REVIEWS 537 



such as I. 5 and I. 47 (to use a forgotten nomenclature), which earlier generations, 

 after surmounting non sine sudore, honoured and remembered for the effort 

 which the ascent had cost. If the reformers had insisted upon preparatory work, 

 in which tables and strings and even men's heads took their proper stations, all 

 might have been well with us to-day ; but revolution was preferred to evolution. 



The first requisite for the pupil learning geometry is either a good teacher 

 or a good text-book. Fortunately we have many good teachers, but unfortunately 

 no Legendre came forward at this juncture to write an English Elements de 

 Ge'omt'iric. The writers of the modern text-books on geometry wrote, not to 

 satisfy the mathematician, but to please the engineer. Even when attempts are 

 made in their books to explain fundamental terms, the explanations often are 

 such that they cannot satisfy either the logician or the mathematician, and every 

 weakness in an explanation is covered by a cloud of illustrations drawn from a 

 knowledge of the outside world of phenomenon which the scholar does not 

 generally possess. The old-fashioned plates of the out-of-date cyclopaedia are 

 pressed into the service of an appeal to the visual organs, while too often the 

 mental vision is dulled rather than sharpened by this process of illumination. 

 The works attest to the ingenuity and the versatility of their writers, but they 

 do not satisfy the geometricians, and they do not even placate the engineers. 

 That the scheme inaugurated by the new text-book is failing is the verdict of 

 competent observers, and especially of those who examine the scholars trained 

 under it. Many of the teachers who have marked out their school curricula 

 by the new charts are justly alarmed by soundings now being taken, and know 

 that their ships are in dangerous waters. 



The human factor was neglected in the old system of geometrical education ; 

 in the new system it is outraged. Formerly only a half, though the more 

 important half, of the subject was taught ; to-day the two parts are crammed 

 down indiscriminately. Practical geometry, with its wholesome fare, attracts 

 every boy and girl ; but it is easy to have too much of it — besides, it is not 

 geometry, except in the narrower philological meaning of the word. Geometry, 

 in spite of the writers of our text-books, remains an abstract science, the type 

 indeed to which all sciences which are not merely descriptive or classificatory 

 will ever strive to conform. The continual intrusion of the crude imperfections 

 of practical illustration does not, after a certain point is reached, aid in the 

 development of abstract ideas ; indeed, such a proceeding often hinders the 

 delicate and subtle mental growth which must accompany the study of the subject. 

 How the modern method of unfolding the theory confuses the student may be 

 shown by analysing a problem which appears in one of the well-known new 

 text-books ; perhaps it is reproduced in them all, for some forms of error are 

 highly infectious. The problem is, To find the locus of a man's head climbing 

 a ladder. The concocter of this precious example intends the answer to be a 

 line parallel to the ladder, and clever persons who are " in the know" might give 

 the required answer. But let us consider what the words of the problem will 

 suggest to a conscientious boy of average intelligence who is ignorant of the 

 tricks of the trade. First he will attempt to settle the shape of the man's 

 head, and will be fortunate if he fixes at once upon a perfectly bald specimen ; 

 his second difficulty will be in deciding where the head finishes and the body 

 begins. On getting over these anatomical details he will be confronted by the 

 irregular motion of the action of climbing, the ups and downs due to the bending 

 and straightening of the knees, also by the swinging motion from right to left 

 which is inseparable from the shifting of the balance of the body from one leg 



