Commentary on Euclid I, Axioms
I. Things which are equal to the same, are equal to one another.
II. If equals be added to equals, the wholes are equal.
III. If equals be taken from equals, the remainders are equal.
IV. If equals are added to unequals, the wholes are unequal.
V. If equals be taken from unequals, the remainders are unequal.
VI. Things which are double of the same, are equal to one another.
VII. Things which are halves of the same, are equal to one another.
VIII. Things which coincide with each other, are mutually equal.
IX. The whole is greater than its part.
X. Two right lines cannot comprehend space.
These are the things which, according to the opinion of all men, are called indemonstrable axioms, so far as their certainty is admitted by all, and no one disputes their evidence. For propositions also are often simply called axioms, of whatever kind they may be, whether they are immediately proper, or require some declaration; and the Stoics, indeed, are accustomed to call every simple enunciative speech an axiom: and when they write on dialectic arts, they say that they discourse on axioms. But some, distinguishing more accurately axioms from other propositions, give this appellation to a proposition immediate, and producing credibility of itself, on account of its evidence: as also Aristotle and geometricians themselves affirm. For, according to the opinion of these, an axiom is the same as a common conception. By no means, therefore, must we praise Apollonius the geometrician, who writ (as it appears) demonstrations of axioms, because he performs the very opposite to Euclid: for he, indeed, enumerates that which is demonstrable among Petitions; but Apollonius endeavours to find out demonstrations of indemonstrables. But these naturally differ from each other, and the genus of the sciences is different: I mean of the things which take place about immediate propositions, which are entirely subject to our knowledge, on account of their evidence; and of things which use demonstrations, which receive principles from them; and which, when received, they orderly employ in their proper conclusions. But that the demonstration of the first axiom, which Apollonius persuades himself he has invented, possesses a medium, not more known, but more dubious than the conclusion may be known by any one from a slight inspection. For let (says he) a be equal to b, and b to c, I say that a also is equal to c.
For since a is equal to b, it occupies the same place as b. And because b is equal to c, it occupies the same place as c; and so a occupies the same place as c, they are therefore equal. Now in this demonstration it is requisite that two things must be previously assumed; one, that things occupying the same place, are mutually equal; but the other, that things occupying the same place, with the same thing, mutually occupy the same place: but these are evidently more obscure than the present axiom. For it is proper to enquire how are things, which fill the same place equal, according to the whole, or according to a part; or according to a figure of speech: hence we must by no means admit a transition to place, which is more unknown than the natures it contains; for the invention of its essence is difficult and ambiguous. That we may avoid prolixity, therefore, all axioms are to be delivered as things immediate and self-manifest, since they are of themselves known and credible; for he who brings demonstration to things the most manifest, does not confirm their truth, but diminishes the evidence we possess in the untaught and innate conceptions of the soul: but this is to be received concerning axioms, as a judgment of their peculiarity; and that all of them are of the common kind of the mathematical sciences; and that each of them is said to be verified, not only in magnitudes, but also in numbers, and motions, and times: and this indeed is necessary. For equal and unequal, the whole and part, and the more and the less, are common to discreet and continued quantities. The contemplation, therefore, which is conversant with times and motions, numbers and magnitudes, requires all these, as things evident by their own intrinsic light; and in all of them both that is true, which says, things equal to the same, are equal to one another; as likewise each of the axioms we have assumed: but as they exist in common, each science uses them according to its proper matter, and one indeed, as in magnitudes; but another, as in numbers; and another, as in times; and after this manner in each science, the conclusions become peculiar and apposite, though the axioms are common. Besides, it is likewise requisite not to contract the number of these to the least, as is done by Heron, who only establishes three axioms; for this also is an axiom, the whole is greater than its part, and the geometrician every where assumes this in his demonstrations; as also, that things which mutually coincide, are equal;
for this is employed with advantage in the solution of the fourth Proposition. Nor is it proper to join some with others, of which some are proper to the geometric matter, as that two right lines cannot comprehend space, (since axioms are, as we have said, of a common kind); but others are consequent to things established, as that which says, things double of the same, are equal. For this is consequent to the axiom, affirming, that if to equals you add equals, the wholes are equal, since things equal to the half, because they assume the half, become double to the same, and mutually equal, on account of an equal addition: and according to this reason, not only the doubles, but also the triples, and all multiples of the same quantity will appear equal. But with these axioms, Pappus says, that certain others are to be classed, as if unequals are added to equals, the excess of the whole, will be equal to the excess of the adjuncts. And on the contrary, if equals are added to unequals, the excess of the wholes is equal to the excess or difference of the unequals themselves. And these also are manifest from themselves, yet they may be made manifest as follows. Let a be equal to b, and add to each the unequals c d, but let c be greater than d by e, and the remainder be f; because, therefore, a is equal to b, and also f to d; a f will be equal to b d. For if equals are added to equals, the wholes are equals: a c, therefore, exceeds b d, by e only, by which alone c exceeds d. Again, c and d are unequals, to which, let the equals a and b be added, and let e be the excess of c, above d, and the remainder be f; because, therefore, a is equal to b, and f to d a f will be equal to b d; the whole, therefore, a c, will exceed b d, by e only, by which c also exceeds d. These, therefore, are consequent to the aforesaid axioms, and are, not undeservedly, in many copies, omitted. But whatever others he adds to these, have been previously assumed by definitions, to which they are consequent. As for example: that all the parts of a plane and a right line mutually agree; for things placed in their extremities, possess a nature of this kind; and that a point divides a line, but a line a superficies, and a superficies a solid. For all things are divided by the natures by which they are proximately bounded;
and that infinite subsists in magnitudes, by addition and diminution, but according to capacity only, in both these respects: for every thing continuous may be infinitely divided and increased. But, as we have summarily spoken concerning these, it remains that we consider things consequent to principles; for thus far principles extend themselves. But of those who oppose geometry, some very much doubt concerning principles, endeavouring to shew that the terms have no subsistence, whose arguments, indeed, are known in common, who endeavour to take away all science, and, like hostile foes from a foreign region, demolish the fruits and fecundity of philosophy, as is the case with the Pyrrhonian philosophers; but others only propose to themselves the subversion of geometrical principles, as the Epicureans. Others, again, admitting the principles, affirm, that things consequent to the principles cannot be demonstrated, unless something else is granted, which was not previously assumed in the principles. Zeno exercised this mode of contradiction, who was a Sidonian by birth, but of the Epicurean sect, against whom Possidonius wrote an entire book, exhibiting the whole of his imbecile opinion; and thus much may suffice for the difference of opinions concerning principles. We shall shortly consider the troublesome objection of Zeno: but now, after we have briefly resumed the consideration of theorems and problems, their difference, and the divisions they receive, we shall proceed, to an exposition of the things exhibited by the institutor of the elements, gathering the more beautiful observations upon the propositions found in the writings of the antients, and contracting the infinite prolixity of their discourses; but delivering such things as are more artificial, and full of methods producing science, dwelling more on an accurate treatise of things than on the variety of cases and assumptions, to which young men, for the most part, eagerly incline.