April 8, 1886] 



NA TURE 



545 



instant the notion of time, or rather of cha}i<ie in time, we have 1 

 linear statics, which consists of little more than the single pro- 

 jjosition — the ** tug-of war " proposition — that the resultant of 

 any number of forces along the same line is their scalar sum. 

 Linear kinetics, however, covers a wide field — the relations of 

 force, mass, and acceleration, their measures and their applica- 

 tions to simple cases of linear motion ; the time integral of force, 

 momentum ; the space integral, work ; energy, kinetic and 

 potential ; the relations of force applied to resistance overcome 

 in simple machines working steadily ; impulsive actions in col- 

 lisions and explosions ; and other simple developments — would 

 here be stulied in their simplest forms apart from any greater 

 mathematical difficulties than arithmetic and very rudimentary 

 algebra, and yet involving almost every truly dynamical principle 

 needed for the highest problems in dynamics. Here, with per- 

 haps a few applications to other branches of physics, the range of 

 the one-dimensional stage ends. 



Proceeding next to two-dimensional magnitudes, we commence 

 of course with elementary plane geometry, in which the proposi- 

 tions, which ae not purely descriptive, deal with the magni- 

 tudes considered in purely metric relations. 



The introduction of the notion of sense for lines and angles, 

 denoted by the signs -I- and - , leads in one direction to ele- 

 mentary trigonometry, and in another to co-ordinate geometry. 



Kinematics is now extended to motion in two dimensions, and 

 this should lead at once to the notion of velocity, acceleration, 

 &c., as vector magnitudes, and with this the general notion of a 

 vector and vector addition. In dynamics force emerges as a 

 vector, and the composition offerees regarded as the addition of 

 vectors hays the foundation of statics, or the relations of forces 

 independent of the element of time, to be developed on the one 

 side with the aid of pure geometry and graphical methods, on 

 the other by the application of trigonometry. This is naturally 

 succeeded l^y uniplanar kinetics, developed more or less fully till 

 it extends to regions beyond the range of elementary mathe- 

 matics. Algebra will have been carried on fari passu to meet 

 the increasing requirements of the special subjects, but will still 

 remain scalar with its impossible or uninterpreted symbols. 



The next step is to complete the algebra of vectors in one 

 plane. The notion of a vector and vector addition will already 

 have been grasped and will need only some further application 

 and development, but the extension of the notion of multiplica- 

 tion to vectors in one plane at once leads to the already familiar 

 algebra, but with wider meaning and without impossible 

 quantities or uninterpretable symbols. The immediate result is 

 a complete trigonometry, of w'lich De Moivre's theorem, now 

 completely intelligible and not a mere formula, forms the basis, 

 and the higher developments of ordinary elementary algebra. It 

 will then appear that ordinary algebra receives its full explana- 

 tion in vectors limited to one plane, and it will naturally be 

 anticipated that the algebra of vectors in any directions in three- 

 dimensional sjiace will be different from the ordinary algebra, 

 an expectation which will be amply justified by the study of the 

 algebra or calculus of quaternions, the grand discovery of Sir 

 VV. R. Hamilton, but to pass on to this would be to pass 

 beyond the limits of what is, in the sense of our Association, 

 elementary mathematics. 



If the correlation of the elementai-y branches of mathematics, 

 which I have now sketched out, is accepted as based on true 

 principles, I cannot doubt but that it will lead to important 

 practical consequences, the development of which I may safely 

 leave in the hands of those who so accept it. There are, how- 

 ever, a few immediate deductions from it, which occur to me as 

 naturally cnlling for expression before I close this paper. 



In the first place I would observe that while I believe the 

 several stages in the foregoing scheme to be natural and such as 

 every teacher would do well to have in his own mind in 

 arranging the course of instruction for his pupils, I do not at all 

 regard it as marking out the exact order to be followed by each 

 individual student. There is room here, still in subordination 

 to the general scheme, for wide variation according to the 

 different requirements of different students. It would in almost 

 all cases, I think, be very unwise that any one of the stages 

 should be completed before the next was commenced. For 

 instance, though the theoiy of ratio is purely one-dimensional 

 and metric, no one, I suppose, would think of dealing uith it 

 otherwise than in the incomplete form sufficient for arithmetic 

 before commencing the study of the simple two-dimensional 

 geometry of Euclid or our own text-book. And again, how far 

 scalar or linear kinematics and dynamics should be studied (or 



whether at all) before proceeding further in the two-dimensional 

 stage to trigonometry, &c. , is a question which may fairly be 

 answered in different ways according to the different objects 

 aimed at in the study of mathematics by different classes of 

 students. 



It appears to me too to follow from our scheme that, whatever 

 may be true for the select few who aim at becoming mathemati- 

 cians, for the great mass of those with whom the chief object is, 

 or ought to be, intellectual training, algebra should be studied 

 at first, not as a subject for its own sake, but as an instrument 

 for use in other subjects. I hold that, unless pur.- ued into its 

 higher developments, algebra /«• se is not a valuable instrument 

 of mental training. Can it be said that such algebra as is 

 required (say) at the Previous Examination at Cambridge, a 

 large part of which has had no application for the student in 

 any other subject, is of any value at all proportional to the time 

 it has taken him to acqui'-e it ? I think, then, that algebra 

 should be studied piecemeal : first just that small quantity which 

 is necessary for one-dimensional magnitudes treated as scalars ; 

 then, when the need was felt from the occurrence of problems 

 requiring more knowledge of algebra, adding more, and so on 

 continually, keeping up the study of algebra concurrently with, 

 and only slightly in advance of, the requirements of the subjects 

 to which it is applied. 



Again, our scheme suggests, I think, a definite answer to the 

 question : — What minimum of mathematics is it reasonable to 

 expect every educated man, not professing to be a mathe- 

 matician, to have acquired ? 



I think there are few who are satisfied with the answer 

 practically given to this question by our Universities in their 

 first examinations for matriculation or degrees. At Cam- 

 bridge, the question with reference to the " Little Go" Exami- 

 nation is even now under consideration. I would submit that 

 the subjects included within our one-dimensional scalar stage 

 together with elementary geometry, and statics, treated geo- 

 metrically, or by graphical methods only, from the two-dimen- 

 sional, would constitute a far more satisfactory minimum than 

 the present. This would exclude a good deal of the algebra 

 now expected and the trigonometry, but would add linear kine- 

 matics and kinetics. The student, who had gone through such 

 a course, would not probably be able to effect any but the 

 simpler algebraical reductions or solve any but the simplest 

 kinds of equations ; but he would have gained some notion of 

 what an algebraical formula means as the expression of a law, 

 and be able to deduce from it numerical consequences and to 

 follow out the simpler general results obtainable from it, and he 

 would have acquired a clearer conception and higher apprecia- 

 tion than is common with people otherwise well educated of the 

 part which mathematics plays in its application to the physical 

 sciences, and with it that sound dynamical basis which is the 

 essential condition of a fruitful study of physics. I feel sure, 

 too, that the consciousness of the student that he was dealing 

 with actual living laws and not with the dry bones of algebraical 

 processes or trigonometrical formula; leading to no results, and 

 that his mathematical studies were meant to be, and were, more 

 than a mere mental gymn.astic, would add life and interest to 

 those studies which would react on the whole of his mental 

 training. 



I may note, further, that our scheme seems to give the best 

 answer to the question which has frequently been mooted of 

 late, in our Association as well as elsewhere, whether statics 

 should precede kinetics, or whether it should be treated as a 

 particular case of the more general science. Linear kinematics 

 and kinetics, being one-dimensional and scalar, may well pre- 

 cede the study of statics, which deals with vectors, though not 

 of necessity in the case of one who has attained sufficient know- 

 ledge of elementary geometry not to be stopped by mere geo- 

 metrical difficidties ; but vector or uniplanar kinetics, on 

 account of its much greater complexity and its consequent larger 

 demands on mathematical attainments, would in general natu- 

 rally follow a somewhat detailed study of statics. 



1 will take this opportunity of making one other remark, 

 which, if it does not directly arise out of the present discussion, 

 is closely akin to it, and that is on the importance of our teach- 

 ing of the several branches of mathematics being prolepti<, or 

 looking forward in one stage to what will be required in a higher 

 stage. In definitions for instance, of two that are equally good 

 for the immediate purpose, that one is to be preferred which will 

 be intelligible and useful when the term defined comes to be 

 extended to higher matter. 



