THE "Forirni dimI'Xsiox."- -a Kl•IM,^' 



By JOHN JOIINnToN. M.A.. LL.I'.. 



It may be well in the first place to state briefly the arguments 

 of Mr. Aiinisoii in his article in the June (1911) issue of 

 " Knowi.edgi;," and these will be slated in his own words: 

 " The basis of our arfjiunent depends on the relation between 

 algebraical e(|nations of two and three variables and geometrical 

 figuresof two and three dimensions. It is known, for instance, 

 that the locus of a point inovint,' according to the equation 

 x'+y' = r' is the circumference of a circle, or, in other words, 

 the equation is the ' law of the circle ' ; and similarly 

 x^+y'-i-z' = r'' is that of the sphere; but there are an infinite 

 number of equations of this type, each containing one more 

 variable than the preceding one ; and arguing by analogy from 

 the first two. the next one, x''' + y'" + z'"+u''' = r'^, is the law of a 

 four-dimensioned figure .... The fact that we are unable to 

 form any mental image of such figures cannot be ascribed to 

 the equations themselves, which obviously contain no reason 

 either why they should or should not be capable of graphical 

 representation. Prima facie, if some equations can be so 

 treated, the remainder should equally admit of such treatment, 

 and our inability to accomplish this must obviously be due to 

 the absence of any mental picture that will satisfy the require- 

 ments of the equations." 



There is no necessary connection or relationship between 

 the numbers or symbols which we use in arithmetical or 

 algebraical calculations — or any combinations of these — and 

 dimensions of any kind. 1, 2, 3, 10, 10", x, x", .\^ + y'' = r% 

 x" + y" + z' = r'-', and any other numbers or symbols or 

 combinations of these are simply tools which we use in 

 arithmetical or algebraical investigations. These tools were 

 invented long ago; throughout the ages many improvements 

 and additions have been made ; and the process is still going 

 on. It is in a totally different sphere of thought and action 

 that we have got our knowledge of dimensions. We, as well 

 as our forefathers in the remote past, have observed that 

 everything has, and must have, length, breadth and thickness. 

 Our observation of this, and our knowledge of this, have 

 nothing to do with calculations of any kind. We might know 

 about dimensions though we could do no calculations, and we 

 might be able to calculate though we knew nothing about 

 dimensions and though there were none. 



There are certain kinds of calculations which have to do 

 with dimensions. Carpenters and engineers make calculations 

 in reference to wood and iron, in order to manufacture what 

 is wished. These calculations are based on measurements in 

 the three dimensions which they have made, and the data of 

 these calculations are what they have observed with their 

 eyes. The fact that many of their calculations have to do with 

 the three dimensions is not because there is any necessary 

 relationship between calculations and dimensions, but simply 

 because these calculations are in reference to the length, 

 breadth and thickness of objects existing or to be made. 

 Whether calculations have to do with dimensions or not, the 

 numbers or symbols which we use are nothing more than tools 

 in our hands. If we are raising numbers to powers far above 

 the third in order to calculate compound interest by means of 

 logarithms we do this, not because these high powers have 

 anything in actual existence corresponding to them, but simply 

 because this is a convenient way of making the calculation. 



The equations x--|-y'^ = r^ and x- + y^+^- = r'- may be 

 simply numerical, or they may be, respectively, the law of 

 the circle and that of the sphere. It is easy with a pen to 

 insert u* and any number of additional symbols which we 

 may wish. But we have not a particle of evidence that the 

 equation with u- is other than simply numerical — that it has 

 anything corresponding to it in real existence. It is not an 

 argument by analogy that the equation with u- is the law of 

 something in four dimensions ; it is a pure assumption. Our 

 knowledge that x'-+y'^ = r'-* is the law of the circle and 

 x'+y^ + z''=r^ is that of the sphere is got by experience of 

 the circle and the sphere, and by that alone. .Vnd we have no 



experience of anything in four dimensions. A man in the 

 presence of three mountains may make sketches of them on 

 paper. He will probably be able to make a fourth sketch — 

 somewhat similar to the others. Hut it would not do for him 

 to conclude that because he made it there must be a fourth 

 mountain. A boy may be making calculations up to the 

 number three and may be illustrating these calculations by 

 making three crosses on paper and by moving about three 

 apples on a table. He will no doubt be able to make a few 

 more crosses on the paper, but it would not do for him to 

 conclude, "arguing by analogy," that there must be more 

 apples on the table, and that his inability to see them must 

 be due to some physical or mental defect. The apples have 

 no necessary relationship with the crosses nor the crosses 

 with the apples ; nor have dimensions any necessary relation- 

 ship with equations or equations with dimensions. 



A correspondent in the August (1911) issue of " Knowledge " 

 says that he believes he has been able to demonstrate, " assuming 

 the truth .... of the principle of the continuity of mathematical 

 law, that the fourth and higher dimensions do actually exist : 

 the existence of a third dimension implying that of a fourth. 

 and so on, to infinity." It is not a case of mathematical law 

 at all. We have no reason to believe that, because certain 

 mathematical expressions correspond to existing things, and 

 because we can arbitrarily add to these expressions, the new- 

 ones must have something in existence corresponding to them. 

 Mathematical law is consistent and continuous within its own 

 sphere — that of calculation — but it will never enable us to 

 discover what is or is not in actual existence. In so far as 

 our calculations can give us knowledge as to things existing. 

 this knowledge must be involved in the data of these calcula- 

 tions, and these data must be got by experience. A third 

 dimension does not imply a fourth any more than a third apple 

 implies a fourth on the boy's table. 



Mr. Hinton — referred to by another correspondent — bases 

 his arguments on descriptions of objects moving about or 

 turning round. It is surely self-evident that whatever objects 

 we take, and whatever motion we give them — backwards, 

 forwards, rotary, or any other motion, or any combination of 

 motions — the appearances which these moving objects present 

 to us can give no proof, or no evidence, of the existence of 

 another dimension. This point need not be laboured ; it can 

 be left to the readers of " Knowledge." 



It may be mentioned incidentally that if Mr. .•\nnison did not 

 confine himself to the circle and the sphere, he would not get 

 his basis. .\'"+y'=r'" may represent a locus in three dimensions 

 as well as one in two. It may represent a right cylinder. 

 Also x" = r' in three dimensions represents two planes parallel 

 to each other and each distant r from origin. 



Of course, there may be a fourth dimension and many more 

 dimensions. There may be invisible apples — of extreme 

 tenuity — on our tables, invisible trees in our gardens, invisible 

 cats at our firesides and invisible planets in our solar system. 

 All that we can say is that we have no evidence of the existence 

 of these apples or these trees or these cats or these planets or 

 of the fourth dimension. 



There is much about us that is beyond our grasp. We 

 cannot conceive of space as being either limited or unlimited. 

 We cannot think of a boundary beyond which space is no 

 more. Nor can we think of space as going on — far beyond 

 the furthest star, if there is one — for ever. Our belief in 

 cause and effect — that everything occurring or existing must 

 have a cause — is inconsistent with our belief in the existence 

 of the world. We are on safe ground only when we keep 

 within the limits of experience, or of what follows closely from 

 the known facts of experience. If we reason too far from these 

 facts — though our reasonings may be quite logical — we land 

 ourselves in inconsistencies. It is worse if we leave our 

 reason and follow our imagination instead. 



