182 LECTURE XX. 



However erroneous the opinion may be, that Pythagoras was acquainted 

 with the laws of gravitation, it is certain that he made considerable im- 

 provements both in mathematics and in mechanics, and in particular that 

 he discovered the well known relation between the hypotenuse and the 

 sides of a right angled triangle, and demonstrated that the square of the 

 hypotenuse is always equal to the sum of the squares of the sides. This 

 theorem is more essential to the perfection of geometry than any other pro- 

 position that can be named ; and if we may judge by the story of his 

 having sacrificed a hecatomb to the Muses on occasion of the discovery, he 

 seems to have had a foresight of the magnificence of the edifice that was in 

 subsequent times to-be built on this foundation. 



Democritus of Abdera lived about a century after Pythagoras, whose 

 works he studied and whose principles he adopted. He appears to have 

 been possessed of very extensive knowledge and profound learning ; but 

 little remains of his works excepting their titles. Some have attributed to 

 him the invention of the method of arranging stones so as to form an arch. 

 Seneca thinks that so simple an invention must have been practised in 

 earlier ages : but Mr. King has endeavoured to show that its general intro- 

 duction in building was of much later date. Architecture and other 

 mechanical arts had however been considerably advanced some time before 

 this period, if it is true that Ctesiphon or Chersiphron, who built the 

 temple of Ephesus, was contemporary with Croesus and Thales. It is un- 

 certain at what time bridges of stone were first built ; and it is doubtful 

 whether the art of building bridges of wood was very well understood in 

 those ages : for according to Herodotus, it was commonly believed that 

 Thales avoided the necessity of procuring a passage over the Halys for the 

 army of Croesus, by encamping them on its banks, and cutting a channel 

 for the river in their rear, although the historian himself is of opinion, that 

 they passed over bridges which already existed. Curtius speaks of a bridge 

 of stone over the Euphrates at Babylon, which appears to have been built 

 long before the time of Alexander, whose expedition he relates ; and it is 

 scarcely probable that a stone bridge could have withstood the impulse of 

 so rapid a river, if it had been supported by columns only, without arches, 

 since they must have left too small a space for the passage of the water. If 

 however, we may believe Herodotus, whom Mr. King has quoted, this was 

 in reality a kind of drawbridge. According to this author, it was built by 

 Nitocris, the immediate successor of Semiramis : the stones were united by 

 iron and lead, and beams were laid across them which were removed at 

 night, in order to prevent the mutual depredations of the inhabitants of dif- 

 ferent parts of the city. We are informed by Pliny that Ctesiphon lowered 

 his large blocks of stone by placing them on heaps of sand bags, and 

 letting out the sand by degrees ; it does not appear how he raised them, but 

 the inclined plane seems to afford the simplest and most obvious method. 



Archytas of Tarentum and Eudoxus of Cnidus were also Pythagoreans. 

 They were the first that attempted to make the mathematical sciences 

 familiar by popular illustrations ; and Archytas is said by some to have 

 invented the pulley and the screw. They lived nearly 150 years after 

 Pythagoras, and geometry had made in the mean time very rapid advances, 



