N^o 1962.
Lord Brouncker à H. Oldenburg.
18 octobre 1673.
Appendice II au No. 1959.

La lettre a été imprimée dans les Phil. Trans.1).

Sir,

It is very sure, that Mr. William Neil had in the year 1657 found out and demonstrated a Streight line equal to a Paraboloeid; and did then communicate and publish the same (though not in print) to my self and others, who used to meet at Gresham Colledge, and it was there received with good approbation; and the same was, presently afterwards, otherwise demonstrated by my self and others: And therefore ancienter than that of Monsieur Heurat, which (as it seems) is not pretended to have been done before the year 16592); and ancienter too than that of Sr. Ch. Wren, finding a Streight line equal to a Cycloid in the year 1658; 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. Hugens, 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 16593), and not by M. Hugens till now, 1673: And yet M. Hugens thinks, the may well claim that invention to be his own, because he now tells us4), that he found it out about the end of the year 1657, and did (some time after) communicate it privately to some friends. And whereas, he doth suppose, 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 M. 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 Sr. Ch. Wren having found a Steright equal to another Curve the year before: Or, if it were necessary to know their symbolization between the Parabolical Line and the Hyperbolical Space, he might have had it earlier from Dr. Wallis. For, when

he had demonstrated (Schol. pr op. 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 √ ∶ A² + b²: 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 consisteth,) are so also, viz. √∶ 1/4 T² + T/L h²: (where A, T, L, are permanent quantities, and b, h, taken successively in Progression Arithmetical;) It was easie (for M. Heuraet, or M. Hugens, or any other,) to infer, That, if we can Rectifie the one, we may Square the other, & vice versa. But from whence soever M. Heuraet had it; we may, as before, reasonably conclude, that Mr. Neil had it before him: And M. Hugens 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.