4 REPORT 1873. 



work of construction. The difficulties of tlie inquiry, the pleasure of grappling 

 with the unknown, the love of abstract truth, can alone have furnished the charm 

 which attracted some of the most powerful minds in antiquity to this research. 

 If Euclid and Apollonius had been told by any of their contemporaries that they 

 were giving a wholly wrong direction to their energies, and that, instead of 

 dealing with the problems presented to them by nature, they were applying their 

 minds to inquiries which not only were of no use, but which never could come to 

 be of any use, I do not know what answer they could have given which might not 

 now be given with equal or even greater justice to the similar reproaches which 

 it is not uncommon to address to those mathematicians of our own day who study 

 quantics of n indeterminates, curves of the nth. order, and, it may be, spaces of 

 n dimensions. And not only so, but for pretty near two thousand years the 

 experience of mankind would have justified the objection ; for there is no record 

 that during that long period which intervened between the first invention of the 

 conic sections and the time of Galileo and Kepler the knowledge of these curves 

 possessed by geometers was of the slightest use to natural science. And yet, 

 when the fulness of time was come, these seeds of knowledge, that had waited 

 so long, bore plentiful fruit in the discoveries of Kepfer. If we may use the 

 great names of Kepler and Newton to signify stages in the progress of human 

 discovery, it is not too much to say that without the treatises of the Greek 

 geometers on the conic sections there could have been no Kepler, without Kepler 

 no Newton, cand without Newton no science in our modern sense of the term, 

 or at least no such conception of nature as now lies at the basis of all our 

 science, of nature as subject in its smallest as well as in its greatest phenomena, 

 to exact quantitative relations, and to definite numerical laws. 



Tills is an old story ; but it has always seemed to me to convey a lesson, occa- 

 sionally needed even in our own time, against a species of scientific utilitarianism 

 which urges the scientific man to devote himself to the less abstract parts of 

 science as being more likely to bear immediate fruit in the augmentation of our 

 knowledge of the world without. I admit, howevei", that the ultimate good fortune 

 of the Greek geometers can hardly be expected by all the abstract speculations 

 which, in the form of mathematical memoirs, crowd the transactions of the learned 

 societies ; and I would venture to add that, on the part of the mathematician, 

 there is room for the exercise of good sense and, I would almost say, of a kind of 

 tact, in the selection of those branches of mathematical inquiry which are likely 

 to be conducive to the advancement of his own or any other science. 



I pass to my second example, of which I may treat very briefly. In the course 

 of the present year a treatise on electricity has been published by Professor Max- 

 well, giving a complete account of the mathematical theory of that science, as we 

 owe it to the labours of a long series of distinguished men, beginning with Coulomb, 

 and ending with our own contemporaries, including Professor Maxwell himself. 

 No mathematician can turn over the pages of these volumes without very speedily 

 convincing himself that they contain the first outlines (and something more than 

 the first outlines) of a theory which has already added largely to the methods and 

 resources of pure mathematics, and which may one day render to that abstract 

 science services no less than those which it owes to astronomy. For electricity 

 now, like astronomy of old, has placed before the mathematician an entirely new 

 set of questions, requiring the creation of entirely new methods for their solution, 

 while the great practical importance of telegraphy has enabled the methods of 

 electrical measurement to be rapidly perfected to an extent which renders their 

 accuracy comparable to that of astronomical observations ; and this makes it possi- 

 ble to bring the most abstract deductions of theory at every moment to the test of 

 fact. It must be considered fortunate for the mathematicians that such a vast field 

 of research in the application of mathematics to physical inquiries should be thrown 

 open to them at the very time when the scientific interest in the old mathematical 

 astronomy has for the moment flagged, and when the very name of physical astro- 

 nomy, so long appropriated to the mathematical development of the theory of gravi- 

 tation, appears likely to be handed over to that wonderful series of discoveries 

 which have already taught us so much concerning the physical constitution of the 

 heavenly bodies themselves. 



