larger distance had a greater velocity, but to the circumstance that motion which in ordinary language is called slow, has been interrupted by more moments of rest, while the motion which ordinarily is called quick has been interrupted by fewer moments of rest. When it is shown that the motion of an arrow, which is shot from a powerful bow, is in contradiction to their theory, they declare that in this case too the motion is interrupted by moments of rest. They believe that it is the fault of man's senses if he believes that the arrow moves continuously, for there are many things which cannot be perceived by the senses, as they assert in the twelfth proposition. But we ask them: "Have you observed a complete revolution of a millstone? Each point in the extreme circumference of the stone describes a large circle in the very same time in which a point nearer the centre describes a small circle: the velocity of the outer circle is therefore greater than that of the inner circle. You cannot say that the motion of the latter was interrupted by more moments of rest; for the whole moving body, i.e., the millstone, is one coherent body." They reply, "During the circular motion, the parts of the millstone separate from each other, and the moments of rest interrupting the motion of the portions nearer the centre are more than those which interrupt the motion of the outer portions." We ask again, "How is it that the millstone, which we perceive as one body, and which cannot be easily broken, even with a hammer, resolves into its atoms when it moves, and becomes again one coherent body, returning to its previous state as soon as it comes to rest, while no one is able to notice the breaking up [of the stone]?" Again their reply is based on the twelfth proposition, which is to the effect that the perception of the senses cannot be trusted, and thus only the evidence of the intellect is admissible. Do not imagine that you have seen in the foregoing example the most absurd of the inferences which may be drawn from these three propositions: the proposition relating to the existence of a vacuum leads to more preposterous and extravagant conclusions. Nor must you suppose that the aforegoing theory concerning motion is less irrational than the proposition resulting from this theory, that the diagonal of a square is equal to one of its sides, and some of the Mutakallemim go so far as to declare that the square is not a thing of real existence. In short, the adoption of the first proposition would be tantamount to the rejection of all that has been proved in Geometry. The propositions in Geometry would, in this respect, be divided into two classes: some would be absolutely rejected: e.g., those which relate to properties of the incommensurability and the commensurability of lines and planes, to rational and irrational lines, and all other propositions contained in the tenth book of Euclid, and in similar works. Other propositions would appear to be only partially correct: e.g., the solution of the problem to divide a line into two equal parts, if the line consists of an odd number of atoms: according to the theory of the Mutakallemim such a line cannot be bisected. Furthermore, in the well-known book of problems by the sons of Shakir are contained more than a hundred problems, all solved and practically demonstrated: but if there really were a vacuum, not one of these problems could be solved, and many of the waterworks [described in that book] could not have been constructed. The refutation of such propositions is a mere waste of time. I will now proceed to treat of the other propositions mentioned above.
FOURTH PROPOSITION.
