not as extended, and the extended cannot consist of the unextended; nor indivisible lines, surfaces, and solids, because these are themselves all segments of space potentially divisible into ever-divisible segments to infinity, and are yet the real objects of pure geometry, which the geometer only imitates by his physical diagrams to the best of his ability. Šimilarly, the same body is not only continuous and enduring time, but as continuous is also potentially divisible to infinity into enduring moments of continuous time; and these moments are not physical discrete parts; nor are they instants, because an instant is a substance as dated but not as enduring, and the enduring cannot consist of the unenduring; nor are they indivisible moments, because continuous moments of time are potentially divisible into ever-divisible moments to infinity. Space and time, then, are two infinite continua, potentially divisible to infinity, the one into segments of space and the other into moments of time, but both continuous attributes of the body of the universe.

On the other hand, the body of the universe as discrete and consisting of all finite bodies is divisible into physical parts, as many as there are-how many we know not. Its quantitative attribute number is also discrete; and, as Aristotle well said, while multitude or number is calculable and divisible into things discontinuous, magnitude or space is measurable and divisible into things continuous. It follows that the continuous is measurable by the discrete when we use a mode of measurement which has been practised from the most primitive times, whenever anybody has measured a continuous surface with a rule marked with discrete degrees. This simple method of measurement became, especially from the time of Kepler and Cavalieri to that of Newton and Leibniz, the foundation of the method of fluxions and of the differential calculus. It is,

however, a good thing to use measures, but a bad thing to confuse the measures with the thing measured; but unfortunately there has arisen an arithmetizing school which has established an arithmetic continuum, and changed the very meaning of the word 'continuous', by applying it to a continual discrete series.

To this school belongs Professor Einstein, who defines space as a three-dimensional continuum, composed of an indefinite number of points near enough to one another to give a kind of apparent continuity. But it is plain that he confuses the continuous as the triply but continuously extended with a continual series of discrete points placed close to one another but not really continuously extended. It is as if a carpenter mistook his measuring rule and its degrees with the continuous surface which he is measuring.

Einstein proceeds to define not exactly time, but Minkowski's world of physical phenomena. He defines it as a four-dimensional continuum composed of as many neighbouring individual events realized or at least thinkable as one cares to choose. This continuum of somewhat scattered and imaginary accidents of bodies does not sound promising. In reality there is no such thing in the world as an event about which all that really exists is some body or other substance happening to be affected in some way or other somewhere at some time. Marriage, for example, is called the event of one's life; but what it really is is a bride and bridegroom marrying. Without the bride and bridegroom what would be

the event?

The consequence, however, of introducing events is that Professor Einstein wishes to add time to the three-dimensional space so as to make the world a fourfold continuum. There are many objections to this new-fangled figment which requires the

addition of a fourth co-ordinate. Time is not something extended, but something enduring which is closely connected with, but by no means the same as something extended: space and time are so different that they cannot be the space-time supposed by Minkowski. Time, past, present, and future is neither co-ordinate nor commensurate with space extended long, broad, and deep. Time is not a line; and, as there are only three rectangular co-ordinates corresponding to three dimensions, and three straight lines at right angles to one another, no place can be found for a fourth straight line for time in any diagram, and still less in the world.

Space and time, though different attributes of the same substance, are nevertheless closely connected. The same substance which as spatial is extended long, broad, and deep, as temporal can be enduring before now and afterwards. Also it enters in both ways into motion, which is change of position in space during time, and is regulated by three laws of motion:

(1) the space (s) is equal to the velocity (2) multiplied by the time (t) of the motion:

s = vt

(2) the velocity (v) is equal to the space (s) divided by the time (t) of the motion:

(3) the time (t) is equal to the space (s) divided by the velocity (v) of the motion:

t

ν

But however closely connected and that, too, in the same moving substance the space remains different from the time: space is not equal to t but vt. Space and time, however close, remain different; for to be

extended as space in three dimensions long, broad, and deep is one thing, but to be enduring as time before, now, and hereafter is different. Minkowski rashly predicted that space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. But in reality it is the substance of the whole world which on the one hand as extended long, broad, and deep, is the real space, and on the other hand as enduring before, now, and hereafter, is the real time; both of which are independent of one another but both dependent on the substance of the world, of which they are different attributes closely connected. Minkowski's hypothesis that space and time are doomed to fade away into mere shadows has led to a habit of reducing them into one single term 'space-time'. But the nearest to which we can reduce the close relation of space and time is space during time, as without time there could be no space.

V. FROM EUCLID TO EINSTEIN

SPACE DURING TIME

FROM The Times, JANUARY 7, 1922.

In The Times of the 4th inst., Mr. Clive E. Temperley has taken me to task over what appear to him some of the errors in my letter of December 24. Before I defend myself I must premise that, in my opinion, space is extended, long, broad, and deep, and is continuous without any intervals such as exist between numbers, or points, or finite bodies, which are discrete; that the immutable space common to the whole body of the universe and in which all finite bodies are, and rest or move, is to be distinguished from the mutable private spaces of finite bodies; and that Professor Einstein, in his book on

Relativity, translated by Dr. Lawson, is not so convincing about common space as he is about private or local spaces. I proceed to meet my errors in the order of Mr. Temperley.

(a) While in my letter I said, in agreement with Professor Einstein, that, in spite of the negative result of the Michelson-Morley experiment, FitzGerald's suggestion of the contraction of a body in the direction of its motion has been calculated by Lorentz, Mr. Temperley, on the other hand, accuses me of reversing the order, on the ground that the negative result follows directly from Lorentz's equation. I answer by quoting from the translation of Professor Einstein's book (p. 53) the following statement of the order, which is the same as mine: "The experiment gave a negative result―a fact very perplex'ing to physicists. Lorentz and FitzGerald rescued the theory 'from this difficulty by assuming that the motion of the body ' relative to aether produces a contraction of the body in the direction of motion.'

(b) While in my letter I said, in opposition to Professor Einstein, that the sun could not deflect the space between itself and the light, and that, even if it could, the deflected space could not deflect the light, because space is not a force, Mr. Temperley replies that, according to Einstein's mathematical conception of the universe, no force is required to cause this deflection. I answer that, in Appendix III, added to his book at the request of Dr. Lawson (p. 127), Professor Einstein did require the deflected space to produce this deflection in the following

sentence:

'It may be added that, according to the theory, half of this 'deflection is produced by the Newtonian field of attraction of the sun, and the other half by the geometrical modifica'tion ("curvature ") of space caused by the sun.'

Professor Einstein then did, perhaps unconsciously, require the supposed curvature of space to be a cause