114 PHILOSOPHICAL TRANSACTIOXS. [anNO 1673. 



is not pretended to have been done before the year 1659 ; and earlier too 

 than that of Sir Ch. Wren, finding a straight line equal to a cycloid in the 

 year l658; and by him admitted so to be. Nor ought it at all to prejudice Mr. 

 Neil, that M. Heuraet's was somewhat sooner abroad in print, than that of M. 

 Neil, though both in the same year 1659 ; since it is well known to many of us, 

 that Mr. Neil's was done before. Otherwise M. Huygens, by the same reason, 

 will grant the precedency to Heuraet, of that which he now claims to be his 

 own invention (that rectifying the parabolical line, and squaring the hyperbolical 

 space, do mutually depend on each other:) for this was published in print by M. 

 Heuraet (or M. Schooten for him) in the year 1659, ^"^1 not by M. Huygens till 

 now, 1673 : and yet M. Huygens thinks, he may well claim that invention to 

 be his own, because he now tells us, that he found it out about the end of the 

 year ]657, and did, some time after, communicate it privately to some friends. 

 And whereas, he supposes that this invention of his might give occasion to that 

 other of Heuraet; we may also as well suppose, that he might have taken such 

 occasion from hearing of Mr. Neil having done the like, for this had been then 

 commonly known for a great while: or might have taken occasion, as well as 

 Mr. Neil, from that of Dr. Wallis Schol. prop. 38. Arith. Infin. or from that of 

 Sir Ch. Wren having found a straight equal to another curve the year before: 

 or if it were necessary to know the symbolization between the parabolical line 

 and the hyperbolical space, he might have had it earlier from Dr. Wallis. For 

 when he had demonstrated (Schol. prop. 38. Ar. Infin.) that the particles which 

 compose the parabolical line, are in power equal to a series of squares increased 

 by a series of equals, suppose s/ : A^ + Z? -: and (prop. 35, 41, conic, sect.) that 

 c the ordinates to the conjugate diameter of an hyperbola, that is, the particles 

 of which that hyperbolical space consists, are so also, viz. y':-i-T^ +| A'^: where 

 A, T, L, are permanent quantities, and h, h, taken successively in arithmetical 

 progression ; it was easy for M. Heuraet, or M. Huygens, or any other, to 

 infer, that if we can rectify the one, we may square the other, and vice versa. 

 But from whencesoever M. Heuraet had it, we may, as before, reasonably con- 

 clude, that Mr. Neil had it before him ; and M. Huygens is a person of that 

 ingenuity, that when he shall better consider of it, he will I doubt not be of the 

 same mind. London, Oct. 8, 1673. 



The other Letter is of Sir Chr. Wren, Knt. Surveyor General of his Majesty's 



Buildings, &c. 



Sir, That I did, in the year l658, find a straight line equal to that of a 

 cycloid, and the parts thereof, was then very well known, not in England only, 

 but in France and Holland. And I have not yet heard of any, who do pretend 



