to JSTon- Metrical Vector Algebra-. 14 T 



once by noticing that the T-points or termini of our pairs of 

 orthogonal lines (such as 1\, Tj) form on the 7-line an 

 involution, with imaginary double points *. Now, this with 

 the role played throughout by the 2-line is the well-known 

 characteristic of the parabolic system of measurement 

 ("absolute involution,'' " circular points'")-. Only the way 

 in which it was here set up is. if 1 may judge, a much more 

 natural one and easier to follow than that based on the usual 

 application of Caylev's ideas. 



Such, however, being the nature of the measurement 

 system equivalent to the proposed Vector Algebra, it is 

 important to notice that it can be set up and used as well in 

 Euclid's as in Lobatchevsky's or Riemann's (elliptic) space, 

 i. e. no matter whether Euclid's parallel-axiom is valid or 

 whether there is a whole pencil of non-intersecting lines, or 

 whether there are none at all. (In order to avoid confusion, 

 it would be well to speak in this connexion of euclidean, 

 lobatchevskyan, and riemannian spaces and to reserve the 

 names parabolic, hyperbolic, and elliptic to the metrical 

 systems themselves.) 



To make these remarks more plain let us take the case of 

 two dimensions, which will suffice, and let us imagine an 

 ordinary spherical surface in ordinary euclidean space. In 

 order to avoid antipodal complications take an appropriately 

 limited portion a of this surface and consider only such 

 figures which, together with all their auxiliaries, do not 

 surpass the limits of a. Let the geodesies of a (great circles) 

 be substituted for the straight lines of our fundamental 

 definitions (P.Y.A.). Thus an arrowed segment OX of such 

 a line will be called a vector, X. One of these geodesies being 

 chosen as the T-line, the sum X+Ywill be defined as the 

 vector whose origin is O and whose end-point is the cross of 

 the geodesies XT y and YT X . Thus vector addition will be 

 commutative. And since Desargues' theorem is obviously 

 valid for spherical triangles in perspective f, the addition of 



* The easiest way to see this is to draw from O. the centre of k } a 

 pencil of lines / and to cut it by any transversal s. Let A be the foot of 

 the normal h drawn from O to s, and let a line I and its corresponding 

 (orthogonal) I' cut s in L and L ', respectively. Then, writing y and — ,V 

 for the coordinates of L',L with respect to A, we have .r.r'=—/r. bv 

 the generalized Pythagorean theorem. Whence also the correspondence 

 /, /' in the pencil of lines is seen to he an involution of the said kind. 



t As far as I can gather this property is not mentioned in text-books. 

 but it is, none the less, a most immediate consequence of the well-known 

 theorem on perspective trikedra. The usual theorem of Desargues is 

 obtained by taking of the space figure a plane section. In order to 

 obtain the above theorem, it is enough to intersect it with our spherical 

 surface. 



