PROFESSOR AT HEIDELBERG 257 



determine what other properties of space besides that of a 

 magnitude of several dimensions were logically conceivable, 

 or, since the question is one of relative magnitudes, algebraically 

 possible, if we set aside the axioms of geometry as hitherto 

 accepted. 



It had been of essential importance in Helmholtz's investiga- 

 tions that he had, in his work on physiological optics, met with 

 two other cases of magnitudes with several variables, which in 

 their system of measurement exhibited certain fundamental 

 differences as compared with spatial measurements. Whereas 

 in space there is a relation of magnitude between any two 

 points, comparable with that existing between any two others 

 i. e. the numerical ratio of the distances ab : be, of the three 

 points a, b, c in the region of colour, when the differences of 

 brightness are taken into consideration, the simplest relation is 

 that between four colours, a, b, c, d, when these can each be 

 made by mixing two of them, when they lie in a straight 

 line in the colour table i.e. the ratio of the two proportions 

 in which a and c must be mixed, in order to produce on the one 

 hand b, and on the other d. He had further found, on investi- 

 gating the formation of our visual measurements in the two- 

 dimensional field of vision, that the measurements very probably 

 depended on the fact that the retina was carried by the move- 

 ments of the eye as a fixed circle past the retinal image ; with 

 this difference, however, from measurements in external space 

 that we practically cannot utilize this circle in our measure- 

 ments of the comparison of the lines in different directions. 

 This drew his attention to the influence exerted by the means 

 of measurement upon the system of measurement as a whole, 

 and the form of the results, and these considerations led him 

 to investigations not only of space but of all other poly-dimen- 

 sional regions, in which a magnitude (distance) given by only 

 two points can be compared by measurement with another 

 corresponding to it, relating to any other given pair of points. 

 Helmholtz showed that it is entirely a question of the formula- 

 tion of special postulates, under which the square of the distance 

 of two infinitely near points is brought under the more general 

 form of the Pythagorean proposition, i. e. it is given by a homo- 

 geneous function of the second degree of the differentials of any 

 three magnitudes used for determining the position of the points. 



