reply, I think Mr. Gregory hath done his work, and hath the better end of the staff. After this is done, I think it will not be necessary for others to appear as seconds in the business, unless upon further provocation; because it may occasion further heats and animosities between the two societies, (that of ours and that of theirs,) than the business doth deserve, and seeking occasions of revenge and disparaging each other, which may prove of ill consequence. What I have said to M. Huygens myself may possibly have as good an influence on him, as if I should more reproach him in public; yet, if my Lord Brounker think fit, (who in the first testimony is as much or more concerned as I,) that something briefly to this purpose be inserted, That the persons concerned in the former testimony of that book, notwithstanding the objections made by M. Huygens, to which Mr. Gregory hath made his reply, do not see any cause to recede from their former favourable opinion of it, I shall not be averse from it. But I think there will be no need of it, and perhaps the issue may be as well without it.

About the segment of circles; it's true I give no other measure of them, than by the versed sine, the right sine, and the arch and semidiameter. You say that you take nothing as known (besides the radius, for so I understand you) but the versed sine. But this being given, the right sine is known also (s2 = 2vR -v2) and then the arch by the tables of sines. But if you would have a table computed for segments of spheres and circles, according to the several proportions of the versed sines, I know of no more expedient way at present than this. Supposing the radius CA divided into any number of equal parts. Suppose 10, 100, 1000, 10000, or as many as you please, CV, VV, &c. and consequently CV, CV, CV. &c, are 1,

2, 3, &c. of such parts, and therefore squares of VP,

represents the whole hemisphere, or such a part thereof as you need: that is, they are thereunto as the square of radius to the circlem. And the square roots of them√R2-1, √R2-4, R2-9, &c. are the lines VP, and the aggregates of them, or so many of them as you need, (multiplied by the altitude of one such part,) gives you the area of the whole quadrant, or such a part as you need. The table is easily calculated, wanting for each place but one extraction of the square root, and one addition of two numbers, after a subduction of the successive squares out of the square of radius. And the result of the work will be either greater or less than the just, according as you take the inscribed or circumscribed figure; and the middle between both will be truer than either. I add here a specimen.

The remainder of this letter does not appear in the collection; it was probably written on the part which contained the address, which is also missing; but there can be little doubt of its having been intended for Collins.

The meaning of the writer is by no means clearly expressed, and there was the more difficulty in ascertaining it from inaccuracy in the MS. This has been cor

rected; and he seems to wish to convey, that the sum of the solids is to the volume of the hemisphere as 1:π.

Sir,

CCCVII.

WALLIS TO COLLINS.

Oxford, Novemb. 3, 1668.

I thank you for yours of Octob. 26, and the book you sent with it, which I received last Saturday toward night. I thought to have sent you the enclosed by the last post, but before I had transcribed it, (thinking fit to keep a copy of it,) I was otherwise diverted. Mr. Gregory is certainly in the wrong, and therefore I am sorry to see him write at that rate he doth. And I could have wished, but that it is now too late, that his angry preface and the first leaf of his book had been suppressed. I wrote him a large letter of Octob. 22, when I knew not of this preface, enclosed in one to my Lord Brounker. Whether it came to his hands before he went out of town I know not. If not, I desire you will send it after him, with this enclosed, which you may, if you have opportunity, shew my Lord before it goes. I would be content to have his Lordship's sense upon the former, and upou this whole matter.

The series for the circle, answerable to that of the hyperbola, is the same which I sent you a while since. For that, which in the ellipse answers to the asymptotes of the hyperbola, are the two equal diameters, which in the circle are any two cross diameters intersecting at right angles, answering to asymptotes so crossing. And the ordinates to one of these diameters answer to the ordinates to the asymptotes, terminated in the curve.

I was not against printing a comment on the Clavis Math., if any think fit to do it; but only that I thought it not necessary, and would swell a manual into a volume. What piece I shall else print I have not yet determined; but an Introduction to Algebra I have not yet ready. That at London gets on so slowly, that, if I had been aware of it, I would never have given way to print it there; and I doubt I must yet be forced to have it finished here. The post is going, and I can add no more, but that I am

What I suppose Mr. Oldenburg intends in the next Transactions, though it contain divers of the principles on which I proceed in my hypothesis of motion, is not intended as any summary of my book now printing; nor is it at all in that method, however upon the same principles. To the other question, the chapter De centro gravitatis must stand where it doth, and is not to be removed. The next, De calculo centri gravitatis, though I once thought of taking it out there, and putting it by itself; yet considering that cannot be done without altering the numbers of the figures, and, in pursuance of that, going over the whole work anew, in which the figures are cited over and over again many times in the same page, it would make so much work to make that alteration all along, and would be subject

to so many mistakes, that I think there will be a necessity of letting it stand where it doth, and proceeding in some following chapters as they were at first designed, without dividing it into two parts. But what you speak of putting out these three or four chapters alone cannot at all be; they being necessarily connected with what is to follow, de vecte, cochlea, trochea, tympano, &c., and that de motuum acceleratione et percussione, which must all go together, because they frequently cite propositions out of the precedent chapters. If that De calculo centri gravitatis be taken out, it is all [that] can be done, and that not without much trouble for the reasons mentioned. The series of which you inquire in pag. 398 of Lalovera de cycloide I have looked upon, but it's complicated so with other things, that I see not how to give an account of it without reading over most of the book, nor can it well be otherwise understood; which at present, having many things on my hands at once, I am not in a capacity to do. And I should rather give an account of the things from my own principles than study to be perfect in his; his whole method all along being somewhat perplexed. Though (because I find the results for the most part agree with mine) I take it to be sound though dark. But the general design [of] these I take to be to shew how by having the sum of lines, making the plane of those figures, the circle and hyperbola, he proceeds to a sum of squares to find the solid ungula, or the moment of that plane; and so to the sums of cubes, to find the moment of that ungle, and so on. Or, which is equivalent, from the squaring of a plane, whose lines are as the lines in an hyperbola or circle, to the squaring of a second, third, fourth, &c. plane, whose lines are in the duplicate, triplicate, quadruplicate, &c. proportion of those lines, which is