538 SCIENCE PROGRESS 



to the other. The proposer of the problem will protest against my analysis as 

 far-fetched. I know that it is not. There is a large class of boys, not the really 

 clever ones, but a set who think carefully, and who are usually classified as 

 duffers by persons who are not sharp enough to understand the working of their 

 pupils' minds, and it is these boys, the backbone of the school and the future 

 makers of the nation, who are honestly perplexed by the slipshod nonsense 

 which is nowadays mixed up with geometry. 



Here is a second example, chosen this time from the text-book to which this 

 long harangue is attached : it is an example found in a section in which parallel 

 lines are treated carefully and thoroughly ; at such a point in his geometrical 

 studies surely the student needs no practical conundrums to distract him from 

 important considerations which require his undivided attention. The problem 

 runs thus, " Two parallel pipes for hot and cold water lie flat along the same wall ; 

 at the ends of each of them an elbow is screwed on which turns the pipe through 

 a right angle. If the pipes connected to these angles also lie flat against the wall, 

 will they be parallel?" As if all this was not enough, a diagram is supplied so 

 complete in its details that one regrets that the hot and cold pipes are not 

 distinguished. It would be useless to analyse this problem, it suffices to refer 

 to the preface, from which one surmises that this may be " an illuminating diagram 

 drawn from the Arts." From the same source we learn that " the function of 

 such problems is not to train the students in the technique of the Arts " (for which 

 at least we are grateful, remembering how dependent we are upon the Art of 

 Plumbing), " rather it is to illuminate the geometric facts and to make clear their 

 importance and their significance." How can an ugly drawing of two cast iron 

 pipes with a knobby elbow make clear the subtle theory of parallels ? Every 

 infant knows the answer required ; it is so obvious that most students will 

 cunningly spend time in trying to find out where the catch is. At any rate the 

 elements of Euclid were not open to the reproach of being too obvious. It is 

 not my wish to suggest that the book by Messrs. Ford and Ammerman is not 

 a good book ; on the lines on which it is constructed it is excellent. The defects 

 to which I have drawn attention are inherent in its avowed design. 



Besides the objective of the practical, the modern text-book writer also aims 

 at smoothing down the difficulties of the subject, and in this he carries the 

 sympathy of all. But authors who sacrifice the subject to its simplification 

 cannot earn the lasting gratitude of their readers. It must be remembered, too, 

 that it may be a sound policy to give the longer proof when it fits more naturally 

 into the development of the subject. Students have told me that in the older 

 Euclid system it was I. 5 which gave them their first real insight into geometrical 

 method. The ambition of a writer on geometry should be to give a connected 

 and logically coherent view of the subject, and not to frame a system of pro- 

 positions, each of which is expressed in the fewest possible number of words. 

 Difficulties must occur in any geometrical system, and the test of a system comes 

 when the student faces these difficulties : if he has been trained to think, he will 

 master them ; if he has not, he will not be helped by practical illustration. It is 

 of importance in judging a book to examine how the writers deal with crucial 

 points ; two such tests suggest themselves, the theory of parallels and the theory 

 of proportion. In Ford and Ammerman parallels are clearly and well explained, 

 whatever one may think of the illustrations appended, but the same cannot be 

 said of their discussion of proportions. The criticisms made upon this part of 

 the book apply, however, to other authors who have had the temerity to run 

 counter to the teaching of Euclid and de Morgan. 



