18 REPORT— 1878. 



back to the original direction of measurement or of reckoning, or to the 

 original kind of operation. 



Suppose, however, that at some stage of a calculation our formulas 

 indicate an alteration in the mode of measurement such that, if the 

 alteration be repeated, a condition of things, not the same as, but the 

 reverse of the original, will be produced. Or suppose that, at a certain 

 stage, our transformations indicate that time is to be reckoned in some 

 manner different from future or past, but still in a way having definite 

 algebraical connexion with time which is gone and time which is to come. 

 It is clear that in actual experience there is no process to which such 

 measurements correspond. Time has no meaning except as future or past ; 

 and the present is but the meeting point of the two. Or, once more, 

 suppose that we are gravely told that all circles pass through the same 

 two imaginary points at an infinite distance, and that every line drawn 

 through one of these points is perpendicular to itself. On hearing the 

 statement, we shall probably whisper, with a smile or a sigh, that we hope 

 it is not true ; but that in any case it is a long way off, and perhaps, after 

 all, it does not very much signify. If, however, as mathematicians we are 

 not satisfied to dismiss the question on these terms, we ourselves must 

 admit that we have here reached a definite point of issue. Our science 

 must either give a rational account of the dilemma, or yield the position 

 as no longer tenable. 



Special modes of explaining this anomalous state of things have 

 occurred to mathematicians. But, omitting details as unsuited to the 

 present occasion, it will, I think, be sufficient to point out in general terms 

 that a solution of the difficulty is to be found in the fact that the formula? 

 which give rise to these results are more comprehensive than the significa- 

 tion assigned to them ; and when we pass out of the condition of things 

 first contemplated they cannot (as it is obvious they ought not) give us 

 any results intelligible on that basis. But it does not therefore by any 

 means follow that upon a more enlarged basis the formulae are incapable 

 of interpretation ; on the contrary, the difficulty at which we have 

 arrived indicates that there must be some more comprehensive state- 

 ment of the problem which will include cases impossible in the more 

 limited, but possible in the wider view of the subject. 



A very simple instance will illustrate the matter. If from a point out- 

 side a circle we draw a straight line to touch the curve, the distance 

 between the starting point and the point of contact has certain geometrical 

 properties. If the starting point be shifted nearer and nearer to the circle 

 the distance in question becomes shorter, and ultimately vanishes. But 

 as soon as the point passes to the interior of the circle the notion of a 

 tangent and distance to the point of contact cease to have any meaning ; 

 and the same anomalous condition of things prevails as long as the point 

 remains in the interior. But if the point be shifted still further until it 

 emerges on the other side, the tangent and its properties resume their 



