is a movement that leaves a and b fixed and transforms c into another point of the plane abc. XIV. If a, b and c are three non-collinear points, and if d and e are points of the plane abc common to the spheres ca and c, and different from c, then d and e coincide. XV. If a, b and c are distinct non-collinear points, there exists at least one point outside the plane abc. XVI. If a, b, c and d are four non-complanar points, there exists a movement that leaves a and b fixed and transforms d into a point of the plane abc. XVII. If a, b, c and d are four distinct collinear points, the point d cannot be upon only one of the segments ab, ac, bc. XVIII. If a, b and c are three collinear points, and if c is between a and b, no point can be at once between a and c and between b and c. XIX. If a, b and c are three non-collinear points, every straight line of the plane abc that has a point in the segment ab has a point in the segment ac or in the segment bc, or it contains one of the points a, b, c. XX. If k is a class of points in the segment ab, there exists in the segment, or coincides with b, a point x, such that no point of k is between x and b, and that for every point y between a and x there is a point k between y and x or coincident with x. Two figures (classes of points) coincide when and only when they are composed of the same points. IV means that a movement is a one-to-one relation between two figures. The movements μ and u (V) are each the other's converse; they are mutually converse biuniform relations. By VI the relative product of the movements μ and is a movement. The relative product μu leaves every point fixed, or, as we say, transforms all points each into itself. In contradistinction from such movements, others are described as effective. VII provides for rotation of a figure about two of its points. A straight line ab is defined to be the class of all points that remain fixed in case of every movement leaving a and b fixed. It is a matter of proof that a straight line is determined by any two distinct points of it. VIII is not valid in space of four or more dimensions, and hence no special postulate restricting our geometry to three dimensions is necessary. It is readily proved that any movement whatever transforms any and every triplet of collinear points into such a triplet; in other words, a movement is a collineation. By plane abc is meant the figure composed of the points of the lines joining a to points of bc, or b to points of ac, or c to points of ab, it being assumed that a, b and c are non-collinear points. It is a theorem that every movement converts a plane into a plane. Postulate IX is necessary to prove that a plane is determined by any three non-collinear points of it. By the sphere b. is meant the class of points such that for each of them there is a movement transforming it into b while leaving a fixed. The point a is the center of the sphere. It is demonstrable that every movement transforms spheres into spheres; that any movement that leaves the center of a sphere fixed transforms the sphere into itself; and that, if two spheres have but one common point, that point is collinear with the centers of the spheres. X, XI, and XII provide for transforming a line into itself; and XIII and XIV make the like provision for the plane. A circle is the logical product of a sphere and a plane containing its center. The center of the circle is that of the sphere. The notion of perpendicularity is introduced by the definition: the pair (a, c) of points is said to be perpendicular to the pair (a, b) when and only when there is a movement that leaves a and b fixed and transforms c into another point of the straight line ac. The notion is readily extensible to straight lines. XV provides for a plurality of planes, and XVI for the transformation of one plane upon another. The notion of equidistance is introduced by the definition: a point a is equidistant from two points b and c when and only when it is the center of a sphere containing b and c. It is demonstrable that, in a plane containing the distinct points a and b, the class of points equidistant from a and b is the straight line perpendicular to the straight line ab and containing the mid-point of the segment ab; that a straight line perpendicular to two straight lines ab and ac is perpendicular to every straight line that contains a and is contained in the plane abc; and other theorems respecting perpendicularity are readily proved. A point is interior to a sphere if it is the mid-point of two distinct points of the sphere. If not, it is exterior, or else is a point of the sphere. A point of a plane containing a circle is interior or exterior to the circle according as it is interior or exterior to the sphere having the same center as the circle and containing the circle. A sphere having for center the mid-point of two points a and b, and containing them, is called the polar sphere of the points a and b. The notion between is introduced by the definition: a point x is between points a and b if it is contained in the straight line ab and is interior to the polar sphere of a and b. The class of points between two points a and b is named segment ab. The segment ab is less than the segment cd when and only when there exists a movement that transforms a into c and b into a point between c and d. Two segments (or other figures) are congruent if there exists a movement transforming one of them into the other. It is demonstrable that if two segments are not congruent, one of them is less than the other. The notion angle is defined and to it are extended the ideas of less than and congruence. If a, b and c are non-collinear points, the triangle abc is the figure composed of the points of the segments, each joining a and a point of the segment bc. The three theorems regarding congruence are proved; and so on and on. By XX, which provides for continuity, is deduced the Archi median "axiom" as a theorem. Thence follows the idea of measurability of segments. General Remarks.-No geometry involves ideas not found in logic or definable in terms of logical constants, and no geometry contains other undemonstrated propositions than the primitive propositions of logic. The name point is merely that of a class of things (if there be such things) that satisfy a certain set of postulates, but geometry does not assert the actual existence of any such class and does not assert the truth of the postulates. What it does assert is that, if such a class exists, then such and such a body of theorems are valid regarding the class. Geometry is thus a body of implications. It says merely "if so and so, then so and so." This important fact is somewhat disguised by the categorical form in which postulates are often stated. Bibliography. Instead of giving a list of the works constituting the vast and rapidly growing modern literature dealing with the foundations of mathematics in general, with the foundations of special branches, and with modern logic, it will be sufficient to refer the reader to Russell's 'Principles of Mathematics,' Vol. I (Cambridge, University Press) and to Couturat's 'Les Principes des Mathématiques' (Paris, Félix Alcan) and 'Traité de Logistique' (Alcan), wherein nearly all the important works are cited in connections showing the bearings of them. Most of the works are too technical for the general reader, who will naturally begin with the mentioned treatises of Russell and Couturat, extending his reading gradually according to increasing ability and interest. CASSIUS J. KEYSER, Adrain Professor of Mathematics, Columbia University. |