[ 5-^5 I 



than what a confideration of the properties of the fphere Would 

 have furnifhed. By proceeding upon too general principles the 

 eafieft cafe efcaped him. This cafe has lately been difcovered and 

 announced in the fecond mathematical volume of the French Nati- 

 onal ilnftitute, by.M. Boffut, from which it appears that Viviani's 

 conflruclion of- the Florentine problem determines at the fame 

 time a portion. of the furface of the fphere, and a portion of its 

 folidity, both exadly affignable. M. Boffut tells us, that the 

 analyfis of his folution is much more difficult than that of Vi- 

 viani's conftru£lion, and promifes to communicate it hereafter, 

 lath© mean time, Mr. Woodhoufe has given a very ingenious 

 demonflration of it in the fir ft part of ' the Phil. Tranf. for 1801. 

 This demonftration, however, being performed by the method 

 o£ coordinates and double integrals, is lefs fimple than the prob- 

 lem admits. 



What feems accidental in Viviani's conftru£lion will take place 



in an innumerable variety of inftances ; and it is the objed of 



the prefent Eflay to ihew that " there are innumerable conflruBions 



by which portions of a fphere may be obtained fo that the fpherical 



fuperfcies and folidity of each portion are accurately affignable. 



3 T 2 Lemma. 



