A Letter from Edmund Stone, F. R. S. to--concerning Two Species of Lines of the Third Order, Not Mentioned by Sir Isaac Newton, Nor Mr. Sterling

I preserved some Plants with their natural Earth, and brought them to London alive; and I observed the remarkable Change produced by the Sun's Heat on them; for the saline Matter in Greenland, which certainly was analogous to a fix'd Salt, became, in a Month's time, almost to the same Volatility as that which naturally grows in England. This I make mention of, in case other Gentlemen, who have had the same Opportunity, have been remiss in their Curiosity. David Nicolson. VI. A Letter from Edmund Stone, F. R. S. to —— concerning two Species of Lines of the Third Order, not mentioned by Sir Isaac Newton, nor Mr. Sterling. SIR, HAVING for some time past been reading and considering the little Treatise of Sir Isaac Newton, intituled, Enumeratio Linearum tertii Ordinis, as also the ingenious Piece of Mr. Sterling, called, Illustratio Tractatus Domini Newtoni Linearum tertii Ordinis; I have observed, that they have neither of them taken Notice of the two following Species of Lines of the Third Order; and venture to affirm, that the Seventy-two Species mentioned by Sir Isaac, together with the Four more of Mr. Sterling, and these Two, making in all Seventy-eight, is the exact Number of the different Species of of the Lines of the Third Order, according to what Sir Isaac has thought fit to constitute a different Species. The two Species I mean, are to be reckoned amongst the Hyperbolo-parabolical Curves, having one Diameter, and one Asymptote, at No. 8. of Newton's Treatise, or Page 104. of Mr. Sterling's; whose Equation is $xy = bx^2 + cx + d$; which will give, not Four, as in these Authors, but Six Species of these Curves: For, I. If the Equation $bx^2 + cx + d = 0$, has two impossible Roots, the Equation $xy = bx^2 + cx + d$, will (as they say) give two Hyperbolo-parabolical Figures equally distant on each side the Diameter $AB$. See the 57th Figure in Newton's Treatise, and this is his 53rd Species, and Sterling's 57th. II. If the Equation $bx^2 - cx + d = 0$, has two equal Roots both with the Sign $+$; the Equation $xy = bx^2 - cx + d$, will (as they say) give two Hyperbolo-parabolical Curves crossing each other at the Point $\tau$ in the Diameter. See Fig. the 58th in Newton; and this is his 54th Species, and Sterling's 58th. III. But if the Equation $bx^2 + cx + d = 0$, has two possible unequal negative Roots $A\rho$ and $A\tau$, the Curve given by the Equation $xy = bx^2 + cx + d$, will consist of two Hyperbolo-pa- rabolical Parts, as also of an Oval on the contrary Side the Asymptote or principal Absciss. And this is one of the Species omitted by Sir Isaac and Mr. Sterling, which is really the 59th Species. IV. Also if the Equation \( bx^2 + cx + d = 0 \), has two equal negative Roots \( A \rho \) and \( A \tau \); the Curve given by the Equation \( xyy = bx + cx + d \), will consist of two Hyperbolo-parabolical Parts, and also of a Conjugate Point on the contrary Side the Asymptote or principal Ordinate: And this is the other Species of these Curves omitted by Sir Isaac and Mr. Sterling, which is really the 60th Species. V. If the Roots of the Equation \( bx^2 - cx + d = 0 \), are real, and unequal, having both the Sign \( + \); the Curve given by the Equation \( xyy = bx^2 - cx + d \), will (as they say) consist of a conchoidal Hyperbola and a Parabola, on the same Side the Asymptote or principal Ordinate. See Fig. the 59th in Newton; and this is really the 61st Species. VI. If the Roots of the Equation \( bx^2 + cx - d = 0 \), have contrary Signs, the Equation \( xyy = bx^2 + cx - d \), will (as they say) give a conchoidal Hyperbola with a Parabola on the contrary Side the Asymptote or principal Ordinate. See Fig. the 60th in Newton; and this is really the 62d Species. I remain Your humble Servant, Edmund Stone. VII. An