392 THE SCIENTIFIC MONTHLY 



Their clumsy notation for numbers handicapped them in arithmetic, 

 but they knew such properties as that the difference of the squares of 

 two consecutive numbers is always an odd number, they defined pro- 

 gressions of different kinds, and they had some acquaintance with irra- 

 tional numbers. This was perhaps the first school which devoted itself 

 to investigation for its own sake without any special reference to pos- 

 sible applications to physical problems. 



Then followed the golden age of Greece with Hippocrates of Chios, 

 Euclid, Archimedes, Appolonius, Hipparchus, and a number of others 

 less well known. Plato and Aristotle must also be included for their 

 contributions to the forms of logical reasoning which should be 

 adopted, and Plato in particular contributed in this respect to an extent 

 which has lasted until our own times. Euclid, whose personality is de- 

 cidedly nebulous, wrote a text book on the geometry of the line and 

 circle by which most mathematicians for the next two thousand years 

 were introduced to the subject: it is indeed only during the last few 

 decades that it has been replaced by modern texts. Appolonius did 

 much the same thing for the curves of the second degree — the ellipse, 

 parabola and hyperbola. While we know little of Euclid's own con- 

 tributions as a discoverer, it is fairly certain that Appolonius had not 

 only mastered all that was previously known, but greatly extended that 

 knowledge himself. But of all those who lived up to the time of Isaac 

 Newton, there can be little doubt that Archimedes is the chief. He is 

 recognized as the founder of mechanics, theoretical and practical. His 

 work on the lever alone entitles him to fame: "give me a fulcrum 

 and I will move the world." He initiated the sound study of hydro- 

 statics, advanced geometry by discovering how to find the area of a 

 sphere and that cut off from a parabola by a straight line. He also dis- 

 cussed spiral curves, finding many of their properties. In arithmetic 

 he seems to have had methods for dealing with very great numbers 

 similar to those we now use by the index of the power of ten which 

 can represent the number. These brief statements are only smbolic of 

 what he achieved. His activities were very extended, and he preferred 

 the modern custom of writing essays, or memoirs as we now call them, 

 rather than treatises or text books. The tradition runs that he was 

 killed at the fall of Syracuse to the Romans because, absorbed in a 

 geometrical diagram, he insulted the Roman soldier who was spoiling 

 it. 



Hipparchus was mainly an astronomer, and indeed one of the 

 founders of the art, and it was in the course of his work that he initiated 

 Trigonometry as an aid when angles had to be measured as well as 

 lines. 



During the following centuries up to the Fall of Rome in A. D. 

 476, there were no great advances, the principal name of the period 



