Febkuary 3, 1905.] 



SCIENCE. 



183 



voted to their work; and the members of 

 the scientific staff have carried out most 

 faithfully their duties of preparing and 

 preserving the collections thus far made. 



We hoped to be docked at Callao, but 

 owing to the prolonged occupation of the 

 dock by a disabled steamer and the uncer- 

 tainty of its becoming free within reason- 

 able time, we decided to proceed without 

 further delay to Easter Island and con- 

 tinue the expedition as we are. 



AijExander Agassiz. 



SCIENTIFIC BOOKS. 

 Rational Geometry. By George Bruce Hal- 



STED. New York and London, John Wiley 



and Sons. 1904. Pp. viii + 285. 



For over two thousand years there has been 

 only one authoritative text-book in geometry. 

 ' No text-book,' says the British Association, 

 ' that has yet been produced is fit to succeed 

 Euclid in the position of authority ! ' There 

 is, in fact, little improvement to be made in 

 Euclid's work along the lines which he adopt- 

 ed, and among the multitude of modern text- 

 books, each has fallen under the weight of 

 criticism in proportion to its essential devia- 

 tion from that ancient author. 



This does not mean that Euclid is without 

 defect, but starting from the discussion of his 

 famous parallel postulate, the modern develop- 

 ment has been in the direction of the exten- 

 sion of geometrical science, with the place of 

 that author so definitely fixed that the system 

 which he developed is called Euclidean geom- 

 etry, to distinguish it from new developments. 

 The defects of Euclid arise out of a new view 

 of rigorous logic whose objections seem finely 

 spun to the average practical man, but which 

 are based upon sound thought. The key to 

 this modern criticism is the doubt which the 

 mind casts upon the reliability of the intui- 

 tions of our senses, and the tendency to make 

 pure reason the court of last resort. Thus, 

 the sense of point between points, the percep- 

 tion of greater and less and many other tacit 

 assumptions of the geometrical diagram, are 

 the vitiating elements on which modern criti- 

 cism concentrates its objections. 



As an evidence of the ease with which the 

 senses can be made to deceive, take a triangle 

 ABC, in which AO is slightly greater than 

 BC. Erect a perpendicular to AB at its 

 middle point to meet the bisector of the angle 

 C in the point D. From D draw perpen- 

 diculars to AC, BC, meeting them respectively 

 in the points E, F. Let the senses admit, as 

 they readily will in a free-hand diagram, that 

 E is between A and C, and F between B and 

 C; then from the equal right triangles AED 

 = BFD, DEC = DFC, we find AE = BF, 

 EC = FC, and, by adding, AC = BC, whereas 

 AC is in fact greater than BC. 



Are we to take our eyes as evidence that 

 one point lies between two other points, or 

 how are we to establish that fact? This 

 query alone lets in a flood of criticism on all 

 established demonstrations. The aim of mod- 

 ern rational geometry is to pass from premise 

 to conclusion solely by the force of reason. 

 Points, lines and planes are the names of 

 things which need not be physically conceived. 

 The object is to deduce the conclusions which 

 follow from certain assumed relations between 

 these things, so that if the relations hold the 

 conclusions follow, whatever these things may 

 be. Space is the totality of these things; its 

 properties are solely logical, and varied in 

 character according to the assumed funda- 

 mental relations. Those assumed relations 

 which develop space concepts that are appar- 

 ently in accord with vision constitute the mod- 

 ern foundations of Euclidean space. 



Mr. Halsted is the first to write an element- 

 ary text-book which adopts the modern view, 

 and in this respect, his ' Eational Geometry ' 

 is epoch-making. It is based upon founda- 

 tions which have been proposed by the German 

 mathematician, Hilbert. In point of fact, the 

 book contains numerous diagrams, and is not 

 to be distinguished in this respect from or- 

 dinary text-books, but these are simply gi-atui- 

 tous and not necessary accompaniments of the 

 argument, designed especially for elementary 

 students whose minds would be unequal to the 

 task of reveling in the domain of pure reason. 

 Also, in opening the book at random, one does 

 not recognize any great difference from an 

 ordinary geometry. In other words, those as- 



