Euclid, Elements (ed. Thomas L. Heath)

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[p. 114]

CHAPTER IX.

§ 1. ON THE NATURE OF ELEMENTS.

It would not be easy to find a more lucid explanation of the terms element and elementary, and of the distinction between them, than is found in Proclus¹ , who is doubtless, here as so often, quoting from Geminus. There are, says Proclus, in the whole of geometry certain leading theorems, bearing to those which follow the relation of a principle, all-pervading, and furnishing proofs of many properties. Such theorems are called by the name of elements; and their function may be compared to that of the letters of the alphabet in relation to language, letters being indeed called by the same name in Greek (stoicheia).

The term elementary, on the other hand, has a wider application: it is applicable to things “which extend to greater multiplicity, and, though possessing simplicity and elegance, have no longer the same dignity as the elements, because their investigation is not of general use in the whole of the science, e.g. the proposition that in triangles the perpendiculars from the angles to the transverse sides meet in a point.”

“Again, the term element is used in two senses, as Menaechmus says. For that which is the means of obtaining is an element of that which is obtained, as the first proposition in Euclid is of the second, and the fourth of the fifth. In this sense many things may even be said to be elements of each other, for they are obtained from one another. Thus from the fact that the exterior angles of rectilineal figures are (together) equal to four right angles we deduce the number of right angles equal to the internal angles (taken together)² , and vice versa. Such an element is like a lemma. But the term element is otherwise used of that into which, being more simple, the composite is divided; and in this sense we can no longer say that everything is an element of everything, but only that things which are more of the nature of principles are elements of those which stand to them in the relation of results, as postulates are elements of theorems. It is [p. 115] according to this signification of the term element that the elements found in Euclid were compiled, being partly those of plane geometry, and partly those of stereometry. In like manner many writers have drawn up elementary treatises in arithmetic and astronomy.

“Now it is difficult, in each science, both to select and arrange in due order the elements from which all the rest proceeds, and into which all the rest is resolved. And of those who have made the attempt some were able to put together more and some less; some used shorter proofs, some extended their investigation to an indefinite length; some avoided the method of reductio ad absurdum, some avoided proportion; some contrived preliminary steps directed against those who reject the principles; and, in a word, many different methods have been invented by various writers of elements.

“It is essential that such a treatise should be rid of everything superfluous (for this is an obstacle to the acquisition of knowledge); it should select everything that embraces the subject and brings it to a point (for this is of supreme service to science); it must have great regard at once to clearness and conciseness (for their opposites trouble our understanding); it must aim at the embracing of theorems in general terms (for the piecemeal division of instruction into the more partial makes knowledge difficult to grasp). In all these ways Euclid's system of elements will be found to be superior to the rest; for its utility avails towards the investigation of the primordial figures³ , its clearness and organic perfection are secured by the progression from the more simple to the more complex and by the foundation of the investigation upon common notions, while generality of demonstration is secured by the progression through the theorems which are primary and of the nature of principles to the things sought. As for the things which seem to be wanting, they are partly to be discovered by the same methods, like the construction of the scalene and isosceles (triangle), partly alien to the character of a selection of elements as introducing hopeless and boundless complexity, like the subject of unordered irrationals which Apollonius worked out at length⁴ , and partly developed from things handed down (in the elements) as causes, like the many species of angles and of lines. These things then have been omitted in Euclid, though they have received full discussion in other works; but the knowledge of them is derived from the simple (elements).”

Proclus, speaking apparently on his own behalf, in another place distinguishes two objects aimed at in Euclid's Elements. The first has reference to the matter of the investigation, and here, like a good Platonist, he takes the whole subject of geometry to be concerned with the “cosmic figures,” the five regular solids, which in Book XIII. [p. 116] 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 principles⁵ ”

Aristotle too speaks of elements of geometry in the same sense. Thus: “in geometry it is well to be thoroughly versed in the elements⁶ ”; “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 terms⁷ ”; “among geometrical propositions we call those ‘elements’ the proofs of which are contained in the proofs of all or most of such propositions⁸ .”; “(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 purposes⁹ ”;

§ 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 Medma, 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 serviceable¹⁰ . Leon was a little older than Eudoxus (about 408-355 B.C.) and a little younger than Plato (428/7-347/6 B.C.), but did not belong to the latter's school. The [p. 117] 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 general¹¹ .” Eudemus mentions no text-book after that of Theudius, only adding that Hermotimus of Colophon “discovered many of the elements¹² .” 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 Aristotle¹³ , 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, AND AXIOMS.

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 principles¹⁴ , he proceeds¹⁵ :

“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 [p. 118] 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 [p. 119] 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 [p. 120] 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 (tithetai) 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 things¹⁶ .”

Aristotle uses as an alternative term for axioms “common (things),” ta koina, or “common opinions” (koinai doxai), as in the following 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 (apolabousa) and directs its investigation to some portion of its proper subject-matter, as e.g. lines or angles, numbers, or any of the other quantities¹⁷ .” “The common (principles), e.g. that one of two contradictories must be true, that equals taken from equals etc., and the like¹⁸ ....” “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 be¹⁹ .” Similarly “every demonstrative (science) investigates, with regard to some subject-matter, the essential attributes, starting from the common opinions²⁰ .” We have then here, as Heiberg says, a sufficient explanation of Euclid's term for axioms, [p. 121] viz. common notions (koinai ennoiai), and there is no reason to suppose it to be a substitution for the original term due to the Stoics: cf. Proclus' remark that, according to Aristotle and the geometers, axiom and common notion are the same thing²¹ .

Aristotle discusses the indemonstrable character of the axioms in the Metaphysics. Since “all the demonstrative sciences use the axioms²² ,” the question arises, to what science does their discussion belong²³ ? The answer is that, like that of Being (ousia), it is the province of the (first) philosopher²⁴ . It is impossible that there should be demonstration of everything, as there would be an infinite series of demonstrations: if the axioms were the subject of a demonstrative science, there would have to be here too, as in other demonstrative sciences, a subject-genus, its attributes and corresponding axioms²⁵ ; thus there would be axioms behind axioms, and so on continually. The axiom is the most firmly established of all principles²⁶ . It is ignorance alone that could lead any one to try to prove the axioms²⁷ ; the supposed proof would be a petitio principii²⁸ . If it is admitted that not everything can be proved, no one can point to any principle more truly indemonstrable²⁹ . If any one thought he could prove them, he could at once be refuted; if he did not attempt to say anything, it would be ridiculous to argue with him: he would be no better than a vegetable³⁰ . The first condition of the possibility of any argument whatever is that words should signify something both to the speaker and to the hearer: without this there can be no reasoning with any one. And, if any one admits that words can mean anything to both hearer and speaker, he admits that something can be true without demonstration. And so on³¹ .

It was necessary to give some sketch of Aristotle's view of the first principles, if only in connexion with Proclus' account, which is as follows. As in the case of other sciences, so “the compiler of elements in geometry must give separately the principles of the science, and after that the conclusions from those principles, not giving any account of the principles but only of their consequences. No science proves its own principles, or even discourses about them: they are treated as self-evident....Thus the first essential was to distinguish the principles from their consequences. Euclid carries out this plan practically in every book and, as a preliminary to the whole enquiry, sets out the common principles of this science. Then he divides the common principles themselves into hypotheses, postulates, and axioms. For all these are different from one another: an axiom, a postulate and a hypothesis are not the same thing, as the inspired Aristotle somewhere says. But, whenever that which is assumed and ranked as a principle is both known to the learner and convincing in itself, such a thing is an axiom, e.g. the statement that things which are equal to the same thing are also equal to one another. When, on [p. 122] the other hand, the pupil has not the notion of what is told him which carries conviction in itself, but nevertheless lays it down and assents to its being assumed, such an assumption is a hypothesis. Thus we do not preconceive by virtue of a common notion, and without being taught, that the circle is such and such a figure, but, when we are told so, we assent without demonstration. When again what is asserted is both unknown and assumed even without the assent of the learner, then, he says, we call this a postulate, e.g. that all right angles are equal. This view of a postulate is clearly implied by those who have made a special and systematic attempt to show, with regard to one of the postulates, that it cannot be assented to by any one straight off. According then to the teaching of Aristotle, an axiom, a postulate and a hypothesis are thus distinguished³² .”

We observe, first, that Proclus in this passage confuses hypotheses and definitions, although Aristotle had made the distinction quite plain. The confusion may be due to his having in his mind a passage of Plato from which he evidently got the phrase about “not giving an account of” the principles. The passage is³³ : “I think you know that those who treat of geometries and calculations (arithmetic) and such things take for granted (hupothemenoi) odd and even, figures, angles of three kinds, and other things akin to these in each subject, implying that they know these things, and, though using them as hypotheses, do not even condescend to give any account of them either to themselves or to others, but begin from these things and then go through everything else in order, arriving ultimately, by recognised methods, at the conclusion which they started in search of.” But the hypothesis is here the assumption, e.g. ‘that there may be such a thing as length without breadth, henceforward called a line³⁴ ,’ and so on, without any attempt to show that there is such a thing; it is mentioned in connexion with the distinction between Plato's ‘superior’ and ‘inferior’ intellectual method, the former of which uses successive hypotheses as stepping-stones by which it mounts upwards to the idea of Good.

We pass now to Proclus' account of the difference between postulates and axioms. He begins with the view of Geminus, according to which “they differ from one another in the same way as theorems are also distinguished from problems. For, as in theorems we propose to see and determine what follows on the premisses, while in problems we are told to find and do something, in like manner in the axioms such things are assumed as are manifest of themselves and easily apprehended by our untaught notions, while in the postulates we assume such things as are easy to find and effect (our understanding suffering no strain in their assumption), and we require no complication of machinery³⁵ .” ...“Both must have the characteristic of being simple [p. 123] and readily grasped, I mean both the postulate and the axiom; but the postulate bids us contrive and find some subject-matter (hulê) to exhibit a property simple and easily grasped, while the axiom bids us assert some essential attribute which is self-evident to the learner, just as is the fact that fire is hot, or any of the most obvious things³⁶ .”

Again, says Proclus, “some claim that all these things are alike postulates, in the same way as some maintain that all things that are sought are problems. For Archimedes begins his first book on Inequilibrium³⁷ with the remark ‘I postulate that equal weights at equal distances are in equilibrium,’ though one would rather call this an axiom. Others call them all axioms in the same way as some regard as theorems everything that requires demonstration³⁸ .”

“Others again will say that postulates are peculiar to geometrical subject-matter, while axioms are common to all investigation which is concerned with quantity and magnitude. Thus it is the geometer who knows that all right angles are equal and how to produce in a straight line any limited straight line, whereas it is a common notion that things which are equal to the same thing are also equal to one another, and it is employed by the arithmetician and any scientific person who adapts the general statement to his own subject³⁹ .”

The third view of the distinction between a postulate and an axiom is that of Aristotle above described⁴⁰ .

The difficulties in the way of reconciling Euclid's classification of postulates and axioms with any one of the three alternative views are next dwelt upon. If we accept the first view according to which an axiom has reference to something known, and a postulate to something done, then the 4th postulate (that all right angles are equal) is not a postulate; neither is the 5th which states that, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, will meet on that side on which are the angles less than two right angles. On the second view, the assumption that two straight lines cannot enclose a space, “which even now,” says Proclus, “some add as an axiom,” and which is peculiar to the subject-matter of geometry, like the fact that all right angles are equal, is not an axiom. According to the third (Aristotelian) view, “everything which is confirmed (pistoutai) by a sort of demonstration [p. 124] will be a postulate, and what is incapable of proof will be an axiom⁴¹ .” This last statement of Proclus is loose, as regards the axiom, because it omits Aristotle's requirement that the axiom should be a selfevident truth, and one that must be admitted by any one who is to learn anything at all, and, as regards the postulate, because Aristotle calls a postulate something assumed without proof though it is “matter of demonstration” (apodeikton on), but says nothing of a quasi-demonstration of the postulates. On the whole I think it is from Aristotle that we get the best idea of what Euclid understood by a postulate and an axiom or common notion. Thus Aristotle's account of an axiom as a principle common to all sciences, which is self-evident, though incapable of proof, agrees sufficiently with the contents of Euclid's common notions as reduced to five in the most recent text (not omitting the fourth, that “things which coincide are equal to one another”). As regards the postulates, it must be borne in mind that Aristotle says elsewhere⁴² that, “other things being equal, that proof is the better which proceeds from the fewer postulates or hypotheses or propositions.” If then we say that a geometer must lay down as principles, first certain axioms or common notions, and then an irreducible minimum of postulates in the Aristotelian sense concerned only with the subject-matter of geometry, we are not far from describing what Euclid in fact does. As regards the postulates we may imagine him saying: “Besides the common notions there are a few other things which I must assume without proof, but which differ from the common notions in that they are not self-evident. The learner may or may not be disposed to agree to them; but he must accept them at the outset on the superior authority of his teacher, and must be left to convince himself of their truth in the course of the investigation which follows. In the first place certain simple constructions, the drawing and producing of a straight line, and the drawing of a circle, must be assumed to be possible, and with the constructions the existence of such things as straight lines and circles; and besides this we must lay down some postulate to form the basis of the theory of parallels.” It is true that the admission of the 4th postulate that all right angles are equal still presents a difficulty to which we shall have to recur.

There is of course no foundation for the idea, which has found its way into many text-books, that “the object of the postulates is to declare that the only instruments the use of which is permitted in geometry are the rule and compass⁴³ .”

§ 4. THEOREMS AND PROBLEMS.

“Again the deductions from the first principles,” says Proclus, “are divided into problems and theorems, the former embracing the [p. 125] generation, division, subtraction or addition of figures, and generally the changes which are brought about in them, the latter exhibiting the essential attributes of each⁴⁴ .”

“Now, of the ancients, some, like Speusippus and Amphinomus, thought proper to call them all theorems, regarding the name of theorems as more appropriate than that of problems to theoretic sciences, especially as these deal with eternal objects. For there is no becoming in things eternal, so that neither could the problem have any place with them, since it promises the generation and making of what has not before existed, e.g. the construction of an equilateral triangle, or the describing of a square on a given straight line, or the placing of a straight line at a given point. Hence they say it is better to assert that all (propositions) are of the same kind, and that we regard the generation that takes place in them as referring not to actual making but to knowledge, when we treat things existing eternally as if they were subject to becoming: in other words, we may say that everything is treated by way of theorem and not by way of problem⁴⁵ (panta theôrêmatikôs all ou problêmatikôs lambanesthai).

“Others on the contrary, like the mathematicians of the school of Menaechmus, thought it right to call them all problems, describing their purpose as twofold, namely in some cases to furnish (porisasthai) the thing sought, in others to take a determinate object and see either what it is, or of what nature, or what is its property, or in what relations it stands to something else.

“In reality both assertions are correct. Speusippus is right because the problems of geometry are not like those of mechanics, the latter being matters of sense and exhibiting becoming and change of every sort. The school of Menaechmus are right also because the discoveries even of theorems do not arise without an issuing-forth into matter, by which I mean intelligible matter. Thus forms going out into matter and giving it shape may fairly be said to be like processes of becoming. For we say that the motion of our thought and the throwing-out of the forms in it is what produces the figures in the imagination and the conditions subsisting in them. It is in the imagination that constructions, divisions, placings, applications, additions and subtractions (take place), but everything in the mind is fixed and immune from becoming and from every sort of change⁴⁶ .”

“Now those who distinguish the theorem from the problem say that every problem implies the possibility, not only of that which is predicated of its subject-matter, but also of its opposite, whereas every theorem implies the possibility of the thing predicated but not of its opposite as well. By the subject-matter I mean the genus which is the subject of inquiry, for example, a triangle or a square or a circle, and by the property predicated the essential attribute, as equality, section, position, and the like. When then any one [p. 126] enunciates thus, To inscribe an equilateral triangle in a circle, he states a problem; for it is also possible to inscribe in it a triangle which is not equilateral. Again, if we take the enunciation On a given limited straight line to construct an equilateral triangle, this is a problem; for it is possible also to construct one which is not equilateral. But, when any one enunciates that In isosceles triangles the angles at the base are equal, we must say that he enunciates a theorem; for it is not also possible that the angles at the base of isosceles triangles should be unequal. It follows that, if any one were to use the form of a problem and say In a semicircle to describe a right angle, he would be set down as no geometer. For every angle in a semicircle is right⁴⁷ .”

“Zenodotus, who belonged to the succession of Oenopides, but was a disciple of Andron, distinguished the theorem from the problem by the fact that the theorem inquires what is the property predicated of the subject-matter in it, but the problem what is the cause of what effect (tinos ontos ti estin). Hence too Posidonius defined the one (the problem) as a proposition in which it is inquired whether a thing exists or not (ei estin ê mê), the other (the theorem⁴⁸ ) as a proposition in which it is inquired what (a thing) is or of what nature (ti estin ê poion ti); and he said that the theoretic proposition must be put in a declaratory form, e.g., Any triangle has two sides (together) greater than the remaining side and In any isosceles triangle the angles at the base are equal, but that we should state the problematic propositi\on as if inquiring whether it is possible to construct an equilateral triangle upon such and such a straight line. For there is a difference between inquiring absolutely and indeterminately (haplôs te kai aoristôs) whether there exists a straight line from such and such a point at right angles to such and such a straight line and investigating which is the straight line at right angles⁴⁹ .”

“That there is a certain difference between the problem and the theorem is clear from what has been said; and that the Elements of Euclid contain partly problems and partly theorems will be made manifest by the individual propositions, where Euclid himself adds at the end of what is proved in them, in some cases, ‘that which it was required to do,’ and in others, ‘that which it was required to prove,’ the latter expression being regarded as characteristic of theorems, in spite of the fact that, as we have said, demonstration is found in problems also. In problems, however, even the demonstration is for the purpose of (confirming) the construction: for wė bring in the demonstration in order to show that what was enjoined has been done; whereas in theorems the demonstration is worthy of study for its own sake as being capable of putting before us the nature of the thing sought. And you will find that Euclid sometimes interweaves theorems with problems and employs them in turn, as in the first [p. 127] book, while at other times he makes one or other preponderate. For the fourth book consists wholly of problems, and the fifth of theorems⁵⁰ .”

Again, in his note on Eucl. 1. 4, Proclus says that Carpus, the writer on mechanics, raised the question of theorems and problems in his treatise on astronomy. Carpus, we are told, “says that the class of problems is in order prior to theorems. For the subjects, the properties of which are sought, are discovered by means of problems. Moreover in a problem the enunciation is simple and requires no skilled intelligence; it orders you plainly to do such and such a thing, to construct an equilateral triangle, or, given two straight lines, to cut off from the greater (a straight line) equal to the lesser, and what is there obscure or elaborate in these things? But the enunciation of a theorem is a matter of labour and requires much exactness and scientific judgment in order that it may not turn out to exceed or fall short of the truth; an example is found even in this proposition (1. 4), the first of the theorems. Again, in the case of problems, one general way has been discovered, that of analysis, by following which we can always hope to succeed; it is this method by which the more obscure problems are investigated. But, in the case of theorems, the method of setting about them is hard to get hold of since ‘up to our time,’ says Carpus, ‘no one has been able to hand down a general method for their discovery. Hence, by reason of their easiness, the class of problems would naturally be more simple.’ After these distinctions, he proceeds: ‘Hence it is that in the Elements too problems precede theorems, and the Elements begin from them; the first theorem is fourth in order, not because the fifth⁵¹ is proved from the problems, but because, even if it needs for its demonstration none of the propositions which precede it, it was necessary that they should be first because they are problems, while it is a theorem. In fact, in this theorem he uses the common notions exclusively, and in some sort takes the same triangle placed in different positions; the coincidence and the equality proved thereby depend entirely upon sensible and distinct apprehension. Nevertheless, though the demonstration of the first theorem is of this character, the problems properly preceded it, because in general problems are allotted the order of precedence⁵² .’”

Proclus himself explains the position of Prop. 4 after Props. 1-3 as due to the fact that a theorem about the essential properties of triangles ought not to be introduced before we know that such a thing as a triangle can be constructed, nor a theorem about the equality of sides or straight lines until we have shown, by constructing them, that there can be two straight lines which are equal to one another⁵³ . It is plausible enough to argue in this way that Props. 2 and 3 at all events should precede Prop. 4. And Prop. 1 is used in [p. 128] Prop. 2, and must therefore precede it. But Prop. I showing how to construct an equilateral triangle on a given base is not important, in relation to Prop. 4, as dealing with the “production of triangles” in general: for it is of no use to say, as Proclus does, that the construction of the equilateral triangle is “common to the three species (of triangles)⁵⁴ ,” as we are not in a position to know this at such an early stage. The existence of triangles in general was doubtless assumed as following from the existence of straight lines and points in one plane and from the possibility of drawing a straight line from one point to another.

Proclus does not however seem to reject definitely the view of Carpus, for he goes on⁵⁵ : “And perhaps problems are in order before theorems, and especially for those who need to ascend from the arts which are concerned with things of sense to theoretical investigation. But in dignity theorems are prior to problems....It is then foolish to blame Geminus for saying that the theorem is more perfect than the problem. For Carpus himself gave the priority to problems in respect of order, and Geminus to theorems in point of more perfect dignity,” so that there was no real inconsistency between the two.

Problems were classified according to the number of their possible solutions. Amphinomus said that those which had a unique solution (monachôs) were called “ordered” (the word has dropped out in Proclus, but it must be tetagmena, in contrast to the third kind, atakta); those which had a definite number of solutions “intermediate” (mesa); and those with an infinite variety of solutions “unordered” (atakta)⁵⁶ . Proclus gives as an example of the last the problem To divide a given straight line into three parts in continued proportion⁵⁷ . This is the same thing as solving the equations . Proclus' remarks upon the problem show that it was solved, like all quadratic equations, by the method of “application of areas.” The straight line a was first divided into any two parts, (x+z) and y, subject to the sole limitation that (x+z) must not be less than 2y, which limitation is the diorismos, or condition of possibility. Then an area was applied to (x+z), or (a-y), “falling short by a square figure” (elleipon eidei tetragônôi) and equal to the square on y. This determines x and z separately in terms of a and y. For, if z be the side of the square by which the area (i.e. rectangle) “falls short,” we have whence And y may be chosen arbitrarily, provided that it is not greater than a/3. Hence there are an infinite number of solutions. If then, as Proclus remarks, the three parts are equal.

Other distinctions between different kinds of problems are added by Proclus. The word “problem,” he says, is used in several senses. In its widest sense it may mean anything “propounded” (proteinomenon), whether for the purpose of instruction (mathêseôs) or construction (poiêseôs). (In this sense, therefore, it would include a theorem.) [p. 129] But its special sense in mathematics is that of something “propounded with a view to a theoretic construction⁵⁸ .”

Again you may apply the term (in this restricted sense) even to something which is impossible, although it is more appropriately used of what is possible and neither asks too much nor contains too little in the shape of data. According as a problem has one or other of these defects respectively, it is called (1) a problem in excess (pleonazon) or (2) a deficient problem (ellipes problêma). The problem in excess (1) is of two kinds, (a) a problem in which the properties of the figure to be found are either inconsistent (asumbata) or non-existent (anuparkta), in which case the problem is called impossible, or (b) a problem in which the enunciation is merely redundant: an example of this would be a problem requiring us to construct an equilateral triangle with its vertical angle equal to two-thirds of a right angle; such a problem is possible and is called “more than a problem” (meizon ê problêma). The deficient problem (2) is similarly called “less than a problem” (elasson ê problêma), its characteristic being that something has to be added to the enunciation in order to convert it from indeterminateness (aoristia) to order (taxis) and scientific determinateness (horos epistêmonikos): such would be a problem bidding you “to construct an isosceles triangle,” for the varieties of isosceles triangles are unlimited. Such “problems” are not problems in the proper sense (kuriôs legomena problêmata), but only equivocally⁵⁹ .

§ 5. THE FORMAL DIVISIONS OF A PROPOSITION.

“Every problem,” says Proclus⁶⁰ , “and every theorem which is complete with all its parts perfect purports to contain in itself all of the following elements: enunciation (protasis), setting-out (ekthesis), definition or specification (diorismos), construction or machinery (kataskeuê), proof (apodeixis), conclusion (sumperasma). Now of these the enunciation states what is given and what is that which is sought, the perfect enunciation consisting of both these parts. The setting-out marks off what is given, by itself, and adapts it beforehand for use in the investigation. The definition or specification states separately and makes clear what the particular thing is which is sought. The construction or machinery adds what is wanting to the datum for the purpose of finding what is sought. The proof draws the required inference by reasoning scientifically from acknowledged facts. The conclusion reverts again to the enunciation, confirming what has been demonstrated. These are all the parts of problems and theorems, but the most essential and those which are found in all are enunciation, proof, conclusion. For it is equally necessary to know beforehand what is sought, to prove this by means of the intermediate steps, and to state the proved fact as a conclusion; it is impossible to dispense with any of these three things. The remaining parts are often brought in, but are often left out as serving no purpose. [p. 130] Thus there is neither setting-out nor definition in the problem of constructing an isosceles triangle having each of the angles at the base double of the remaining angle, and in most theorems there is no construction because the setting-out suffices without any addition for proving the required property from the data. When then do we say that the setting-out is wanting? The answer is, when there is nothing given in the enunciation; for, though the enunciation is in general divided into what is given and what is sought, this is not always the case, but sometimes it states only what is sought, i.e. what must be known or found, as in the case of the problem just mentioned. That problem does not, in fact, state beforehand with what datum we are to construct the isosceles triangle having each of the equal angles double of the remaining angle, but (simply) that we are to find such a triangle.... When, then, the enunciation contains both (what is given and what is sought), in that case we find both definition and setting-out, but, whenever the datum is wanting, they too are wanting. For not only is the setting-out concerned with the datum, but so is the definition also, as, in the absence of the datum, the definition will be identical with the enunciation. In fact, what could you say in defining the object of the aforesaid problem except that it is required to find an isosceles triangle of the kind referred to? But that is what the enunciation stated. If then the enunciation does not include, on the one hand, what is given and, on the other, what is sought, there is no setting-out in virtue of there being no datum, and the definition is left out in order to avoid a mere repetition of the enunciation.”

The constituent parts of an Euclidean proposition will be readily identified by means of the above description. As regards the definition or specification (diorismos) it is to be observed that we have here only one of its uses. Here it means a closer definition or description of the object aimed at, by means of the concrete lines or figures set out in the ekthesis instead of the general terms used in the enunciation; and its purpose is to rivet the attention better, as Proclus indicates in a later passage (tropon tina prosecheias estin aitios ho diorismos)⁶¹ .

The other technical use of the word to signify the limitations to which the possible solutions of a problem are subject is also described by Proclus, who speaks of diorismoi determining “whether what is sought is impossible or possible, and how far it is practicable and in how many ways⁶² ”; and the diorismos in this sense appears in Euclid as well as in Archimedes and Apollonius. Thus we have in Eucl. 1. 22 the enunciation “From three straight lines which are equal to three given straight lines to construct a triangle,” followed immediately by the limiting condition (diorismos). “Thus two of the straight lines taken together in any manner must be greater than the remaining one.” Similarly in VI. 28 the enunciation “To a given straight line to apply a parallelogram equal to a given rectilineal [p. 131] figure and falling short by a parallelogrammic figure similar to a given one” is at once followed by the necessary condition of possibility: “Thus the given rectilineal figure must not be greater than that described on half the line and similar to the defect.”

Tannery supposed that, in giving the other description of the diorismos as quoted above, Proclus, or rather his guide, was using the term incorrectly. The diorismos in the better known sense of the determination of limits or conditions of possibility was, we are told, invented by Leon. Pappus uses the word in this sense only. The other use of the term might, Tannery thought, be due to a confusion occasioned by the use of the same words (dei dê) in introducing the parts of a proposition corresponding to the two meanings of the word diorismos⁶³ . On the other hand it is to be observed that Eutocius distinguishes clearly between the two uses and implies that the difference was well known⁶⁴ . The diorismos in the sense of condition of possibility follows immediately on the enunciation, is even part of it; the diorismos in the other sense of course comes immediately after the setting-out.

Proclus has a useful observation respecting the conclusion of a proposition⁶⁵ . “The conclusion they are accustomed to make double in a certain way: I mean, by proving it in the given case and then drawing a general inference, passing, that is, from the partial conclusion to the general. For, inasmuch as they do not make use of the individuality of the subjects taken, but only draw an angle or a straight line with a view to placing the datum before our eyes, they consider that this same fact which is established in the case of the particular figure constitutes a conclusion true of every other figure of the same kind. They pass accordingly to the general in order that we may not conceive the conclusion to be partial. And they are justified in so passing, since they use for the demonstration the particular things set out, not quâ particulars, but quâ typical of the rest. For it is not in virtue of such and such a size attaching to the angle which is set out that I effect the bisection of it, but in virtue of its being rectilineal and nothing more. Such and such size is peculiar to the angle set out, but its quality of being rectilineal is common to all rectilineal angles. Suppose, for example, that the given angle is a right angle. If then I had employed in the proof the fact of its being right, I should not have been able to pass to every species of rectilineal angle; but, if I make no use of its being right, and only consider it as rectilineal, the argument will equally apply to rectilineal angles in general.”

§ 6. OTHER TECHNICAL TERMS.

1. Things said to be given.

Proclus attaches to his description of the formal divisions of a proposition an explanation of the different senses in which the word given or datum (dedomenon) is used in geometry. “Everything that is given is given in one or other of the following ways, in position, in ratio, in magnitude, or in species. The point is given in position only, but a line and the rest may be given in all the senses⁶⁶ .”

The illustrations which Proclus gives of the four senses in which a thing may be given are not altogether happy, and, as regards things which are given in position, in magnitude, and in species, it is best, I think, to follow the definitions given by Euclid himself in his book of Data. Euclid does not mention the fourth class, things given in ratio, nor apparently do any of the great geometers.

(1) Given in position really needs no definition; and, when Euclid says (Data, Def. 4) that “Points, lines and angles are said to be given in position which always occupy the same place,” we are not really the wiser.

(2) Given in magnitude is defined thus (Data, Def. 1): “Areas, lines and angles are called given in magnitude to which we can find equals.” Proclus' illustration is in this case the following: when, he says, two unequal straight lines are given from the greater of which we have to cut off a straight line equal to the lesser, the straight lines are obviously given in magnitude, “for greater and less, and finite and infinite are predications peculiar to magnitude.” But he does not explain that part of the implication of the term is that a thing is given in magnitude only, and that, for example, its position is not given and is a matter of indifference

(3) Given in species. Euclid's definition (Data, Def. 3) is: “Rectilineal figures are said to be given in species in which the angles are severally given and the ratios of the sides to one another are given” And this is the recognised use of the term (cf. Pappus, passim) Proclus uses the term in a much wider sense for which I am not aware of any authority. Thus, he says, when we speak of (bisecting) a given rectilineal angle, the angle is given in species by the word rectilineal, which prevents our attempting, by the same method, to bisect a curvilineal angle! On Eucl. 1. 9, to which he here refers, he says that an angle is given in species when e.g. we say that it is right or acute or obtuse or rectilineal or “mixed,” but that the actual angle in the proposition is given in species only. As a matter of fact, we should say that the actual angle in the figure of the proposition is given in magnitude and not in species, part of the implication of given in species being that the actual magnitude of the thing given in species is indifferent; an angle cannot be given in species in this sense at all. The confusion in Proclus' mind is shown when, after saying that a right angle is given in species, he describes a third of a right angle as given in magnitude. [p. 133]

No better example of what is meant by given in species, in its proper sense, as limited to rectilineal figures, can be quoted than the given parallelogram in Eucl. VI. 28, to which the required parallelogram has to be made similar; the former parallelogram is in fact given in species, though its actual size, or scale, is indifferent.

(4) Given in ratio presumably means something which is given by means of its ratio to some other given thing. This we gather from Proclus' remark (in his note on 1. 9) that an angle may be given in ratio “as when we say that it is double and treble of such and such an angle or, generally, greater and less.” The term, however, appears to have no authority and to serve no purpose. Proclus may have derived it from such expressions as “in a given ratio” which are common enough.

2. Lemma.

“The term lemma,” says Proclus⁶⁷ , “is often used of any proposition which is assumed for the construction of something else: thus it is a common remark that a proof has been made out of such and such lemmas. But the special meaning of lemma in geometry is a proposition requiring confirmation. For when, in either construction or demonstration, we assume anything which has not been proved but requires argument, then, because we regard what has been assumed as doubtful in itsėlf and therefore worthy of investigation, we call it a lemma⁶⁸ , differing as it does from the postulate and the axiom in being matter of demonstration, whereas they are immediately taken for granted, without demonstration, for the purpose of confirming other things. Now in the discovery of lemmas the best aid is a mental aptitude for it. For we may see many who are quick at solutions and yet do not work by method; thus Cratistus in our time was able to obtain the required result from first principles, and those the fewest possible, but it was his natural gift which helped him to the discovery. [p. 134] Nevertheless certain methods have been handed down. The finest is the method which by means of analysis carries the thing sought up to an acknowledged principle, a method which Plato, as they say, communicated to Leodamas⁶⁹ , and by which the latter, too, is said to have discovered many things in geometry. The second is the method of division⁷⁰ , which divides into its parts the genus proposed for consideration and gives a starting-point for the demonstration by means of the elimination of the other elements in the construction of what is proposed, which method also Plato extolled as being of assistance to all sciences. The third is that by means of the reductio ad absurdum, which does not show what is sought directly; but refutes its opposite and discovers the truth incidentally.”

3. Case.

“The case⁷¹ (ptôsis),” Proclus proceeds⁷² , “announces different ways of construction and alteration of positions due to the transposition of points or lines or planes or solids. And, in general, all its varieties are seen in the figure, and this is why it is called case, being a transposition in the construction.”

4. Porism.

“The term porism is used also of certain problems such as the Porisms written by Euclid. But it is specially used when from what has been demonstrated some other theorem is revealed at the same time without our propounding it, which theorem has on this very account been called a porism (corollary) as being a sort of incidental gain arising from the scientific demonstration⁷³ .” Cf. the note on I. 15. [p. 135]

5. Objection.

“The objection (enstasis) obstructs the whole course of the argument by appearing as an obstacle (or crying ‘halt,’ apantôsa) either to the construction or to the demonstration. There is this difference between the objection and the case, that, whereas he who propounds the case has to prove the proposition to be true of it, he who makes the objection does not need to prove anything: on the contrary it is necessary to destroy the objection and to show that its author is saying what is false⁷⁴ .”

That is, in general the objection endeavours to make it appear that the demonstration is not true in every case; and it is then necessary to prove, in refutation of the objection, either that the supposed case is impossible, or that the demonstration is true even for that case. A good instance is afforded by Eucl. 1. 7. The text-books give a second case which is not in the original text of Euclid. Proclus remarks on the proposition as given by Euclid that the objection may conceivably be raised that what Euclid declares to be impossible may after all be possible in the event of one pair of stiaight lines falling completely within the other pair. Proclus then refutes the objection by proving the impossibility in that case also. His proof then came to be given in the text-books as part of Euclid's proposition.

The objection is one of the technical terms in Aristotle's logic and its nature is explained in the Prior Analytics⁷⁵ . “An objection is a proposition contrary to a proposition.... Objections are of two sorts, general or partial.... For when it is maintained that an attribute belongs to every (member of a class), we object either that it belongs to none (of the class) or that there is some one (member of the class) to which it does not belong.”

6. Reduction.

This is again an Aristotelian term, explained in the Prior Analytics⁷⁶ . It is well described by Proclus in the following passage:

“Reduction (apagôgê) is a transition from one problem or theorem to another, the solution or proof of which makes that which is propounded manifest also. For example, after the doubling of the cube had been investigated, they transformed the investigation into another upon which it follows, namely the finding of the two means; and from that time forward they inquired how between two given straight lines two mean proportionals could be discovered. And they say that the first to effect the reduction of difficult constructions was Hippocrates of Chios, who also squared a lune and discovered many other things in geometry, being second to none in ingenuity as regards constructions⁷⁷ .” [p. 136]

7. Reductio ad absurdum.

This is variously called by Aristotle “reductio ad absurdum” (hê eis to adunaton apagôgê)⁷⁸ , “proof per impossibile” (hê dia tou adunatou deixis or apodeixis)⁷⁹ , “proof leading to the impossible” (hê eis to adunaton agousa apodeixis)⁸⁰ . It is part of “proof (starting) from a hypothesis⁸¹ ” (ex hupotheseôs). “All (syllogisms) which reach the conclusion per impossibile reason out a conclusion which is false, and they prove the original contention (by the method starting) from a hypothesis, when something impossible results from assuming the contradictory of the original contention, as, for example, when it is proved that the diagonal (of a square) is incommensurable because, if it be assumed commensurable, it will follow that odd (numbers) are equal to even (numbers)⁸² .” Or again, “proof (leading) to the impossible differs from the direct (deiktikês) in that it assumes what it desires to destroy [namely the hypothesis of the falsity of the conclusion] and then reduces it to something admittedly false, whereas the direct proof starts from premisses admittedly true⁸³ .”

Proclus has the following description of the reductio ad absurdum. “Proofs by reductio as absurdum in every case reach a conclusion manifestly impossible, a conclusion the contradictory of which is admitted. In some cases the conclusions are found to conflict with the common notions, or the postulates, or the hypotheses (from which we started); in others they contradict propositions previously established⁸⁴ ” ...“Every reductio ad absurdum assumes what conflicts with the desired result, then, using that as a basis, proceeds until it arrives at an admitted absurdity, and, by thus destroying the hypothesis, establishes the result originally desired. For it is necessary to understand generally that all mathematical arguments either proceed from the first principles or lead back to them, as Porphyry somewhere says. And those which proceed from the first principles are again of two kinds, for they start either from common notions and the clearness of the self-evident alone, or from results previously proved; while those which lead back to the principles are either by way of assuming the principles or by way of destroying them. Those which assume the principles are called analyses, and the opposite of these are syntheses-- for it is possible to start from the said principles and to proceed in the regular order to the desired conclusion, and this process is synthesis--while the arguments which would destroy the principles are [p. 137] called reductiones ad absurdum. For it is the function of this method to upset something admitted as clear⁸⁵ .”

8. Analysis and Synthesis.

It will be seen from the note on Eucl. XIII. I that the MSS. of the Elements contain definitions of Analysis and Synthesis followed by alternative proofs of XIII. 1-5 after that method. The definitions and alternative proofs are interpolated, but they have great historical interest because of the possibility that they represent an ancient method of dealing with these propositions, anterior to Euclid. The propositions give properties of a line cut “in extreme and mean ratio,” and they are preliminary to the construction and comparison of the five regular solids. Now Pappus, in the section of his Collection dealing with the latter subject⁸⁶ , says that he will give the comparisons between the five figures, the pyramid, cube, octahedron, dodecahedron and icosahedron, which have equal surfaces, “not by means of the so-called analytical inquiry, by which some of the ancients worked out the proofs, but by the synthetical method⁸⁷ ....” The conjecture of Bretschneider that the matter interpolated in Eucl. XIII. is a survival of investigations due to Eudoxus has at first sight much to commend it⁸⁸ . In the first place, we are told by Proclus that Eudoxus “greatly added to the number of the theorems which Plato originated regarding the section, and employed in them the method of analysis⁸⁹ .” It is obvious that “the section” was some particular section which by the time of Plato had assumed great importance; and the one section of which this can safely be said is that which was called the “golden section,” namely, the division of a straight line in extreme and mean ratio which appears in Eucl. II. 11 and is therefore most probably Pythagorean. Secondly, as Cantor points out⁹⁰ , Eudoxus was the founder of the theory of proportions in the form in which we find it in Euclid V., VI., and it was no doubt through meeting, in the course of his investigations, with proportions not expressible by whole numbers that he came to realise the necessity for a new theory of proportions which should be applicable to incommensurable as well as commensurable magnitudes. The “golden section” would furnish such a case. And it is even mentioned by Proclus in this connexion. He is explaining⁹¹ that it is only in arithmetic that all quantities bear “rational” ratios (rhêtos logos) to one another, while in geometry there are “irrational” ones (arrêtos) as well. “Theorems about sections like those in Euclid's second Book are common to both [arithmetic and geometry] except that in which the straight line is cut in extreme and mean ratio⁹² .” [p. 138]

The definitions of Analysis and Synthesis interpolated in Eucl. XIII. are as follows (I adopt the reading of B and V, the only intelligible one, for the second).

“Analysis is an assumption of that which is sought as if it were admitted through its consequences to something admitted (to be) true.

“Synthesis is an assumption of that which is admitted through its consequences to the finishing or attainment of what is sought.”

The language is by no means clear and has, at the best, to be filled out.

Pappus has a fuller account⁹³ :

“The so-called analuomenos (‘Treasury of Analysis’) is, to put it shortly, a special body of doctrine provided for the use of those who, after finishing the ordinary Elements, are desirous of acquiring the power of solving problems which may be set them involving (the construction of) lines, and it is useful for this alone. It is the work of three men, Euclid the author of the Elements, Apollonius of Perga, and Aristaeus the elder, and proceeds by way of analysis and synthesis.

“Analysis then takes that which is sought as if it were admitted and passes from it through its successive consequences to something which is admitted as the result of synthesis: for in analysis we assume that which is sought as if it were (already) done (gegonos), and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the class of first principles, and such a method we call analysis as being solution backwards (anapalin lusin).

“But in synthesis, reversing the process, we take as already done that which was last arrived at in the analysis and, by arranging in their natural order as consequences what were before antecedents, and successively connecting them one with another, we arrive finally at the construction of what was sought; and this we call synthesis.

“Now analysis is of two kinds, the one directed to searching for the truth and called theoretical, the other directed to finding what we are told to find and called problematical. (1) In the theoretical kind we assume what is sought as if it were existent and true, after which we pass through its successive consequences, as if they too were true and established by virtue of our hypothesis, to something admitted: then (a), if that something admitted is true, that which is sought will also be true and the proof will correspond in the reverse order to the analysis, but (b), if we come upon something admittedly false, that which is sought will also be false. (2) In the problematical kind we assume that which is propounded as if it were known, after which we pass through its successive consequences, taking them as true, up to something admitted: if then (a) what is admitted is possible and obtainable, that is, what mathematicians call given, what was originally proposed will also be possible, and the proof will again correspond in [p. 139] reverse order to the analysis, but if (b) we come upon something admittedly impossible, the problem will also be impossible.”

The ancient Analysis has been made the subject of careful studies by several writers during the last half-century, the most complete being those of Hankel, Duhamel and Zeuthen; others by Ofterdinger and Cantor should also be mentioned⁹⁴ .

The method is as follows. It is required, let us say, to prove that a certain proposition A is true. We assume as a hypothesis that A is true and, starting from this we find that, if A is true, a certain other proposition B is true; if B is true, then C; and so on until we arrive at a proposition K which is admittedly true. The object of the method is to enable us to infer, in the reverse order, that, since K is true, the proposition A originally assumed is true. Now Aristotle had already made it clear that false hypotheses might lead to a conclusion which is true. There is therefore a possibility of error unless a certain precaution is taken. While, for example, B may be a necessary consequence of A, it may happen that A is not a necessary consequence of B. Thus, in order that the reverse inference from the truth of K that A is true may be logically justified, it is necessary that each step in the chain of inferences should be unconditionally convertible. As a matter of fact, a very large number of theorems in elementary geometry are unconditionally convertible, so that in practice the difficulty in securing that the successive steps shall be convertible is not so great as might be supposed. But care is always necessary. For example, as Hankel says⁹⁵ , a proposition may not be unconditionally convertible in the form in which it is generally quoted. Thus the proposition “The vertices of all triangles having a common base and constant vertical angle lie on a circle” cannot be converted into the proposition that “All triangles with common base and vertices lying on a circle have a constant vertical angle”; for this is only true if the further conditions are satisfied (1) that the circle passes through the extremities of the common base and (2) that only that part of the circle is taken as the locus of the vertices which lies on one side of the base. If these conditions are added, the proposition is unconditionally convertible. Or again, as Zeuthen remarks⁹⁶ , K may be obtained by a series of inferences in which A or some other proposition in the series is only apparently used; this would be the case e.g. when the method of modern algebra is being employed and the expressions on each side of the sign of equality have been inadvertently multiplied by some composite magnitude which is in reality equal to zero.

Although the above extract from Pappus does not make it clear that each step in the chain of argument must be convertible in the case taken, he almost implies this in the second part of the definition of Analysis where, instead of speaking of the consequences B, C... [p. 140] successively following from A, he suddenly changes the expression and says that we inquire what it is (B) from which A follows (A being thus the consequence of B, instead of the reverse), and then what (viz. C) is the antecedent cause of B; and in practice the Greeks secured what was wanted by always insisting on the analysis being confirmed by subsequent synthesis, that is, they laboriously worked backwards the whole way from K to A, reversing the order of the analysis, which process would undoubtedly bring to light any flaw which had crept into the argument through the accidental neglect of the necessary precautions.

Reductio ad absurdum a variety of analysis.

In the process of analysis starting from the hypothesis that a proposition A is true and passing through B, C... as successive consequences we may arrive at a proposition K which, instead of being admittedly true, is either admittedly false or the contradictory of the original hypothesis A or of some one or more of the propositions B, C... intermediate between A and K. Now correct inference from a true proposition cannot lead to a false proposition; and in this case therefore we may at once conclude, without any inquiry whether the various steps in the argument are convertible or not, that the hypothesis A is false, for, if it were true, all the consequences correctly inferred from it would be true and no incompatibility could arise. This method of proving that a given hypothesis is false furnishes an indirect method of proving that a given hypothesis A is true, since we have only to take the contradictory of A and to prove that it is false. This is the method of reductio ad absurdum, which is therefore a variety of analysis. The contradictory of A, or not-A, will generally include more than one case and, in order to prove its falsity, each of the cases must be separately disposed of: e.g., if it is desired to prove that a certain part of a figure is equal to some other part, we take separately the hypotheses (1) that it is greater, (2) that it is less, and prove that each of these hypotheses leads to a conclusion either admittedly false or contradictory to the hypothesis itself or to some one of its consequences.

Analysis as applied to problems.

It is in relation to problems that the ancient analysis has the greatest significance, because it was the one general method which the Greeks used for solving all “the more abstruse problems” (ta asaphestera tôn problêmatôn)⁹⁷ .

We have, let us suppose, to construct a figure satisfying a certain set of conditions If we are to proceed at all methodically and not by mere guesswork, it is first necessary to “analyse” those conditions. To enable this to be done we must get them clearly in our minds, which is only possible by assuming all the conditions to be actually fulfilled, in other words, by supposing the problem solved. Then we have to transform those conditions, by all the means which practice in such cases has taught us to employ, into other conditions which are necessarily fulfilled if the original conditions are, and to continue this [p. 141] transformation until we at length arrive at conditions which we are in a position to satisfy⁹⁸ . In other words, we must arrive at some relation which enables us to construct a particular part of the figure which, it is true, has been hypothetically assumed and even drawn, but which nevertheless really requires to be found in order that the problem may be solved. From that moment the particular part of the figure becomes one of the data, and a fresh relation has to be found which enables a fresh part of the figure to be determined by means of the original data and the new one together. When this is done, the second new part of the figure also belongs to the data; and we proceed in this way until all the parts of the required figure are found⁹⁹ . The first part of the analysis down to the point of discovery of a relation which enables us to say that a certain new part of the figure not belonging to the original data is given, Hankel calls the transformation; the second part, in which it is proved that all the remaining parts of the figure are “given,” he calls the resolution. Then follows the synthesis, which also consists of two parts, (1) the construction, in the order in which it has to be actually carried out, and in general following the course of the second part of the analysis, the resolution; (2) the demonstration that the figure obtained does satisfy all the given conditions, which follows the steps of the first part of the analysis, the transformation, but in the reverse order. The second part of the analysis, the resolution, would be much facilitated and shortened by the existence of a systematic collection of Data such as Euclid's book bearing that title, consisting of propositions proving that, if in a figure certain parts or relations are given, other parts or relations are also given. As regards the first part of the analysis, the transformation, the usual rule applies that every step in the chain must be unconditionally convertible; and any failure to observe this condition will be brought to light by the subsequent synthesis. The second part, the resolution, can be directly turned into the construction since that only is given which can be constructed by the means provided in the Elements.

It would be difficult to find a better illustration of the above than the example chosen by Hankel from Pappus¹⁰⁰ .

Given a circle ABC and two points D, E external to it, to draw straight lines DB, EB from D, E to a point B on the circle such that, if DB, EB produced meet the circle again in C, A, AC shall be parallel to DE.

Analysis.

Suppose the problem solved and the tangent at A drawn, meeting ED produced in F.

(Part I. Transformation.)

Then, since AC is parallel to DE, the angle at C is equal to the angle CDE.

But, since FA is a tangent, the angle at C is equal to the angle FAE.

Therefore the angle FAE is equal to the angle CDE, whence A, B, D, F are concyclic. [p. 142]

Therefore the rectangle AE, EB is equal to the rectangle FE, ED.

[Figure]

(Part II. Resolution.)

But the rectangle AE, EB is given, because it is equal to the square on the tangent from E.

Therefore the rectangle FE, ED is given; and, since ED is given, FE is given (in length). [Data, 57.]

But FE is given in position also, so that F is also given. [Data, 27.]

Now FA is the tangent from a given point F to a circle ABC given in position; therefore FA is given in position and magnitude. [Data, 90.]

And F is given; therefore A is given.

But E is also given; therefore the straight line AE is given in position. [Data, 26.]

And the circle ABC is given in position; therefore the point B is also given. [Data, 25.]

But the points D, E are also given; therefore the straight lines DB, BE are also given in position.

Synthesis.

(Part I. Construction.)

Suppose the circle ABC and the points D, E given.

Take a rectangle contained by ED and by a certain straight line EF equal to the square on the tangent to the circle from E.

From F draw FA touching the circle in A; join ABE and then DB, producing DB to meet the circle at C. Join AC.

I say then that AC is parallel to DE.

(Part II. Demonstration.)

Since, by hypothesis, the rectangle FE, ED is equal to the square on the tangent from E, which again is equal to the rectangle AE, EB, the rectangle AE, EB is equal to the rectangle FE, ED.

Therefore A, B, D, F are concyclic, whence the angle FAE is equal to the angle BDE.

But the angle FAE is equal to the angle ACB in the alternate segment; therefore the angle ACB is equal to the angle BDE.

Therefore AC is parallel to DE.

In cases where a diorismos is necessary, i.e. where a solution is only possible under certain conditions, the analysis will enable those conditions to be ascertained. Sometimes the diorismos is stated and proved at the end of the analysis, e.g. in Archimedes, On the Sphere and Cylinder, II. 7; sometimes it is stated in that place and the proof postponed till after the end of the synthesis, e.g. in the solution of the problem subsidiary to On the Sphere and Cylinder, II. 4, preserved in Eutocius' commentary on that proposition. The analysis should also enable us to determine the number of solutions of which the problem is susceptible.

§ 7. THE DEFINITIONS.

General. “Real” and “Nominal” Definitions.

It is necessary, says Aristotle, whenever any one treats of any whole subject, to divide the genus into its primary constituents, those which are indivisible in species respectively: e.g. number must be divided into triad and dyad; then an attempt must be made in this way to obtain definitions, e.g. of a straight line, of a circle, and of a right angle¹⁰¹ .

The word for definition is horos. The original meaning of this word seems to have been “boundary,” “landmark.” Then we have it in Plato and Aristotle in the sense of standard or determining principle (“id quo alicuius rei natura constituitur vel definitur,” Index Aristotelicus)¹⁰² ; and closely connected with this is the sense of definition. Aristotle uses both horos and horismos for definition, the former occurring more frequently in the Topics, the latter in the Metaphysics.

Let us now first be clear as to what a definition does not do. There is nothing in connexion with definitions which Aristotle takes more pains to emphasise than that a definition asserts nothing as to the existence or non-existence of the thing defined. It is an answer to the question what a thing is (ti esti), and does not say that it is (hoti esti). The existence of the various things defined has to be proued, except in the case of a few primary things in each science, the existence of which is indemonstrable and must be assumed among the first principles of each science; e.g. points and lines in geometry must be assumed to exist, but the existence of everything else must be proved. This is stated clearly in the long passage quoted above under First Principles¹⁰³ . It is reasserted in such passages as the following. “The (answer to the question) what is a man and the fact that a man exists are different things¹⁰⁴ .” “It is clear that, even according to the view of definitions now current, those who define things do not prove that they exist¹⁰⁵ .” “We say that it is by demonstration that we must show that everything exists, except essence (ei mê ousia eiê). But the existence of a thing is never essence; for the existent is not a genus. Therefore there must be demonstration that a thing exists. Thus, what is meant by triangle the geometer assumes, but that it exists he has to prove¹⁰⁶ .” “Anterior knowledge of two sorts is necessary: for it is necessary to presuppose, with regard to some things, that they exist; in other cases it is necessary to understand what the thing described is, and in other cases it is necessary to do both. Thus, with the fact that one of two contradictories must be true, we must know that it exists (is true); [p. 144] of the triangle we must know that it means such and such a thing; of the unit we must know both what it means and that it exists¹⁰⁷ .” What is here so much insisted on is the very fact which Mill pointed out in his discussion of earlier views of Definitions, where he says that the so-called real definitions or definitions of things do not constitute a different kind of definition from nominal definitions, or definitions of names; the former is simply the latter plus something else, namely a covert assertion that the thing defined exists. “This covert assertion is not a definition but a postulate. The definition is a mere identical proposition which gives information only about the use of language, and from which no conclusion affecting matters of fact can possibly be drawn. The accompanying postulate, on the other hand, affirms a fact which may lead to consequences of every degree of importance. It affirms the actual or possible existence of Things possessing the combination of attributes set forth in the definition: and this, if true, may be foundation sufficient on which to build a whole fabric of scientific truth¹⁰⁸ .” This statement really adds nothing to Aristotle's doctrine¹⁰⁹ : it has even the slight disadvantage, due to the use of the word “postulate” to describe “the covert assertion” in all cases, of not definitely pointing out that there are cases where existence has to be proued as distinct from those where it must be assumed. It is true that the existence of a definiend may have to be taken for granted provisionally until the time comes for proving it; but, so far as regards any case where existence must be proved sooner or later, the provisional assumption would be for Aristotle, not a postulate, but a hypothesis. In modern times, too, Mill's account of the true distinction between real and nominal definitions had been fully anticipated by Saccheri¹¹⁰ , the editor of Euclides ab omni naevo vindicatus (1733), famous in the history of non-Euclidean geometry. In his Logica Demonstrativa (to which he also refers in his Euclid) Saccheri lays down the clear distinction between what he calls definitiones quid nominis or nominales, and definitiones quid rei or reales, namely that the former are only intended to explain the meaning [p. 145] that is to be attached to a given term, whereas the latter, besides declaring the meaning of a word, affirm at the same time the existence of the thing defined or, in geometry, the possibility of constructing it. The definitio quid nominis becomes a definitio quid rei “by means of a postulate, or when we come to the question whether the thing exists and it is answered affirmatively¹¹¹ . ” Definitiones quid nominis are in themselves quite arbitrary, and neither require nor are capable of proof; they are merely provisional and are only intended to be turned as quickly as possible into definitiones quid rei, either (1) by means of a postulate in which it is asserted or conceded that what is defined exists or can be constructed, e.g. in the case of straight lines and circles, to which Euclid's first three postulates refer, or (2) by means of a demonstration reducing the construction of the figure defined to the successive carrying-out of a certain number of those elementary constructions, the possibility of which is postulated. Thus definitiones quid rei are in general obtained as the result of a series of demonstrations. Saccheri gives as an instance the construction of a square in Euclid I. 46. Suppose that it is objected that Euclid had no right to define a square, as he does at the beginning of the Book, when it was not certain that such a figure exists in nature; the objection, he says, could only have force if, before proving and making the construction, Euclid had assumed the aforesaid figure as given. That Euclid is not guilty of this error is clear from the fact that he never presupposes the existence of the square as defined until after I. 46.

Confusion between the nominal and the real definition as thus described, i.e. the use of the former in demonstration before it has been turned into the latter by the necessary proof that the thing defined exists, is according to Saccheri one of the most fruitful sources of illusory demonstration, and the danger is greater in proportion to the “complexity” of the definition, i.e. the number and variety of the attributes belonging to the thing defined. For the greater is the possibility that there may be among the attributes some that are incompatible, i.e. the simultaneous presence of which in a given figure can be proved, by means of other postulates etc. forming part of the basis of the science, to be impossible.

The same thought is expressed by Leibniz also. “If,” he says, “we give any definition, and it is not clear from it that the idea, which we ascribe to the thing, is possible, we cannot rely upon the demonstrations which we have derived from that definition, because, if that idea by chance involves a contradiction, it is possible that even contradictories may be true of it at one and the same time, and thus our demonstrations will be useless. Whence it is clear that definitions are not arbitrary. And this is a secret which is hardly sufficiently known¹¹² .” Leibniz' favourite illustration was the “regular polyhedron with ten faces,” the impossibility of which is not obvious at first sight. [p. 146]

It need hardly be added that, speaking generally, Euclid's definitions, and his use of them, agree with the doctrine of Aristotle that the definitions themselves say nothing as to the existence of the things defined, but that the existence of each of them must be proved or (in the case of the “principles”) assumed. In geometry, says Aristotle, the existence of points and lines only must be assumed, the existence of the rest being proved. Accordingly Euclid's first three postulates declare the possibility of constructing straight lines and circles (the only “lines” except straight lines used in the Elements). Other things are defined and afterwards constructed and proved to exist: e.g. in Book I., Def. 20, it is explained what is meant by an equilateral triangle; then (I. 1) it is proposed to construct it, and, when constructed, it is proved to agree with the definition. When a square is defined (I. Def. 22), the question whether such a thing really exists is left open until, in I. 46, it is proposed to construct it and, when constructed, it is proved to satisfy the definition¹¹³ . Similarly with the right angle (I. Def. 10, and I. 11) and parallels (I. Def. 23, and I. 27-29). The greatest care is taken to exclude mere presumption and imagination. The transition from the subjective definition of names to the objective definition of things is made, in geometry, by means of constructions (the first principles of which are postulated), as in other sciences it is made by means of experience¹¹⁴ .

Aristotle's requirements in a definition.

We now come to the positive characteristics by which, according to Aristotle, scientific definitions must be marked.

First, the different attributes in a definition, when taken separately, cover more than the notion defined, but the combination of them does not. Aristotle illustrates this by the “triad,” into which enter the several notions of number, odd and prime, and the last “in both its two senses (a) of not being measured by any (other) number (hôs mê metreisthai arithmôi) and (b) of not being obtainable by adding numbers together” (hôs mê sunkeisthai ex arithmôn), a unit not being a number. Of these attributes some are present in all other odd numbers as well, while the last [primeness in the second sense] belongs also to the dyad, but in nothing but the triad are they all present¹¹⁵ .” The fact can be equally well illustrated from geometry. Thus, e.g. into the definition of a square (Eucl. I., Def. 22) there enter the several notions of figure, four-sided, equilateral, and right-angled, each of which covers more than the notion into which all enter as attributes¹¹⁶ .

Secondly, a definition must be expressed in terms of things which are prior to, and better known than, the things defined¹¹⁷ . This is [p. 147] clear, since the object of a definition is to give us knowledge of the thing defined, and it is by means of things prior and better known that we acquire fresh knowledge, as in the course of demonstrations. But the terms “prior” and “better known” are, as usual susceptible of two meanings; they may mean (1) absolutely or logically prior and better known, or (2) better known relatively to us. In the absolute sense, or from the standpoint of reason, a point is better known than a line, a line than a plane, and a plane than a solid, as also a unit is better known than number (for the unit is prior to, and the first principle of, any number). Similarly, in the absolute sense, a letter is prior to a syllable. But the case is sometimes different relatively to us; for example, a solid is more easily realised by the senses than a plane, a plane than a line, and a line than a point. Hence, while it is more scientific to begin with the absolutely prior, it may, perhaps, be permissible, in case the learner is not capable of following the scientific order, to explain things by means of what is more intelligible to him. “Among the definitions framed on this principle are those of the point, the line and the plane; all these explain what is prior by means of what is posterior, for the point is described as the extremity of a line, the line of a plane, the plane of a solid.” But, if it is asserted that such definitions by means of things which are more intelligible relatively only to a particular individual are really definitions, it will follow that there may be many definitions of the same thing, one for each individual for whom a thing is being defined, and even different definitions for one and the same individual at different times, since at first sensible objects are more intelligible, while to a better trained mind they become less so. It follows therefore that a thing should be defined by means of the absolutely prior and not the relatively prior, in order that there may be one sole and immutable definition. This is further enforced by reference to the requirement that a good definition must state the genus and the differentiae, for these are among the things which are, in the absolute sense, better known than, and prior to, the species (tôn haplôs gnôrimôterôn kai proterôn tou eidous estin). For to destroy the genus and the differentia is to destroy the species, so that the former are prior to the species; they are also better known, for, when the species is known, the genus and the differentia must necessarily be known also, e.g. he who knows “man” must also know “animal” and “land-animal,” but it does not follow, when the genus and differentia are known, that the species is known too, and hence the species is less known than they are¹¹⁸ . It may be frankly admitted that the scientific definition will require superior mental powers for its apprehension; and the extent of its use must be a matter of discretion. So far Aristotle; and we have here the best possible explanation why Euclid supplemented his definition of a point by the statement in I. Def. 3 that the extremities of a line are points and his definition of a surface by I. Def. 6 to the effect that the extremities of a surface are lines. The supplementary explanations [p. 148] do in fact enable us to arrive at a better understanding of the formal definitions of a point and a line respectively, as is well explained by Simson in his note on Def. 1. Simson says, namely, that we must consider a solid, that is, a magnitude which has length, breadth and thickness, in order to understand aright the definitions of a point, a line and a surface. Consider, for instance, the boundary common to two solids which are contiguous or the boundary which divides one solid into two contiguous parts; this boundary is a surface. We can prove that it has no thickness by taking away either solid, when it remains the boundary of the other; for, if it had thickness, the thickness must either be a part of one solid or of the other, in which case to take away one or other solid would take away the thickness and therefore the boundary itself: which is impossible. Therefore the boundary or the surface has no thickness. In exactly the same way, regarding a line as the boundary of two contiguous surfaces, we prove that the line has no breadth; and, lastly, regarding a point as the common boundary or extremity of two lines, we prove that a point has no length, breadth or thickness.

Aristotle on unscientific definitions.

Aristotle distinguishes three kinds of definition which are unscientific because founded on what is not prior (mê ek proterôn). The first is a definition of a thing by means of its opposite, e.g. of “good” by means of “bad”; this is wrong because opposites are naturally evolved together, and the knowledge of opposites is not uncommonly regarded as one and the same, so that one of the two opposites cannot be better known than the other. It is true that, in some cases of opposites, it would appear that no other sort of definition is possible: e.g. it would seem impossible to define double apart from the half and, generally, this would be the case with things which in their very nature (kath hauta) are relative terms (pros ti legetai), since one cannot be known without the other, so that in the notion of either the other must be comprised as well¹¹⁹ . The second kind of definition which is based on what is not prior is that in which there is a complete circle through the unconscious use in the definition itself of the notion to be defined though not of the name¹²⁰ . Trendelenburg illustrates this by two current definitions, (1) that of magnitude as that which can be increased or diminished, which is bad because the positive and negative comparatives “more” and “less” presuppose the notion of the positive “great,” (2) the famous Euclidean definition of a straight line as that which “lies evenly with the points on itself” (ex isou tois eph heautês sêmeiois keitai), where “lies evenly” can only be understood with the aid of the very notion of a straight line which is to be defined¹²¹ . The third kind of vicious definition from that which is not prior is the definition of one of two coordinate species by means of its coordinate (antidiêirêmenon), e.g. a definition of “odd” as that which exceeds the even by a unit (the second alternative in Eucl. VII. Def. 7); for “odd” and “even” are coordinates, being differentiae of [p. 149] number¹²² . This third kind is similar to the first. Thus, says Trendelenburg, it would be wrong to define a square as “a rectangle with equal sides.”

Aristotle's third requirement.

A third general observation of Aristotle which is specially relevant to geometrical definitions is that “to know what a thing is (ti estin) is the same as knowing why it is (dia ti estin)¹²³ .” “What is an eclipse? A deprivation of light from the moon through the interposition of the earth. Why does an eclipse take place? Or why is the moon eclipsed? Because the light fails through the earth obstructing it. What is harmony? A ratio of numbers in high or low pitch. Why does the high-pitched harmonise with the low-pitched? Because the high and the low have a numerical ratio to one another¹²⁴ .” “We seek the cause (to dioti) when we are already in possession of the fact (to hoti). Sometimes they both become evident at the same time, but at all events the cause cannot possibly be known [as a cause] before the fact is known¹²⁵ .” “It is impossible to know what a thing is if we do not know that it is¹²⁶ ” Trendelenburg paraphrases: “The definition of the notion does not fulfil its purpose until it is made genetic. It is the producing cause which first reveals the essence of the thing.... . The nominal definitions of geometry have only a provisional significance and are superseded as soon as they are made genetic by means of construction.” E.g. the genetic definition of a parallelogram is evolved from Eucl. I. 31 (giving the construction for parallels) and I. 33 about the lines joining corresponding ends of two straight lines parallel and equal in length. Where existence is proved by construction, the cause and the fact appear together¹²⁷ .

Again, “it is not enough that the defining statement should set forth the fact, as most definitions do; it should also contain and present the cause; whereas in practice what is stated in the definition is usually no more than a conclusion (sumperasma). For example, what is quadrature? The construction of an equilateral right-angled figure equal to an oblong. But such a definition expresses merely the conclusion. Whereas, if you say that quadrature is the discovery of a mean proportional, then you state the reason¹²⁸ .” This is better understood if we compare the statement elsewhere that “the cause is the middle term, and this is what is sought in all cases¹²⁹ ,” and the illustration of this by the case of the proposition that the angle in a semicircle is a right angle. Here the middle term which it is sought to establish by means of the figure is that the angle in the semi-circle is equal to the half of two right angles. We have then the syllogism: Whatever is half of two right angles is a right angle; the angle in a semi-circle is the half of two right angles; therefore (conclusion) the angle in a semi-circle is a right angle¹³⁰ . As with the demonstration, so [p. 150] it should be with the definition. A definition which is to show the genesis of the thing defined should contain the middle term or cause; otherwise it is a mere statement of a conclusion. Consider, for instance, the definition of “quadrature” as “making a square equal in area to a rectangle with unequal sides.” This gives no hint as to whether a solution of the problem is possible or how it is solved: but, if you add that to find the mean proportional between two given straight lines gives another straight line such that the square on it is equal to the rectangle contained by the first two straight lines, you supply the necessary middle term or cause¹³¹ .

Technical terms not defined by Euclid.

It will be observed that what is here defined, “quadrature” or “squaring” (tetragônismos), is not a geometrical figure, or an attribute of such a figure or a part of a figure, but a technical term used to describe a certain problem. Euclid does not define such things; but the fact that Aristotle alludes to this particular definition as well as to definitions of deflection (keklasthai) and of verging (neuein) seems to show that earlier text-books included among definitions explanations of a number of technical terms, and that Euclid deliberately omitted these explanations from his Elements as surplusage. Later the tendency was again in the opposite direction, as we see from the much expanded Definitions of Heron, which, for example, actually include a definition of a deflected line (keklasmenê grammê)¹³² . Euclid uses the passive of klan occasionally¹³³ , but evidently considered it unnecessary to explain such terms, which had come to bear a recognised meaning.

The mention too by Aristotle of a definition of verging (neuein) suggests that the problems indicated by this term were not excluded from elementary text-books before Euclid. The type of problem (neusis) was that of placing a straight line across two lines, e.g. two straight lines, or a straight line and a circle, so that it shall verge to a given point (i.e. pass through it if produced) and at the same time the intercept on it made by the two given lines shall be of given length. [p. 151] In general, the use of conics is required for the theoretical solution of these problems, or a mechanical contrivance for their practical solution¹³⁴ . Zeuthen, following Oppermann, gives reasons for supposing, not only that mechanical constructions were practically used by the older Greek geometers for solving these problems, but that they were theoretically recognised as a permissible means of solution when the solution could not be effected by means of the straight line and circle, and that it was only in later times that it was considered necessary to use conics in every case where that was possible¹³⁵ . Heiberg¹³⁶ suggests that the allusion of Aristotle to neuseis perhaps confirms this supposition, as Aristotle nowhere shows the slightest acquaintance with conics. I doubt whether this is a safe inference, since the problems of this type included in the elementary text-books might easily have been limited to those which could be solved by “plane” methods (i.e. by means of the straight line and circle). We know, e.g., from Pappus that Apollonius wrote two Books on plane neuseis¹³⁷ . But one thing is certain, namely that Euclid deliberately excluded this class of problem, doubtless as not being essential in a book of Elements.

Definitions not afterwards used.

Lastly, Euclid has definitions of some terms which he never afterwards uses, e.g. oblong (heteromêkes), rhombus, rhomboid. The “oblong” occurs in Aristotle; and it is certain that all these definitions are survivals from earlier books of Elements.

¹ Proclus, Comm. on Eucl. I., ed. Friedlein, pp. 72 sqq.

² to plêthos tôn entos orthais isôn. If the text is right, we must apparently take it as “the number of the angles equal to right angles that there are inside,” i.e. that are made up by the internal angles

³ tôn archikôn schêmatôn, by which Proclus probably means the regular polyhedra (Tannery, P. 143 n.).

⁴ We have no more than the most obscure indications of the character of this work in an Arabic MS. analysed by Woepcke, Essai d'une restitution de travaux perdus d'Apollonius sur les quantités irrationelles d'après des indications tirées d'un manuscrit arabe in Mémoires présentés à l'académie des sciences, XIV. 658-720, Paris, 1856. Cf. Cantor, Gesch. d. Math. I_{3}, pp. 348-9: details are also given in my notes to Book X.

⁵ Proclus, pp. 70, 19-71, 21.

⁶ Topics VIII. 14, 163 b 23.

⁷ Topics VIII. 3, 158 b 35.

⁸ Metaph. 998 a 25

⁹ Metaph. 1014 a 35-b 5.

¹⁰ Proclus, p. 66, 20 ôste ton Leonta kai ta stoicheia suntheinai tôi te plhêthei kai tê chreiai tôn deiknumenôn epimelesteron.

¹¹ Proclus, p. 67, 14 kai gar ta stoicheia kalôs sunetaxen kai polla tôn merikôn [horikôn (?) Friedlein] katholikôtera epoiêsen.

¹² Proclus, p. 67, 22 tôn stoicheiôn polla aneure.

¹³ Mathematisches zu Aristoteles in Abhandlungen zur Gesch. d. math. Wissenschaften, XVIII. Heft (1904), pp. 1-49.

¹⁴ Anal. post. 1. 6, 74 b 5.

¹⁵ ibid. 1. 10, 76 a 31-77 a 4.

¹⁶ Anal. post. 1. 2, 72 a 14-24.

¹⁷ Metaph. 1061 b 19-24.

¹⁸ Anal. post. 1. 11, 77 a 30.

¹⁹ Metaph. 996 b 26-30.

²⁰ Metaph. 997 a 20-22.

²¹ Proclus, p. 194, 8.

²² Metaph. 997 a 10.

²³ ibid. 996 b 26.

²⁴ ibid. 1005 a 21--b 11.

²⁵ ibid. 997 a 5-8.

²⁶ ibid. 1005 b 11-17.

²⁷ ibid. 1006 a 5.

²⁸ ibid. 1006 a 17.

²⁹ ibid. 1006 a 10.

³⁰ ibid. 1006 a 11-15.

³¹ ibid. 1006 a 18 sqq.

³² Proclus, pp. 75, 10-77, 2.

³³ Republic, VI. 510 c. Cf. Aristotle, Nic. Eth. 1151 a 17.

³⁴ H. Jackson, Journal of Philology, vol. x. p. 144.

³⁵ Proclus, pp. 178, 12-179, 8. In illustration Proclus contrasts the drawing of a straight line or a circle with the drawing of a “single-turn spiral” or of an equilateral triangle, the spiral requiring more complex machinery and even the equilateral triangle needing a certain method. “For the geometrical intelligence will say that by conceiving a straight line fixed at one end but, as regards the other end, moving round the fixed end, and a point moving along the straight line from the fixed end, I have described the single-turn spiral; for the end of the straight line describing a circle, and the point moving on the straight line simultaneously, when they arrive and meet at the same point, complete such a spiral. And again, if I draw equal circles, join their common point to the centres of the circles and draw a straight line from one of the centres to the other, I shall have the equilateral triangle. These things then are far from being completed by means of a single act or of a moment's thought” (p. 180, 8-21).

³⁶ Proclus, p. 181,4-11.

³⁷ It is necessary to coin a word to render anisorropiôn, which is moreover in the plural. The title of the treatise as we have it is Equilibria of planes or cenires of gravity of planes in Book I and Equilibria of planes in Book II.

³⁸ Proclus, p. 181, 16-23.

³⁹ ibid. p. 182, 6-14.

⁴⁰ Pp. 118, 119.

⁴¹ Proclus, pp. 182, 21-183, 13.

⁴² Anal. post. 1. 25, 86 a 33-35.

⁴³ Cf. Lardner's Euclid: also Todhunter.

⁴⁴ Proclus, p. 77, 7-12.

⁴⁵ ibid. pp. 77, 15-78, 8.

⁴⁶ ibid. pp. 78, 8-79, 2.

⁴⁷ Proclus, pp. 79, 11-80, 5.

⁴⁸ In the text we have to de problêma answering to to men without substantive: problêma was obviously inserted in error.

⁴⁹ Proclus, pp. 80, 15-81, 4.

⁵⁰ Proclus, p. 81, 5-22.

⁵¹ to pempton. This should apparently be the fourth because in the next words it is implied that none of the first three propositions are required in proving it.

⁵² Proclus, pp. 241, 19-243, 11.

⁵³ ibid. pp. 233, 21-234, 6.

⁵⁴ Proclus, p. 234, 21.

⁵⁵ ibid. p. 243, 12-25.

⁵⁶ ibid. p. 220, 7-12.

⁵⁷ ibid. pp. 220, 16-221, 6.

⁵⁸ Proclus, p. 221, 7-11.

⁵⁹ ibid. pp. 221, 13-222,14.

⁶⁰ ibid. pp. 203, 1-204, 13; 204, 23-205, 8.

⁶¹ Proclus, p. 208, 21.

⁶² ibid. p. 202, 3.

⁶³ La Géométrie grecque, p. 149 note. Where dei dê introduces the closer description of the problem we may translate, “it is then required” or “thus it is required” (to constructetc.): when it introduces the condition of possibility we may translate “thus it is necessary etc.” Heiberg originally wrote dei de in the latter sense in 1. 22 on the authority of Proclus and Eutocius, and against that of the MSS. Later, on the occasion of XI. 23, he observed that he should have followed the MSS. and written dei dê which he found to be, after all, the right reading in Eutocius (Apollonius, ed. Heiberg, II. p. 178). dei dê is also the expression used by Diophantus for introducing conditions of possibility.

⁶⁴ . See the passage of Eutocius referred to in last note.

⁶⁵ Proclus, p. 207, 4-25.

⁶⁶ Proclus, p. 205, 13-15.

⁶⁷ Proclus, pp. 211, 1-212, 4.

⁶⁸ It would appear, says Tannery (p. 151 n.), that Geminus understood a lemma as being simply lambanomenon, something assumed (cf. the passage of Proclus, p. 73, 4, relating to Menaechmus' view of elements): hence we cannot consider ourselves authorised in attributing to Geminus the more technical definition of the term here given by Proclus, according to which it is only used of propositions not proved beforehand. This view of a lemma must be considered as relatively modern. It seems to have had its origin in an imperfection of method. In the course of a demonstration it was necessary to assume a proposition which required proof, but the proof of which would, if inserted in the particular place, break the thread of the demonstration: hence it was necessary either to prove it beforehand as a preliminary proposition or to postpone it to be proved afterwards (hôs hexês deichthêsetai). When, after the time of Geminus, the progress of original discovery in geometry was arrested, geometers occupied themselves with the study and elucidation of the works of the great mathematicians who had preceded them. This involved the investigation of propositions explicitly quoted or tacitly assumed in the great classical treatises; and naturally it was found that several such remained to be demonstrated, either because the authors had omitted them as being easy enough to be left to the reader himself to prove, or because books in which they were proved had been lost in the meantime. Hence arose a class of complementary or auxiliary propositions which were called lemmas. Thus Pappus gives in his Book VII a collection of lemmas in elucidation of the treatises of Euclid and Apollonius included in the so-called “Treasury of Analysis” (topos analnomenos). When Proclus goes on to distinguish three methods of discovering lemmas, analysis, division, and reductio ad absurdum, he seems to imply that the principal business of contemporary geometers was the investigation of these auxiliary propositions.

⁶⁹ This passage and another from Diogenes Laertius (III. 24, p. 74 ed. Cobet) to the effect that “He [Plato] explained (eisêgêsato) to Leodamas of Thasos the method of inquiry by analysis” have been commonly understood as ascribing to Plato the invention of the method of analysis; but Tannery points out forcibly (pp. 112, 113) how difficult it is to explain in what Plato's discovery could have consisted if analysis be taken in the sense attributed to it in Pappus, where we can see no more than a series of successive, reductions of a problem until it is finally reduced to a known problem. On the other hand, Proclus' words about carrying up the thing sought to “an acknowledged principle” suggest that what he had in mind was the process described at the end of Book VI of the Republic by which the dialectician (unlike the mathematician) uses hypotheses as stepping-stones up to a principle which is not hypothetical, and then is able to descend step by step verifying every one of the hypotheses by which he ascended. This description does not of course refer to mathematical analysis, but it may have given rise to the idea that analysis was Plato's discovery, since analysis and synthesis following each other are related in the same way as the upward and the downward progression in the dialectician's intellectual method. And it may be that Plato's achievement was to observe the importance, from the point of view of logical rigour, of the confirmatory synthesis following analysis, and to regulariśe in this way and elevate into a completely irrefragable method the partial and uncertain analysis upon which the works of his predecessors depended.

⁷⁰ Here again the successive bipartitions of genera into species such as we find in the Sophist and Republic have very little to say to geometry, and the very fact that they are here mentioned side by side with analysis suggests that Proclus confused the latter with the philosophical method of Rep. VI.

⁷¹ Tannery rightly remarks (p. 152) that the subdivision of a theorem or problem into several cases is foreign to the really classic form; the ancients preferred, where necessary, to multiply enunciations. As, however, some omissions necessarily occurred, the writers of lemmas naturally added separate cases, which in some instances found their way into the text. A good example is Euclid 1. 7, the second case of which, as it appears in our text-books, was interpolated. On the commentary of Proclus on this proposition Th. Taylor rightly remarks that “Euclid everywhere avoids a multitude of cases.”

⁷² Proclus, p. 212, 5-11.

⁷³ Tannery notes however that, so far from distinguishing his corollaries from the conclusions of his propositions, Euclid inserts them before the closing words “(being) what it was required to do” or “to prove.” In fact the porism-corollary is with Euclid rather a modified form of the regular conclusion than a separate proposition.

⁷⁴ Proclus, p. 212, 18-23.

⁷⁵ Anal. prior. II. 26, 69 a 37.

⁷⁶ ibid. II. 25, 69 a 20.

⁷⁷ Proclus, pp. 212, 24-213, 11. This passage has frequently been taken as crediting Hippocrates with the discovery of the method of geometrical reduction: cf. Taylor (Translation of Proclus, II. p. 26), Allman (p. 41n., 59), Gow (pp. 169, 170). As Tannery remarks (p. 110), if the particular reduction of the duplication problem to that of the two means is the first noted in history, it is difficult to suppose that it was really the first; for Hippocrates must have found instances of it in the Pythagorean geometry. Bretschneider, I think, comes nearer the truth when he boldly (p. 99) translates: “This reduction of the aforesaid construction is said to have been first given by Hippocrates.” The words are prôton de phasi tôn aporoumenôn diagrammatôn tên apagôgên poiêsasthai, which must, literally, be translated as in the text above; but, when Proclus speaks vaguely of “difficult constructions,” he probably means to say simply that “this first recorded instance of a reduction of a difficult construction is attributed to Hippocrates.”

⁷⁸ Aristotle, Anal. prior. I. 7, 29 b 5; 1. 44, 50 a 30.

⁷⁹ ibid. I. 21, 39 b 32; I. 29, 45 a 35.

⁸⁰ Anal. post. I. 24, 85 a 16 etc.

⁸¹ Anal. prior. I. 23, 40 b 25.

⁸² Anal. prior. I. 23, 41 a 24.

⁸³ ibid. II. 14, 62 b 29.

⁸⁴ Proclus, p. 254, 22-27.

⁸⁵ Proclus, p. 255, 8-26.

⁸⁶ Pappus, v. p. 410 sqq.

⁸⁷ ibid. pp. 410, 27-412, 2.

⁸⁸ Bretschneider, p. 168. See however Heiberg's recent suggestion (Paralipomena zu Euklid in Hermes, XXXVIII., 1903) that the author was Heron. The suggestion is based on a comparison with the remarks on analysis and synthesis quoted from Heron by an-Nairĩzĩ (ed. Curtze, p. 89) at the beginning of his commentary on Eucl. Book II. On the whole, this suggestion commends itself to me more than that of Bretschneider.

⁸⁹ Proclus, p. 67, 6.

⁹⁰ Cantor, Gesch. d. Math. I_{3}, p. 241.

⁹¹ Proclus, p. 60, 7-9.

⁹² ibid. p. 60, 16-19.

⁹³ Pappus, VII. pp. 634-6.

⁹⁴ Hankel, Zur Geschichte der Mathematik in Alterthum und Mittelalter, 1874, pp. 137-150; Duhamel, Des'méthodes dans les sciences de raisonnement, Part I., 3 ed., Paris, 1885, pp. 39-68; Zeuthen, Geschichte der Mathematik im Altertnm und Mittelalter, 1896, pp. 92-104; Ofterdinger, Beiträge zur Geschichte der griechischen Mathematik, Ulm, 1860; Cantor, Geschichte der Mathematik, I_{3}, pp. 220-2.

⁹⁵ Hankel, p. 139.

⁹⁶ Zeuthen, p. 103.

⁹⁷ Proclus, p. 242, 16, 17.

⁹⁸ Zeuthen, p. 93.

⁹⁹ Hankel, p. 141.

¹⁰⁰ Pappus, VII. pp. 830-2.

¹⁰¹ Anal. post. II. 13, 96 b 15.

¹⁰² Cf. De anima, I. 2, 404 2 9, where “breathing” is spoken of as the oros of “life,” and the many passages in the Politics where the word is used to denote that which gives its special character to the several forms of government (virtue being the horos of aristocracy, wealth of oligarchy, liberty of democracy, 1294 a 10); Plato, Republic, VIII. 551 c.

¹⁰³ Anal. past. I. 10, 76 a 31 sqq.

¹⁰⁴ ibid. II. 7, 92 b 10.

¹⁰⁵ ibid. 92 b 19.

¹⁰⁶ ibid. 92 b 12 sqq.

¹⁰⁷ Anal. post. I. 1, 71 a II sqq.

¹⁰⁸ Mill's System of Logic, Bk. I. ch. Viii.

¹⁰⁹ It is true that it was in opposition to “the ideas of most of the Aristotelian logicians” (rather than of Aristotle himself) that Mill laid such stress on his point of view. Cf. his observation: “We have already made, and shall often have to repeat, the remark, that the philosophers who overthrew Realism by no means got rid of the consequences of Realism, but retained long afterwards, in their own philosophy, numerous propositions which could only have a rational meaning as part of a Realistic system. It had been handed down from Aristotle, and probably from earlier times, as an obvious truth, that the science of geometry is deduced from definitions. This, so long as a definition was considered to be a proposition ‘unfolding the nature of the thing,’ did well enough. But Hobbes followed and rejected utterly the notion that a definition declares the nature of the thing, or does anything but state the meaning of a name; yet he continued to affirm as broadly as any of his predecessors that the archai, principia, or original premisses of mathematics, and even of all science, are definitions; producing the singular paradox that systems of scientific truth, nay, all truths whatever at which we arrive by reasoning, are deduced from the arbitrary conventions of mankind concerning the signification of words.” Aristotle was guilty of no such paradox; on the contrary, he exposed it as plainly as did Mill.

¹¹⁰ This has been fully brought out in two papers by G. Vailati, La teoria Aristotelica della definizione (Riuista di Filosofia e scienze affini, 1903), and Di un' opera dimenticata del P. Gerolamo Saccheri (“Logica Demonstrativa,” 1697) (in Rivista Filosofica, 1903).

¹¹¹ “Definitio quid nominis nata est evadere definitio quid rei per postulatum vel dum venitur ad quaestionem an est et respondetur affirmative.”

¹¹² Opuscules et fragments inédits de Leibniz, Paris, Alcan, 1903, p. 431. Quoted by Vailati.

¹¹³ Trendelenburg, Elementa Logices Aristoteleae, § 50.

¹¹⁴ Trendelenburg, Erläuterungen zu den Elementen der aristotelischen Logik, 3 ed. p. 107. On construction as proof of existence in ancient geometry cf. H. G. Zeuthen, Die geometrische Construction als “Existenzbeweis” in der antiken Geometrie (in Mathematische Annalen, 47. Band).

¹¹⁵ Anal. post. II. 13, 96 a 33--b 1.

¹¹⁶ Trendelenburg, Erläuterungen, p. 108.

¹¹⁷ Topics VI. 4, 141 a 26 sqq.

¹¹⁸ Topics VI. 4, 141 b 25-34.

¹¹⁹ Topics VI. 4, 142 a 22-31.

¹²⁰ ibid. 142 a 34--b 6.

¹²¹ Trendelenburg, Erläuterungen, p. 115.

¹²² Topics VI. 4, 142 b 7-10.

¹²³ Anal. post. II. 2, 90 a 31.

¹²⁴ Anal. post. II. 2, 90 a 15-21.

¹²⁵ ibid. II. 8, 93 a 17.

¹²⁶ ibid. 93 a 20.

¹²⁷ Trendelenburg, Erläuterungen, p. 110.

¹²⁸ De anima II. 2, 413 a 13-20,

¹²⁹ Anal. post. II. 2, 90 a 6,

¹³⁰ ibid. II. 11, 94 a 28.

¹³¹ Other passages in Aristotle may be quoted to the like effect: e.g. Anal. post. I. 2, 71 b 9 “We consider that we know a particular thing in the absolute sense, as distinct from the sophistical and incidental sense, when we consider that we know the cause on account of which the thing is, in the sense of knowing that it is the cause of that thing and that it cannot be otherwise,” ibid. I. 13, 79 a 2 “For here to know the fact is the function of those who are concerned with sensible things, to know the cause is the function of the mathematician; it is he who possesses the proofs of the causes, and often he does not know the fact.” In view of such passages it is difficult to see how Proclus came to write (p. 202, 11) that Aristotle was the originator (Aristotelous katarxantos) of the idea of Amphinomus and others that geometry does not investigate the cause and the why (to dia ti). To this Geminus replied that the investigation of the cause does, on the contrary, appear in geometry. “For how can it be maintained that it is not the business of the geometer to inquire for what reason, on the one hand, an infinite number of equilateral polygons are inscribed in a circle, but, on the other hand, it is not possible to inscribe in a sphere an infinite number of polyhedral figures, equilateral, equiangular, and made up of similar plane figures? Whose business is it to ask this question and find the answer to it if it is not that of the geometer? Now when geometers reason per impossibile they are content to discover the property, but when they argue by direct proof, if such proof be only partial (epi merous), this does not suffice for showing the cause; if however it is general and applies to all like cases, the why (to dia ti) is at once and concurrently made evident.”

¹³² Heron, Def. 12 (vol. IV. Heib. pp. 22-24).

¹³³ e.g. in III. 20 and in Data 89.

¹³⁴ Cf. the chapter on neuseis in The Works of Archimedes, pp. c--cxxii.

¹³⁵ Zeuthen, Die Lehre von den Kegelschnitten im Altertum, ch. 12, p. 262.

¹³⁶ Heiberg, Mathematisches zu Aristoteles, p. 16.

¹³⁷ Pappus VII. pp. 670-2.

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