TOL. XVIII.] PHILOSOPHICAL TRANSACTIONS. 600 



exhibited, he sunk under the oppression of those cruel symptoms. The body 

 was not suffered to be opened. But the abdomen was perceived to be excessive- 

 ly inflated, his limbs convulsed, and the surface of the body of a livid colour ; 

 the muscles of the face were contracted into such a form as nearly represented 

 a spasmus cinicus. 



Solution of the Florentine Problem, concerning the Testudo Feliformis Quadra- 



bilis* By David Gregory, M. D. and F. R. S. N° 207, p. 25. Translated 



from the Latin. 



The author of this enigmatical problem has now given an ingenious and easy 

 construction of it, in an Italian treatise on the formation and measurement of 

 all vaults and cupolas, dedicated to the Grand Duke of Tuscany; and there 

 giving the initials of his name V. V.-f- the last disciple of Galileo. But here the 

 author changes the enigma into the following problem: "On the surface of a 

 hemisphere to assign a portion equal to a given square." Which problem he 

 thus constructs : 



Let a sphere, whose axis is equal to the side of the given square, be denoted 

 by the circle ACBD (fig. 7>pl- l-i) which is vertical in the proposed sphere, 

 the horizontal diameter being AB, and the centre E. Let the sphere be per- 

 forated by two right cylinders, whose sections with the plane ACBD are the 

 circles AHEI, BLEG, on the diameters A E, EB: then the thing is done; 

 that is, from any hemisphere, as for instance the upper ACB, four bilinear 

 figures are taken away by the perforating cylinders, two on the anterior side, 

 and two on the posterior, which are similar and similarly posited, so as that the 

 remaining hemispherical superficies is equal to the square on the line AB. 

 And because the hemispherical superficies, when the said 4 bilinear spaces are 

 taken away, resembles a sail filled and extended by the wind, and also a hemi- 

 spherical cupola admitting light by 4 windows, which being placed on a circular 

 base AEB, rests on it at the points A,E, E, B, this he calls " the quadrable 

 Florentine and veliform cupola." The author then delivers several things re- 



* Se« N" 196, p. 479, of this volume of the Abridgment. 



f Vincent Viviani, mathematician to the Grand Duke, was born at Florence in 1()22, and died in 

 1703, in the 82th year of his age. He was an excellent mathematician, and particularly excelled in 

 the branch of the ancient and pure geometry, on which he left some ingenious specimens : viz 

 ]."DeMaxirais et Minimis Geometrica Divinatio; &c. in fol. l659;" in which Viviani not oolr 

 guessed what ApoUonius had written, in a work of his which is lost, but also extended the subject 

 much further. 2. " Enodatio Problematum universii Geometris propositorum ^ Claudio Com- 

 miers, in 4-to, l677." 3. " De Ix)ci8 Solidis secunda Divinatio Geometrica in quinque libios in- 

 juria temporum amissos Aristea senioris Geometric, in fol, 1701:" a work full of deep rem rks on 

 conies, &c. 



VOL. III. 4 I 



