1915-16.] The “Geometria Organica ” of Colin Maclaurin. 129 
§ 41. Prop. III. 
If P lies on a curve of degree n, Q traces out a curve of degree 2 n. 
For let Q lie on a straight line l ; then P will generate a conic which 
cuts the n - ic in 2 n points Pp P 2 , . . . P 2n , to which correspond on l 2 n 
points, Q p Q 2 , • , . Q 2n . etc. 
Cor. 1. Construct A0 1 0 2 0 3 , and let 0 1 = a, 0 2 = /3, as in figure. 
Then each side of 0 1 0 2 0 3 cuts the n - ic in n points, to which corre- 
sponds the unique point the opposite vertex. The curve therefore has 
Ti-ple points at Op 0 2 , 0 3 . 
Cor. 2. There cannot be four ?i-ple points on the new curve ; for through 
these and a fifth point on the curve we could describe a conic cutting the 
curve in 4^ + 1 points, which is impossible. 
Cor. 4. It has been assumed that Oj and 0 2 do not lie on the given 
curve of degree n. If O x is an ordinary point on the latter the curve 
obtained is of degree 2n — 1 ; and if 0 1 is an r-ple point the curve is of 
degree 2 n — r. 
[We might, of course, have established this proposition by showing 
that the co-ordinates of P and Q are connected by an ordinary Cremona 
quadratic transformation. We therefore have before us established, for 
the first time, the fundamental features of a Cremona transformation 
more than a century before it was to become the property of all mathe- 
maticians through Cremona’s research 3S.] 
Remark. 
“Newton has given Props. I and II and indicated their generalisation. 
“ This generalisation we have attempted to effect in Prop. III. We have 
to this end made use of a given curve and two constant angles. In the 
following we shall attempt to generalise all the propositions of Part I, 
just as we have generalised the Proposition I of Part I.” 
VOL. xxxvi. 
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