VOL. XXXIX.] PHILOSOPHICAL TRANSACTIONS. 43 



lines through given points, to the higher orders of lines. Some of the tiieo- 

 rems he ft)un(l at that time accompany this. In November 1722, looking into 

 Sir Isaac's Principia, he saw that the description of the conic sections by three 

 riglit lines, moving as above, about three poles, could be immediately drawn 

 from his 20th lemma, which itself is a case of this description. This gradually 

 led him to seek geometrical demonstrations for the whole, as far as it related to 

 the conic sections. He sent some leaves of this paper, dated at Nancy, No- 

 vember 17 22. Since that time, he had not added much to this subject, but 

 what relates to drawing tangents, determining the asymptotes, and the puncta 

 duplica, or multiplicia of these curves. He considered it the less, as he did 

 not find it more advantageous in any respect, than the method he had consi- 

 dered in tiis book, nor more general. 



In 1727 lie added to a chapter in his algebra, an algebraic demonstration of 

 the locus, when three poles are employed; and the method of describing a 

 conic section through five given points, subjoining at the same time, that if 

 more poles are employed, and angles or right lines, the locus was still a conic 

 section ; which he thought was a remarkable property of the conic sections, not 

 observed before. 



These things he intended to put in order, and publish in the supplement to 

 his book, a part of which had been printed since the year 1 721. He intended 

 also to give several other things in that supplement; two of which he only just 

 mentions at present, as they are foreign to the present affair. He subjoins a 

 problem determining the figure of a fluid, whose parts are supposed to be attracted 

 to two or more centres; and a solution of a general problem about the collision 

 of bodies. 



The author of the papers given in to the Royal Society will not deny, that 

 Mr. M. showed him the theorems, now sent, in 1727. He owned it last sum- 

 mer at least; Mr. M. intended to publish these very soon. Whether he has 

 carried the subject farther, he leaves to the judgment of the gentlemen to whom 

 they were referred. As to the demonstrations, it would take some time to put 

 them in a proper form to be published. He could send those that are algebraic 

 easily; but did not care to send those that are geometrical, till more leisure. 



An Abstract of ivhat has been printed since the Year 1721, as a Supplement to a 

 Treatise concerning the Description of Curve Lines, published in 1719, and 

 of what the ^-Juuhor proposes to add to that Supplement. By Mr. Maclaurin. 

 N°439, p. 148. 



I. In the first part of the supplement, a general demonstration is given of 

 the theorem, that if two lines of the orders or dimensions, expressed by the 



G 2 



