1915-16.] The “ Geometria Organica” of Colin Maclaurin. 147 
or * 
(6) 
.-. Pl = K'rS (7) 
Thus Maclaurin’s theorem is established by the important converse that 
only such curves (7) obey this law. 
§ 62. The curves of Maclaurin are the so-called sine spirals, an account 
of which will be found in chap, xviii of Loria’s Ebene Kurven. From 
Maclaurin’s thorough discussion of them it might have been better to have 
called them the Curves of Maclaurin. 
The sine spirals are defined in polar co-ordinates by an equation 
of the form 
r n — A sin nO. 
It is easy to see that for any curve 
or 
Hence, when 
or 
dO _ p 
7 dr -y/ (r 2 — p 2 ) 
d$=— sh 
r V(1 -p*/r 2 ) 
, — = Gr n 
* r 
cW = _Cdrr^_ 
V(i-cv-j) 
. n{6 + a) = sin _1 (Cr”) 
Cr n = sin n{6 + a) . 
.\ etc. 
• ( 1 ) 
• (2) 
• (3) 
• W 
• (5) 
Note. — Maclaurin’s Theory of Pedals (including the Theorem for the Sine 
Spirals) was originally published in 1718 in the Philosophical Transactions . 
In substance it is the same as in the Geometria Organica , but the method 
of fluxions is used more freely in the earlier work. 
His “New Universal Method of describing curves of any order by the 
sole use of given angles and straight lines” appeared in 1719, likewise in 
the Philosophical Transactions. The account given is very brief, and 
there is inaccuracy in the theory of double points. 
SECTION IV. 
63. This section is concerned with applications to mechanics. 
