392 PHILOSOPHICAL TRANSACTIONS. [aNNO 17 IQ. 



problem, when the given series of curves is defined by a finite equation, ex- 

 pressing the relation between the absciss and ordinate. The same solution 

 holds when the equation is a converging series, or when the property of the 

 curve to be cut can be reduced to such an equation, by the Analysis per Series 

 numero terminorum infinitas. But Mr. Leibnitz was for solving the problem 

 without converging series. 



Pars reliqua Dissertationis de Potentid Cordis, yiuthore Jacobo Jurin, M.D. 

 et R. S. S. N*^ 359, p. 929. [See p. 375 of this vol.] 



A New Universal Method of describing all Kinds of Curves by means of Right 

 Lines and Angles only. By Colin Maclaurin^ Profes. of Math, in the New 

 College of Aberdeen. N° 359, P* 939- Translated from the Latin. 



As the great Newton has not extended his method of describing curves to 

 those of the 3d order which are without a double point, nor to those of a 

 higher order wanting a multiple point; and pronounces their description to be 

 counted among the more difficult problems in geometry; I hope the following 

 method will not be unacceptable to geometricians, by which geometrical curves 

 of any order are constructed, though they may be without a punctum duplex 

 or multiplex. 



1. Lines of the first order are only right lines themselves, which can meet 

 one another only in one point. Lines of the 2d order are the conic sections, 

 which cannot be cut by a right line in more than 2 points. And all these may 

 be thus constructed, according to lemma 21, lib. I, of Newton's Principia : 

 Let two given angles mcr, lsn (fig. 3, pi. 10) move round two given points 

 c and s, so that a the concourse of the legs cm, sl, may always describe the 

 indefinite right line ae given in position; then the concourse p of the other 

 legs CR and sn will describe a line of the 2d order, or a conic section. 



2. Let the angle mcr move as before about the given point c (fig. 4); and 

 the given angle lnq have its angular point n always carried along the given 

 right line ae, so that the leg gn may always pass through the given point s. 

 Then, 1st, if the concourse a of the legs cr and sn be drawn along the infinite 

 line AB, the concourse of the legs cm and nl will describe a curve line of the 

 3d order having a double point at c. 2dly, Other things as before, if the con- 

 course of the legs cm and nl (fig. 5) be drawn along the indefinite line ab; 

 then the concourse p of the legs cr and sn will describe a curve of the 3d 

 order having a double point at s. 



Example of Case 1. Let the angles mcr, lns be right ones (fig. 6), and 

 AE, DB, OS be parallels; also let sa and sd be perpendicular to ae and db 



