January 17, 1901] 



NATURE 



275 



Now I think there are several points in the above sentences 

 liable to misconstruction. Mathematics is purely a form of 

 reasoning, and, as in the case of all forms of logic, it is merely 

 an instrument, and the product depends upon the material dealt 

 with. This may be the result of observation or of experiment, 

 either of which may or may not be statistical in character. 

 Prof. Howes, in contrasting "statistical, experimental and 

 mathematical tests" with the "observational method," seems 

 to be looking upon mathematical reasoning as something which 

 has more relation to experiment than to observation. I fail to see 

 why as an instrument it is less applicable to the gigantic over- 

 thrust of the geologist than to the test-piece in the laboratory, less 

 applicable to an observation on the mottling of birds' eggs than to 

 an experiment on the breeding of mice. It is perfectly true, as 

 Huxley said, that what you get out of the machine depends 

 entirely on what you put into it. Such a platitude in its 

 right context may be a useful reminder. But wUhotit yaiir 

 machine you may be able to get nothing at all out of your 

 material ; and I venture to think that this is the case, not with 

 a few, but with many branches of biological inquiry. 



The reason thereof is easy to find. In vital phenomena we 

 are never able to repeatedly observe or to experiment, as we can 

 very closely do in physics, under exactly the same conditions 

 with the same quantities of the same substances. The reader 

 will probably interject, "No, and this is the very reason 

 why mathematics can be applied to the one and not to the 

 other ! " On the contrary, because in biological mvestigation 

 an exact A cannot be associated with an exact B, and an 

 exact C observed (as we can do in physics), biology jrequires a 

 much more refined logic, much more subtle mathematics than 

 the simplest branches, at any rate, of physical inquiry do. There 

 is nothing more full of pitfalls than " ordinary reasoning " ap- 

 plied to the problems of association. The biologist observes that 

 some A is associated with B, and that some C is associated with 

 B. But if he wishes to discover whether the relation between 

 A and C is causal, he will need all the refinements of symbolic 

 logic, a mathematical analysis, which is analogous to the 

 geometry of hyperspace, before he can come to a definite logical 

 conclusion on the possible relationship of A and C. He may 

 observe as much as he will, but he will not find out 

 whether the association is confirmable or non confirmable 

 without this higher logic. It is the all-pervading law of 

 vital phenomena that no two individuals are identical among 

 living forms, that variation exists in every organ and every 

 character, which, so far from disqualifying biological phenomena 

 for mathematical treatment, enforces a need for the most 

 generalised forms of mathematical reasoning. Prof. Howes 

 tells us that the mystery of life can never be solved by mathe- 

 matical treatment. If he had said that the mystery of life can- 

 not be solved by any treatment whatever, I should have heartily 

 concurred with him. But if he means that observation, rather 

 than observation plus the higher logic, is likely to discover the 

 most comprehensive formula under which the phenomena of life 

 can be described, then I am quite sure he is in error. Observa- 

 tion, for example, has collected a mass of most valuable facts 

 during the past thirty years, but can any one by merely verbal 

 generalising upon these facts venture to assert that evolution by 

 natural selection is more than a probable hypothesis ? The very 

 nature of such ideas as variation, whether continuous or discon- 

 tinuous, as inheritance, whether exclusive or blended, as 

 selection, whether natural or sexual, leads us to the idea of 

 number, of statistics, of frequency, of association, and enforces 

 upon us an appeal to mathematical logic. If we are to feel that 

 evolution by natural selection is as sure a formula as that of gravi- 

 tation, it will be because mathematics steps in and reasons on the 

 data provided by the Tycho Brahes and Keplers of biological 

 observation. 



Prof. Howes must not for a moment suppose I claim biology 

 for the mathematician. I do not even want the mathematician 

 to have a biological training, conscious as I am personally of 

 the disadvantages of its absence. The mathematician who turns 

 physicist is rarely so valuable a discoverer as the born and 

 trained physicist who knows mathematics so far as he needs 

 them. I believe the day must come when the biologist will — 

 without being a mathematician — not hesitate to use mathe- 

 matical analysis when he requires it. The increasing amount of 

 work being turned out, both in America and Germany, by the 

 younger biologists with a mathematical training, is a sign of the 

 times. In England, I suppose (where, as usual, an Englishman, 

 Mr. Francis Galton, first indicated the great possibilities of a 



NO. 1629. VOL. 63] 



new method), we shall be left behind, and let other nations 

 gather the fruits of our sowing. Prof Howes, indeed, leaves a 

 field for mathematical investigation ; but it was only a few 

 weeks ago, at a discussion at the Roval Society, that another 

 distinguished biologist asserted that in living forms there was no 

 such thing as number ! 



Et Verbum interrogabat Vitam : Quod tibi nomen est ? Et 

 dicit ei : Legio, id est Numeri, mihi nomen est, quia multi sumus. 

 Et deprecahatur eum multum, ne se expelleret extra regionem. 



I doubt whether the demon can now be exorcised conjure 

 Verbum ever so cunningly. Karl Pearson. 



Education in Science. 



Some discussion has recently arisen as to the methods of 

 teaching mathematics. Euclid has been condemned on the 

 score of its advancement and its antiquity. An infusion of 

 more modern geometry has been recommended, with correspond- 

 ing arithmetic and algebra. In science, at the same time, there 

 has been a tendency to recognise the historic method. Prof. 

 Perry considers it unnecessary for pupils to traverse the course 

 of their ancestors. But let us ask why this course has been 

 recommended. On account of the successive growth of faculties 

 in a historical sequence. Is this a fact or not ? It is an un- 

 doubtable fact, and it is not sufficiently realised by any teachers. 

 Prof. Perry has two saving principles, first to teach by practice, 

 and second to satisfy the pupils' instincts. These being the 

 same reasons which are used by advocates of historical methods 

 secure a certain amount of agreement. We ought to arrive at 

 the same result whether we study the natural methods of pupils, 

 or the methods of primitive peoples. But Mr. Herbert Spencer 

 has well pointed out somewhere that we ought not to go to the 

 Greeks for examples of primitive peoples. They were highly 

 and very specially developed. Hence arises a very great danger 

 in the historic method. 



With regard to practical and rational methods, it must have 

 often been noticed by teachers that a great number of pupils 

 have an inherent objection to carrying out rules without some 

 kind of reason for them. It is also to be observed that a very 

 vague, or even a verbal reason, will be more satisfactory than a 

 real one. This is surely in accord with the studies of the history 

 of science. Although it is somewhat misleading to reason from 

 the experience of men of genius, it may be worth while to call 

 to mind the intense satisfaction of Darwin with Euclid's con- 

 catenation, and the disgust of Huxley at the irrational rule and 

 rote method of mathematics under which schoolboys grow up. 

 It is the exceptional boy who delights in carrying out enigmatic 

 rules, although all have a temporary taste for that work as sauce 

 to the rest. It is treacherous to reason from one lesson of this 

 kind to a regular course of it. 



It is customary to speak of the activity, observation, ingenuity 

 of children. But it is not found, either in the history of children 

 or of primitive peoples, that they are capable of continued mental 

 application, observation or contrivance. We might just as well 

 speak of the great reasoning powers of children on account of 

 their perpetual " why." It is also improper to underestimate 

 the value of this tendency. By it children acquire and cement 

 their knowledge, although a chain of real reasoning will abso- 

 lutely exhaust them. 



From this kind of reasoning we conclude that the time-taught 

 method now pursued in mathematics is a reasonable one, that 

 Euclid with the algebra and arithmetic corresponding is in the 

 main advantageous. But why ? Because it is conformable to 

 the instincts of pupils, and also because it is historic. But is it 

 conformable enough ? Is it historic enough ? I think not. 

 Euclid was a grown man in a grown community of very special 

 bent of mind. Where he does not agree with the reason for his 

 inclusion in school curricula he should be neglected. But 

 instead of supplementing him from more recent geometry, it 

 should be from more antique writers and from study of pupils' 

 methods. 



Now we come to the bearing of this on the teaching of 

 science. We are comparatively new at this game. We are 

 finding that we have started in too high a key, and we are being 

 recommended to go back. I have not yet seen a recommenda- 

 tion to exactly imitate the mathematical teachers, and go back 

 to Pliny, Geber, Gilbert and Pallissy. But several have advised 

 Boyle and Black. Along with this advice is an insistence on 

 quantitative work from the very start. It appears to me this 

 is a very grave mistake. The use of a rough balance and rough 

 methods of measurement is all that should be aimed at in 



