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Viviani demonftrated his conftrudion. It may be readily and ac- 

 curately done by methods familiar to geometricians before the 

 difcovery of fluxions and of the differential calculus. Yet Vi- 

 viani fays of the p^roblenfi " Cujias divinatio, a fecretis aftibus 

 " illuftrium analyftartim vigentis aevi, expeflatur, quod in geometriae 

 " pur« hifloria tantommodo verfatus ad tarn recondita videatur inva- 

 " lidus." By *' fecretae artes" there can be no doubt he defigned the 

 method of fluxions and the differential calculus (at that time in 

 manner concealed by Newton and Leibnitz) ; it therefore may 

 be fuppofed that Viviani's demonftration was not the fimpleft one 

 of which the conftrudion of his problem admits. Undoubtedly 

 the demonftration, independent of fluxions, is much lefs diflicult 

 than the problems propofed by Pafcal concerning the cycloidal 

 folids, in which the mathematicians of the middle of the fe- 

 venteenth century were fo much engaged. Euler long after- 

 wards propofed to determine a portion of a fphere, which inftead 

 of its furface fliould have its folidity accurately afllgnable. EukV 

 has given a very ingenious folution of this indeterminate problem, 

 but not one that merits the epithet of elegant, which he juftly 

 gives to the problem. His folution, being derived from the 

 method of coordinates and double integrals, is much tefs fimple 



than 



