354 PHILOSOPHICAL TRANSACTIONS. [ANNO \7Q8 . 



Of this solution it may be remarked, that it does not assume as a postulate the 

 description of the cycloid ; but gives a simple construction of that curve, flowing 

 from a curious property, by which it is related to a given circle. This cycloid too 

 gives, by its intersection with the ellipse, the point required, directly, and not by a 

 subsequent construction, as Sir I. Newton's does. I was induced to give the de- 

 monstration, from a conviction that it is a good instance of the superiority of mo- 

 dern over ancient analysis ; and in itself perhaps no inelegant specimen of algebraic 

 demonstration. 



Prop. 21. Problem. Fig. 20. — To find the curve whose tangent is always of 

 the same magnitude. 



Analysis. Let mn be the curve required, ab the given axis, sm a tangent at any 

 point m, and let d be the given magnitude ; then, sm . q . = sp . q . -f- pm . q. = cP; 



or> f + St = <?> and y = ^r^ J therefore, i = ^X V(P - y 2 ). In order 



to integrate this equation, divide - ^(d? — y*) into its 2 parts, - — -— ^ — - and 



— 'ZM ■ ; to find the fluent of the former. 

 V(* - y 1 ) 



a 



(1 + ) 



*y fi * </(* -/') m d v / # *} x 



y^-y l ) ~~ y d+ Vi^-y 2 ) ~" " > y 2 y* W - y 1 ) ' 



d + V(d*-f) 



d x fluxion of 



= d+W-fa 1 therefore the fluent of - yV( ff_ y) is - d X hyp. 



log. ** + V(<*'-y a ) ^ and the fluent of the other partj yy [ s + V (d 2 — y*); 



therefore the fluent of the aggregate - y/{tf — y 2 ), is /(^ - /) - rf X h. 1. 



l±-£ ( *^2, or V (# -/)+^Xh. 1. rf + -^A _ ; a final equation to the 



curve required, a. e. i. 



I shall throw together, in a few corollaries, the most remarkable things that 

 have occurred to me concerning this curve. 



Corol. 1. The subtangent of this curve is </ (d 2 — y 2 ). 



Corol. 2. In order to draw a tangent to it, from a given point without it ; 

 from this point as pole, with radius equal to d, and the curve's axis as directrix, de- 

 scribe a concoid of Nicomedes : to its intersections with the given curve draw 

 straight lines from the given point ; these will touch the curve. 



Corol. 3. This curve may be described, organically, by drawing one end of a 

 given flexible line or thread along a straight line, while the other end is urged by 

 a weight towards the same straight line. It is consequently the curve of traction 

 to a straight line. 



