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at this stage The theory of distance will be considered after the principles of descriptive geometry have been developed.

Descriptive Geometry.

Descriptive geometry is essentially the science of multiple order for open series. The first satisfactory system of axioms was given by M. Pasch. An improved version is due to G. Peano. Both these authors treat the idea of the class of points constituting the segment lying between two points as an undefined fundamental idea. Thus in fact there are in this system two fundamental ideas, namely, of points and of segments. It is then easy enough to define the prolongations of the segments, so as to form the complete straight lines. D. Hilbert's formulation of the axioms is in this respect practically based on the same fundamental ideas. His work is justly famous for some of the mathematical investigations contained in it, but his exposition of the axioms is distinctly inferior to that of Peano. Descriptive geometry can also be considered as the science of a class of relations, each relation being a two-termed serial relation, as considered in the logic of relations, ranging the points between which it holds into a linear open order. Thus the relations are the straight lines, and the terms between which they hold arc the points. But a combination of these two points of view yields the simplest statement of all. Descriptive geometry is then conceived as the investigation of an undefined fundamental relation between three terms (points); and when the relation holds between three points A, B, C, the points are said to be" in the [linear] order ABC."

O. Veblen's axioms and definitions, slightly modified, are as follows:

1. If the points A, B, C are in the order ABC, they are in the order CBA.

2. If the points A, B, C are in the order ABC, they are not in the order BCA.

3. If the points A, B, C are in the order ABC, A is distinct from C.

4. If A and B are any two distinct points, there exists a point C such that A, B, C are in the order ABC.

The

Definition. The line AB (AB) consists of A and B, and of all points X in one of the possible orders, ABX, AXB, XAB. points X in the order AXB constitute the segment AB.

5. If points C and D (CD) lie on the line AB, then A lies on the line CD.

6. There exist three distinct points A, B, C not in any of the orders ABC, BCA, CAB.

7. If three distinct points A, B, C (fig. 73) do not lie on the same line, and D and E are two distinct points in the orders BCD and CEA, then a point F exists in the order AFB, and such that D, E, F are collinear.

C

Definition.-If A, B, C are three non-collinear points, the plane ABC is the class of points which lie on any one of the lines joining any two of the D points belonging to the boundary of FIG. 73. the triangle ABC, the boundary being formed by the segments BC, CA and AB. The interior of the triangle ABC is formed by the points in segments such as PQ, where P and Q are points respectively on two of the segments BC, CA, AB.

8. There exists a plane ABC, which does not contain all the points.

Definition-If A, B, C, D are four non-coplanar points, the space ABCD is the class of points which lie on any of the lines containing two points on the surface of the tetrahedron ABCD, the surface being formed by the interiors of the triangles ABC, BCD, DCA,

DAB.

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10. The Dedekind property holds for the order of the points on any straight line.

It follows from axioms 1-9 that the points on any straight line are arranged in an open serial order. Also all the ordinary theorems respecting a point dividing a straight line into two parts, a straight line dividing a plane into two parts, and a plane dividing space into two parts, follow.

Again, in any plane a consider a line and a point A (fig. 74). Let any point B divide into two half-lines 4 and 4. Then it can be proved that the set of half-lines, emanating from A and intersecting (such as m), are bounded by two half-lines, of which ABC is one. Let r be the other. Then it can be proved that r does not intersect 4. Similarly for the half-line, such as n, intersecting 4. Let s be its bounding half-line. Then two cases are possible. (1) The half-lines and s are collinear, and together form one complete line. In this case, there is one and only one line (viz. r+s) through A and lying in a which does not intersect 1. This is the Euclidean case, and the assumption that this case holds is the Euclidean parallel axiom. But (2) the half-lines and s may not be collinear. In this case there will be an infinite number of lines, such as k for instance, containing A and lying in a which do not intersect. Then the lines through A in a are divided into two classes by reference to I, namely, the secant lines which intersect, and the non-secant lines which do not intersect l. The halves, may be called the two parallels to / through A. two boundary non-secant lines, of which and s are respectively

FIG. 74.

The perception of the possibility of case 2 constituted the startingpoint from which Lobatchewsky constructed the first explicit coherent theory of non-Euclidean geometry, and thus created a revolution in the philosophy of the subject. For many centuries the speculations of mathematicians on the foundations of geometry were almost confined to hopeless attempts to prove the parallel axiom" without the introduction of some equivalent axiom.

Associated Projective and Descriptive Spaces.-A region of a projective space, such that one, and only one, of the two supplementary segments between any pair of points within it lies entirely within it, satisfies the above axioms (1-10) of descriptive geometry, where the points of the region are the descriptive. points, and the portions of straight lines within the region are the descriptive lines. If the excluded part of the original projective space is a single plane, the Euclidean parallel axiom also holds, otherwise it does not hold for the descriptive space of the limited region. Again, conversely, starting from an original descriptive space an associated projective space can be constructed by means of the concept of ideal points. These are also called projective points, where it is understood that the simple points are the points of the original descriptive space. An ideal point is the class of straight lines which is composed of two coplanar lines a and b, together with the lines of intersection of all pairs of intersecting planes which respectively contain a and b, together with the lines of intersection with the plane ab of all planes containing any one of the lines (other than a or b) already specified as belonging to the ideal point. It is evident that, if the two original lines a and b intersect, the corresponding ideal point is nothing else than the whole class of lines which are concurrent at the point ab. But the essence of the definition is that an ideal point has an existence when the lines a and b do not intersect, so long as they are coplanar. An ideal point is termed proper, if the lines composing it intersect; otherwise it is improper.

A theorem essential to the whole theory is the following: if any two of the three lines a, b, c are coplanar, but the three lines are not all coplanar, and similarly for the lines a, b, d, then c and d are coplanar. It follows that any two lines belonging to an ideal point can be used as the pair of guiding lines in the definition. An ideal point is said to be coherent with a plane, if any of the lines composing it lie in the plane. An ideal line is the class of ideal points each of which is coherent with two given planes. ♦ Cf. P. Stäckel and F. Engel, Die Theorie der Parallellinien von Euklid bis auf Gauss (Leipzig, 1895).

7Cf Pasch, loc. cit., and R. Bonola, "Sulla introduzione degli enti improprii in geometria projettive," Giorn. di mat. vol. xxxviii. (1900); and Whitehead, Axioms of Descriptive Geometry (Cambridge, 1907).

If the planes intersect, the ideal line is termed proper, otherwise it is improper. It can be proved that any two planes, with which any two of the ideal points are both coherent, will serve as the guiding planes used in the definition. The ideal planes are defined as in projective geometry, and all the other definitions (for segments, order, &c.) of projective geometry are applied to the ideal elements. If an ideal plane contains some proper ideal points, it is called proper, otherwise it is improper. Every ideal plane contains some improper ideal points.

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cessive displacements of a rigid body from position A to position B, and from position B to position C, are the same in effect as one displacement from A to C. But this is the characteristic "group" property. Thus the transformations of space into itself defined by displacements of rigid bodies form a group.

Call this group of transformations a congruence-group. Now according to Lie a congruence-group is defined by the following characteristics:

1. A congruence-group is a finite continuous group of one-one transformations, containing the identical transformation.

2. It is a sub-group of the general projective group, i.e. of the group of which any transformation converts planes into planes, and straight lines into straight lines.

3. An infinitesimal transformation can always be found satis

It can now be proved that all the axioms of projective geometry hold of the ideal elements as thus obtained; and also that the order of the ideal points as obtained by the projective method agrees with the order of the proper ideal points as obtained from that of the associated points of the descriptive geometry. Thus a projective space has been 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 descriptive axioms hold. Accordingly, by substituting ideal elements, a descriptive space can always be considered as a region within a projective space. This is the justification for the ordinary use of the " points at infinity "in the ordinary Euclidean geometry; the reasoning has been transferred from the original descriptive space to the associated projective space of ideal elements; and with the Euclidean parallel axiom the improper ideal elements reduce to the ideal points on a single improper ideal plane, namely, the plane at infinity.1

4. No infinitesimal transformation of the group exists, such that, at least in the region for which (3) holds, a straight line, a point on it, and a plane through it, shall all be latent. The property enunciated by conditions (3) and (4), taken together, is named by Lie "Free mobility in the infinitesimal." Lie proves the following theorems for a projective space:1. If the above four conditions are only satisfied by a group throughout part of projective space, this part either (a) must be the region enclosed by a real closed quadric, or (8) must be the whole of the projective space with the exception of a single plane. In case (a) the corresponding congruence group is the continuous group for which the enclosing quadric is latent; and in case (8) an imaginary. conic (with a real equation) lying in the latent plane is also latent, and the congruence group is the continuous group for which the plane and conic are latent.

2. If the above four conditions are satisfied by a group throughout the whole of projective space, the congruenoc group is the continuous group for which some imaginary quadric (with a real equation) is latent.

of any quadrics of the types considered, either in theorem 1(a), or in By a proper choice of non-homogeneous co-ordinates the equation theorem 2, can be written in the form 1+c(x2+ y2+2)=o, where is negative for a real closed quadric, and positive for an imaginary quadric. Then the general infinitesimal transformation is defined by the three equations: "" dx/dt=u-way+w23+cx(ux+vy+wz), ) dy/dt = v12+w3x+cy(ux+vy+wz), dz/dt=w-x+w1y+cz(ux+vy+uz).

Congruence and Measurement.-The property of physical space which is expressed by the term "measurability" has now to be considered. This property has often been considered as essential to the very idea of space. For example, Kant writes, "Space is represented as an infinite given quantity." This quantitative aspect of space arises from the measurability of distances, of angles, of surfaces and of volumes. These four types of quantity depend upon the two first among them as fundamental. The measurability of space is essentially connected with the idea of congruence, of which the simplest examples are to be found in the proofs of equality by the method of superposition, as used in elementary plane geometry. The mere concepts of " part and of "whole must of necessity be inadequate as the foundation of measurement, since we require the comparison as to quantity of regions of space which have no portions in common. The idea of congruence, as exemplified by the method of superposition in geometrical reasoning, appears to be founded upon that of the "rigid body," which moves from one position to another with its internal spatial relations unchanged. But unless there is a previous concept of the metrical relations between the parts of the body, there can be no basis from which to deduce that they are unchanged.

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It would therefore appear as if the idea of the congruence, or metrical equality, of two portions of space (as empirically suggested by the motion of rigid bodies) must be considered as a fundamental idea incapable of definition in terms of those geometrical concepts which have already been enumerated. This was in effect the point of view of Pasch. It has, however, been proved by Sophus Lie that congruence is capable of definition without recourse to a new fundamental idea. This he does by means of his theory of finite continuous groups (see GROUPS, THEORY OF), of which the definition is possible in terms of our established geometrical ideas, remembering that coordinates have already been introduced. The displacement of a rigid body is simply a mode of defining to the senses a oneone transformation of all space into itself. For at any point of space a particle may be conceived to be placed, and to be rigidly connected with the rigid body; and thus there is a definite correspondence of any point of space with the new point occupied by the associated particle after displacement. Again two sucThe original idea (confined to this particular case) of ideal points is due to von Staudt (loc. cit.).

Cf. Critique, "Trans. Aesth." Sect. 1.

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(A)

In the case considered in theorem 1 (8), with the proper choice of
co-ordinates the three equations defining the general infinitesimal
transformation are:
dx/dl-u-y+wq Z,
dy/dt=v-z + wxx,
dz/dl=w-wqx+w1y.

(B)

In this case the latent plane is the plane for which at least one of x, y, z are infinite, that is, the plane o.x+o.y+o.z+a=0; and the latent conic is the conic in which the cone x2+y+22=0 intersects the latent plane.

It follows from theorems 1 and 2 that there is not one unique congruence-group, but an indefinite number of them. There is one congruence-group corresponding to each closed real quadric, one to each imaginary quadric with a real equation, and one to each imaginary conic in a real plane and with a real equation. The quadric thus associated with each congruence-group is called the absolute for that group, and in the degenerate case of 1 (B) the absolute is the latent plane together with the latent imaginary conic. If the absolute is real, the congruence-group is hyperbolic; if imaginary, it is elliptic; if the absolute is a plane and imaginary conic, the group is parabolic. Metrical geometry is simply the theory of the properties of some particular congruence-group selected for study.

The definition of distance is connected with the corresponding
congruence-group by two considerations in respect to a range of five
points (A1, A2, P1, P2, P3), of which A, and A, are on the absolute.
Let (A,PAP stand for the cross ratio (as defined above) of the
range (APAP1), with a similar notation for the other ranges.
Then
(1) ̧ log(A,PA2P2} + log{A,P2A2P3} =log{A,P‚Å‚P3},
and

(2), if the points A1, A2, P1, P; are transformed into A', A'2, P', P'2
by any transformation of the congruence-group, (a) {A,P,A,P;} =
{Á'P'A'1⁄2P'), since the transformation is projective, and (8) A',. A',
are on the absolute since A, and A, are on it. Thus if we define

the distance PP, to be k log (A,P,A,P2), where A, and A, are the points in which the line PP, cuts the absolute, and k is some constant, the two characteristic properties of distance, namely, (1) the addition of consecutive lengths on a straight line, and (2) the invariability of distances during a transformation of the congruencegroup, are satisfied. This is the well-known Cayley-Klein projective definition of distance, which was elaborated in view of the addition property alone, previously to Lie's discovery of the theory of congruence-groups. For a hyperbolic group when P, and P, are in the region enclosed by the absolute, log A,PAP) is real, and therefore must be real. For an elliptic group A, and A, are conjugate imaginaries, and log (APAP) is a pure imaginary, and k is chosen Similarly the angle between two planes, p, and pa, is defined to be (1/2) log (php), where and are tangent planes to the absolute through the line pip. The planes and are imaginary for an elliptic group, and also for an hyperbolic group when the planes P and p intersect at points within the region enclosed by the absolute. The development of the consequences of these metrical definitions is the subjct of non-Euclidean geometry.

to be x, where x is real and = √~;

The definitions for the parabolic case can be arrived at as limits of those obtained in either of the other two cases by making k ultimately to vanish. It is also obvious that, if P. and P, be the points (x, y, z) and (x, y, z), it follows from equations (B) above that (x-x1)+(y1 — y2)2 + (51 — 22 ) }} is unaltered by a congruence transformation and also satisfies the addition property for collinear distances. Also the previous definition of an angle can be adapted to this case, by making and ↳ to be the tangent planes through the line pip to the imaginary conic. Similarly if pi and p are intersecting lines, the same definition of an angle holds, where and are now the lines from the point pipa to the two points where the plane Pip cuts the imaginary conic. These points are in fact the circular points at infinity on the plane. The development of the consequences of these definitions for the parabolic case gives the ordinary Euclidean metrical geometry.

Thus the only metrical geometry for the whole of projective space is of the elliptic type. But the actual measure-relations (though not their general properties) differ according to the elliptic congruence-group selected for study. In a descriptive space a congruence-group should possess the four characteristics of such a group throughout the whole of the space. Then form the associated ideal projective space. The associated congruencegroup for this ideal space must satisfy the four conditions throughout the region of the proper ideal points. Thus the boundary of this region is the absolute. Accordingly there can be no metrical geometry for the whole of a descriptive space unless its boundary (in the associated ideal space) is a closed quadric or a plane. If the boundary is a closed quadric, there is one possible congruence-group of the hyperbolic type. If the boundary is a plane (the plane at infinity), the possible congruence-groups are parabolic; and there is a congruencegroup corresponding to each imaginary conic in this plane, together with a Euclidean metrical geometry corresponding to each such group. Owing to these alternative possibilities, it would appear to be more accurate to say that systems of quantities can be found in a space, rather than that space is a quantity.

Lie has also deduced the same results with respect to congruence-groups from another set of defining properties, which explicitly assume the existence of a quantitative relation (the distance) between any two points, which is invariant for any transformation of the congruence-group.3

The above results, in respect to congruence and metrical geometry, considered in relation to existent space, have led to the doctrine that it is intrinsically unmeaning to ask which system of metrical geometry is true of the physical world. Any one of these systems can be applied, and in an indefinite number of ways. The only question before us is one of convenience in respect to simplicity of statement of the physical laws. This point of view seems to neglect the consideration that science is to be relevant to the definite perceiving minds of men; and that (neglecting the ambiguity introduced by the invariable slight inexactness of observation which is not relevant to this special doctrine) 'Cf. A. Cayley, "A Sixth Memoir on Quantics," Trans. Roy. Soc., 1859, and Coll. Papers, vol. ii.; and F. Klein, Math. Ann. vol. iv., 1871.

* Cf. loc. cit.

'For similar deductions from a third set of axioms, suggested in essence by Peano, Riv. mat. vol. iv. loc. cit. cf. Whitehead, Desc. Geom. loc. cit.

Cf. H. Poincaré, La Science et l'hypothèse, ch. iii.

Į we have, in fact, presented to our senses a definite set of transformations forming a congruence-group, resulting in a set of measure relations which are in no respect arbitrary. Accordingly our scientific laws are to be stated relevantly to that particular congruence-group. Thus the investigation of the type (elliptic, hyperbolic or parabolic) of this special congruence-group is a perfectly definite problem, to be decided by experiment. The consideration of experiments adapted to this object requires some development of non-Euclidean geometry (see section VI., Non-Euclidean Geometry). But if the doctrine means that, assuming some sort of objective reality for the material universe, beings can be imagined, to whom either all congruence-groups are equally important,or some other congruence-group is specially important, the doctrine appears to be an immediate deduction from the mathematical facts. Assuming a definite congruencegroup, the investigation of surfaces (or three-dimensional loci in space of four dimensions) with geodesic geometries of the form of metrical geometries of other types of congruence-groups forms an important chapter of non-Euclidean geometry. Arising from this investigation there is a widely-spread fallacy, which has found its way into many philosophic writings, namely, that the possibility of the geometry of existent three-dimensional space being other than Euclidean depends on the physical existence of Euclidean space of four or more dimensions. The foregoing exposition shows the baselessness of this idea.

BIBLIOGRAPHY.-For an account of the investigations on the axioms of geometry during the Greek period, see M. Cantor, Vorlesungen über die Geschichte der Mathematik, Bd. i. and iii.; T. L. Heath, The Thirteen Books of Euclid's Elements, a New Translation from the Greek, with Introductory Essays and Commentary, Historical, Critical, and Explanatory (Cambridge, 1908)-this work is the standard source of information; W. B. Frankland, Euclid, Book I., with a Commentary (Cambridge, 1905)-the commentary contains copious extracts from the ancient commentators. The next period of really. authors are: G. Saccheri, S.J., Euclides ab omni naevo vindicatus The leading substantive importance is that of the 18th century, (Milan, 1733). Saccheri was an Italian Jesuit who unconsciously discovered non-Euclidean geometry in the course of his efforts to prove its impossibility. J. H. Lambert, Theorie der Parallellinien (1766); A. M. Legendre, Eléments de géométrie (1794). An adequate account of the above authors is given by P. Stäckel and F. Engel, Die Theorte der Parallellinien von Euklid bis auf Gauss (Leipzig, 1895). The next period of time (roughly from 1800 to 1870) contains two streams of thought, both of which are essential to the modern analysis of the subject. The first stream is that which produced the discovery and investigation of non-Euclidean geometries, the second stream is that which has produced the geometry of position, comprising both projective and descriptive geometry not very accurately discriminated. The leading authors on non-Euclidean geometry Engel, loc. cit.; N. Lobatchewsky, rector of the university of Kazan, are K. F. Gauss, in private letters to Schumacher, cf. Stäckel and to whom the honour of the effective discovery of non-Euclidean geometry must be assigned. His first publication was at Kazan in 1826. His various memoirs have been re-edited by Engel; cf. Urkunden zur Geschichte der nichteuklidischen Geometrie by Stackel and Engel, vol. i. “Lobatchewsky." J. Bolyai discovered non-Euclidean geometry apparently in independence of Lobatchewsky. His memoir was published in 1831 as an appendix to a work by his father W. Bolyai, Tentamen juventutem. memoir has been separately edited by J. Frischauf, Absolute Geometrie nach J. Bolyai (Leipzig, 1872); B. Riemann, Über die Hypothesen, welche der Geometrie zu Grunde liegen (1854); cf. Gesamte Werke, a translation in The Collected Papers of W. K. Clifford. This is a fundamental memoir on the subject and must rank with the work of Lobatchewsky. Riemann discovered elliptic metrical geometry, and Lobatchewsky hyperbolic geometry. A full account of Ricmann's ideas, with the subsequent developments due to Clifford, F. Klein and W. Killing, will be found in The Boston Colloquium for 1903 (New York, 1905), article "Forms of Non-Euclidean Space," by F. S. Woods. A. Cayley, loc. cit. (1859), and F. Klein," Über die sogenannte nichteuklidische Geometrie," Math. Annal. vols. iv. and vi. (1871 and 1872), between them elaborated the projective theory of distance; H. Helmholtz, "Über die thatsächlichen Grundlagen der Geometric" (1866), and " Über die Thatsachen, die der Geometrie zu Grunde liegen" (1868), both in his Wissenschaftliche Abhandlungen, vol. íi., and S. Lie, loc. cit. (1890 and 1893), between them elaborated the group theory of congruence.

This

The numberless works which have been written to suggest equivalent alternatives to Euclid's parallel axioms may be neglected as being of, trivial importance, though many of them are marvels of geometric ingenuity.

The second stream of thought confined itself within the circle of ideas of Euclidean geometry. Its origin was mainly due to a

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 projectives des figures (1822); M. Chasles, Aperçu historique sur l'origine et le développement des méthodes en géométrie (Bruxelles, 1837), mented the labours of others by practical experience gained and Traité 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 influence on thought as a store-house of ideas relevant to the foundations of during the early empire Julius Graecinus and Julius Atticus on on farming and bee-keeping, Sabinus Tiro on horticulture, and geometry are K. G. C. von Staudt's two works, Geometrie der Lage (Nürnberg, 1847); and Beiträge zur Geometrie der Lage (Nürnberg, the culture of vines, and Cornelius Celsus (best known for his 1856, 3rd ed. 1860). De medicina) on farming. The chief work of the kind, however, The final period is characterized by the successful production of is that of Lucius Junius Moderatus Columella (q.v.). About the exact systems of axioms, and by the final solution of problems middle of the 2nd century the two Quintilii, natives of Troja, which have occupied mathematicians for two thousand years. The 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 math. vol. 2. Complete systems of axioms have been stated by M. Pasch, loc. subjects of instruction in the superior educational establishments cit.; 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 Geometric, Leipzig, 1894). Most of the leading memoirs on special questions involved have been cited in the text; in addition a Geoponica in which medical botany and the veterinary art there may be mentioned M. Pieri, "Nuovi principii di geometria were included. The De re rustica of Palladius (4th century), in 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; O. 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. Kempe, "On the Relation between the Logical Theory of Classes are arranged in order of the months. The fourteenth book 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; A. Schoenflies, Über die Möglichkeit einer projectiven Geometrie (1794-1797), and the whole subject is exhaustively treated by The best edition of the Scriptores rei rusticae is by J. G. Schneider bei transfiniter (nichtarchimedischer) Massbestimmung," Deutsch. A. Magerstedt, Bilder aus der römischen Landwirtschaft (1858M. V. Jahresb., 1906. C. F. Bähr in Ersch and Gruber's Allgemeine Encyklopädie. 1863); see also Teuffel-Schwabe, Hist. of Roman Literature, 543

Mill.

"

For general expositions of the bearings of the above investigations, cf. Hon. Bertrand Russell, loc. cit.; L. Couturat, Les Principes des mathématiques (Paris, 1905); H. Poincaré, loc. cit.; Russell and Whitehead, Principia mathematica (Cambridge, Univ. Press). The philosophers whose views on space and geometric truth deserve especial study are Descartes, 'Leibnitz, Hume, Kant and J. S. (A. N. W.) GEOPONICI,' or Scriptores rei rusticae, the Greek and Roman writers on husbandry and agriculture. On the whole the Greeks paid less attention than the Romans to the scientific study of these subjects, which in classical times they regarded as a branch of economics. Thus Xenophon's Oeconomicus (see also Memorabilia, ii. 4) contains a eulogy of agriculture and its beneficial ethical effects, and much information is to be found in the writings of Aristotle and his pupil Theophrastus. About the same time as Xenophon, the philosopher Democritus of Abdera wrote a treatise Ilepi Tewpyias, frequently quoted and much used by the later compilers of Geoponica (agricultural treatises). Greater attention was given to the subject in the Alexandrian period; a long list of names is given by Varro and Columella, amongst them Hiero II. and Attalus III. Philometor. Later, Cassius Dionysius of Utica translated and abridged the great work of the Carthaginian Mago, which was still further condensed by Diophanes of Nicaea in Bithynia for the use of King Deiotarus. From these and similar works Cassianus Bassus (q.v.) compiled his Geoponica. Mention may also be made of a little work Περὶ Γεωργικῶν by Michael Psellus (printed in Boissonade, Anecdota Gracca, i.).

The Romans, aware of the necessity of maintaining a numerous and thriving order of agriculturists, from very early times endeavoured to instil into their countrymen both a theoretical and a practical knowledge of the subject. The occupation of the farmer was regarded as next in importance to that of the soldier, and distinguished Romans did not disdain to practise it. In furtherance of this object, the great work of Mago was translated into Latin by order of the senate, and the elder Cato wrote his De agri cultura (extant in a very corrupt state), a simple record in homely language of the rules observed by the old Roman landed proprietors rather than a theoretical treatise. He was followed by the two Sasernae (father and son) and Gnaeus Tremellius Scrofa, whose works are lost. The learned Marcus Terentius Varro of Reate, when eighty years of age, composed his Rerum rusticarum, libri tres, dealing with agriculture, the The latinized form of a non-existent rewToPikol, used for

convenience.

GEORGE, SAINT (d. 303), the patron saint of England, Aragon and Portugal. According to the legend given by Metaphrastes the Byzantine hagiologist, and substantially repeated in the Roman Acta sanctorum and in the Spanish breviary, he was born in Cappadocia of noble Christian parents, from whom he received Lydda, but preserve his Cappadocian parentage. Having ema careful religious training. Other accounts place his birth at braced the profession of a soldier, he rapidly rose under Diocletian to high military rank. In Persian Armenia hc organized and energized the Christian community at Urmi (Urumiah), and even visited Britain on an imperial expedition. Diocletian had begun to manifest a pronounced hostility towards When Christianity, George sought a personal interview with him, in remonstrating against the persecution which had begun, resigned which he made deliberate profession of his faith, and, earnestly his commission. He was immediately laid under arrest, and after various tortures, finally put to death at Nicomedia (his body being afterwards taken to Lydda) on the 23rd of April 303. festival is observed on that anniversary by the entire Roman Catholic Church as a semi-duplex, and by the Spanish Catholics celebrated as a principal feast in the Orthodox Eastern Church, as a duplex of the first class with an octave. The day is also where the saint is distinguished by the titles μeyadóμaprup and τροπαιοφόρος.

His

The historical basis of the tradition is particularly unsound, there being two claimants to the name and honour. Eusebius, of the edict (of Diocletian) a certain man of no mean origin, but Hist. eccl. viii. 5, writes: "Immediately on the promulgation highly esteemed for his temporal dignities, as soon as the decree by a divine zeal and excited by an ardent faith, took it as it was was published against the churches in Nicomedia, stimulated openly placed and posted up for public inspection, and tore it done when the two Caesars were in the city, the first of whom to shreds as a most profane and wicked act. This, too, was the imperial dignity after him. was the eldest and chief of all and the other held fourth grade of But this man, as the first that was distinguished there in this manner, after enduring what and serene, until the moment when his spirit fled." Rivalling was likely to follow an act so daring, preserved his mind, calm this anonymous martyr, who is often supposed to have been St George, is an earlier martyr briefly mentioned in the Chronicon Pascale: "In the year 225 of the Ascension of our Lord a persecution of the Christians took place, and many

suffered martyrdom, among whom also the Holy George was martyred."

Two Syrian church inscriptions bearing the name, one at Ezr'a and the other at Shaka, found by Burckhardt and Porter, and discussed by J. Hogg in the Transactions of the Royal Literary Society, may with some probability be assigned to the middle of the 4th century. Calvin impugned the saint's existence altogether, and Edward Reynolds (1599–1676), bishop of Norwich, like Edward Gibbon a century later, made him one with George of Laodicea, called "the Cappadocian," the Arian bishop of Alexandria (see GEORGE of Laodicea).

Modern criticism, while rejecting this identification, is not unwilling to accept the main fact that an officer named Georgios, of high rank in the army, suffered martyrdom probably under Diocletian. In the canon of Pope Gelasius (494) George is mentioned in a list of those “whose names are justly reverenced among men, but whose acts are known only to God," a statement which implies that legends had already grown up around his name. The caution of Gelasius was not long preserved; Gregory of Tours, for example, asserts that the saint's relics actually existed in the French village of Le Maine, where many miracles were wrought by means of them; and Bede, while still explaining that the Gesta Georgii are reckoned apocryphal, commits himself to the statement that the martyr was beheaded under Dacian, king of Persia, whose wife Alexandra, however, adhered to the Christian faith. The great fame of George, who is reverenced alike by Eastern and Western Christendom and by Mahommedans, is due to many causes. He was martyred on the eve of the triumph of Christianity, his shrine was reared near the scene of a great Greek legend (Perseus and Andromeda), and his relics when removed from Lydda, where many pilgrims had visited them, to Zorava in the Hauran served to impress his fame not only on the Syrian population, but on their Moslem conquerors, and again on the Crusaders, who in 'grateful memory of the saint's intervention on their behalf at Antioch built a new cathedral at Lydda to take the place of the church destroyed by the Saracens. This cathedral was in turn destroyed by Saladin.

The connexion of St George with a dragon, familiar since the Golden Legend of Jacobus de Voragine, can be traced to the close of the 6th century. At Arsuf or Joppa-neither of them far from Lydda-Perseus had slain the sea-monster that threatened the virgin Andromeda, and George, like many another Christian saint, entered into the inheritance of veneration previously enjoyed by a pagan hero. The exploit thus attaches itself to the very common Aryan myth of the sun-god as the conqueror of the powers of darkness.

The popularity of St George in England has never reached the height attained by St Andrew in Scotland, St David in Wales or St Patrick in Ireland. The council of Oxford in 1222 ordered that his feast should be kept as a national festival; but it was not until the time of Edward III. that he was made patron of the kingdom. The republics of Genoa and Venice were also under his protection.

See P. Heylin, The History of... S. George of Cappadocia (1631); S. Baring-Gould, Curious Myths of the Middle Ages; Fr. Gorres, "Der Ritter St Georg in der Geschichte, Legende und Kunst" (Zeit schrift für wissenschaftliche Theologie, xxx., 1887, Heft i.); E. A. W. Budge, The Martyrdom and Miracles of St George of Cappadocia: the Coptic texts edited with an English translation (1888); Bolland, Acta Sancti, iii. 101; E. O. Gordon, Saint George (1907); M. H. Bulley, St George for Merrie England (1908).

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IG. A. Smith (Hist. Geog. of Holy Land, p. 164) points out another coincidence. The Mahommedans who usually identify St George with the prophet Elijah, at Lydda confound his legend with one about Christ himself. Their name for Antichrist is Dajjal, and they have a tradition that Jesus will slay Antichrist by the gate of Lydda. The notion sprang from an ancient bas-relief of George and the Dragon on the Lydda church. But Dajjal may be derived, by a very common confusion between # and 1, from Dagon, whose name two neighbouring villages bear to this day, while one of the gates of Lydda used to be called the Gate of Dagon. It is a curious process by which the monster that symbolized heathenism conquered by Christianity has been evolved out of the first great rival of the God of Israel

GEORGE I. [George Louis] (1660-1727), king of Great Britain and Ireland, born in 1660, was heir through his father Ernest Augustus to the hereditary lay bishopric of Osnabrück, and to the duchy of Calenberg, which formed one portion of the Hanoverian possessions of the house of Brunswick, whilst he secured the reversion of the other portion, the duchy of Celle or Zell, by his marriage (1682) with the heiress, his cousin Sophia Dorothea. The marriage was not a happy one. The morals of German courts in the end of the 17th century took their tone from the splendid profligacy of Versailles. It became the fashion for a prince to amuse himself with a mistress or more frequently with many mistresses simultaneously, and he was often content that the mistresses whom he favoured should be neither beautiful nor witty. George Louis followed the usual course. Count Königsmark-a handsome adventurer--seized the opportunity of paying court to the deserted wife. Conjugal infidelity was held at Hanover to be a privilege of the male sex. Count Königsmark was assassinated. Sophia Dorothea was divorced in 1694, and remained in seclusion till her death in 1726. When George IV., her descendant in the fourth generation, attempted in England to call his wife to account for sins of which he was himself notoriously guilty, free-spoken public opinion reprobated the offence in no measured terms. But in the Germany of the 17th century all free-spoken public opinion had been crushed out by the misery of the Thirty Years' War, and it was understood that princes were to arrange their domestic life according to their own pleasure.

The prince's father did much to raise the dignity of his family. By sending help to the emperor when he was struggling against the French and the Turks, he obtained the grant of a ninth electorate in 1692. His marriage with Sophia, the youngest daughter of Elizabeth the daughter of James I. of England, was not one which at first seemed likely to confer any prospect of advancement to his family. But though there were many persons whose birth gave them better claims than she had to the English crown, she found herself, upon the death of the duke of Gloucester, the next Protestant heir after Anne. The Act of Settlement in 1701 secured the inheritance to herself and her descendants. Being old and unambitious she rather permitted herself to be burthened with the honour than thrust herself forward to meet it. Her son George took a deeper interest in the matter. In his youth he had fought with determined courage in the wars of William III. Succeeding to the electorate on his father's death in 1698, he had sent a welcome reinforcement of Hanoverians to fight under Marlborough at Blenheim. With prudent persistence he attached himself closely to the Whigs and to Marlborough, refusing Tory offers of an independent command, and receiving in return for his fidelity a guarantee by the Dutch of his succession to England in the Barrier treaty of 1709. In 1714 when Anne was growing old, and Bolingbroke and the more reckless Torics were coquetting with the son of James II., the Whigs invited George's eldest son, who was duke of Cambridge, to visit England in order to be on the spot in case of need. Neither the elector nor his mother approved of a step which was likely to alienate the queen, and which was specially distasteful to himself, as he was on very bad terms with his son. Yet they did not set themselves against the strong wish of the troubles would have arisen from any attempt to carry out the party to which they looked for support, and it is possible that plan, if the deaths, first of the electress (May 28) and then of the queen (August 1, 1714), had not laid open George's way to the succession without further effort of his own.

In some respects the position of the new king was not unlike that of William III. a quarter of a century before. Both sovereigns were foreigners, with little knowledge of English politics and little interest in English legislation. Both sovereigns arrived at a time when party spirit had been running high, and when the task before the ruler was to still the waves of contention. In spite of the difference between an intellectually great man and an intellectually small one, in spite too of the difference between the king who began by choosing his ministers from both parties and the king who persisted in choosing his minister.

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