CCCXXI. WALLIS TO COLLINS. Oxford, March 27, 1672. Sir, How I had newly sent away my letter to you by the last post, when upon a review of yours to me I began to suspect an error [of] mine by misapprehending the nature of the curve; which therefore if you please to return me, that I may a little better consider of it, I will see to mend it by the next. Meanwhile I have been reading Poterius de Ponderibus, &c., which you sent to Mr. Bernard, who imparted it to me. accurate he hath been in his collections I know not, for we have for the most part but his own assertions, not his authorities. His reductions are mostly to French measures, not to English, which makes it more proper to have been printed in France. And in our English measure he is grossly out, making our foot less than it ought to be by at least an inch and a half, supposing the French foot to be truly taken. For he makes the proportion of the Paris foot to ours to be as 1560 to 1302, which I have myself found by comparing them to be as 16 to 15; theirs containing of ours 12 inches proxime, which by him should be more than 14. And I fear therefore that he may be alike mistaken [in] others. He makes the height of Goliath about 15 French feet, which is above a perch, or 5 English yards, by his computation; about three times the height of an ordinary person, and must therefore be strangely disproportionate in bigness to his height, by what we have in Galileo's Dialogues de Motu, pag. 129. But enough of this at present. I am, Sir, yours to serve you, JOHN WALLIS. CCCXXII. WALLIS TO COLLINS. Stoke juxta Guilford, Maii 13, 1672. Ad tuas Maii 9 datas quo respondeam, hæc habe. Figuram tangentium CA a O, pl. 6, fig. 2, (conchoidi congenerem) complere intelligantur æqualibus intervallis dissitæ tangentes Va, sinibus versis AV (adeoque et complementorum rectis CV = x) arithmetice proportionalibus convenientes; adeoque, posito radio CA =r, et sinu recto VB=s=√r2-x2, erit Va=b 2-2; et posito VD-a, adeoque = AS= sr = r CD=x+a, DO = = √r2x2 + 2xa x + a x + a √2+2xa-. Curvam AaO tangat aTF, occurrens in F recta CA, abscindens VF =ƒ, adeoque DF =ƒ±a, et (propter FV : FD : : Va: DT,) DT=ƒ±a 's2 + 2xa — . Et (sumptis quadratis) 222 + 2 fr2s2 a ; adeoque f2 r2 s2 x2 ± 2ƒr2 s2 x2 a + Qƒ2 r2 s2 x a ± ƒ2r2s2x2 + Qƒ2r2x3a-. Hoc est ±2ƒr2s2x2a+2ƒ2r2s2xa± ± 2ƒ2 r2 x3a - ; et di $2 visis omnibus per + 2fr2xa, positoque D in V, s2x —ƒs2=ƒx2, seu s2x=ƒs2 +ƒx2=fr2, et ƒ ==~; unde punctum F determinatur. Atque hactenus in epistola de tangentibus jam edita. Petis jam ut velim punctum contrarii flexus eadem methodo determinare: nempe quo ita sumatur Va, ut infra hanc sit DT <DO, sed supra DT> DO. Est autem (per jam ostensa) FV: FD: :ƒƒ±a :: : : s2 x : s2 x + r2 a : : Va (=3) : DT ( s2x + r2a s2xr+r3a < 8x2 > x + a et (dividendo utrinque per < sx2 > x + a = $2x 82x +a s3xr+sr3a 82 x2 's2 + 2xa-a2; et (multiplicando per $2 x = a,) s2x2 + r2 xa + s2 xa— r2 a2 ( = s2 x2 + x3 a − r2 a2) < sx2 > 2 √ s2+2xa-a2, et (sumptis quadratis) s1x1 ± 2s2x3a < +x6a2-2s2x2r2 a2 + 2 x3 r2 a3 + r1 a1 − s1x1 + 2 s2 x3 a 土 -s2x4a2; hoc est x6a2 - 2s2x2r2 a2 + 2x3r2a3 +r1a1 < < > −s2x1a2, seu æo — 2s2x2r2 + 2x3r2a +r1a2 —— s2x1, hoc est x6+s2x4 - 2s2r2x2 ± 2x3r2a —r1a2. Et propterea (posito D in V, quo fiat a = 0, adeoque evanescat æquationis pars posterior, simulque excessus defectusve,) erit x® + s2x1 — 2s2r2x2=0, adeoque x1 + s2x2 = 2s2r2, seu (propter a2 + s2 = r2,) x2r2 = 2s2r2, seu x2 = 2s2, adeoque 2 r2 = s2 + x2 = 3s2 == x2, et 2r2 = 3x2, seu r2 = x2. 3 2 3 Intelligatur jam Figura tangentium, non (ut prius) ad radium AC, sed ad arcum ABQ in rectam expansum, applicata (pl. 6, fig. 2, and 3). Hujusque tum tangentes tum punctum contrarii flexus inquirantur. Figura hæc a præcedente in hoc differt, quod ordinatarum tangentium Va, DO, (nunc Ba, BO,) intervallum, quod prius fuerat VD, jam erit æqualis respectivo arcui æque alto Bß, hoc est (in partibus exiguis) ВT. Est autem (per Prop. cap. 1, De Motu) TT. BV = DD. CA, puta ts=or, adeoque =T=B. Sed DD:TT::VF:Bộ, hoc est o $2 p2 or :: s: r :: VF (= f === x) : Bp (= p = = Atque hinc determinabitur punctum . or S x= x). S 7.2° r The conclusion of this letter is omitted, as being recalled in Letter CCCXXIV. CCCXXIII. WALLIS TO COLLINS. June 8, 1672. Oxford. Sir, I send you, here inclosed, those heads of a letter, (which you desired, put into Latin, with a postscript of my own; and, in the other part of this sheet, the equivalent designations by sines, tangents, secants, &c. as you desired. That the figure of tangents applied to the arch stretched out into a straight line, hath no contrary flexure, I am well satisfied, and can demonstrate it; so that in the last of those four operations in my letter of May 13, 1672, there is a mistake; but the three first I take to be sound. I am yours to serve you, JOHN WALLIS. Sinuum rectorum et versorum, tangentiumque et secantium, pro arcubus angulisve expositis, eorumque complementis Ισοδυναμία. Esto R radius, S sinus rectus, Σ sinus rectus complementi; Ttangens, 7 tangens complementi; s secans, ☛ secans complementi; V sinus versus, v sinus versus |