Commentary on Book I Definitions¶
Back to Book I Definitions.
DEFINITION 1.¶
An exactly parallel use of μέρος (ἐστί) in the singular is found in Aristotle, Metaph. 1035 b 32 μέρος μὲν οὖν ἐστὶ καὶ τοῦ εἴδους, literally “There is a part even of the form”; Bonitz translates as if the plural were used, “Theile giebt es,” and the meaning is simply “even the form is divisible (into parts).” Accordingly it would be quite justifiable to translate in this case “A point is that which is indivisible into parts.”
Martianus Capella (5th c. A.D.) alone or almost alone translated differently, “Punctum est cuius pars nihil est,” “a point is that a part of which is nothing.” Notwithstanding that Max Simon (Euclid und die sechs planimetrischen Bücher, 1901) has adopted this translation (on grounds which I shall presently mention), I cannot think that it gives any sense. If a part of a point is nothing, Euclid might as well have said that a point is itself “nothing,” which of course he does not do.
Pre-Euclidean definitions.¶
It would appear that this was not the definition given in earlier text-books ; for Aristotle (Topics VI. 4, 141 b 20), in speaking of “the definitions” of point, line, and surface, says that they all define the prior by means of the posterior, a point as an extremity of a line, a line of a surface, and a surface of a solid.
The first definition of a point of which we hear is that given by the Pythagoreans (cf. Proclus, p. 95, 21), who defined it as a “monad having position” or “with position added” (μονὰς προσλαβοῦσα θέσιν). It is frequently used by Aristotle, either in this exact form (cf. De anima I. 4, 409 a 6) or its equivalent : e.g. in Metaph. 1016 b 24 he says that that which is indivisible every way in respect of magnitude and quâ magnitude but has not position is a monad, while that which is similarly indivisible and has position is a point.
Plato appears to have objected to this definition. Aristotle says (Metaph. 992 a 20) that he objected “to this genus [that of points] as being a geometrical fiction (γεωμετρικὸν δόγμα), and called a point the beginning of a line (ἀρχὴ γραμμῆς), while again he frequently spoke of ‘indivisible lines.’” To which Aristotle replies that even “indivisible lines” must have extremities, so that the same argument which proves the existence of lines can be used to prove that points exist. It would appear therefore that, when Aristotle objects to the definition of a point as the extremity of a line (πέρας γραμμῆς) as unscientific (Topics VI. 4, 141 b 21), he is aiming at Plato. Heiberg conjectures (Mathematisches zu Aristoteles, p. 8) that it was due to Plato’s influence that the word for “point” generally used by Aristotle (στιγμή) was replaced by σημεῖον (the regular term used by Euclid, Archimedes and later writers), the latter term (= nota, a conventional mark) probably being considered more suitable than στιγμή (a puncture) which might appear to claim greater reality for a point.
Aristotle’s conception of a point as that which is indivisible and has position is further illustrated by such observations as that a point is not a body (De caelo II. 13, 296 a 17) and has no weight (ibid. III. 1, 299 a 30); again, we can make no distinction between a point and the place (τόπος) where it is (Physics IV. 1, 209 a 11). He finds the usual difficulty in accounting for the transition from the indivisible, or infinitely small, to the finite or divisible magnitude. A point being indivisible, no accumulation of points, however far it may be carried, can give us anything divisible, whereas of course a line is a divisible magnitude. Hence he holds that points cannot make up anything continuous like a line, point cannot be continuous with point (οὐ γάρ ἐστιν ἐχόμενον σημεῖον σημείου ἢ στιγμὴ στιγμῆς, De gen. et corr. I. 2, 317 a 10), and a line is not made up of points (οὐ σύγκειται ἐκ στιγμῶν, Physics IV. 8, 215 b 19). A point, he says, is like the now in time : now is indivisible and is not a part of time, it is only the beginning or end, or a division, of time, and similarly a point may be an extremity, beginning or division of a line, but is not part of it or of magnitude (cf. De caelo III. 1, 300 a 14, Physics IV. 11, 220 a 1—21, VI. 1, 231 b 6 sqq.). It is only by motion that a point can generate a line (De anima I. 4, 409 a 4) and thus be the origin of magnitude.
Other ancient definitions.¶
According to an-Nairīzī (ed. Curtze, p. 3) one “Herundes” (not so far identified) defined a point as “the indivisible beginning of all magnitudes,” and Posidonius as “an extremity which has no dimension, or an extremity of a line.”
Criticisms by commentators.¶
Euclid’s definition itself is of course practically the same as that which Aristotle’s frequent allusions show to have been then current, except that it omits to say that the point must have position. Is it then sufficient, seeing that there are other things which are without parts or indivisible, e.g. the now in time, and the unit in number? Proclus answers (p. 93, 18) that the point is the only thing in the subject-matter of geometry that is indivisible. Relatively therefore to the particular science the definition is sufficient. Secondly, the definition has been over and over again criticised because it is purely negative. Proclus’ answer to this is (p. 94, 10) that negative descriptions are appropriate to first principles, and he quotes Parmenides as having described his first and last cause by means of negations merely. Aristotle too admits that it may sometimes be necessary for one framing a definition to use negations, e.g. in defining privative terms such as “blind”; and he seems to accept as proper the negative element in the definition of a point, since he says (De anima III. 6, 430 b 20) that “the point and every division [e.g. in a length or in a period of time], and that which is indivisible in this sense, is exhibited as privation (δηλοῦται ὡς στέρησις).”
Simplicius (quoted by an-Nairīzī) says that “a point is the beginning of magnitudes and that from which they grow ; it is also the only thing which, having position, is not divisible.” He, like Aristotle, adds that it is by its motion that a point can generate a magnitude : the particular magnitude can only be “of one dimension,” viz. a line, since the point does not “spread itself” (dimittat). Simplicius further observes that Euclid defined a point negatively because it was arrived at by detaching surface from body, line from surface, and finally point from line. “Since then body has three dimensions it follows that a point [arrived at after successively eliminating all three dimensions] has none of the dimensions, and has no part.” This of course reappears in modern treatises (cf. Rausenberger, Elementar-geometrie des Punktes, der Geraden und der Ebene, 1887, p. 7).
An-Nairīzī adds an interesting observation. “If any one seeks to know the essence of a point, a thing more simple than a line, let him, in the sensible world, think of the centre of the universe and the poles.” But there is nothing new under the sun : the same idea is mentioned, in an Aristotelian treatise, in controverting those who imagine that the poles have some influence in the motion of the sphere, “when the poles have no magnitude but are extremities and points” (De motu animalium 3, 699 a 21).
Modern views.¶
In the new geometry represented by the excellent treatises which start from new systems of postulates or axioms, the result of the profound study of the fundamental principles of geometry during recent years (I need only mention the names of Pasch, Veronese, Enriques and Hilbert), points come before lines, but the vain effort to define them a priori is not made ; instead of this, the nearest material things in nature are mentioned as illustrations, with the remark that it is from them that we can get the abstract idea. Cf. the full statement as regards the notion of a point in Weber and Wellstein, Encyclopädie der elementaren Mathematik, II., 1905, p. 9. “This notion is evolved from the notion of the real or supposed material point by the process of limits, i.e. by an act of the mind which sets a term to a series of presentations in itself unlimited. Suppose a grain of sand or a mote in a sunbeam, which continually becomes smaller and smaller. In this way vanishes more and more the possibility of determining still smaller atoms in the grain of sand, and there is evolved, so we say, with growing certainty, the presentation of the point as a definite position in space which is one and is incapable of further division. But this view is untenable ; we have, it is true, some idea how the grain of sand gets smaller and smaller, but only so long as it remains just visible ; after that we are completely in the dark, and we cannot see or imagine the further diminution. That this procedure comes to an end is unthinkable ; that nevertheless there exists a term beyond which it cannot go, we must believe or postulate without ever reaching it… . It is a pure act of will, not of the understanding.” Max Simon observes similarly (Euclid, p. 25) “The notion ‘point’ belongs to the limit-notions (Grenzbegriffe), the necessary conclusions of continued, and in themselves unlimited, series of presentations.” He adds, “The point is the limit of localisation ; if this is more and more energetically continued, it leads to the limit-notion ‘point,’
better ‘position,’ which at the same time involves a change of notion. Content of space vanishes, relative position remains. ‘Point’ then, according to our interpretation of Euclid, is the extremest limit of that which we can still think of (not observe) as a spatial presentation, and if we go further than that, not only does extension cease but even relative place, and in this sense the ‘part’ is nothing.” I confess I think that even the meaning which Simon intends to convey is better expressed by “it has no part” than by “the part is nothing,” since to take a “part” of a thing in Euclid’s sense of the result of a simple division, corresponding to an arithmetical fraction, would not be to change the notion from that of the thing divided to an entirely different one.
DEFINITION 2.¶
This definition may safely be attributed to the Platonic School, if not to Plato himself. Aristotle (Topics VI. 6, 143 b 11) speaks of it as open to objection because it “divides the genus by negation,” length being necessarily either breadthless or possessed of breadth ; it would seem however that the objection was only taken in order to score a point against the Platonists, since he says (ibid. 143 b 29) that the argument is “of service only against those who assert that the genus [sc. length] is one numerically, that is, those who assume ideas,” e.g. the idea of length (αὐτὸ μῆκος) which they regard as a genus : for if the genus, being one and self-existent, could be divided into two species, one of which asserts what the other denies, it would be self-contradictory (Waitz).
Proclus (pp. 96, 21—97, 3) observes that, whereas the definition of a point is merely negative, the line introduces the first “dimension,” and so its definition is to this extent positive, while it has also a negative element which denies to it the other “dimensions” (διαστάσεις). The negation of both breadth and depth is involved in the single expression “breadthless” (ἀπλατές), since everything that is without breadth is also destitute of depth, though the converse is of course not true.
Alternative definitions.¶
The alternative definition alluded to by Proclus, μέγεθος ἐφ’ ἓν διαστατόν “magnitude in one dimension” or, better perhaps, “magnitude extended one way” (since διάστασις as used with reference to line, surface and solid scarcely corresponds to our use of “dimension” when we speak of “one,” “two,” or “three dimensions”), is attributed by an-Nairīzī to “Heromides,” who must presumably be the same as “Herundes,” to whom he attributes a certain definition of a point. It appears however in substance in Aristotle, though Aristotle does not use the adjective διαστατόν, nor does he apparently use διάστασις except of body as having three “dimensions” or “having dimension (or extension) all ways (πάντῃ),” the “dimensions” being in his view (1) up and down, (2) before and behind, and (3) right and left, and “up” being the principle or beginning of length, “right” of breadth, and “before” of depth (De caelo II. 2, 284 b 24). A line is, according to Aristotle, a magnitude “divisible in one way only” (μοναχῇ διαιρετόν), in contrast to a magnitude divisible in two ways (διχῇ διαιρετόν), or a surface, and a magnitude divisible “in all or in three ways” (πάντῃ καὶ τριχῇ διαιρετόν), or a body (Metaph. 1016 b 25—27) ; or it is a magnitude “continuous one way (or in one direction),” as compared with magnitudes continuous two ways or three ways, which curiously enough he describes as “breadth” and “depth” respectively (μέγεθος δὲ τὸ μὲν ἐφ’ ἓν συνεχὲς μῆκος, τὸ δ’ ἐπὶ δύο πλάτος, τὸ δ’ ἐπὶ τρία βάθος, Metaph. 1020 a 11), though he immediately adds that “length” means a line, “breadth” a surface, and “depth” a body.
Proclus gives another alternative definition as “flux of a point” (ῥύσις σημείου), i.e. the path of a point when moved. This idea is also alluded to in Aristotle (De anima I. 4, 409 a 4 above quoted) : “they say that a line by its motion produces a surface, and a point by its motion a line.” “This definition,” says Proclus (p. 97, 8—13), “is a perfect one as showing the essence of the line : he who called it the flux of a point seems to define it from its genetic cause, and it is not every line that he sets before us, but only the immaterial line ; for it is this that is produced by the point, which, though itself indivisible, is the cause of the existence of things divisible.”
Proclus (p. 100, 5—19) adds the useful remark, which, he says, was current in the school of Apollonius, that we have the notion of a line when we ask for the length of a road or a wall measured merely as length ; for in that case we mean something irrespective of breadth, viz. distance in one “dimension.” Further we can obtain sensible perception of a line if we look at the division between the light and the dark when a shadow is thrown on the earth or the moon ; for clearly the division is without breadth, but has length.
Species of “lines.”¶
After defining the “line” Euclid only mentions one species of line, the straight line, although of course another species appears in the definition of a circle later. He doubtless omitted all classification of lines as unnecessary for his purpose, whereas, for example, Heron follows up his definition of a line by a division of lines into (1) those which are “straight” and (2) those which are not, and a further division of the latter into (a) “circular circumferences,” (b) “spiral-shaped” (ἑλικοειδεῖς) lines and (c) “curved” (καμπύλαι) lines generally, and then explains the four terms. Aristotle tells us (Metaph. 986 a 25) that the Pythagoreans distinguished straight (εὐθύ) and curved (καμπύλον), and this distinction appears in Plato (cf. Republic X. 602 c) and in Aristotle (cf. “to a line belong the attributes straight or curved,” Anal. post. I. 4, 73 b 19 ; “as in mathematics it is useful to know what is meant by the terms straight and curved,” De anima I. 1, 402 b 19). But from the class of “curved” lines Plato and Aristotle separate off the περιφερής or “circular” as a distinct species often similarly contrasted with straight. Aristotle seems to recognise broken lines forming an angle as one line : thus “a line, if it be bent (κεκαμμένη), but yet continuous, is called one” (Metaph. 1016 a 2) ; “the straight line is more one than the bent line” (ibid. 1016 a 12). Cf. Heron, Def. 12, “A broken line (κεκλασμένη γραμμή) so-called is a line which, when produced, does not meet itself.”
When Proclus says that both Plato and Aristotle divided lines into those which are “straight,” “circular” (περιφερής) or “a mixture of the two,” adding, as regards Plato, that he included in the last of these classes “those which are called helicoidal among plane (curves) and (curves) formed about solids, and such species of curved lines as arise from sections of solids” (p. 104, 1—5), he appears to be not quite exact. The reference as regards Plato seems to be to Parmenides 145 B : “At that rate it would seem that the one must have shape, either straight or round (στρογγύλον) or some combination of the two” ; but this scarcely amounts to a formal classification of lines. As regards Aristotle, Proclus seems to have in mind the passage (De caelo I. 2, 268 b 17) where it is stated that “all motion in space, which we call translation (φορά), is (in) a straight line, a circle, or a combination of the two ; for the first two are the only simple (motions).”
For completeness it is desirable to add the substance of Proclus’ account of the classification of lines, for which he quotes Geminus as his authority.
Geminus’ first classification of lines.¶
This begins (p. 111, 1—9) with a division of lines into composite (σύνθετος) and incomposite (ἀσύνθετος). The only illustration given of the composite class is the “broken line which forms an angle” (ἡ κεκλασμένη καὶ γωνίαν ποιοῦσα) ; the subdivision of the incomposite class then follows (in the text as it stands the word “composite” is clearly an error for “incomposite”). The subdivisions of the incomposite class are repeated in a later passage (pp. 176, 27—177, 23) with some additional details. The following diagram reproduces the effect of both versions as far as possible (all the illustrations mentioned by Proclus being shown in brackets).
lines
│
┌──────────────────────────┴──────────────────────────┐
composite incomposite
(broken line forming an angle) │
┌─────────────────────────────────┴─────────────────────────────────┐
forming a figure not forming a figure
σχηματοποιοῦσαι or
or determinate indeterminate
ὡρισμέναι ἀόριστοι
(circle, ellipse, cissoid) and
extending without limit
ἐπ' ἄπειρον ἐκβαλλόμεναι
(straight line, parabola, hyperbola, conchoid)
The additional details in the second version, which cannot easily be shown in the diagram, are as follows :
Of the lines which extend without limit, some do not form a figure at all (viz. the straight line, the parabola and the hyperbola) ; but some first “come together and form a figure” (i.e. have a loop), “and, for the rest, extend without limit” (p. 177, 8).
As the only other curve, besides the parabola and the hyperbola, which has been mentioned as proceeding to infinity is the conchoid (of Nicomedes), we can hardly avoid the conclusion of Tannery[1] that the curve which has a loop and then proceeds to infinity is a variety of the conchoid itself. As is well known, the ordinary conchoid (which was used both for doubling the cube and for trisecting the angle) is obtained in this way. Suppose any number of rays passing through a fixed point (the pole) and intersecting a fixed straight line ; and suppose that points are taken on the rays, beyond the fixed straight line, such that the portions of the rays intercepted between the fixed straight line and the point are equal to a constant distance (διάστημα), the locus of the points is a conchoid which has the fixed straight line for asymptote. If the “distance” a is measured from the intersection of the ray with the given straight line, not in the direction away from the pole, but towards the pole, we obtain three other curves according as a is less than, equal to, or greater than b, the distance of the pole from the fixed straight line, which is an asymptote in each case. The case in which a > b gives a curve which forms a loop and then proceeds to infinity in the way Proclus describes. Now we know both from Eutocius (Comm. on Archimedes, ed. Heiberg, III. p. 98) and Proclus (p. 272, 3—7) that Nicomedes wrote on conchoids (in the plural), and Pappus (IV. p. 244, 18) says that besides the “first” (used as above stated) there were “the second, the third and the fourth which are useful for other theorems.”
Proclus next observes (p. 177, 9) that, of the lines which extend without limit, some are “asymptotic” (ἀσύμπτωτοι), namely “those which never meet, however they are produced,” and some are “symptotic,” namely “those which will meet sometime” ; and, of the “asymptotic” class, some are in one plane, and others not. Lastly, of the “asymptotic” lines in one plane, some preserve always the same distance from one another, while others continually “lessen the distance, like the hyperbola with reference to the straight line, and the conchoid with reference to the straight line.”
Geminus’ second classification.¶
This (from Proclus, pp. 111, 9—20 and 112, 16—18) can be shown in a diagram thus :
Incomposite lines
ἀσύνθετοι γραμμαί
│
┌─────────────────────────┴─────────────────────────┐
simple, ἁπλῆ mixed, μικτή
│ │
┌────────┴────────┐ ┌────────────────┴────────────────┐
making a figure indeterminate lines in planes lines on solids
σχῆμα ποιοῦσα ἀόριστος │ αἱ ἐν τοῖς στερεοῖς
(e.g. circle) (straight line) │ │
┌──────────────┴──────────────┐ │
line meeting itself extending without limit │
ἡ ἐν αὐτῇ συμπίπτουσα ἡ ἐπ' ἄπειρον ἐκβαλλόμενη │
(e.g. cissoid) │
┌────────────────────────────────┴────────────────┐
lines formed by sections lines round solids
αἱ κατὰ τὰς τομάς αἱ περὶ τὰ στερεά
(e.g. conic sections, spiric curves) (e.g. helix about a sphere or about a cone)
│ │
└────────────────────────┬────────────────────────┘
│
┌──────────────┴──────────────┐
homoeomeric not homoeomeric
(cylindrical helix) (all others)
Notes on classes of “lines” and on particular curves.¶
We will now add the most interesting notes found in Proclus with reference to the above classifications or the particular curves mentioned.
1. Homoeomeric lines.¶
By this term (ὁμοιομερεῖς) are meant lines which are alike in all parts, so that in any one such curve any part can be made to coincide with any other part. Proclus observes that these lines are only three in number, two being “simple” and in a plane (the straight line and the circle), and the third “mixed,” (subsisting) “about a solid,” namely the cylindrical helix. The latter curve was also called the cochlias or cochlion, and its homoeomeric property was proved by Apollonius in his work περὶ τοῦ κοχλίου (Proclus, p. 105, 5). The fact that there are only three homoeomeric lines was proved by Geminus, “who proved, as a preliminary proposition, that, if from a point (ἀπό του σημείου, but on p. 251, 4 ἀφ’ ἑνὸς σημείου) two straight lines be drawn to a homoeomeric line making equal angles with it, the straight lines are equal” (pp. 112, 1—113, 3, cf. p. 251, 2—19).
2. Mixed lines.¶
It might be supposed, says Proclus (p. 105, 11), that the cylindrical helix, being homoeomeric, like the straight line and the circle, must like them be simple. He replies that it is not simple, but mixed, because it is generated by two unlike motions. Two like motions, said Geminus, e.g. two motions at the same speed in the directions of two adjoining sides of a square, produce a simple line, namely a straight line (the diagonal) ; and again, if a straight line moves with its extremities upon the two sides of a right angle respectively, this same motion gives a simple curve (a circle) for the locus of the middle point of the straight line, and a mixed curve (an ellipse) for the locus of any other point on it (p. 106, 3—15).
Geminus also explained that the term “mixed,” as applied to curves, and as applied to surfaces, respectively, is used in different senses. As applied to curves, “mixing” neither means simple “putting together” (σύνθεσις) nor “blending” (κρᾶσις). Thus the helix (or spiral) is a “mixed” line, but (1) it is not “mixed” in the sense of “putting together,” as it would be if, say, part of it were straight and part circular, and (2) it is not mixed in the sense of “blending,” because, if it is cut in any way, it does not present the appearance of any simple lines (of which it might be supposed to be compounded, as it were). The “mixing” in the case of lines is rather that in which the constituents are destroyed so far as their own character is concerned, and are replaced, as it were, by a chemical combination (ἔστιν ἐν αὐτῇ συνεφθαρμένα τὰ ἄκρα καὶ συγκεχυμένα). On the other hand “mixed” surfaces are mixed in the sense of a sort of “blending” (κατά τινα κρᾶσιν). For take a cone generated by a straight line passing through a fixed point and passing always through the circumference of a circle : if you cut this by a plane parallel to that of the circle, you obtain a circular section, and if you cut it by a plane through the vertex, you obtain a triangle, the “mixed” surface of the cone being thus cut into simple lines (pp. 117, 22—118, 23).
3. Spiric curves.¶
These curves, classed with conics as being sections of solids, were discovered by Perseus, according to an epigram of Perseus’ own quoted by Proclus (p. 112, 1), which says that Perseus found “three lines upon (or, perhaps, in addition to) five sections” (τρεῖς γραμμὰς ἐπὶ πέντε τομαῖς). Proclus throws some light on these in the following passages :
“Of the spiric sections, one is interlaced, resembling the horse-fetter (ἵππου πέδη) ; another is widened out in the middle and contracts on each side (of the middle), a third is elongated and is narrower in the middle, broadening out on each side of it” (p. 112, 4—8).
“This is the case with the spiric surface ; for it is conceived as generated by the revolution of a circle remaining at right angles [to a plane] and turning about a point which is not its centre [in other words, generated by the revolution of a circle about a straight line in its plane not passing through the centre]. Hence the spire takes three forms, for the centre [of rotation] is either on the circumference, or within it, or without it. And if the centre of rotation is on the circumference, we have the continuous spire (συνεχής), if within, the interlaced (ἐμπεπλεγμένη), and if without, the open (διεχής). And the spiric sections are three according to these three differences” (p. 119, 8—17).
“When the hippopede, which is one of the spiric curves, forms an angle with itself, this angle also is contained by mixed lines” (p. 127, 1—3).
“Perseus showed for spirics what was their property (σύμπτωμα)” (p. 356, 12).
Thus the spiric surface was what we call a tore, or (when open) an anchor-ring. Heron (Def. 97) says it was called alternatively spire (σπεῖρα) or ring (κρίκος) ; he calls the variety in which “the circle cuts itself,” not “interlaced,” but “crossing-itself” (ἐπαλλάττουσα).
Tannery[2] has discussed these passages, as also did Schiaparelli[3]. It is clear that Proclus’ remark that the difference in the three curves which he mentions corresponds to the difference between the three surfaces is a slip, due perhaps to too hurried transcribing from Geminus : all three arise from plane sections of the open anchor-ring. If $r$ is the radius of the revolving circle, $a$ the distance of its centre from the axis of rotation, $d$ the distance of the plane section (supposed to be parallel to the axis) from the axis, the three curves described in the first extract correspond to the following cases :
$d = a - r$. In this case the curve is the hippopede, of which the lemniscate of Bernoulli is a particular case, namely that in which $a = 2r$.
The name hippopede was doubtless adopted for this one of Perseus’ curves on the ground of its resemblance to the hippopede of Eudoxus, which seems to have been the curve of intersection of a sphere with a cylinder touching it internally.
$a + r > d > a$. Here the curve is an oval.
$a > d > a - r$. The curve is now narrowest in the middle.
Tannery explains the “three lines upon (in addition to) five sections” thus. He points out that with the open tore there are two other sections corresponding to
$d = a$ : transition from (2) to (3).
$a - r > d > 0$, in which case the section consists of two symmetrical ovals.
He then shows that the sections of the closed or continuous tore, corresponding to $a = r$, give curves corresponding to (2), (3) and (4) only. Instead of (1) and (5) we have only a section consisting of two equal circles touching one another.
On the other hand, the third spire (the interlaced variety) gives three new forms, which make a group of three in addition to the first group of five sections.
The difficulty which I see in this interpretation is the fact that, just after “three lines on five sections” are mentioned, Proclus describes three curves which were evidently the most important ; but these three belong to three of the five sections of the open tore, and are not separate from them.
4. The cissoid.¶
This curve is assumed to be the same as that by means of which, according to Eutocius (Comm. on Archimedes, III. p. 66 sqq.), Diocles in his book περὶ πυρείων (On burning-glasses) solved the problem of doubling the cube. It is the locus of points which he found by the following construction. Let AC, BD be diameters at right angles in a circle with centre O.
Let E, F be points on the quadrants BC, BA respectively such that the arcs BE, BF are equal.
Draw EG, FH perpendicular to CA. Join AE, and let P be its intersection with FH.
The cissoid is the locus of all the points P corresponding to different positions of E on the quadrant BC and of F at an equal distance from B along the arc BA.
A is the point on the curve corresponding to the position C for the point E, and B the point on the curve corresponding to the position of E in which it coincides with B.
It is easy to see that the curve extends in the direction AB beyond B, and that CK drawn perpendicular to CA is an asymptote. It may be regarded also as having a branch AD symmetrical with AB, and, beyond D, approaching KC produced as asymptote.
If OA, OD are coordinate axes, the equation of the curve is obviously
$$y^2 (a + x) = (a - x)^3,$$
where a is the radius of the circle.
There is a cusp at A, and it agrees with this that Proclus should say (p. 126, 24) that “cissoidal lines converging to one point like the leaves of ivy—for this is the origin of their name—form an angle.” He makes the slight correction (p. 128, 5) that it is not two parts of a curve, but one curve, which in this case makes an angle.
But what is surprising is that Proclus seems to have no idea of the curve passing outside the circle and having an asymptote, for he several times speaks of it as a closed curve (forming a figure and including an area) : cf. p. 152, 7, “the plane (area) cut off by the cissoidal line has one bounding (line), but it has not in it a centre such that all (straight lines drawn to the curve) from it are equal.” It would appear as if Proclus regarded the cissoid as formed by the four symmetrical cissoidal arcs shown in the figure.
Even more peculiar is Proclus’ view of the
5. “Single-turn Spiral.”¶
This is really the spiral of Archimedes traced by a point starting from the fixed extremity of a straight line and moving uniformly along it, while simultaneously the straight line itself moves uniformly in a plane about its fixed extremity. In Archimedes the spiral has of course any number of turns, the straight line making the same number of complete revolutions. Yet Proclus, while giving the same account of the generation of the spiral (p. 180, 8—12), regards the single-turn spiral as actually stopping short at the point reached after one complete revolution of the straight line : “it is necessary to know that extending without limit is not a property of all lines ; for it neither belongs to the circle nor to the cissoid, nor in general to lines which form figures ; nor even to those which do not form figures. For even the single-turn spiral does not extend without limit—for it is constructed between two points—nor does any of the other lines so generated do so” (p. 187, 19—25). It is curious that Pappus (VIII. p. 1110 sqq.) uses the same term μονόστροφος ἕλιξ to denote one turn, not of the spiral, but of the cylindrical helix.
DEFINITION 3.¶
It being unscientific, as Aristotle said, to define a point as the “extremity of a line” (πέρας γραμμῆς), thereby explaining the prior by the posterior, Euclid defined a point differently ; then, as it was necessary to connect a point with a line, he introduced this explanation after the definitions of both had been given. This compromise is no doubt his own idea ; the same thing occurs with reference to a surface and a line as its extremity in Def. 6, and with reference to a solid and a surface as its extremity in XI. Def. 2.
We miss a statement of the facts, equally requiring to be known, that a “division” (διαίρεσις) of a line, no less than its “beginning” or “end,” is a point (this is brought out by Aristotle : cf. Metaph. 1060 b 15), and that the intersection of two lines is also a point. If these additional explanations had been given, Proclus would have been spared the difficulty which he finds in the fact that some of the lines used in Euclid (namely infinite straight lines on the one hand, and circles on the other) have no “extremities.” So also the ellipse, which Proclus calls by the old name θυρεός (“shield”). In the case of the circle and ellipse we can, he observes (p. 103, 7), take a portion bounded by points, and the definition applies to that portion. His rather far-fetched distinction between two aspects of a circle or ellipse as a line and as a closed figure (thus, while you are describing a circle, you have two extremities at any moment, but they disappear when it is finished) is an unnecessarily elaborate attempt to establish the literal universality of the “definition,” which is really no more than an explanation that, if a line has extremities, those extremities are points.
DEFINITION 4.¶
The only definition of a straight line authenticated as pre-Euclidean is that of Plato, who defined it as “that of which the middle covers the ends” (relatively, that is, to an eye placed at either end and looking along the straight line). It appears in the Parmenides 137 E : “straight is whatever has its middle in front of (i.e. so placed as to obstruct the view of) both its ends”
(εὐθύ γε οὗ ἂν τὸ μέσον ἀμφοῖν τοῖν ἐσχάτοιν ἐπίπροσθεν ᾖ). Aristotle quotes it in equivalent terms (Topics VI. 11, 148 b 27), οὗ τὸ μέσον ἐπιπροσθεῖ τοῖς πέρασιν ; and, as he does not mention the name of its author, but states it in combination with the definition of a line as the extremity of a surface, we may assume that he used it as being well known. Proclus also quotes the definition as Plato’s in almost identical terms, ἧς τὰ μέσα τοῖς ἄκροις ἐπιπροσθεῖ (p. 109, 21). This definition is ingenious, but implicitly appeals to the sense of sight and involves the postulate that the line of sight is straight. (Cf. the Aristotelian Problems 31, 20, 959 a 39, where the question is why we can better observe straightness in a row, say, of letters with one eye than with two.) As regards the straightness of “visual rays,” ὄψεις, cf. Euclid’s own Optics, Deff. 1, 2, assumed as hypotheses, in which he first speaks of the “straight lines” drawn from the eye, avoiding the word ὄψεις, and then says that the figure contained by the visual rays (ὄψεις) is a cone with its vertex in the eye.
As Aristotle mentions no definition of a straight line resembling Euclid’s, but gives only Plato’s definition and the other explaining it as the “extremity of a surface,” the latter being evidently the current definition in contemporary textbooks, we may safely infer that Euclid’s definition was a new departure of his own.
Proclus on Euclid’s definition.¶
Coming now to the interpretation of Euclid’s definition, εὐθεῖα γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐφ’ ἑαυτῆς σημείοις κεῖται, we find any number of slightly different versions, but none that can be described as quite satisfactory ; some authorities, e.g. Savile, have confessed that they could make nothing of it. It is natural to appeal to Proclus first ; and we find that he does in fact give an interpretation which at first sight seems plausible. He says (p. 109, 8 sq.) that Euclid “shows by means of this that the straight line alone [of all lines] occupies a distance (κατέχειν διάστημα) equal to that between the points on it. For, as far as one of the points is distant from another, so great is the length (μέγεθος) of the straight line of which they are the extremities ; and this is the meaning of lying ἐξ ἴσου to (or with) the points on it” [ἐξ ἴσου being thus, apparently, interpreted as “at” (or “over”) “an equal distance”]. “But if you take two points on the circumference (of a circle) or any other line, the distance cut off between them along the line is greater than the interval separating them. And this is the case with every line except the straight line. Hence the ordinary remark, based on a common notion, that those who journey in a straight line only travel the necessary distance, while those who do not go straight travel more than the necessary distance.” (Cf. Aristotle, De caelo I. 4, 271 a 13, “we always call the distance of anything the straight line” drawn to it.) Thus Proclus would interpret somewhat in this way : “a straight line is that which represents extension equal with (the distances separating) the points on it.” This explanation seems to be an attempt to graft on to Euclid’s definition the assumption (it is a λαμβανόμενον, not a definition) of Archimedes (On the sphere and cylinder I. ad init.) that “of all the lines which have the same extremities the straight line is least.” For this purpose ἐξ ἴσου has apparently to be taken as meaning “at an equal distance,” and again “lying at an equal distance” as equivalent to “extending over (or representing) an equal distance.” This is difficult enough in itself, but is seen to be an impossible interpretation when applied to the similar definition of a plane by Euclid (Def. 7) as a surface “which lies evenly with the straight lines on itself.” In that connexion Proclus tries to make the same words ἐξ ἴσου κεῖται mean “extends over an equal area with.” He says namely (p. 117, 2) that, “if two straight lines are set out” on the plane, the plane surface “occupies a space equal to that between the straight lines.” But two straight lines do not determine by themselves any space at all ; it would be necessary to have a closed figure with its boundaries in the plane before we could arrive at the equivalent of the other assumption of Archimedes that “of surfaces which have the same extremities, if those extremities are in a plane, the plane is the least [in area].” This seems to be an impossible sense for ἐξ ἴσου even on the assumption that it means “at an equal distance” in the present definition. The necessity therefore of interpreting ἐξ ἴσου similarly in both definitions makes it impossible to regard it as referring to distance or length at all. It should be added that Simplicius gave the same explanations as Proclus (an-Nairīzī, p. 5).
The language and construction of the definition.¶
Let us now consider the actual wording and grammar of the phrase ἥτις ἐξ ἴσου τοῖς ἐφ’ ἑαυτῆς σημείοις κεῖται. As regards the expression ἐξ ἴσου we note that Plato and Aristotle (whose use of it seems typical) commonly have it in the sense of “on a footing of equality” : cf. οἱ ἐξ ἴσου in Plato’s Laws 777 D, 919 D ; Aristotle, Politics 1259 b 5 ἐξ ἴσου εἶναι βούλεται τὴν φύσιν, “tend to be on an equality in nature,” Eth. Nic. VIII. 12, 1161 a 8 ἐνταῦθα πάντες ἐξ ἴσου, “there all are on a footing of equality.” Slightly different are the uses in Aristotle, Eth. Nic. X. 8, 1178 a 25 τῶν μὲν γὰρ ἀναγκαίων χρεία καὶ ἐξ ἴσου ἔστω, “both need the necessaries of life to the same extent, let us say” ; Topics IX. 15, 174 a 32 ἐξ ἴσου ποιοῦντα τὴν ἐρώτησιν, “asking the question indifferently” (i.e. without showing any expectation of one answer being given rather than another). The natural meaning would therefore appear to be “evenly placed” (or balanced), “in equal measure,” “indifferently” or “without bias” one way or the other. Next, is the dative τοῖς ἐφ’ ἑαυτῆς σημείοις constructed with ἐξ ἴσου or with κεῖται ? In the first case the phrase must mean “that which lies evenly with (or in respect to) the points on it,” in the second apparently “that which, in (or by) the points on it, lies (or is placed) evenly (or uniformly).” Max Simon takes the first construction to give the sense “die Gerade liegt in gleicher Weise wie ihre Punkte.” If the last words mean “in the same way as (or in like manner as) its points,” I cannot see that they tell us anything, although Simon attaches to the words the notion of distance (Abstand) like Proclus. The second construction he takes as giving “die Gerade liegt für (durch) ihre Punkte gleichmässig,” “the straight line lies symmetrically for (or through) its points” ; or, if κεῖται is taken as the passive of τίθημι, “die Gerade ist durch ihre Punkte gleichmässig gegeben worden,” “the straight line is symmetrically determined by its points.” He adds that the idea is here direction, and that both direction and distance (as between two different given points simply) would be to Euclid, as later to Bolzano (Betrachtungen über einige Gegenstände der Elementargeometrie, 1804, quoted by Schotten, Inhalt und Methode des planimetrischen Unterrichts, II. p. 16), primary irreducible notions.
While the language is thus seen to be hopelessly obscure, we can safely say that the sort of idea which Euclid wished to express was that of a line which presents the same shape at and relatively to all points on it, without any irregular or unsymmetrical feature distinguishing one part or side of it from another. Any such irregularity could, as Saccheri points out (Engel and Stäckel, Die Theorie der Parallellinien von Euklid bis Gauss, 1895, p. 109), be at once made perceptible by keeping the ends fixed and turning the line about them right round ; if any two positions were distinguishable, e.g. one being to the left or right relatively to another, “it would not lie in a uniform manner between its points.”
A conjecture as to its origin and meaning.¶
The question arises, what was the origin of Euclid’s definition, or, how was it suggested to him ? It seems to me that the basis of it was really Plato’s definition of a straight line as “that line the middle of which covers the ends.” Euclid was a Platonist, and what more natural than that he should have adopted Plato’s definition in substance, while regarding it as essential to change the form of words in order to make it independent of any implied appeal to vision, which, as a physical fact, could not properly find a place in a purely geometrical definition ? I believe therefore that Euclid’s definition is simply an attempt (albeit unsuccessful, from the nature of the case) to express, in terms to which a geometer could not object as not being part of geometrical subject-matter, the same thing as the Platonic definition.
The truth is that Euclid was attempting the impossible. As Pfleiderer says (Scholia to Euclid), “It seems as though the notion of a straight line, owing to its simplicity, cannot be explained by any regular definition which does not introduce words already containing in themselves, by implication, the notion to be defined (such e.g. are direction, equality, uniformity or evenness of position, unswerving course), and as though it were impossible, if a person does not already know what the term straight here means, to teach it to him unless by putting before him in some way a picture or a drawing of it.” This is accordingly done in such books as Veronese’s Elementi di geometria (Part I., 1904, p. 10) : “A stretched string, e.g. a plummet, a ray of light entering by a small hole into a dark room, are rectilineal objects. The image of them gives us the abstract idea of the limited line which is called a rectilineal segment.”
Other definitions.¶
We will conclude this note with some other famous definitions of a straight line. The following are given by Proclus (p. 110, 18—23).
A line stretched to the utmost, ἐπ’ ἄκρον τεταμένη γραμμή. This appears in Heron also, with the words “towards the ends” (ἐπὶ τὰ πέρατα) added. (Heron, Def. 4).
Part of it cannot be in the assumed plane while part is in one higher up (ἐν μετεωροτέρῳ). This is a proposition in Euclid (XI. 1).
All its parts fit on all (other parts) alike, πάντα αὐτῆς τὰ μέρη πᾶσιν ὁμοίως ἐφαρμόζει. Heron has this too (Def. 4), but instead of “alike” he says παντοίως, “in all ways,” which is better as indicating that the applied part may be applied one way or the reverse way, with the same result.
That line which, when its ends remain fixed, itself remains fixed, ἡ τῶν περάτων μενόντων καὶ αὐτὴ μένουσα. Heron’s addition to this, “when it is, as it were, turned round in the same plane” (οἷον ἐν τῷ αὐτῷ ἐπιπέδῳ στρεφομένη), and his next variation, “and about the same ends having always the same position,” show that the definition of a straight line as “that which does not change its position when it is turned about its extremities (or any two points in it) as poles” was no original discovery of Leibniz, or Saccheri, or Krafft, or Gauss, but goes back at least to the beginning of the Christian era. Gauss’ form of this definition was : “The line in which lie all points that, during the revolution of a body (a part of space) about two fixed points, maintain their position unchanged is called a straight line.” Schotten
(I. p. 315) maintains that the notion of a straight line and its property of being determined by two points are unconsciously assumed in this definition, which is therefore a logical “circle.”
That line which with one other of the same species cannot complete a figure, ἡ μετὰ τῆς ὁμοειδοῦς μιᾶς σχῆμα μὴ ἀποτελοῦσα. This is an obvious ὕστερον-πρότερον, since it assumes the notion of a figure.
Lastly Leibniz’ definition should be mentioned : A straight line is one which divides a plane into two halves identical in all but position. Apart from the fact that this definition introduces the plane, it does not seem to have any advantages over the definition last but one referred to.
Legendre uses the Archimedean property of a straight line as the shortest distance between two points. Van Swinden observes (Elemente der Geometrie, 1834, p. 4), that to take this as the definition involves assuming the proposition that any two sides of a triangle are greater than the third and proving that straight lines which have two points in common coincide throughout their length (cf. Legendre, Éléments de Géométrie I. 3, 8).
The above definitions all illustrate the observation of Unger (Die Geometrie des Euklid, 1833) : “Straight is a simple notion, and hence all definitions of it must fail…. But if the proper idea of a straight line has once been grasped, it will be recognised in all the various definitions usually given of it ; all the definitions must therefore be regarded as explanations, and among them that one is the best from which further inferences can immediately be drawn as to the essence of the straight line.”
DEFINITION 5.¶
The word ἐπιφάνεια was used by Euclid and later writers to denote surface in general, while they appropriated the word ἐπίπεδον for plane surface, thus making ἐπίπεδον a species of the genus ἐπιφάνεια. A solitary use of ἐπιφάνεια by Euclid when a plane is meant (XI. Def. 11) is probably due to the fact that the particular definition came from an earlier textbook. Proclus (p. 116, 17) remarks that the older philosophers, including Plato and Aristotle, used the words ἐπιφάνεια and ἐπίπεδον indifferently for any kind of surface. Aristotle does indeed use both words for a surface, with perhaps a tendency to use ἐπιφάνεια more than ἐπίπεδον for a surface not plane. Cf. Categories 6, 5 a 1 sq., where both words are used in one sentence : “You can find a common boundary at which the parts fit together, a point in the case of a line, and a line in the case of a surface (ἐπιφάνεια) ; for the parts of the surface (ἐπιπέδου) do fit together at some common boundary. Similarly also in the case of a body you can find a common boundary, a line or a surface (ἐπιφάνεια), at which the parts of the body fit together.” Plato however does not use ἐπιφάνεια at all in the sense of surface, but only ἐπίπεδον for both surface and plane surface. There is reason therefore for doubting the correctness of the notice in Diogenes Laertius, III. 24, that Plato “was the first philosopher to name, among extremities, the plane surface” (ἐπίπεδος ἐπιφάνεια).
ἐπιφάνεια of course means literally the feature of a body which is apparent to the eye (ἐπιφανής), namely the surface.
Aristotle tells us (De sensu 3, 439 a 31) that the Pythagoreans called a surface χροιά, which seems to have meant skin as well as colour. Aristotle explains the term with reference to colour (χρῶμα) as a thing inseparable from the extremity (πέρας) of a body.
Alternative definitions.¶
The definitions of a surface correspond to those of a line. As in Aristotle a line is a magnitude “(extended) one way, or in one ‘dimension’” (ἐφ’ ἕν), “continuous one way” (ἐφ’ ἓν συνεχές), or “divisible in one way” (μοναχῇ διαιρετόν), so a surface is a magnitude extended or continuous two ways (ἐπὶ δύο), or divisible in two ways (διχῇ). As in Euclid a surface has “length and breadth” only, so in Aristotle “breadth” is characteristic of the surface and is once used as synonymous with it (Metaph. 1020 a 12), and again “lengths are made up of long and short, surfaces of broad and narrow, and solids (ὄγκοι) of deep and shallow” (Metaph. 1085 a 10).
Aristotle mentions the common remark that a line by its motion produces a surface (De anima I. 4, 409 a 4). He also gives the a posteriori description of a surface as the “extremity of a solid” (Topics VI. 4, 141 b 22), and as “the section (τομή) or division (διαίρεσις) of a body” (Metaph. 1060 b 14).
Proclus remarks (p. 114, 20) that we get a notion of a surface when we measure areas and mark their boundaries in the sense of length and breadth ; and we further get a sort of perception of it by looking at shadows, since these have no depth (for they do not penetrate the earth) but only have length and breadth.
Classification of surfaces.¶
Heron gives (Def. 74, p. 50, ed. Heiberg) two alternative divisions of surfaces into two classes, corresponding to Geminus’ alternative divisions of lines, viz. into (1) incomposite and composite and (2) simple and mixed.
Incomposite surfaces are “those which, when produced, fall into (or coalesce with) themselves” (ὅσαι ἐκβαλλόμεναι αὐταὶ καθ’ ἑαυτῶν πίπτουσιν), i.e. are of continuous curvature, e.g. the sphere.
Composite surfaces are “those which, when produced, cut one another.” Of composite surfaces, again, some are (a) made up of non-homogeneous (elements) (ἐξ ἀνομοιογενῶν) such as cones, cylinders and hemispheres, others (b) made up of homogeneous (elements), namely the rectilineal (or polyhedral) surfaces.
Under the alternative division, simple surfaces are the plane and the spherical surfaces, but no others ; the mixed class includes all other surfaces whatever and is therefore infinite in variety.
Heron specially mentions as belonging to the mixed class (a) the surface of cones, cylinders and the like, which are a mixture of plane and circular (μικταὶ ἐξ ἐπιπέδου καὶ περιφερείας) and (b) spiric surfaces, which are “a mixture of two circumferences” (by which he must mean a mixture of two circular elements, namely the generating circle and its circular motion about an axis in the same plane).
Proclus adds the remark that, curiously enough, mixed surfaces may arise by the revolution either of simple curves, e.g. in the case of the spire, or of mixed curves, e.g. the “right-angled conoid” from a parabola, “another conoid” from the hyperbola, the “oblong” (ἐπιμήκης, in Archimedes παραμάκες) and “flat” (ἐπιπλατύ) spheroids from an ellipse according as it revolves about the major or minor axis respectively (pp. 119, 6—120, 2). The homoeomeric surfaces, namely those any part of which will coincide with any other part, are two only (the plane and the spherical surface), not three as in the case of lines (p. 120, 7).
DEFINITION 6.¶
It being unscientific, as Aristotle says, to define a line as the extremity of a surface, Euclid avoids the error of defining the prior by means of the posterior in this way, and gives a different definition not open to this objection. Then, by way of compromise, and in order to show the connexion between a line and a surface, he adds the equivalent of the definition of a line previously current as an explanation.
As in the corresponding Def. 3 above, he omits to add what is made clear by Aristotle (Metaph. 1060 b 15) that a “division” (διαίρεσις) or “section” (τομή) of a solid or body is also a surface, or that the common boundary at which two parts of a solid fit together (Categories 6, 5 a 2) may be a surface.
Proclus discusses how the fact stated in Def. 6 can be said to be true of surfaces like that of the sphere “which is bounded (πεπέρασται), it is true, but not by lines.” His explanation (p. 116, 8—14) is that, “if we take the surface (of a sphere), so far as it is extended two ways (διχῇ διαστατή), we shall find that it is bounded by lines as to length and breadth ; and if we consider the spherical surface as possessing a form of its own and invested with a fresh quality, we must regard it as having fitted end on to beginning and made the two ends (or extremities) one, being thus one potentially only, and not in actuality.”
DEFINITION 7.¶
The Greek follows exactly the definition of a straight line mutatis mutandis, i.e. with ταῖς…εὐθείαις for τοῖς…σημείοις. Proclus remarks that, in general, all the definitions of a straight line can be adapted to the plane surface by merely changing the genus. Thus, for instance, a plane surface is “a surface the middle of which covers the ends” (this being the adaptation of Plato’s definition of a straight line). Whether Plato actually gave this as the definition of a plane surface or not, I believe that Euclid’s definition of a plane surface as lying evenly with the straight lines on itself was intended simply to express the same idea without any implied appeal to vision (just as in the corresponding case of the definition of a straight line).
As already noted under Def. 4, Proclus tries to read into Euclid’s definition the Archimedean assumption that “of surfaces which have the same extremities, if those extremities are in a plane, the plane is the least.” But, as I have stated, his interpretation of the words seems impossible, although it is adopted by Simplicius also (see an-Nairīzī).
Ancient alternatives.¶
The other ancient definitions recorded are as follows.
The surface which is stretched to the utmost (ἐπ’ ἄκρον τεταμένη) : a definition which Proclus describes as equivalent to Euclid’s definition (on Proclus’ own view of that definition). Cf. Heron, Def. 9, “(a surface) which is right (and) stretched out” (ὀρθὴ οὖσα ἀποτεταμένη), words which he adds to Euclid’s definition.
The least surface among all those which have the same extremities. Proclus is here (p. 117, 9) obviously quoting the Archimedean assumption.
A surface all the parts of which have the property of fitting on (each other) (Heron, Def. 9).
A surface such that a straight line fits on all parts of it (Proclus, p. 117, 8), or such that the straight line fits on it all ways, i.e. however placed (Proclus, p. 117, 20).
With this should be compared :
“(A plane surface is) such that, if a straight line pass through two points on it, the line coincides wholly with it at every spot, all ways,” i.e. however placed (one way or the reverse, no matter how), ἧς ἐπειδὰν δύο σημείων ἅψηται εὐθεῖα, καὶ ὅλη αὐτῇ κατὰ πάντα τόπον παντοίως ἐφαρμόζεται (Heron, Def. 9). This appears, with the words κατὰ πάντα τόπον παντοίως omitted, in Theon of Smyrna (p. 112, 5, ed. Hiller), so that it goes back at least as far as the 1st c. A.D. It is of course the same as the definition commonly attributed to Robert Simson, and very widely adopted as a substitute for Euclid’s.
This same definition appears also in an-Nairīzī (ed. Curtze, p. 10) who, after quoting Simplicius’ explanation (on the same lines as Proclus’) of the meaning of Euclid’s definition, goes on to say that “others defined the plane surface as that in which it is possible to draw a straight line from any point to any other.”
Difficulties in ordinary definitions.¶
Gauss observed in a letter to Bessel that the definition of a plane surface as a surface such that, if any two points in it be taken, the straight line joining them lies wholly in the surface (which, for short, we will call “Simson’s” definition) contains more than is necessary, in that a plane can be obtained by simply projecting a straight line lying in it from a point outside the line but also lying on the plane ; in fact the definition includes a theorem, or postulate, as well. The same is true of Euclid’s definition of a plane as the surface which “lies evenly with (all) the straight lines on itself,” because it is sufficient for a definition of a plane if the surface “lies evenly” with those lines only which pass through a fixed point on it and each of the several points of a straight line also lying in it but not passing through the point. But from Euclid’s point of view it is immaterial whether a definition contains more than the necessary minimum provided that the existence of a thing possessing all the attributes contained in the definition is afterwards proved. This however is not done in regard to the plane. No proposition about the nature of a plane as such appears before Book XI., although its existence is presupposed in all the geometrical Books I.—IV. and VI. ; nor in Book XI. is there any attempt to prove, e.g. by construction, the existence of a surface conforming to the definition. The explanation may be that the existence of the plane as defined was deliberately assumed from the beginning like that of points and lines, the existence of which, according to Aristotle, must be assumed as principles unproved, while the existence of everything else must be proved ; and it may well be that Aristotle would have included plane surfaces with points and lines in this statement had it not been that he generally took his illustrations from plane geometry (excluding solid).
But, whatever definition of a plane is taken, the evolution of its essential properties is extraordinarily difficult. Crelle, who wrote an elaborate article Zur Theorie der Ebene (read in the Academie der Wissenschaften in 1834) of which account must be taken in any full history of the subject, observes that, since the plane is the field, as it were, of almost all the rest of geometry, while a proper conception of it is necessary to enable Eucl. I. 1 to be understood, it might have been expected that the theory of the plane would have been the subject of at least the same amount of attention as, say, that of parallels. This however was far from being the case, perhaps because the subject of parallels (which, for the rest, presuppose the notion of a plane) is much easier than that of the plane. The nature of the difficulties as regards the plane have also been pointed out recently by Mr Frankland (The First Book of Euclid’s Elements, Cambridge, 1905): it would appear that, whatever definition is taken, whether the simplest (as containing the minimum necessary to determine a plane) or the more complex, e.g. Simson’s, some postulate has to be assumed in addition before the fundamental properties, or the truth of the other definitions, can be established. Crelle notes the same thing as regards Simson’s definition, containing more than is necessary. Suppose a plane in which lies the triangle ABC. Let AD join the vertex A to any point D on BC, and BE the vertex B to any point E on CA. Then, according to the definition, AD lies wholly in the plane of the triangle; so does BE. But, if both AD and BE are to lie wholly in the one plane, AD, BE must intersect, say at F: if they did not, there would be two planes in question, not one. But the fact that the lines intersect and that, say, AD does not pass above or below BE, is by no means self-evident.
Mr Frankland points out the similar difficulty as regards the simpler definition of a plane as the surface generated by a straight line passing always through a fixed point and always intersecting a fixed straight line. Let OPP', OQQ' drawn from O intersect the straight line X at P, Q respectively. Let R be any third point on X: then it needs to be proved that OR intersects P'Q' in some point, say R'. Without some postulate, however, it is not easy to see how to prove this, or even to prove that P'Q' intersects X.
Crelle’s essay. Definitions by Fourier, Deahna, Becker.¶
Crelle takes as the standard of a good definition that it shall be, not only as simple as possible, but also the best adapted for deducing, with the aid of the simplest possible principles, further properties belonging to the thing defined. He was much attracted by a very lucid definition, due, he says, to Fourier, according to which a plane is formed by the aggregate of all the straight lines which, passing through one point on a straight line in space, are perpendicular to that straight line. (This is really no more than an adaptation from Euclid’s proposition XI. 5, to the effect that, if one of four concurrent straight lines be at right angles to each of the other three, those three are in one plane, which proposition is also used in Aristotle, Meteorologica III. 3, 373 a 13.) But Crelle confesses that he had not been able to deduce the necessary properties from this and had had to substitute the definition, already mentioned, of a plane as the surface containing, throughout their whole length, all the straight lines passing through a fixed point and also intersecting a straight line in space; and he only claims to have proved, after a long series of propositions, that the “Fourier”- or “perpendicular”-surface and the plane of the other definition just given are identical, after which the properties of the “Fourier”-surface can be used along with those of the plane. The advantage of the Fourier definition is that it leads easily, by means of the two propositions that triangles are equal in all respects (1) when two sides and the included angle are respectively equal and (2) when all three sides are respectively equal, to the property expressed in Simson’s definition. But Crelle uses to establish these two congruence-theorems a number of propositions about equal angles, supplementary angles, right angles, greater and less angles; and it is difficult to question the soundness of Schotten’s criticism that these notions in themselves really presuppose that of a plane. The difficulty due to Fourier’s use of the word “perpendicular,” if that were all, could no doubt be got over. Thus Deahna in a dissertation (Marburg, 1837) constructed a plane as follows. Presupposing the notions of a straight line and a sphere, he observes that, if a sphere revolve about a diameter, all the points of its surface which move describe closed curves (circles). Each of these circles, during the revolution, moves along itself, and one of them divides the surface of the sphere into two congruent parts. The aggregate then of the lines joining the centre to the points of this circle forms the plane. Again, J. K. Becker (Die Elemente der Geometrie, 1877) pointed out that the revolution of a right angle about one side of it produces a conical surface which differs from all other conical surfaces generated by the revolution of other angles in the fact that the particular cone coincides with the cone vertically opposite to it: this characteristic might therefore be taken in order to get rid of the use of the right angle.
W. Bolyai and Lobachewsky.¶
Very similar to Deahna’s equivalent for Fourier’s definition is the device of W. Bolyai and Lobachewsky (described by Frischauf, Elemente der absoluten Geometrie, 1876). They worked upon a fundamental idea first suggested, apparently, by Leibniz. Briefly stated, their way of evolving a plane and a straight line was as follows. Conceive an infinite number of pairs of concentric spheres described about two fixed points in space, $O$, $O'$, as centres, and with equal radii, gradually increasing: these pairs of equal spherical surfaces intersect respectively in homogeneous curves (circles), and the “Inbegriff” or aggregate of these curves of intersection forms a plane. If A be a point on one of these circles (k say), suppose points $M$, $M'$ to start simultaneously from A and to move in opposite directions at the same speed till they meet at B, say: B then is “opposite” to A, and A, B divide the circumference into two equal halves. If the points A, B be held fast and the whole system be turned about them until $O$ takes the place of $O'$, and $O'$ of $O$, the circle k will occupy the same position as before (though turned a different way). Two opposite points, $P$, $Q$ say, of each of the other circles will remain stationary during the motion as well as A, B: the “Inbegriff” or aggregate of all such points which remain stationary forms a straight line. It is next observed that the plane as defined can be generated by the revolution of the straight line about $OO'$, and this suggests the following construction for a plane. Let a circle as one of the curves of intersection of the pairs of spherical surfaces be divided as before into two equal halves at A, B. Let the arc ADB be similarly bisected at D, and let C be the middle point of AB. This determines a straight line CD which is then defined as “perpendicular” to AB. The revolution of CD about AB generates a plane. The property stated in Simson’s definition is then proved by means of the congruence-theorems proved in Eucl. I. 8 and I. 4. The first is taken as proved, practically by considerations of symmetry and homogeneity. If two spherical surfaces, not necessarily equal, with centres $O$, $O'$ intersect, A and its “opposite” point B are taken as
before on the curve of intersection (a circle) and, relatively to $OO'$, the point A is taken to be convertible with B or any other point on the homogeneous curve. The second (that of Eucl. I. 4) is established by simple application. Rausenberger objects to these proofs on the grounds that the first assumes that the two spherical surfaces intersect in one single curve, not in several, and that the second compares angles : a comparison which, he says, is possible only in a plane, so that a plane is really presupposed. Perhaps as regards the particular comparison of angles Rausenberger is hypercritical; but it is difficult to regard the supposed proof of the theorem of Eucl. I. 8 as sufficiently rigorous (quite apart from the use of the uniform motion of points for the purpose of bisecting lines).
Simson’s property is proved from the two congruence-theorems thus. Suppose that AB is “perpendicular” (as defined by Bolyai) to two generators CM, CN of a plane, or suppose CM, CN respectively to make with AB two angles congruent with one another. It is enough to prove that, if P be any point on the straight line MN, then CP, just as much as CM, CN respectively, makes with AB two angles congruent with one another and is therefore a generator. We prove successively the congruence of the following pairs of triangles :
whence the angles ACP, BCP are congruent.
Other views.¶
Enriques and Amaldi (Elementi di geometria, Bologna, 1905), Veronese (in his Elementi) and Hilbert all assume as a postulate the property stated in Simson’s definition. But G. Ingrami (Elementi di geometria, Bologna, 1904) proves it in the course of a remarkable series of closely argued proposition based upon a much less comprehensive postulate. He evolves the theory of the plane from that of a triangle, beginning with a triangle as a mere three-side (trilatero), i.e. a frame, as it were. His postulate relates to the three-side and is to the effect that each “(rectilineal) segment” joining a vertex to a point of the opposite side meets every segment similarly joining each of the other two vertices to the points of the sides opposite to them respectively, and, conversely, if a point be taken on a segment joining a vertex to a point of the opposite side, and if a straight line be drawn from another vertex to the point on the segment so taken, it will if produced meet the opposite side. A triangle is then defined as the figure formed by the aggregate of all the segments joining the respective vertices of a three-side to points on the opposite sides. After a series of propositions, Ingrami evolves a plane as the figure formed by the “half straight-lines” which project from an internal point of the triangle the points of the perimeter, and then, after two more theorems, proves that a plane is determined by any three of its points which are not in a straight line, and that a straight line which has two points in a plane has all its points in it.
The argument by which Bolyai and Lobachewsky evolved the plane is of course equivalent to the definition of a plane as the locus of all points equidistant from two fixed points in space.
Leibniz in a letter to Giordano defined a plane as that surface which divides space into two congruent parts. Adverting to Giordano’s criticism that you could conceive of surfaces and lines which divided space or a plane into two congruent parts without being plane or straight respectively, Beez (Über Euklidische und Nicht-Euklidische Geometrie, 1888) pointed out that what was wanted to complete the definition was the further condition that the two congruent spaces could be slid along each other without the surfaces ceasing to coincide, and claimed priority for his completion of the definition in this way. But the idea of all the parts of a plane fitting exactly on all other parts is ancient, appearing, as we have seen, in Heron, Def. 9.
DEFINITIONS 8, 9.¶
Ἐπίπεδος δὲ γωνία ἐστὶν ἡ ἐν ἐπιπέδῳ δύο γραμμῶν ἁπτομένων ἀλλήλων καὶ μὴ ἐπ’ εὐθείας κειμένων πρὸς ἀλλήλας τῶν γραμμῶν κλίσις.
Ὅταν δὲ αἱ περιέχουσαι τὴν γωνίαν γραμμαὶ εὐθεῖαι ὦσιν, εὐθύγραμμος καλεῖται ἡ γωνία.
A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.
And when the lines containing the angle are straight, the angle is called rectilineal.
The phrase “not in a straight line” is strange, seeing that the definition purports to apply to angles formed by curves as well as straight lines. We should rather have expected continuous (συνεχής) with one another; and Heron takes this to be the meaning, since he at once adds an explanation as to what is meant by lines not being continuous (οὐ συνεχεῖς). It looks as though Euclid really intended to define a rectilineal angle, but on second thoughts, as a concession to the then common recognition of curvilineal angles, altered “straight lines” into “lines” and separated the definition into two.
I think all our evidence suggests that Euclid’s definition of an angle as inclination (κλίσις) was a new departure. The word does not occur in Aristotle; and we should gather from him that the idea generally associated with an angle in his time was rather deflection or breaking of lines (κλάσις) : cf. his common use of κεκλάσθαι and other parts of the verb κλᾶν, and also his reference to one bent line forming an angle (τὴν κεκαμμένην καὶ ἔχουσαν γωνίαν, Metaph. 1016 a 13).
Proclus has a long and elaborate note on this definition, much of which (pp. 121, 12—126, 6) is apparently taken direct from a work by his master Syrianus (ὁ ἡμέτερος καθηγεμών). Two criticisms contained in the note need occasion no difficulty. One of these asks how, if an angle be an inclination, one inclination can produce two angles. The other (p. 128, 2) is to the effect that the definition seems to exclude an angle formed by one and the same curve with itself, e.g. the complete cissoid [at what we call the “cusp”] or the curve known as the hippopede (horse-fetter) [shaped like a lemniscate]. But such an “angle” as this belongs to higher geometry, which Euclid may well be excused for leaving out of account in any case.
Other ancient definitions : Apollonius, Plutarch, Carpus.¶
Proclus’ note records other definitions of great interest. Apollonius defined an angle as a contracting of a surface or a solid at one point under a broken line or surface (συναγωγὴ ἐπιφανείας ἢ στερεοῦ πρὸς ἑνὶ σημείῳ ὑπὸ κεκλασμένῃ γραμμῇ ἢ ἐπιφανείᾳ), where again an angle is supposed to be formed by one broken line or surface. Still more interesting, perhaps, is the definition by “those who say that the first distance under the point (τὸ πρῶτον διάστημα ὑπὸ τὸ σημεῖον) is the angle. Among these is Plutarch, who insists that Apollonius meant the same thing; for, he says, there must be some first distance under the breaking (or deflection) of the including lines or surfaces, though, the distance under the point being continuous, it is impossible to obtain the actual first, since every distance is divisible without limit” (ἐπ’ ἄπειρον). There is some vagueness in the use of the word “distance” (διάστημα); thus it was objected that “if we anyhow separate off the first” (distance being apparently the word understood) “and draw a straight line through it, we get a triangle and not one angle.” In spite of the objection, I cannot but see in the idea of Plutarch and the others the germ of a valuable conception in infinitesimals, an attempt (though partial and imperfect) to get at the rate of divergence between the lines at their point of meeting as a measure of the angle between them.
A third view of an angle was that of Carpus of Antioch, who said “that the angle was a quantity (ποσόν), namely a distance (διάστημα) between the lines or surfaces containing it. This means that it would be a distance (or divergence) in one sense (ἐφ’ ἓν διεστώς), although the angle is not on that account a straight line. For it is not everything extended in one sense (τὸ ἐφ’ ἓν διαστατόν) that is a line.” This very phrase “extended one way” being held to define a line, it is natural that Carpus’ idea should have been described as the greatest possible paradox (πάντων παραδοξότατον). The difficulty seems to have been caused by the want of a different technical term to express a new idea; for Carpus seems undoubtedly to have been anticipating the more modern idea of an angle as representing divergence rather than distance, and to have meant by ἐφ’ ἓν in one sense (rotationally) as distinct from one way or in one dimension (linearly).
To what category does an angle belong?¶
There was much debate among philosophers as to the particular category (according to the Aristotelian scheme) in which an angle should be placed; is it, namely, a quantum (ποσόν), quale (ποιόν) or relation (πρός τι)?
Those who put it in the category of quantity argued from the fact that a plane angle is divided by a line and a solid angle by a surface. Since, then, it is a surface which is divided by a line, and a solid which is divided by a surface, they felt obliged to conclude that an angle is a surface or a solid, and therefore a magnitude. But homogeneous finite magnitudes, e.g. plane angles, must bear a ratio to one another, or one must be capable of being multiplied until it exceeds the other. This is, however, not the case with a rectilineal angle and the horn-like angle (κερατοειδής), by which latter is meant the “angle” between a circle and a tangent to it, since (Eucl. III. 16) the latter “angle” is less than any rectilineal angle whatever. The objection, it will be observed, assumes that the two sorts of angles are homogeneous. Plutarch and Carpus are classed among those who, in one way or other, placed an angle among magnitudes; and, as above noted, Plutarch claimed Apollonius as a supporter of his view, although the word contraction (of a surface or solid) used by the latter does not in itself suggest magnitude much more than Euclid’s inclination. It was this last consideration which doubtless led “Aganis,” the “friend” (socius) apparently of Simplicius, to substitute for Apollonius’ wording “a quantity which has dimensions and the extremities of which arrive at one point” (an-Nairīzī, p. 13).
Eudemus the Peripatetic, who wrote a whole work on the angle, maintained that it belonged to the category of quality. Aristotle had given as his fourth variety of quality “figure and the shape subsisting in each thing, and,
besides these, straightness, curvature, and the like” (Categories 8, 10 a 11). He says that each individual thing is spoken of as quale in respect of its form, and he instances a triangle and a square, using them again later on (ibid. 11 a 5) to show that it is not all qualities which are susceptible of more and less; again, in Physics I. 5, 188 a 25 angle, straight, circular are called kinds of figure. Aristotle would no doubt have regarded deflection (κεκλάσθαι) as belonging to the same category with straightness and curvature (καμπυλότης). At all events, Eudemus took up an angle as having its origin in the breaking or deflection (κλάσις) of lines: deflection, he argued, was quality if straightness was, and that which has its origin in quality is itself quality. Objectors to this view argued thus. If an angle be a quality (ποιότης) like heat or cold, how can it be bisected, say? It can in fact be divided; and, if things of which divisibility is an essential attribute are varieties of quantum and not qualities, an angle cannot be a quality. Further, the more and the less are the appropriate attributes of quality, not the equal and the unequal; if therefore an angle were a quality, we should have to say of angles, not that one is greater and another smaller, but that one is more an angle and another less an angle, and that two angles are not unequal but dissimilar (ἀνόμοιοι). As a matter of fact, we are told by Simplicius, 538, 21, on Arist. De caelo that those who brought the angle under the category of quale did call equal angles similar angles; and Aristotle himself speaks of similar angles in this sense in De caelo 296 b 20, 311 b 34.
Euclid and all who called an angle an inclination are held by Syrianus to have classed it as a relation (πρός τι). Yet Euclid certainly regarded angles as magnitudes; this is clear both from the earliest propositions dealing specifically with angles, e.g. I. 9, 13, and also (though in another way) from his describing an angle in the very next definition and always as contained (περιεχομένη) by the two lines forming it (Simon, Euclid, p. 28).
Proclus (i.e. in this case Syrianus) adds that the truth lies between these three views. The angle partakes in fact of all those categories: it needs the quantity involved in magnitude, thereby becoming susceptible of equality, inequality and the like; it needs the quality given it by its form, and lastly the relation subsisting between the lines or planes bounding it.
Ancient classification of “angles.”¶
An elaborate classification of angles given by Proclus (pp. 126, 7—127, 16) may safely be attributed to Geminus. In order to show it by a diagram it
Angles
┌───┴────┐
on surfaces in solids
│ (ἐν στερεοῖς)
┌───────┴───────────┐
on simple surfaces on mixed surfaces
│ (e.g. cones, cylinders)
┌──────┴─────────┐
on planes on spherical surfaces
┌────────────┴─────────────────────────┬────────────────────────────┐
made by simple lines made by "mixed" lines by one of each
│ e.g. the angle made by a (e.g. the angle formed by an
│ curve, such as the cissoid ellipse and its axis or by
│ and hippopede, with itself) an ellipse and a circle)
┌────┴─────────┬──────────────────────────────┐
line-line line-circumf. circumf.-circumf.
┌─────┴───────┐ ┌─────────┴──────────┬────────────────┐
line-convex line-concave convex-convex concave-concave mixed, or
(e.g. angle of a e.g. horn-like (ἀμφίκυρτοι) (ἀμφίκοιλοι) convex-concave
semicircle) (κερατοειδής) or "scraper-like" (e.g. those of
(ξυστροειδεῖς) lunes)
Angles
on surfaces
on simple surfaces
on planes
made by simple lines
- line-line
line-circumf.
- line-convex (e.g. angle of a semicircle)
line-concave e.g. horn-like (κερατοειδής)
circumf.-circumf.
- convex-convex (ἀμφίκυρτοι)
concave-concave (ἀμφίκοιλοι) or “scraper-like” (ξυστροειδεῖς)
mixed, or convex-concave (e.g. those of lunes)
- made by “mixed**” lines** e.g. the angle made by a curve, such as the cissoid and hippopede, with itself)
by one of each (e.g. the angle formed by an ellipse and its axis or by an ellipse and a circle)
on spherical surfaces
on mixed surfaces (e.g. cones, cylinders)
in solids (ἐν στερεοῖς)
will be necessary to make a convention about terms. Angles are to be understood under each class, “line-circumference” means an angle contained by a straight line and an arc of a circle, “line-convex” an angle contained by a straight line and a circular arc with convexity outwards, and so on in every case.
Definitions of angle classified.¶
As for the point, straight line, and plane, so for the angle, Schotten gives a valuable summary, classification and criticism of the different modern views up to date (Inhalt und Methode des planimetrischen Unterrichts, II., 1893, pp. 94—183); and for later developments represented by Veronese reference may be made to the third article (by Amaldi) in Questioni riguardanti le matematiche elementari, I. (Bologna, 1912).
With one or two exceptions, says Schotten, the definitions of an angle may be classed in three groups representing generally the following views :
The angle is the difference of direction between two straight lines. (With this group may be compared Euclid’s definition of an angle as an inclination.)
The angle is the quantity or amount (or the measure) of the rotation necessary to bring one of its sides from its own position to that of the other side without its moving out of the plane containing both.
The angle is the portion of a plane included between two straight lines in the plane which meet in a point (or two rays issuing from the point).
It is remarkable however that nearly all of the text-books which give definitions different from those in group 2 add to them something pointing to a connexion between an angle and rotation : a striking indication that the essential nature of an angle is closely connected with rotation, and that a good definition must take account of that connexion.
The definitions in the first group must be admitted to be tautologous, or circular, inasmuch as they really presuppose some conception of an angle. Direction (as between two given points) may no doubt be regarded as a primary notion; and it may be defined as “the immediate relation of two points which the ray enables us to realise” (Schotten). But “a direction is no intensive magnitude, and therefore two directions cannot have any quantitative difference” (Bürklen). Nor is direction susceptible of differences such as those between qualities, e.g. colours. Direction is a singular entity : there cannot be different sorts or degrees of direction. If we speak of “a different direction,” we use the word equivocally; what we mean is simply “another” direction. The fact is that these definitions of an angle as a difference of direction unconsciously appeal to something outside the notion of direction altogether, to some conception equivalent to that of the angle itself.
Recent Italian views.¶
The second group of definitions are (says Amaldi) based on the idea of the rotation of a straight line or ray in a plane about a point : an idea which, logically formulated, may lead to a convenient method of introducing the angle. But it must be made independent of metric conceptions, or of the conception of congruence, so as to bring out first the notion of an angle, and afterwards the notion of equal angles.
The third group of definitions satisfy the condition of not including metric conceptions; but they do not entirely correspond to our intuitive conception of an angle, to which we attribute the character of an entity in one dimension (as Veronese says) with respect to the ray as element, or an entity in two dimensions with reference to points as elements, which may be called an angular sector. The defect is however easily remedied by considering the angle as “the aggregate of the rays issuing from the vertex and comprised in the angular sector.”
Proceeding to consider the principal methods of arriving at the logical formulation of the first superficial properties of the plane from which a definition of the angle may emerge, Amaldi distinguishes two points of view (1) the genetic, (2) the actual.
From the first point of view we consider the cluster of straight lines or rays (the aggregate of all the straight lines in a plane passing through a point, or of all the rays with their extremities in that point) as generated by the movement of a straight line or ray in the plane, about a point. This leads to the postulation of a closed order, or circular disposition, of the straight lines or rays in a cluster. Next comes the connexion subsisting between the disposition of any two clusters whatever in one plane, and so on.
Starting from the point of view of the actual, we lay the foundation of the definition of an angle in the division of the plane into two parts (half-planes) by the straight line. Next, two straight lines (a, b) in the plane, intersecting at a point O, divide the plane into four regions which are called angular sectors (convex); and finally the angle (ab) or (ba) may be defined as the aggregate of the rays issuing from O and belonging to the angular sector which has a and b for sides.
Veronese’s procedure (in his Elementi) is as follows. He begins with the first properties of the plane introduced by the following definition.
The figure given by all the straight lines joining the points of a straight line r to a point P outside it and by the parallel to r through P is called a cluster of straight lines, a cluster of rays, or a plane, according as we consider the element of the figure itself to be the straight line, the ray terminated at P, or a point.
[It will be observed that this method of producing a plane involves using the parallel to r. This presents no difficulty to Veronese because he has previously defined parallels, without reference to the plane, by means of reflex or opposite figures, with respect to a point $O$ : “two straight lines are called parallel, if one of them contains two points opposite to (or the reflex of) two points of the other with respect to the middle point of a common transversal (of the two lines).” He proves by means of a postulate that the parallel r' does belong to the plane Pr. Ingrami avoids the use of the parallel by defining a plane as “the figure formed by the half straight lines which project from an internal point of a triangle (i.e. a point on a line joining any vertex of a three-side to a point of the opposite side) the points of its perimeter,” and then defining a cluster of rays as “the aggregate of the half straight lines in a plane starting from a given point of the plane and passing through the points of the perimeter of a triangle containing the point.”]
Veronese goes on to the definition of an angle. “We call an angle a part of a cluster of rays, bounded by two rays (as the segment is a part of a straight line bounded by two points).
“An angle of the cluster, the bounding rays of which are opposite, is called a flat angle.”
Then, after a postulate corresponding to postulates which he lays down for a rectilineal segment and for a straight line, Veronese proves that all flat angles are equal to one another.
Hence he concludes that “the cluster of rays is a homogeneous linear system in which the element is the ray instead of the point. The cluster being a homogeneous linear system, all the propositions deduced from [Veronese’s] Post. 1 for the straight line apply to it, e.g. that relative to the sum and difference of the segments: it is only necessary to substitute the ray for the point, and the angle for the segment.”
DEFINITIONS 10, 11, 12.¶
Ὅταν δὲ εὐθεῖα ἐπ’ εὐθεῖαν σταθεῖσα τὰς ἐφεξῆς γωνίας ἴσας ἀλλήλαις ποιῇ, ὀρθὴ ἑκατέρα τῶν ἴσων γωνιῶν ἐστί, καὶ ἡ ἐφεστηκυῖα εὐθεῖα κάθετος καλεῖται, ἐφ’ ἣν ἐφέστηκεν.
Ἀμβλεῖα γωνία ἐστὶν ἡ μείζων ὀρθῆς.
Ὀξεῖα δὲ ἡ ἐλάσσων ὀρθῆς.
When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
An obtuse angle is an angle greater than a right angle.
An acute angle is an angle less than a right angle.
ἐφεξῆς is the regular term for adjacent angles, meaning literally “(next) in order.” I do not find the term used in Aristotle of angles, but he explains its meaning in such passages as Physics VI. 1, 231 b 8: “those things are (next) in order which have nothing of the same kind (συγγενές) between them.”
κάθετος, perpendicular, means literally let fall: the full expression is perpendicular straight line, as we see from the enunciation of Eucl. I. 11, and the notion is that of a straight line let fall upon the surface of the earth, a plumb-line. Proclus (p. 283, 9) tells us that in ancient times the perpendicular was called gnomon-wise (κατὰ γνώμονα), because the gnomon (an upright stick) was set up at right angles to the horizon.
The three kinds of angles are among the things which according to the Platonic Socrates (Republic VI. 510 C) the geometer assumes and argues from, declining to give any account of them because they are obvious. Aristotle discusses the priority of the right angle in comparison with the acute (Metaph. 1084 b 7): in one way the right angle is prior, i.e. in being defined (ὅτι ὥρισται) and by its notion (τῷ λόγῳ), in another way the acute is prior, i.e. as being a part, and because the right angle is divided into acute angles; the acute angle is prior as matter, the right angle in respect of form; cf. also Metaph. 1035 b 6, “the notion of the right angle is not divided into that of an acute angle, but the reverse; for, when defining an acute angle, you make use of the right angle.” Proclus (p. 133, 15) observes that it is by the perpendicular that we measure the heights of figures, and that it is by reference to the right angle that we distinguish the other rectilineal angles, which are otherwise undistinguished the one from the other.
The Aristotelian Problems (16, 4, 913 b 36) contain an expression perhaps worth quoting. The question discussed is why things which fall on the ground and rebound make “similar” angles with the surface on both sides of the point of impact; and it is observed that “the right angle is the limit (ὅρος) of the opposite angles,” where however “opposite” seems to mean, not “supplementary” (or acute and obtuse), but the equal angles made with the surface on opposite sides of the perpendicular.
Proclus, after his manner, remarks that the statement that an angle less than a right angle is acute is not true without qualification, for (1) the horn-like angle (between the circumference of a circle and a tangent) is less than a right angle, since it is less than an acute angle, but is not an acute angle, while (2) the “angle of a semicircle” (between the arc and a diameter) is also less than a right angle, but is not an acute angle.
The existence of the right angle is of course proved in I. 11.
DEFINITION 13.¶
Ὅρος ἐστίν, ὅ τινός ἐστι πέρας.
A boundary is that which is an extremity of anything.
Aristotle also uses the words ὅρος and πέρας as synonymous. Cf. De gen. animal. II. 6, 745 a 6, 9, where in the expression “limit of magnitude” first one and then the other word is used.
Proclus (p. 136, 8) remarks that the word boundary is appropriate to the origin of geometry, which began from the measurement of areas of ground and involved the marking of boundaries.
DEFINITION 14.¶
Σχῆμά ἐστι τὸ ὑπό τινος ἤ τινων ὅρων περιεχόμενον.
A figure is that which is contained by any boundary or boundaries.
Plato in the Meno observes that roundness (στρογγυλότης) or the round is a “figure,” and that the straight and many other things are so too; he then inquires what there is common to all of them, in virtue of which we apply the term “figure” to them. His answer is (76 A): “with reference to every figure I say that that in which the solid terminates (τοῦτο, εἰς ὃ τὸ στερεὸν περαίνει) is a figure, or, to put it briefly, a figure is an extremity of a solid.” The first observation is similar to Aristotle’s in the Physics I. 5, 188 a 25, where angle, straight, and circular are mentioned as genera of figure. In the Categories 8, 10 a 11, “figure” is placed with straightness and curvedness in the category of quality. Here however “figure” appears to mean shape (μορφή) rather than “figure” in our sense. Coming nearer to “figure” in our sense, Aristotle admits that figure is “a sort of magnitude” (De anima III. 1, 425 a 18), and he distinguishes plane figures of two kinds, in language not unlike Euclid’s, as contained by straight and circular lines respectively: “every plane figure is either rectilineal or formed by circular lines (περιφερόγραμμον), and the rectilineal figure is contained by several lines, the circular by one line” (De caelo II. 4, 286 b 13). He is careful to explain that a plane is not a figure, nor a figure a plane, but that a plane figure constitutes one notion and is a species of the genus figure (Anal. post. II. 3, 90 b 37). Aristotle does not attempt to define figure in general, in fact he says it would be useless: “From this it is clear that there is one definition of soul in the same way as there is one definition of figure; for in the one case there is no figure except the triangle, quadrilateral, and so on, nor is there any soul other than those above mentioned. A definition might be constructed which should apply to all figures but not specially to any particular figure, and similarly with the species of soul referred to. [But such a general definition would serve no purpose.] Hence it is absurd here as elsewhere to seek a general definition which will not be properly a definition of anything in existence and will not be applicable to the particular irreducible species before us, to the neglect of the definition which is so applicable” (De anima II. 3, 414 b 20—28).
Comparing Euclid’s definition with the above, we observe that by introducing boundary (ὅρος) he at once excludes the straight which Aristotle classed as figure; he doubtless excluded angle also, as we may judge by (1) Heron’s statement that “neither one nor two straight lines can complete a figure,” (2) the alternative definition of a straight line as “that which cannot with another line of the same species form a figure,” (3) Geminus’ distinction between the line which forms a figure (σχηματοποιοῦσα) and the line which extends indefinitely (ἐπ’ ἄπειρον ἐκβαλλομένη), which latter term includes a hyperbola and a parabola. Instead of calling figure an extremity as Plato did in the expression “extremity (or limit) of a solid,” Euclid describes a figure as that which has a boundary or boundaries. And lastly, in spite of Aristotle’s objection, he does attempt a general definition to cover all kinds of figure, solid and plane. It appears certain therefore that Euclid’s definition is entirely his own.
Another view of a figure, recalling that of Plato in Meno 76 A, is attributed by Proclus (p. 143, 8) to Posidonius. The latter regarded the figure as the confining extremity or limit (πέρας συγκλεῖον), “separating the notion of figure from quantity (or magnitude) and making it the cause of definition, limitation, and inclusion (τοῦ ὡρίσθαι καὶ πεπεράσθαι καὶ τῆς περιοχῆς)… Posidonius thus seems to have in view only the boundary placed round from outside, Euclid the whole content, so that Euclid will speak of the circle as a figure in respect of its whole plane (surface) and of its inclusion (from) without, whereas Posidonius (makes it a figure) in respect of its circumference… Posidonius wished to explain the notion of figure as itself limiting and confining magnitude.”
Proclus observes that a logical and refining critic might object to Euclid’s definition as defining the genus from the species, since that which is enclosed by one boundary and that which is enclosed by several are both species of figure. The best answer to this seems to be supplied by the passage of Aristotle’s De anima quoted above.
DEFINITIONS 15, 16.¶
Κύκλος ἐστὶ σχῆμα ἐπίπεδον ὑπὸ μιᾶς γραμμῆς περιεχόμενον [ἣ καλεῖται περιφέρεια], πρὸς ἣν ἀφ’ ἑνὸς σημείου τῶν ἐντὸς τοῦ σχήματος κειμένων πᾶσαι αἱ προσπίπτουσαι εὐθεῖαι [πρὸς τὴν τοῦ κύκλου περιφέρειαν] ἴσαι ἀλλήλαις εἰσίν.
A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another;
And the point is called the centre of the circle.
The words ἣ καλεῖται περιφέρεια, “which is called the circumference,” and πρὸς τὴν τοῦ κύκλου περιφέρειαν, “to the circumference of the circle,” are bracketed by Heiberg because, although the MSS. have them, they are omitted in other ancient sources, viz. Proclus, Taurus, Sextus Empiricus and Boethius, and Heron also omits the second gloss. The recently discovered papyrus Herculanensis No. 1061 also quotes the definition without the words in question, confirming Heiberg’s rejection of them (see Heiberg in Hermes XXXVIII., 1903, p. 47). The words were doubtless added in view of the occurrence of the word “circumference” in Deff. 17, 18 immediately following, without any explanation. But no explanation was needed. Though the word περιφέρεια does not occur in Plato, Aristotle uses it several times (1) in the general sense of contour without any special mathematical signification, (2) mathematically, with reference to the rainbow and the circumference, as well as an arc, of a circle. Hence Euclid was perfectly justified in employing the word in Deff. 17, 18 and elsewhere, but leaving it undefined as being a word universally understood and not involving in itself any mathematical conception. It may be added that an-Nairīzī had not the bracketed words in his text; for he comments on and tries to explain Euclid’s omission to define the circumference.
The definition itself contained nothing new in substance. Plato (Parmenides 137 E) says: “Round is, I take it, that the extremes of which are every way equally distant from the middle” (στρογγύλον γέ πού ἐστι τοῦτο, οὗ ἂν τὰ ἔσχατα πανταχῇ ἀπὸ τοῦ μέσου ἴσον ἀπέχῃ). In Aristotle we find the following expressions: “the circular (περιφερόγραμμον) plane figure bounded by one line” (De caelo II. 4, 286 b 13—16); “the plane equal (i.e. extending equally all ways) from the middle” (ἐπίπεδον τὸ ἐκ τοῦ μέσου ἴσον), meaning a circle (Rhetoric III. 6, 1407 b 27); he also contrasts with the circle “any other figure which has not the lines from the middle equal, as for example an egg-shaped figure” (De caelo II. 4, 287 a 19). The word “centre” (κέντρον) was also regularly used: cf. Proclus’ quotation from the “oracles” (λόγια), “the centre from which all (lines extending) as far as the rim are equal.”
The definition as it stands has no genetic character. It says nothing as to the existence or non-existence of the thing defined or as to the method of constructing it. It simply explains what is meant by the word “circle,” and is a provisional definition which cannot be used until the existence of circles is proved or assumed. Generally, in such a case, existence is proved by actual construction; but here the possibility of constructing the circle as defined, and consequently its existence, are postulated (Postulate 3). A genetic definition might state that a circle is the figure described when a straight line, always remaining in one plane, moves about one extremity as a fixed point until it returns to its first position (so Heron, Def. 27).
Simplicius indeed, who points out that the distance between the feet of a pair of compasses is a straight line from the centre to the circumference, will have it that Euclid intended by this definition to show how to construct a circle by the revolution of a straight line about one end as centre; and an-Nairīzī points to this as the explanation (1) of Euclid’s definition of a circle as a plane figure, meaning the whole surface bounded by the circumference, and not the circumference itself, and (2) of his omission to mention the “circumference,” since with this construction the circumference is not drawn separately as a line. But it is not necessary to suppose that Euclid himself did more than follow the traditional view; for the same conception of the circle as a plane figure appears, as we have seen, in Aristotle. While, however, Euclid is generally careful to say the “circumference of a circle” when he means the circumference, or an arc, only, there are cases where “circle” means “circumference of a circle,” e.g. in III. 10: “A circle does not cut a circle in more points than two.”
Heron, Proclus and Simplicius are all careful to point out that the centre is not the only point which is equidistant from all points of the circumference. The centre is the only point in the plane of the circle (“lying within the figure,” as Euclid says) of which this is true; any point not in the same plane which is equidistant from all points of the circumference is a pole. If you set up a “gnomon” (an upright stick) at the centre of a circle (i.e. a line through the centre perpendicular to the plane of the circle), its upper extremity is a pole (Proclus, p. 153, 3); the perpendicular is the locus of all such poles.
DEFINITION 17.¶
Διάμετρος δὲ τοῦ κύκλου ἐστὶν εὐθεῖά τις διὰ τοῦ κέντρου ἠγμένη καὶ περατουμένη ἐφ’ ἑκάτερα τὰ μέρη ὑπὸ τῆς τοῦ κύκλου περιφερείας, ἥτις καὶ δίχα τέμνει τὸν κύκλον.
A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.
The last words, literally “which (straight line) also bisects the circle,” are omitted by Simson and the editors who followed him. But they are necessary even though they do not “belong to the definition” but only express a property of the diameter as defined. For, without this explanation, Euclid would not have been justified in describing as a semi-circle a portion of a circle bounded by a diameter and the circumference cut off by it.
Simplicius observes that the diameter is so called because it passes through the whole surface of a circle as if measuring it, and also because it divides the circle into two equal parts. He might however have added that, in general, it is a line passing through a figure where it is widest, as well as dividing it equally: thus in Aristotle τὰ κατὰ διάμετρον κείμενα, “things diametrically situated” in space, are at their maximum distance apart. Diameter was the regular word in Euclid and elsewhere for the diameter of a square, and also of a parallelogram; diagonal (διαγώνιος) was a later term, defined by Heron (Def. 67) as the straight line drawn from an angle to an angle.
Proclus (p. 157, 10) says that Thales was the first to prove that a circle is bisected by its diameter; but we are not told how he proved it. Proclus gives as the reason of the property “the undeviating course of the straight line through the centre” (a simple appeal to symmetry), but adds that, if it is desired to prove it mathematically, it is only necessary to imagine the diameter drawn and one part of the circle applied to the other; it is then clear that they must coincide, for, if they did not, and one fell inside or outside the other, the straight lines from the centre to the circumference would not all be equal: which is absurd.
Saccheri’s proof is worth quoting. It depends on three “Lemmas” immediately preceding, (1) that two straight lines cannot enclose a space, (2) that two straight lines cannot have one and the same segment common, (3) that, if two straight lines meet at a point, they do not touch, but cut one another, at it.
“Let MDHNKM be a circle, A its centre, MN a diameter. Suppose the portion MNKM of the circle turned about the fixed points M, N, so that it ultimately comes near to or coincides with the remaining portion MNHDM.
“Then (i) the whole diameter MAN, with all its points, clearly remains in the same position, since otherwise two straight lines would enclose a space (contrary to the first Lemma).
“(ii) Clearly no point K of the circumference NKM falls within or outside the surface enclosed by the diameter MAN and the other part, NHDM, of the circumference, since otherwise, contrary to the nature of the circle, a radius as AK would be less or greater than another radius as AH.
“(iii) Any radius MA can clearly be rectilineally produced only along a single other radius AN, since otherwise (contrary to the second Lemma) two lines assumed straight, e.g. MAN, MAH, would have one and the same common segment.
“(iv) All diameters of the circle obviously cut one another in the centre (Lemma 3 preceding), and they bisect one another there, by the general properties of the circle.
“From all this it is manifest that the diameter MAN divides its circle and the circumference of it just exactly into two equal parts, and the same may be generally asserted for every diameter whatsoever of the same circle; which was to be proved.”
Simson observes that the property is easily deduced from III. 31 and 24; for it follows from III. 31 that the two parts of the circle are “similar segments” of a circle (segments containing equal angles, III. Def. 11), and from III. 24 that they are equal to one another.
DEFINITION 18.¶
Ἡμικύκλιον δέ ἐστι τὸ περιεχόμενον σχῆμα ὑπό τε τῆς διαμέτρου καὶ τῆς ἀπολαμβανομένης ὑπ’ αὐτῆς περιφερείας. κέντρον δὲ τοῦ ἡμικυκλίου τὸ αὐτό, ὃ καὶ τοῦ κύκλου ἐστίν.
A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle.
The last words, “And the centre of the semicircle is the same as that of the circle,” are added from Proclus to the definition as it appears in the MSS. Scarburgh remarks that a semicircle has no centre, properly speaking, and thinks that the words are not Euclid’s, but only a note by Proclus. I am however inclined to think that they are genuine, if only because of the very futility of an observation added by Proclus. He explains, namely, that the semicircle is the only plane figure that has its centre on its perimeter (!), “so that you may conclude that the centre has three positions, since it may be within the figure, as in the case of a circle, or on the perimeter, as with the semicircle, or outside, as with some conic lines (the single-branch hyperbola presumably)”!
Proclus and Simplicius point out that, in the order adopted by Euclid for these definitions of figures, the first figure taken is that bounded by one line (the circle), then follows that bounded by two lines (the semicircle), then the triangle, bounded by three lines, and so on. Proclus, as usual, distinguishes different kinds of figures bounded by two lines (pp. 159, 14—160, 9). Thus they may be formed
by circumference and circumference, e.g. ($a$) those forming angles, as a lune (τὸ μηνοειδές) and the figure included by two arcs with convexities outward, and ($b$) the angle-less (ἀγώνιον), as the figure included between two concentric circles (the coronal);
by circumference and straight line, e.g. the semicircle or segments of circles (ἁψῖδες is a name given to those less than a semicircle);
by “mixed” line and “mixed” line, e.g. two ellipses cutting one another;
by “mixed” line and circumference, e.g. intersecting ellipse and circle;
by “mixed” line and straight line, e.g. half an ellipse.
Following Def. 18 in the MSS. is a definition of a segment of a circle which was obviously interpolated from III. Def. 6. Proclus, Martianus Capella and Boethius do not give it in this place, and it is therefore properly omitted.
DEFINITIONS 19, 20, 21.¶
Σχήματα εὐθύγραμμά ἐστι τὰ ὑπὸ εὐθειῶν περιεχόμενα, τρίπλευρα μὲν τὰ ὑπὸ τριῶν, τετράπλευρα δὲ τὰ ὑπὸ τεσσάρων, πολύπλευρα δὲ τὰ ὑπὸ πλειόνων ἢ τεσσάρων εὐθειῶν περιεχόμενα.
Τῶν δὲ τριπλεύρων σχημάτων ἰσόπλευρον μὲν τρίγωνόν ἐστι τὸ τὰς τρεῖς ἴσας ἔχον πλευράς, ἰσοσκελὲς δὲ τὸ τὰς δύο μόνας ἴσας ἔχον πλευράς, σκαληνὸν δὲ τὸ τὰς τρεῖς ἀνίσους ἔχον πλευράς.
Ἔτι δὲ τῶν τριπλεύρων σχημάτων ὀρθογώνιον μὲν τρίγωνόν ἐστι τὸ ἔχον ὀρθὴν γωνίαν, ἀμβλυγώνιον δὲ τὸ ἔχον ἀμβλεῖαν γωνίαν, ὀξυγώνιον δὲ τὸ τὰς τρεῖς ὀξείας ἔχον γωνίας.
Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.
Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.
Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.
19.¶
The latter part of this definition, distinguishing three-sided, four-sided and many-sided figures, is probably due to Euclid himself, since the words τρίπλευρον, τετράπλευρον and πολύπλευρον do not appear in Plato or Aristotle (only in one passage of the Mechanics and of the Problems respectively does even τετράπλευρον, quadrilateral, occur). By his use of τετράπλευρον, quadrilateral, Euclid seems practically to have put an end to any ambiguity in the use by mathematicians of the word τετράγωνον, literally “four-angled (figure),” and to have got it restricted to the square. Cf. note on Def. 22.
20.¶
Isosceles (ἰσοσκελής, with equal legs) is used by Plato as well as Aristotle. Scalene (σκαληνός, with the variant σκαληνής) is used by Aristotle of a triangle with no two sides equal: cf. also Tim. Locr. 98 B. Plato, Euthyphro 12 D, applies the term “scalene” to an odd number in contrast to “isosceles” used of an even number. Proclus (p. 168, 24) seems to connect it with σκάζω, to limp; others make it akin to σκολιός, crooked, aslant. Apollonius uses the same word “scalene” of an oblique circular cone.
Triangles are classified, first with reference to their sides, and then with reference to their angles. Proclus points out that seven distinct species of triangles emerge: (1) the equilateral triangle, (2) three species of isosceles triangles, the right-angled, the obtuse-angled and the acute-angled, (3) the same three varieties of scalene triangles.
Proclus gives an odd reason for the dual classification according to sides and angles, namely that Euclid was mindful of the fact that it is not every triangle that is trilateral also. He explains this statement by reference (p. 165, 22) to a figure which some called barb-like (ἀκιδοειδής) while Zenodorus called it hollow-angled (κοιλογώνιος). Proclus mentions it again in his note on I. 22 (p. 328, 21 sqq.) as one of the paradoxes of geometry, observing that it is seen in the figure of that proposition. This “triangle” is merely a quadrilateral with a re-entrant angle; and the idea that it has only three angles is due to the non-recognition of the fourth angle (which is greater than two right angles) as being an angle at all. Since Proclus speaks of the four-sided triangle as “one of the paradoxes in geometry,” it is perhaps not safe to assume that the misconception underlying the expression existed in the mind of Proclus alone; but there does not seem to be any evidence that Zenodorus called the figure in question a triangle (cf. Pappus, ed. Hultsch, pp. 1154, 1206).
DEFINITION 22.¶
Τῶν δὲ τετραπλεύρων σχημάτων τετράγωνον μέν ἐστιν, ὃ ἰσόπλευρόν τέ ἐστι καὶ ὀρθογώνιον, ἑτερόμηκες δέ, ὃ ὀρθογώνιον μέν, οὐκ ἰσόπλευρον δέ, ῥόμβος δέ, ὃ ἰσόπλευρον μέν, οὐκ ὀρθογώνιον δέ, ῥομβοειδὲς δὲ τὸ τὰς ἀπεναντίον πλευράς τε καὶ γωνίας ἴσας ἀλλήλαις ἔχον, ὃ οὔτε ἰσόπλευρόν ἐστιν οὔτε ὀρθογώνιον · τὰ δὲ παρὰ ταῦτα τετράπλευρα τραπέζια καλείσθω.
Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.
τετράγωνον was already a square with the Pythagoreans (cf. Aristotle, Metaph. 986 a 26), and it is so most commonly in Aristotle; but in De anima II. 3, 414 b 31 it seems to be a quadrilateral, and in Metaph. 1054 b 2, “equal and equiangular τετράγωνα,” it cannot be anything else but quadrilateral if “equiangular” is to have any sense. Though, by introducing τετράπλευρον for any quadrilateral, Euclid enabled ambiguity to be avoided, there seem to be traces of the older vague use of τετράγωνον in much later writers. Thus Heron (Def. 100) speaks of a cube as “contained by six equilateral and equiangular τετράγωνα” and Proclus (p. 166, 10) adds to his remark about the “four-sided triangle” that “you might have τετράγωνα with more than the four sides,” where τετράγωνα can hardly mean squares.
ἑτερόμηκες, oblong (with sides of different length), is also a Pythagorean term.
The word right-angled (ὀρθογώνιον) as here applied to quadrilaterals must mean rectangular (i.e., practically, having all its angles right angles); for, although it is tempting to take the word in the same sense for a square as for a triangle (i.e. “having one right angle”), this will not do in the case of the oblong, which, unless it were stated that three of its angles are right angles, would not be sufficiently defined.
If it be objected, as it was by Todhunter for example, that the definition of a square assumes more than is necessary, since it is sufficient that, being equilateral, it should have one right angle, the answer is that, as in other cases, the superfluity does not matter from Euclid’s point of view; on the contrary, the more of the essential attributes of a thing that could be included in its definition the better, provided that the existence of the thing defined and its possession of all those attributes is proved before the definition is actually used; and Euclid does this in the case of the square by construction in I. 46, making no use of the definition before that proposition.
The word rhombus (ῥόμβος) is apparently derived from ῥέμβω, to turn round and round, and meant among other things a spinning-top. Archimedes uses the term solid rhombus to denote a solid figure made up of two right cones with a common circular base and vertices turned in opposite directions. We can of course easily imagine this solid generated by spinning; and, if the cones were equal, the section through the common axis would be a plane rhombus, which would also be the apparent form of the spinning solid to the eye. The difficulty in the way of supposing the plane figure to have been named after the solid figure is that in Archimedes the cones forming the solid are not necessarily equal. It is however possible that the solid to which the name was originally given was made up of two equal cones, that the plane rhombus then received its name from that solid, and that Archimedes, in taking up the old name again, extended its signification (cf. J. H. T. Müller, Beiträge zur Terminologie der griechischen Mathematiker, 1860, p. 20). Proclus, while he speaks of a rhombus as being like a shaken, i.e. deformed, square, and of a rhomboid as an oblong that has been moved, tries to explain the rhombus by reference to the appearance of a spinning square (τετράγωνον ῥομβούμενον).
It is true that the definition of a rhomboid says more than is necessary in describing it as having its opposite sides and angles equal to one another. The answer to the objection is the same as the answer to the similar objection to the definition of a square.
Euclid makes no use in the Elements of the oblong, the rhombus and the rhomboid. The explanation of his inclusion of definitions of these figures is no doubt that they were taken from earlier text-books. From the words “let quadrilaterals other than these be called trapezia” we may perhaps infer that trapezium was a new name or a new application of an old name.
As Euclid has not yet defined parallel lines and does not anywhere define a parallelogram, he is not in a position to make the more elaborate classification of quadrilaterals attributed by Proclus to Posidonius and appearing also in Heron’s Definitions. It may be shown by the following diagram, distinguishing seven species of quadrilaterals.
Quadrilaterals
__________________|__________________
| |
parallelograms non-parallelograms
_________|_________ _________|_________
| | | |
rectangular non-rectangular two sides parallel no sides parallel
____|____ ____|____ (trapezium) (trapezoid)
| | | | _______|_______
square oblong rhombus rhomboid | |
isosceles scalene
trapezium trapezium
It will be observed that, while Euclid in the above definition classes as trapezia all quadrilaterals other than squares, oblongs, rhombi, and rhomboids, the word is in this classification restricted to quadrilaterals having two sides (only) parallel, and trapezoid is used to denote the rest. Euclid appears to have used trapezium in the restricted sense of a quadrilateral with two sides parallel in his book περὶ διαιρέσεων (on divisions of figures). Archimedes uses it in the same sense, but in one place describes it more precisely as a trapezium with its two sides parallel.
DEFINITION 23.¶
Παράλληλοί εἰσιν εὐθεῖαι, αἵτινες ἐν τῷ αὐτῷ ἐπιπέδῳ οὖσαι καὶ ἐκβαλλόμεναι εἰς ἄπειρον ἐφ᾽ ἑκάτερα τὰ μέρη ἐπὶ μηδέτερα συμπίπτουσιν ἀλλήλαις.
Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.
Παράλληλος (alongside one another) written in one word does not appear in Plato; but with Aristotle it was already a familiar term.
εἰς ἄπειρον cannot be translated “to infinity” because these words might seem to suggest a region or place infinitely distant, whereas εἰς ἄπειρον, which seems to be used indifferently with ἐπ᾽ ἄπειρον, is adverbial, meaning “without limit,” i.e. “indefinitely.” Thus the expression is used of a magnitude being “infinitely divisible,” or of a series of terms extending without limit.
In both directions, ἐφ᾽ ἑκάτερα τὰ μέρη, literally “towards both the parts” where “parts” must be used in the sense of “regions” (cf. Thuc. II. 96).
It is clear that with Aristotle the general notion of parallels was that of straight lines which do not meet, as in Euclid: thus Aristotle discusses the question whether to think that parallels do meet should be called a geometrical or an ungeometrical error (Anal. post. I. 12, 77 b 22), and (more interesting still in relation to Euclid) he observes that there is nothing surprising in different hypotheses leading to the same error, as one might conclude that parallels meet by starting from the assumption, either ($a$) that the interior (angle) is greater than the exterior, or ($b$) that the angles of a triangle make up more than two right angles (Anal. prior. II. 17, 66 a 11).
Another definition is attributed by Proclus to Posidonius, who said that “parallel lines are those which, (being) in one plane, neither converge nor diverge, but have all the perpendiculars equal which are drawn from the points of one line to the other, while such (straight lines) as make the perpendiculars less and less continually do converge to one another; for the perpendicular is enough to define (ὁρίζειν δύναται) the heights of areas and the distances between lines. For this reason, when the perpendiculars are equal, the distances between the straight lines are equal, but when they become greater and less, the interval is lessened, and the straight lines converge to one another in the direction in which the less perpendiculars are” (Proclus, p. 176, 6—17).
Posidonius’ definition, with the explanation as to distances between straight lines, their convergence and divergence, amounts to the definition quoted by Simplicius (an-Nairīzī, p. 25, ed. Curtze) which described straight lines as parallel if, when they are produced indefinitely both ways, the distance between them, or the perpendicular drawn from either of them to the other, is always equal and not different. To the objection that it should be proved that the distance between two parallel lines is the perpendicular to them Simplicius replies that the definition will do equally well if all mention of the perpendicular be omitted and it be merely stated that the distance remains equal, although “for proving the matter in question it is necessary to say that one straight line is perpendicular to both” (an-Nairīzī, ed. Besthorn-Heiberg, p. 9). He then quotes the definition of “the philosopher Aganis”: “Parallel straight lines are straight lines, situated in the same plane, the distance between which, if they are produced indefinitely in both directions at the same time, is everywhere the same.” (This definition forms the basis of the attempt of “Aganis” to prove the Postulate of Parallels.) On the definition Simplicius remarks that the words “situated in the same plane” are perhaps unnecessary, since, if the distance between the lines is everywhere the same, and one does not incline at all towards the other, they must for that reason be in the same plane. He adds that the “distance” referred to in the definition is the shortest line which joins things disjoined. Thus, between point and point, the distance is the straight line joining them; between a point and a straight line or between a point and a plane it is the perpendicular drawn from the point to the line or plane; “as regards the distance between two lines, that distance is, if the lines are parallel, one and the same, equal to itself at all places on the lines, it is the shortest distance and, at all places on the lines, perpendicular to both” (ibid. p. 10).
The same idea occurs in a quotation by Proclus (p. 177, 11) from Geminus. As part of a classification of lines which do not meet he observes: “Of lines which do not meet, some are in one plane with one another, others not. Of those which meet and are in one plane, some are always the same distance from one another, others lessen the distance continually, as the hyperbola (approaches) the straight line, and the conchoid the straight line (i.e. the asymptote in each case). For these, while the distance is being continually lessened, are continually (in the position of) not meeting, though they converge to one another; they never converge entirely, and this is the most paradoxical theorem in geometry, since it shows that the convergence of some lines is non-convergent. But of lines which are always an equal distance apart, those which are straight and never make the (distance) between them smaller, and which are in one plane, are parallel.”
Thus the equidistance-theory of parallels (to which we shall return) is very fully represented in antiquity. I seem also to see traces in Greek writers of a conception equivalent to the vicious direction-theory which has been adopted in so many modern text-books. Aristotle has an interesting, though obscure, allusion in Anal. prior. II. 16, 65 a 4 to a petitio principii committed by “those who think that they draw parallels” (or “establish the theory of parallels,” which is a possible translation of τὰς παραλλήλους γράφειν): “for they unconsciously assume such things as it is not possible to demonstrate if parallels do not exist.” It is clear from this that there was a vicious circle in the then current theory of parallels; something which depended for its truth on the properties of parallels was assumed in the actual proof of those properties, e.g. that the three angles of a triangle make up two right angles. This is not the case in Euclid, and the passage makes it clear that it was Euclid himself who got rid of the petitio principii in earlier text-books by formulating and premising before I. 29 the famous Postulate 5, which must ever be regarded as among the most epoch-making achievements in the domain of geometry. But one of the commentators on Aristotle, Philoponus, has a note on the above passage purporting to give the specific character of the petitio principii alluded to; and it is here that a direction-theory of parallels may be hinted at, whether Philoponus is or is not right in supposing that this was what Aristotle had in mind. Philoponus says: “The same thing is done by those who draw parallels, namely begging the original question; for they will have it that it is possible to draw parallel straight lines from the meridian circle, and they assume a point, so to say, falling on the plane of that circle and thus they draw the straight lines. And what was sought is thereby assumed; for he who does not admit the genesis of the parallels will not admit the point referred to either.” What is meant is, I think, somewhat as follows. Given a straight line and a point through which a parallel to it is to be drawn, we are to suppose the given straight line placed in the plane of the meridian. Then we are told to draw through the given point another straight line in the plane of the meridian (strictly speaking it should be drawn in a plane parallel to the plane of the meridian, but the idea is that, compared with the size of the meridian circle, the distance between the point and the straight line is negligible); and this, as I read Philoponus, is supposed to be equivalent to assuming a very distant point in the meridian plane and joining the given point to it. But obviously no ruler would stretch to such a point, and the objector would say that we cannot really direct a straight line to the assumed distant point except by drawing it, without more ado, parallel to the given straight line. And herein is the petitio principii. I am confirmed in seeing in Philoponus an allusion to a direction-theory by a remark of Schotten on a similar reference to the meridian plane supposed to be used by advocates of that theory. Schotten is arguing that direction is not in itself a conception such that you can predicate one direction of two different lines. “If any one should reply that nevertheless many lines can be conceived which all have the direction from north to south,” he replies that this represents only a nominal, not a real, identity of direction.
Coming now to modern times, we may classify under three groups practically all the different definitions that have been given of parallels (Schotten, op. cit. II. p. 188 sqq.).
Parallel straight lines have no point common, under which general conception the following varieties of statement may be included:
($a$) they do not cut one another,
($b$) they meet at infinity, or
($c$) they have a common point at infinity.
Parallel straight lines have the same, or like, direction or directions, under which class of definitions must be included all those which introduce transversals and say that the parallels make equal angles with a transversal.
Parallel straight lines have the distance between them constant; with which group we may connect the attempt to explain a parallel as the geometrical locus of all points which are equidistant from a straight line.
But the three points of view have a good deal in common; some of them lead easily to the others. Thus the idea of the lines having no point common led to the notion of their having a common point at infinity, through the influence of modern geometry seeking to embrace different cases under one conception; and then again the idea of the lines having a common point at infinity might suggest their having the same direction. The “non-secant” idea would also naturally lead to that of equidistance (3), since our observation shows that it is things which come nearer to one another that tend to meet, and hence, if lines are not to meet, the obvious thing is to see that they shall not come nearer, i.e. shall remain the same distance apart.
We will now take the three groups in order.
The first observation of Schotten is that the varieties of this group which regard parallels as (a) meeting at infinity or (b) having a common point at infinity (first mentioned apparently by Kepler, 1604, as a “façon de parler” and then used by Desargues, 1639) are at least unsuitable definitions for elementary text-books. How do we know that the lines cut or meet at infinity? We are not entitled to assume either that they do or that they do not, because “infinity” is outside our field of observation and we cannot verify either. As Gauss says (letter to Schumacher), “Finite man cannot claim to be able to regard the infinite as something to be grasped by means of ordinary methods of observation.” Steiner, in speaking of the rays passing through a point and successive points of a straight line, observes that as the point of intersection gets further away the ray moves continually in one and the same direction (“nach einer und derselben Richtung hin”); only in one position, that in which it is parallel to the straight line, “there is no real cutting” between the ray and the straight line; what we have to say is that the ray is “directed towards the infinitely distant point on the straight line.” It is true that higher geometry has to assume that the lines do meet at infinity: whether such lines exist in nature or not does not matter (just as we deal with “straight lines” although there is no such thing as a straight line). But if two lines do not cut at any finite distance, may not the same thing be true at infinity also? Are lines conceivable which would not cut even at infinity but always remain at the same distance from one another even there? Take the case of a line of railway. Must the two rails meet at infinity so that a train could not stand on them there (whether we could see it or not makes no difference)? It seems best therefore to leave to higher geometry the conception of infinitely distant points on a line and of two straight lines meeting at infinity, like imaginary points of intersection, and, for the purposes of elementary geometry, to rely on the plain distinction between “parallel” and “cutting” which average human intelligence can readily grasp. This is the method adopted by Euclid in his definition, which of course belongs to the group (1) of definitions regarding parallels as non-secant.
It is significant, I think, that such authorities as Ingrami (Elementi di geometria, 1904) and Enriques and Amaldi (Elementi di geometria, 1905), after all the discussion of principles that has taken place of late years, give definitions of parallels equivalent to Euclid’s: “those straight lines in a plane which have not any point in common are called parallels.” Hilbert adopts the same point of view. Veronese, it is true, takes a different line. In his great work Fondamenti di geometria, 1891, he had taken a ray to be parallel to another when a point at infinity on the second is situated on the first; but he appears to have come to the conclusion that this definition was unsuitable for his Elementi. He avoids however giving the Euclidean definition of parallels as “straight lines in a plane which, though produced indefinitely, never meet,” because “no one has ever seen two straight lines of this sort,” and because the postulate generally used in connexion with this definition is not evident in the way that, in the field of our experience, it is evident that only one straight line can pass through two points. Hence he gives a different definition, for which he claims the advantage that it is independent of the plane. It is based on a definition of figures “opposite to one another with respect to a point” (or reflex figures). “Two figures are opposite to one another with respect to a point $O$, e.g. the figures $ABC\dots$ and $A'B'C'\dots$, if to every point of the one there corresponds one sole point of the other, and if the segments $OA$, $OB$, $OC$, $\dots$ joining the points of one figure to $O$ are respectively equal and opposite to the segments $OA'$, $OB'$, $OC'$, $\dots$ joining to $O$ the corresponding points of the second”: then, a transversal of two straight lines being any segment having as its extremities one point of one line and one point of the other, “two straight lines are called parallel if one of them contains two points opposite to two points of the other with respect to the middle point of a common transversal.” It is true, as Veronese says, that the parallels so defined and the parallels of Euclid are in substance the same; but it can hardly be said that the definition gives as good an idea of the essential nature of parallels as does Euclid’s. Veronese has to prove, of course, that his parallels have no point in common, and his “Postulate of Parallels” can hardly be called more evident than Euclid’s: “If two straight lines are parallel, they are figures opposite to one another with respect to the middle points of all their transversal segments.”
The direction-theory.
The fallacy of this theory has nowhere been more completely exposed than by C. L. Dodgson (Euclid and his modern Rivals, 1879). According to Killing (Einführung in die Grundlagen der Geometrie, I. p. 5) it would appear to have originated with no less a person than Leibniz. In the text-books which employ this method the notion of direction appears to be regarded as a primary, not a derivative notion, since no definition is given. But we ought at least to know how the same direction or like directions can be recognised when two different straight lines are in question. But no answer to this question is forthcoming. The fact is that the whole idea as applied to non-coincident straight lines is derived from knowledge of the properties of parallels; it is a case of explaining a thing by itself. The idea of parallels being in the same direction perhaps arose from the conception of an angle as a difference of direction (the hollowness of which has already been exposed); sameness of direction for parallels follows from the same “difference of direction” which both exhibit relatively to a third line. But this is not enough. As Gauss said (Werke, IV. p. 365), “If it [identity of direction] is recognised by the equality of the angles formed with one third straight line, we do not yet know without an antecedent proof whether this same equality will also be found in the angles formed with a fourth straight line” (and any number of other transversals); and in order to make this theory of parallels valid, so far from getting rid of axioms such as Euclid’s, you would have to assume as an axiom what is much less axiomatic, namely that “straight lines which make equal corresponding angles with a certain transversal do so with any transversal” (Dodgson, p. 101).
In modern times the conception of parallels as equidistant straight lines was practically adopted by Clavius (the editor of Euclid, born at Bamberg, 1537) and (according to Saccheri) by Borelli (Euclides restitutus, 1658) although they do not seem to have defined parallels in this way. Saccheri points out that, before such a definition can be used, it has to be proved that “the geometrical locus of points equidistant from a straight line is a straight line.” To do him justice, Clavius saw this and tried to prove it: he makes out that the locus is a straight line according to the definition of Euclid, because “it lies evenly with respect to all the points on it”; but there is a confusion here, because such “evenness” as the locus has is with respect to the straight line from which its points are equidistant, and there is nothing to show that it possesses this property with respect to itself. In fact the theorem cannot be proved without a postulate.