APPENDIX II. Is Euclid's Axiom true? § 1. [Pg 40] Infinite and Finite Magnitudes. The answer I propose to give to this alarming question is that, though true for Finite Magnitudes—the sense in which, as I believe, Euclid meant it to be taken—it is not universally true. Will the gentle Reader be so kind as to join me in contemplating, for a few minutes, the Infinite Space which surrounds our tiny planet? We believe—those of us, at least, who answer fully to the ancient definition of Man, 'animal rationale'—that it is infinite. And that, not because we profess to have grasped the conception of Infinity, but because the contrary hypothesis contradicts Reason: and what contradicts Reason we feel ourselves authorised to deny. Both conceptions—that Space has a limit, and that it has none—are beyond our Reason: but the former is also against our Reason: for we may fairly say "When we have reached the limit, what then? What do we come to? There must be either Something, or Nothing. If Something, it is full Space, 'plenum': if Nothing, it is empty Space, 'vacuum.' That there should be neither of these is a logical impossibility. Such an hypothesis would be—in the[Pg 41] words of Master Constable Dogberry—'most tolerable and not to be endured.'" I propose to show, by certain considerations which begin with Infinite Space, but will speedily condescend to Finite Magnitudes, that it is possible for two homogeneous Magnitudes to be so related to each other that no multiple of the lesser will exceed the greater. (It is of course assumed that a 'multiple' of a Magnitude is the result produced by the use of a 'multiplier,' and that a 'multiplier' is a nameable—and therefore a finite—number.) "Yet surely," the gentle Reader will protest, "Euclid has assumed the exact contrary of this? Does he not, in Book X, Prop. 1, tacitly assume the Axiom that the lesser of two Magnitudes may be so multiplied as to exceed the greater?" Gentle Reader, he does! But my contention is that, in so doing, he excludes from his view both Infinities and Infinitesimals, and is contemplating Finite Magnitudes only. For consider Euclid's Definitions of the word 'Ratio' and of the phrase 'to have a Ratio to.' (Book V. Def. 3, 4.) (3) λόγος ἐστὶ δύο μεγεθῶν ὁμογενῶν ἡ κατὰ πηλικότητα πρὸς ἄλληλα ποιὰ σχέσις. (4) λόγον ἔχειν πρὸς ἄλληλα μεγέθη λέγεται, ἅ δύναται πολλαπλασιαζόμενα ἀλλήλων ὑπερέχειν. Quite literally, these are:— (3) "Ratio is a certain relationship, as to size, of two homogeneous Magnitudes, each to the other." (4) "Magnitudes, which can, (on) being multiplied, exceed each the other, are said to have a Ratio, each to the other." But they become more intelligible when less literally translated:— (3) "Ratio is a certain relationship, as to size, borne, by a Magnitude, to another Magnitude homogeneous with it." [Pg 42]