6 PHILOSOPHICAL TRANSACTIONS. [anNO J 735, 



section of right lines moved round poles, Mr. Braikenridge thinks is much more 

 simple than that of Sir I. Newton, and will give a solution of many difficult pro- 

 blems, which he doubts if they can be found by any other principles. The 

 author gave only one particular case of this in his Geometrical Exercitation, 

 printed at London in 1733, not thinking it convenient to explain the whole 

 method at that time, though he was then, he says, well acquainted with that 

 method. It is now, he says, 3 years since he fell upon the general theorem, 

 which he had many reasons for concealing ; being determined to let 2 years at 

 least pass, after the publication of that Exercitation, before disclosing this 

 general method : for he doubted not, that if any others were possessed of this 

 invention, they would, on the publication of a particular case, especially as they 

 were provoked to it, embrace the opportunity to publish their general method, 

 if they had really discovered one. 



About 3 given points, as poles, a, b, c, fig. 1, pi. 1, in any plane, let there 

 be turned 3 right lines, ans, bos, cno, which may intersect each other in the 

 points s, n, o ; and let the two points of intersection s and n be drawn along 

 the right lines dks, knk, given by position ; then the remaining point o will 

 describe a conic section ; as is demonstrated in the Exercit. prop. 1. — If through 

 the points a, b, c, be drawn the right lines ab, ac, meeting each other in a, 

 and the right lines rk, dk, given by position, in r and m ; then the figure de- 

 scribed will pass through the 5 points b, c, k, m, r. And hence appears a new 

 method of describing a conic section through 5 given points, much easier than 

 any yet invented. See Exercit. prop. 3. 



Let there be moved around 4 points a, b, c, d, fig. 2, as poles, in any plane, 

 as many right lines ans, bos, cno, dpo, three of which, ans, bos, cno, may 

 intersect each other in three points, s, n, o ; and let the two points of inter- 

 section s, N, be drawn along the right lines dK, rk, given in position ; and at 

 the same time let the right line dpo, drawn from the 4th pole d, pass through 

 the remaining point o, and cut the right line ans in p : then that point p 

 will describe a line of the 3d order : as is demonstrated prop. 1 1 of the exer- 

 citation. 



Through the poles a, b, d, let there be drawn the right lines abr, bdh, 

 meeting each other in b, and the right lines kr, Kd, given by position, in r 

 and H : then the figure described by the motion of the point p, will pass 

 through the 3 points a, d, h, k, r, of which a will be double. Hence is de- 

 duced a method of describing a line of the 3d order through 7 given points, 

 one of which may be double. For let a, d, h, k, p, m, r be given, fig. 3, one 

 of which A is to be double. Through the two points h, r, and another k, let 

 the right lines hk, rk pass ; also join the points a, r, and h, d, and produce ar, 



