CHAPTER XLVII¶
Part VI — Space — Metrical Geometry (§§ 388–404, pp. 404–418)
§ 388. Metrical geometry presupposes projective or descriptive geometry¶
388. The subject of the present chapter is elementary Geometry, as treated by Euclid or by any other author prior to the nineteenth century. This subject includes the usual analytical Geometry, whether Euclidean or non-Euclidean; it is distinguished from projective and descriptive Geometry, not by any opposition corresponding to that of Euclid and non-Euclid, but by its method and its indefinables. The question whether its indefinables can, or cannot, be defined in terms of those of projective and descriptive Geometry, is a very difficult one, which I postpone to the following chapter. For the present, I shall develop the subject straightforwardly, in a manner as similar to Euclid’s as is consistent with the requisite generality and with the avoidance of fallacies. Metrical Geometry is logically subsequent to the two kinds which we have examined, for it necessarily assumes one or other of these two kinds, to which it merely adds further specifications. I shall, as a rule, assume descriptive Geometry, mentioning projective Geometry only in connection with points in which it shows important metrical differences from descriptive Geometry. In the former case, all the first twenty-six propositions of Euclid will hold. In the latter, the first, seventh, sixteenth, and seventeenth require modification; for these propositions assume, in one form or another, that the straight line is not a closed series. Propositions after the twenty-sixth—or, with a suitable definition of parallels, after the twenty-eighth—depend, with few exceptions, upon the postulate of parallels, and are therefore not to be assumed generally.
§ 389. Errors in Euclid¶
389. Since Euclid still has popularly, and even with mathematicians, a reputation for rigour, in virtue of which his circumlocution and long-windedness are condoned, it may be worth while to point out, to begin with, a few of the errors in his first twenty-six propositions [1]. To begin with the first proposition. There is no evidence whatever that the circles which we are told to construct intersect, and if they do not, the whole proposition fails. Euclid’s problems are often regarded as existence-theorems, and from this point of view, it is plain, the assumption that the circles in question intersect is precisely the same as the assumption that there is an equilateral triangle on a given base. And in elliptic space, where the straight line is a closed series, the construction fails when the length of the base exceeds half the length of the whole straight line. As regards the second and third propositions, there is nothing to be said, except that they are not existence-theorems. The corresponding existence-theorem—i.e. on any straight line, in either direction from a given point on the line, there is a point whose distance from the given point is equal to a given distance—is equivalent to the postulate concerning the circle, and is thus prior to the second and third propositions. With regard to the fourth, there is a great deal to be said; indeed Euclid’s proof is so bad that he would have done better to assume this proposition as an axiom [2]. As the issues raised by this proof are of great importance, both to mathematics and to philosophy, I shall set forth its fallacies at some length.
§ 390. Superposition is not a valid method¶
390. The fourth proposition is the first in which Euclid employs the method of superposition—a method which, since he will make any détour to avoid it [3], he evidently dislikes, and rightly, since it has no logical validity, and strikes every intelligent child as a juggle. In the first place, to speak of motion implies that our triangles are not spatial, but material. For a point of space is a position, and can no more change its position than the leopard can change his spots. The motion of a point of space is a phantom directly contradictory to the law of identity: it is the supposition that a given point can be now one point and now another. Hence motion, in the ordinary sense, is only possible to matter, not to space. But in this case superposition proves no geometrical property. Suppose that the triangle ABC is by the window, and the side AB consists of the column of mercury in a thermometer; suppose also that DEF is by the fire. Let us apply ABC to DEF as Euclid directs, and let AB just cover DE. Then we are to conclude that ABC and DEF, before the motion, were equal in all respects. But if we had brought DEF to ABC, no such result would have followed. But how foolish! I shall be told; of course ABC and DEF are to be both rigid bodies. Well and good. But two little difficulties remain. In the first place—and for my opponent, who is an empirical philosopher, this point is serious—it is as certain as anything can be that there are no rigid bodies in the universe. In the second place—and if my opponent were not an empiricist, he would find this objection far more fatal—the meaning of rigidity presupposes a purely spatial metrical equality, logically independent of matter. For what is meant by a rigid body? It is one which, throughout a continuous portion of time, preserves all its metrical properties unchanged. Hence we incur a most fatally vicious circle if we attempt to define metrical properties by rigidity. If αβγ be a material triangle, which occupies at one time the space ABC, at another the space $A'B'C'$, to say that αβγ is rigid means that, however the two times be chosen (within some assigned period), the triangles ABC, $A'B'C'$ are equal in all respects. If we are to avoid this conclusion, we must define rigidity in some wholly non-geometrical manner. We may say, for example, that a rigid body means one which is made of steel, or of brass. But then it becomes a logical error to regard brass eternal as slave to mortal rage; and if we define equal spaces as those which can be occupied by one and the same rigid body, the propositions of metrical Geometry will be one and all false.
The fact is that motion, as the word is used by geometers, has a meaning entirely different from that which it has in daily life, just as a variable, in mathematics, is not something which changes, but is usually, on the contrary, something incapable of change. So it is with motion. Motion is a certain class of one-one relations, each of which has every point of space for its extension, and each of which has a converse also belonging to the class. That is, a motion is a one-one relation, in which the referent and the relatum are both points, and in which every point may appear as referent and again as relatum. A motion is not this only: on the contrary, it has this further characteristic, that the metrical properties of any class of referents are identical with those of the corresponding class of relata. This characteristic, together with the other, defines a motion as used in Geometry, or rather, it defines a motion or a reflexion; but this point need not be elucidated at present. What is clear is, that a motion presupposes the existence, in different parts of space, of figures having the same metrical properties, and cannot be used to define those properties. And it is this sense of the word motion, not the usual material sense, which is relevant to Euclid’s use of superposition.
§ 391. Errors in Euclid (continued)¶
391. Returning now to Euclid’s fourth proposition, we see that the superposition of ABC on DEF involves the following assumptions. (1) On the line DE there is a point E, on either side of D, such that DE = AB. This is provided for by the postulate about the circle. (2) On either side of the ray DE, there is a ray DF such that the angle EDF is equal to the angle BAC. This is required for the possibility of a triangle DEF such as the enunciation demands, but no axiom from which this follows can be found in Euclid. The problem, to construct an angle EDF equal to BAC, does not occur till I. 23, and there I. 4 is used in the proof. Hence the present assumption must be added to Euclid’s axioms. It now follows that on DF there is a point F such that DF = AC. Hence the possibility of two such triangles as the enunciation demands is established. But in order to prove that DEF is equal in all respects to ABC, we need a further axiom, namely: With one angle at D, one side along the ray DE, and the other side to the right (or left) of DE, there exists a triangle which is equal in all respects to the triangle ABC [4]. This is, in fact, the exact assumption which is concealed in the method of superposition. With this assumption, it finally becomes possible to prove that DEF is the triangle satisfying the above conditions and equal in all respects to ABC.
The next remark concerns I. 6. Here Euclid first employs an axiom of which he is wholly unconscious, though it is very essential to his system, namely: If OA, OB, OC be three rays which meet a straight line not passing through O in A, B, C respectively, and if B be between A and C, then the angle AOB is less than the angle AOC. This axiom, it will be seen, is not applicable in projective space, since it presupposes that the line is not a closed series. In I. 7, if this proposition is to apply to hyperbolic space, we require further the axiom: If three non-intersecting lines in one plane meet two lines in A, B, C; $A'$, $B'$, $C'$, respectively; and if B be between A and C; then $B'$ is between $A'$ and $C'$. Also it may be observed that Euclid gives no definition of the two sides of a line, a notion which again presupposes that the straight line is not a closed series. And with regard to angles, I. 7 requires sufficient axioms to show that they are a series of the kind explained in Part IV, Chapter xxiv; or else we must assume the descriptive axiom of the last chapter, to the effect that, if A, B, C, D be coplanar points, no three of which are collinear, there is a point common to the stretches AB, CD, or to AC, BD, or to AD, BC. All these assumptions will be found implicit in I. 7, as may be seen by attempting a symbolic proof in which no figure is used.
Similar remarks apply to I. 16. In I. 12 it is assumed that a circle must meet a line in two points, if at all. But enough has been said to show that Euclid is not faultless, and that his explicit axioms are very insufficient. Let us, then, make an independent examination of metrical Geometry.
§ 392. Axioms of distance¶
392. Metrical Geometry is usually said to be distinguished by the introduction of quantity. It is sufficient for the characterization of metrical Geometry to observe that it introduces, between every pair of points, a relation having certain properties in virtue of which it is numerically measurable—i.e. such that numbers can be given a one-one correspondence with the various relations of the class in question. The class of relations is called distance, and will be regarded, though this is not strictly necessary, as a class of magnitudes. Some of the properties of distance are as follows.
(1) Every pair of points has one and only one distance.
(2) Distances are symmetrical relations.
(3) On a given straight line through a given point, there are two and only two points at a given distance from the given point.
(4) There is no maximum distance.
(5) The distance of a point from itself is zero [5].
(6) There is no minimum to the distance between distinct points.
(7) If d, δ be two given distances, and $A_{0}$, $A_{1}$, $A_{2}$, … $A_{n}$, … be distinct points on a straight line, whose distances one from the next are all δ, then for some value of n, $A_{0}A_{n}$ is greater than d.
(8) If $A_{0}$, $A_{n}$ be any two points, there exist n − 1 distinct points (whatever integer n may be) on the straight line $A_{0}A_{n}$, such that the distances of each from the next, of $A_{0}$ from the first, and of $A_{n}$ from the last, are all equal [6].
§ 393. Stretches¶
393. It may be observed that, if we admit the axiom that the whole is greater than the part, the properties (1), (4), (5) and (6) belong to stretches, while (2) becomes admissible by abstracting from the sense of a stretch. With regard to the remaining properties, (3), (7) and (8), there is nothing in descriptive Geometry to show whether or not they belong to stretches. Hence we may, if we choose, regard these three properties as axioms regarding stretches, and drop the word distance altogether. I believe that this represents the simplest course, and, as regards actual space, the most correct. At the same time, there is no contradiction in regarding distances as new relations distinct from stretches [7]. If we identify distance and stretch, what distinguishes metrical from descriptive Geometry is primarily the three additional axioms (3), (7) and (8), applied to a new indefinable, namely, the magnitude of divisibility of a stretch. This is not properly a notion of pure mathematics, since it cannot be derived from our original apparatus of logical notions. On the other hand, distance is not indefinable, being a class of one-one relations with certain assignable properties. On this point either course is logically permissible, but only distance can be introduced into pure mathematics in the strict sense in which the word is used in this work.
The above axioms are required for showing that all distances are numerically measurable in terms of any standard distance [8]. It is not necessary that distances should be magnitudes, or even relations; all that is essential is that distances should form a series with certain properties. If the points of a line form a continuous series, then distances do so also, in virtue of (3); thus all signless real numbers will be required for their measurement.
§ 394. Order as resulting from distance alone¶
394. Assuming that distance and stretch are distinct, it may be asked whether distances do not suffice for generating order on the straight line, without the need of any asymmetrical transitive relation of points. This represents, I think, the usual view of philosophers; but it is by no means easy to decide whether it represents a tenable view. It might perhaps be thought that (2) might be dropped, and distance regarded as an asymmetrical relation. So long as we confine our attention to one line, this view seems unobjectionable. But as soon as we consider the fact that distances on different lines may be equal, we see that the difference of sense between AB and BA is not relevant to distance, since there is no such difference between distances on different lines. Thus if CD be a distance on another line, CD may be equal both to AB and BA, and hence AB and BA must be equal, not opposite, distances. And the same thing may be made evident by considering a sphere. For this certainly consists of points at a given distance from the centre; and thus points at opposite ends of a diameter must have the same distance from the centre. Distance, then, is symmetrical; but it does not follow that the order on a line cannot be generated by distance. Let A, B be given points on a line, and let C, $C'$ be two points on AB whose distances from A are equal, and less than AB. If we now set up the axiom that either BC or $BC'$ is less than AB, while the other, $BC'$ or BC, is greater than AB, we shall, I think, after some further axioms, be able to generate order without any other relation than distance. If A, B, C be three collinear points such that the distances AC, CB are both less than AB, then we shall say that C is between A and B. If A, B, $C'$ be points such that $AC'$, AB are both less than $BC'$, then we shall say that A is between B and $C'$. If, finally, A, B, $C''$ be points such that AB, $BC''$ are both less than $AC''$, we shall say that B is between A and $C''$. It remains to see whether, as the generation of a series requires, one of these always happens. Let A, B, C be any three collinear points. First suppose, if possible, that the distances AB, BC, CA are all equal. This case is not excluded by anything hitherto assumed; we require, therefore, the further axiom that, if AB, BC be equal, AC is not equal to either of them; and I think it will be prudent to assume that AC is greater than either. Thus the case of two equal distances and one less than either is excluded. Of the three distances AB, BC, AC, therefore, one must be the greatest: let this be AC. Then in virtue of the definition, B will be between A and C. But our difficulties are not at an end. For we require further that any point between A and B shall be between A and C; and that, if A be between D and C, B shall be between D and C. With regard to the first point, if E be between A and B, AE and EB are less than AB, and therefore less than AC. But nothing assures us that EC is less than AC. For this purpose we need a new axiom, which will be just what we set out to prove, namely: If AE, EB be both less than AB, and AB, BC be both less than AC, then EC is less than AC. Finally, to prove that, if A be between D and C, and B between A and C, then B is between D and C. Here DA, AC are less than DC, and AB, BC are less than AC. Hence BC is less than DC; but nothing proves BD less than DC. For this we shall need a new axiom, and then at last our order will be definite. But the process, as is evident, is extremely complicated.
§ 395. Geometries which derive the straight line from distance¶
395. Moreover we still need a method of defining the straight line. Pieri has shown, in an admirable memoir [9], how to deduce metrical geometry by taking point and motion as the only indefinables. In §390, we objected to the introduction of motion, as usually effected, on the ground that its definition presupposes metrical properties; but Pieri escapes this objection by not defining motion at all, except through the postulates assumed concerning it. The straight line joining two points is the class of points that are unchanged by a motion which leaves the two points fixed. The sphere, the plane, perpendicularity, the order of points on a line, etc. are easily defined. This procedure is logically unimpeachable, and is probably the simplest possible for elementary geometry. But we must now return to the consideration of other suggested systems.
There is a method, invented by Leibniz [10] and revived by Frischauf [11] and Peano [12], in which distance alone is fundamental, and the straight line is defined by its means. In this method distances are given to begin with as a class of relations which are the field of a certain transitive asymmetrical relation (greater and smaller); if we assume this relation to be continuous, distances will be measurable; all distances have the same domain and the same converse domain, namely all the points of the space in question; the locus of points equidistant from two fixed points is called a plane, and the intersection of two non-coincident planes, when it is not null, is called a straight line. (The definition of the straight line given by Peano [13] is as follows: The straight line ab is the class of points x such that any point y, whose distances from a and b are respectively equal to the distances of x from a and b, must be coincident with x.) Leibniz, who invented this method, failed, according to Couturat, to prove that there are straight lines, or that a straight line is determined by any two of its points. Peano has not, so far as I am aware, succeeded in proving either of these propositions, but it is of course possible to introduce them by means of axioms. Frischauf professes to demonstrate them, but his proofs are very informal, and it is difficult to know what axioms he is assuming. In any case, however, the definitions prove that, by a sufficient use of axioms, it is possible to construct a geometry in which distance is fundamental, and the straight line derivative. The method is so complicated as to be not practically desirable; but its logical possibility is nevertheless important.
§ 396. In most spaces, magnitude of divisibility can be used instead of distance¶
396. It is thus plain that the straight line must be independent of distance, while distance may be independent of the straight line. Taking both as symmetrical relations, we can, by a very complicated series of axioms, succeed in generating order on the straight line and in explaining the addition and measurement of distances. But this complication, in most spaces [14], is logically unnecessary, and is wholly avoided by deriving distances from stretches. We now start, as in descriptive Geometry, with an asymmetrical transitive relation by which the straight line is both defined and shown to be a series. We define as the distance of two points A and B the magnitude of divisibility of the stretch from A to B or B to A—for divisibility is a signless magnitude. Divisibility being a kind of magnitude, any two distances will be equal or unequal. As with all divisibilities, the sum of the divisibilities of AB and EF is the divisibility of the logical sum of the classes AB and EF, provided these classes have no common part. If they have a common part, we substitute for EF a stretch $E'F'$ equal to it and having no part in common with AB. The difference of the distances AB, EF (supposing AB the greater) is the divisibility of a stretch CD which, added logically to EF, and having no part in common with EF, produces a stretch equal to AB. It follows at once that, if A, B, C be collinear, and B be between A and C, AB + BC = AC and AC − AB = BC. No further axiom is required for these propositions. For the proposition that, if AB = $A'B'$ , and CD = $C'D'$, then AB + CD = $A'B'$ + $C'D'$, we require only the general axiom, applicable to all divisibilities, that the sums of equals are equal. Thus by the help of the axioms (3), (7), (8) above, we have everything that is required for the numerical measurement (theoretically speaking) of all distances in terms of any given distance, and for the proof that change of unit involves multiplication throughout by a common factor.
§ 397. Meaning of magnitude of divisibility¶
397. With regard to magnitude of divisibility, in the sense in which this is relevant to metrical Geometry, it is important to realize that it is an ordinal notion, expressing a property of relations, not of their fields. We wish to say that a stretch of two inches has twice as much divisibility as a stretch of one inch, and that an area is infinitely more divisible than a stretch. Now, if we are dealing (as will be assumed in this discussion) with a continuous space, every stretch, area or volume is a class of $2^{\alpha_0}$ terms; and considered as a class, it is the field of an infinite number of relations beside that (or those) belonging to it in respect of the space we are considering. The habit of allowing the imagination to dwell upon actual space has made the order of points appear in some way intrinsic or essential, and not merely relative to one of many possible ordering relations. But this point of view is not logical: it arises, in regard to actual space, only from the fact that the generating relations of actual space have a quite peculiar connection with our perceptions, and, through the continuity of motion, with time. From the standpoint of logic, no one of the relations having a given field has any preeminence, and the points of actual space, like any other class of $2^{\alpha_0}$ terms, form, with regard to other sets of generating relations, other sorts of continuous spaces—indeed any other continuous space, having any finite number of dimensions, or even ω dimensions, can be formed of the points of a Euclidean space by attending to other generating relations.
From this it follows that magnitude of divisibility, if it is to distinguish a long stretch from a short one, or an area from a stretch, must be a property of the relations involved, not of the class of points composing the area or the stretch. It is not quite easy to define the exact property which is required; for any two stretches are ordinally similar. We require some sense for the equality or inequality of the relations whose fields are the given stretches. Where coordinates (i.e. a correlation of the points of a line with the real numbers) have been already introduced, we may define the magnitude of a stretch as the difference of the coordinates of its end-points or its limits (according as the stretch has ends or not); but if this is done, the magnitudes of stretches will depend upon the necessarily more or less arbitrary plan upon which we have introduced our coordinates. This is the course adopted in the projective theory of distance—a course which has the merit of making metrical Geometry a logical development from projective axioms alone (see next chapter). The other course that may be adopted is, to assume that the generating relations of any two stretches have either a symmetrical transitive relation (equality), or an asymmetrical transitive relation or its converse (greater or less). Certain axioms will be required, as, for example, that if the points A, B, C, D are collinear, and AC is greater than AD, then BC is greater than BD [15]. The relations of equal, greater and less may be regarded as defined by these axioms, and the common property of the generating relations of those stretches that are equal to a given stretch may be defined as the magnitude of divisibility of the said generating relations. The sense in which an area has infinitely more divisibility than a stretch is that, if n be any finite integer, and n stretches equal to a given stretch be removed from an area, there always remains an area, however great n may be. What is important to observe, in the above discussion, is that the logical parity of all the orders of which a class of terms is capable makes it necessary to regard the magnitudes with which metrical Geometry deals as belonging to relations or classes of relations, not, as is commonly supposed, to the class of points forming their fields.
§ 398. Difficulty of making distance independent of stretch¶
398. In elliptic space, where the straight line is a closed series, the attempt to make distance independent of stretch leads to still further complications. We now no longer have the axiom that, if A, B, C be collinear, we cannot have AB = BC = CA; and we have to recognize two distances between every pair of points, which, when distance is taken as fundamental, becomes extremely awkward. We may however avoid admitting two distances by refusing to regard the greater of the two as properly a distance. This will then be only a stretch. If two distances are admitted, one is always greater than the other, except in a limiting case, when both are the lower limit of the greater distances and the upper limit of the lesser distances. Further if a, b, c, d be any four distinct points, the greater of the two distances ab is always greater than the lesser of the two distances cd. Thus the whole class of greater distances may be banished, and only greater stretches be admitted.
We must now proceed as follows. Distances are a class of symmetrical relations, which are magnitudes of one kind, having a maximum, which is a one-one relation whose field is all points, and a minimum, which is the distance of any point from itself. Every point on a given line has a given distance other than the maximum or minimum from two and only two other points on the line. If a, b, c, d be four distinct points on one line, we shall say that a and c are separated by b and d in the following four cases, of which (1) and (2) and also (3) and (4) are not mutually exclusive:
(1) If ab < ac . bc < ac . ad > ac.
(2) If ab < ac . bc < ac . dc > ac.
(3) If ab > ac . ad < ac . dc < ac.
(4) If bc > ac . ad < ac . dc < ac.
We then need Vailati’s five axioms enumerated in Part IV, Chap. xxiv, in order to generate a closed series from the separation of couples so defined. Thus it is possible, though by a somewhat complicated process, to generate a closed series of points on a line by means of the symmetrical relation of distance.
I shall not work out in further detail the consequences of this hypothesis in elliptic space, but proceed at once to the hypothesis that distances are the magnitudes of stretches. When the number of dimensions exceeds two, the polar form of elliptic space is merely projective space together with the necessary metrical axioms; the antipodal form is a space in which two antipodal points together have the properties of a single projective point. Neglecting the latter, to which similar remarks will apply, I shall confine myself to the polar form. Since this is a projective space, every pair of points determines two segments on the line joining the points. The sum of these two segments, together with the two points, is the whole line, and therefore constant. It is an axiom that all complete straight lines have the same divisibility. The divisibility of either segment is a distance between the two points: when the two distances are equal, either may be called the distance; when they are unequal, it will be convenient to call the smaller the distance, except in special problems. The whole theory then proceeds as in the case of descriptive space. But it is important to observe that, in elliptic space, the quadrilateral construction and the generation of order, being prior to stretches, are prior to distances, and are presupposed in metrical Geometry.
§ 399. Theoretical meaning of measurement¶
399. So far, therefore, metrical Geometry introduces three new axioms, and one new indefinable. The stretch in every series is a quantity, and metrical Geometry merely introduces such axioms as make all stretches of points measurable. A few words may be useful as to the sense in which, in a theoretical discussion, the word measurement is to be understood. The actual application of the foot-rule is here not in question, but only those properties of pure space which are presupposed in the use of the foot-rule. A set of magnitudes is theoretically measurable when there is a one-one relation between them and some or all numbers; it is practically measurable when, given any magnitude, we can discover, with a certain margin of error, what the number is to which our magnitude has the relation in, question. But how we are to discover this is a subsequent question, presupposing that there is such a proposition to be discovered, and soluble, if at all, by empirical means to be invented in the laboratory. With practical measurement, then, we are not at all concerned in the present discussion.
§ 400. Definition of angle¶
400. I come now to a more difficult question than distance, namely the question as to the definition of angle. Here, to begin with, we must deal with rays, not with whole straight lines. The ray may be taken either as an asymmetrical relation, or as the half-line on one side of a given point on a line. The latter usage is very convenient, and I shall frequently employ it. Elementary Geometry assumes that two rays starting from the same point determine a certain magnitude, called the angle between them. This magnitude may, however, be defined in various ways. In the first place, we must observe that, since the rays in a plane through a point form a closed series, every pair of rays through a point defines two stretches of rays. Of these, however, one stretch contains the opposites of both rays, while the other stretch contains the opposites of neither—except, indeed, in the one case where the two rays are each other’s opposites. This case is met by Euclid’s postulate that all right angles are equal—a postulate, however, which is now known to be demonstrable [16]. Omitting this case, the angle between two rays may be defined as that stretch of rays through their intersection which is bounded by the two rays and does not contain the opposite of either, i.e. if A, B be the rays, and $\breve{A}$, $\breve{B}$ their opposites, the angle is the class of rays C which are separated from $\breve{A}$ or $\breve{B}$ by A and B. We might also, but for an objection to be mentioned shortly, define the angle as all the points on such rays. A definition equivalent to this last, but simpler in form, and avoiding the mention of the opposite rays, is the following [17]. Let a, b be any two points of the rays A, B, and let c be any point of the stretch ab. Then the class of points c, for all possible positions of a and b on their respective rays, is the angle between A and B. That is, every pair of intersecting rays divides the plane of the rays into two parts: the part defined as above is the angle. Or rather, the part so defined is the angle as a quantity: the angle as a magnitude is the divisibility of this part. But to these latter definitions we shall find fatal objections, and we shall find it necessary to adhere to the definition as a stretch of rays.
§ 401. Axioms concerning angles¶
401. Thus angle, like distance, is not a new indefinable, but like distance, it requires some new axioms. The angle between a ray A and its opposite $A'$ cannot be defined as above, but may be defined as the logical sum of the angles between A and B, B and $A'$ respectively. This limiting angle is greater than any other at the point, being in fact the whole half of the plane on one side of the straight line $AA'$. If the angles between A and B, B and $A'$ are equal, each is called a right angle. (That there are such angles, can be proved if we assume continuity.) Two intersecting straight lines make four angles, which are equal in pairs. The order of a collection of rays through a point in a plane may be obtained by correlation with the points where these rays intersect a given straight line, provided there is any straight line which all of them intersect. But since rays through a point in a plane form a closed series, while the points on a line do not, we require a four-term relation for the former order. The following definition seems adequate. Given four rays OA, OB, OC, OD through a point O and in one plane, if these all meet a certain straight line in A, B, C, D respectively, and A and C are separated by B and D, then OA and OC are said to be separated by OB and OD. In projective space this suffices. But in descriptive space we must provide for other cases. Thus if OA, OB, OC meet the given line, and B is between OA and OC, while OD does not meet the given line, then OA and OC are again said to be separated by OB and OD. If, finally, $OA'$ and $OB'$ be the opposites of OA and OB, then OA and $OA'$ are separated by OB and $OB'$. In virtue of the descriptive axioms of the preceding chapter, the order among the rays so obtained will be unambiguous, i.e. independent of our choice of the line ABC, and will cover all cases.
But now we need axioms analogous to those which, in the case of distance, were numbered (3), (7) and (8). At any given point in a given ray, there must be, in a given plane, two and only two rays, on opposite sides of the given ray (i.e. separated from each other by the given ray and its opposite), which make a given angle with the given ray; and angles must obey the axioms of Archimedes and of linearity. But in addition to these axioms, which insure that angles shall be numerically measurable, we must have some method of connecting the measure of angles with that of distances, such as is required for the solution of triangles. Does this require a new axiom? Euclid appears to obtain this, by means of I. 47, II. 12, and II. 13, without any fresh axiom. For this result we depend upon the propositions on the congruence of triangles (I. 4, 8, 26), which demand only, as we saw, the axiom that, with one angle at a given point, and one side along a given ray through that point, there exist two and only two triangles in a given plane through the ray (one on each side of the given ray), which are equal in all respects to a given triangle. Thus it would seem that no fresh axioms are required for angles in a plane.
§ 402. An angle is a stretch of rays, not a class of points¶
402. With regard to the definition of an angle as a portion of a plane, it is necessary (as in many other cases), if we retain this definition, somewhat to restrict the axiom that the whole is greater than the part. If a whole A has two parts B, C, which together constitute A, and if C be infinitesimal with respect to A, then B will be equal to A. This case occurs in a plane under the following circumstances. Let O, $O'$ be any two points, OP, $O'P'$ lines in one plane and making equal angles with the ray $OO'$ [18]. Then in Euclidean or hyperbolic space these lines OP, $O'P'$ will not intersect; thus the angle between $OO'$ and $O'P'$ will be part of the angle $O'$OP. Hence the above restriction is necessary as regards the axiom that the whole is greater than the part.
In Euclidean space this answer is sufficient, since, if OP makes with $OO'$ a less angle than $O'P'$ does, OP and $O'P'$ will intersect. But in hyperbolic space, OP and $O'P'$ may not intersect even then. Hence if we adhere to the above definition of angle, we shall have to hold that the whole may be less than the part. This, however, is intolerable, and shows that the definition in question must be rejected. We may, however, still regard angle as the stretch of rays; for the rays in the angle at $O'$ are not part of the rays in the angle at O. Hence it is only as a stretch of rays, or as the magnitude of such a stretch, that an angle can be properly defined.
As showing, in a curious manner, the increased power of deduction which results from the above axioms concerning distances and angles, we may remark that the uniqueness of the quadrilateral construction, which before could not be proved without three dimensions, can now be proved, as regards all constructions in one plane, without any assumption of points outside that plane. Nothing is easier than to prove this proposition by the methods of elementary coordinate Geometry. Thus although projective Geometry, as an independent science, requires three dimensions, any projective proposition concerning plane figures can be metrically proved, if the above axioms hold, for a two-dimensional space.
§ 403. Areas and volumes¶
403. As regards figures of three dimensions, angles between planes and solid angles can be defined exactly as rectilinear angles were defined. Moreover fresh axioms will not be required, for the measurement of such angles can be deduced from the data we already possess.
With regard to areas and volumes some remarks seem necessary. Areas and volumes, like angles, are classes of points when taken as quantities, and divisibilities when taken as magnitudes. For areas and volumes we do not require afresh the axioms of Archimedes and of linearity, but we require one axiom apiece to give a criterion of equal areas and volumes, i.e. to connect their equality with that of distances and angles. Such an axiom is supplied, as regards areas, by the axiom that two congruent triangles have the same area, and as regards volumes, by the corresponding axiom concerning tetrahedra. But the existence of congruent tetrahedra, like that of congruent triangles, demands an axiom. For this purpose, Pasch [19] gives the following general axiom: If two figures are congruent, and a new point be added to one of them, a new point can be added to the other so that the two new figures are congruent. This axiom allows us to infer congruent tetrahedra from congruent triangles; and hence the measurement of volumes proceeds smoothly.
§ 404. Right and left¶
404. In three dimensions, a curious fact has to be taken account of, namely, the disjunction of right and left-handedness, or of clockwise and counter-clockwise. This fact is itself of a descriptive nature, and may be defined as follows. Between two non-coplanar rays, or between four non-coplanar points taken in an assigned order, there is always one of two opposite relations, which may be called right and left. The formal properties of these relations have been explained in Part IV (§222); for the present I am concerned with their geometrical consequences. In the first place, they cause volumes to become magnitudes with sign, in exactly the way in which distances on a straight line have sign when compounded with their sense. But in the case of distances, since not all are on one straight line, we could not thus compound distance and sense generally: we should require, for a compound, some more general notion than sense, such as vectors supply. Here, on the contrary, since, in a three-dimensional space, all volumes have one or other of two senses, the compound can be made for all volumes. Thus if the volume of the tetrahedron abcd has one sign, that of bacd will have the opposite sign. This is the familiar geometrical fact that the determinant giving the volume of a tetrahedron abcd has one or other sign according as the sense of abcd is the same as or different from that of OXYZ, where O is the origin and X, Y, Z any positive points on the axes. It is this fact, also, which gives signs to angular momentum in Dynamics. The importance of the fact (which itself seems to be an independent axiom) is this, that it makes a distinction between two figures whose metrical properties are all identical. It is this distinction which puzzled Kant, who, like most of his contemporaries, supposed all geometrical facts to be metrical. In itself, the fact would be no more puzzling than, the distinction between the stretches AB and BA, which are metrically indistinguishable. But it becomes puzzling when metrical equality is supposed to result from motion and superposition. In our former definition of motion (§390) we omitted (as was then observed) a condition essential to its definition. Not only must two congruent figures be metrically equal, but there must be a continuous series of equal figures leading from the one to the other. Or, what amounts to the same thing, if a, b, c, d and $a'$, $b'$, $c'$, $d'$ be homologous non-coplanar points in the two figures, the tetrahedra abcd, $a'b'c'd'$ must have the same sense. In the case of equal and opposite tetrahedra, these conditions fail. For there is no gradual transition from clockwise to counter-clockwise; thus at some point in the series a sudden jump would be necessary. No motion will transform abcd into a tetrahedron metrically equal in all respects, but with the opposite sense. In this fact, however, there seems, to my mind, to be nothing mysterious, but merely a result of confining ourselves to three dimensions. In one dimension, the same would hold of distances with opposite senses; in two dimensions, of areas. It is only to those who regard motion as essential to the notion of metrical equality that right and left-handedness form a difficulty; in our theory, they are rather a confirmation than a stumbling-block.
With this we may end our brief review of metrical Geometry, leaving it to the next chapter to discuss its relation to projective Geometry and the projective theory of distance and angle.
Author’s Footnotes (1903 Edition)