are constructed, inscribed in a sphere and compared with one another. The second object is relative to the learner; and, from this standpoint, the elements may be described as "a means of perfecting the learner's understanding with reference to the whole of geometry. For, starting from these (elements), we shall be able to acquire knowledge of the other parts of this science as well, while without them it is impossible for us to get a grasp of so complex a subject, and knowledge of the rest is unattainable. As it is, the theorems which are most of the nature of principles, most simple, and most akin to the first hypotheses are here collected, in their appropriate order; and the proofs of all other propositions use these theorems as thoroughly well known, and start from them. Thus Archimedes in the books on the sphere and cylinder, Apollonius, and all other geometers, clearly use the theorems proved in this very treatise as constituting admitted principles'." Aristotle too speaks of elements of geometry in the same sense. Thus: "in geometry it is well to be thoroughly versed in the elements"; "in general the first of the elements are, given the definitions, e.g. of a straight line and of a circle, most easy to prove, although of course there are not many data that can be used to establish each of them because there are not many middle terms"; "among geometrical propositions we call those 'elements' the proofs of which are contained in the proofs of all or most of such propositions"; "(as in the case of bodies), so in like manner we speak of the elements of geometrical propositions and, generally, of demonstrations; for the demonstrations which come first and are contained in a variety of other demonstrations are called elements of those demonstrations... the term element is applied by analogy to that which, being one and small, is useful for many purposes." § 2. ELEMENTS ANTERIOR TO EUCLID'S. The early part of the famous summary of Proclus was no doubt drawn, at least indirectly, from the history of geometry by Eudemus; this is generally inferred from the remark, made just after the mention of Philippus of Mende, a disciple of Plato, that "those who have written histories bring the development of this science up to this point." We have therefore the best authority for the list of writers of elements given in the summary. Hippocrates of Chios (fl. in second half of 5th c.) is the first; then Leon, who also discovered diorismi, put together a more careful collection, the propositions proved in it being more numerous as well as more serviceable. Leon was a little older than Eudoxus (about 390-337 B.C.) and a little younger than Plato (429-348 B.C.), but did not belong to the latter's school. The 1 Proclus, pp. 70, 19—71, 21. 2 Topics VIII. 14, 163 b 23. Metaph. 1014 a 35-b 5. • Proclus, p. 66, 2ο ὥστε τὸν Λέοντα καὶ τὰ στοιχεῖα συνθεῖναι τῷ τε πλήθει καὶ τῇ χρείᾳ τῶν δεικνυμένων ἐπιμελέστερον. geometrical text-book of the Academy was written by Theudius of Magnesia, who, with Amyclas of Heraclea, Menaechmus the pupil of Eudoxus, Menaechmus' brother Dinostratus and Athenaeus of Cyzicus consorted together in the Academy and carried on their investigations in common. Theudius "put together the elements admirably, making many partial (or limited) propositions more general." Eudemus mentions no text-book after that of Theudius, only adding that Hermotimus of Colophon "discovered many of the elements?" Theudius then must be taken to be the immediate precursor of Euclid, and no doubt Euclid made full use of Theudius as well as of the discoveries of Hermotimus and all other available material. Naturally it is not in Euclid's Elements that we can find much light upon the state of the subject when he took it up; but we have another source of information in Aristotle. Fortunately for the historian of mathematics, Aristotle was fond of mathematical illustrations; he refers to a considerable number of geometrical propositions, definitions etc., in a way which shows that his pupils must have had at hand some textbook where they could find the things he mentions; and this text-book must have been that of Theudius. Heiberg has made a most valuable collection of mathematical extracts from Aristotle, from which much is to be gathered as to the changes which Euclid made in the methods. of his predecessors; and these passages, as well as others not included in Heiberg's selection, will often be referred to in the sequel. § 3. FIRST PRINCIPLES: DEFINITIONS, POSTULATES, On no part of the subject does Aristotle give more valuable information than on that of the first principles as, doubtless, generally accepted at the time when he wrote. One long passage in the Posterior Analytics is particularly full and lucid, and is worth quoting in extenso. After laying it down that every demonstrative science starts from necessary principles, he proceeds": "By first principles in each genus I mean those the truth of which it is not possible to prove. What is denoted by the first (terms) and those derived from them is assumed; but, as regards their existence, this must be assumed for the principles but proved for the rest. Thus what a unit is, what the straight (line) is, or what a triangle is (must be assumed); and the existence of the unit and of magnitude must also be assumed, but the rest must be proved. Now of the premisses used in demonstrative sciences some are peculiar to each science and others common (to all), the latter being common by analogy, for of course they are actually useful in so far as they are applied to the subject-matter included under the particular science. Instances of first i Proclus, p. 67, 14 καὶ γὰρ τὰ στοιχεῖα καλῶς συνέταξεν καὶ πολλὰ τῶν μερικῶν [ὁρικῶν (?) Friedlein] καθολικώτερα ἐποίησεν. 2 Proclus, p. 67, 22 τῶν στοιχείων πολλὰ ἀνεῦρε. Mathematisches zu Aristoteles in Abhandlungen zur Gesch. d. math. Wissenschaften, XVIII. Heft (1904), pp. 1—49. 5 ibid. 1. 10, 76 a 31—77 a 4. • Anal. post. 1. 6, 74b 5. principles peculiar to a science are the assumptions that a line is of such and such a character, and similarly for the straight (line); whereas it is a common principle, for instance, that, if equals be subtracted from equals, the remainders are equal. But it is enough that each of the common principles is true so far as regards the particular genus (subject-matter); for (in geometry) the effect will be the same even if the common principle be assumed to be true, not of everything, but only of magnitudes, and, in arithmetic, of numbers. "Now the things peculiar to the science, the existence of which must be assumed, are the things with reference to which the science investigates the essential attributes, e.g. arithmetic with reference to units, and geometry with reference to points and lines. With these things it is assumed that they exist and that they are of such and such a nature. But, with regard to their essential properties, what is assumed is only the meaning of each term employed: thus arithmetic assumes the answer to the question what is (meant by) 'odd' or 'even,' 'a square' or 'a cube,' and geometry to the question what is (meant by) 'the irrational' or 'deflection' or (the so-called) . 'verging' (to a point); but that there are such things is proved by means of the common principles and of what has already been demonstrated. Similarly with astronomy. For every demonstrative science has to do with three things, (1) the things which are assumed to exist, namely the genus (subject-matter) in each case, the essential properties of which the science investigates, (2) the common axioms so-called, which are the primary source of demonstration, and (3) the properties with regard to which all that is assumed is the meaning of the respective terms used. There is, however, no reason why some sciences should not omit to speak of one or other of these things. Thus there need not be any supposition as to the existence of the genus, if it is manifest that it exists (for it is not equally clear that number exists and that cold and hot exist); and, with regard to the properties, there need be no assumption as to the meaning of terms if it is clear: just as in the common (axioms) there is no assumption as to what is the meaning of subtracting equals from equals, because it is well known. But none the less is it true that there are three things naturally distinct, the subject-matter of the proof, the things proved, and the (axioms) from which (the proof starts). "Now that which is per se necessarily true, and must necessarily be thought so, is not a hypothesis nor yet a postulate. For demonstration has not to do with reasoning from outside but with the reason dwelling in the soul, just as is the case with the syllogism. It is always possible to raise objection to reasoning from outside, but to contradict the reason within us is not always possible. Now anything that the teacher assumes, though it is matter of proof, without proving it himself, is a hypothesis if the thing assumed is believed by the learner, and it is moreover a hypothesis, not absolutely, but relatively to the particular pupil; but, if the same thing is assumed when the learner either has no opinion on the subject or is of a contrary opinion, it is a postulate. This is the difference between a hypothesis and a postulate; for a postulate is that which is rather contrary than otherwise to the opinion of the learner, or whatever is assumed and used without being proved, although matter for demonstration. Now definitions are not hypotheses, for they do not assert the existence or non-existence of anything, while hypotheses are among propositions. Definitions only require to be understood: a definition is therefore not a hypothesis, unless indeed it be asserted that any audible speech is a hypothesis. A hypothesis is that from the truth of which, if assumed, a conclusion can be established. Nor are the geometer's hypotheses false, as some have said: I mean those who say that 'you should not make use of what is false, and yet the geometer falsely calls the line which he has drawn a foot long when it is not, or straight when it is not straight.' The geometer bases no conclusion on the particular line which he has drawn being that which he has described, but (he refers to) what is illustrated by the figures. Further, the postulate and every hypothesis are either universal or particular statements; definitions are neither" (because the subject is of equal extent with what is predicated of it). Every demonstrative science, says Aristotle, must start from indemonstrable principles: otherwise, the steps of demonstration would be endless. Of these indemonstrable principles some are (a) common to all sciences, others are (b) particular, or peculiar to the particular science; (a) the common principles are the axioms, most commonly illustrated by the axiom that, if equals be subtracted from equals, the remainders are equal. Coming now to (b) the principles peculiar to the particular science which must be assumed, we have first the genus or subject-matter, the existence of which must be assumed, viz. magnitude in the case of geometry, the unit in the case of arithmetic. Under this we must assume definitions of manifestations or attributes of the genus, e.g. straight lines, triangles, deflection etc. The definition in itself says nothing as to the existence of the thing defined: it only requires to be understood. But in geometry, in addition to the genus and the definitions, we have to assume the existence of a few primary things which are defined, viz. points and lines only: the existence of everything else, e.g. the various figures made up of these, as triangles, squares, tangents, and their properties, e.g. incommensurability etc., has to be proved (as it is proved by construction and demonstration). In arithmetic we assume the existence of the unit: but, as regards the rest, only the definitions, e.g. those of odd, even, square, cube, are assumed, and existence has to be proved. We have then clearly distinguished, among the indemonstrable principles, axioms and definitions. A postulate is also distinguished from a hypothesis, the latter being made with the assent of the learner, the former without such assent or even in opposition to his opinion (though, strangely enough, immediately after saying this, Aristotle gives a wider meaning to "postulate" which would cover "hypothesis" as well, namely whatever is assumed, though it is matter for proof, and used without being proved). Heiberg remarks that there is no trace in Aristotle of Euclid's Postulates, and that "postulate" in Aristotle has a different meaning. He seems to base this on the alternative description of postulate, indistinguishable from a hypothesis; but, if we take the other description in which it is distinguished from a hypothesis as being an assumption of something which is a proper subject of demonstration without the assent or against the opinion of the learner, it seems to fit Euclid's Postulates fairly well, not only the first three (postulating three constructions), but eminently also the other two, that all right angles are equal, and that two straight lines meeting a third and making the internal angles on the same side of it less than two right angles will meet on that side. Aristotle's description also seems to me to suit the "postulates" with which Archimedes begins his book On the equilibrium of planes, namely that equal weights balance at equal distances, and that equal weights at unequal distances do not balance but that the weight at the longer distance will prevail. Aristotle's distinction also between hypothesis and definition, and between hypothesis and axiom, is clear from the following passage: "Among immediate syllogistic principles, I call that a thesis which it is neither possible to prove nor essential for any one to hold who is to learn anything; but that which it is necessary for any one to hold who is to learn anything whatever is an axiom: for there are some principles of this kind, and that is the most usual name by which we speak of them. But, of theses, one kind is that which assumes one or other side of a predication, as, for instance, that something exists or does not exist, and this is a hypothesis; the other, which makes no such assumption, is a definition. For a definition is a thesis thus the arithmetician posits (Tiberal) that a unit is that which is indivisible in respect of quantity; but this is not a hypothesis, since what is meant by a unit and the fact that a unit exists are different things'." Aristotle uses as an alternative term for axioms "common (things)," тà κowά, or "common opinions" (xoivaì dókai), as in the following τὰ κοινά, passages. "That, when equals are taken from equals, the remainders are equal is (a) common (principle) in the case of all quantities, but mathematics takes a separate department (aroλaßovoa) and directs its investigation to some portion of its proper subject-matter, as e.g. lines or angles, numbers, or any of the other quantities?" "The common (principles), e.g. that one of two contradictories must be true, that equals taken from equals etc., and the like3...." "With regard to the principles of demonstration, it is questionable whether they belong to one science or to several. By principles of demonstration I mean the common opinions from which all demonstration proceeds, e.g. that one of two contradictories must be true, and that it is impossible for the same thing to be and not be." Similarly "every demonstrative (science) investigates, with regard to some subject-matter, the essential attributes, starting from the common opinions." We have then here, as Heiberg says, a sufficient explanation of Euclid's term for axioms, 1 Anal. post. I. 2, 72 a 14—24. Anal. post. I. II, 77 a 30. 2 Metaph. 1061 b 19-24. |