472 PHILOSOPHICAL TRANSACTIONS. [aNNO I67O, 



and the solids made by their conversion about an axe: In many of which, the 

 magnitude of the ungula or solid is shown to be but finite, where the magni- 

 tude of the respective plain (on which they stand, or by whose <:onversion they 

 be made) is infinite : and, contrarywise, in others, the magnitude of the ungula 

 or solid to be infinite, where that of the plain is but finite. 



He gives also general methods, from the centre of gravity of a plain, or of 

 lines, to find the magnitude of the ungulas and solids made by rotation about 

 assigned axes. He shows particularly, that the scalene cylindric surface, is to 

 the erect, as the perimeter of an ellipse to that of a circle. He shows the centre 

 of gravity of all arches of circles, with their superficial ungulas, and the surfaces 

 made by the conversion of such arches about assigned axes. And the like of 

 the sectors, segments, and other portions of circles, which are applicable also 

 to those of ellipses ; with the ungulas and solids made by such conversion ; and 

 their centres of gravity. He does the like in the cycloid, showing the length 

 of the curve, ^nd of the portions thereof, with their centres of gravity, and 

 their several superficial ungulas, and surfaces, made by conversion and the 

 centres of gravity of all these, &c. 



He does the like in the figure of right sines, in the figure of versed sines, 

 and of arches; assigning the magnitude of those figures, and of their segments 

 and portions; with their ungulas, and solids by conversion; and the centres of 

 gravity of all these. Whence (amongst many other things) are deduced the 

 sums of the right sines, versed sines and arches, appertaining to any assigned 

 portion of a circle; and the sums of their squares, cubes, or other powers. 



He does the like in spiral figures, as well that of Archimedes, as an infinite 

 number of other spirals; showing the magnitude of the several parts or sectors; 

 with their centres of gravity ; and the respective paraboloids. He prosecutes 

 the same in part, but more briefly, in the cissoid and conchoid, and the figure 

 of tangents ; as to the magnitude of those figures, and the parts thereof; their 

 ungulas, solids, and centres of gravity. 



He shows also the quadrature of the hyperbola, and parts thereof; their 

 ungulas, solids, and centres of gravity: As also an hyperbolical solid, made by 

 the conversion of a straight line about an axis not in the same plain; showing 

 the magnitude of that solid, and of its parts, and their centres of gravity ; and 

 the several sections of that solid made by plains in any assigned position ; being 

 parabolas, hyperbolas, ellipses, circles, parallelograms, and triangles; accord- 

 ing to the different positions of the cutting plains, &c. 



II. Exercitationes Mechanicae, Alexandri Marchetti. Pisi, 1669, in 4to. 



This author declares, that though many eminent men have already treated of 

 the subject of his book, as Aristotle, Archimedes, Lucas Valerius, Guldinus, 



