Elements. The compilation became popular almost immediately 

 in Alexandria, and, for long centuries, formed the basis of all geo- 

 metry. It is interesting however to note that what are called in 

 our text books axioms were called by Euclid "common notions" 

 and that, though there has been much controversy upon the point, 

 the "common notions" of Euclid, himself, seem to have been not 

 twelve in number but ten. The axioms about right angles and paral- 

 lels were probably placed by Euclid among the postulates, a class 

 to which they better belong. One postulate, as mathematicians 

 have recently pointed out, the compiler of the Elements certainly 

 overlooked entirely, though he frequently made use of it in his proofs, 

 namely that of superposition, according to which figures can be 

 moved about in space without any alteration in form or magnitude. 

 The work contained in Euclid's Elements had a tremendous influ- 

 ence upon succeeding generations. Geometricians have not been 

 alone in their admiration for the results achieved and the apparent 

 infallibility of the method used, and it is only in comparatively 

 recent times since the writings of Lobatchewsky, 1793-1856, that 

 further developments along the line of a non-Euclidean geometry 

 have detracted somewhat from the interest attaching to this great 

 work. But the mathematics of the Greeks had not yet reached 

 its zenith, for, shortly before the death of Euclid, there was born, 

 most probably of very humble parents, the greatest mathematician 

 of antiquity, as he has been called, Archimedes of Syracuse. The 

 fame of this splendid scholar depends not only upon his achieve- 

 ments in geometry but also upon his attainments in mechanics and 

 arithmetic, and his success in mechanical inventions. These latter 

 Archimedes himself is reported to have belittled. He prized far 

 more highly his discoveries in what some would call pure science, and 

 those which were by him considered most valuable are contained in 

 his "Sphere and Cylinder". "In it are proved the new theorems, 

 that the surface of a sphere is equal to four times a great circle; 

 that the surface of a segment of a sphere is equal to a circle whose 

 radius is the straight line drawn from the vertex of the segment to 

 the circumference of its basal circle ; that the volume and the sur- 

 face of a sphere are two-thirds of the volume and surface respec- 

 tively of the cylinder circumscribed about the sphere. Archimedes 

 desired that the figure to the last proposition be inscribed on his 



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