FRANCESCO ( 1598-1647 ). _\n Italian mathemati cian and astronomer. lie was educated at Pisa, and was a pupil of Castelli. In 1629 he was made professor at Bologna, where he died. llis chief contribution to mathematics is the method of indivisibles. first conceived in 16•9 and pub lished in 1635. method forms a connecting link between the Greek method of exhaustion,. :171(1 the methods of Newton and Leibnitz. The basal idea of the method consists in considering a line as composed of a series of points (or small line-segments of equal length), a surface as com posed of a series of adjacent lines (or strips of area of equal width), and a solid as composed of a series of planes (or laminae of equal thickness). In general, however, a summation of such ele ments, if they are finite (no matter how small), only approximates, but does not equal, the length, area, or volume of a given magnitude. E.g. consider a triangle as composed of a series of very narrow rectangles constructed on its base; the sum of such elements will differ the less from tile area of the triangle, the smaller the width of the rectangles; but as long as that width remains a finite quantity, the difference in area will, evidently, likewise remain finite. Nevertheless, with the aid of limits, the method may be used to deter mine the ratio of the area of a given figure to that of an other figure whose area is known. In fact, it was thus actually employed for measuring areas and volumes for more than half a century before the introduction of the integral calculus. For example. it was used to prove t h e proposition that two solids ly ing between two parallel planes, and such that the two sections made by any plane parallel to the given planes are equal, are themselves equal: as. for example, S
and S' in the ac companying Fig. I. Such solids are called Cavalieri bodies. This forms one of the best bases for proving that the volume of a sphere is ar for, ,as may be seen from Fig. 2. the area of the ring CD is easily shown to be 7r and this is also easily shown to be the area of the circle AB. Renee the sphere and the difference between the one and cylinder are two Cavalieri bodies. :nut are therefore equal. Hence r = r r 2 r—w gr= This method solved many ditlieult !problems and enabled Cavalieri to a satisfactory demon stration of 1:iddin's theorem. published in the Exereitafioncs Utoinetrio• sex (1647). By Ill•:111s of it. also, Torricelli proved that the area of a cycloid (q.v.) is three times the area of the circle. Since the method of Cavalicri, combined with the modern theory of limits, offers an easy and correct way of expressing the areas and volumes of several elementary forms, and since it is a natural stepping-stone to the meth ods of integral calculus, it is desirable material for elementary instruction. Cava lieri's chief are: 17, ono trio ladirisibilibus Continuo rum Nora Quadam Rations Promota (Bologna, 16:35) ; Excreitationes Geotne:ri( St.:, (Bologna, 1647): Specchio ustorio ovr, ro trattato dells set tioni coniehe (Bologna, 1632) ; and Trigononte trio Plana et Sphcrica (Bologna, 1635).