'b'c'd' describe the line u in the same direction; if, however, O, were placed within the acute angle contained between the lines u and u1, the two systems would describe the line in opposite directions. Homologous systems, whether of points or rays, proceeding in the same direction are said to be concurrent; and homologous systems proceeding in contrary directions are said to be opposed.

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COR. III. If two lines u and u, be in perspective from the same centre O; and if their homologous points be then projected from a second centre O, situate in the same plane; the point O, will be the centre of two homologous pencils having the two rays which join O to O and to the point of intersection of u with u, in common. These two rays coincide when 0,0 traverses the point of intersection of u with u.

COR. IV. Two projective or homologous systems (such as u and u2) can always be regarded as first and last of a series of projective and homologous systems, of which each (u) is perspective of those immediately preceding and following it.

4. If in the anharmonic ratio or pencil O (Fig. 7) a transversal u be drawn parallel to a ray Oc, and a second transversal

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u' be drawn parallel to a second ray Oa of the same pencil; and if the third ray Obb' be made to revolve round O so as to describe the lines u and u', the rectangle ab. b'c' will remain constant and By the definition of an anharmonic ratio

equal to ad. d'c'.

we have

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COR. By subtracting each side of equation (2) from

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bd ad ab

or

b'd'

b'c' = d'c'

eq. 2);

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so that, if we make ab d'c', we shall also have bd = b'd'. Wherefore, if in the similar case (Fig. 8) we set off from points a and c', correlative of infinity on lines u and u', the equal segments af and c'h'; the segments fh and f'h', cut off upon lines u and u' by the rays Of and Oh', will be equal. Similarly, if we set off from the same points the equal external segments ag and c'k'; the corresponding segments gk and g'k', intercepted by the rays Og and Ok', will be equal.

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5. If a transversal abc (Fig. 9) be drawn across the sides of

a triangle, the products of the alternate segments are equal.

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COR. If the transversal be parallel to AB, that is, to one of the sides, the point c passes to infinity on AB; so

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6. PROBLEM.

Ba Ca: Ab: Cb.

(Compare Euclid VI. 2.)

To locate a given triangle DEF with its three angular points DEF upon three given lines AB, AC, and BC (Figs. 10 and 11). Upon DE, DF, and FE (Fig. 10) as chords describe three circles containing respectively the angles A, B, and C, on that side of their respective chords

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Let the circles DGF and DRE, whose centres lie at Q and P respectively, meet in G. Join GP and PQ, and make Ga of such a length that, a being a point on the circumference DRE,

Ga GP
AB PQ

C

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