MATHEMATICAL AND PHYSICAL SCIENCE. 



[Diss. VI. 



(20.) 

 Pure ma- 



their 

 progress 

 and tech- 

 nicality. 



before him and since by Galileo, Newton, and their 

 disciples. 



With regard to Pure Mathematics, and their pro- 

 gress during the last seventy years, to the difficulty 

 arising from the extent to which the review must have 

 enlarged this Essay, and the enormous and dispropor- 

 tionate labour it must have cost (a labour far greater 

 to the present writer than to one by taste and habit 

 more addicted to the study of merely abstract Ana- 

 lysis), a conclusive argument against their systematic 

 introduction into this historical sketch is to be found 

 in the very nature of these modern improvements. 

 All sciences but especially the abstract sciences 

 tend to become more intensely technical the farther 

 they are pursued. These especially are incapable of 

 popular treatment, although in their applications to 

 physical science they occasionally admit of it in a 

 very remarkable manner. The progress of analysis 

 cannot be even enunciated or expressed but in the 

 language of analysis, and the History becomes almost 

 a Treatise, or, if not a Treatise, something nearly as 

 technical. It is partly for this reason that the his- 

 tory of the Pure Mathematics has so seldom been 

 even attempted to be written. 1 Mathematicians have, 

 since the time of D'Alembert, been noted for being 

 more ready themselves to publish than to become 

 acquainted with what others have done ; and one con- 

 sequence of this has been the formation of a mathe- 

 matical literature, able, profound, and original, but 

 cumbrous, fragmentary, and full of repetitions. 2 

 Besides, the seventeenth century had attained the 

 vantage-ground of those grand and striking im- 

 provements in methods to which no subsequent im- 

 provements, however real and ingenious, can by 

 possibility compare. We shall never have inventions 

 comparable, in universality and importance, to the 

 application of Algebra to Geometry, and the dis- 

 covery of Fluxions. These also admit of being at 

 least partly explained in language not obtrusively 

 technical, and have been so explained by the facile pen 

 of Playfair ; but all subsequent discoveries have been 

 but enlargements and improvements on these pri- 

 mary and distinguishing ones ; and before the date 

 at which our present discourse properly opens, even 

 the larger generalizations of Newton's fertile calcu- 

 lus the method, namely, of Variations, and the in- 

 tegration of partial differential equations, had been 



established, by the genius of Euler and Lagrange, on 

 an impregnable basis. 



The intense, and praiseworthy, and successful ( 21 -) 

 labours of their followers have been, then, chiefly de- 

 voted to the occupation of the fields of conquest thus 

 summarily opened; or, rather, to storming, one by one, 

 fortresses still unreduced, after the main resisting army 

 had been first routed in the open field. To quit meta- 

 phor, the efforts of mathematicians have for many 

 years been chiefly applied to rendering possible the 

 solution of problems involving quantities which ac- 

 tually occurred in the course of the rapid simulta- 

 neous advances of physical science. They are in a 

 manner inseparable from the branches of physics in 

 aid of which they were originally called forth, and 

 will therefore be most properly noticed, however 

 briefly, in the chapters of this Dissertation where 

 their application is considered. Some farther obser- 

 vations on this subject will be found in the imme- 

 diately succeeding section of the present Essay. 



In reviewing the progress of science physical (22.) 

 science in particular during the last seventy or s . ubdivi " 

 eighty years, I have thought it advisable not to sico-Mathe" 

 subdivide the subjects too minutely, and, following matical 

 nearly the arrangements of Dr Whewell's excellent Sciences, 

 treatise, already quoted, 3 1 shall treat, in successive 

 chapters, of Analytical Mechanics including Physical 

 Astronomy as their loftiest and most successful ap- 

 plication ; of Astronomy as a science of observation ; 

 of Mechanics, with reference to the intimate consti- 

 tution of matter, including Hydrodynamics, Acous- 

 tics, and Civil Engineering* ; of Optics, or Light ; of 

 Heat, including the Daltonian theory of the gases 

 and chemical elements ; and, finally, of the large and 

 comprehensive science of Electro-magnetism, includ- 

 ing ordinary and Voltaic Electricity, Terrestrial Mag- 

 netism, and Diamagnetism the discovery of Dr Fa- 

 raday. 



The arrangement of the chapters is thus strictly (23.) 

 Methodical ; but in the subdivision into sections, I Arran S e - 

 have allowed the Biographical principle to predomi- methodical, 

 nate, thus giving as much as possible a historical partly bio- 

 character to the whole, and endeavouring to intro- graphical, 

 duce the reader to the intellectual acquaintance of 

 the eminent men who are selected for notice on the 

 principles which have been already detailed. In some 



1 Specimens of what a history of pure mathematics would be, and must be, are to be found in the able " Reports " of Dr 

 Peacock and Mr Leslie Ellis, in the Transactions of the British Association for 1833 and for 1846. A glance at these profound 

 and very technical essays will show the impossibility of a popular mode of treatment, whilst the difficulty and labour of pro- 

 ducing such summaries may be argued from their exceeding rarity in this or any other language. 



2 The celebrated Lagrange, in his later years, contrasting the mathematical works of his own generation with those which he 

 studied when a youth, is said to have observed : " I pity the young mathematicians who have so many theories to wade through. 

 If I were to begin, I would not study ; these large quartos frighten me too much." (Thomson's Annals, vol. iv.) And it is stated 

 that whilst his own most abstruse investigations were conducted in Paris, he kept the perusal of M. Gauss's writings for the tran- 

 quil retirement of the country, a distinction intelligible enough between the intense effort of invention more than sustained by 

 the vis viva of genius which prompts it, and the strain required to master the dead weight of reasoning imposed upon the mind 

 by the discoveries of another. Dr Young, in his biography of Lagrange, observes upon the voluminous mathematical literature 

 of his time, that " unless something be done to check the useless accumulation of weighty materials, the fabric of science will sink 

 in a few ages under its own insupportable bulk." The fact is that a large proportion of the mathematical writings of even 

 eminent authors are in a few years forgotten, or only casually consulted on some matter of history. 



3 History and Philosophy of the Inductive Sciences. 



