126 Proceedings of the Royal Society of Edinburgh. [Sess. 
Curves. He points out the desirability of having a general method of 
generating curves of all degrees. The method employed does not give all 
curves, but it may serve to pave the way for future perfection of the 
theory. 
In the part just completed only straight lines and constant angles have 
been employed. In Part II other curve loci are utilised from which to 
obtain more complicated curves of higher degree. 
Part II. 
Wherein Curves of all Higher Orders are described by the Use of 
Curves of Lower Order. 
SECTION I. 
§ 39. Newton’s Organic Description of Curves. 
Prop. I. 
Round the fixed points 0 X and 0 2 are rotated the constant angles 
POfii^a, P0 2 Q = (3. 
P 
If P traces out a conic through 0 V Q generates a cubic having a double 
point at 0 X and an ordinary point at 0 2 . 
Maclaurin’s proof runs thus. Find in how many points a straight line 
l can cut the curve, i.e. how many points Q can lie on l. 
Let Q trace out the line l, P being left free. P will generate a conic 
through O x and 0 2 cutting the given conic in four points O x , P 15 P 2 , P 3 . To 
Pj, P 2 , P 3 correspond three points Q p Q 2 , Q 3 on l : so that the locus cuts l in 
three points and is therefore a cubic. 
