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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.

VII. THEORY OF MOTION

WAS ARISTOTLE A RELATIVIST?

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

of one object has no meaning unless there be other objects to which to refer it'.

But if he looks again at the only passage on which he relies in Chapter 3 of the tract concerning Xenophanes, Zeno, and Gorgias, imputed to Aristotle, he will find that the author of the tract, whoever he may have been, expresses no opinion of his own about motion, but only quotes what Zeno says for the Pantheism of the Eleatics who believed only in one universe always at rest against the Pluralism of those who believed in many moving things. Nevertheless, though his argument fails, Mr. Temperley is right in calling attention to Aristotle's theory of motion. But is Aristotle's theory of motion that of a Relativist?

Aristotle in his Physics rightly recognized that a finite body continuously moves through continuous space during continuous time; and thereby he anticipated the first principle of motion in modern Kinematics, that motion is space during time. In order also to explain how a body moves through space, he distinguished between the private space of each finite body and the common space in which all bodies are and move. He says, in Physics, iv. 2:

'Space is of one kind common space in which all bodies are, ' and of another kind private space in which a body primarily 'is: I mean, for example, you are now in the universe, because in the air which is in the universe, and in the air, 'because in the earth, and similarly in this earth because in 'this individual space which contains nothing more than you.'

Having thus provided containing space for all finite bodies to be and to move therein, he connected space and time by proving that they are different, but similar, because neither of them is discrete or composed of discrete parts or points, or instants, but both are continuous yet divisible, space into continuous segments of space, time into continuous moments of time, and both to infinity.

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Having also established the kinematic law of motion in space during time on clear and distinct principles, he used the continuous motion of a finite body moving in a continuous common space through a continuous common time to destroy the Pantheistic arguments advanced against motion by Zeno, who had confused the measures of space with space itself, and supposed that a body could not move because it would have to move through infinite discrete parts between the terminus a quo and the terminus ad quem in a finite time, which is impossible, or rather would be impossible if motion were really what Zeno supposes it to be.

Aristotle's solution of Zeno's paradox is that a body does not have to move by leaps through a series of discrete parts or points, because its motion is not discrete but continuous through continuous space during continuous time; nor does it even have to move through infinite actual segments of continuous space; because this continuous space is only potentially divisible into its continuous segments: in fact all the body has to do is to move through a finite continuous space during a finite continuous time. He wittily adds that even if a body had to move through infinite spaces it would nevertheless have infinite times to arrive at the terminus, because both continuous space and continuous time are alike divisible ad infinitum.

Moreover, Aristotle completed this acute and subtle analysis of the motion of bodies through space during time without saying one word about relative motion, in which the body moves in relation to other bodies, and without showing any indication that he saw any fault in the direct method of investigating the motion of bodies by space during time.

It is clear, therefore, that he did not, as Mr. Temperley supposes, anticipate Einstein in supposing that every motion must only be considered as a

relative motion, which is the motion of one body relatively to another, as, for example, that of a carriage moving alongside of an embankment. On the contrary, he would adhere and rightly adhere to the principle that a body always moves through space during time, though in consequence it may also move or not move in relation to this or that other body. Aristotle's theory of motion is therefore not that of Professor Einstein, nor that of any other Relativist. Oxford, May 12.

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