Page:Philosophical Transactions - Volume 003.djvu/255

 tamen cauteque usurpata) acutissini Cavalerii; per Indivisibilium doctrinam nobis amicissimam.

And when thus carefully, to apply it, of that see Lalovera's Elementa Tetragonismica Tolosæ 1651, where more Archimedo he demonstrates the truth of this Method: which Book if Angeli had seen, he would certainly have quoted it, and admired the Author.

For want of this Method, it was, saith Angeli, by way of complaint, of Tacquet, that he omitted some Theorems, which by aid thereof he might easily have found out. See him in his Preface to his Infinite spirals; but especially at Schol. 3. Prop. 15. l. 2.

Siergo Tacquet recepisset doctrinam Cavalerii, potuisset non solum Cubare portionem Cylindrici Parabolisi super quæcunque Infinitarum Parabolarum per Basin Parabolæ & Punctum in latere; sed etiam ex iis, quæ in Exercitat. 4. Cavalerii tradunt ipse & Beaugrand, potuisset Cubare Segmenta portionis cujuscunque Cylindrici Parabolici reseclæ planis sectioni maximæ parallelis: Imo ex doctrinæ Cavalerii potuisset etiam Cubare, & portionem Cylindrici super Hyperbola per basin Hyperbolæ & Punctum in latere, & segmenta hujus portionis resectæ planis sectioni maximæ parallelis (supposita tamen Hyperbolæ Quadratura.)

Angeli finds afterwards another deservedly famous Man, viz. Dr John Wallis, owning and using the Method of Indivisibles, and advancing it to admiration in his Arithmetica Infinitorum; who in his Book de Cycloide at Oxford 1659, saith thus, Pag. 9.

Supponimus enim (quodet facile, si opus est, probabitur) Planum quodvis tantundem hujusmodi Conversatione (seu Rotatione) producere, quantum est quod fit ex eodem Plano in lineam ipsius Centro grsvitatis, descripta ducto; quod & de linea quavis sive recta sive curva, in eo Plano descripta, pariter intelligendum est: Quod quidem enim ipse olim me primum invenisse putaverim, monitus moxeram, nonnihil apud Guldinum ''exstare quod huc spectet. Id autem si animadvertisset Tacquteus, dum de Cylindricis & Annularibus acutum Opus conscripsit, non patum illi fuisset adjumento, multique quæ illic extant, tum Universalius tum contractius forte faissent edita.''

All which is not recited here, to disparage our Author, but to take off the prejudice, which he may beget in his Readers against the Method of Indivibles, which hath been owned by other famous Men, besides those already recited; viz. by Mengolus, who from the Excellencies of this Method, Archimeds Method, and Vietas Specious Algebra, compos'd his Geometria Speciosa; by Antimo Farby, alias (as 'tis suggested) Hon. Fabri in Tract. De Linea Sinuum & Cycloide; by Pascal, alias Dettonville; by Des Cartes himself Vol. 3. of Letters, who saith, that by it he squared the Cycloid, and lately by the excellent Siuius, &c. 2. To remove the other prejudice that may be against this Author as defective: for the 5th Book Cylindricorum & Annularium (now printed with the rest) the Prefacer asserts to be first extant in 1659. And because we presume, the rest of these Books are already known and common, and that this hath not formerly been expos'd to sale in England; and because also it supplies and compensates those defects, we think fit to acquaint the Reader with the Argument thereof. The Author divides this Fifth Book into six parts:

1. In the first he demonstrates (in 6 Lemma's and 9 Propositions) That, if any Plain Surface have allocation about its Axis in any Situation whatsoever, and at any distance whatsoever, or none, it produceth a Round Solid equal to an Upright Solid, whole Base is the begetting Figure, and Height is equal to the Circumference described by its Center of Gravity. (This Universal Rule was invented by Guldin, and is the Basis of most of his Doctrine; but be could not demonstrate the same, though 'twas much desired.)

2. In like manner, If any Perimeter have a Rotation about its Axis in any Situation whatsoever, it begets a round Surface, equal to aright Surface, made by the same Perimeter as a Base (which may be evolv'd and made a Plain Surface) whole height is the way of circumference described by its Center of Gravity. This by 5 Lemma's and 10 Propositions. Rh