curve either to measure that curve, or to compare it with another curve?

§. 3. I answer, that the hour itself can certainly be pursued by the imagination to the end of it, but not the innumerable fubdivifions, which Philalethes pretends to have made by his motion. Perhaps I may be easiest understood by comparing the prefent point with the old argument against motion from Achilles and the Tortoife. It is impoffible to pursue in the imagination their motion by the means proposed in that argument to the point of their meeting; because the motion of cach is described by the terms of an infinite progreffion; but if we feek after the place of their meeting by the method propofed in that argument, we must have recourse to the doctrine of prime and ultimate ratios, and find the nearest limit of each of thofe progreffions.

§. 4. Does Philalethes here mean, that it is not neceffary in every geometrical demonstration to form in the imagination a diftinct conception of the subjects under confideration? to do otherwife is not only contrary to the practice of the ancients, but to that of every other just reafoner.

THUS We have gone through the Confiderations of Philalethes published last month, paragraph by paragraph; and defign to examine the reft after the fame manner, as foon as they fhall be published. But to prevent the difpute running into an unmeasurable length, we fhall afterwards reduce our Remarks into as narrow a compass as poffible; and for that end intend to confine ourselves to the difcuffion of one fingle point, by which we apprehend the merits of the queftion will in a manner be wholly decided. This is the examining whose interpretation of Sir Ifaac Newton's first Lemma is agreeable to the nature of vanishing

G4

quantities,

quantities, as foon as Philaletbes fhall have rendred his interpretation a true and confiftent propofition. But after this point is fettled, we shall be ready to explain ourselves upon any other, Philalethes fhall defire.

ARTI

The REMAINDER of the Paper begun in our laft, entituled, Confiderations upon fome paffages of a Differtation concerning the Doctrine of Fluxions, publifhed by Mr. Robins in the Republick of Letters for A pril laft. By Philalethes Cantabrigienfis.

XVII.

HR

Οι Owever, fince Mr. Robins is pleased to talk fo much about ftraining our imagination for fome involved and perplexed kind of motion, let us fee, if we cannot find fome plain and eafy way, of reprefenting to the imagination, that actual equality, at which the infcribed and circumfcribed figures will arrive with each other, and with the curvilineal figure, at the expiration of the finite time.

For this purpose, perhaps the best method we can take, will be to imitate that judicious expedient, which Sir Ifaac Newton has made use of in the feventh Lemma, to reprefent to the imagination the last proportion of decreafing quantities.

As thofe quantities, by a conftant decrease, are diminished ad infinitum, and at laft vanish, they arrive at their laft ratio at the inftant of their evanescence. And as, at that inftant, they flip away and withdraw themfelves from our conception; for this reafon, Sir Ifaac Newton, to help our imagination, teaches us to contemplate the variable ratio of thofe quantities, and particularly their laft ratio, not in themselves, but in other quantities proportional to them, which do not vanish, but continue and fubfift at the inftant that they arrive at this laft proportion; as we have more

particu

particularly observed in the Republick of Letters for November last.*

So here, as the inscribed and circumfcribed figures, at the inftant of their coincidence with the curvilineal figure, do, with refpect to their rectilineal form, likewife flip away and withdraw themselves from our conception; it will be of use to confider their variable ratio, and particularly their last ratio, not in themselves, but in other quantities, which undergo no alteration of their form, when they arrive at this last proportion.

Let us therefore fuppofe the curvilineal figure ABE to be equal to the rectangle contained under the base EA and fome conftant line, as AF.

And

*Pag. 378, 379.

And at the inftant that the point, which we have supposed to describe the base EA in a finite time, arrives at any point as C, let the rectangle under the fame base EA and fome variable line, as Cd, parallel to AB, be equal to the fum of the infcribed parallelograms, at that inftant conceived as ftanding upon CA the remaining undescribed part of the bafe, and upon as many other parts of the base E A, as can be taken equal to CA and adjoining to it, i. e. let the rectangle under EA and Cd be equal to the infcribed figure: And let Edd be the curve, which the point d continually touches or describes.

A

In like manner, let the rectangle under the fame base E A and some other variable line, as CD, alfo parallel to A B, be equal to the fum of the cir cumfcribed parallelograms, at the fame inftant conceived to be standing upon the fame parts of the base as above, and upon Ee the fmaller remainder of the base, if any fuch there be, adjoining to the point E, after the manner described in the Republick of Letters for January last; * i. e. let the rectangle under E A and CD be equal to the circumfcribed figure: And let GDD be the curve, which the point D continually touches or describes.

Then, as the curvilineal, the infcribed, and eircumfcribed figures are refpectively equal to thefe, three rectangles EAXAF, EAxСd, EAxCD, having all the fame common base EA; it is manifeft that those figures will be refpectively proportional to the lines AF, Cd, and CD; and may confequently be always represented by these lines. Now, fince the two lines Cd, CD, do

1. Con + Republick of Letters for November, p. 376.

* Page 99.