18 PHILOSOPHICAL TRANSACTIONS. [aNNO 1703. 



Nor shall I scruple to call it a new one, since none of the celebrated geometri- 

 cians, as far as I know, have ever published any thing of the kind. I shall here 

 produce only a specimen in a few instances. 



Let AGH (fig. 13, pi. l) be a curve, having the vertex A, axis ak, ordinate 

 PD, and centre (if it have one) the point k. Taking l a point in the axis, 

 make al = n, ad = a:, fd=^, fl = z. Of these quantities the last three 

 are flowing quantities, and n is a constant quantity ; for this being always the 

 same, answers to the others which are always variable. From the right-angled 

 triangle fdl, we have this equation, zz =zyy-{'nn — 2nx-{-xx; and deter- 

 mining z to be an extreme, there arises 2j/i/ — 2n.r -j- 2xj: = O; whence by 

 interpreting lyy according to the particular nature of the curve, the quantity n 

 will l»e left, expressed in terms that will also be proper to the curve. 



And now by this means having z determined to its extreme value, that is, 

 having the line fl either the greatest or least of all those which can be drawn 

 to the curve from the point l, and therefore perpendicular to the curve in the 

 point F ; it is evident that dl is the subnormal, from whence the subtangent is 

 easily derived. 



For an example, let us first take the Apollonian parabola, which curve we 

 will suppose to be here delineated. Therefore we have 2t/j/ = nv^ supposing r 

 to be the parameter ; whence rJ; — 2nx + 2xr = 0, and n =: -^r -\- x. 

 Therefore the subnormal dl = -j-r. Now the meaning of this theorem is this :. 

 if beyond the limit d of the absciss ad, there is taken dl equal to the semi 

 parameter, and from the point l be drawn lf straight to the point f ; the right 

 line so drawn will be perpendicular to the parabola in the point f, and the least 

 of all the lines that can be drawn to the curve from the point l. I say it is the 

 least ; for to any one that considers the nature of the curve, it is evident it 

 cannot be the greatest, (which I would have observed in what follows ;) but it 

 is necessarily either the greatest or least, and therefore the latter. And this is 

 the first part of theor. 5, lib. 7, of De la Hire's Conies. 



Let the ordinate eb be drawn, and join the points e, l, make the intercepted 



tries which was copied into several foreign works. His other publications were, 1. General Laws 

 of Nature and Motion, 1705 ; 2. An Institution of Ructions, 17O6"; 3. Alexander's Synopsis of 

 Algebra, with many additions and corrections, 1709; 4. Treatise on Perspective, 1712 j 5. The 

 New Law of Fluids, 171+} 6. The same year several theological pieces, particularly his discourse 

 on the Resurrection of Jesus Christ : all which works met with very distinguished applause from the 

 learned of several countries. Mr. Ditton was connected with the celebrated Mr, Whiston in a scheme 

 for the longitude, viz, by the firing of great guns at a distance ; but the plan being rejected by the 

 Board of Longitude, it is said the grief at the disappointment, and the ridicule thrown upon it, in a 

 ludicrous poem, by Dr. Swift, had the unhappy effect of occasioning his death. See a more parti* 

 cular account of Mr. Ditton, and his writings, in Dr. Mutton's Dictionary, vol. i. p. 388^ 



