544 



NA TURE 



[April Z, 1886 



and their consequences in geometry and mechanics ought 

 assuredly to be considered as within tliat range, as ought also, 

 for a complete view of ordinary algebra, vector products and 

 quotients in one plane. 



If we inquire in what manner we should expect the idea of a 

 Vector and its attendant ideas to affect our elementary teaching, 

 I think the answer would be that it would naturally lead t > a 

 different grouping, or arrangement in order, of the various 

 branches taught. It would lead us to group them not according 

 to their subject-matter — arithmetic and algebra, the sciences of 

 number, particular and general ; geometry, the science of space ; 

 trigonometry, in one aspect treating of space and number in 

 combination, in another as a development of algebra ; statics, 

 dealing with forces in equilibrium ; dynamics, with forces pro- 

 ducing motion ; and so on — but according to their form, as 

 dependent on the nature of the magnitudes dealt with. One- 

 dimensional magnitudes, that is, magnitudes defined by one 

 element only, whether such as are completely defined by one 

 element, or more complex magnitudes regarded for the moment 

 in respect of one of their elements only, would naturally form 

 the first stage, with subdivisions according as the treatment is 

 purely qitantitaiive or metj'k-, or scalar, that is, metric with the 

 addition of the notion of sign or sense. Then would follow 

 two-dimensional magnitudes, or magnitudes defined by two 

 elements, treated with respect to both elements in subdivisions 

 metric and scalar as in the first stage, and also finally as com- 

 plete Vectors. 



If we further inquire how far these notions have in fact 

 affected our elementary text-books, I think we shall find that 

 the extent to which they have done so is very small. A com- 

 parison of the text books of the present day (I speak of them in 

 the gross, not forgetting that there are some important excep- 

 tions) with those that were current at the time when my own 

 mathematical studies began, an interval of some f irty years, 

 produces the impression of likeness rather than that of contrast. 

 Changes, which are welcome improvements, have doubtless 

 been made in matters of detail, and in various ways the paths 

 have been smoothed for the student ; but the general treatment 

 is essentially the same, and shuws very little sign of independent 

 thought, informed by more extended views, having been exer- 

 cised with regard to the old traditional modes of presenting the 

 subject as a whole. 



The algebra, for instance, of our ordinary text-books is (if I 

 may venture to give it a nickname which every brother Johnian 

 at any rate will understand) lief'tadiaholic,^ or that whose highest 

 outcome, in the mind of the pupil who has studied it, is the 

 solution {so called) of a hard equation or equation problem — 

 little more in fact than a series of rules of operation, which 

 skilfully used (and how many fail to attain even this amount of 

 skill) will solve a few puzzles at the end, but very barren of any 

 intellectual result in the way of mental training : — an algebra 

 in which the interpretation of negative results and the use of the 

 negative sign as a sign of affection has been ignored, or so 

 lightly dwelt on, that the notion of the signs + and - as 

 appropriately expressing opposite senses along a line, has to be 

 elaborately explained as almost a new idea in commencing tri- 

 gonometry ; and further, an algebra which, as Prof. Chrystal has 

 observed in his address to the British Association, is useless as 

 an instrument for application to co-ordinate geometry, so that 

 the student has at this stage practically to study the subject 

 again, and only then obtains something of a true notion of what 

 algebra really is. 



With the foregoing general considerations as a guide, I will 

 now examine in some detail the correlation or affiliation of the 

 several branches of elementary mathematics to which they seem 

 to lead. 



Mathematics naturally begins by treating magnitudes with 

 reference to the single element of quantity. The answers to the 

 simple questions. How many ? How much ? How much greater ? 

 How many times greater ? lead up to the arithmetic of abstract 

 and concrete number, and the doctrines of ratio and proportion, 

 and the development of these with the use of the signs -^ , - , 

 &c., as signs of the elementary operations, and letters to denote 

 numbers or ratios, naturally leads to generalised arithmetic or 



' The allusion is to a paper which used to 

 Examinations at St. John's College, Cambridge, 

 hard equations and equation problems, familiarly kr 

 devils." As a test of a certain kind of skill in algebraical operation and of 

 ingenuity and clearheadedness it was not without considerable value, but it 

 tended to produce false notions of algebra in its relations to mathematics 

 generally. 



arithmetical algebra. At this stage a-l>, where b is greater 

 th.an a, is an impossible quantity, and a negative quantity has 

 by itself no meaning. In this earliest stage the magnitudes 

 dealt with are either pure numbers or concrete one-dimensional 

 magnitudes, value, time, length, weight, &c. , whose measure- 

 ments are assumed as known. There are few magnitudes which 

 are metric or quantitative only, but all magnitudes have quan- 

 titative relations which may be regarded apart from their other 

 relations, and so may be the matter or subject of arithmetic, if 

 they are such that their quantity can be estimated definitely or 

 measured. Purely metric magnitudes are such as can be con- 

 ceived to be reduced in quantity down to zero or annihilated, 

 but of which the negative is inconceivable, so that at zero the 

 process must stop. Such are many magnitudes that are mea- 

 sured by integral numbers — as population, numbers of an army 

 or a flock, a pile of shot, &c. , or continuous magnitudes, such 

 as mass, energy, quantity of heat or light, the moisture of the 

 atmosphere, the siltness of water, &c. But there is a far larger 

 class of magnitudes, of which it is true that not only the oppo- 

 site or negative can be conceived, but that they cannot be fully 

 treated without regard to such opposite. For these, reduction 

 to zero, or annihilation, is only a stage in passing from the mag- 

 nitude to its opposite, e.g. time after antl time before a given 

 epoch, lengths forward and backward along a line, receipts and 

 payments, gain and loss, and so forth. The consideration of 

 such magnitudes at once leads to the scalar subdivision of the 

 one-dimensional stage. In this, magnitudes which are them- 

 selves purely scalar, or the scalar elements of more complex 

 magnitudes, are alone considered. But to the quantitative ele- 

 ment is now superadded the notion of sign or sense, appro- 

 priately denoted by the signs + and - , which, without ceasing 

 to be signs of operation, are now regarded also as signs of affec- 

 tion. The introduction of this notion leads at once to scalar 

 algebra, in which a- b, where li is greater than a, is no longer 

 an impossible quantity, and a negative result has a definite 

 meaning, so long as the magnitudes dealt with are not purely 

 metric. The step from arithmetical to scalar algebra, though 

 very simple and almost insensibly made, should, I think, be 

 much more distinctly emphasised in our teaching and our text- 

 books than is usually the case. Exercises in metric and scalar 

 readings of the same simple expressions should be frequent, and 

 negative results, whenever they occur, examined and shown to 

 be impossible only if the magnitude in the question is purely 

 metric, but interpretable if it is scalar. Thus the idea would 

 be gradually evolved that the impossibles or imaginaries of 

 algebra are so in a purely relative sense and with regard to the 

 particular subject-matter treated of, and it would become readily 

 conceivable that the remaining impossible quantity a + b -.J - I, 

 to which form scalar algebra, working on the basis of its laws 

 of combination, would show that all expressions may be reduced, 

 may be completely interpretable when ultra-scalar magnitudes 

 form the subject of investigation. 



Passing now to the consideration of special magnitudes and 

 how far their discussion can be carried in the one-dimensional 

 stage, I think we shall arrive at some important practical 

 results. 



The scalar element of space is length measured forwards or 

 backwards along a line, and the resuhmg geometry is the simple 

 geometry of points on the same line. Starting from the defini- 

 tion that - AB is BA, the fundamental proposition is that 

 AB + BC = AC, whatever be the positions of A, B, C on the 

 line, and this with a few simple consequences completes all that 

 is necessary to be considered in linear geometiy. 



Combine with this the notion of time, and the science oflinear 

 or scalar kinematics emerges This includes the measurement 

 of the motion of a point along a given line by the scalar magni- 

 tudes, spced^ and acceleration, and the discussion of different 

 kinds of linear motion, uniform and uniformly accelerated, and 

 so the laws of falling bodies. When the notion of a variable 

 rate became firmly grasped, the investigation might be extended 

 to sone simple cases of variable acceleration without any large 

 demand on algebraical skill, and so the fundamental notions of 

 the fluxional or differential calculus and some idea of its scope 

 and aim be attained. 



Introduce now the notions of force and mass and the axioms 

 of force or motion as contained in Newton's laws, and the 

 science of linear or scalar dynamics results. If we drop for an 



* The term speed has been happily appropriated for the scalar element of 

 velocity. A corresponding term is wanted for the scalar element of accelera- 

 tion ; no better word than qitickenittg suggests itself to me. 



