CHAPTER I

ANHARMONIC PENCILS AND RATIOS

1. DEFINITION. -If a series of three or more points FPC situate on any line uu (Fig. 1) be projected from a centre of perspective G upon a second line u'u' in the points F'P'C', the lines uu and u'u' are said to be in perspective; and the points FF, PP', CC' are termed correlative or homologous points.

LEMMA I.-If the correlative or homologous points I and I' of the punctuated lines uu and u'u' coincide with their point of intersection, forming a double point, the lines uu and u'u' will be in perspective. For let FF' and CC' be any other two pairs of homologous points, and let the lines FF' and CC' meet in G, then G will be the centre of perspective of the two lines; because in a harmonic or other system three of the four members of a compound proportion suffice to determine the fourth. Then the fourth point, being taken in conjunction with any two of the other three known points, will serve to determine a fifth; and so on indefinitely.

LEMMA II. Similarly, if any pair of homologous rays GP' and HP' of two pencils G and H be coincident or coperspective, the pencils will be perspective of the same line, and consequently perspective of each other. For let any other pair of homologous rays GF' and HF" meet in

B

F', and any third pair in C', then the line F'C' will be perspective of both pencils; because, since three pairs of homologous rays meet upon the line F'C', it necessarily follows that any fourth pair will meet upon the same line.

THEOREM.-If from any point I (Fig. 1) three lines be drawn, and if upon those three lines a double trio of points FFF" and CC'C" be chosen; the intersections of the corresponding pairs of sides, namely, FF' and CC', FF" and CC", F'F" and C'C", will all lie upon the same straight line GKH.

Conversely, if from any three points GKH upon a given

straight line rays, such as GFF' and GCC', KFF" and KCC", HF'F" and HC'C", be drawn, so as to determine by their mutual intersections the corresponding triangles FF'F" and CC'C"; then the corresponding lines FC, F'C', and F"C" produced will meet in the same point I. For, let FF' and CC' meet in G; then, since the point I of intersection of the lines uu and u'u' is a homologous point common to those lines, the lines uu and u'u' are in perspective from the centre G (Lemma I.). For similar reasons the lines u'u' and u"u" are in perspective from the centre H, the point of intersection of the sides F'F" and C'C". But, the three pairs of points FF", CC", and PP" being in perspective of the three points F'C'P' respectively, the fourth pair, or double point of intersection II", must be in perspective of the point of intersection I'; or, in other words, the three points of intersection coincide.

Again, if the points II'I" coincide and the pairs FF' and CC', FF" and CC", F'F" and C'C" be drawn in perspective from GKH respectively, we have three pairs of points on each pair of lines in perspective from GKH. Hence the fourth pairs PP', PP", and P'P" must be in perspective from the same three centres. Now let PP" be taken on the common ray KH. Then, since P' must lie on HP", the points PP'P" will fall on the common ray KH. But at the same time G must fall on PP'. Therefore GHK lie

on the same line.

DEF. A transversal is a line drawn across the sides or sides produced of a closed figure or a pencil of rays.

COR.-From any pole O (Fig. 2) draw four rays, upon three of which choose any three points abc. Produce ba and be to intersect any transversal 54 in the points 1 and 2. Next take any point b' on the ray Ob, and join b'1 and 6'2 intersecting the rays Oa and Oc in a' and c' respectively. Then by Art. 1 the triangles abc and a'b'c' are correlative, because their corresponding lines meet in the same transversal, and the rays through their corresponding

F', and any third pair in C', then the line F'C' will be perspective of both pencils; because, since three pairs of homologous rays meet upon the line F'C', it necessarily follows that any fourth pair will meet upon the same line.

THEOREM.-If from any point I (Fig. 1) three lines be drawn, and if upon those three lines a double trio of points FFF" and CC'C" be chosen; the intersections of the corresponding pairs of sides, namely, FF' and CC', FF" and CC", F'F" and C'C", will all lie upon the same straight line GKH.

Conversely, if from any three points GKH upon a given