1915-16.] The “ Geometria Organica ” of Colin Maclaurin. 119 
SECTION IV. 
Wherein are demonstrated General Theorems regarding the De- 
scription of J f C urves of any Order by the Use only of 
Linear Loci and Constant Angles. 
§ 29. This section takes up the discussion from a more general point of 
view, and, while Maclaurin’s theorems are adhered to in their order, their 
demonstrations, when analytical, are frequently altered. Before we pro- 
ceed to these it will be convenient, just as Maclaurin does, to pave the 
way by some preliminary theorems. 
Instead of an ordinary angle he makes use of what may be termed a 
serrate angle consisting of a broken line OP 1 P 2 . . . PJP, i n which the 
component angles at the teeth are of constant magnitude, while the 
segments of the line are freely variable. The vertices P X P 2 . . . P w lie on 
linear loci l v l 2 , . . . l n , and 0 is a fixed point. 
Let the equation to OP x depend on a parameter t, and be given by 
L 0 + ^L 1 = 0 . . . . . . (1) 
Then the equation to P-,P 2 is of the form 
M 0 + tM 1 + £ 2 M 2 =■• 0 (2) 
Similarly, for P 2 P 3 we in general find an equation of the form 
No + ^ + ^'+^N 3 = 0 (3) 
etc., etc. 
The lines P X P 2 , P 2 P S , etc., envelop unicursal curves of class 2 (parabola), 
3, etc , having a special relation to the line at infinity. 
Also the co-ordinates of P n are rational functions of t of degree n. 
§ 30. Prop. XX. 
Let 0 1 P l P 2 ... P Q be a serrate angle (n lines l v 1 2 , . . . l n ), 0 2 a 
second fixed point through which 0 2 Q is drawn such that 0 2 QP n is a 
constant angle , then the locus of Q is a curve of degree n + 2. 
