42 I'HILOSOPHICAL TRANSACTIONS. [aNNO 1735. 



lines, which he hud not published, having been engaged for the most part in 

 business of a different nature, and in pursuits on other subjects since that time. 

 He first gives an abstract of that supplement, as far as it was then printed, and 

 subjoins an account of some theorems he added to it the following year, viz. 

 in 1722. He was led into those new theorems by Mr. Robert Simson's giving 

 him at that time a hint of the ingenious paper, which has been since published 

 in the Philosophical Transactions. Mr. M. had tried, in the year 17 ig, what 

 could be done by the rotation of angles on more than two poles; and had ob- 

 served, that if the intersections of the legs of the jingles were carried oyer 

 right lines, as in Sir Isaac Newton's description, the dimensions of the curve 

 were not raised by this increase of the number of poles, angles, and right lines; 

 and therefore he neglected this at that time, as of no use to him, confining 

 hiuiself to two poles only, and varying the motions of the angles as in his 

 book. He found this by inquiring in how many points the locus could cut a 

 right line drawn in its plane, and found, by a method often used in his book, 

 that it could meet it in two points only. 



Having found then, that three or more poles, were of no more service than 

 two, while the intersections were carried over fixed right lines; he thought it 

 needless to prosecute that matter then, since by increasing the number of poles, 

 his descriptions would become more complex, without any advantage. But in 

 June or July, J 722, on the hint he got from Mr. Simson of Pappus's porisms, 

 he saw that what he has there ingeniously demonstrated, might be considered 

 as a case of the abovementioned description of a conic section, by the rotation 

 of any number of angles about as many poles; the intersections of their legs in 

 the mean time being carried over fixed right lines, excepting that of two of 

 them which describes the locus. For by substituting right lines instead of the 

 angles, in certain situations of the poles and of the fixed right lines, the locus 

 becomes a right line; as for example, in the case of three poles, when these 

 three are in one right line, in which case the locus is a right line, which is a 

 case of the porism. 



It was this that led him to consider this subject anew; and first he demon- 

 strated the locus to be a conic section algebraically ; and found theorems for 

 drawing tangents to it, and determining its asymptotes. He also drew from it 

 at that time a method of describing a conic section through five given points.* 

 This encouraged him to substitute curves for the right lines, to see if by this 

 method he could be enabled to carry on his theorems, about the descriptions of 



* The paper on this subject I have, says Mr. M. is dated July 31, 1722, at sea, being then in my 

 way to London, going for Cambray. Orig. 



