218 Proceedings of Philosophical Societies. [Makch, 



of the inclinations, which was beyond the geometry of Euclid, 

 but it also supplies the most simple relation to determine the 

 nature and the number of the regular solids which can be 

 inscribed in the same sphere; so that a single triangle, a single 

 formula, serves for all. This will be demonstrated by one of us 

 in the ' History of Modern Astronomy,' at the Article Kepler, 

 who wished to prove, by means of the five regular solids, that no 

 other planets than those known from time immemorial could 

 exist. 



" Another observation, not less curious or new, is, that the 

 general trigonometrical expressions (the most expeditious that 

 can be imagined for logarithmic calculation) may be transformed 

 with wonderful facility into those irrational expressions which 

 the. Greeks call major, minor, and cpofomc. In fact all the 

 primitive angles are of 30°, 36°, 45°, 54°, 60°, and 90°, whose 

 trigonometrical lines are irrational, and lead directly to the con- 

 structions of Euchd and of Isidorus. The consequence of this 

 is, that the unknown parts of each problem may be expressed at 

 our option by the sines, cosines, and tangents, either of the arc 

 or of its half, so that we have always six dirferent expressions 

 for every one, and among so many expressions, we may always 

 select the most convenient ; the calculation is also shortened 

 still more by the cii'cumstance that there is scarcely one of these 

 quantities which is not to be found again in another of these 

 solids ; so that there are never more than four calculations in 

 the whole to make for the 15 unknown quantities of the general 

 problem. In this way, after having completed and simplified 

 the constructions of Euclid for the five edges, we have 

 succeeded in forming more easy and uniform constructions than 

 those of Isidorus, in rectilinear isosceles triangles, whose 

 common basis is the diameter of the sphere. 



" We, therefore, think we may differ in opinion from the trans- 

 lator, and consider the two Books of Hypsicles as a curious 

 remnant of the ancient geometry, inasmuch as they contain 

 notions that are not to be found elsewhere. The most important 

 points are, to obtain true theorems, and faultless constructions. 

 As for the demonstrations, they likewise are of importance, 

 doubtless; but should we be dissatisfied with them, we may, 

 without much difficulty, find others. The principal defect of 

 those of Hypsicles has already been mentioned ; it is, that the 

 first half is in every one of them quite useless. 



" It is true that the demonstration of the second proposition in 

 the second Book was perfectly unintelligible ; but we may be 

 allowed to conjecture that only the copyists are to blame. 

 M. Peyrard has given a new one, which may probably have 

 been originally the author's. There is also a demonstration of 

 Euclid which all commentators had agreed to look upon as 

 altered, or as quite deficient. It is one of the most important 

 propositions in the book of data, and may be reduced to aa 



