ALEXANDRIAN SCIENCE. 35 



that of the mathematician EU^LJJD (B.C. 323 283). In his well-known 

 "Elements of Geometry" he collected the truths of elementary geo- 

 metry which were known in his time, and his work became everywhere 

 among the Greeks the text-book of the science. Its use as such has 

 been continued down to the present day, when it still retains its pre- 

 eminence as the model of exact reasoning, and clear demonstration. 

 Its propositions are so arranged that each one is related to those that 

 precede it and to those that follow, forming a necessary link in a long 

 chain of demonstrations. Some moderns have supposed that by cer- 

 tain alterations they could ioiniQe upon the arrangement of Euclid, 

 but no attempt of this kind has yet been acknowledged as successful. 

 The superiority in philosophical system of Euclid's train of cfemonstra 1 

 tion is admitted even by those who have proposed to initiate youth 

 into the science of geometry by some method less severe than the in- 

 flexible logic of the Alexandrian mathematician. The work of Euclid 

 is so immediately accessible to every reader that it would be quite 

 superfluous to even mention the subjects of which it treats. But if 

 the reader should perchance be unacquainted with the Euclidean 

 geometry, we would recommend him to turn to the forty-seventh pro- 

 position of the first book of Euclid's " Elements," and observe the 

 specimen of geometrical demonstration employed in proof of that 

 theorem, in which he will recognize the discovery of Pythagoras al- 

 ready mentioned on page 13. The elegance of the demonstration will 

 be understood only by such as have made sufficient progress in the 

 science to be able to reason out the conclusion for themselves from the 

 data which are supplied in the definitions and axioms with which Eu- 

 clid sets out. But any one may directly by practical methods convince 

 himself of the truth so irrefragably established by the elaborate chain of 

 deductive reasonings. He need only construct the triangle and squares 

 on a sheet of paper, and he may then easily discover by actual super- 

 position that the equality of the areas is invariably true. Each village 

 bricklayer is almost daily taking practical advantage of the truth which 

 Pythagoras discovered and Euclid demonstrated. When required to 

 set out the rectangular sides of a building, he will tie to a peg one cord 

 of three yards length, and another of four yards in length, and when 

 he has stretched these so that their other extremities are exactly five 

 yards distant from each other, he will be confident that he has a right 

 angle between the cords. The connection between the theorem of 

 Pythagoras and the practice of the mechanic will not fail to manifest 

 itself to the intelligent reader. 



Euclid's geometry enables us to compare the areas of any two plane 

 figures bounded by straight lines. He compares also curvilinear figures 

 with each other in certain cases. The problem of comparing the areas 

 of curvilinear figures with those of rectilinear figures is a much more 

 difficult one, and one case of it has attracted more general attention 

 than perhaps any other mathematical problem that could be named. 



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