November 25, 1909] 



NATURE 



109 



I'lic regions of Virginia surrounding tlie Chesapeake 

 Hay probably produce more early potatoes than any other 

 part of the eastern States, the annual value of the crop 

 approaching 6,000,000 dollars. Little damage is caused 

 by blight, but the Colorado potato-beetle {Lcptinotarsa 

 iliicmlineala, Say) is a serious pest, and only very crude 

 methods are adopted for keeping it in check, because of 

 the prevalence of negro labour and the scarcity of capable 

 white help. Mr. Popenoe gives a description of the pest 

 and of the damage it does, and describes experiments in 

 which three applications of lead arsenate mi.xcd with 

 Mordeaux mixture, the first about the time the eggs begin 

 10 hatch, and the others at intervals of three weeks, 

 sulliced to control it. 



Some new breeding records of the coffee-bean weevil 

 (Aracccnis fasciciilaliis, Dc Gcer) are published by Mr. 

 Tucker. He found the larval and pupal stages in some 

 dried maize stalks, and obtained evidence that the insect 

 causes injury to the maize plant. The attacks begin in 

 the green stalks before the corn matures, and thus cause 

 stunted ears. This weevil has also been found in the 

 berries of the China berry tree. 



Stringent laws are in operation in most of the Slates 

 with regard to the importation of nursery stock. It is 

 commonly necessary to notify the State entomologist 

 within twenty-four hours of the arrival of the stock, and 

 to fumigate satisfactorily. The laws of the different 

 States are not all alike, and Mr. Burgess has collected in 

 a short pamphlet the requirements which must be com- 

 plied with by those making inter-State shipments of 

 nursery stock. The pamphlet will form an interesting 

 study for those who are agitating for some State supervision 

 in this country. 



77//!; METHODS OF MATHEMATICS.' 

 'yillj position assigned to mathematics in the educational 

 system of every civilised country seems to mark it 

 out as an essential element of mental culture, but an ex- 

 amination of the arguments that have been put forward 

 from time to time to justify this position reveals a diversity 

 ol view that is at first sight discjuieting. 



Of those who acknowledge the value of mathematics 

 there are many who see th.at value almost solely in its 

 usr'fulness, in the help it brings to other sciences. Not 

 unnaturally, those who are absorbed in the work of applied 

 science are apt to turn away from the more abstract 

 developments of modern mathematics ; even the men whose 

 special pur.suits call for constant applications of mathe- 

 matical processes, as in physics and engineering, can 

 hardly be blamed if they lay special emphasis on those 

 eliinents of a mathematical training that are of immediate 

 application to their daily work. Yet it is not this aspect 

 of mathematics that is usually present to the professional 

 mathematician when he seeks to uphold the position of 

 lii>i subject in an educational system. 



Mathematics m.ay be assigned its place for a different 

 reason. To those who reject the argument from utility, 

 mathematics is not the humble auxiliary of other sciences, 

 but is itself the one genuine science ; it often comes to 

 the aid of other sciences, but does not depend for the 

 justification of its existence on the help it may be able to 

 l)ring. From the adherents of this view come the familiar 

 arguments for the disciplinary value of a mathematical 

 tr.iining in which deductive logic is given a prominent 

 place. 



The question naturally arises whether these two aspects 

 of mathematics are incompatible. To the teacher, whether 

 in school or in college, the question is of prime import- 

 ance ; for the whole scheme of study and the methods of 

 instruction will be found in the long run to be deter- 

 mined by the general attitude that is taken up with respect 

 to the value of the subject. At the present time there is 

 considerable uncertainty in the minds of te.achers regard- 

 ing the methods of school mathematics, and many of the 

 older men are disposed to look unfavourably on recent 

 changes as tending to impair the disciplinary effects of a 

 mathematical training. 



It may help us tc understand more clearly the points 



1 From the inouguml .icMrew ilelivered on October n by Dr. George A. 

 • Gibson, Professor of Mathematics in the University of Glasgow. 



NO. 20QT, VOL. 82] 



at issue if we consider for a little the trend of mathe- 

 matical inquiry during the nineteenth century. It is not 

 necessary that' I should sketch even in the roughest out- 

 line the' development of mathematical science in that 

 period; it will be suflkient for my purpose to indicate 

 one dominant feature of the mathematical methods that 

 were introduced in the early years of that century and 

 that revolutionised the treatment of pure mathematics 

 before it had reached its close. 



During the eighteenth century the infinitesimal calculus 

 and the doctrine of infinite series enabled mathem.aticians 

 to investigate problems, intractable by the older methods, 

 with a lacility that led to a wide extension of the field 

 of mathematical inquiry and to an enormous accumulation 

 of results. In this period interest was centred less in 

 demonstrations than in results, which were often reached 

 by methods of a strange character, and sometimes, indeed, 

 seem so absurd in theinselvcs that we find it hard to under- 

 stand how they were ever promulgated. Induction played 

 a most important part in the discovery of theorems, and 

 these inductions were often made froin insufficient data 

 and too seldom verified by subsequent tests. When the 

 novelty of the processes had worn o(I, the necessity for 

 a critical examination of their legitimacy became evident, 

 and this examination was one of the tasks of the nine- 

 teenth century. It should be noted, however, that the 

 great critics were also great creators ; the criticism of the 

 methods of mathematics was accompanied by a wide 

 extension of its domain. 



Of those who first saw the necessity for criticism and 

 set themselves to the task were Gauss, Cauchy, and Abel. 

 Gauss was first in the field, but, for various reasons, his 

 work was long neglected. It was not until the publication 

 in 1821 of Cauchy's " Cours d'Analyse " that the attention 

 of mathematicians was effectively directed to the question. 



Geometry in the hands of the Greek mathematicians had 

 been reduced to a system of logically consistent truth ; 

 from assumed definitions, axioms, and postulates the 

 various theorems of geometry were derived by the methods 

 of formal logic, and Kuclid's " Elements " were for 

 centuries the standard of mathematical rigour. Algebra, 

 or, in modern terminology, analysis, was of much later 

 growth, and Cauchy's reference to the rigour _ that is 

 demanded in geometry simply means that the time had 

 come when the revision of principles and inethods that 

 the Greek mathematicians had effected in geometry should 

 be carried out for algebra or analysis. The eighteenth 

 century was a period of great activity in the development 

 of analysis, and it is not surprising that the pioneers of 

 this development should have been more interested in the 

 resources of the country they were opening up than in 

 the roads they followed. Their methods of mathematical 

 inquiry were not limited by the traditional canons of Greek 

 geometry; they included induction as well as deduction, 

 there was constant appeal to intuition, and general 

 theorems in mathematics were often est.ablished from 

 physical considerations. The usefulness of mathematics 

 as an aid in the investigation of the phenomena of the 

 material world was the predominating feature of the 

 period. The aim of Gauss, Cauchy, Abel, and their 

 coadjutors was, in general terms, to do for analysis what 

 the Greeks had done for geometry, and to make _ mathe- 

 matics an independent science by clearly defining its pro- 

 vince, stating the postulates from which the science st.arts 

 and developing the consequences by the laws of logical 

 operation without appeal to extraneous considerations. 



The work of scrutinising the methods of analysis was 

 vigorously pursued throughout the nineteenth century, and 

 exerted a far-reaching infiuence. The notion of continuity, 

 which seems so naturally to attach to geometrical 

 quantity, required to be formulated in such a way that it 

 would be amenable to calculation. Current conceptions of 

 nuinher were too vague, and it was found necessary to 

 analyse more carefully the notion of nuinerical quantity 

 so as to frame definitions and to establish rules of operation 

 for the continuous variable of analysis. The so-called 

 im.iginary numbers h.ad been long in use, but their exist- 

 ence was of a precarious nature, and the right to use 

 such numbers h.ad to be justified. 



As will bo easily understood, many of these discussions 

 are of a very abstract nature, but they have provided a 



