94 A SHORT HISTORY OF SCIENCE 



"(5) In general the one-sided emphasis on the logical makes 

 it difficult to understand the subject-matter as a whole, and its 

 internal relations." 



The Elements of the great Alexandrian remain for all time the 

 first, and one may venture to assert, the only perfect model of logical 

 exactness of principles, and of rigorous development of theorems. 

 If one would see how a science can be constructed and developed to 

 its minutest details from a very small number of intuitively per- 

 ceived axioms, postulates, and plain definitions, by means of rigorous, 

 one would almost say chaste, syllogism, which nowhere makes use 

 of surreptitious or foreign aids, if one would see how a science may 

 thus be constructed, one must turn to the Elements of Euclid. 

 - Hankel. 



Euclid always contemplates a straight line as drawn between two 

 definite points, and is very careful to mention when it is to be pro- 

 duced beyond this segment. He never thinks of the line as an entity 

 given once for all as a whole. This careful definition and limitation, 

 so as to exclude an infinity not immediately apparent to the senses, 

 was very characteristic of the Greeks in all their many activities. 

 It is enshrined in the difference between Greek architecture and Gothic 

 architecture, and between the Greek religion and the modern religion. 

 The spire on a Gothic cathedral and the importance of the unbounded 

 straight line in modern geometry are both emblematic of the trans- 

 formation of the modern world. - - Whitehead. 



The universally admired perfection of the work of Euclid is re- 

 vealed to the historians as the natural product of a long criticism 

 which was developed in the constructive period of rational geometry, 

 from Pythagoras to Eudoxus. Then commenced to appear the signifi- 

 cation of those methods and principles by means of which the Greeks 

 themselves attempted to interpret and conquer the paradoxes con- 

 cerning infinity. These are the same difficulties which reappeared 

 at the time the infinitesimal calculus was founded, and are now again 

 asserting themselves in the most refined analysis. Enriques. 



OTHER WORKS OF EUCLID. - - Besides the " Elements " Euclid 

 wrote several other mathematical treatises, including one on 

 Porisms, a special type of geometrical proposition; and one on 

 Data, containing such theorems as the following : 



