Here _s_₀ has real values only as long as _r_ = R; for _r_ = ℛ the distance _s_₀ in pseudospherical space is infinite. The image in plane space is, on the contrary, contained in the sphere of radius _R_, and every point of this sphere forms only one point of the infinite pseudospherical space. The extension in the direction of _r_ is _ds_₀ ℛ² ----- = ----------- _dr_ ℛ² - _r_² For linear elements, on the contrary, whose direction is at right angles to _r_, and for which _t_ is unchanged, we have in both cases _____________________ √_dx_² + _dy_² + _dz_² _t_ τ _σ_ ---------------------- = ----- = ---- = ----- _____________________ _R_ ℛ _r_ √_d_χ² + _d_ϒ² + _d_ζ² __________________ √_x_² + _y_² + _z_² = ------------------- _____________ √χ² + ϒ² + ζ² ON THE RELATION OF OPTICS TO PAINTING. _Being the substance of a series of Lectures delivered in Cologne, Berlin, and Bonn._ I fear that the announcement of my intention to address you on the subject of plastic art may have created no little surprise among some of my hearers. For I cannot doubt that many of you have had more frequent opportunities of viewing works of art, and have more thoroughly studied its historical aspects, than I can lay claim to have done; or indeed have had personal experience in the actual practice of art, in which I am entirely wanting. I have arrived at my artistic studies by a path which is but little trod, that is, by the physiology of the senses; and in reference to those who have a long acquaintance with, and who are quite at home in the beautiful fields of art, I may compare myself to a traveller who has entered upon them by a steep and stony mountain path, but who, in doing so, has passed many a stage from which a good point of view is obtained. If therefore I relate to you what I consider I have observed, it is with the understanding that I wish to regard myself as open to instruction by those more experienced than myself.