1. Conftantly tend to equality with each other, and with the conftant line A F,

2. During a finite time,

3. And approach within less than any given difference,

4. Before the end of that finite time; agreeably to the four fuppofitions of the first Lemma; These three lines will, by that Lemma, be equal to one another at the end of the finite time.

This we apprehend not only to be undeniably true, but now to be acknowledged even by Mr. Robins; fince that Gentleman is at laft brought to confefs, that * whenever the "quantities com"pared in this lemma are capable of an actual "equality, they muft really become fo;" and alfo that "There is no ultimate fum of these pa❝rallelograms, nor no ultimate figure compound"ed of them, diftinct from the very curve itself." "This," he fays, "Sir Ifaac Newton, in the co"rollaries annexed to the third lemma exprefsly "declares." Here, it is manifeft, that, by the ultimate fum of these parallelograms, can only be understood the fumma ultima parallelogrammorum evanefcentium, which is faid coincidere omni ex parte cum figura curvilinea, in the first of thofe corollaries; and that by the ultimate figure compounded of them, nothing elfe can be meant, than the figure ultime, of which it is faid, non funt rectilinea, fed rectilinearum limites curvilinei, in the fourth corollary; both which corollaries had been expressly || quoted and infifted on by me against Mr. Robins.

With this allowance therefore of Mr. Robins, we may fafely affert, that, at the end of the finite time, the variable lines Cd, CD, fince they

* Republick of Letters for April, p. 309. + Ibid. p. 311.

are

Rep. of Let. for Jan. p. 84, 85.

are capable of an actual equality, muft really become equal to each other, and to the conftant line A F

And, if there were occafion, the fame thing might also be proved after another manner. For, fince the relation between Cd, the ordinate to the curve Edd, and the abfcifs EC, may always be expreffed by an equation to the curve Edd; and fince the relation between CD, the ordinate to the curve GDD, and the same abfcifs EC, may always be expreffed by an equation to the curve GDD: by means of those two equations it may be made to appear, that, when the common abfcifs becomes equal to the whole base E A, these ordinates Cd, CD, will be equal to each other, and to the conftant line AF.

This

This being premifed, and the proportion between the curvilineal, the infcribed and circumfcribed figures, being always reprefented to the imagination by means of the lines AF, Cd, and CD; it is eafy to conceive, and as it were to fee, that the circumfcribed figure, from being at first equal to the rectangle ABGE, and represented by the line GE, does by a conftant diminution, while represented by the line CD, tend to an equality with the infcribed and curvilineal figures, represented by the lines Cd, AF, and does at laft actually arrive at that equality; as likewife that the infcribed figure, beginning from nothing, does by a conftant increase, while represented by the line Cd, tend to an equality with the circumfcribed and curvilineal figures represented by the lines CD, AF, and does at laft actually arrive at that equality, when the defcribing point arrives at A the end of the bafe.

It may not be improper to illuftrate what we have been faying, by an easy example; and for my own and my reader's ease, I fhall chufe the fimpleft example I can think of.

Instead

Instead of a curvilineal figure, let us take a rectilineal one, viz. the rectangular triangle ABE, whose base A E is equal to the altitude AB. Let us imagine two figures, one infcribed, and the other circumfcribed about this triangle, in the fame manner as about the curvilineal figure in Sir Ifaac Newton's fecond lemma: And let these three figures be refpectively reprefented, as before, by the conftant line AFAB, and the variable lines, or ordinates, Cd, CD.

Then, if EA, or A B be called a, and the part of the bafe, as EC, which at any inftant of time is already described by the motion of the point C, be called x; it will eafily be found, that, when CA the remainder of the bafe, is any aliquot part of the whole bafe, the ordinate Cd, reprefenting the inscribed figure, is equal to EC, or x. Confequently, all these ordinates will be termina

ted

ted by the right line EF, drawn from E to the point F bifecting the altitude AB.

Likewife, it will be found, that the ordinate CD, representing the circumfcribed figure, will be equal to a-x. And if, instead of CD, we take Kd, which is always equal to it, these ordinates Kd to the bafe BG, will be all terminated by the fame right line EF already described.

But when CA, the remainder of the base E A, is not an aliquot part of the whole bafe; if we divide the base into as many parts as may be, feverally equal to CA and adjoining to it; there will be left, adjoining to the point E, a portion as Ee, less than any of the reft, which let us call And in this cafe it will be found, that the ordinate Cdxrxa-r; and all thefe ordinates

7.

2a

will be bounded by the continued curve Edd F, rifing a little above the right line EF, and touching it in every point, where an ordinate can be drawn to that curve from the end of any aliquot part of the bafe, lying between A and the middle of the base.

Alfo, it will be found, that the ordinate CD, *+*xar, and all these ordi

or Kda

*+rxa—r

ར

nates Kd will be bounded by the fame curve. EddF.

Therefore, when xa, and r vanishes, Cd

a, and Kda, i. e. Cd and K d do then become equal to each other, and to the line A F. Confequently the infcribed and circumfcribed figures do then become equal to each other, and to the triangle ABE.