An Account of a Book

An account of a Book. *Methodus Figurarum lineis rectis & curvis comprehensarum quadraturas determinandi* Authore J. Craige. 4to. Londini 1635. The great use of drawing the Tangents of Curve Lines, has made the most famous amongst the Modern Mathematicians endeavour to find out General Methods of finding the Tangents of Curve Lines, as may be seen from the several ways invented by Des Cartes, Morsieur Fermat, Slusius, Dr. Barrow, Dr. Willis, Tschurneyms, and Leibnitz; But as yet none has attempted to invert this problem generally, that is, having the Tangent to find the Curve Line whose tangent it is. Therefore the Author of this Treatise perceiving that the doing of this would give a General Method of determining the Quadrature of any Curvilinear space, has laid down a rule for inverting Slusius his method mentioned in the Philosophick Transactions Num. 90. He has illustrated his Method of Quadratures by several Figures which have been already considered by Geometers. As for the Circle & Hyperbola, he asserts that their indefinite Quadratures are impossible, and therefore in these & such like cases, he expresses the Area by an infinite Series, which is easily done by his Method, except the Series consist of irrational terms, for in these he has recourse to Leibnitz's method of finding Tangents, where the Calculation will be more tedious. By his resolving the Area of the Hyperbola into an infinite series, he comes to the same expression with that of N. Mercator: And in measuring the Zone of a Circle, his expression falls in with that invented by Mr. Isaac Newton, as Mr. David Gregory relates in his Treatise. He has subjoined a Method of measuring the Curve Superficies made by the rotation of any Curve upon its Axis, with a small Animadversion on the Method of Quadratrices, published in the Acta Lipsiensia Eruditorum of October, 168. Since the Publication of this Treatise, the Author is pleased to make the following Addition. Z Addi Additio ad Methodum Figurarum Quadraturas Determinandi. Autore Johanne Craige. Quoniam omnium Figurarum Quadratura ex perfecta nostri primi problematis Solutione determinantur; propterca utile judicabam nonnulla adiere, qua solutionem meam non modo plenius illustrant sed omnino perficiunt: non alco tamen sunt obscura, quin facile quisquam in istiummodi reus ve satus, ex iis qua jam exposui, omnia suppleere possit. Problema sic se habet. Data expressione Analytica linea inter ordinatam & Curva perpendiculari designatae, Invenire equationem naturam illius Curva desinientem. Hoc problema tres casus includit. 1. Cum expressio istius lineae talis est, qualis a vulgaribus tangentium Methodis exhibetur. 2. Cum ad simpliciorem reducitur, fasta divisione numeratoris & denominatoris per communem simplicem divisorem. 3. Cum expressio fit simplicitissima, dividendo per divisorem compositum. Duos priores casus Regula, prout eam explicui, universaliiter comprehendit; superest tantum ut ostendam quo pacto tertium pariter casum comprehendat. Postquam expressio data per y multiplicatur, apponantur omnes termini qui sub maximo continentur (Terminorum magnitudinem e dimensionibus quantitatis y mensurans) & connectantur signo affirmativo vel negativo, prout libuerit; adaquentur omnes illi termini (prius in coefficientes incognitas multiplicati) Quadrato quantitatis per x designatae: eritq; inde resultans aquatio quaestia, vel quaestam includet; & determinationes coefficientium terminos equationem constituentes a reliquis distinguunt. Sit in apposito schemate abscissa AM = y, ordinata MC = z, & Curva ACE proprietas \( z^2 = \frac{a_2y^2 + y^4}{p^2} \), & invenienda sit quadratura Area a lineis rectis & illa Curva comprehensa. Querenda est alia. Curva A G H, in qua PM = \( \sqrt{\frac{a_2y^2 + y^4}{p^2}} = z \); ubi P G Curve quaestae perpendiculararem, & MG = x illius ordinatam denotat. Cumq; hac expressio linea PM in y multiplicata continet sextim quantitatis y dimen- sionem, ideo appono omnes terminos sub illa sexta dimensione contentos, unde resultans equatio est. na6 + ma5y + la4y2 + ha3y3 + ka2y4 + gay5 + fy6 = x4. \[ PM = \frac{ma5 + 2la4y + 3ha3y2 + 4ka2y3 + 5gay4 + 6fy5}{4p\sqrt{na6 + ma5y + la4y2 + ha3y3 + ka2y4 + gay5 + fy6}} \] \( = \sqrt{\frac{a_2y^2 + y^4}{p^2}} \) auferantur fractiones & Signa radicalia, & deter- minentur coefficientes n, m, l &c: (ut in prob: 2 tractatus nos- tri:) ijectis his quarum determinationes absurdum involvunt: & cetera, in quibus nihil tale contingit, equationem constituent. Sic in exemplo proposito, erit f = \(\frac{4}{9}\), k = \(\frac{12}{9}\), l = \(\frac{12}{9}\), n = \(\frac{4}{9}\). sed dum g determino, invenio 24og = 144g, quod absurdum involvit; & sic p o a comparatio erit 16h = 16h unde nullus illius valor, & pro m erit 4 m = 4 m, quod itidem est absurdum: quapropter ter- mini a quantitatis g, h, m affecti ad equationem non pertine- bunt; unice reliqui a literis n, l, k, f affecti equationem naturam Curva definitem constituent. sc. \(\frac{4a6 + 12a4y2 + 12a2y4 + 4y6}{9p^2} = x4\) \[ x_4, \text{ adeoq}; AMC = \sqrt{\frac{a^6 + 3a^4y^2 + 3a^2y^4 + y^6}{6p^2}} = \frac{x^2}{2}. \text{ Exem. 2} \] Sit Curva Linea ACE talis proprietas \( z^2 = \frac{qy^2 + y^3}{q} \) & invenienda sit Quadratura spatii AMC. Querenda est Curva AGH in qua sit PM = \(\sqrt{\frac{qy^2 + y^3}{p}} = Z\) & quoniam hic valor in y multiplicatus continet quintam quantitatis y dimensionem, apponantur omnes termini suae illa quinta dimensione, & sequitur quadrato quantitatis per x designata; unde aquatio resultans est. \[ nq_5 + mq_4y + lq_3y^2 + kq_2y^3 + hq_1y^4 + fy_5 = x_4. \text{ atq; sola coefficientis (m) determinatio absurdum involvet, eruntq; reliquae, } \] \[ n = \frac{64}{225}, k = \frac{16}{15}, h = \frac{6}{45}, f = \frac{16}{15}, \text{ unde aquatio curvam quasi tam definiens est. } \] \[ \frac{64q_5}{225p} + \frac{16q_3y^2}{15p} + \frac{16q_2y^3}{45p} + \frac{16q_4y^4}{15p} + \frac{16y_5}{25p} = x_4 \text{ adeoq; } \] \[ AMC = \sqrt{\frac{16q_5}{225p} - \frac{4q_3y^2}{15p} - \frac{4q_2y^3}{45p} + \frac{4q_4y^4}{1p} + \frac{4y_5}{2p^2}} = xx \] Exem. 3. Inuenienda sit Quadratura spatii AMC, definita natura Curva ACE hac Equatione \( z^2 = \frac{2a}{4y + 4a} \). Queratur alia Curva AGH, in qua PM = \(\sqrt{\frac{a^3}{4y + a}} = z\). Ex premissis constat Equationem primum fore \[ na_y + ma_4y + 16a_5 = 4x \] & determinationes Coefficientium \( n = 1^4, m = 32, l = 16 \). Quibus substitutis, erit aquatio \[ \frac{1a_3y + 1a_2y + 16a_5}{4y + 4a} = x_4 \] \[ = a_3y + 4a_4: \text{ adeoq; AMC} = \sqrt{a_3y + a_2} = \frac{x^2}{2}. \] Notatu dignissimum est, has tres (sic ut infinitas alias) Quadraturas abscisse AM (sive y) non convenire. Quoniam in istiusmodi modi Figuris, simplicissima Area expressio huic portioni non respondet: attamen Quadratura abscisse conveniens exinde parvo labore deducitur. Ut in Exem. 3. ut Area est $\sqrt{a_3y+a_4}$; fiat $y = 0$, erit Area $\sqrt{a_4} = a_2$, & subducatur hoc ex generali; proveniet Quadratura portionis abscissa respondentis, sc. $\sqrt{a_3y + a_4 - a_2}$. Quam observatiunculam mihi primus significavit Vir celeberrimus D. Isaacus Newton. Tentetur jam idem processus in Circulo ACE, cujus diameter sit r, ac proinde $Z = \sqrt{ry-y^2}$, Quarenda est Curva AGH in qua $PM = \sqrt{ry-y^2} = z$, sed ex dictis constat equationem primam fore $mr^4 + mr^3y + hr^2y^2 + hry^3 - ky^4 = x^4$: & singula coefficientium determinationes crunt impossibiles; adeoque nulla datur Curva AGH in qua $PM = \sqrt{ry-y^2}$, ac proinde Circuli Quadratura indefinita est impossibilis. Fieri tamen potest ut sit aliqua hujusmodi Curva AGH, sed ex earum numero, quas post Cartesium Mechanicas Geometrae communiter appellant: sed quia harum usus non libenter admittunt Mathematici, praestat hujusmodi Quadraturas per series infinitas exhibere. Benevole Lector Ob inopiam Typorum Numeraliun minusculorum, qui ad designandas quantitatum potestates supra Symbola dextrorsum apponi solent, festinante prato, Typographus paulo majoribus usus est in eadem linea immediate sequentibus; ubicunque itaque offenderis a3, vel x2, &c. cubum vel quadratum, &c. e quantitate, cui suffigitur numerus, intelligas. LONDON, Printed by Joseph Streater, and are to be sold by Samuel Smith, at the Princes Arms in St. Pauls Church Yard.