3l6 PHILOSOPHICAL TRANSACTIONS. [aNNO l66g. 



parage our author, but to take ofF the prejudice which he may beget in his 

 readers against the method of indivisibles, which has been owned by other 

 famous men, besides those already named, viz. by Mengoli, who from the ex- 

 cellencies of this method, Archimedes' method, and Vieta's Specious Algebra, 

 composed his Geometria Speciosa; by Antimo Farby, alias Hon. Fabri, in 

 tract De Linea Sinuum et Cycloide; by Pascal, alias Dettonville; by Descartes 

 himself, vol. 3 of Letters, who says, that by it he squared the Cycloid ; and 

 lately by the excellent Sluse, &c. 2. To remove the other prejudice that may be 

 against this author as defective: for the 5th book Cylindricorum et Annularium, 

 now printed with the rest, the prefacer asserts to be first extant in 1659. The 

 author divides this fifth book into six parts : In the first he demonstrates, that 

 if any plane surface have a rotation about its axis in any situation whatsoever, 

 and at any distance whatsoever or none, it produces a round solid equal to an 

 upright solid, whose base is the generating figure, and height equal to the cir- 

 cumference described by its centre of gravity. This universal rule was invented 

 by Guldin, and is the basis of most of his doctrine ; but he could not demon- 

 strate the same, though it was much desired. In like manner, if any perimeter 

 have a rotation about its axis, in any situation whatsoever, it generates a round 

 surface, equal to a right surface, made by the same perimeter as a base 

 (which may be evolved and made a plane surface) whose height is the way 

 or circumference described by its centre of gravity. These being two ad- 

 mirable universal rules in geometry, the reader will find the same (with many 

 others) demonstrated by Dr. Wallis in his treatise De Calculo Centri Gra- 

 vitatis, which, together with his other tracts, De Motu, Statica, Me- 

 chanica, are now at the press in London. The same rules are likewise de- 

 monstrated in Geometriae pare Universali Jacobi Gregorii Scoti, Patavii, 1668. 

 The methods of these learned men are different, and good arguments might be 

 given, that they have not communicated nor seen the works of each other. 



Guldin, 1. 1, c. 12, shows a mechanical way to find the centre of gravity of a 

 surface or curved line, by two free suspensions, from the points of which, per- 

 pendiculars being drawn, do cross each other at the centre of gravity. This we 

 mention to keep the reader from taking the centre of gravity of a curved line as 

 such, which is intended in this second rule, to be the same with the centre of 

 gravity of the figure thereby terminated in the first rule. 3. Considers the 

 affections of round solids, generated from a parabola, in ten propositions ; where- 

 of the 21st and 23d give the hoof, required by Angeli, which was formerly 

 cubed by Greg, de S. Vincentio. In the 27th proposition he gives the proportion 

 of the parabolical conoid to the spindle made of the same parabola by rotation 

 about its base, to be as the base of the parabola is to -ff of the axis ; showing. 



