762 PROCEEDINGS OF THE AMERICAN ACADEMY. 



one point being fixed, is an irreducible quadratic form. Our distance 

 and angle are more like Euclidean angle tban Euclidean distance. 



Metric representation of points and lines. 



25. We wish to obtain a representation of point and line which dis- 

 tinguishes A and AA in the case of point addition. Then AA is coinci- 

 dent in position with A but diflFers from it in something that we have 

 called magnitude. Grassman replaced a line by a segment joining two 

 of its points. Then a and Aa give segments differing in length. We 

 wish in this section to determine, if possible, a representation in which 

 a line is replaced by a segment beginning at an arbitrary point of the 

 line, and a point by a sector beginning at an arbitrary line through the 

 point and to determine an addition of these segments or sectors such 

 that their addition relations shall be the same as those of the corre- 

 sponding lines or points. 



Consider the lines through a point A not on f We assume that to 

 a line through A, corresponds a segment AB, to a segment AB a point 

 B, and conversely. Since there is a [1,1] correspondence between the 

 lines and the points B, it follows that to an addition of lines corresponds 

 an addition of points, and that the two additions have the same formal 

 laws. Since a and Aa are represented by different segments, B and AB 

 are different points. The addition of points thus suggests the vector 

 addition of § 1. The segments corresponding to lines through a point 

 should then add like vectors from that point as origin. Distances along 

 a line being proportional to the corresponding vectors, a and Aa should 

 be represented at A by segments whose lengths have the ratio A. 



We thus represent a line of magnitude A by a segment of length A 

 joining two points A, B of the line. Unit lines are represented by 

 segments of unit length. These segments are like localized vectors in 

 physics. Segments on the same line are proportional to their lengths. 

 Segments on different lines may be added by moving them to the point 

 of intersection of the two lines and there adding them like vectors. 



To add two lines Ac, /xd, of magnitudes A and fi, we construct, at 

 their point of intersection A, segments of lengths A, ju, ending in the 

 points C, D. We then draw through A and the harmonic of f with 

 respect to C, D the line h required. Our earlier method of constructing 

 the sum Ac -\- fid. was to draw a line h through A such that 



(hd|cF) = -^. 



That the two constructions are consistent follows since F and P (Fig- 



