September 5, 19 12] 



NATURE 



5 



scientific training;. On the other hand, much good 

 work was done by French mathematicians in the 

 eighteenth century in laying the foundations of 

 naval architecture. The discussions of recent 

 years have tended to the conclusion that the 

 mathematical portion of an engineer's training is 

 best given in the regular manner by a mathe- 

 matician, rather than in a selected course by an 

 engineer. There can be no doubt as to the value 

 of mathematics, both in indicating the lines along 

 which experiments must be made and in framing 

 a theory from their results. Many problems, 

 such as that of the design of ship propellers, stand 

 urgently. in need of the mathematician's help. 



.\t an extra meeting of Section 1\', Mr. P. J. 

 Harding lectured on "The History and Evolution 

 of Arithmetic Division." The two methods of 

 calculation prevalent in Europe previous to the 

 intioduction of the Arabic numerals were that of 

 the algorists, who used counting-boards ruled with 

 lines representing successive powers of ten on 

 which counters were placed, and that of the 

 abacus. Arabic numerals followed the trend of 

 commerce from India through Arabia and Italy 

 into northern Europe ; so far as we know, they 

 first appeared in Itah' in 1202. Subtraction was 

 first performed from the left by scratching out the 

 digits successively, a method evolved from the 

 sand-board used in the East, which was small 

 compared with the size of the numerals, so that 

 successive deletion was necessary. From this 

 followed the method of division by scratching, 

 known as the galleon method owing to a fancied 

 resemblance of the resulting disposition of digits 

 to the form of a ship. The modern method of 

 division first appeared in print in Italy in 1494, 

 but it only superseded the galleon method after a 

 struggle which lasted more than a century. In 

 England its ultimate triumph was largely due to 

 the writing-master Cocker, who advocated it to 

 the exclusion of the older method. 



At a special meeting of Section 111 (n), Sir J. J. 

 Thomson (Cambridge) gave a lecture illustrated 

 by experiments on " Multiply Charged Atoms," in 

 which he described some recent investigations on 

 positive charges. He explained the parabolic 

 grouping effected by the simultaneous action of 

 electric and magnetic fields, and showed photo- 

 graphs of the parabolic arcs obtained in various 

 particular cases. In the case of mercury atoms, 

 eight such arcs were obtained, due to one or more 

 of the charges originallv carried being lost in 

 transit, so that the particles arrived at the screen 

 with their original energy but with reduced charge. 



ht an ordinary meeting of Section IV, Dr. 

 h.. N. Whitehead (London) read a paper on the 

 principles of mathematics in relation to elementary 

 teaching. The only justification for the inclusion 

 of mathematics in a liberal education is the power 

 of abstraction and deductive reasoning fostered 

 thereby. These powers can only be acquired by 

 constant practice, and no short-cuts are possible. 

 But this does not imply that such powers are to 

 be assumed in the pupil from the outset. On the 

 contrary, no generalisation can be made bv the 

 XO. 2236, VOL. 90] 



pupil until he is familiar with the raw material 

 from which it is to be made. There is no final 

 degree of rigour in deduction, and the degree to 

 be adopted is a matter for the teacher to decide. 

 His personal choice would be approximately the 

 degree of rigour, though not necessarily the con- 

 tent of Euclid's Elements. No compromise is 

 desirable between the purely utilitarian procedure 

 of looking up a formula in an engineering pocket- 

 book and the acquisition of a mathematical habit 

 of mind by years of practice in abstraction and 

 deduction. 



Mr. G. E. St. L. Carson (Tonbridge) read a 

 paper on the place of deduction in elementary 

 mechanics. He suggested that, besides the old 

 method of teaching mechanics in which a structure 

 of deduction was raised on a few postulated laws, 

 and the new method in which principles are demon- 

 strated independently by experiment, there is 

 a third method possible in which the logical inter- 

 dependence of the principles demonstrated is 

 discussed. Not only is this an aid to understand- 

 ing the foundations of the subject, but they are 

 shown to constitute a broad inductive basis. 



A paper by Dr. T. P. Nunn (London) was read 

 on the proper scope and method of instruction 

 in the calculus in schools. He advocated the 

 teaching of integration by means of graphical 

 illustration on the lines originally adopted by 

 Wallis. This should be followed by a considera- 

 tion of differentiation as the converse geometrical 

 problem. The teacher should avoid all use of 

 such mystic phrases as " infinite " and " ultimately 

 become," keeping carefully to the definition of 

 limit in terms of finite quantities. 



At meetings of Section IV (b), in conjunction 

 with the International Commission on the Teach- 

 ing of Mathematics, reports were presented, with 

 a "few explanatory remarks, by delegates from 

 twenty-one countries. The reports exceeded 280 

 in number, forming an aggregate of more than 

 gooo octavo pages. These may be obtained from 

 Messrs. Georg et Cie. , of Geneva ; the English 

 reports have recently been issued in two volumes 

 by the Board of Education. The commission was 

 reappointed for a further period of four years, in 

 order that a digest of these reports may be pre- 

 pared for the use of teachers in each country. 

 The commission has also conducted special investi- 

 gations, and reports were presented on the results 

 of two of these. 



Prof. C. Range (Gottingen) presented a report 

 on the mathematical training of the physicist 

 in the university. The need for the closer co- 

 operation of the mathematician and the physicist 

 is strongly felt. It would be of benefit not only 

 to the future physicist or engineer, but also to the 

 student of pure mathematics, if in mathematical 

 lectures theoretical solutions were followed up by 

 numerical computations and applications to 

 material problems. It is also felt that mathe- 

 matical teaching in the university would be 

 improved if the lecturer were assisted by demon- 

 strators who could keep in personal touch with the 

 student, and aid him as difficulties arise. In com- 



