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 (1. 1) it is proposed to construct it, and, when constructed, it is proved to agree with the definition. When a square is defined (1. 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 definition'. Similarly with the right angle (1. Def. 10, and I. 11) and parallels (1. 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 experience'.

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 (s μn μетρеîobaι apieμg) and (b) of not being obtainable by adding numbers together” (ὡς μὴ συγκεῖσθαι ἐξ ἀριθμῶν), 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 present." 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 às attributes".

Secondly, a definition must be expressed in terms of things which are prior to, and better known than, the things defined. This is 1 Trendelenburg, Elementa Logices Aristoteleae, § 50.

? 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).

Anal. post. II. 13, 96 a 33-b 1.

♦ Trendelenburg, Erläuterungen, p. 108.

Topics VI. 4, 141 a 26 sqq.

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 (τῶν ἁπλῶς γνωριμωτέρων καὶ προτέρων τοῦ eidovs coτív). 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 are1. 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 expla

1 Topics VI. 4, 141 b 25—34.

nations 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. I. 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 (uỶ Èx πρOTÉρWV). 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 (καθ' αυτά) are relative terms (πρός τι λέγεται), since one cannot be known without the other, so that in the notion of either the other must be comprised as well1. 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 name. 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" (ἐξ ἴσου τοῖς ἐφ' ἑαυτῆς σημείοις κείται), where “lies evenly " can only be understood with the aid of the very notion of a straight line which is to be defined. 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 (avridiņpnμévov), e.g. a definition of "odd which exceeds the even by a unit (the second alternative in Eucl. VII. Def. 7); for "odd" and "even" are coordinates, being differentiae of sibid. 142 a 34-b 6.

1 Topics VI. 4, 142 a 22-31.
'Trendelenburg, Erläuterungen, p. 115.

as that

7

number'. 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 (Tí ÉσTW) is the same as knowing why it is (dià tí ẻσti)?.” "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 another." "We seek the cause (Tò dióTI) when we are already in possession of the fact (Tò OT). 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 known." "It is impossible to know what a thing is if we do not know that it is." 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. 1. 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 together.

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 (ovμméρaoμa). 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 reason?." 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 cases," 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 angle'. As with the demonstration, so

1 Topics VI. 4, 142 b 7—10.

• Anal. post. II. 2, 90 a 15-21.

• ibid. 93 a 20.

7 De anima II. 2, 413 a 13—20, • ibid. II. 11, 94 a 28.

- Anal. post. II. 2, 90 a 31.

4 ibid. 11. 8, 93 a 17.

Trendelenburg, Erläuterungen, p. 116.

• Anal, post. 11. 2, 90 a 6,

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 cause1.

Technical terms not defined by Euclid.

It will be observed that what is here defined, "quadrature" or "squaring" (Tетрayшvioμós), 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 (Keкλáobai) and of verging (vevewv) 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 (Keкλаoμévη ypaμun). Euclid uses the passive of λâv occasionally, 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 (veveiv), suggests that the problems indicated by this term were not excluded from elementary text-books before Euclid. The type of problem (vevois) 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.

1 Other passages in Aristotle may be quoted to the like effect: e.g. Anal. post. 1. 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 ('Αριστοτέλους κατάρξαντος) of the idea of Amphinomus and others that geometry does not investigate the cause and the why (Tò dià Tí). Tỏ 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 (el μépovs), this does not suffice for showing the cause; if however it is general and applies to all like cases, the why (rò did Tí) is at once and concurrently made evident."

2 Heron, ed. Hultsch, Def. 14, p. 11.

3e.g. in III. 20 and in Data 89.