493 



NA TURE 



\_Sept. 20, 1S83 



I may mention here especially four men of whom each invented 

 a new method and established a new system of geometry. Two 

 of these, Mbbius and Pliicker, still use algebra, but in perfectly 

 new and original manners, which, although very different froui 

 each other, have this in common, that in both Ave have not 

 algebra interpreted geometrically, but rather geometry veiled in 

 an algebraic garb. The geometrical meaning is never lost 

 sight of. 



But perfectly independent of algebra was the great Steiner, 

 the greatest geometrician since the times of Euclid, Appolonius, 

 and Archimedes. In his celebrated " Systematische Entwicke- 

 lungen " he has laid the foundation of a pure geometry, on which 

 a wonderful edifice has since been raised. His treatment of the 

 principle of duality, and his method of generating conies by 

 projective, or hornographic, rows of pencils which have been 

 extended to curves of all degrees, have given to geometrical 

 reasoning a generality never before dreamed of He is in one 

 respect the opposite of Lagrange, hating and despising analysis 

 as much as ever Lagrange didiked pure geometry. Steiner 

 started from the geomeiry of the Greeks, Euclid's elements, and 

 a few other metrical properties he takes for granted ; but then 

 he goes on with essentially modern methods of hi* own to 

 investigate what are now called projective properties of curves 

 and surfaces. 



This metrical foundation Von Staudt changed. In his 

 " Geometrie der Lage," published fifteen years after Steiner's 

 "Entvvickelungen," he established a most remarkable and com- 

 plete system, into which the notion of a magnitude does not 

 enter at all. He shows that projective properties of figures, 

 which have no relation whatsoever to measurements, can be 

 established without any mention of them. He goes so far as 

 even to give a geometrical definition of a number in its relation to 

 geometry as determining the position of a point, in his theory of 

 what he calls " Wiirfe" ; and one of the most interesting parts 

 of his work is the purely geometrical treatment of imaginary 

 points, lines, and planes. 



In the hands of these men, and since their time, pure geo- 

 metry has become a most important instrument for research, 

 rivalling in power the more or less algebraical methods, and sur- 

 passing them all in the manner in which they raise before the 

 mind's eye a clear realisation of the forms and figures which are 

 the object of the investigation. 



In close connection with these methods stand descriptive 

 geometry and geometrical drawing, which teach how to represent 

 figures on a plane or other surface. These have been treated as 

 arts unknown at English universities, and relegated to the 

 drawing office. Instead of this they ought to be an essential 

 and integral part of the teaching of geometry in connection with 

 the purely geometrical methods. 



As far as the progress of science is concerned, this neglect of 

 pure geometry in England has been of little consequence — per- 

 haps it has rather been a gain. For science itself it is often an 

 advantage that a centre of learning becomes one-sided, neglects 

 many parts in order to concentrate all its energy on some parti- 

 cular points and make rapid progress in the directions in which 

 these lie. At present, when mathematics nourishes as never 

 before, when almost every nation, however small, has its eminent 

 mathematician, there are so many such centres that what i- 

 neglected at one place is pretty surely taken up and advanced at 

 another. But what may suffer if one side of a science is not 

 cultivated in a country is the industry which would have gained 

 by its applications. 



In considering the teaching of any mathematical or other 

 scientific subject, we cannot at the present time neglect the 

 wants of the ever-increasing class of men who require what has 

 been called technical education. Among these the large number 

 who want mathematics at all require geometry much more than 

 algebra and analysis, and geometry as applied to drawing and 

 mensuration. 



This want has been supplied by the numerous science classes 

 spread over the country, with their head-quarters at the Science 

 and Art Departmental South Kensington, whose examinations — 

 now, however, put in competition with those of the City and 

 Guilds of London Institute, and others — have pretty much 

 guided and regulated the teaching. A great deal of good lias 

 thus been done, but there is still much room for improvement. 

 The teaching of geometry especially, as judged by the text-books 

 which have c >me before me, is somewhat deplorable. And this 

 is so, principally, because the spirit of Euclid and the methods of 

 the ancient Egyptians and Greeks, rather than the fundamentally 

 different ideas and methods of modern geometry, still rule 



supreme ; though the latter have had their origin partly in 

 technical wants. 



In what is called geometrical drawing or practical geometry, 

 for instance, there are first given a number of elementary con- 

 structions — such as drawing parallels and perpendiculars, or 

 bisecting the distance between two given points. They are 

 solved by aid of those instruments only which Euclid knew — 

 viz. the pair of compasses for drawing circles, and the straight 

 edge for drawing straight lines. But there is no draughtsman 

 who would not, as a matter of course, use set squares for the 

 former problem, and solve the latter by trial rather than by 

 coustruction. Then again ther- come constructions like the 

 division of the circumference of a circle into seven parts, which 

 cannot be solved accurately, but which is very easily solved by 

 trial. Instead of that, a construction is given which takes much 

 more time, and is by no means more accurate. For, after all, 

 our lines drawn on the paper are not without thickness, so that, 

 for this reason alone, every part of the construction is affected by 

 some small error ; and it is absurd to employ a construction, 

 though theoretically true for ideal figures as conceived in our 

 mind, in preference to a much simpler one which, within our 

 practical limits, is equally, or perhaps more, correct. 



This is very much like the manner in which I found problems 

 on decimal fractions treated by the candidates for the Matricula- 

 tion Examination at the London University, and which reflected 

 little credit on the manner in which the important subject of 

 decimals is handled at our schools. It is so characteristic that I 

 may be excused for giving it here. The problem, for instance, 

 being to give the product of two decimal frac ions, exact to, say, 

 four decimal,-, each of the factors having the same number of 

 places. This was almost regularly performed as follows. First, 

 the decimals are converted into vulgar fractions, then these are 

 duly multiplied, numerator by numerator, and denominator by 

 denominator, and then the resulting fraction is again converted 

 to a decimal, with as many places as it may yield, and, lastly, of 

 these the first four are taken and put down, duly marked Answer. 

 Or a candidate, standing however on a far higher level, multi- 

 plies both decimal ; out in the proper fashion, but to eight places, 

 and cuts off four places at the end. No wonder that the public 

 at large will hear nothing of the decimal system of weights and 

 measures if the very essence of the decimal sy-tem of numbeis is 

 s > little understood by the men who have to train the minds of 

 the young generation ! 



I need scarcely sny that I do not mean to blame the Science 

 and Art Department, far less the teachers who have simply to 

 follow suit. They act up to their light, and cannot be expected 

 to introduce methods which are practically unknown at Cam- 

 bridge, and of which the only good text-books are in foreign 

 languages — books which are probably not at all suitable for 

 introduction into our schools without considerable change. 



It is satisfactory to learn that an association has recently been 

 formed under the presidency of Prof. Huxley "to effect the 

 general a tvancement of the profession of science and art teaching 

 by securing improvements in the schemes of study, and the estab- 

 lishment of satisfactory relations between teachers and the Science 

 and Art Department, the City and Guilds of London Institute, 

 and other public authorities. " 



The good wishes of all who have the cause of sound education 

 at heart must go with such an undertaking, one of the principal 

 aims of which seems to be to save teaching from being any 

 longer enslaved by examinations, and to promote greater accora 

 between the teacher and the examiner. It is to be hoped that 

 this association will con-ider geometry as one of the subjects 

 included under the designation of science. 



It is by the neglect of pure geometry and its applications to 

 geometrical drawing that Cambridge has lost, or rather has never 

 had, contact with the practical needs of the nation. All the 

 marvels of modern engineering have sprung into existence with- 

 out its help. The great engineers have had to depend to a 

 degree, now unheard of, upon costly experiments, until they 

 themselves gradually discovered mathematical methods adapted 

 to their purposes. 



Only the electrical engineer found ready to his hands a com- 

 plete theory of which the mathematical part has been to a very 

 great extent developed at Cambridge, or by men who have had 

 their mathematical training there. This theory is, however, in 

 its very nature less geometrical. One at least of the great men 

 to whom the present theory of electricity is due, the late Cleric 

 Maxwell, had the keenest appreciation of the value of modern 

 geometry. I remember a characteristic letter of his being read 

 to the Council of the London Mathematical Society, in which 



