1915-16.] The “ Geometria Organica ” of Colin Maclaurin. 125 
In how many points can the curve given by the intersection of (3) and 
(4) be cut by the straight line 
Ax + By + 0 = 0^ (5) 
At such points we must have 
ABC 
A r (\) B f .(A) C r (X) =0 . . . . (6) 
A§) B s (t) C s (t) 
taken along with (2). 
By the theory of equations the £-eliminant of (2) and (6) is of degree 
(m+ l)-s + (n + l)r, 
and this number therefore represents the number of intersections of the 
curve with a straight line, and so the degree of the curve. 
Cor. 2. The theorem may be extended as in Cor. 2 of Prop. XXIV. 
§ 37. Prop. XXVII. 
Suppose that , in addition to the data of Prop. XXIV, there are given 
the serrate angles 
PlOiPi • • • Pr-lP 
QA q 1 q s -iq, 
then the intersection of p r -ip and q s -p[ is on a curve of degree ms + nr+s + r. 
For the datum that P m P and Q n Q intersect on a straight line leads to 
(2) of preceding. 
There is a 1 — 1 correspondence between P 1 0 1 and 0 go 1 so that p r ^p has 
an equation like (3). Similarly, qs-fls has an equation like (4). .*. etc. 
Scholium. 
§ 38. In the scholium Maclaurin gives credit to Fermat, Varignon, De 
la Plire, Nicole for special curves : and to Newton’s great work on Cubic 
