producing half of the deflection, and thus converted space into being a force. In this sentence he contradicts the whole science of classical mechanics, which has, in the course of some twenty-four centuries, been the gradual work of Archimedes, Galileo, Newton, Maxwell, Thomson, and Tait, and has at length proved that space and time are the kinematic conditions, but force and energy are the kinetic causes which produce motion. Which of them would have admitted the statement that curved space is a cause of the deflection of light? Sad to say, Professor Einstein has added one more to the many examples of the modern tendency to supersede old truths by new paradoxes-without sufficient evidence.

(c) and (d) Under these two heads, which are really parts of one argument, Mr. Temperley, without quoting Professor Einstein's definition, either from the translation of his book (page 55) or from my letter, attacks my criticism of that definition out of other parts of Professor Einstein's book, and thus falls into the fallacy of ignoratio elenchi. In selfdefence, therefore, I must quote the definition for him from my letter, in which I said that Professor Einstein defines space as a three-dimensional continuum, composed of an indefinite number of points near enough one another to give a kind of apparent continuity. I answer that space cannot be both a three-dimensional continuum in the old and proper sense of being continuously extended and a continuum in the new and improper sense of being a continual series of discrete points, however close together; but that space can, on the one hand, be a three-dimensional continuum as continuously extended long, broad, and deep, and, on the other hand, can be measured by, but not composed of, a continual series of discrete points, which are not segments of extended space potentially divisible to infinity.

(e) While in my letter, in opposition to the hypothesis of a fourfold continuum and a space-time advocated by Minkowski and Einstein, I said that, in addition to the three straight lines of space at right angles to one another, no place could be found for a fourth straight line of time as a fourth coordinate in any diagram of the supposed fourfold continuum, Mr. Temperley replies that lengths in a time-co-ordinate appear in Einstein's equation, indicating the impossibility of including them in a diagram. I answer first, that it is no argument against me to point out the very defect which I pointed out-namely, the absence of time from a diagram when it ought ex hypothesi to be present; secondly, that he takes no notice of another fault, which I also pointed out, that time is not a line, and therefore has no lengths to lie in a space-time-coordinate; and thirdly, that he overlooks the consequence that the presence of a length of time in Professor Einstein's equation does not excuse its absence from a diagram of a supposed fourfold continuum, or guarantee the presence of a fourfold continuum of space-time in what Minkowski calls the world, but I call the body of the world.

Lastly, I submit that, whatever it may be called, the world, so far as it is continuous, is neither space plus time, nor space-time, nor any kind of union of space and time except space during time and divided by time, without which space would have no time to endure even for an instant.



FROM The Times, MAY 2, 1922.

A hope has been expressed in your columns by Mr. Max Judge that Sir Oliver Lodge's letter of the 13th inst. will not be neglected. I share this hope, if

only an attempt is made to go to the bottom of the subject before Professor Einstein is allowed to confound pure mathematics and concrete physics, although they have been generally distinguished since the time of Aristotle, who long ago was the first to say, 'things mathematical are from abstraction, things physical from addition'.

Aristotle's primary philosophy, or' metaphysics', as his commentators called it, is the science of things as things, and not of mere ideas or names. Its first principle is that all things as things are substances, or whole things, and reduces itself to three main propositions:

1. All things are substances, each of which is an individual, independent, determinate, whole thing, and a real subject of real predicates or attributes, each of which is a thing, yet not a whole thing, but something which, being quantitative, or qualitative, or relative, &c., is only something which a substance is, or, in other words, is a substance so far as that substance is quantitative, or qualitative, or relative, &c. As Aristotle pithily says, "The same thing in a way is Socrates and Socrates musical.' Everything, therefore, is a substance both as itself a whole thing, and also as something quantitative, or qualitative, or relative, &c.

2. The most evident, the most knowable, and the most acknowledged substances are natural substances or bodies-e. g. earth and water, plants and animals, sun, moon, and stars, which are all investigated by what Aristotle was the first to call natural science or natural philosophy.

3. Over and above all other substances is God, the supernatural substance, who is animate, eternal, and best, and therefore, according to Aristotle, the prime mover as motive of Nature.

Such is Aristotle's science of things as things, containing the realism without materialism, which

governed the intellect of nations down to the Reformation. Aristotle's Realism has nothing to do with latter-day New Realism, which is built on no such sound foundations, but tends to start not from things but from minds.

Aristotle proceeded to apply the universal science of things as things to things mathematical. In concrete arithmetic and concrete geometry things mathematical are substances, such as men's fingers and toes and feet with which they originally calculated and measured; and there is no difference between those two sciences except that arithmetic applies to all substances, but geometry only to bodies.

But when, in the sixth century B.C., the genius of the Greeks discovered pure mathematics, nobody successfully explained what the objects of pure mathematics are, until Aristotle discovered that pure mathematical things are real predicates, or attributes as we say, which are substances as quantitative and only quantitative. Hence pure arithmetical things are substances as quantitatively one and numberable which are discrete, whereas pure geometrical things are bodies as quantitatively extended, long, broad, and deep, which are continuous. Not Plato but Aristotle was the first metaphysician to apply this analysis of things as purely quantitative to the objects of pure mathematics.

Aristotle, the founder both of the metaphysics of things and of the logic of reasoning, was the first to analyse the process of logical abstraction by which the mathematician purifies his object: it is a double process of elimination and concentration, elimination of the non-quantitative, concentration on the quantitative. A pure arithmetician even eliminates the geometrical, and only attends to any substance as one and numbered: a pure geometer, going further and availing himself of pure arithmetic, eliminates everything mechanical or physical, but attends

to any body as quantitatively extended long, broad, and deep. The mathematician having thus concentrated himself on these pure mathematical objects in arithmetic and in geometry by elimination of what is not, and by concentrating himself on what is, mathematical, henceforth as the data for his further purely mathematical investigations, and is the more scientific the more he considers the quantitatively discrete things one and numerous, and the quantitatively continuous, extended long, broad, and deep, without regard to the substance which is quantitative being also qualitative, or relative, or in any other way complicated. Not that the pure mathematician despises physics: on the contrary he knows full well that the purer his mathematics the clearer and more distinct will be the help it gives when applied to physics.

Mathematics and physics are different sciences, both concerned with things, and neither with ideas, but in a different way. But, so far from pure mathematics requiring physics to give it reality and truth as Einstein supposes, the pure mathematician has its real and true quantitative objects without regard to things physical, which he eliminates in order to attend to things mathematical, because first comes the mathematical and afterwards the physical. This was the order of Aristotle; and, if Aristotle was right, Einstein is wrong. Oxford, April 29, 1922.



FROM The Times, MAY 17, 1922.

Mr. C. E. Temperley, in answer to my letter of the 2nd inst., has, in your issue of the 8th inst., tried to prove that Aristotle anticipated Professor Einstein's theory of relative motion in the sense that ' motion

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