December 7, 1888.] 



SCIENCE. 



273 



tions of Euclid, Archimedes, Ptolemy, and perhaps of other Greek 

 writers, together with the works of the Arabian algebraists, were 

 read and commented on at the three great Moorish universities or 

 schools of Granada, Cordova, and Seville. It seems probable that 

 these works represent the extent of Moorish learning; but, as all 

 knowledge was jealously guarded from any Christians, it is impos- 

 sible to speak with certainty either on this point or on that of the 

 time when the Arab books were first introduced into Spain " (p. 



157). 



A good summary of the condition of mathematical knowledge at 

 the close of the renaissance is given at p. 228 : " By the beginning 

 of the seventeenth century we may say that the fundamental prin- 

 ciples of arithmetic, algebra, theory of equations, and trigonometry 

 had been laid down, and the outlines of the subjects as we know 

 them had been traced. It must, however, be remembered that 

 there were no good elementary text-books on these subjects ; and 

 a knowledge of them was thus confined to those who could extract 

 it from the ponderous treatises in which it lay buried. Though 

 much of the modern algebraical and trigonometrical notation had 

 been introduced, it was not familiar to mathematicians, nor was it 

 even universally accepted ; and it was not until the end of the sev- 

 enteenth century that the language of the subject was definitely 

 fixed. ... If we turn to applied mathematics, we find, on the 

 other hand, that the science of statics had made but little ad- 

 vance in the eighteen centuries that had elapsed since the time of 

 Archimedes, while the foundations of dynamics were only laid by 

 Galileo at the close of the sixteenth century. In fact, it was not 

 until the time of Newton that the science of mechanics was placed 

 on a satisfactory basis. The fundamental conceptions of mechan- 

 ics are difficult, but the ignorance of the principles of the subject 

 shown by the mathematicians of this time is greater than would 

 have been anticipated from their knowledge of pure mathematics. 

 With this exception, we may say that the principles of analytical 

 geometry and of the infinitesimal calculus were needed before 

 there was likely to be much further progress. The former was 

 employed by Descartes in 1637 ; the latter was invented by New- 

 ton (and possibly independently by Leibnitz) some thirty or forty 

 years later: and their introduction may be taken as marking the 

 commencement of the period of the modern mathematics." 



That which follows is more familiar, and the feature of Mr. Ball's 

 chapters on the modern period is his full and clear analysis of 

 Newton's contributions to mathematical science. Descartes, Pas- 

 cal, Barrow, Huygens, Newton, Leibnitz, the Bernoullis, Euler, 

 Lagrange, Laplace, Legendre, Poisson, and others less important, 

 are treated in turn and with excellent judgment. Their successors 

 are very briefly mentioned, and no attempt is made to follow out in 

 detail the researches of Abel, Gauss, Sir William Rowan Hamil- 

 ton, Henry J. S. Smith, Weierstrass, Cayley, Sylvester, and Klein. 

 But from this history, or historical sketch, the intelligent reader can 

 gain a Very complete view of the progress of mathematical science 

 from its beginnings until its contemporary differentiation into nu- 

 merous specialties, — each of them important and difficult enough 

 to detain for a lifetime a brilliant mind, — all of which are fruitful 

 in their applications to the various phases of modern science and 

 modern industry. 



A Brie/ History of Greek Philosophy. By B. C. Burt. Boston, 

 Ginn. 12°. $1:25. 

 This work had its origin, the author tells us, in a series of articles 

 in a religious newspaper, but has been expanded so as to cover the 

 whole history of Greek speculation from Thales to Proclus. The 

 result is a volume of three hundred pages, in which the leading 

 doctrines of the various schools are concisely yet for the most part 

 clearly presented. Mr. Burt's style is plainer than that of most 

 writers of the school to which he belongs ; and his readers will 

 seldom have any difficulty in understanding what he says, except 

 where the theories he is trying to explain are themselves obscure. 

 The main fault in the book, according to our view, is the author's 

 Hegelianism. This leads him not only to look in the ancient 

 thinkers for anticipations of his own views, but also to give too 

 much attention to some theories of the earlier philosophers and of 

 the Neo-Platonists which can only be regarded as products of 

 imagination. What we want to learn about the ancient philoso- 



phers is their contributions to the real philosophies of the world ; 

 while their visionary theories, which they themselves in many cases 

 put forth as only conjectural, ought to be passed over in silence, 

 or with a bare mention. In the main, however, Mr. Burt has con- 

 fined himself to the best portions of Greek thought, the great 

 names of Socrates, Plato, and Aristotle necessarily occupying the 

 foremost place, yet without excluding what is most important in 

 the works of others. The exposition of Aristotle is excellent, 

 though the treatise on ' Politics ' is accorded rather too much at- 

 tention ; but the account of Plato is hardly so satisfactory. The 

 author's view of the periods of Greek thought is essentially that 

 now commonly held. The first period was that of naturalism, or 

 the attempt to explain the physical world ; the second, that of ra- 

 tionalism, or the endeavor to understand human nature and dis- 

 cover the basis of morals ; while in the third or Neo-Platonic 

 period theological speculation held the leading place. Of these 

 different phases of thought, the second is so much the most im- 

 portant that the exposition of it rightly occupies the greater part of 

 the volume ; yet the others receive all the notice that is necessary 

 in so compendious a treatise. Mr. Burt is careful also to trace the 

 connection of each period with the preceding one, and also of one 

 individual thinker with another, thus exhibiting the course of philo- 

 sophical development. On the whole, the book is well adapted to 

 its purpose, and will undoubtedly be useful to young students, in 

 college and elsewhere, for whom it is more especially intended. 



Paradoxes 0/ a Philistine. By William S. WALSH. Philadel- 

 phia, Lippincott. 16°. $1. 



This volume consists of a number of short essays, most of which 

 had previously appeared in certain periodicals. The author re- 

 marks that the word ' paradox ' " is usually held to be a synonyme 

 for flippant smartness," but that it really means a proposition that 

 seems absurd but is nevertheless true in fact. His own paradoxes, 

 however, hardly answer to either of these definitions ; for only a 

 perverted intelligence could regard the mass of them as true, and, 

 though they are flippant enough, we fail to see any ' smartness ' in 

 them. The book is a continuous sneer at men of genius and at in- 

 tellectual and moral superiority of every kind. Mr. Walsh main- 

 tains that " men are more nearly equal than we suppose," and 

 that " there is no such great difference between a genius and a 

 dunce." " The great historian, the great poet, the great statesman, 

 the great philosopher, . . . are as fallible and as foolish ... as 

 you and I are. The intellectual feats that they perform only hap- 

 pen to be more difficult to the average man, that is all." He en- 

 deavors to sustain this view by citing examples of follies and sins 

 committed by men of genius ; but most of the men he refers to 

 were not geniuses at all, but commonplace men who followed the 

 profession of literature. The author adopts a cynical tone through- 

 out, which adds to the disagreeableness of what he says. More- 

 over, there is hardly any thing of a different character in the book, 

 except some fantastic remarks on ' The Sense of Pre-existence,' 

 and a few pages about ' Mother Goose.' Mr. Walsh says that he 

 has collected these papers into a volume, " because the author 

 likes them," and the world will probably be willing to grant him 

 the exclusive enjoyment of them. 



The Critical Period of American History. By JOH.V FISK.E. 

 New York, Houghton, Mifflin, cS: Co. 12". $2. 



This book consists of lectures delivered first in the Old South 

 Meeting- House at Boston, and afterwards in other cities. It re- 

 lates to the time between the close of the revolutionar>' war and 

 the adoption of the Federal Constitution, which Mr. Fiske justly 

 regards as the most important period in our national history-. Our 

 popular historians are apt to give altogether too much attention to 

 wars and other stirring events, and too little to the quieter but 

 more influential movements of political and constitutional reform. 

 Mr. Fiske, however, has a much better idea of the historian's duty, 

 and has accordingly given us a much better book than most of his 

 predecessors have done. His philosophical studies have given him 

 the comprehensiveness that good historical writing requires, and 

 have fitted him to trace with clearness the chain of causes and 

 effects which is the fundamental fact in historical development. 

 His style, too, with its clear and easy flow, is well fitted for the 



