traced upon the Surface of the Sphere. 328 



both numerical and geometrical, we might not have been at this day in 

 possession of his three celebrated laws nor, therefore, of the theory of gra- 

 vitation of the Principia or of the Mdcanique Cele'ste. Our theory of 

 the Planets, and especially of the Moon, and consequently our whole art of 

 navigation, and apparatus for conducting it and hence, our commerce, and 

 our mechanical arts would have been in a much less advanced state than 

 they now are. Is it not easy to see, that upon the high degree of perfection of 

 the mechanical arts, which are produced by extended commercial relations, 

 the perfection of our philosophical instruments depends that our data for test- 

 ing hypotheses already framed, and to serve as conditions to be fulfilled by 

 future theories, depends entirely upon that excellence of workmanship which 

 a high state of commerce alone can call into being ? Where, indeed, would 

 have been our precision in all the sciences and what probability is there 

 that we should have entertained correct and legitimate views upon any one 

 of them ? It would be a curious and interesting amusement to speculate 

 upon the probable state, at the present time, of physical and practical astro- 

 nomy of the doctrines of sound and light of meteorology of chemistry 

 of electro-magnetism and even of mathematical science itself had the 

 spherical epicycloid possessed a numerous and interesting assemblage of geo- 

 metrical properties. 



There is, however, one point of view in which it possesses considerable 

 interest that individuals of this curve traced on the surface of the sphere 

 are capable of rectification. The Florentine paradox proposed to find qua- 

 drable portions of the sphere ; and it could scarcely escape the geometers 

 who considered VIVIANI'S problem, to inquire into the possibility of finding 

 curves whose contours should be equal to given straight lines. The ques- 

 tion does not appear to have been publicly hazarded till more than twenty 

 years afterward, when it was proposed by M. OFFENBERG, in the Leipsic 

 Acts for 1718. I do not find that the proposer ever offered any solution, 

 and it is probable, either that he was not in possession of one, or that he 

 afterwards discovered the fallacy of that which he supposed himself to have 

 found before he proposed the question. HERMANN, howev.r, was led, by 

 the property of the plane epicycloid being in certain cases rectifiable, to sup- 

 pose the same thing held in spherical ones, under similar circumstances ; 



