Philosophical Transactions. For the months of September and October, 1715 - Part IV.

Ⅳ. A ready description and quadrature of a curve of the third order, resembling that commonly call'd the Foliate. Communicated by Mr. Abr. de Moivre, F.R.S.

I have look'd a little farther into that curve which fell lately under my consideration. It is not the Foliate as I did at first imagine, but I believe it ought not to make a species distinct from it. AEB (Fig. I.) is the curve I thus describe. Let AB and BK be perpendicular to each other. From the point A draw AR cutting BK in R, and make RE=BR, the point E belongs to the curve draw BC making an angle of 45 grad. with AB, this line BC touches the curve in B; from the point E draw ED perpendicular to BC, and calling BD, x; DE, y;AB, a; and making √8aa=n the equation belonging to that curve is x3+xxy+xyy+y3=nxy or x4−y4x−y=nxy. Taking BG=AB, and drawing GP perpendicular to BG, PG is an asymptote. In the Foliate the equation is x3+y3=½nxy, in which the two terms xxy+xyy of the former equation are wanting; and its asymptote is distant from B by ⅓ BA. Again draw EF perpendicular to AB; let BF be called z and FE v; the equation belonging to the curve AEB is vv= azz−z3a+z. In the Foliate the equation is vv= azz−z3a+3z
From these two last equations it seems that these curves differ no more from one another than the circle from the ellipsis. I should be very glad to know your opinion thereupon.
The quadrature of the curve here described has something of simplicity with which I was well pleased. With the radius BA and center B describe a circle AKG, let the square HPST circumscribe it, so that HP be parallel to AG; prolong FE till it meet the circumference of the circle in M, and through M draw LMQ parallel to HP. The area BFE is equal to the area KHLM, comprehended by KH, HL, LM and the arc KM. And the area BFe is equal to the area KmLH or KMPQ. Therefore if BF and Bf are equal to the rectangle HQ and therefore the whole space comprehended by BEAXBeTGZ (supposing T and Z to be at an infinite distance) is equal to the circumscribed square HS.
N.B. This quadrature is easily demonstrated from the equation; for by it a+z:a−z::zz:vv, that is AF:EF::MF:FB, and so of the fluxion of AF to Ll the fluxion of MF. Hence the areola EFoe will be always equal to the areola MLlu, and therefore the area AEF always equal to the area MAL.
Hence it appears that this curve requires the quadrature of the circle to square it; whereas the Foliate is exactly quadrable, the whole leaf thereof being but one third of the square of AB, which in this is above three sevenths of the same. Again in our curve, the greatest breadth is when the point F divides the line AB in extream and mean proportion; whereas in the Foliate it is when AB is triple in power to BF. And the greatest EF or ordinate in the Foliate is to that of our curve nearly as 3 to 4, or exactly as ⁄√⅔√−⅓ to ⁄√5√5⁄4−5½.
But still these differences are not enough to make them two distinct species, they being both defined by a like equation, if the asymptote SGP be taken for the diameter. And they are both comprehended under the fortieth kind of the curves of the third oder, as they stand enumerate by Sir Isaac Newton, in his incomparable treatise on that subject.


A pdf version of the entire text of the Philosophical Transactions of the Royal Society of London can be found here (this article begins on page 329).

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