Mabch 7, 1902.] 



SCIENCE. 



385 



day and for his purposes, and continued so 

 for many a century. 



The educational wants of our age are far 

 more numerous, pressing and diverse than 

 when the Greek geometer taught in the schools 

 of Alexandria 2,300 years ago. We must re- 

 member, too, that it was before admiring 

 throngs of men that he unfolded book after 

 book of his 'Elements,' men who were at- 

 tracted round his cathedra more by the 

 knowledge that he had to impart than by the 

 mental training to be derived from his teach- 

 ing. The geometrical truths that Euclid 

 discussed before his mature scholars have to 

 be placed to-day before the undeveloped minds 

 of mere tyros and placed before them as only 

 one of the many elements of their daily pabu- 

 lum.* 



The necessities of our times call for less 

 verbosity, prolixity and iteration than we 

 find in Simson's rendering of Euclid, which is 

 the orthodox text in English schools modified 

 somewhat by Todhunter, Hamblin Smith, Mrs. 

 Sophia Bryant and others. The whole presen- 

 tation of the subject is decidedly hard and 



* Here are the latest from Nature of January 

 16, 1902: "Mr. W. C. Fletcher, headmaster of 

 Liverpool Institute, says that ' six years' expe- 

 rience in teaching geometry has led him to be- 

 lieve that Euclid is a great hindrance to ninety- 

 nine boys out of every hundred in training and 

 knowledge. A great deal of damage is done by 

 insistence, not only upon the particular method, 

 but on the particular order of Euclid" (p. 262). 



A letter signed by mathematical teachers in the 

 great public schools of England such as Eton, 

 Harrow, Rugby and Winchester contains the fol- 

 lowing momentous acknowledgments: "It may 

 be felt convenient to retain Euclid; but perhaps 

 the amount to be memorized might be curtailed 

 by omitting all propositions except such as may 

 serve for landmarks. We can well dispense with 

 many propositions in the first book. The second 

 book, or whatever part of it we may think essen- 

 tial, should be postponed till it is needed for 

 Book III., 35. The third book is easy and inter- 

 esting; but Euclid proves several propositions 

 whose truth is obvious to all but the most unin- 

 tellectual. The fourth book is a collection of 

 pleasant problems for geometrical drawing; and 

 in many cases, the proofs are tedious and unin- 

 Btructive. No one teaches Book V." (p. 258). 



deterrent. Think of Book II. and especially 

 Book v., also Books XL and XII., and try 

 to realize the hardship, the worry and vex- 

 ation of spirit they have caused ! A different 

 sequence of propositions is needed, as well as 

 shorter demonstrations and modern methods. 

 When we want to cross a stream, we do not go 

 up one bank to its source and down the other; 

 but avail ourselves of a bridge, a ferry or an 

 electric launch if handy. 



Nor is prolixity the only salient fault of 

 Euclid's 'Elements'; it lacks suggestive- 

 ness and fails to adequately stimulate the in- 

 ventive faculty of the student. It leaves him 

 but too often unable to think out a simple 

 matter for himself, to originate a plan of 

 attack or even to act on a hint. He can ap- 

 preciate a neat solution to a rider; he is 

 receptive, but not creative enough; a good 

 consumer but poor producer. 



On the other hand, we have found boys very 

 different who had worked out their Davies or 

 Robinson or other simplifier or improver of 

 Legendre's great work. They had a clearer 

 apprehension of the meaning of geometrical 

 truths and firmer grasp of them. They usually 

 showed marked readiness in applying their 

 knowledge, in extending conclusions, detecting 

 flaws and attacking problems. There was a 

 resourcefulness and a vigor about their ways 

 that bespoke the benefit which they derived 

 from the subject. It was less a task imposed 

 upon them than a congenial study. Here, then, 

 we have the full realization of the twofold 

 end of all undergraduate study, viz., culture 

 and utility ; or, to put it otherwise, the awaken- 

 ing of the faculties and the acquisition of 

 knowledge. 



Euclid might well say to Ptolemy that 

 there was no royal road to geometry; but all 

 the same, such a work as the 'Plane and Solid 

 Geometry ' of Dr. Sevenoak and Dr. Schultze 

 will go far toward smoothing the way for the 

 young geometer and inspiring him with a 

 liking for the subject. The letterpress and 

 figures leave nothing to be desired; the dem- 

 onstrations are well spaced, short and sug- 

 gestive. The arrangement of the propositions 

 follows a logical and pedagogical order, and 

 the exercises, which form an integral part of 



