Commentary on Proposition III.16

Back to Proposition III.16.

This proposition is historically interesting because of the controversies to which the last part of it gave rise from the 13th to the 17th centuries. History was here repeating itself, for it is certain that, in ancient Greece, both before and after Euclid’s time, there had been a great deal of the same sort of contention about the nature of the “angle of a semicircle” and the “remaining angle” between the circumference of the semicircle and the tangent at its extremity. As we have seen (note on I. Def. 8), the latter angle had a recognised name, κερατοειδὴς γωνία, horn-like or cornicular angle; though this term does not appear in Euclid, it is often used by Proclus, evidently as a term well understood. While it is from Proclus that we get the best idea of the ancient controversies on this subject, we may, I think, infer their prevalence in Euclid’s time from this solitary appearance of the two “angles” in the $Elements$. Along with the definition of the angle of a segment, it seems to show that, although these angles are only mentioned to be dropped again immediately, and are of no use in elementary geometry, or even at all, Euclid thought that an allusion to them would be expected of him; it is as if he merely meant to guard himself against appearing to ignore a subject which the geometers of his time regarded with interest. If this conjecture is right, the mention of these angles would correspond to the insertion of definitions of which he makes no use, e.g. those of a rhombus and a rhomboid.

Proclus has no hesitation in speaking of the “angle of a semicircle” and the “horn-like angle” as true angles. Thus he says that “angles are contained by a straight line and a circumference in two ways; for they are either contained by a straight line and a convex circumference, like that of the semicircle, or by a straight line and a concave circumference, like the κερατοειδής” (p. 127, 11—14). “There are mixed lines, as spirals, and angles, as the angle of a semicircle and the κερατοειδής” (p. 104, 16—18). The difficulty which the ancients felt arose from the very fact which Euclid embodies in this proposition. Since an angle can be divided by a line, it would seem to be a magnitude; “but if it is a magnitude, and all homogeneous magnitudes which are finite have a ratio to one another, then all homogeneous angles, or rather all those on surfaces, will have a ratio to one another, so that the cornicular will also have a ratio to the rectilineal. But things which have a ratio to one another can, if multiplied, exceed one another. Therefore the cornicular angle will also sometime exceed the rectilineal; which is impossible, for it is proved that the former is less than any rectilineal angle” (Proclus, p. 121, 24—122, 6). The nature of contact between straight lines and circles was also involved in the question, and that this was the subject of controversy before Euclid’s time is clear from the title of a work attributed to Democritus (fl. 420—400 B.C.) περὶ διαφορῆς γνώμονος ἢ περὶ ψαύσιος κύκλου καὶ σφαίρης, On a difference in a gnomon or on contact of a circle and a sphere. There is, however, another reading of the first words of this title as given by Diogenes Laertius (IX. 47), namely περὶ διαφορῆς γνώμης, On a difference of opinion, etc. May it not be that neither reading is correct, but that the words should be περὶ διαφορῆς γωνίης ἢ περὶ ψαύσιος κύκλου καὶ σφαίρης, On a difference in an angle or on contact with a circle and a sphere? There would, of course, hardly be any “angle” in connexion with the sphere; but I do not think that this constitutes any difficulty, because the sphere might easily be tacked on as a kindred subject to the circle. A curiously similar collocation of words appears in a passage of Proclus, though this may be an accident. He says (p. 50, 4) πῶς δὲ γωνιῶν διαφορὰς λέγομεν καὶ αὐξήσεις αὐτῶν… and then, in the next line but one, πῶς δὲ τὰς ἁφὰς τῶν κύκλων ἢ τῶν εὐθειῶν, “In what sense do we speak of differences of angles and of increases of them… and in what sense of the contacts (or meetings) of circles or of straight lines?” I cannot help thinking that this subject of cornicular angles would have had a fascination for Democritus as being akin to the question of infinitesimals, and very much of the same character as the other question which Plutarch (On Common Notions, XXXIX. 3) says that he raised, namely that of the relation between the base of a cone and a section of it by a plane parallel to the base and apparently, to judge by the context, infinitely near to it: “if a cone were cut by a plane parallel to its base, what must we think of the surfaces of the sections, that they are equal or unequal? For, if they are unequal, they will make the cone irregular, as having many indentations like steps, and unevennesses; but, if they are equal, the sections will be equal, and the cone will appear to have the property of the cylinder, as being made up of equal and not unequal circles, which is the height of absurdity.”

The contributions by Democritus to such investigations are further attested by a passage in the Method of Archimedes discovered by Heiberg in 1906 (Archimedes, ed. Heiberg, Vol. II. 1913, p. 430; T. L. Heath, The Method of Archimedes, 1912, p. 13), which says that, though Eudoxus was the first to discover the scientific proof of the propositions (attributed to him) that the cone and the pyramid are one-third of the cylinder and prism respectively which have the same base and equal height, they were first stated, without proof, by Democritus.

A full history of the later controversies about the cornicular “angle” cannot be given here; more on the subject will be found in Camerer’s Euclid (Excursus IV. on III. 16) or in Cantor’s Geschichte der Mathematik, Vol. II. (see Contingenzwinkel in the index). But the following short note about the attitude of certain well-known mathematicians to the question will perhaps not be out of place. Johannes Campanus, who edited Euclid in the 13th century, inferred from III. 16 that there was a flaw in the principle that the transition from the less to the greater, or vice versâ, takes place through all intermediate quantities and therefore through the equal. If a diameter of a circle, he says, be moved about its extremity until it takes the position of the tangent to that circle, then, as long as it cuts the circle, it makes an acute angle less than the “angle of a semicircle”; but the moment it ceases to cut, it makes a right angle greater than the same “angle of a semicircle.” The rectilineal angle is never, during the transition, equal to the “angle of a semicircle.” There is therefore an apparent inconsistency with X. 1, and Campanus could only observe (as he does on that proposition), in explanation of the paradox, that “these are not angles in the same sense (univoce), for the curved and the straight are not things of the same kind without qualification (simpliciter).” The argument assumes, of course, that the right angle is greater than the “angle of a semicircle.”

Very similar is the statement of the paradox by Cardano (1501—1576), who observed that a quantity may continually increase without limit, and another diminish without limit; and yet the first, however increased, may be less than the second, however diminished. The first quantity is of course the angle of contact, as he calls it, which may be “increased” indefinitely by drawing smaller and smaller circles touching the same straight line at the same point, but will always be less than any acute rectilineal angle however small.

We next come to the French geometer, Peletier (Peletarius), who edited the Elements in 1557, and whose views on this subject seem to mark a great advance. Peletier’s opinions and arguments are most easily accessible in the account of them given by Clavius (Christoph Klau [?], 1537—1612) in the 1607 edition of his Euclid. The violence of the controversy between the two will be understood from the fact that the arguments and counter-arguments (which sometimes run into other matters than the particular question at issue) cover, in that book, 26 pages of small print. Peletier held that the “angle of contact” was not an angle at all, that the “contact of two circles,” i.e. the “angle” between the circumferences of two circles touching one another internally or externally, is not a quantity, and that the “contact of a straight line with a circle” is not a quantity either; that angles contained by a diameter and a circumference whether inside or outside the circle are right angles and equal to rectilineal right angles, and that angles contained by a diameter and the circumference in all circles are equal. The proof which Peletier gave of the latter proposition in a letter to Cardano is sufficiently ingenious. If a greater and a less semicircle be placed with their diameters terminating at a common point and lying in a straight line, then (1) the angle of the larger obviously cannot be less than the angle of the smaller. Neither (2) can the former be greater than the latter; for, if it were, we could obtain another angle of a semicircle greater still by drawing a still larger semicircle, and so on, until we should ultimately have an angle of a semicircle greater than a right angle: which is impossible. Hence the angles of semicircles must all be equal, and the differences between them nothing. Having satisfied himself that all angles of contact are not-angles, not-quantities, and therefore nothings, Peletier holds the difficulty about X. 1 to be at an end. He adds the interesting remark that the essence of an angle is in cutting, not contact, and that a tangent is not inclined to the circle at the point of contact but is, as it were, immersed in it at that point, just as much as if the circle did not diverge from it on either side.

The reply of Clavius need not detain us. He argues, evidently appealing to the eye, that the angle of contact can be divided by the arc of a circle greater than the given one, that the angles of two semicircles of different sizes cannot be equal, since they do not coincide if they are applied to one another, that there is nothing to prevent angles of contact from being quantities, it being only necessary, in view of X. 1, to admit that they are not of the same kind as rectilineal angles; lastly that, if the angle of contact had been a nothing, Euclid would not have given himself so much trouble to prove that it is less than any acute angle. (The word is desudasset, which is certainly an exaggeration as applied to what is little more than an obiter dictum in III. 16.)

Vieta (1540—1603) ranged himself on the side of Peletier, maintaining that the angle of contact is no angle; only he uses a new method of proof. The circle, he says, may be regarded as a plane figure with an infinite number of sides and angles; but a straight line touching a straight line, however short it may be, will coincide with that straight line and will not make an angle. Never before, says Cantor ($\text{II}_1$, p. 540), had it been so plainly declared what exactly was to be understood by contact.

Galileo Galilei (1564—1642) seems to have held the same view as Vieta and to have supported it by a very similar argument derived from the comparison of the circle and an inscribed polygon with an infinite number of sides.

The last writer on the question who must be mentioned is John Wallis (1616—1703). He published in 1656 a paper entitled De angulo contactus et semicirculi tractatus in which he also maintained that the so-called angle was not a true angle, and was not a quantity. Vincent Leotaud (1595—1672) took up the cudgels for Clavius in his Cyclomathia which appeared in 1663. This brought a reply from Wallis in a letter to Leotaud dated 17 February, 1667, but not apparently published till it appeared in A defense of the treatise of the angle of contact which, with a separate title-page, and date 1684, was included in the English edition of his Algebra dated 1685. The essence of Wallis’ position may be put as follows. According to Euclid’s definition, a plane angle is an inclination of two lines; therefore two lines forming an angle must incline to one another, and, if two lines meet without being inclined to one another at the point of meeting (which is the case when a circumference is touched by a straight line), the lines do not form an angle. The “angle of contact” is therefore no angle, because at the point of contact the straight line is not inclined to the circle but lies on it ἀκλινῶς, or is coincident with it. Again, as a point is not a line but a beginning of a line, and a line is not a surface but a beginning of a surface, so an angle is not the distance between two lines, but their initial tendency towards separation: Angulus (seu gradus divaricationis) Distantia non est sed Inceptivus distantiae. How far lines, which at their point of meeting do not form an angle, separate from one another as they pass on depends on the degree of curvature (gradus curvitatis), and it is the latter which has to be compared in the case of two lines so meeting. The arc of a smaller circle is more curved as having as much curvature in a lesser length, and is therefore curved in a greater degree. Thus what Clavius called angulus contactus becomes with Wallis gradus curvitatis, the use of which expression shows that curvature and curvature can be compared according to one and the same standard. A straight line has the least possible curvature; but of the “angle” made by it with a curve which it touches we cannot say that it is greater or less than the “angle” which a second curve touching the same straight line at the same point makes with the first curve; for in both cases there is no true angle at all (cf. Cantor $\text{III}_1$, p. 24).

The words usually given as a part of the corollary “and that a straight line touches a circle at one point only, since in fact the straight line meeting it in two points was proved to fall within it” are omitted by Heiberg as being an undoubted addition of Theon’s. It was Simson who added the further remark that “it is evident that there can be but one straight line which touches the circle at the same point.”