Helmholtz. by Ndx+pox, where the parameter :p may have any value. distance of u, v from the origin, we have, for a geodesic through the This pencil generates a two-dimensional series of points, which origin, --may be regarded as a surface, and for which we may apply dp=Rads/(a?—-4), p=fR 1060 v= a tanh Gauss's formula for the measure of curvature at any point. Thus points on the surface corresponding to points in the plane Thus at every point of our manifold thercisa mcasurcof curvature on the limiting circle r =ą, are all at an infinite distance from the corresponding to cvery such pencil; but all these can be found origin. Again, considering r constant, the arc of a geodesic circle when n.n-1/2 of them are known. If figures are to be freely subtending an angle at the origin is movable, it is necessary and sufficient ihat the measure of o=Roulu (al-5) =rR sinh (0/R), curvature should be the same for all points and all directions whence the circumference of a circle of radius p is 2TR sinh (/R). at cach point. Where this is the case, it a be the measure of Again, il a be the angle between any two geodesics V-o=m(Uu), V-o=n(U-u), curvature, the linear element can be put into the form then tan a = a(n - m)wll(i+mn)a-(v-mu) (o- nu)). ds =V (Edx)/(1+1a8x*). Thus a is imaginary when u, v is outside the limiting circle, and If a be positive, space is finite, though still unbounded, and is zero when, and only when, u, v is on the limiting circle. every straight line is closed-a possibility first recognized by maximuin triangle, whose angles are all zero, is represented in the these results agree with those of Lobatchewsky and Bolyai. The Riemann. It is pointed out that, since the possible values of auxiliary plane by a triangle inscribed in the limiting circle. The a form a continuous series, observations cannot prove that our angle of parallelism is also easily obtained. The perpendicular space is strictly Euclidean. It is also regarded as possible that, to oo at a distance o from the origin is u =a tanh (8/R), and the in the infinitesimal, the measure of curvature of our space should parallel to this through the origin is 1 = y sinh (8/R). Hence 0 (). the angle which this parallel niakes with v=o, is given by be variable. tan [1(8) .sinh (8/R) = 1, or tan {[(8) =e-8/R There are four points in which this profound and epoch-makingwhich is Lobatchewsky's formula. We also obtain easily for the work is open to criticism or development-(1) the idea of a mani arca of a triangle the formula R’(T-A-B-C). fold requires more precise determination; (2) the introduction Beltrami's treatment connects two curves which, in the earlier treatment, had no connexion. These are limit-lines and curves of coordinates is entirely unexplained and the requisitc pre of constant distance from a straight line. suppositions are unanalysed; (3) the assumption that ds is the may be regarded as circles, the first having an infinite, the second an imaginary square root of a quadratic function of dx1, dx2, ... is arbitrary; radius. The equation to a circle of radius p and centre usto is (4) the idea of superposition, or congruence, is not adequately (a? - 440-vv.)? =cosh (0/Rw;w? =C2208 (say). analysed. The modern solution of these difficulties is properly | This equation remains real when p is a pure imaginary, and remains considered in connexion with the general subject of the axioms finite when wo = 0, provided o becomes infinite in such a way that of geometry, con cosh (p/R) remains finite. In the latter case the equation repre. The publication of Ricmann's dissertation was closely followed sents a limit-line. In the former case, by giving different values to C, by iwo works of Hermann von Helmholtz,' again undertaken these, obtained by putting C=0, is the straight line a -40, -, = 0. in ignorance of the work of predecessors. In these a Hence the others are cach throughout at a constant distance from proof is attempted that ds must be a rational integral this line. (It may be shown that all motions in a hyperbolic plane quadratic function of the increments of the coordinates. This consist, in a general sense, of rotations; but threc types must proof has since been shown by Lie to stand in need of correction infinity. "All points describc, accordingly, one of the three types of be distinguished according as the centre is real, imaginary or at (see VII. Axioms of Gcometry). Helmholtz's remaining works circles.) on the subjeci? are of almost exclusively philosophical interest. The above Euclidean interpretation fa:ls for three or more dimenWe shall return to them later. sions. In the Teoria fondamentale, accordingly, where a dimensions The only other writer of importance in the second period is spirit. The paper shows that Lobatchewsky's space of any number are considered, Beltrami treats hyperbolic space in a purely analytical Eugenio Beltrami, by whom Riemann's work was brought into of dimensions has, in Riemann's sense, a constant negative measure connexion with that of Lobatchewsky and Bolyai. of curvature. Beltrami starts with the formula (analogous to that of the Saggio) As he gave, by an elegant method, a convenient ds:= R?r?(dx?+dri?+dx: + ... +dx,"> Euclidean interpretation of hyperbolic plane geometry, his **+*+x: +...+x,l=?. results will be stated at some length. The Saggio shows that He shows that geodesics are represented by lincar equations beLobatchewsky's plane geometry holds in Euclidean geometry tween x1, x2, ..., xn, and that the geodesic distance p betwcen two on surfaces of constant negative curvature, straight lines being points x and x' is given by replaced by geodesics. Such surfaces are capable of a conformal al-xx':-xx'-... --Xnx' representation on a plane, by which gcodesics are represented {(a2-x-x; -... - x)(02-**-* -43) by straight lines. Hence if we take, as coordinates on the surface, (a formula practically identical with Cayley's, though obtained by the Cartesian coordinates of corresponding points on the plane, a very different method). In order to show that the measure of curvature is constant, we make the substitutions the geodesics must have linear equations. 41=, Xg=s/z...n=rda, where EX=1, Hence it follows that Hence ds? =(Radrlaz-.77)? +RudA/(0). ds? = R?w-'[(ai – v*)du: +2uvdudv+(a?--?)dve}. DA*-da? '. where we=Qi-4° -2, and -1/R2 is the measure of curvature Also calling p the geodesic distance from the origin, we have of our sursace (note that kuyas used above). The angle between two geodesics u =const., v = const. is 0, where cosh (/R) = Tanya) sinh (w/R) = Viax) Cos 0 = uv/v sla?u?) (a: ---2)], sin 0 =ow!V (al- ui)(a?-o?)]. Hence dsi= dp +(R sinh (/R))'DA!. Thus u=o is orthogonal to all geodesics v=const., and vice versa. Putting 2 = pds. 3 = pdg. ...in = plan we obtain In order that sin o may be real, w? must be positive; thus geodesics have no real intersection when the corresponding straight ds=?d'+{ cm sinh ß)-1}=(2,den-zuds.). lines intersect outside the circle uitv?=0?. When they intersect on this circle, O=o. Thus Lobatchewsky's parallels are represented Hence when p is small, we have approximately by straight lines intersecting on the circle. Again, transforming to polar coordinates u=1 cosä, v=r sin , and calling the geodesic ds? = Ed=+3R:2 (s.dza – zadz.): (1). Considering a surface element through the origin, we may choose Wiss. Abh. vol. ii. pp. 610, 618 (1866, 1868). our axes so that, for this clement, : Mind; O.S., vols. i. and iii.; Vorträge und Reden, vol. ii. pp. 1, * = 5= ... =:=0. 256. His papers are " Saggio di interpretazione della geometria non. Thus dse = d.14d2+3RI(dods.) (2). Euclidea," Giornale di matematiche, vol. vi. (1868); " Teoria fondamentale degli spazi di curvatura costante." Annali di matematica, Now the area of the triangle whose vertices are (0, 0), (2, s.), vol. ii. (1868-1869). Both were translated into French by ). Hoüel, (do. dz) is } (21. 02. ). Hence the quotient when the terms of Annales scientifiques de l'École Normale supérieure, vol. vi. (1869). the fourth order in (2) are divided by the square of this triangle is Beltrami. where coshk where a 1 to the method. 4/3R'; hence, returning to general axes, the same is the quotient question as to the best available evidence concerning the nature when the terms of the fourth order in (1) are divided by the square of this geometry is one beset with difficulties of a peculiar kind. (da, das, dzs...ds). But - f of this quotient is defined by Riemann We are obstructed by the fact that all existing physical science as the measure of curvature. Hence the measure of curvature is assumes the Euclidean hypothesis. This hypothesis has been - 1/R?, i.e. is constant and negative. The properties of parallels, involved in all actual measurements of large distances, and in all triangles, &c., arc as in the Saggio. It is also shown that the ana; the laws of astronomy and physics. The principle of simplicity radius p have constant positive curvature 1/R: sinh(/R), so that would therefore lead us, in general, where an observation conspherical geometry may be regarded as contained in the pseudo- flicted with one or more of those laws, to ascribe this anomaly, spherical (as Beltrami calls Lobatchewsky's system). not to the falsity of Euclidcan geometry, but to the falsity of the The Saggio, as we saw, gives a Euclidean interpretation laws in question. This applies especially to astronomy. On the confined to two dimensions. But a consideration of the auxiliary earth our means of measurement are many and direct, and so Traeskioa plane suggests a different interpretation, which may be long as no great accuracy is sought they involve few scientific extended to any number of dimensions. If, instead laws. Thus we acquire, from such direct measurements, a projective of referring to the pseudosphere, we merely define very high degree of probability that the space-constant, if not distance and angle, in the Euclidean plane, as those infinite, is yet large as compared with terrestrial distances. But functions of the coordinates which gave us distance and angle astronomical distances and triangles can only be measured by on the pseudosphere, we find that the geometry of our plane has means of the received laws of astronomy and optics, all of which become Lobatchewsky's. All the points of the limiting circle have been established by assuming the truth of the Euclidean are now at infinity, and points beyond it are imaginary. If we hypothesis. It therefore remains possible (until a detailed proof give our circle an imaginary radius the geometry on the plane of the contrary is forthcoming) that a large but finite spacebecomes elliptic. Replacing the circle by a sphere, we obtain constant, with different laws of astronomy and optics, would an analogous representation for three dimensions. Instead of have cqually explained the phenomena. We cannot, therefore, a circle or sphere we may take any conic or quadric. With this accept the measurements of stellar parallaxes, &c., as conclusive definition, if the fundamental quadric be Enr=0, and if Ers' evidence that the space-constant is large as compared with stellar be the polar form of Ear, the distance p between # and x' is distances. For the present, on grounds of simplicity, we may given by the projective formula righely adopt this view; but it must remain possible that, in cos (p/k) = Esz'l (229.23'.11. view of some hitherto undiscovered discrepancy, a slight correcThat this formula is projective is rendered evident by observing tion of the sort suggested might prove the simplest alternative. that criplk is the anharmonic ratio of the range consisting of But conversely, a finite parallax for very distant stars, or a the two points and the intersections of the line joining them with negative parallax for any star, could not be accepted as conclusive the fundamental quadric. With this we are brought to the third evidence that our geometry is non-Euclidean, unless it were or projective period. The method of this period is due to Cayley; shown-and this seems scarcely possible that no modification its application to previous non-Euclidean geometry is due to of astronomy or optics could account for the phenomenon. Klein. The projective method contains a generalization of dis-Thus although we may admit a probability that the spacecoveries already made by Laguerrein 1853 as regards Euclidean constant is large in comparison with stellar distances, a conclusive geometry. The arbitrariness of this procedure of deriving proof or disproof seems scarcely possible. metrical geometry from the properties of conics is removed by Finally, it is of interest to note that, though it is theoretically Lie's theory of congruence. We then arrive at the stage of possible to prove, by scientific methods, that our geometry is thought which finds its expression in the modern treatment of non-Euclidean, it is wholly impossible to prove by such methods the axioms of geometry. that it is accurately Euclidean. For the unavoidable errors of The projective method leads to a discrimination, first made observation must always leave a slight margin in our measureby Klein, of two varieties of Riemann's space; Klein calls ments. A triangle might be found whose angles were certainly these elliptic and spherical. They are also called the greater, or certainly less, than two right angles; but to prove polar and antipodal forms of elliptic space. The latter them exactly equal to two right angles must always be beyond our elliptic names will here be used. The difference is strictly powers. If, therefore, any man cherishes a hope of proving the analogous to that between the diameters and the points exact truth of Euclid, such a hope must be based, not upon of a sphere. In the polar form two straight lines in a plane scientific, but upon philosophical considerations. always intersect in one and only one point; in the antipodal BIBLIOGRAPHY:--The bibliography appended to section VII. should form they intersect always in two points, which are antipodes. be consulted in this connexion. Also, in addition to the citations According to the definition of geometry adopted in section VII. already made, the following works may be mentioned. For Lobatchewsky's writings, cf. Urkunden zur Geschichte der (Axioms of Geometry), the antipodal form is not to be termed nichteuklidischen Geometrie, i, Nikolaj Iwanovilsch Lobalschefsky, geometry,” since any pair of coplanar straight lines intersect by F. Engel and P. Stäckel (Leipzig, 1898). For John Bolyai's each other in two points. It may be called a “ quasi-geometry.” | Appendix, cf. Absolute Geometrie nach Johann Bolyai, by J. Frischauf Similarly in the antipodal form two diameters always determine (Leipzig, 1872), and also the new edition of his father's large work, a plane, but two points on a sphere do not determine a great the second volume contains the appendix. Cl. also J. Frischauf, circle when they are antipodes, and two great circles always Elemente der absoluten Geometric (Leipzig, 1876); M. L. Gérard, Sur intersect in two points. Again, a plane does not form a boundary la géométrie non-Euclidienne (thesis for doctorate) (Paris, 1892); among lines through a point: we can pass from any one such de Tilly, Essai sur les principes fondamentales de la géométrie et de la mécanique (Bordeaux, 1879); Sir R. S. Ball, “On the Theory of line to any other without passing through the plane. But a great Content," Trans. Roy. Irish Acad. vol. xxix. (1889); F. Lindemann, circle does divide the suriace of a sphere. So, in the polar form,Mechanik bei projectiver Maasbestimmung," Math. Annal, vol. a complete straight line does not divide a plane, and a plane does vii.; W. K. Clifford, " Preliminary Sketch of Biquaternions," Proc. not divide space, and does not, like a Euclidean plane, have two of Lond. Math. Soc. (1873), and Coll. Works; A. Buchheim, "On the sides. But, in the antipodal form, a plane is, in these respects, Theory of Screws in Elliptic Space," Proc. Lond. Math. Soc. vols. xv., On the Application of Quaternions and like a Euclidean plane. Grassmann's Algebra to different kinds of Uniform Space," Trans. It is explained in section VII. in what sense the metrical Camb. Phil. Soc. (1882); M. Dehn, Die Legendarischen Sätze über geometry of the material world can be considered to be deter- die Winkelsumme im Dreieck," Math. Ann. vol. 53 (1900), and minate and not a matter of arbitrary choice. Über den Rauminhalt," Math. Annal. vol. 55 (1902). For expositions of the whole subject, cf. F. Klein, Nicht-Euklidische 1 Beltrami shows also that this definition agrees with that of Gauss. Geometrie (Göttingen, 1893); R. Bonola, La Geometria non-Euclidca 3 "Sur la théorie des foyers," Nouv. Ann. vol. xii. (Bologna, 1996); P. Barbarin, La Géométrie non-Euclidienne (Paris, • Math. Annalen, iv. vi., 1871-1872. 1902); W. Killing. Die nicht-Euklidischen Raumformen in analytischer • For an investigation of these and similar properties, see White Behandlung (Leipzig, 1885). The last-named work also deals with head, Universal Algebra (Cambridge, 1898), bk. vi. ch. ii. The polar geometry of more than three dimensions; in this connexion c. also form was independently discovered by Simon Newcomb in 1877. Ĉ. Veronese, Fondamenti di geometria a più dimensioni ed a più specie 1 The two klads of space. The scientific Theories of space. di unitd rellilinee (Padua, 1891, German translation, Leipzig, of our perception of existent space from our various types of 1894); G. Fontené, L'Hyperespace d (n-1) dimensions (Paris, 1892); and A. N. Whitehead, loc. cit. Cr. also E. Study, “ Über nicht sensation. This is a question for psychology." Euklidische und Liniengeometrie,". Jahr. d. Deutsch. Moth. Ver. Definition of Abstract Geomelry.-Existent space is the subject vol. xv. (1906); W. Burnside, “On the Kinematics of non-Euclidean matter of only one of the applications of the modern science of Space," Proc. Lond. Math. Soc. vol. xxvi. (1894). A bibliography abstract geometry, viewed as a branch of pure mathematics. on the subject up to 1878. has been published by G. B. Halsted, Geometry has been defined as “the study of series of two or more Amer. Journ. of Mathvols. i. and ii.; and one up to 1900 by R. dimensions." It has also been defined as “the science of cross Bonola, Index operum ad geometriam absolutam spectantium (1902, and Leipzig, 1903). (B. A. W. R.; A. N. W.) classification.” These definitions are founded upon the actual practice of mathematicians in respect to their use of the term VII. AXIOMS OF GEOMETRY "Geometry." Either of them brings out the fact that geometry Until the discovery of the non-Euclidean geometries (Lobat- is not a science with a determinate subject matter. It is concerned chewsky, 1826 and 1829; J. Bolyai, 1832; B. Riemann, 1854), with any subject matter to which the formal axioms may apply. geometry was universally considered as being ex- Geometry is not peculiar in this respect. All branches of pure clusively the science of existent space. (See section mathematics deal merely with types of relations. Thus the VI. Non-Euclidean Geomelry.) In respect to the fundamental ideas of geometry (e.g. those of points and of science, as thus conceived, two controversies may be noticed. straight lines) are not ideas of determinate entities, but of any First, there is the controversy respecting the absolute and entities for which the axioms are true. And a set of formal relational theories of space. According to the absolute theory, geometrical axioms cannot in themselves be true or false, since which is the traditional view (held explicitly by Newton), space they are not determinate propositions, in that they do not refer has an existence, in some sense whatever it may be, independent to a determinate subject matter. The axioms are propositional of the bodies which it contains. The bodies occupy space, and functions. When a set of axioms is given, we can ask (1) it is not intrinsically unmeaning to say that any definite body whether they are consistent, (2) whether their “existence occupies this part of space, and not that part of space, without theorem” is proved, (3) whether they are independent. Axioms reference to other bodies occupying space. According to the are consistent when the contradictory of any axiom cannot be relational theory of space, of which the chief exponent was deduced from the remaining axioms. Their existence theorem Leibnitz,' space is nothing but a certain assemblage of the rela- is the proof that they are true when the fundamental ideas are tions between the various particular bodies in space. The idea of considered as denoting some determinate subject matter, so space with no bodies in it is absurd. Accordingly there can be that the axioms are developed into determinate propositions. no meaning in saying that a body is here and not there, apart It follows from the logical law of contradiction that the proof from a reference to the other bodies in the universe. Thus, on of the existence theorem proves also the consistency of the this theory, absolute motion is intrinsically unmeaning. It is axioms. This is the only method of proof of consistency. The admitted on all hands that in practice only relative motion is axioms of a set are independent of each other when no axiom directly measurable. Newton, however, maintains in the can be deduced from the remaining axioms of the set. The Principia (scholium to the 8th definition) that it is indirectly independence of a given axiom is proved by establishing the measurable by means of the effects of “ centrifugal force" as consistency of the remaining axioms of the set, together with the it occurs in the phenomena of rotation. This irrelevance of contradictory of the given axiom. The enumeration of the absolute motion (if there be such a thing) to science has led to axioms is simply the enumeration of the hypotheses (with the general adoption of the relational theory by modern men respect to the undetermined subject matter) of which some at of science, But no decisive argument for either view has at least occur in cach of the subsequent propositions. present been elaborated. Kant's view of space as being a form Any science is called a geometry" if it investigates the of perception at first sight appears to cut across this controversy. theory of the classification of a set of entities (the points) into But he, saturated as he was with the spirit of the Newtonian classes (the straight lines), such that (1) there is one and only physics, must (at least in both editions of the Critique) be classed one class which contains any given pair of the entities, and (2) with the upholders of the absolute theory. The form of per- every such class contains more than two members. In the two ception has a type of existence proper to itself independently geometries, important from their relevance to existent space, of the particular bodies which it contains. For example he axioms which secure an order of the points on any line also writes: 3 “Space does not represent any quality of objects by occur. These geometries will be called “ Projective Geometry" themselves, or objects in their relation to one another, i.e. space and “Descriptive Geometry." In projective geometry any does not represent any determination which is inherent in the two straight lines in a plane intersect, and the straight lines objects themselves, and would remain, even if all subjective are closed series which return into themselves, like the circumconditions of intuition were removed." ference of a circle. In descriptive geometry two straight lines in The second controversy is that between the view that the a plane do not necessarily intersect, and a straight line is an open axioms applicable to space are known only from experience, series without beginning or end. Ordinary Euclidean geometry and the view that in some sense thesc axioms are is a descriptive geometry; it becomes a projective geometry given a priori. Both these views, thus broadly stated, when the so-called "points at infinity” are added. are capable of various subtle modifications, and a discussion of them would merge into a general treatise on epistemology. Projective Geometry. The cruder forms of the a priori view have been made quite fundamental ideas, namely, that of a point ” and that of a Projective geometry may be developed from two undefined untenable by the modern mathematical discoveries. Geometers "straight line.” These undetermined ideas take different now profess ignorance in many respects of the exact axioms which apply to existent space, and it seems unlikely that a specific meanings for the various specific subject matters to profound study of the question should thus obliterate a priori which projective geometry can be applied. The number of the intuitions. axioms is always to some extent arbitrary, being dependent Another question irrelevant to this article, but with some upon the verbal forms of statement which are adopted. They will relevance to the above controversy, is that of the derivation *Cf. Ernst Mach, Erkenntniss und Irrtum (Leipzig); the relevant chapters are translated by T. J. McCormack, Space and Geometry 'For an analysis of Leibnitz's ideas on space, cf. B. Russell, The (London, 1906); also A. Meinong. Über die Stellung der GegensiandsPhilosopry of Leibnitz, chs. viii.-X. Theorie im System der Wissenschaften (Leipzig, 1907). : Cf. 'Hon. Bertrand Russell, Is Position in Time and Space Cl. Russell, Principles of Mathematics, $ 352 (Cambridge, 1903). Absolute or Relative?" Mind, n.s. vol. 10 (1901), and A. N. White- •Cf. A. N. Whitehead, The Axioms of Projeclive Geometry, 3 head," Mathematical Concepts of the Material World," Phil. Trans. (Cambridge, 1906). (1906), P: 205. Cl. Russell, Princ. of Math., ch. i. Cl. Critique of Pure Reason, ist section: Of Space," con- Cf. Russell , loc. cit., and G. Frege, Ober die Grundlagen der clusion A, Max Müller's translation. Geometrie," Jahresber. der Deutsch. Math. Ver. (1906). Axloms. the concurrent. be presented here as twelve in number, eight being “axioms are said to be projectively related. Any property of a plane fgure of classification," and four being “ axioms of order." which necessarily also belongs to any projectively related figure, is called a projective property. Arioms of Classification. The eight axioms of classification The following theorem, known from its importance as are as follows: fundamental theorem of projective geometry," cannot be proved 1. Points form a class of entities with at least two members. from axioms 1.8. The enunciation is: “A projective correspond2. Any straight line is a class of points containing at least ence between the points on two straight lines is completely deter mined when the correspondents of three distinct points on one line three members. are determined on the other." This theorem is equivalent 3. Any two distinct points lie in one and only one straight (assuming axioms 1-8) to another theorem, known as Pappus's line. Thcorem, namely: "If I and l' are two distinct coplanar lines, and 4. There is at lea one straight line which does not contain points on ?, then the three points of intersection of AA' and B'c, A, B, C are three distinct points on l, and A', B', C'are three distinct all the points. of A'B and CC', of BB' and C'A, are collinear." This theorem is 5. If A, B, C are non-collinear points, and A' is on the straight obviously Pascal's well-known theorem respecting a hexagon line BC, and B' is on the straight line CA, then the straight lines inscribed in a conic, for the special case when the conic has deAA' and BB' possess a point in common. generated into the two lines 1 and l'. Another theorem also plane ABC is the class of points lying on the straight lines joining Band B”, C and"C', are such that the three pairs of opposite sides Definition:- A, B, C are any three non-collinear points, the squivalent (assuming axioms 1-8) to the fundamental theorem is A with the various points on the straight line BC. 6. There is at least one plane which does not contain all the pair through A and A' respectively, and so on, and is also the three points. sides of the quadrangle which pass through A, B, and C, are con7. There exists a plane a, and a point A not incident in a, current in one of the corners of the quadrangle, then another quadsuch that any point lies in some straight line which contains rangle can be found with the same relation to the three pairs of points, both A and a point in a. except that its thrce sides which pass through A, B, and C, are not Definition. -Harm. (ABCD) symbolizes the following conjoint statements: (1) that the points A, B, C, D are collinear, and (2) axiom, all the theorems of projective geometry which do not require Thus, if we choose to take any one of these three theorems as an that a quadrilateral can be found with one pair of opposite sides ordinal or metrical ideas for their enunciation can be proved. Also intersorting at A, with the other pair intersecting at C, and with its diagon Is passing through B and D respectively. Then B and D are construction, based upon Pascal's theorem, for points on the conic a conic can be defined as the locus of the points found by the usual said to be" harmonic conjugates" with respect to A and C. through five given points. But it is unnecessary to assume here 8. Harm. (ABCD) implies that B and D are distinct points. any one of the suggested axioms; for the fundamental theorem can In the above axioms 4 secures at least two dimensions, axiom be deduced from the axioms of order together with axioms 1-8. 5 is the fundamental axiom of the plane, axiom 6 secures at Axioms of Order. It is possible to define (cf. Pieri, loc. cit.) least three dimensions, and axiom 7 secures at most three the property upon which the order of points on a straight line dimensions. From axioms 1-5 it can be proved that any two depends. But to secure that this properly does in fact range distinct points in a straight line determine that line, that any the points in a serial order, some axioms are required. A straight three non-collinear points in a plane determine that plane, that line is to be a closed series; thus, when the points are in order, the straight line containing any two points in a plane lies wholly it requires two points on the line to divide it into two distinct in that plane, and that any two straight lines in a plane intersect. complementary segments, which do not overlap, and together From axioms 1-6 Desargues's well-known theorem on triangles form the whole line. Accordingly the problem of the definition in perspective can be proved. of order reduces itself to the definition of these two segments The enunciation of this theorem is as follows: If ABC and formed by any two points on the line; and the axioms are A'B'C' are two coplanar triangles such that the lines AA', BB', CC' are concurrent, then the three points of intersection of BC and stated relatively to these segments. B'C' of CA and C'A', and of AB and A'B' are collincar; and Definition.-II A, B, C are three collinear points, the points on the conversely if the three points of intersection are collinear, the three segment ABC are defined to be those points such as X, for which The proof which can be applied is the usual (AYCY') and Harm. (BYXY') both hold. The supplementary there èxist two points Y and Y', with the property that Harm. projective proof by which a third triangle A'B'C' is constructed not coplanar with the other two, but in perspective with each segment ABC is defined to be the rest of the points on the line. of them. It has been proved that Desargues's theorem cannot be deduced geometrical ideas, if B and X are any two points between A and C, from axioms 1-5, that is, is the geometry be confined to two then the two pairs of points, A and C, B and X, define an involution dimensions. All the proofs proceed by the method of producing a with real double points, namely, the Y and Y' of the above definition. The property of belonging to a segment ABC is projective, since specification of " points straight lines which satisfies the harmonic relation is projective. axioms 1-5, and such that Desargues's theorem does not hold. It follows from axioms 1-5 that Harm. (ABCD) implies Harm. The first three axioms of order (cf. Pieri, loc. cil.) are: (ADCB) and Harm. (CBAD), and that, if A, B, C be any three 9. If A, B, C are three distinct collinear points, the suppledistinct collincar points, there exists at least one point D such that Harm. (ABCD). ' But it requires Desargues's theorem, and hence mentary segment ABC is contained within the segment BCA. axiom 6, to prove that Harm. (ABCD) and Harm. (ABCD') imply 10. If A, B, C are three distinct collinear points, the common the identity of D and D'. part of the segments BCA and CAB is contained in the suppleThe necessity for axiom 8 has been proved by G. Fano, who mentary segment ABC. has produced a three dimensional geometry of fifteen points, 11. II A, B, C are three distinct collinear points, and D lies i.e. a method of cross classification of fifteen entities, in which in the segment ABC, then the segment ADC is contained each straight line contains three points, and each plane contains within the segment ABC. seven straight lines. In this geometry axiom 8 does not hold. From these axioms all the usual properties of a closed order Also from axioms 1-6 and 8 it follows that Harm. (ABCD) follow. It will be noticed that, if A, B, C are any three collinear implies Harm. (BCDA). points, C is necessarily traversed in passing from A to B by one Definitions.--When two plane figures can be derived from one route along the line, and is not traversed in passing from A to B another by a single projection, they are said to be in perspective. along the other route. Thus there is no meaning, as referred When two plane figures can be derived one from the other by a finite series of perspective relations between intermediate figures, they !o closed straight lines, in the simple statement that C lies between A and B. But there may be a relation of separation *This formulation-though not in respect to number-is in a!! between two pairs of collinear points, such as A and C, and essentials that of M. Pieri, cf. " ! principii della Geometria di Posizione." Accad. R. di Torino (1898); also cf. Whitehcad, loc. cil. B and D. The couple B and D is said to separate A and C, if : C1. G. Peano, "Sui fondamenti della Geometria," p. 73, Rivisla • Cf. Hilbert, loc. cit.; for a fuller exposition of Hilbert's proof di matemalica, vol. iv. (1894), and D. Hilbert, Grundlagen der Geo- c. K. T. Vahlen, Abstrakte Geometrie (Leipzig, 1905), also Whitehead, melric (Leipzig, 1899); and R. F. Moulton. " A Simple non-Desar- loc. cit. guesian Plane Geometry," Trans. Amer. Math. Soc., vol. iii. (1902). Cl. H. Wiener, Jahresber. der Deutsch. Math. Ver. vol. i. (1890); Cl. "Sui postulati fondamentali della geometria projettiva, and F. Schur, "Uber den Fundamentalsatz der projectiven GeoGiorn. di malemalica, vol. xxx. (1891); also of Pieri, loc. cil., and metrie,” Math. Ann. vol. li. (1899). Whitehead, loc. cit. •Cl. Hilbert, loc. cit., and Whitehead, loc. cit. lines are concurrent. and А The pro SR: O EP the four points are collinear and D lies in the segment comple- 1 relations, however m, n, Sand S be varied, are called "prosper mentary to the segment ABC. The property of the separation tivities and is the double point of the prospectivity. il a point of pairs of points by pairs of points is projective. Also it can be which (1) have the same double point O on l is related to A by a prospectivity, then all prospectivities, proved that Harm. (ABCD) implies that B and D separate U, and (2) relate O to A, give the same A and C. correspondent (Q. in figure) to any or closed, is said to be compact, if the series contains no immediately m, n, S, and smay have been varied Definitions.-A series of entities arranged in a serial order, open point P on the line b; in fact they are Such entity to any other entity it is necessary to pass through entities subject to these conditions. It was the mcrit of R. Dedekind and of a prospectivity will be denoted by G. Cantor explicitly to formulate another fundamental property of rou). series. The Dedekind property' as applied to an open series can The sum of two prospectivities, be defined thus: An open series possesses the Dedekind property, written (OAUP) +(OBU), is defined Fig. 69. if, however, it be divided into two mutually exclusive classes and i into itself which is obtained by first applying the prospectivity to be that transformation of the line such that every member of u precedes in the serial order every (OAU2) and then applying the prospectivity (OBU). member of v, there is always a member of the series, belonging to one transformation, when the two summands have the same double of the two, u or y, which precedes every member of v (other than point, is itself a prospectivity with that double point. With this definition of addition it can be proved that prospecitself if it belong to v), and also succeeds every member of u (other than itself if it belong to u). Accordingly in an open series with the tivities with the same double point satisfy all the axioms of magDedekind property there is always a member of the series marking nitude. Accordingly they can be associated in a one-one corre the junction of two classes such as u and v. An open series is con spondence with the positive and negative real numbers. Let E tinuous if it is compact and possesses the Dedekind property. A (fig. 70) be any point on l, distinct from Q and U. Then the closed series can always be transformed into an open series by taking prospectivity (OEU2) is associated with unity, the prospectivity any arbitrary member as the first term and by taking one of the two ool', is associated with zero, ways round as the ascending order of the series. Thus the definitions and OWU?) with co. of compactness and of the Dedekind property can be at once trans- spectivities of the type (OPU?). ferred to a closed series. where P is any point on the seg. ment OEU, correspond to the posi12. The last axiom of order is that there exists at least one tive numbers; also if P' is the straight line for which the point order possesses the Dedekind harmonic conjugate of p with property. respect to and U, the prospecIt follows from axioms 1-12 by projection that the Dedekind tivity (OP'U?) is associated with property is true for all lines. Again the harmonic syslem ABC, (The subjoined hgure explains this the corresponding negative number.? where A, B, C are collinear points, is defined ? thus: take the relation of the positive and nega Fig. 70. harmonic conjugates A', B', C' of each point with respect to tive prospectivities.) Then any the other two, again take the harmonic conjugates of each of point pont is associated with the same number as is the prospecthe six points A, B, C, A', B', C' with respect to each pair of the tivity (OPU). It can be proved that the order of the numbers in algebraic order remaining five, and proceed in this way by an unending series of magnitude agrees with the order on the line of the associated of steps. The set of points thus obtained is called the harmonic points. Let the numbers, assigned according to the preceding system ABC. It can be proved that a harmonic system is specification, be said to be associated with the points according to compact, and that every segment of the line containing it the “ numeration-system (OEU)." The introduction of a coordinate system for a plane is now managed possesses members of it. Furthermore, it is easy as follows: Take any triangle OUV the fundamental theorem holds for harmonic systems, in the in the plane, and on the lines OU sense that, if A, B, C are three points on a line l, and A', B', C and OV, establish the numeration are three points on a line l', and if by any two distinct series systems (OE,U) and (OEV), where of projections A, B, C are projected into A’, B', C', then any point | Then (cf. fig. 71) if M and N are of the harmonic system ABC corresponds to the same point of associated with the numbers x and the harmonic system A'B'C' according to both the projective y according to these systems, the relations which are thus established between 1 and l'. It now coordinates of P are x and y. It then FIG. 71. follows that the equation of a straight follows immediately that the fundamental theorem must hold for line is of the form ax +by+c=0. Both coordinates of any point on all the points on the lines l and I', since (as has been pointed out) the line UV are infinite. This can be avoided by introducing harmonic systems are “everywhere dense" on their containing homogeneous coordinates X, Y, Z, where x-X/2, and y=Y/2, and lines. Thus the fundamental theorem follows from the axioms 2=o is the equation of UV. The procedure for three dimensions is similar. Let OUVW of order. (fig. 72) be any tetrahedron, and associate points on OU, OV, OW A system of numerical coordinates can now be introduced, with numbers according to the numera possessing the property that linear equations represent planes tion systems (OE,y), (OEV), and and straight lines. The outline of the argument by which this coe:W). Let the planes VWP, 'WUP, remarkable problem in that " distance " is as yet undefined) is tively; and let x, y, z be the numbers solved, will now be given. It is first proved that the points on associated with L, M, N respectively. any line can in a certain way be definitely associated with all Then P is the point (x, y, b). the positive and negative real numbers, so as to form with them homogeneous coordinates can be ina one-one correspondence. The arbitrary elements in the produced as before, thus avoiding the infinities on the planc UVW. establishment of this relation are the points on the line associated The cross ratio of a range of four with o, 1 and co. collinear points can now be defined This association' is most easily effected by considering a as a number characteristic of that range. Let the coordinates of any class of projective relations of the line with itself, called hy point P. of the range P. P. P. P.be 4a+w+8% 1,6+ub 1,C+uc'. (p= 1, 2, 3, 4) F. Schur (loc. cit.) prospectivities. Atur + Hp Xter Let I (fig. 69) be the given line, m and n any two lines intersecting and let thek) be written for Arte - Metro Then the cross ratio at U on?, S and S' two points on n. Then a projective relation | P, P, P. P. is defined to be the number (A.per) (gr.)/(A) (Ass); between l'and itself is formed by projecting ! from S on to m, and The equality of the cross ratios of the ranges (P. P, P, ) and then by projecting m from S' back on to l. All such projective (Q & Q. Q) is proved to be the necessary and sufficient condition 'Cf. Dedekind, Steligkeit und irrationale Zahlen (1872). for their mutual projectivity: The cross ratios of all harmonic ? Cf. v. Staudt, Geometrie der Lage (1847). ranges are then easily seen to be all cqual to - I, by comparing with * Cf. Pasch, Vorlesungen über neuere Geometrie (Leipzig, 1882), a the range (OE,UE') on the axis of x. classic work: also Fiedler, Die darstellende Geometrie (Ist ed., 1871, Thus all the ordinary propositions of geometry in which distance 3rd ed., 1888); Clebsch. Vorlesungen über Geometrie, vol. i.; and angular measure do not enter otherwise than in cross ratios Hilbert, loc. cit.: F. Schur, Math. Ann. Bd. Iv. (1902); Vahlen can now be enunciated and proved. Accordingly the greater part of loc. cit.; Whitehead, loc. cil. the analytical theory of conics and quadrics belongs to geometry prove that E, M M W Also E, FIG. 72. |