December 30, 1892. 



SCIENCIi. 



371 



infinite and undounded as applied to space, vre shall content our 

 selves with a few remarks on the niiinber of dimensions of space. 



In ordinary geometry we say that the limit or boundary of 

 a solid is a surface, the limit of a surface is a line, the limit of 

 a line is a point, while the point is imiivisible. The same thought 

 is expressed in other words when we say a solid has three dimen 

 sions, a surface two. a line one. while a point has no dimensions. 

 Although the question of three dimensions of space has engaged 

 the attention of many philosophers, no one has succeeded, to the 

 present, to give a deep reason which is not based upon our ex- 

 periences why after three passages over the limits (beginning 

 with a solid) we should arrive at the indivisible.' Our inability 

 to coDceive solids or figures of more than three dimensions does 

 not disprove their existence. If we imagine a world of two di- 

 mensions, in which all things consist of two dimensional figures 

 in which the mhabitants are so constituted that they can receive 

 impressions only from things ii. the surface which constiuites 

 their universe, and if we consider how unthinkable to such beings 

 might appear figures of three dimensions, we may perhaps be 

 prepared to admit the po-si lility of a space of more than three 

 dimensions.^ 



The relations of algebra and geometry are such that an equation 

 involving » unknowns (n^ 3) finds its geometric interpretation in 

 a space whose dimensions are equal to the number of unknowns 

 in the equation. The dual (algebraic and geometric) solution of 

 algebraic equations enhances greatly iheir value and interest 

 Algebra does not restrict itself to a fixed number of unknowns. 

 The question whether there is a corresponding practical geometry 

 of a space whose dimensions are not fixed is of the greatest inter- 

 est. We shall designate such a space by E , (0 '^ ?i ^ oc), hence 



E contains all the points of this space. 



In constructing a geometry for E it is necessary to select a set 



of axioms. These axioms must be so chosen that when E be 



n 



comes an £ (0 < a ^ 3) this geometry will lead to results har- 

 monizing with our experiences. We proceed to give a few of the 

 assumptions from an approved work on ??.-dimensional space.' 

 Through each point pass many E _ ^ having the following 



properties: — 



Through point of an E _ . pass many E _ „ on wUich the 



E may be moved ; by this motion the E , may be made to 



occupy completely its first position, while the individual points 

 have changed their positions. 



In each E _ „ there are many E _ q on which the £ . 



may be rotated in itself. By this rotation each point will de- 

 scribe a closed curve. 



Starting with such assumptions, a geometry is constructed by 

 collecting and classifying theorems which rest ultimately upon 

 them. It i'; perhaps worthy of remark that attempts have been 

 made to prove the impossibility of a fourth dimension.' 



As the main object of this article is the presentation of the non- 

 Euclidean geometry of two dimensions, we proceed to develop 

 the foundations on which rests a still more general two-dimen- 

 sional geometry than the one noted in the fore-part of this paper. 

 The understanding of the following processes will demand some 

 mathematical attainments beyond what is required to appreciate 

 the preceding. The formula which we desire to use is given in 

 Killing's Nicht-Euklidische Raumformen, p 14. We shall here 

 give a simple outline of its development, referring the reader to 

 that work for the rigorous proof^ of some of our statements. We 

 give here two almost axiomatic theorems which we shall need 

 later. 



To a triangle whose sides are all infinitesimals all the priiici- 



' Killing's Nicht-Euklidlschen Raumformen, p. 64. 



'' A short romance, entitled " Flatland," depicts the dlfflculty an inhabitant 

 "Df a two-dimension world la square) bad to conceive of three-dimensional 

 apace, even after he had acquired some idea of a one-dimensiOD, or line, 

 "world. The book is published by Roberts Brothers, Boston, Mass. 



■'' iiiillDg's Nlcht-Euklidlschen Raumformen, p. 65. 



' Max Simon, Zii doa Qruudlagen der nicht-eukildischen Geometrle, p S6. 



pies of the ordinary geometry and plane trigonometry apply, in- 

 dependent of the axiom of parallels. If one angle of a triangle 

 becomes an infinitesimal while the others remain finite, the ratio 

 of the sides including the infinitesimal angle has unit_v for its 

 limit. 



Given a triangle with a constant finite side (c) and a constant 

 adjacent angle /3. while the other adjacent angle (a) is infinitesi- 

 mal. The side (a) opposite a must also be infinitesimal. The 

 lines which divide this a into n equal parts also divide a into n 



equal parts. Hence the ratio — depends not upon n (a remain- 

 ing infinitesimal) bit upon c and ,3. The limit of this ratio when 

 P = — and a is in the act of vanishing is denoted by /fc). For 



Let e increase bv an 



any other value of /3 this limit is ^ ^ ' - 



sin 13 



infinitesimal/!, and ^ = -^J^±^. Since '^ x ^l^lJ^ has a 

 a sin (i ah 



finite limit when h is in the act of vanishing, its equal, /'(c). must 

 have the same limit. We may suppose a triangle formed by 

 keeping /3 and e constant while a increases to a finite angle. We 

 thus obtain a triangle in which a, /3, a, c are finite. We will call 

 the third side and the third angle b and y, re.spectively. In this 

 triangle we may let a undergo an infinitesimal increase, da, at 

 the same time a, b, y will increase by da, db, dy, respectively. 

 This increase of the triangle is a triangle like the one just con- 

 sidered, and the formulas obtained are directly applicable to it. 

 The following formulas can easily be proved : — 



da _ f (b) dy _ 



da siny da " ^ " da 



The first has been found in the preceding triangle. From it we 

 also obtain — 



sin(7T-y) ( 



(1) 



= cos y. 



h 



= /'{&); 



but 



a — a _ sin [ir - (- - ;■ + y -f dy) ] 

 h sm y 



hence the second equation. The third follows directly after 

 drawing perpendicular from y upon the side b 4- db. 

 From equations (1) we obtain easily — 

 /" (b) db _ _ cos y dy 



f [b) Si7l y 



Intergrating this — 



log f{b) = — log sin y + log C. 

 When a - 0. it follows that & = e. y — - — .3. From this we find 

 log C, and the equation takes the foim 



f{b) sin y =f{c) sin j3. 

 Differentiating, a and c being regarded constant, we obtain 



/' (b) sin, y db +f (b) cos y dy —f (c) cos (3 da. 

 Substitutin:: from (1), remembering that 3. b takes the places of 

 a, a, there results 



f{a)f\b) -f(b)f\a) cos y=f{c) cos ft. 

 Hence, cyclically, 



na)f(c) -f(c)f{a) cos ft=f(b) cos ;. 

 Multiplying the last by /'(a) and subtracting from the preceding 

 there results. 



/(«) [/(&)-/(a)/(c)]=/(c)[l - ; f(a)\'-]cosft. 

 Combining with this the analogous equation 



f(c) [/■(6)-/(a)/(c)] =/(«) [I - ! /W''] <'osf^- 

 By division, • 



l-!/'(a)t« 1 



i/wr 



|/ (Op 



Since a and c are independent, the members of this equation must 

 he constant. Hence the important equation — 



' .f (c 



;/'(C);- 



. = k-' 



