June 25, 1891] 



NATURE 



171 



would appeal to Prof. Greenhill. When a man, so able 

 and unconventional as he, writes a book of 455 pages on 

 the infinitesimal calculus, is it too much to expect that he 

 will everywhere give a thorough discussion of its few 

 fundamental principles, that he will rigorously prove what 

 he professes to demonstrate, and honestly point out what 

 he assumes without demonstration ? We certainly ex- 

 pect him to root out of the subject every trace of the 

 sham demonstration — that wily artifice of the coaching 

 and examining days of our dear old abna ;«rt/d'r— which 

 used sometimes to be dignified by the name of the 

 "short proof." This used, to be employed when we 

 had on hand the establishment of some proposition which 

 was not universally true (although usually so enunciated), 

 or which had exceptions too tedious to enumerate in an 

 examination. The method was to make a kind oi precis 

 containing as few words of intelligible English as pos- 

 sible, but a considerable sprinkling of ingeniously con- 

 structed but unexplained symbols and formula; ; so that 

 an examiner of average conscience, suspecting that the 

 truth was not there, might nevertheless, without mental 

 distress, make believe that it luas there, and award the 

 coveted marks. 



We complain that Prof. Greenhill should countenance 

 the slipshod exposition of elementary principles which is 

 the bad feature of so many of our English mathematical 

 text-books. Having started his furrow, he should have 

 ploughed to the end. He may retort that he has adhered 

 to the traditional usage out of consideration for the weak- 

 ness of the practical man, who abhors sound logic quite 

 as much as his academic brother. Cruel consideration 

 for the practical man ! for what he wants above all is a 

 firm grasp of the fundamental principles of the calculus ; 

 he has rarely any use for the analytical house of cards, 

 composed of complicated and curious formulae, which the 

 academic tyro builds with such zest upon a slippery 

 foundation. 



It would take up too much of the columns of Nature 

 to give all the examples that might be adduced of the 

 laxity we complain of. A few must suffice. We are told 

 in § I that the " calculus to be developed is the method 

 of reasoning applicable to variable quantities in a state of 

 continuous change ; " yet no definition or discussion of 

 " continuity " is given : the word, so far as we can find, 

 does not occur again in the first chapter, although it is 

 the keynote of the subject. " Newton's microscope," for 

 example, is quoted in § 9, as a proof of the theorem 

 Zi'(chord arc) = i ; but the essential condition, " in medio 

 curvaturae continuae," which makes it a proof (if proof be 

 the word that describes its purpose) is omitted. Although 

 the differential calculus is merely a piece of machinery 

 for calculating, and calculating with limiting values, a 

 limiting value is not defined ; nor is there any discussion 

 of the algebra of limiting values —a matter which has 

 puzzled beginners in all ages, and which has stopped many 

 on the threshold of the calculus. It is true that we are 

 referred to Hall and Knight's " Algebra," but what we 

 find there is little to the purpose, and certainly could 

 never have been meant by its authors as a foundation 

 for the difi'erential calculus. 



In § 16 we are given a quantity of elementary instruc- 

 tion, in the middle of which the trigonometrical functions 

 are inadequately defined ; but nothing adequate is said 

 NO. II 30, VOL. 44] 



regarding the sense in which the many-valued functions 

 sin~'a', cos~'.r, &c., are continuous : and in § 25 the 

 beginner is led by implication to believe that </(sin"'.r) vir 

 is always -f \l J{i - x-), and <i{co%~^x)!dx always 

 - i/x'(i - x'^)\ although this is not so, and the point 

 is one that is of the greatest importance in the integral 

 calculus, and is a standing rock of offence for learners. 

 In § 28 we have, reproduced "for the sake of complete- 

 ness," the time-honoured " short proof" of the existence 

 of the exponential limit, which proof is half the real 

 proof p/us a suggestio falsi. If the proper proof (a very 

 simple matter) was thought too much for the reader, then 

 it would have been better simply to tell him the fact, and 

 not to corrupt his intellectual honesty by demanding his 

 assent to a piece of reasoning which is not conclusive. 

 § 31 is no better ; what, for instance, does Prof. Greenhill 

 mean, after proving that exp n — e", where n is a positive 

 integer, by saying, " and thence generally by induction^ 

 exp X = e^ for all values of a-." It would scarcely be 

 possible to write down a statement to which more excep- 

 tions could be taken unless " induction " is a misprint for 

 " assumption." 



The chapter on the expansion of functions is not satis- 

 factory. We are first introduced to " a general theorem 

 called Taylor s theorem, by means of which any function 

 whatever can be expanded [in ascending powers of .f]." 

 Prof. Greenhill knows as well as we that there is no such 

 theorem. No theorem ever to be discovered will expand 

 in ascending powers of x, ijx, Jx, log x, or any function 

 which has x — o for a critical point. Why does our 

 author hide his light from the reader ? Does it make the 

 apprehension of Taylor's theorem any easier to enunciate 

 it falsely.'' We are told in § 114 that "some functions? 



for instance sec-U", cannot be expanded in an 



infinite series in ascending powers of x, because x must 

 be greater than unity, and the expansion by Taylor's or 

 Maclaurin's theorem would h^ divergent, and the theorem 

 is then said to fail." 



" This difficulty will be avoided if we can make the 

 series terminate after a finite number of terms." 



We would not advise the practical man to try to over- 

 come the difficulty of expanding sec'^r by the method 

 thus indicated (use of Maclaurin's theorem with the 

 remainder), because the result might be that the bond of 

 amity struck in the preface between him and the author 

 would be broken. All the king's horses and all the king's 

 men will not get over this difficulty. Incidentally we are 

 told in § 112 that a rigorous proof is given in treatises on 

 trigonometry of the resolution into factors of sin 6 and 

 cos 6. If standard English treatises, such as Todhunter, 

 Locke, and Johnson, are meant, this is not true : the 

 demonstrations they give are unsound. Mr. Hobson's 

 article on trigonometry in the " Encyclopaedia Britan- 

 nica " is the only separate English treatise on trigono- 

 metry of which we are aware where a sound proof can be 

 found. 



When so many novelties of less importance are noticed, 

 surely our author might have found a place for a reference 

 to the theorem that puts the expansibility of a function 

 in ascending powers of x in its true position, viz. Cauchy's 

 theorem that every function is so expansible within a 

 certain region surrounding x — o, provided x = o be 

 not a critical value. Considering the great importance 



