PROFESSOR IN BERLIN 299 



His brief stay in Florence only permitted him further to visit 

 the Galleries of the Uffizi, which he left, after a five hours' 

 visit, almost fainting with hunger and fatigue, to take the 

 magnificent walk round the city, at sunset, by the southern 

 heights. After meeting Beltrami on the return journey at 

 Bologna, as had been arranged, for the discussion of a number 

 of geometrical speculations, and problems in mathematical 

 physics, he joined his wife in Vienna (where she was staying 

 with her sister, the wife of the Sectionschef von Schmidt- 

 Zabierow, who was afterwards Governor of Carinthia), and 

 returned to Berlin via Munich. 



At the end of the summer session, and during the autumn 

 holidays, Helmholtz had been occupied, in addition to his 

 electrodynamic researches, with some very abstruse problems 

 in physical optics, which he communicated to the Academy on 

 October 20, 1873, in a brief note ' On the Limits of the Efficiency 

 of the Microscope', afterwards published at length in the jubilee 

 volume of Poggendorff's Annalen for 1874, as ' The Theoretical 

 Limits to the Efficiency of the Microscope '. His researches 

 and results were on the same lines as those of the great master 

 in that branch of optics, Herr Abbe of Jena. Helmholtz 

 attacked the question which is so important for all branches of 

 science, how much it was possible to increase the efficiency 

 of the microscope, and points out that its development had 

 already reached a point at which each minute improvement 

 could be attained only by a disproportionate outlay of mental 

 and mechanical labour. The reason for this was generally held 

 to' lie in the fact that the spherical aberration of small lenses 

 with high curvatures is difficult to overcome ; while Helmholtz 

 considered diffraction and brightness as the essential factors. 

 The ratio of brightness and magnification he finds to be wholly 

 independent of the special construction of the instrument, so 

 that increased magnification only becomes possible with the 

 application of much stronger light ; the dimness of the micro- 

 scopic image thus increases with increasing magnification. 



Helmholtz, moreover, finds that with compound microscopes 

 diffraction produces far more pronounced deviations of the rays 

 from their focal point than do chromatic and spherical aberra- 

 tion, and he therefore subjects it to exact investigation. If the 

 size of the smallest perceptible object is judged by the distance 



