solid; and if ye like this, I shall give you with the next a series for the second segments of an hyperbolic spindle, which I imagine is of greater consequence than anything else for gauging. Having now room and leisure, I resolve to perfect some things I formerly sent you. Ye know the series I gave you for the hyperbolic curve serveth only near to the vertex; and therefore for the complete measure of that curve, Sit hyperbola GEF cujus asymptoti AI, AB; ducantur asymptoto AI I parallelæ DG, CE, BF, quarum DG ipsi AD sit æqualis, reliquæ vero CE, BF, ad libitum; fiatque GH asymptoto AB perpendicularis. Sint AD=r, GH =c, واح E F r C D H C B f AC=g, AB=f,√4r2-4c2=b. Si angulus IAB fuerit acutus, erit curva EF =ƒ−g + br br + 2f 2g 4 p4-b2 p2 b2 r2 - 4r4 4br5 — b3μ3 b3 g3 — 4 br5 24 g3 + 24f3 + 40 b3 r7 — 48 br9 −7 b5 r5, 48 br9 +7b5 r5 – 40 b3 rī 64 12-240 b2 10 + 140 b4 μ8 — 21 b6 μ6 11264 gll 240 b2 10-64 r12 + 21 b6 μ6 — 140 b4 p8 11264 fl + + &c. + + Si vero (reliquis manentibus) angulus IAB fuerit + b2 p2 - 4p4 b3 73 -4br5 4 br5 — b3 p3 24f3 + + 80f5 24 b2 p6 — 16 78 – 5b1μa, 16μ8 + 5 b1μa — 24 b2 2.6 896f7 + +7 b5 r5 – 40 b3μ7 40 b3μ7 – 48 br9 — 7 b5 p5 896g7 48 br9 2304g9 2304f9 64 12-240 b2 p10 + 140 b4 μ8 – 21 b6 μ6 11264 gl 240 b2 10-64 r12+21 b6 μ6 — 140 b4 p8 11264 fll + + + &c. Si de nique angulus IAB fuerit rectus, foret b=0, et proinde evanescerent omnes quantitates in quibus reperitur b. This series is the more exact the further CE be taken from the vertex, contrary to that other I gave [you], which was also applicable, by a little alteration, unto the ellipsis. I gave you also only one series for the measure of the logarithmic curve, which I do not remember; and therefore, in case I give you the same over again, I shall give you two to complete the measure of it. Sit igitur ML curva logarithmica, cujus recta asymptota HK, in quam sint ordinatæ MH, LK; sitque MA ipsi HK M Erit itidem eadem curva ML=b-c+ 7610-7c10 + + &c. 1024 r7 2560,9 The second series helpeth the defect of the first; so that either the one or the other is abundantly sufficient for the measure of the curve. I know that the segments of circles are of great use in practice, but if the segment be little, Mr. Newton's series, which ye sent me, is not of ready use, and therefore ye may make use of this: Sit radius=r, sagitta segmenti circularis = a, et -&c., ejusque arcus integer = 2b+ a2 3a4 5a6 35a8 63a 10 + + + + 3b 2063 56b5 576b7 140869 + &c. I have received lately another of yours dated 6th May, in which ye continue still engaging me with excellent extracts and information of the best mathematical books. I am heartily sorry that Doctor Wallis is so troubled with a quartan ague, yet it may prove healthful to him. It is a disease that I am very happily acquaint[ed] with, for thirteen years ago I had it a whole year and a half, and since that time I never had the least indisposition; nevertheless that I was of a very tender and sickly constitution formerly. As to the rest of your letter, I have not so much vanity as to persuade myself that ye are serious, I having never heard any thing relating to that formerly. I have had sufficient experience of the uncertainty of things of that nature before now, which maketh me, since I came to Scotland, how mean and despicable soever my condition be, to rest contented, and satisfy myself with that, that I am at home in a settled condition, by which I can live. I have known many learned men, far above me upon every account, with whom I would not change my condition. No more at present, A considerable time ago I received one from you by Mr. Sinclair his hand, for which I acknowledge myself much obliged to you and him. I think [it] strange that so many eminent men have failed of their promises. I am confident Huddenius is able to perform what he hath promised, for his two epistles, in my opinion, go beyond all who ever did write in Algebra, yea, [Des] Cartes himself not being excepted: only in Epist. 1ma, pag. 492, he seemeth to fail in his assertion "neque quod sedulo observo," &c.; for by his two equations any of these unknown quantities x or z can be taken away. It is true ye may perchance have ground enough at present to think as much of me, yet, albeit I know myself not to be comparable to the least of these ye mention, God willing, with the first occasion, I shall perform the utmost of what I ever promised. As for Bartholini Dioristice, if I be not mistaken, it is infinitely outdone, even in the highest equations, by what may be easily deduced from Huddenii Epist. 2da. As for the second segments of round solids, I have no other but what I sent to you; I expected that ye would have advertised me, if Newton had done any thing more. I am confident that the tables of logarithms and sines cannot resolve all equations, albeit they can very many, yet not without great preparation, so as a sursolid equation, which can be reduced to a pure one, must first ascend to the twentieth potestas, not without extraordinary work. The only universal method I know is the infinite serieses. There can be given one which will serve for all cubic equations, another for all biquadratics, another for all sursolids, &c. and I suppose that tables of these serieses were the best of any. I should be glad to see how Ferguson gives the roots of all cubic equations in intelligible surds. I suppose he doth it only in some particular examples, (and not indefinitely,) which is no great matter. I believe ye will find a great abbreviation of Cardan's method in Vieta himself, (I know not if it be that ye ascribe to Dulaurens,) as also in De Beaune de Natura Æquationum, pag. 113; and in Schotenius' Appendix de Cubic. Æquat. resolutione, pag. 367; as also in Huddenii Epist. 1ma, pag. 499. There is no great mystery in what ye mention from Dr. Pell and Mons. Dulaurens; for ye may know by this that there is no great affinity betwixt equations and sines or logarithms, the curve of sines and that of logarithms have no contrary turnings, but these of equations have many. It is true, ye can take them all away from the side where the true roots are, but when this is done, nothing can be applicable to the side of the true roots, which is not also applicable to the other, suo modo. I wonder how ye speak yet of interpolation by the help of figurate numbers, seeing I sent you, a long time ago, a method much shorter and readier by a series. To let you see that there is a plus ultra (as ye tell me) in these things:-in omni æquatione indefinita, si desit secundus terminus, in ejus radice nulla erit quantitas |