112 PHILOSOPHICAL TRANSACTIONS. [aNNO iS/S. 



are plenty. I shall add only a construction by a parabola, and that in two ways ; 

 which though it may seem more operose than by an hyperbola, yet the simpli- 

 city of this curve compensates for the labour, as in that respect the parabola has 

 an advantage over the other conic sections. 



The same things then being given; join AI (fig. 9), and produce it to S, till 

 AS be equal to AH; then joining HS, and bisecting IS in M, through M 

 draw RMQ perpendicular to HS, on which demit the perpendicular AQ, 

 parallel to which draw the radius AC. Then, forming the three proportionals 

 AI, AC, AE, make as SA to AE so MQ to AD and so RS to AP (towards 

 Q in the line AQ) ; and in the same line, on the other side, take DO = DC. 

 Then, bisecting PD in X, let VXL, drawn through X, make with AX half a 

 right angle, meeting in V the perpendicular erected at D, and on which demit 

 the perpendicular OB. I say, if there be made as VX to XB, so XB to BL . 

 then L is the vertex, LV the axis, and XV the latus rectum of the parabola, 

 which satisfies the problem in every case; viz. cutting the given circle in the 

 points K, of which the highest and lowest belong to Alhazen's problem, and the 

 rest to another problem, concerning which I lately wrote to you. 



There may also be given another parabola, as was hinted above, which will 

 perform the same thing as this, and the description of which is so easily de- 

 duced from this, that there is no occasion for a new one. For let A^be 

 taken = DA and in that direction, and Aw = OA and in that direction. Then 

 bisecting P(^ in ^, draw 8^(3 perpendicular to XB, meeting ^a, perpendicular to 

 OA, in 8, and on it demit the perpendicular w|3; also make as «^ to ^f3, so ^(3 

 to (3x : Then x will be the vertex, x^ the axis, and ag the lastus rectum of the 

 parabola, which will cut the given circle in the same points with the former. 



On thejirst Invention and Demonstration of a Right Line equal to a Curve. In a 

 Letter from Dr. IVallis, Oxf. Oct. 4, 1673. Translated from the Latin. 

 N°98, p. 6146. 



Learned Sir — As to the rectification of that curve, which I have used to call 

 the semicubical paraboloid, M. Huygens must be under a great mistake, when 

 (in his horologium oscillatorium, p. 71? 72.,) he ascribes the first invention of 

 it to John Heuraet of Harlem, in the year 1659. For it is certain, that it was 

 discovered and demonstrated two years before, by Mr. Wm. Neil,* an English- 



* This ingenious young gentleman was the eldest son of Sir Paul Neil, Knt. one of the ushers of 

 the privy chamber to King Cha. I. and was grandson of Dr. Rd. Neil, aixhbishop of York. He was 

 born Dec. 7 , l637, and was educated at Oxford, where he became gentleman commoner of Wad- 

 ham-college, in l6"52, for the sake of Dr. Wilkins the warden 3 by whose instructions, and those of 



