The basis of the theory of projections must be the investigation of properties which, being true of a figure, are therefore true of its projections. Some of these are evident enough : thus the projection of an intersection of two lines is the intersection of their projectione ; if two curves touch one another, their projections touch at a point which is the projection of the point of contact. But the following property, which is projective, that is, true of the projections of every figure of which it is true in the first instance, will give a good idea of the facility with which certain properties of the conic sections may ho deduced from the circle.
Let there be any figure in which the product of certain straight lines is equal to, or in any absolutely given ratio to, the product of certain others, each line being denoted by an initial and terminal letter in the ,ieual way. We might say, in more geometrical language, let there be any number of ratios which, compounded together, give either a ratio of equality or a given ratio. Two simple conditions being fulfilled, this property will be as true of the projections as it is of the figure itself. These conditions are, first, that every initial and terminal letter shall occur the same number of times on both sides of the equation; secondly, that for every line on the first side, there shall be a distinct line on the other side, which is in the aline straight line. For example (the reader may draw the diagram for himself), let each of the sides of the triangle A n c cut one circle, namely, A n in P and Q, tic in 11 and 8, CA in T and v; the order of the points being APQ8n8CT v A. Then by the properties of the circle, it is easily seen that A V. A 7. n. (1. r = P. A Q. a 8. 0 T. 0 v.
'In this equation, A, n, and c occur twice on each side, and each of Q, n, s, v, once. Moreover, out of A n there are two segments, A r and A Q, On the first aide, and as many, n Q and Dr, on the second ; and the same of BC and C A. lie then who is acquainted with the theory of projections, immediately knows that this property is true of any projection of a circle, or of any conic section : but ho would be an energetic algebraist who should attempt to prove this (or still new° the equally demonstrable similar property in the case of a polygon of is sides) by the common algebraic methods.
The proof of the preceding general projeclire property is not difficult. Take a point o outside the plane for the centre of projection, and let o • =a, 0 n b, &c.; moreover lot the angle made by a anti b be called (a 6). Let A'. , be the projections of A, II, &C., let 0 = o le= &c., and let (a' !I) be the auglo of a' and which is = (a 1). Moreover let [A n] moan the perpendicular let fall from o upon A n, &c. It is then easily proved that A e. sin A a 1. sin fie [AV] [s vj Substitute these values in the equation, and it will be readily seen that the existence of the two conditions above named amounts to all the quantities except the sines of the angles being climinablo by division. There remains then sin (a r). sin (a t). &e. = sin (ap). sin (a ?). &c.
or sin (a' r'). sin (a' f'). &c. = sin (a'//'). sin (a'g').
In this write a, G', [A' &o., where there were previously a, 1,, CA yr &c., which will amount (by the conditions) to multiplying both sides 'y the s. tme quantities : there will then remain an equation which is obviously A' A' &c.= r'. &co and in the same way any other case may be proved.