466 DR. A. N. WHITEHEAD ON 



allowed of a simpler enunciation of physical laws. But in physical research so much 

 depends upon a trained imaginative intuition, that it seems most unlikely that 

 existing physicists would, in general, gain any advantage from deserting familiar 

 habits of thought. 



Part I. (i) consists of general considerations upon the nature of the problem and 

 the method of procedure. Part I. (ii) contains a short explanation of the symbols 

 used. Part II. is devoted to the consideration of three concepts, which embody the 

 ordinary prevailing ideas upon the subject and slight variants from them. The 

 present investigation has, as a matter of fact, grown out of the Theory of Interpoints, 

 which is presented in Part III. (ii), and of the Theory of Dimensions of Part IV. (i). 

 These contain two separate answers to the question : How can a point be defined in 

 terms of lines ? The well-known definition* of the projective point, as a bundle of 

 lines, assumes the descriptive point. The problem is to define it without any such 

 assumption. By the aid of these answers two concepts, IV. and V., differing very 

 widely from the current concepts, have been elaborated. Concept V., in particular, 

 appears to have great physical possibilities. Indeed, its chief difficulty is the 

 bewildering variety of material which it yields for use in shaping explanations of 

 physical laws. It requires, however, the discovery of some appropriate laws of motion 

 before it can be applied to the ordinary service of physical science. 



The Geometry throughout is taken to be three-dimensional and Euclidean. In 

 Concept V. the definition of parallel lines and the " Euclidean " axiom receive new 

 forms ; also the " points at infinity " are found to have an intimate connection with 

 the theory of the order of points on any straight line. The Theory of Dimensions is 

 based on a new definition of the dimensions of a space. 



The main object of the memoir is the development of the TJieory of Interpoints, of 

 the Theory of Dimensions, and of Concept V. The other parts are explanatory and 

 preparatory to these, though it is hoped that they will be found to have some 

 independent value. 



PART I. (i) GENERAL CONSIDERATIONS. 



Definition. The Material World is conceived as a set of relations and of entities 

 which occur as forming the " fields " of these relations. 



Definition. The Fundamental Relations of the material world are those relations 

 in it, which are not defined in terms of other entities, but are merely particularized 

 by hypotheses that they satisfy certain propositions. 



Definition. The hypotheses, as to the propositions which the fundamental relations 

 satisfy, are called the Axioms of that concept of the material world. 



Definition. Each complete set of axioms, together with the appropriate definitions 

 and the resulting propositions, will be called a Concept of the Material World. 



* Here in "Descriptive Geometry" straight lines are open, and three collinear points have a non- 

 projective relation of order ; in " Projective Geometry " straight lines are closed, and four collinear points 

 have a projective relation of separation. 



