INFLUENCE OF THE PTOLEMAIC EPOCH. 179 



ical observations was admissible. Wbile the hydraulic clock 

 of Ctesibius, an improvement on the earlier clepsydra, must 

 have yielded more exact measurements of time, determinations 

 iu space must likewise have improved in accuracy, in conse- 

 quence of the better modes of measuring angles, which tho 

 Alexandrian astronomers gradually possessed, from the period 

 of the ancient gnomon and the scaphe to the invention of as- 

 trolabes, solstitial armils, and linear dioptrics. It was thus 

 that man, and step by step, as it were, by the acquisition of 

 new organs, arrived at a more exact knowledge of the move- 

 ments of the planetary system. Many centuries, however, 

 elapsed before any advance was made toward a knowledge of 

 the absolute size, form, mass, and physical character of the 

 heavenly bodies. 



Many of the astronomers of the Alexandrian Museum were 

 not only distinguished as geometrician*, but the age of the 

 Ptolemies was, moreover, a most br^iant epoch in the prose- 

 cution of mathematical investigations. In the same century 

 there appeared Euclid, the creator of mathematics as a science, 

 Apollonius of Perga, and Archimedes, who visited Egypt, and 

 was connected through Conon with the school of Alexandria. 

 The long period of time which leads from the so-called geo- 

 metrical analysis of Plato, and the three conic sections of Me- 

 nsechmes,* to the age of Kepler and Tycho Brahe, Euler and 

 Clairaut, D'Afembert and Laplace, is marked by a series of 

 mathematical discoveries, without which the laws of the mo- 

 tion of the heavenly bodies and their mutual relations in the 

 reo-iops of gpace would not have been revealed to mankind. 

 While the telescope serves as a means of penetrating space, 

 and of bringing its remotest regions nearer to us, mathematics, 

 by inductive reasoning, have led us onward to the remotest 

 regions of heaven, and brought a portion of them within the 

 range of our possession ; nay, in our own times so propitious 

 to extension of knowledge the application of all the elements 

 yielded by the present condition of astronomy has even reveal- 

 ed to the intellectual eye a heavenly body, and assigned to it 

 its place, orbit, and mass, before a single telescope had been 

 directed toward it.f 



* Ideler, on Eudoxus, s. 23. 



t The planet discovered by Le Verrier. 



