In the development it is important to ob serve that some theorems depend upon only part of the axioms. Thus from group I alone it follows that two planes having a point in i common necessarily have a line in common, and that a line and a point determine a plane. The property of triangles and polygons stated above follows from group I and II. The theory of proportion may be established without employing group V. The theorem that if two triangles in one plane have their sides re spectively parallel, the lines joining correspond ing vertices are either parallel or concurrent (a special case of Desargues' theorem), can be proved without using axiom III 6 if and only if the spatial axioms in addition to the plane axioms are employed.
An important result which has been ob tained recently is that while the areas of plane polygons may be treated without appealing to the continuity axioms, this is not possible with the volumes of polyhedrons. The dif ference is observed in Euclid's proofs: the proposition that triangles having the same base and altitude are equal in area is demon strated by adding or taking away congruent parts from congruent figures, while the corre sponding proposition concerning triangular pyramids is proved by the method of limits, or the equivalent method of exhaustion. That this difference in treatment is not avoidable was established by Dehn (1901), who showed that there exist polyhedrons of equal volumes which cannot be formed by the addition or sub traction of respectively congruent polyhedrons. In plane geometry this formation applies to any twp polygons with the same area.
The most fundamental question concerning the set of axioms is that of consistency. In the development of geometry no contra diction has thus far presented itself ; but will this always be the case? Can it be shown that no inconsistency can ever arise? The only known method of answering this question depends upon establishing a correspondence between the geometrical elements and certain numerical entities, and showing that any in herent contradiction in geometry would in volve contradictory relations among these entities. The question is thus transferred to the field of arithmetic. Are the axioms of number (commutative, associative, distribu tive, etc.) inconsistent? No perfectly satis factory disproof of this has yet been devised. See however ALEGEBRA : DEFINITIONS AND FUN DAMENTAL CONCEPT& Another question to be considered is the in dependence of the axioms. If any one axiom can be deduced from the others, it may be omitted from the list and introduced as a theorem. It is therefore desirable that the axioms should express mutually independent statements. The standard method employed in proving the independence of an axiom or group of axioms consists in devising a set of objects of any kind which, when considered as ele ments, fulfill the relations expressed in remain ing axioms, but for which the axiom or group in question is not satisfied. Thus the fact that
the axiom of parallels (IV) cannot be de duced from the other axioms is shown by the non-Euclidean geometry of Lobachevsky. Similarly, the independence of axiom VI is proved by means of the non-Archimedean geometries of Veronese and Hilbert. Various apparently artificial systems have been devised in this connection, which, while not amenable to the intuition, are conceivable and mathe matically true because based on assumptions which may be shown to be free from incon sistency.
The set of axioms presented above is of course not the only one which may serve as foundation for ordinary geometry. Thus the axiom of parallels may be replaced by the statement that the sum of the angles of a triangle is two right angles. In general the propositions of a given collection may be derived from various sets selected from the collection. In the present case the possi bilities are endless.
Geometry may also be founded on other primative (undefined) concepts than those introduced above. Thus in the discussions inaugurated by Helmholtz and continued by Lie and Poincare, the principal concept is that of transformation (displacement, rigid motion) and the axioms include the group property (the resultant of two displacements is itself a displacement). The straight line is then no longer, as in Hilbert's system, a primitive concept, but receives definition: if in a dis placement two points are fixed, there are an infinite number of fixed points forming, by definition, a straight line (the axis of rotation).
In the usual intuitional treatment the con cept of general surface is assumed as a starting point and the plane is then defined as a surface such that if any two of its points are joined by a straight line, the latter lies entirely in the surface. This obviously states more than is required for the determination of the surface. To meet this objection the plane is sometimes defined as generated by drawing straight lines from a fixed point A. to all the points of a straight line a. To obtain the entire plane it is necessary to add the line through A parallel to a. This definition is therefore un satisfactory, because parallel lines require in their definition the previous definition of the plane. Peano has met the difficulty by this definition: Consider three fixed points A. B, C not in a straight line; take a fixed point D within the segment BC, and on the segment AD take a fixed point E; a plane is then gen erated by the lines (rays) from E to every point of the perimeter ABC. It may then be proved that a straight line connecting any two points of such a surface lies on the surface.