Sept. 2o, 1883] 



NATURE 



497 



SECTION A 



MATHEMATICAL AND PHYSICAL 



Opening Address by Prof. Olaus Henrici, Ph. D., F.R.S., 

 President of the Section. 



On reading through the addresses delivered by my predecessors 

 in this office, I was struck by the fact that in nearly every case 

 the speaker began with a lamentation over his unfitness for the 

 work before him, and those seemed to me to be the more elo- 

 quent on these points who showed by their addre-s that they 

 least needed an excuse. The amount of excuse given appears 

 in fact to be directly proportional to the gifts of the speaker, 

 and hence inversely proportional to the need of such an excuse. ' 

 Under these circumstances I cannot express my sense of my 

 own unfitness for this post better than by saying nothing about it. 

 I must, however, lie.; your indulgence for my shortcomings, both 

 as regards my address and my manner of conducting the general 

 business of this secii on. 



As the Presidential chair is occupied by one of the most illus- 

 trious of mathematicians, it would be presumptuous for me to 

 attempt to g,ve an account of the recent progress of mathematics. 

 I propose only to speak for a short time on that part of mathe- 

 matics which has always been most attractive to myself— that is 

 pure geometry as apart from algebra, but I shall confine myself 

 to some considerations relating to the teaching of geometry in 

 this country. Pure geometry seems to me to be of the greatest 

 tducational value, and almost indispensable in many applica- 

 tions ; but it has scarcely ever been introduced at Cambridge 

 the centre of mathematics and mathematical education °in 

 England. 



The number of geometrical methods now in use is astonish- 

 ingly great. These differ, on the one hand, according to the 

 nature of the result aimed at, but, on the other, according to the 

 amount of algebra employed, and to the relation in which this 

 algebra stands to the pure " Anschauung." \ use the word 

 Anschauung because I know of no English equivalent • the 

 German word has the philosophic meaning rendered by intuition 

 and retains its original concrete meaning of look-in" at a thine 

 and might perhaps be translated : intuition by inspection. It is 

 the inspection of figures which is of the greatest importance in 

 geometry. It is hereby of little consequence whether the figures are 

 seen by the physical eye or only mentally ; because the conception 

 of that space in which we perceive everything and without which 

 we can perceive nothing, which therefore is, according to Kant 

 a form of our Anschauung, is built up in our mind thiou"h many 

 generations in conformity with sensual impressions. 



It would be of interest, if time permitted, to follow up the 

 gradual development and extension of geometry into the wider 

 science of algebra, from the first introduction of the latter in the 

 theory of proportion to the present state, where there exists 

 really no essential difference between the two, where Geometry is 

 only one manifestation of algebra, but so complete a one that at 

 least within its number of dimensions it again contains algebra 



In some of the methods just referred to no algebra is used at 

 all, whilst others may be distinguished according to the nature 

 of the algebra used, whether equations containing one, two 

 three, or more variables are employed. In such a division, Von 

 Staudt's system, without a vestige of algebra, would occupy the 

 one end, and the purely algebraical theory of invariants' with 

 geometrical interpretation the other. 



There is, however, not only a difference in the amount of 

 algebra used, but, if possible, a greater one in the manner in 

 which the symbols are interpreted. And it is here that algebra 

 has apparently he greater power. One algebraical theorem, by 

 being read indifferent ways, by giving ever different meanings 

 to the symbols, reveals a variety of geometrical and other theo- 

 rems. We have in it the crystallised form, the very essence of 

 the mathematical truth, but in the most abstract form conceiv- 

 able. Now this most abstract form is the highest and the most 

 perfect which mathematical truth as such can assume, and which 

 it p ust assume before a theory is really complete in the eyes of 

 a pure mathematician. It is only in this shape that it is ready 

 to be turned to account in any direction where it may be needed 

 In thus placing algebra on the highest pinnacle, the reasons 

 will be apparent which will make many mathematicians, not to 

 mention others, prefer the truths it reveals cast in a mould which 

 connects them with concrete things rather than with abstract 

 notions. In fact, to be thoroughly at home in the highest 

 theories of pure algebra requires some of the genius of men like 

 Cayley and Sylvester who have founded, and to a great extent 



built up, modern algebra. But even they constantly make use of 

 geometry to assist them in their investigations, and no one could 

 have expressed this more strongly than Prof. Sylvester himself 

 in his brilliant address delivered from this chair at the Exeter 

 meeting of our Association. 



If this is so, surely every progress in the spread of the know- 

 edge of pure geometry should be welcomed and encouraged: 

 but in England pure geometry is almost unknown excepting in 

 the elements as contained in Euclid and in the old-fashioned 

 geometrical conies. The modern methods of synthetic projec- 

 tive geometry as developed on the Continent hare never become 

 generally known here. The few men who have thoroughly 

 made themselves acquainted with them, and who have preferred 

 pure y geometrical reasoning, have not belonged to Cambridge 

 and have thus stood somewhat outside the national system of 

 training mathematical teachers. The late Prof. Smith intro- 

 duced these methods at Oxford, and there was some expectation 

 that he would have written, if he had been spared, a text-book 

 which might have done much to introduce the subject more 

 widely. His principal mathematical work lay, however, in 

 another direction. 



The one English mathematician whose mathematical thousht 

 is purely geometrical is Dr. Hirst, a pupil of Steiner, who in 

 the position which he has just relinquished has been able to 

 introduce, as the first, modern geometncil methods into a regu- 

 lar system of professional education, whilst showing at the sime 

 time by his original work what can be done with these methods 

 Other mathematicians who have studied these methods— and I 

 believe there are many— have made use of them by translating 

 the geometrical into algebraical reasoning. 



Towards the early possibility of such a translation much was 

 done by the labours of the late Mr. Spottiswoode, who years ago 

 wrote the first connected treatise on the theory of determinants 

 and who up to the last few years employed some of his leisure 

 hours in working out geometrical problems, the work consisting 

 always of sjmie beautiful piece of algebra. 



It is not often that our Section has to mourn in one year the 

 loss of two such men as Smiih and Spottiswoode. 



It is easy to see how the ne,dect complained of has come to 

 Pass. In England when mathematics, after having lain dormant 

 for about a century, began to revive, the first necessity was to 

 become acquainted with the enormous amount of work mean- 

 while done on the Continent. This acquaintance was made 

 through France, at that time nearly all the standard works bein° 

 in the French language, which was at the same time the language 

 be-t known to English students. The subjects principally taken 

 up were the calculus and its application to mechanics. And I 

 believe I am not far wrong when I say that the wonderful 

 writings of Lagrange, with their extraordinary analytical ele- 

 gance, had the greatest influence. But in his works anything 

 geometrical was studiously avoided. Lagrange prided himself 

 that there was no figure in his " Mccanique analytique." 



The best analytical methods of the Continent were thus intro- 

 duced into England, rapidly assimilated and made the founda- 

 tion of new theories, so that the mathematical activity in this 

 country is now at least as great a-- it ever has been anywhere. 



but whilst analysis, algebra, and with it analytical Geometry 

 made rapid progress, pare geometry was not equally fortunate.' 

 Here the hold which Euclid had long obtained, strengthened, no 

 doubt, by Newton's example, prevented any change in the 

 methods of teaching. 



Most of all, perhaps, solid geometry has suffered, because 

 Euclid s treatment of it is scanty, and it seems almost incredible 

 that a great part of it— the mensuration of areas of simple curved 

 surfaces and of volumes of simple solids— is not included in 

 ordinary school teaching. The subject is, possibly, mentioned 

 in arithmetic, wdiere, under the name of mensuration, a number 

 of rules are given. But the justification of these rules is not 

 supplied, except to the student who reaches the application of 

 the integral calculus ; and what is almost worse is that the 

 general relations of points, lines, and planes, in space, is scarcely 

 touched upon, instead of being fully impressed on the student's 

 mind. 



The methods for doing this have long been developed in the 

 new geometry which originated in France with Mon^e. But 

 thee have never been thoroughly introduced. 



Works written in the German language naturally received even 

 less attention. But it was in Germany, at the beginning of the 

 second quarter of this century, that geometry received at the 

 hands of several masters an impulse which put the subject on an 

 entirely new footing. 



