De Figurarum Geometricè Irrationalium Quadraturis. Autore Johanne Craig

IV. De Figurarum Geometricè irrationalium Quadraturis. Autore Johanne Craig. SIT ACF Semicirculus, cujus Diameter est AF, ADE Curva Geometricè irrationalis, cujus Ordinatim applicata BD secat semicirculum in C. Quantitates verò sic designentur; Diameter AF = 2a, Abscissa AB = y, Arcus AC = v, Ordinata BD = z: sitque Z = rvyⁿ æquatio generalis exprimens naturas Curvavum Geometricè irrationalium ADE, in qua r denotat quantitatem quamlibet datam & determinatam, & n exponentem indefinitum quantitatis indeterminatæ y. Dico Aream. \[ ABD = \frac{rvy^{n+1}}{n+1} - qv + \sqrt{2aj} - yy x \frac{ra}{n+1} y^n + \frac{znra^2 + ra^2}{nx n+1} y^n + \frac{aAx_{2n-1} y^{n-2}}{n-1} + \frac{aBx_{2n-3} y^{n-3}}{n-2} + \frac{aCx_{2n-5} y^{n-4}}{n-3} + \frac{aDx_{2n-7} y^{n-5}}{n-4} + \frac{aEx_{2n-9} y^{n-6}}{n-5} + \text{&c.} \] De hac Serie Infinitâ hæc sunt notanda: (1.) Quod Literæ majusculæ A, B, C, D, E, &c. designent coefficientes terminorum ipsis immediatè præcedentium, sciz: \( A = \frac{znraa + ra}{nx n+1 xn+1} \) \( B = \frac{aAx_{2n-1}}{n-1}, \quad C = \frac{aBx_{2n-3}}{n-2} \) & sic porro. (2.) Quod si exponentis n sit numerus integer & positivus, aut nihilo æqualis, vel etiam si 2n sit numerus impar, tum Quadratura Spatii ABD exhibeatur per Quantitatem finitam; serie in his casibus abrumpente. (3.) Quod q designet terminum ultimo abrumpentem. (4.) Quod omnes illæ Figureæ in quibus series abrupitur habeant unam portionem Geometricè Quadrabiliem, ex ipsâ serie facillimè assignabilem: Nimirum si capiatur atur Abscissa \( y = r^{1 - \frac{1}{n+1}} \times nq + q^{1/n+1} \); Erit huic Abscissae competens Area Geometri cè quadrabilis. (5.) Quod solus terminus irrationalis \( \sqrt{2ay-y^2} \) in terminos ipsum sequentes sit multiplicandus. Exemplum 1. Sit \( z = v \), quia in hoc casu \( r = 1 \), \( n = 0 \), ideo \( \frac{r^n}{n+1}y \) est terminus ultimo abrumpens; quare \( q = a \), unde \[ ABD = vy - av + a\sqrt{2ay-y^2} \] &c proinde si (per not. 4) capiatur Abscissa \( y = a \), id est, si ordinata transeat per circuli centrum erit portio huic competens Geometricè Quadrabilis, scil. Area \( = aa \), id est, Radii Quadrato. Exemp. 2. Sit \( z = \frac{vy}{a} \), quia in hoc casu \( r = \frac{1}{a} \), \( n = 1 \), ideo \[ \frac{2nr^a + ra}{n x n+1} y \] est terminus ultimo abrumpens, quare \( q = \frac{3a}{4} \); unde \( ABD = \frac{vy^2}{2a} - \frac{3ay}{4} + \frac{y + 2a\sqrt{2ay-y^2}}{4} \) &c proinde si (per not. 4) capiatur \( y = \frac{\sqrt{3a^2}}{2} \), erit huic abscessae competens Area Geometricè Quadrabilis., Scil. Area \( = \sqrt{6a^4 - \frac{3a^2}{2}} \times \sqrt{\frac{3a^2}{2}} + \frac{3a}{4} \). Exemplum 3. Sit \( z = \frac{vy^2}{aa} \); In hoc casu \( r = \frac{1}{aa} \), \( n = 2 \), ideo \[ \frac{aAx^{n-1}}{n-1} y \] est terminus ultimo abrumpens, ergo \( q = \frac{5a}{6} \); unde per Seriem Infinitam erit \[ ABD = \frac{6vy^2 - 15a^3v + 2ay^2 + 5a^2y + 15ai\sqrt{2ay-y^2}}{18a^2} \] Et proinde si (per not. 4) Capiatur \( y = \frac{\sqrt{3a^3}}{2} \), erit abscessae competens Area Geometricè quadrabilis: scil. Area \( = \frac{2ay^2 + 5a^2y + 15a^2}{18a} \times \sqrt{2ay-y^2} \). Secundò. Sit A C F Parabola, cujus Axis A E, Vertex A, & latus rectum (B a). Sitque A D G Curva Geometricè irrationalis, cujus Ordinatim applicata B D secat Parabolam in C. Et vocetur Abscissa AB=y, Ordinata BD=z, Arcus Parabolicus AC=v. Sitque æquatio generalis exprimens Naturas infinitarum Curvarum irrationalium, hæc. Z=rvy^n in qua r denotat quantitatem datam & determinatam, & n exponentem indefinitum quantitatis indeterminatæ y. Dico Aream \[ \frac{ry^{n+1}xv}{n+1} - qv + \sqrt{2ay+yx} - \frac{r}{n+2xn+1}y^{n+1} = \frac{ra}{n+2xn+1}y^n + \frac{raa \times 2n+1}{nxn+2xn+1}y^{n-1} - \frac{aAx2n-1}{n-1}y^{n-2} + \frac{aEx2n-3}{n-2}y^{n-3} \] \[ \frac{C \times n-5}{n-3}y^{n-4} + &c. \] De hac serie hæc sunt notanda: (1.) Quod literæ majuscule, A, B, C, &c. denotent coefficients terminorum ipsis praecedentium. (2.) Quod si exponens n sit integer positivus aut nihilo æqualis, aut etiam si 2 n sit numerus impar, tum Quadratura exhibeatur per numerum Terminorum finitum; serie in his casibus abrupmente. (3.) Quod \( \pm q \) sit æqualis termino ultimo abrupmente. (4.) Quod ex terminis quantitatem \( \sqrt{2ay+yx} \) multiplicantes ultimò abruppens sit duplicandus. (5.) Quod omnes illæ figuræ, in quibus n est numerus integer positivus & impar, vel generalius, omnes illæ Figuræ, in quibus ultimus terminus abruppens habet signum affirmativum seu \( \pm \), habeant unam portionem Geometricè Quadrabilem, & ex ipso serie facile assignabilem, sumendo abscissam ut in not. 4. praecedentis Seriei. Exemplum 1. Sit z=v, quia in hoc casu r=1, n=0, ideo terminus ultimò abruppens est \( \frac{ra}{n+2xn+1}y^n \), unde \( \pm q = \frac{a}{z} \) (per not. 3) & quia in hoc casu \( \frac{a}{z} \) est terminus ultimo ultimo abrumpens, ideo — a est ultimus terminus in \( \sqrt{2ay + yy} \) multiplicandus (per not. 4). Adeoque \[ ABD = vy + \frac{av}{z} + \sqrt{2ay + yx} - \frac{1}{2} y - a. \] Exemp. 2. Sit \( z = \frac{vy}{a} \), quia in hoc casu \( r = \frac{1}{2}, n = 1 \), ideo terminus ultimò abrumpens est \( \frac{raa \times 2n+1}{nxn+2xn+1} \times \frac{n-1}{y} = \frac{a}{4} \), unde \( q = \frac{a}{4} \) &c. \( \frac{a}{2} \) ultimus terminus in \( \sqrt{2ay + yy} \) multiplicandus; adeoque \[ ABD = \frac{vy}{2a} - \frac{av}{4} + \sqrt{2ay + yyx} - \frac{y^2}{6a} - \frac{y}{12} + \frac{a}{2}. \] Et si capiatur \( y = \sqrt{\frac{aa}{2}} \), erit Area competens huic abscessae Geometricè Quadrabilis, scil. Area \( = \frac{1}{12} \sqrt{\frac{a^4}{2a^4} + \frac{a^2}{2}} \times 5a - \sqrt{\frac{a^2}{2}} : \) Plura habeo hujusmodi Theoremata, pro Figuris ex circulo Parabolâ & dependentibus; sed haec duo, speciminis gratia, sufficient ad ostendendum usum Methodi meæ in tractatu nostro de Quadraturis editæ, in determinandis Figurarum irrationalium Quadraturis, ad quas nulla alia (quantum scio) Methodus haætenus porrigitur. --- V. Part of a Letter of Mr. Robert Tredwey, to Dr. Leonard Plukenet, Dated Jamaica, Feb. 12. 1696, giving an Account of a great piece of Ambergriese thrown on that Island; with the Opinion of some there about the way of its Production. I shall only at present let you know the Account I received from Ambergriese Ben, for so the Man is called from the vast Quantity of that valuable Commo-