1915-16.] The “ Geometria Organica” of Colin Maclaurin. 91 
Universal Description of Geometrical Lines. 
Part I. 
Wherein, by a Universal Method, Lines of all Orders are described 
by the sole use of constant given Angles and Straight Lines. 
SECTION I. 
The Conic. 
§ 3. This section gives an analytical demonstration o£ Newton’s Organic 
Description of Conics (Principia, Bk. I ; and Arithmetica Universalis). It 
is the generalisation of this method that gives rise to Maclaurin s treatise. 
Prop. I. 
0 and O' are fixed points : L POQ = a, and L PO'Q = /3, two angles of 
constant magnitude that can be rotated round 0 and O' respectively. If 
the intersection P of OP and OT is conducted along a straight line l, the 
point Q in general traces out a conic section through O and O'. 
To get the conic, therefore, one straight-line locus and two given angles 
are required. In modern terms, if OP and OT are in perspective corre- 
spondence, Q generates a conic. 
For any point P on l may be supposed to have the co-ordinates 
x = at + b 
y = ct+d 
where t is a variable parameter: and the ordinary calculations give the 
equation to OQ in the form 
L| + ^1^2 = 6 
(i) 
