Dico (sumpto quovis in KD puncto E) fore KD2 VD nam KE2 = KV + VE 2 = VD+ VE2 = VD2 + VE2 + 2VD × VE; at VD2 + VE2 — 2VD × VE, (sunt enim VD2, VD × VE, VE2 proportionales,) ergo liquet propositum. Ductâ jam EF ad DB parallelâ, quæ secet hyperbolam in F, et paraboliformem (cujus index 3) in G, EF; ergo EG EF; quare punctum G est intra hyperbolam, et tota proinde paraboliformis VGB. est Not. Quo punctum E propius est ipsi D, eo minus Ita sufficiet utcunque regularum fontes indicâsse. That these rules may be better accommodated to practice, and that it may be known what numbers for m n are fit in each particular case to be taken, I add these two rules. In circulo si sit VD = CV, sumatur m = t-s, et n = 2t-s. In hyperbola si sit VD=2, CV, sumatur m =t+s, et n = 2t+8. And to avoid multiplicity, the former rules may be altered, putting these equal terms every where in the place of m and n respectively, which I leave to your pleasure. To give some experiment of those rules, I shall by them examine that method, which you say the gaugers use; taking an easy case to that purpose. Conus rectus VBA, cujus axis VC, secetur planis parallelis VCH, FEK, et sit BD = BC, conoque circumscribatur cylindrus &c. vero segmentum FVCHKEFV=VC × CHKE; est autem juxta methodum propositam, DE: CA (hoc est in præsenti casu 5: 2):: segm. cyl. : frust. con. :: VC× CEGKE+VC × CHKE CHKE : 3 :: 3VC × CHKE = : CE × GKE + VC × CHKE; ergo 6VC × CHKE = 5CE GKE +5VC × CHKE, vel VC × CHKE 5CE × GKE; et quia VC=5CE, hinc CHKE = GKE. Hoc est segmentum circulare CHKE hyperbolicum spatium GKE adæquabit: hoc ad regulas nostras exigamus. m 2 Sumatur = ergo n 3 2m-n = 1, est autem F cen trum hyperbolæ EGK, et FG-GE: ergo cautioni nostræ hac in parte satisfactum est quoad hunc casum. Est proinde hyperb. GKE 3 n m + n hoc est (CA=r) — rr√3 (nam GE= 1 et (ductâ KL ad HC per Segm. circ. (hoc est 8 16 3rr 3rr √3 (3) 3 x 2 adjungatur rectang. CLKE = 3rr 1 +rr√3HKEC. Hac 8 16 tenus nil occurrit absurditatis aut repugnantiæ. Ita quiddam minus hyperbolico spatio GKE majus est quodam quod segmentum circulare excedit. Et proinde spatia hæc inæqualia sunt, quod adversatur præmonstratis. Methodus igitur indigitata peccare deprehenditur in hoc casu; ac proinde in aliis aberret oportet. Concerning the construction of solid problems, to which your inclosed refers, I have sometime, among other things, observed that all, when the equations are quadrato-quadratical, may be referred to these two problems, and be constructed as they, without any reduction or alteration of the terms. 1. Dato circulo, et recta AE positione data, et in hac assignato puncto A; reperiatur in ea punctum B, a quo ducendo BD ipsi AE perpendicularem, quæ circulum secet in D, sit rectangulum ABD æquale dato plano (P2). 2. Itidem data hyperbola A et recta AE positione, punctoque A in hac; reperiatur punctum B, a quo si ducatur BD ad AE perpendicularis, sit rect. ABD dato P2. A M B E N D C M B E C Nt D Prius ad æquationes spectat, quales a, la3, mmaa, n3a, + p1, in quibus habetur + p4; alterum ad illas a1, la3, mmaa, n3a, — p1 in quibus p1 negatur. Varia positio lineæ AB et puncti A in ea, et quantitas datarum &c. cunctas æquationum varietates efficit. Etiam cubicæ æquationes hic se offerunt, modo detur una radix, ceu MN; vel, modo planum datum æquetur rectangulo AMN. Concerning the imparting those things, although I persist in my opinion, yet I shall leave it to your pleasure, on condition they be first well examined. I writ in haste, and, had I thought of such a thing, might have considered and polished them better: and perhaps there may be divers errors in them; as for instance, after I had sent my last, I presently recollected made DN = EB, &c. which error I pray you to correct, that came from my haste and slip of memory. And others, I suppose, there may be the like; wherefore I would not that any of those things be communicated before they be tried. If that honourable person, whom you mentioned, should think it worth his regard, and condescend to try them, I should be content to have them disposed as he should approve, &c. But I see no need of mentioning names. A member of the society to another inquiring about the dimension of, &c. would be abundantly, to my seeming, sufficient. With my best wishes, hearty love, and service, I rest Your most affectionate friend and obliged servant, Trin. Coll. March 28, 1668. ISAAC BARROW. |