94 Proceedings of the Royal Society of Edinburgh. [Sess. 
conic which passes through D and E and also through C, A, and B, i.e. 
Q generates the conic through the five given points. 
If four points only are given, an infinity of conics can be described 
through them. Thus there are two parabolas through the four points, 
or two hyperbolas whose asymptotes intersect at a given angle. For 
example, let the parabolas through A, B, C, D be sought. Proceed as 
before and find D'. On AB describe a segment of a circle containing an 
angle y such that 
a + /3 + y = 7T (or 2tt). 
Either tangent from D' to this circle will furnish the line l for the 
parabola. 
“ The method employed will furnish the complete system of conic 
sections which were the objects of research of the older geometers- 
Newton was the first to attack the problem to enumerate and classify 
Curves of the Third Order, and thereby added a fresh triumph to his 
genius. We now proceed to delineate curves of this order.” 
§ 7. Newton's Organic Description as a Cremona Transformation. 
Fig. 4. 
Let O be the origin, O' the point ( a , 0), P any point (£ rj). 
is given by 
and O'Q is given by 
where 
y = - a) = y.(x - a), say 
— ct 
y = m(x - a) 
i.e. 
m - fx 
1 = my. 
— tan f 3 , 
m = (/jl + tan /?)/( 1 - /x tan /3) 
_ rj + (£ - a) tan )3 
£ - a - y tan /3 
Then OT 
• • a) 
( 2 ) 
