NATURE 



293 



THURSDAY, JANUARY 29, 1? 



THE FUNDAMENTAL DEFINITIONS AND 

 PROPOSITIONS OF GEOMETRY, WITH ES- 

 PECIAL REFERENCE TO THE SYLLABUS 

 OF THE ASSOCIATION FOR THE IMPROVE- 

 MENT OF GEOMETRICAL TEACHING 



I DESIRE to offer some suggestions respecting the 

 form and arrangement of the elementary definitions 

 and propositions of the Euclidian geometry. It has ap- 

 peared to me that the recent German textbooks upon the 

 subject have made a great improvement upon the older 

 system, as developed in the works of Euclid and Legendre. 

 I have but recently obtained the " Syllabus of the Associa- 

 tion for the Improvement of Geometrical Teaching " and 

 compared it with the corresponding parts of a summary 

 of my own, the latter still in an inchoate state. 



I now take the liberty of making some remarks on a 

 few points on which I should be greatly pleased to know 

 the views of those interested. In making them, however, 

 no attempt will be made to go below the fundamental 

 conceptions of the subject which are taken for granted in 

 ordinary textbooks. It may be assumed that there is a 

 general agreement that these conceptions are to be taken 

 for granted, and that the only question is respecting their 

 form and arrangement. One general remark may not, 

 however, be out of place. The aim of elementary geometry 

 is to present its definitions and propositions in a perfectly 

 logical arrangement, so that each definition shall be a 

 complete description, and nothing more, and each pro- 

 position be founded strictly on definitions and axioms. 

 It may be doubted whether this perfect ideal is attainable. 

 It might be claimed that our elementary conceptions of 

 relations in space have been derived from experience by 

 processes of abstraction and generalisation, in which no 

 logical order was followed, and that it is impossible to 

 arrange them with that perfect unity which logical method 

 aims at. However this may be, it will, I think, be con- 

 ceded on all sides that all our systems have hitherto been 

 mere approximations to an ideal which no one has actually 

 reached. 



In framing a geometrical definition three different ob- 

 jects may be aimed at. 



1. To express our fundamental conceptions of the thing 

 defined in the most accurate form possible. 



2. To specify those qualities which most completely 

 differentiate the thing defined from all other things. 



3. To describe its axiomatic properties, or those which 

 are subsequently used in demonstrating propositions re- 

 lating to it. 



We thus have three tests which we may apply to a 

 definition and which may lead to different judgments of 

 it. In most cases the same definition will be reached 

 which ever object wc have in view. The only concept 

 the definitions of which can be separately classed under 

 all three heads is, so far as I have noticed, that of a 

 straight line. The fundamental quality of a straight line 

 as we conceive of it is, I think, that of symmetry, or 

 similarity of properties with respect to space on all sides 

 of it. A line which is throughout its whole length per- 

 fectly symmetrical, having no properties on o;ie side which 

 Vol. xxi — No. 535 1 



it does not equally possess on all other sides, is a straight 

 line. A curve is concave on one side and convex on 

 another. The definition of Simpson's Euclid that a 

 straight line lies evenly between its extreme points, may 

 be considered as an attempt to formulate this conception 

 of symmetry. 



The definition which most completely differentiates a 

 straight line from all others is that of some editions of 

 Euclid and Legendre as the shortest distance between two 

 points. It is to be remarked, however, that neither of 

 these properties is directly made use of in demonstrating 

 the subsequent theorems of geometry. The axiomatic 

 definition of a straight line, if I may be allowed to use the 

 expression, is that of Playfair's Euclid, as being lines 

 which must coincide throughout if they coincide in two 

 points. 



Quite similar to that is Definition V. of the Syllabus. 

 This class of definitions, or the axioms in which they are 

 embodied, include the only ones which serve as a basis 

 for the subsequent theorems of geometry. 



It is to the definition of plane figures given in the Syl- 

 labus that the attention of those interested in this subject 

 is especially asked. 



The following are extracts from the Syllabus : — 



" Def. VII. — A plane figure is a portion of a plane sur- 

 face inclosed by a line or lines. 



"Def. VIII. — A circle is a plane figure contained by 

 one line, which is called the circumference, &c. 



"Def. XXII. — A plane rectilineal figure is a portion 

 of a plane surface inclosed by straight lines. 



"Def. XXVIII. — A triangle is a figure contained by 

 three straight lines." 



These definitions agree with those of the old geometry 

 in defining plane figures as inclosed portions of a plane 

 surface. It seems to me that in no part of geometry 

 is greater reform needed than in this. 



Figures on a plane surface should, it seems to me, be 

 defined as lines simply, and not as portions of the surface. 

 The following are some of the objections against the old 

 and in favour of the new system of definition : — 



1. By Defi> ition VII., as quoted above, an ellipse is a 

 plane figure because it incloses a portion of a plane sur- 

 face, but a parabola or hyperbola is not. Three straight 

 lines may form a figure, but two cannot. But if we form 

 a figure of three straight lines we must cut off all those 

 portions of each line which lie outside of its intersection 

 with the other two as forming no part of the figure. 



2. In the modern synthetic geometry figures are con- 

 sidered in a more general way as formed of lines. A 

 triangle, for instance, is a combination of three indefinite 

 straight lines. To this we may, if we please, add the 

 restriction that no two shall be parallel, and that all three 

 shall not pass through a point. The quadrilateral is a 

 combination of four such indefinite lines, to which again, 

 if necessary, may be applied the restriction that no three 

 shall be parallel or pass through a point ; the circle also 

 becomes the line, not the inclosed space. Therefore »hen 

 the student, whose ideas of such figures are only those- of 

 the elementary geometry, passes to the study of the higher 

 geometry, he is obliged to form a new set of conceptions 

 for the same terms ; so great a change, for instance, as 

 substituting the conception of three indefinite straight 

 lines for that of a triangular piece of paper. He reads of 



