(4) "A Magnitude is said 'to bear a Ratio to' another Magnitude, homogeneous with it, when either of them, that is not greater than the other, can be made so by multiplication." Some translators introduce the word 'mutual' into No. (3), and tell us that Ratio is 'a mutual relation of two Magnitudes': but this seems to me incorrect, as seeming to imply that the Ratio, borne by to , is identical with that borne by to . But, if were 3-4ths of , would not be 3-4ths, but 4-3ds, of : and the Ratio '4-3ds,' though of the same nature as the Ratio '3-4ths,' is not identical with it. Now it seems to me clear that Euclid does not mean to imply that any two homogeneous Magnitudes bear 'Ratios' to each other: for in No. (4) he gives us a test, by which to know in what cases two such Magnitudes do, and in what cases they do not, bear 'Ratios' to each other. This test would be wholly superfluous if it were true, of any two homogeneous Magnitudes, that one could be said 'to bear a Ratio to' the other. What cases then, does Euclid mean to exclude by this test? My answer is "all cases in which one of the Magnitudes is infinitely greater than the other." Take, as an example, these two Magnitudes—a Cubic Inch, and Infinite Space. It is not possible, by multiplying a Cubic Inch by any finite number, however great, to make it exceed, or even equal, Infinite Space: hence Euclid's test fails in this case, and Euclid would, no doubt, decline to say that either of these Magnitudes, though they are strictly homogeneous, bears a 'Ratio' to the other. My conclusion, then, is that, in Book X. Prop. 1, Euclid is limiting his view to the case of two homogeneous Magnitudes which are such that neither of them is infinitely greater than the other: nay, more—for such a limitation would not exclude the case of two Infinities of the same order—that he is contemplating Finite Magnitudes only. § 2. [Pg 43] Infinitesimal Lines and Strips. We have already seen that, in the case of two homogeneous Magnitudes, one Finite and the other Infinite, no multiple of the lesser will exceed the greater: and I now propose to show that it is possible for the same thing to happen in the case of two homogeneous Magnitudes, neither of them being Infinite.