1915-16.] The “ Geometria Organica” of Colin Maclaurin. 89 
cujus-cumque Ordinis Mechanice describendi sola datorum Angulorum et 
Rectarum Ope, 1719. 
Maclaurin’s imagination had been fired by Newton’s classic Enume- 
ratio Linearum Curvarum Tertii Ordinis, and by the organic description 
of the Conic given in the Principia; and in his attempt to generalise 
the latter so as to obtain curves of all possible degrees by a mechanical 
description he was led to write the Geometria Organica. 
It will appear in the sequel how remarkably successful he was in 
obtaining nearly all the particular curves known in his time (which he is 
careful to ascribe to their inventors), besides a whole host of new curves 
never before discussed, and which have since been named and investigated 
with but scant acknowledgment of their true inventor. His method, 
however, does not furnish all curves, though it may furnish curves of all 
degrees ; and it will be the business of this note to point out some of the 
limitations of the method applied, as well as the rare mistakes Maclaurin 
makes regarding the double points of the curves investigated, — a weakness 
more pronounced in the earlier memoirs. 
In establishing his theorems he frequently employs the method of 
analysis furnished by the Cartesian geometry. The Cartesian geometry 
was then in its infancy, and Maclaurin’s use of it seems to us nowadays 
somewhat cumbersome and certainly tedious. But when Maclaurin 
reasons “more veterum,” he handles geometry with consummate skill ; and 
the impression gains upon the reader that, however imperishable his 
reputation in analysis may be, he was greater as a geometer than as an 
analyst. He occasionally makes petty errors in his analytical demon- 
strations which somewhat mar the interest in his work, but the beauty of 
his synthetic reasoning is untarnished by any such blemish. 
In any analysis that follows, the demonstrations he gives are frequently 
replaced by others that are more in touch with modern methods, but this 
does not apply to the geometrical reasoning proper, which is as fresh to-day 
as when written. The treatise is divided into two parts. In the first part 
the only loci admitted are straight lines along which the vertices of 
constant angles are made to move. In the second part the curves so found 
in the first part are added to the loci to obtain curves of higher order. 
It contains, in particular, the theory of pedals and the epicycloidal 
generation of curves by rolling one curve upon a congruent curve. 
A section is devoted to the application to mechanics ; and the last section 
contains some general theorems in curves forming the foundation of 
the theory of Higher Plane Curves. It also contains what is erroneously 
termed Cramer’s Paradox, the paradox being really Maclaurin’s, for 
