CHAPTER XLVIII¶
Part VI — Space — Relation of Metrical to Projective and Descriptive Geometry (§§ 405–411, pp. 419–428)
§ 405. Non-quantitative geometry has no metrical presuppositions¶
405. In the present chapter I wish to discuss two questions. First, can projective and descriptive Geometry be established without any metrical presuppositions, or even without implying metrical properties? Secondly, can metrical Geometry be deduced from either of the others, or, if not, what unavoidable novelties does it introduce? The previous exposition has already dogmatically assumed certain answers to these questions, but we are now to examine critically the various possible answers.
The distinction between projective and descriptive Geometry is very recent, and is of an essentially ordinal nature. If we adopt the view—which, as we saw, is the simpler of two legitimate views—that the straight line is defined by a certain relation between any two of its points, then in projective Geometry this relation is symmetrical, while in descriptive Geometry it is asymmetrical. Beyond this we have the difference that, in projective Geometry, a line and a plane, two planes, or two lines in a plane, always intersect, while in descriptive Geometry the question whether this is the case or not is left open. But these differences are not very important for our present purpose, and it will therefore be convenient to speak of projective and descriptive Geometry together as non-quantitative Geometry.
The logical independence of non-quantitative Geometry is now scarcely open to question. We have seen, in Chapters xlv and xlvi, how it may be built up without any reference whatever to quantitative considerations. Quantity, in fact, though philosophers appear still to regard it as very essential to mathematics, does not occur in pure mathematics, and does occur in many cases not at present amenable to mathematical treatment. The notion which does occupy the place traditionally assigned to quantity is order; and this notion, we saw, is present in both kinds of non-quantitative Geometry. But the purity of the notion of order has been much obscured by the belief that all order depends upon distance—a belief which, though it is entertained by so excellent a writer as Meinong, we have seen to be false. Distance being essentially quantitative, to admit that series depend upon distance is to admit that order depends upon quantity. But this view leads at once to an endless regress, since distances have an order of magnitude, which would have to be derived from new distances of distances, and so on. And positively, an asymmetrical transitive relation suffices to generate a series, but does not imply distance. Hence the fact that the points of a line form a series does not show that Geometry must have metrical presuppositions, and no such presuppositions appear in the detail of projective or descriptive Geometry.
§ 406. Historical development of non-quantitative geometry¶
406. But although non-quantitative Geometry, as it now exists, is plainly independent of everything metrical, the historical development of the subject has tended greatly to obscure this independence. A brief historical review of the subject may be useful in showing the relation of the more modern to the more traditional methods.
In Euclid, and in Greek geometers generally, hardly any descriptive theorems are to be found. One of the earliest discoveries of an important descriptive theorem was the one named after Pascal [1]. Gradually it was found that propositions which assert points to be collinear or lines to be concurrent, or propositions concerning tangents, poles and polars, and similar matters, were unaltered by projection; that is, any such property belonging to a plane figure would belong also to the projection or shadow of this figure from any point on to any plane. All such properties (as, for instance, those common to all conics) were called projective or descriptive. Among these properties was anharmonic ratio, which was defined as follows. If A, B, C, D be four points on one straight line, their anharmonic ratio is $\frac{AB}{CB} / \frac{AD}{CD}$ ; if OA, OB, OC, OD be four lines through a point, their anharmonic ratio is $\frac{\sin AOB}{\sin COB} / \frac{\sin AOD}{\sin COD}$. In Chasles’s great work on descriptive Geometry, and even in most recent works (such as Cremona’s projective Geometry), this definition will be found at a very early stage in the development of the subject, together with a proof that anharmonic ratio is unaltered by projection. But such a definition is itself metrical, and cannot therefore be used to found a subject independent of metrical Geometry. With other portions of what used to be called descriptive or projective Geometry, the same lack of independence will be found. Consider, for example, the definition of a conic. To define it as a curve of the second degree would require projective coordinates, which there was no known method of introducing. To define it as a curve meeting any straight line in not more than two points would require the distinction of real and imaginary points, for if we confine ourselves to real points there are innumerable curves other than conics which satisfy the definition. But imaginary points are, in ordinary metrical Geometry, imaginary coordinates, for which there is no purely geometrical interpretation; thus without projective coordinates, our definition again fails. To define a conic as the locus of points P for which the anharmonic ratio of PA, PB, PC, PD (where A, B, C, D are fixed points) is constant, again involves metrical considerations, so long as we have no projective definition of anharmonic ratio. And the same dependence upon metrical Geometry appears as regards any other projective or descriptive theorem, so long as the traditional order of ideas is adhered to.
The true founder of non-quantitative Geometry is von Staudt [2]. It was he who introduced the definition of a harmonic range by means of the quadrilateral construction, and who rendered it possible, by repetitions of this construction, to give projective definitions of all rational anharmonic ratios [3]. These definitions indicate the succession of quadrilateral constructions required in order to obtain a fourth point from three given points; thus, though they are essentially numerical, they have no reference whatever to quantity. But there remained one further step, before projective Geometry could be considered complete, and this step was taken by Pieri. In Klein’s account, it remains doubtful whether all sets of four collinear points have an anharmonic ratio, and whether any meaning can be assigned to irrational anharmonic ratios. For this purpose, we require a method of generating order among all the points of a line. For, if there be no order but that obtained from Klein’s method, there is no sense in which we can regard a point not obtained by that method as the limit of a series of points which are so obtained, since the limit and the series which it limits must always both belong to some one series. Hence there will be no way of assigning irrational, coordinates to the points which do not have rational coordinates. There is, of course, no projective reason for supposing that there are such points; but there are metrical reasons, and in any case it is well, if possible, to be able to deal projectively with a continuous space. This is effected by Pieri, with the help of certain new axioms, but without any new indefinables. Thus at last the long process by which projective Geometry has purified itself from every metrical taint is completed.
§ 407. Non-quantitative theory of distance¶
407. Projective Geometry, having achieved its own independence, has, however, embarked upon a career of foreign aggrandisement; and in this we shall, I think, though on the whole favourable, be obliged to make some slight reservations. The so-called projective theory of distance aims at proving that metrical is merely a branch of projective Geometry, and that distances are merely logarithms of certain anharmonic ratios. If this theory be correct, there is not a special subject of metrical Geometry, and the axioms by which, in the preceding chapter, we distinguished this subject, must be consequences of projective axioms. Let us examine the manner in which this result is obtained [4].
We have already seen how to assign coordinates to every point of a line in projective space, and how to define the anharmonic ratio of any four points. We have seen also how to obtain a projective from a descriptive space. In a descriptive space, when an ideal point has a real correlative (i.e. when it is a sheaf of lines which has a vertex), we assign to the real point the coordinate which belongs to the ideal point considered as belonging to a projective space. In this way, the coordinate Geometry of the two spaces becomes very similar, the difference being that, in projective space, every real set of coordinates gives a real point, whereas, in descriptive space, this holds of each coordinate only within certain limits (both of which limits are excluded). In what follows, therefore, remarks concerning projective space will apply also to descriptive space except when the contrary is expressly stated.
Let us consider the anharmonic ratios of all ranges axby, where a, b are fixed points and x, y variable points on our line. Let α, ξ, β, η be the coordinates of these points. Then $\frac{\xi - \alpha}{\xi - \beta} / \frac{\eta - \alpha}{\eta - \beta}$ will be the anharmonic ratio of the four points, which, since a, β are constants, may be conveniently denoted by (ξη). If now ζ be the coordinate of any other point z, we have
(ξη)(ηζ) = (ξζ).Hence log (ξη) + log (ηζ) = log (ξζ).
Thus the logarithm of the anharmonic ratio in question has one of the essential properties of distance, namely additiveness. If xy, yz, xz be the distances of x, y, z taken as having sign, we must have
xy + yz = xz.
We have also log (ξξ) = 0 and log (ξη) = −log (ηξ), which are two further properties of distance. From these properties (of which the third follows from the other two) it is easy to show that all properties of distances which have no reference to the fixed points a, b belong to the logarithm in question. Hence, if the distances of points from a and b can also be made, by a suitable choice of a and b, to agree with those derived from the logarithm, we shall be able to identify distance with this logarithm. In this way—so it is contended—metrical Geometry may be wholly brought under the projective sway; for a similar theory applies to angles between lines or planes.
§ 408. In descriptive geometry¶
408. Let us consider first the case where our projective points are the ideal points of a descriptive space. Let x be considered fixed, and distinct from a and b. Let y be moved so that η becomes more and more nearly equal to β. Then as η approaches β, log (ξη) will be always finite, but will assume values exceeding any that may be assigned. This is mathematically expressed by saying that, if ξ be any number other than α and β, then log (ξβ) is infinite. (If ξ be equal to α or β, log (ξα) and log (ξβ) are indeterminate; this case will therefore be supposed excluded in what follows.) Hence a and b must be at an infinite distance from every point except each other; and their distance from each other is indeterminate. Again x and y must not be separated by a and b, i.e. y must belong to the segment axb, if we wish the distance to be real; for if ξ − α and ξ − β have the same sign, η − α and η − β must also have the same sign, but if ξ − α and ξ − β have different signs, η − α and η − β must also have different signs; and these conditions amount to the same as the condition that y must belong to the segment axb. Hence if we insist that any two real points (i.e. points which are not merely ideal) are to have a real distance (i.e. a distance measured by a number which is not complex or purely imaginary), we shall require a and b to fulfil the following conditions: (1) they must be ideal points to which no real ones correspond; (2) they must be the two limits of the series of those ideal points to which real points do correspond. These two conditions include all that has been said. For, in the first place, there is no real distance of any point from α or β; hence α and β must not be coordinates of real points. In the second place, on one of the two segments defined by a and b, there is a real distance xy however near ξ or η may approach to α or β; hence a and b are the limits of the ideal points to which real ones correspond. In the third place, it follows from the last proposition that all ideal points to which real ones correspond belong to one of the two segments ab, and all ideal points to which no real ones correspond (except a and b themselves) belong to the other of the two segments ab. When these conditions are satisfied, the function log (ξη) will have all the properties which are required for a measure of distance.
The above theory is only applicable to descriptive space, for it is only there that we have a distinction between ideal and actual points. And in descriptive space we begin with an asymmetrical transitive relation by which order is generated on the straight line. Before developing a theory which is applicable to pure projective space, let us examine a little further the above theory, which may be called the descriptive theory of distance.
In the first place, the ideal points to which real ones correspond, which for shortness I shall call proper points, form part of the whole series of ideal points, which is closed. The proper points are a semi-continuous portion of this closed series, i.e. they have all the properties of a continuum except that of having two ends. It may happen that there is only one ideal point which is not proper, or it may happen that there are many. In the former case, the one purely ideal point will be the limit of the proper points in both directions. This is the case of Euclidean space, for in Euclidean space there is only one sheaf of lines to which a given line belongs and which has no vertex, namely the sheaf of lines parallel to the given line. Hence in this case the points a and b must be taken to be identical. The function log (ξη) is then zero for all values of ξ and η, and is therefore useless as a measure of distance. But by a familiar process of proceeding to the limit, we can, in this case, obtain the value ξ − η for the distance [5]. This is the usual measure of elementary Geometry; and for the distance of two points in a plane or in space we should similarly obtain the usual formula in this case. We see here the exact meaning of the common phrase that, in Euclidean space, +∞ is the same as −∞, or that the two ends of a line coincide. The fact is, of course, that the line has no ends, but that it determines only one ideal point which is not proper, and that this is the limit of proper ideal points in both directions: when it is added to the proper ideal points, we obtain a closed continuous series of sheaves to which the line in question belongs. In this way, a somewhat cryptic expression is found to have a very simple interpretation.
But it may happen also—and this is the case of hyperbolic space—that there are many improper ideal points on a line. In this case, the proper ideal points will have two different limits; these will be the sheaves of Lobatchewsky’s parallels in the two directions. In this case, our function log (ξη) requires no modification, but expresses distance as it stands. The ideal points a and b are distinct, which is commonly expressed by saying that our line has two real and distinct points at infinity.
Thus in descriptive space, in which our coordinates are obtained by correlation with those of the derived projective space, it is always possible to define a certain function of our projective coordinates which will fulfil the conditions required for a measure of distance. These conditions may be enumerated as follows [6]. (1) Every pair of real points is to have a distance whose measure is real and finite, and vanishes only when the two points coincide. (2) If x, y, z are collinear, and y lies between x and z, the sum of the measures of xy and yz is to be the measure of xz. (3) As the ideal point corresponding to y approaches the ideal point which is the limit of proper ideal points, while x remains fixed, the absolute value of the measure of xy is to grow without limit.
It may well be asked, however, why we should desire to define a function of two variable points possessing these properties. If the mathematician replies that his only object is amusement, his procedure will be logically irreproachable, but extremely frivolous. He will, however, scarcely make this reply. We have, as a matter of fact, the notion of a stretch, and, in virtue of the general axiom that every class has some magnitude of divisibility, we know that the stretch has magnitude. But we do not know, without a special assumption to that effect, that the stretch fulfils the axioms of Archimedes and of linearity. When once these are assumed, the above properties of the measure of distance become properties which must belong to the measure of stretch. But if these two axioms are not assumed, there is no reason why there should be any magnitude having a measure possessing the above four characteristics. Thus the descriptive theory of distance, unless we regard it as purely frivolous, does not dispense with the need of the above axioms. What it does show—and this fact is extremely remarkable—is that, if stretches are numerically measurable, then they are measured by a constant multiple of the logarithm of the anharmonic ratio of the two ideal points associated with the ends of the stretch together with the two ideal points which limit the series of proper ideal points; or, in case the latter pair are identical, the stretch is measured by a function obtained as the limit of the above when the said pair approach to identity and the constant factor increases without limit. This is a most curious result, but it does not obviate the need for the axioms which distinguish metrical Geometry. The same conclusion follows as regards metrical Geometry in a plane or in three dimensions; but here new complications are introduced, which are irrelevant to the present issue, and will therefore not be discussed.
It is important to realize that the reference to two fixed ideal points, introduced by the descriptive theory of distance, has no analogue in the nature of distance or stretch itself. This reference is, in fact, a convenient device, but nothing more. The stretch, in descriptive space, is completely defined by its end-points, and in no way requires a reference to two further ideal points. And as descriptive Geometry starts with the stretch, it would be a needless complication to endeavour subsequently to obtain a definition of stretch in terms of four points. In short, even if we had a projective theory of distance in descriptive space, this would still be not purely projective, since the whole projective space composed of ideal elements is derived from axioms which do not hold in projective space.
§ 409. And in projective geometry¶
409. It remains to examine the projective theory of distance in projective space. The theory we have hitherto examined, since it used the distinction of real and ideal elements, was descriptive, not projective; we have now to examine the corresponding theory for pure projective Geometry. Here there are no ideal elements of the above sort associated with our line; if, therefore, α and β be real and distinct numbers, they will be the coordinates of real and distinct points. Hence there will he real points x, y which will be separated by a and b, and will have an imaginary measure of distance. To this there could be no objection, but for the fact that we wish our measure to be the measure of a stretch. This is the reason why it is desired that any two real points should have a real measure of distance. In order to insure this result in a pure projective space, it is necessary that α and β should not be the coordinates of points at all, but should be conjugate complex numbers. It is further necessary that the constant multiple of the logarithm should be a pure imaginary. We then find that the distance of two real points always has a real measure, which is an inverse cosine [7]. In a projective space, the condition (2) of p. 424 introduces complications, since between has not, as in descriptive space, a simple meaning. The definition of between in this case is dealt with fully by Mr Whitehead in his Universal Algebra (§206).
§ 410. Geometrical theory of imaginary point-pairs¶
410. But if such a function is to be properly geometrical, and to give a truly projective theory of distance, it will be necessary to find some geometrical entity to which our conjugate complex numbers α and β correspond. This can be done by means of involutions. Although, in a projective space, there are no ideal points, yet there are what may be called ideal point-pairs. In Chapter xlv we considered involutions with real double points: if a, b be two points on a line, all point-pairs x, $x'$ such that x, $x'$ are harmonic conjugates with respect to a, b form an involution. In this case, x and $x'$ are said to be conjugate; a and b are each self-conjugate, and are called the double points of the involution. But there are also involutions without real double points. The general definition of an involution may be given as follows (substituting the relation of x to $x'$ for the pair x, $x'$): An involution of points is a symmetrical one-one relation, other than identity, whose domain and converse domain are the same straight line, and which is such that any class of referents is projectively similar to the corresponding class of relata. Such a relation is either strictly an aliorelative, or is a self-relative as regards two and only two points, namely the double points of the involution. For every pair of distinct points on the line as double points there will be one and only one involution: all point-pairs (using this expression so as to exclude the identity of the two points of the pair) have a one-one correlation with some involutions. Thus involutions may be called ideal point-pairs: those that correspond to an actual point-pair are called hyperbolic, the others elliptic. Thus an ideal point-pair is one and indivisible, being in fact a one-one relation. Two proper ideal point-pairs have an anharmonic ratio defined by their respective double points: two improper ideal point-pairs, or a proper and an improper ideal point-pair, have an analogous projective relation, which is measured by the function obtained as above from the supposition that α and β are conjugate complex numbers. This function may be called the anharmonic ratio of the two ideal point-pairs. If one be fixed and improper, the other variable and proper, an imaginary multiple of the logarithm of the resulting anharmonic ratio has the properties required for a measure of the distance of the actual point-pair corresponding to the proper ideal point-pair. This gives the pure projective theory of distance. But to this theory, as anything more than a technical development, there are the same objections as in the case of descriptive space; i.e. unless there be some magnitude determined by every actual point-pair, there is no reason for the process by which we obtain the above measure of distance; and if there is such a magnitude, then the above process gives merely the measure, not the definition, of the magnitude in question. Thus stretch or distance remains a fundamental entity, of which the properties are such that the above method gives a measure of it, but not a definition [8].
§ 411. New projective theory of distance¶
411. There is however another and a simpler way of introducing metrical notions into a projective space, and in this way distance becomes a natural accompaniment of the introduction of coordinates. Let p, q, r be three fixed points, abc a line not passing through p or q or r but in the plane pqr. Let qr pass through a, rp through b, pq through c. Let $R_{1}$ be the relation which holds between x and y when these are points on abc, and xr, yq meet on ap; and let $R_{2}$, $R_{3}$ be similarly defined. Then a Möbius net may be regarded as constructed by repetitions of the relations $R_{1}$, $R_{2}$, $R_{3}$. We shall have, if xR$_{1}$y, yR$_{1}$z, then xH$_{ay}$x. We can define the square root of $R_{1}$, or any power of $R_{1}$ whose index is a positive or negative power of 2. Further, if s is any point of qr, and xR$_{1}$′y means that x and y are on abc and xr, ys meet on ap, then $R_{1}R_{1}$′ = $R_{1}$′$R_{1}$. From these propositions, which are proved by pure projective methods, it follows that if α and β be numbers, we may define $R_{1}^{\alpha +\beta}$ to mean $R_{1}^{\alpha}R_{1}^{\beta}$, provided $R_{1}^{\alpha}$ and $R_{1}^{\beta}$ have been already defined; whence, since $R_{1}^{2$^{n}$}$ can be defined if n is a positive or negative integer, all rational powers of $R_{1}$ can be defined, and irrational powers can be defined as limits. Hence, if x be any real number, positive or negative, we can define $R_{1}^{x}$, for we may identify $R_{1}^{-x}$ with $\breve{R}_{1}^{x}$. We may now take this relation $R_{1}^{x}$ as the distance of any two points between which it holds, and regard x as the measure of the distance. We shall find that distances so defined have the usual properties of Euclidean distances, except that the distance of a from any other point is infinite. Thus on a projective line any two points do actually have a relation which may be called distance, and in this sense a projective theory of metrical properties can be justified. But I do not know whether this method can be extended to a plane or to space.
To sum up: Although the usual so-called projective theory of distance, both in descriptive and in projective space, is purely technical, yet such spaces do necessarily possess metrical properties, which can be defined and deduced without new indefinables or indemonstrables. But metrical Geometry, as an independent subject, requires the new idea of the magnitude of divisibility of a series, which is indefinable, and does not belong, properly speaking, to pure mathematics. This idea is applied to stretches, angles, areas, etc., and it is assumed that all the magnitudes dealt with obey the axioms of Archimedes and linearity. Without these axioms, many of the usual metrical propositions cannot be proved in the usual metrical manner; with these axioms, the usual kind of elementary Geometry becomes possible, and such results as the uniqueness of the quadrilateral construction can be proved without three dimensions. Thus there is a genuinely distinct science of metrical Geometry, but, since it introduces a new indefinable, it does not belong to pure mathematics in the sense in which we have used the word in this work. It does not, as is often supposed, require distances and angles as new relations between points or lines or planes, but stretches and magnitudes of divisibility suffice throughout. On the other hand, projective and descriptive Geometry are both independent of all metrical assumptions, and allow the development of metrical properties out of themselves; hence, since these subjects belong to pure mathematics, the pure mathematician should adopt their theory of metrical matters. There is, it is true, another metrical Geometry, which does work with distances, defined as one-one relations having certain properties, and this subject is part of pure mathematics; but it is terribly complicated, and requires a bewildering number of axioms. Hence the deduction of metrical properties from the definition of a projective or descriptive space has real importance, and, in spite of appearances to the contrary, it affords, from the point of view of pure mathematics, a genuine simplification and unification of method.
Author’s Footnotes (1903 Edition)