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at this stage The theory of distance will be considered after the 10. The Dedekind property holds for the order of the points principles of descriptive geometry have been developed.

on any straight line.

It follows from axioms 1-9 that the points on any straight line Descriplive Geomelry.

are arranged in an open serial order. Also all the ordinary Descriptive geometry is essentially the science of multiple theorems respecting a point dividing a straight line into two order for open series. The first satisfactory system of axioms parts, a straight line dividing a plane into two parts, and a plane was given by M. Pasch.' An improved version is due to G. dividing space into two parts, follow. Peano. Both these authors treat the idea of the class of points Again, in any plane a consider a line l and a point A (fig. 74). constituting the segment lying belween two points as an undefined Let any point B divide ! into two half-lines li and h. Then it can fundamental idea. Thus in fact there are in this system two secting h (such as m), are bounded by two hall-lines, of which ABC

be proved that the set of half-lines, emanating from A and interfundamental ideas, namely, of points and of segments. It is is one. Let be the other. Then it can be proved that does not then easy enough to define the prolongations of the segments, intersect b. Similarly for the half-line, so as to form the complete straight lines. D. Hilbert's" formulas such as n, intersecting hz. Let s be its tion of the axioms is in this respect practically based on the same possible. (1). The half-lines r and s are fundamental ideas. His work is justly famous for some of the collinear, and together form one com, mathematical investigations contained in it, but his exposition of plete linc. In this case, there is one and the axioms is distinctly inferior to that of Peano. Descriptive only one line (viz. ?+s) through A and geometry can also be considered as the science of a class of lying in a which does not intersect l.

This is the Euclidean case, and the relations, each relation being a two-termed serial relation, as assumption that this case holds is the considered in the logic of relations, ranging the points between Euclidean parallel axiom. But (2) the which it holds into a linear open order. Thus the relations are

hall-lines i and s may not be collinear.

In this case there will be an infinite the straight lines, and the terms between which they hold arc

number of lines, such as k for instance, containing A and lying in a the points. But a combination of these two points of view which do not intersect l. Then the lines through A in a are divided yields 'the simplest statement of all. Descriptive geometry is into two classes by reference to I, nainely, the secant lines which then conceived as the investigation of an undefined fundamental intersect l, and the non-secant lines which do not intersect l. The relation between three terms (points); and when the relation halves, may be called the two parallels to l through A.

two boundary non-secant lines, of which , and s are respectively holds between three points A, B, C, the points are said to be" in The perception of the possibility of case 2 constituted the startingthe (linear) order ABC.”

point from which Lobatchewsky constructed the first explicit 0. Veblen's axioms and definitions, slightly modified, are as coherent theory of non-Euclidean geometry, and thus created a follows:

revolution in the philosophy of the subject. For many centuries 1. If the points A, B, C are in the order ABC, they are in the were almost confined to hopeless attempts to prove the " parallel

the speculations of mathematicians on the foundations of geometry order CBA.

axiom" without the introduction of some equivalent axiom. 2. If the points A, B, C are in the order ABC, they are not Associated Projective and Descriptive Spaces.-A region of a in the order BCA.

projective space, such that one, and only one, of the two supple3. If the points A, B, C are in the order ABC, A is distinct mentary segments between any pair of points within it lies from C.

entirely within it, satisfies the above axioms (1-10) of descriptive 4. If A and B are any two distinct points, therc exists a point geometry, where the points of the region are the descriptive. C such that A, B, C are in the order ABC.

points, and the portions of straight lines within the region are Definition. The line AB (A + B) consists of A and B, and of all the descriptive lines. If the excluded part of the original pro. points X in one of the possible orders, ABX, AXB, XAB.

jective space is a single plane, the Euclidean parallel axiom also points X in the order AXB constitute the segment AB.

holds, otherwise it docs not hold for the descriptive space of the 5. If points C and D (C#D) lie on the line AB, then A lies on limited region. Again, conversely, starting from an original the line CD.

descriptive space an associated projective space can be con6. There exist three distinct points A, B, C not in any of the structed by means of the concept of ideal points. These are also orders ABC, BCA, CAB.

called projective points, where it is understood that the simple 7. If three distinct points A, B, C (fig. 73) do not lic on the points are the points of the original descriptive space. same line, and D and E are two distinct points in the orders ideal point is the class of straight lines which is composed of two

BCD and CEA, then a point Fexists coplanar lines a and b, together with the lines of intersection of in the order AFB, and such that all pairs of intersecting planes which respectively contain a and b, D, E, F are collinear.

together with the lines of intersection with the plane ab of all Definition.-11. A, B, C are three planes containing any one of the lines (other than a or b) already is the class

of points which lie on any specified as belonging to the ideal point. It is evident that, if one of the lines joining any two of the the two original lincs a and b intersect, the corresponding ideal points belonging to the boundary of point is nothing else than the whole class of lines which are The triangle ABC, the boundary being concurrent at the point ab. But the essence of the definition is

formed by the segments BC, CA and AB. The interior of the triangle ABC is formed by the points in

that an ideal point has an existence when the lines a and b do segments such as PQ. where P and Q are points respectively on

not intersect, so long as they are coplanar. An ideal point is twc of the segments BC, CA, AB.

termed proper, if the lines composing it intersect; otherwise it 8. There exists a plane ABC, which does not contain all the | is improper. points.

A theorem essential to the whole theory is the following: if Definition.- A, B, C, D are four non-coplanar points, the space any two of the three lines a, b, c are coplanar, but the three lines ABCD is the class of points which lie on any of the lines containing are not all coplanar, and similarly for the lines a, b, d, then c two points on the surface of the tetrahedron ABCD, the surface and d are coplanar. It follows that any two lines belonging to an being formed by the interiors of the triangles ABC, BCD, DCA, ideal point can be used as the pair of guiding lines in the definition.' DAB.

An ideal point is said to be coherent with a plane, if any of the 9. There exists a space ABCD which contains all the points. lines composing it lie in the plane. An ideal line is the class of

ideal points cach of which is coherent with two given planes. CI. I Principii di geometria (Turin, 1889) and "Sui fondamenti CA. P. Stäckel and F. Engel, Die Theorie der Parallellinien von della geometria." Rivisia di mal. vol. iv. (1894).

Euklid bis auf Gauss (Leipzig, 1895). 1 C1, loc. cil.

C Pasch, loc. cil., and R. Bonola, "Sulla introduzione degli CI. Vailati, Rivista di mot. vol. iv. and Russell, loc. cil. $ 376. cnti improprii in geometria projettive," Giorn. di mal. vol. xxxviii. (f. O. Veblen, " On the Projective Axioms of Geometry," (1900); and Whitehead, Axioms of Descriptive Geometry (Cambridge, Trans. Amer. Math. Soc. vol. ii. (1902).




FIG. 73.

ici. loc. cit.

If the planes intersect, the ideal line is termed proper, otherwise cessive displacements of a rigid body from position A to position it is improper. It can be proved that any two planes, with which | B, and from position B to position C, are the same in effect as one any iwo of the ideal points are both coherent, will serve as the displacement from A to C. But this is the characteristic "group" guiding planes used in the definition. The ideal planes are property. Thus the transformations of space into itself defined defined as in projective geometry, and all the other definitions by displacements of rigid bodies form a group. (for segments, order, &c.) of projective geometry are applied Call this group of transformations a congruence-group. Now to the ideal elements. If an ideal plane contains some proper according to Lie a congruence-group is defined by the following ideal points, it is called proper, otherwise it is improper. Every characteristics: ideal plane contains some improper ideal points.

1. A congruence-group is a finite continuous group of one-one It can now be proved that all the axioms of projective geometry transformations, containing the identical transformation. hold of the ideal elements as thus obtained; and also that the 2. It is a sub-group of the general projective group, i.e. of order of the ideal points as obtained by the projective method the group of which any transformation converts planes into agrees with the order of the proper ideal points as obtained from planes, and straight lines into straight lines. that of the associated points of the descriptive geometry. Thus 3. An infinitesimal transformation can always be found satisa projective space has bcen constructed out of the ideal elements, fying the condition that, at least throughout a certain enclosed and the proper ideal elements correspond element by element with region, any definite line and any definite point on the line are the associated descriptive elements. Thus the proper ideal latent, i.e. correspond to themselves. clements form a region in the projective space within which the 4. No infinitesimal transformation of the group exists, such descriptive axioms hold. Accordingly, by substituting ideal that, at least in the region for which (3) holds, a straight line, elements, a descriptive space can always be considered as a a point on it, and a plane through it, shall all be latent. region within a projective space. This is the justification for the The property enunciated by conditions (3) and (4), taken ordinary use of the points at infinity "in the ordinary Euclidean together, is named by Lie“ Frce mobility in the infinitesimal." geometry; the reasoning has been transferred from the original Lie proves the following theorems for a projective space:

riptive space to the associated projective space of ideal 1. If the above four conditions are only satisfied by a group elements; and with the Euclidean parallel axiom the improper throughout part of projective space, this part either (a) must be the ideal elements reduce to the ideal points on a single improper ideal region enclosed by a rcal closcd quadric, or (B) must be the whole of

the projective space with the exception of a single plane. In case plane, namely, the plane at infinity.'

(a) the corresponding congruence group is the continuous group for Congruence and Measurement.-- The property of physical space which the enclosing quadric is latent; and in case (3) an imaginary which is expressed by the term “measurability " has now to be conic (with a real equation) lying in the latent plane is also latent, considered. This property has often been considered as essential and the congruence group is the continuous group for which the !o the very idea of space. For example, Kant writes,? " Space plane and conic are lateni

2. If the above four conditions are satisfied by a group throughout is represented as an infinite given quanlity.” This quantitative the whole of projective space, the congruence group is the continuous aspect of space arises from the measurability of distances, of group for which some imaginary quadric (with a real equation) is angles, of surfaces and of volumes. These four types of quantity latent. depend upon the two first among them as fundamental. The of any quadrics of the types considered, either in theorem 1(a), or in

By a proper choice of non-homogenous co-ordinates the equation measurability of space is essentially connected with the idea of theorem 2, can be written in the form 1+c(x2 + y2 + 3 ) =0, wherec is congruence, of which the simplest examples are to be found in negative for a real closed quadric, and positive for an imaginary the proofs of equality by the method of superposition, as used quadric. Then the general infinitesimal transformation is defined in elementary plane geometry. The mere concepts of “

by the three equations:

part and of " whole" must of necessity be inadequate as the founda

dx/d!=u-w.y+wa+cx(ux +vy+w2).)

(A) tion of measurement, since we require the comparison as to

da di = w-wx+w.y+c:(ux +vy+ws). ) quantity of regions of space which have no portions in common. In the case considered in theorem 1 (B), with the proper choice of The idea of congruence, as exemplified by the method of super-co-ordinates the three equations defining the general infinitesimal position in geometrical reasoning, appears to be founded upon

dx/dl=u-y+wap, that of the “rigid body," which moves from one position to

dydi = v^,: +wyx, (B) another with its internal spatial relations unchanged. But unless

dz/di = w-wx twiy. there is a previous concept of the metrical relations between the In this case the latent planc is the plane for which at least one of parts of the body, there can be no basis from which to deducet.. y2 are infinite, that is, the plane 0.2 to.y+0.2ta=0; and the

latent conic is the conic in which the cone x? +78 +22= o intersects that they are unchanged.

the latent planc. It would therefore appear as if the idea of the congruence, or It follows from theorems 1 and 2 that there is not one unique metrical cquality, of two portions of space (as empirically sug. congruencc-group, but an indefinite number of them. There is gested by the motion of rigid bodies) must be considered as a

one congruence-group corresponding to cach closed rcal quadric, fundamental idea incapable of definition in terms of those one to cach imaginary quadric with a real equation, and one to geometrical concepts which have already been enumerated. cach imaginary conic in a real plane and with a real equation. This was in effect ihe point of view of Pasch. It has, however, The quadric Thus associated with cach congruencc-group is been proved by Sophus Lies that congruence is capable of called the absolute for that group, and in the degenerate case definition without recourse to a new fundamental idea. This of 1 (B) the absolute is the latent plane together with i he latent he does by means of his theory of finito continuous groups (see imaginary conic. If the absolute is real, the congruence-group GROUPS, THEORY OF), of which the definition is possible in terms is hyperbolic; if imaginary, it is elliptic; is the absolute is a of our established geometrical idcas, remembering that co- plane and imaginary conic, the group is parabolic. Metrical ordinates have already been introduced. The displacement geometry is simply the theory of the properties of suine particular of a rigid body is simply a mode of defining to the senses a one

congruence-group selected for study. one transformation of all space into itself. For at any point of The definition of distance is connected with the corresponding space a particle may be conceived to be placed, and to be rigidly congruencc-group by two considerations in respect to a range of five connected with the rigid body; and thus there is a definite points (A1, A2, P. Pz, Ps), of which A, and Az are on the absolute. correspondence of any point of space with the new point occupied

Let IAP,AzPzl stand for the cross ratio (as defined above) of the

range (AIP,AgP), with a similar nolation for the other ranges, by the associated particle after displacement. Again two suc- Then

* The original idea (confined to this particular case) of ideal (1), log(A.P.A.Px} + log|A, P2A_P:) = log(A.P.A.Ps). points is due to von Staudt (loc. cil.).

and * C1. Critique, " Trans. Aesth." Sect. 1.

(2), if the points A1, A2, P1, P, are transformed into A1, A's, P. P. * Cl. loc. cit.

by any transformation of the congruence-group. (a) 1A,PAPA = Cf. Über die Grundlagen der Geometrie (Leipzig, Ber., 1890); A'P' A',Pzl. since the transformation is projective, and (B) AA, and Theorie der Transformationsgruppen (Leipzig, 1893), vol. iii. are on the absolute since A, and Ag are on it. . Thus if we define

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transformation are:

the distance P.P, to be k log (A,P,A,P2), where A, and A, are the

we have, in fact, presented to our senses a definite set of trans. points in which the line PiPo cuts the absolute, and k is some con- formations forming a congruence-group, resulting in a set of stant, the two characteristic properties of distance, namely, (1) the addition of consecutive lengths on a straight line, and (2) the in

measure relations which are in no respect arbitrary. Accordingly variability of distances during a transformation of the congruence- our scientific laws are to be stated relevantly to that particular group, are satisfied. This is the well-known Cayley-klein projective congruence-group. Thus the investigation of the type (elliptic, definition of distance, which was elaborated in view of the addition hyperbolic or parabolic) of this special congruence-group is a gruence-groups. For a hyperbolic group when P, and P, are in the perfectly definite problem, to be decided by experiment. The region enclosed by the absolute, log|A, P,A,P;} is real, and therefore consideration of experiments adapted to this object requires some k must be real. For an elliptic group A, and A3 are conjugate development of non-Euclidean geometry (see section VI., imaginaries, and log. (A.PLA»P} is a pure imaginary, and k is chosen Non-Euclidean Geometry). But if the doctrine means that, to be xlı, where a is real and .=v-:

Similarly the angle between two planes, pi and P:, is defined to be assuming some sort of objective reality for the material universe, (1/21) log (patzpa), where 1 and 1 are tangent planes to the absolute beings can be imagined, to whom either all congruence-groups through the line podz. The planes 4 and hare imaginary for an are equally important,or some other congruence-group is specially elliptic group, and also for an hyperbolic group when the planes A important, the doctrine appears to be an immediate deduction The development of the consequences of these metrical definitions from the mathematical facts. Assuming a definite congruenceis the subjct of non-Euclidean geometry.

group, the investigation of surfaces (or three-dimensional loci The defnitions for the parabolic case can be arrived at as limits in space of four dimensions) with geodesic geometrics of the form of those obtained in either of the other two cases by, making, k of metrical geometries of other types of congruence-groups forms ultimately to vanish. It is also obvious that, if P, and P, be the points (xi, yu, 21) and (x2. Ya, 22), it follows from equations (B) above

an important chapter of non-Euclidean geometry. Arising that }(x2-x.)+(9:- y)2+(21-9); } is unaltered by a congruence from this investigation there is a widely-spread fallacy, which transformation and also satishes the addition property for collinear has found its way into many philosophic writings, namely, that distances. Also the previous definition of an angle can be adapted the possibility of the geometry of existent three-dimensional the line pipa to the imaginary conic. Similarly it pi and p are inter-space being other than Euclidean depends on the physical secting lines, the same definition of an angle holds, where is and lz existence of Euclidean space of four or more dimensions. The are now the lines from the point pipe to the two points where the foregoing exposition shows the baselessness of this idea. plane pipe cuts the imaginary conic. These points are in fact the circular points at infinity on the plane. The development of

BIBLIOGRAPHY.-For an account of the investigations on the the consequences of these definitions for the parabolic case gives the axioms of geometry during the Greek period, see M. Cantor, Vor. ordinary Euclidean metrical geometry.

lesungen uber die Geschichte der Mathematik, Bd. i. and ii.; T. L. Thus the only metrical geometry for the whole of projective Heath, The Thirteen Books of Euclid's Elements, a New Translation space is of the elliptic type. But the actual measure-relations | Critical, and Explanatory (Cambridge, 1908)-this work is the standard (though not their general properties) difíer according to the source of information; W. B. Frankland, Euclid, Book I., with a elliptic congruence-group selected for study. In a descriptive Commentary (Cambridge, 1905)—the commentary contains copious space a congruence-group should possess the four characteristics substantive importance is that of the 18th century. The leading

The next period of really. of such a group throughout the whole of the space. Then form

authors are: G. Saccheri, S.J., Euclides ab omni naevo vindicatus the associated ideal projective space. The associated congruence-(Milan, 1733). Saccheri was an Italian Jesuit who unconsciously group for this ideal space must satisfy the four conditions discovered non-Euclidean geometry in the course of his efforts to throughout the region of the proper ideal points. Thus the prove its impossibility. . H. Lambert, Theorie der Parallellinien boundary of this region is the absolute. Accordingly there can

(1766); A. M. Legendre, Elements de géométrie (1794). An adequate

account of the above authors is given by P. Ståckel and F. Engel, be no metrical geometry for the whole of a descriptive space Die Theorle der Parallellinien von Euklid bis auf Gauss (Leipzig, unless its boundary (in the associated ideal space) a closed 1895). The next period of time (roughly from 1800 to 1879) contains quadric or a plane. If the boundary is a closed quadric, there two streams of thought, both of which are essential to the modera

analysis of the subject. The first stream is that which produced the is one possible congruence-group of the hyperbolic type. If

discovery and investigation of non-Euclidean geometries, the second the boundary is a plane (the plane at infinity), the possible stream is that which has produced the geometry of position, comcongruence-groups are parabolic; and there is a congruence- prising both projective and descriptive geometry not very accurately group corresponding to each imaginary conic in this plane, are K. F. Gauss, in private letters to Schumacher, cf. Stackel and

discriminated. The leading authors on non-Euclidean geometry together with a Euclidean metrical geometry corresponding to Engel, loc. cit.; N. Lobatchewsky, rector of the university of Kazan, each such group. Owing to these alternative possibilities, it to whom the honour of the effective discovery of non-Euclidean would appear to be more accurate to say that systems of quantities geometry must be assigned. His first publication was at Kazan can be found in a space, rather than that space is a quantity.

in 1826. His various memoirs have been re-edited by Engel;

cf. Urkunden zur Geschichte der nichteuklidischen Geometrie by Lie has also deduced” the same results with respect to con

Stäckel and Engel, vol. i. "Lobatchewsky.". J. Bolyai discovered gruence-groups from another set of defining properties, which non-Euclidean geometry "apparently in independence of Lobatexplicitly assume the existence of a quantitative relation (the chewsky. His memoir, was published in 1839 as an appendix to a distance) between any two points, which is invariant for any work by his father W. Bolyai, Tentamen juventutem.. transformation of the congruence-group.

memoir has been separately edited by J. Frischauf, Absolute Geometrie

nach J. Bolyai (Leipzig, 1872); B. Riemann, Über die Hypothesen, The above results, in respect to congruence and metrical welche der Geometrie zu "Grunde liegen (1854): cf. Gesamle Werke, a geometry, considered in relation to existent space, have led to the translation in The Collected Papers of W. K. Clifford. This is a doctrine that it is intrinsically unmeaning to ask which system fundamental memoir on the subject and must rank with the work of of metrical geometry is true of the physical world. Any one of Lobatchewsky. Riemann discovered elliptic metrical geometry,

and Lobatchewsky hyperbolic geometry. 'A full account of Ricthese systems can be applied, and in an indefinite number of ways.

mann's ideas, with the subsequent developments due to Clificra, The only question before us is one of convenience in respect to F. Klein and W. Killing, will be found in The Boston Colloquium for simplicity of statement of the physical laws. This point of view 1903 (New York, 1905), article " Forms of Non-Euclidean Space, seems to neglect the consideration that science is to be relevant by F. S. Woods. A. Cayley, loc. cit. (1859), and F. Klein,“: Über die

sogenannte nichteuklidische Geometric," Math. Annal. vols. iv. to the definite perceiving minds of men; and that (neglecting and vi. (1871 and 1872), between them claborated the projective the ambiguity introduced by the invariable slight inexactness theory of distance;' H. Helmholtz, Über die that sächlichen of observation which is not relevant to this special doctrine) Grundlagen der Geometrie" (1806), and". Über die Thatsachen, die

der Geometrie zu Grunde liegen" (1868), both in his Wissenschaftliche "Cf. A. Cayley, " A Sixth Memoir on Quantics," Trans. Roy. Soc., Abhandlungen, vol. ii., and S. Lie, loc. cit. (1890 and 1893), between 1859, and Coll. Papers, vol. ii.; and F. Klein, Math. Ann. vol. iv., them claborated the group theory of congruence.

The numberless works which have been written to suggest equi1 CI. loc. cit.

valent alternatives to Euclid's parallel axioms may be neglected as • For similar deductions from a third set of axioms, suggested in being of trivial importance, though many of them are marvels of essence by Peano, Riv. mat. vol. iv. loc. cil. cf. Whitehead, Desc. geometric ingenuity. Geom. loc. cit.

The second stream of thought confined itself within the circle of • Ch. H, Poincaré, La Science et l'hypothèse, ch. ii.

ideas of Euclidean geometry. Its origin was mainly due to a


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succession of great French mathematicians, for example, G. Monge, / rearing of cattle, and the breeding of fishes. He was the first to Géométrie descriptive (1800); J. V. Poncelet, Traité des proprietés systematize what had been written on the subject, and supple Forigine et le développement des méthodes en géométrie (Bruxelles, 1837). mented the labours of others by practical cxperience gained and Traile de géométrie supérieure (Paris, 1852); and many others. during his travels. In the Augustan age Julius Hyginus wrote But the works which have been, and are still,

of decisive influcnce on on farming and bee-keeping, Sabinus Tiro on horticulture, and geometry are K. G. C. von Staudt's two works, Geometrie der Lage the culture of vines, and Cornelius Celsus (best known for his thought as a store-house of ideas relevant to the foundations of during the early empire Julius Graecinus and Julius Atticus on

Nürnberg, 1847); and Beilräge zur Geomelric der Lage (Nürnberg. De medicina) on farming. The chief work of the kind, however, 1856, 3rd ed. 1860)

The final period is characterized by the successful production of is that of Lucius Junius Moderatus Columella (9.8.). About the exact systems of axioms, and by the final solution of problems middle of the and century the two Quintilii, natives of Troja, successful analysis of the ideas involved in serial continuity is due to wrote on the subject in Greek. It is remarkable that Columella's R. Dedekind, Stetigkeit und irrationale Zahlen (1872), and to G. work exercised less influence in Rome and Italy than in southern Cantor, Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Leipzig. Gaul and Spain, where agriculture became one of the principal 1883), and Acta malk. vol. 2.

Complete systems of axioms have been stated by M. Pasch, loc. subjects of instruction in the superior educational establishments cil.; G. Peano, loc. cit.; M. Pieri, loc.

cit.; B. Russell, Principles of that were springing up in those countries. One result of this was Mathematics; O. Veblen, loc. cit.; and by G. Veronese in his treatise, the preparation of manuals of a popular kind for use in the schools. Fondamenti di geometria (Padua, 1891: German transl

. by A. Schepp. In the 3rd century Gargilius Martialis of Mauretania compiled Grundzüge der Geometrie, Leipzig. 1894). Most of the leading memoirs a Geoponica in which medical botany and the veterinary art on special questions involved have been cited in the text; in addition

were included. The De rc rustica of Palladius (4th century), in there may be mentioned M. Pieri, "Nuovi principii di geometria projettiva complessa," Trans. Accad. R. d. Sci. (Turin, 1905); fourteen books, which is almost entirely borrowed from Columella, E. H. Moore, On the Projective Axioms of Geometry," Trans is greatly inferior in style and knowledge of the subject. It is a Amer. Math. Soc., 1902: 0. Veblen and W. H. Bussey. “Finite kind of farmer's calendar, in which the different rural occupations Projective Geometries," Trans. Amer. Math. Soc., 1905; A. B.

are arranged in order of the months. The fourteenth book Kempe, "On the Relation between the Logical Theory of Classes and the Geometrical Theory of Points," Proc. Lond. Math. Soc., (on forestry) is written in elegiacs (85 distichs). The whole of 1890; J. Royce, "The Relation of the Principles of Logic to the Palladius and considerable fragments of Martialis are extant. Foundations of Geometry," Trans. of Amer. Math. Soc., 1905; The best edition of the Scriptores rei rusticae is by J. G. Schneider A. Schoenflies, " Über die Möglichkeit ciner projectiven Geometrie (1794-1797), and the whole subject is exhaustively treated by bei transfiniter (nichtarchimedischer) Massbestimmung," Deutsch. A. Magerstedt, Bilder aus der römischen Landwirtschaft (1858M. V. Jahresb., 1906.


see also Teuffel-Schwabe, Hist. of Roman Literature, 54; For general expositions of the bearings of the above investiga: C. 8. Bähr in Ersch and Gruber's Algemeine Encyklopädie. tions, cf. Hon. Bertrand Russell, loc. cil.;

L. Couturat, Les Principes

GEORGE, SAINT (d. 303), the patron saint of England, Aragon des mathémaliques (Paris, 1905); H. Poincaré, loc. cil.; Russell and Whitehead, Principia malhemalica (Cambridge, Univ. Press). and Portugal. According to the legend given by Metaphrastes The philosophers whose views on space and geometric truth de- the Byzantine hagiologist, and substantially repeated in the serve especial study are Descartes, 'Leibnitz, Hume, Kant and J: S. Roman Adla sanclorum and in the Spanish breviary, he was born Mill.

(A. N. W.)

in Cappadocia of noble Christian parents, from whom he received GEOPONICI,' or Scriplores rei rusticae, the Greek and Roman a careful religious training. Other accounts place his birth at writers on husbandry and agriculture. On the whole the Greeks Lydda, but preserve his Cappadocian parentage. Having enpaid less attention ihan the Romans to the scientific study of braced the profession of a soldier, he rapidly rose under DioThese subjects, which in classical times they regarded as a branch clelian to high military rank. In Persian Armenia hc organized of economics. Thus Xenophon's Oeconomicus (see also Memo and energized the Christian community at Urmi (Urumiah), rabilia, ii. 4) contains a culogy of agriculture and its beneficial and even visited Britain on an imperial expedition. When ethical effects, and much information is to be found in the writings Diocletian had begun to manifest a pronounced hostility towards of Aristotle and his pupil Theophrastus. About the same time Christianity, George sought a personal interview with him, in as Xenophon, the philosopher Democritus of Abdera wrote a which he made deliberate profession of his faith, and, carnestly treatise slepi Tewprias, frequently quoted and much used by remonstrating against the persecution which had begun, resigned the later compilers of Geoponica (agricultural treatises). Greater his commission. He was immediately laid under arrest, and attention was given to the subject in the Alexandrian period; after various tortures, finally put to death at Nicomedia (his body a long list of names is given by Varro and Columella, amongst being afterwards taken to Lydda) on the 23rd of April 303. His them Hiero II. and Attalus III. Philometor. Later, Cassius festival is observed on that anniversary by the entire Roman Dionysius of Utica translated and abridged the great work of Catholic Church as a semi-duplex, and by the Spanish Catholics the Carthaginian Mago, which was still further condensed by as a duplex of the first class with an octave. The day is also Diophanes of Nicaea in Bithynia for the use of King Deiotarus. celebrated as a principal feast in the Orthodox Eastern Church, From these and similar works Cassianus Bassus (9.0.) compiled where the saint is distinguished by the titles weyalópaprup and his Geoponica. Mention may also be made of a little work

τροπαιοφόρος. . Περί Γεωργικών by Michael Psellus (printed in Boissonade,

The historical basis of the tradition is particularly unsound, Anecdota Gracca, i.).

there being two claimants to the name and honour. Eusebius, The Romans, aware of the necessity of maintaining a numerous Hist. eccl. viii. 5, writes: “Immediately on the promulgation and thriving order of agriculturists, from very early times of the edict (of Diocletian) a certain man of no mean origin, but endeavoured to instil into their countrymen both a theoretical highly esteemed for his temporal dignities, as soon as the decree and a practical knowledge of the subject. The occupation of was published against the churches in Nicomedia, stimulated the farmer was regarded as next in importance to that of the by a divine zeal and excited by an ardent faith, look it as it was soldier, and distinguished Romans did not disdain to practise openly placed and posted up for public inspection, and tore it it. In furtherance of this object, the great work of Mago was to shreds as a most profane and wicked act. This, too, was translated into Latin by order of the senate, and the elder Cato done when the two Caesars were in the city, the first of whom wrote his De agri cultura (extant in a very corrupt state), a

was the eldest and chief of all and the other held fourth grade of simple record in homely language of the rules observed by the old the imperial dignity after him. But this man, as the first that Roman landed proprietors rather than a theoretical treatise.

was distinguished there in this manner, after enduring what He was followed by the two Sasernae (father and son) and Gnaeus was likely to follow an act so daring, preserved his mind, calm Tremellius Scrofa, whose works are lost. The learned Marcus and serene, until the moment when his spirit fled.” Rivalling Terentius Varro of Reate, when eighty years of age, composed this anonymous martyr, who is often supposed to have his Rerum rusticarum, libri tres, dealing with agriculture, the been St George, is an earlier martyr briefly mentioned in the

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suffered martyrdom, among whom also the Holy George was GEORGE I. (George Louis) (1660-1727), king of Great Britain martyred."

and Ireland, born in 1660, was heir through his father Ernest Two Syrian church inscriptions bearing the name, one at Ezr'a Augustus to the hereditary lay bishopric of Osnabrück, and to and the other at Shaka, found by Burckhardt and Porter, and the duchy of Calenberg, which formed one portion of the Hanodiscussed by J. Hogg in the Transactions of the Royal Literary verian possessions of the house of Brunswick, whilst he secured Society, may with some probability be assigned to the middle the reversion of the other portion, the duchy of Celle or Zell, of the 4th century. Calvin impugned the saint's existence by his marriage (1682) with the heiress, his cousin Sophia altogether, and Edward Reynolds (1599-1676), bishop of Norwich, Dorothea. The marriage was not a happy one. The morals like Edward Gibbon a century later, made him one with George of German courts in the end of the 17th century took their tone of Laodice F, called "the Cappadocian," the Arian bishop of from the splendid profligacy of Versailles. It became the Alexandria (see GEORGE OF LAODICEA).

fashion for a prince to amuse himself with a mistress or more Modern criticism, while rejecting this identification, is not frequently with many mistresses simultaneously, and he was unwilling to accept the main fact that an officer named Georgios, often content that the mistresses whom he favoured should be of high rank in the army, suffered martyrdom probably under neither beautiful nor wilty. George Louis followed the usual Diocletian. In the canon of Pope Gelasius (494) George is course. Count Königsmark-a handsome adventurer-seized mentioned in a list of those“ whose names are justly reverenced the opportunity of paying court to the deserted wise. Conjugal among men, but whose acts are known only to God," a statement infidelity was held at Hanover to be a privilege of the male sex. which implies that legends had already grown up around his Count Königsmark was assassinated. Sophia Dorothea was name. The caution of Gelasius was not long preserved; Gregory divorced in 1694, and remained in seclusion till her death in of Tours, for example, asserts that the saint's relics actually 1726. When George IV., her descendant in the fourth generaexisted in the French village of Le Maine, where many miracles tion, attempted in England to call his wife to account for sins of were wrought by means of them; and Bede, while still explaining which he was himself notoriously guilty, free-spoken public that the Gesta Georgii are reckoned apocryphal, commits himself opinion reprobated the offence in no measured terms. But in to the statement that the martyr was beheaded under Dacian, the Germany of the 17th century all free-spoken public opinion king of Persia, whose wife Alexandra, however, adhered to the had been crushed out by the misery of the Thirty Years' War, Christian faith. The great fame of George, who is reverenced and it was understood that princes were to arrange their domestic alike by Eastern and Western Christendom and by Mahom- life according to their own pleasure. medans, is due to many causes. He was martyred on the eve The prince's father did much to raise the dignity of his family. of the triumph of Christianity, his shrine was reared near the By sending help to the emperor when he was struggling against scene of a great Greek legend (Perseus and Andromeda), and the French and the Turks, he obtained the grant of a ninth his relics when removed from Lydda, where many pilgrims had clectorate in 1692. His marriage with Sophia, the youngest visited them, to Zorava in the Hauran served to impress his fame daughter of Elizabeth the daughter of James I. of England, not only on the Syrian population, but on their Moslem con- was not one which at first seemed likely to conser any prospect querors, and again on the Crusaders, who in grateful memory of advancement to his family. Bul though there were many of the saint's intervention on their behalf at Antioch built a new persons whose birth gave them better claims than she had to the cathedral at Lydda to take the place of the church destroyed | English crown, she found herself, upon the death of the duke of by the Saracens. This cathedral was in turn destroyed by Gloucester, the next Protestant heir after Anne. The Act of Saladin.

Scutlement in 1701 secured the inheritance to herself and her The connexion of St George with a dragon, familiar since the descendants. Being old and unambitious she rather permitied Golden Legend of Jacobus de Voragine, can be traced 10 the herself to be burthened with the honour than thrust herself close of the 6th century. At Arsus or Joppa-neither of them forward to meet it. Her son George took a deeper interest in far from Lydda--Perseus had slain the sea-monster that the matter. In his youth he had fought with determined courage threatened the virgin Andromeda, and George, like many another in the wars of William III. Succeeding to the electorate on his Christian saint, entered into the inheritance of veneration pre- father's death in 1698, he had sent a welcome reinforcement viously enjoyed by a pagan hero. The exploit thus attaches of Hanoverians to fight under Marlborough at Blenheim. With itself to the very common Aryan myth of ihe sun-god as the prudent persistence he attached himself closely to the Whigs conqueror of the powers of darkness.

and to Marlborough, refusing Tory offers of an independent The popularity of St George in England has never reached command, and receiving in return for his fidelity a guarantee by the height attained by St Andrew in Scotland, St David in Wales | the Dutch of his succession to England in the Barrier treaty of or St Patrick in Ireland. The council of Oxford in 1222 ordered 1709. In 1714 when Anne was growing old, and Bolingbroke that his seast should be kept as a national festival; but it was and the more reckless Tories were coquctling with the son of not until the time of Edward III. that he was made patron of James II., the Whigs invitcd George's eldest son, who was duke the kingdom. The republics of Genoa and Venice were also of Cambridge, to visit England in order to be on the spot in case under his protection.

of need. Neither the elector nor his mother approved of a step See P. Heylin, The History of ... S. George of Cappadocia (1631); which was likely to alienate the queen, and which was specially S. Baring-Gould, Curious Myths of the Middle Ages: Fr. Gorres, distastcful to himself, as he was on very bad terms with his son. Der Ritter St Georg in der Geschichte, Legende und Kunst" (Zeit. Yet they did not set themselves against the strong wish of the schrift für wissenschaftliche Theologie, xxx., 1887, Helt i.): E. A. W. party to which they looked for support, and it is possible that Budge, The Martyrdom and Miracles of St George of Cappadocia: troubles would have arisen from any attempl to carry out the the Coptic texts edited with an English translation (1888); Bolland, Acta Sancti, iii. tor; E. O. Gordon, Saint George (1907); M. H. plan, if the deaths, first of the clectress (May 28) and then of the Bulley, St George for Merrie England (1908).

queen (August 1, 1714), had not laid open George's way to the !G. A. Smith (Hist. Geog. of Holy Land, p. 164) point's out another succession without further effort of his own. coincidence. ". The Mahom medans who usually identify St George

In some respects the position of the new king was not unlike with the prophet Elijah, at Lydda confound his legend with one that of William III, a quarter of a century before. Both about Christ himself. Their name for Antichrist is Dajjal, and they sovereigns were foreigners, with little knowledge of English have a tradition that Jesus will slay Antichrist by the gate of Lydda. politics and little interest in English legislation. Both sovereigns The notion sprang from an ancient bas-relics of George and the Dragon on the Lydda church. But Dajjal may be derived, by a

arrived at a time when party spirit had been running high, and very common confusion between n and 1, from Dagon, whose name when the task before the ruler was to still the waves of contention. two neighbouring villages bear to this day, while one of the gates of In spite of the difference between an intellectually great man Lydda used to be called the Gate of Dagon. It is a curious process by which the monster that symbolized heathenism conquered by

and an intellectually small one, in spite too of the difference Christianity has been evolved out of the first great rival of the God of between the king who began by choosing his ministers from Israel

both parties and the king who persisted in choosing his minister.'

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