The Distance of the Visible Horizon

Abstract

HAS not Prof. Lodge in his enthusiasm, which I fully share, for an absolute system of measurement rather overstepped the mark when in the equation 2 R h = d2 for the distance of the visible horizon, he says that “h is not the number of feet, or of metres, or anything else, it is the actual height; d is not the number of miles or of inches to the horizon, but it is the distance itself; and similarly 2 R is the diameter of the earth, and not any numerical specification of that diameter (see NATURE, vol. lv. page 125). Surely the equation as written is an algebraical equation, and, as such, the symbols it contains express numbers and not things. The multiplication as he implies of one length (2 R) by another length (h), is abhorrent to the mind of “the Cambridge mathematician.” The superiority of the formula over the mutilated apology for it which Prof. Lodge quotes, lies in the fact that the equation is true in terms of any conceivable unit of length in which the three lengths involved in it are measured. I am of course aware that the particular formula given may be regarded as an abbreviated statement of the approximate geometrical proposition that the rectangle contained by the diameter of the earth and the height of the observer above its surface equals the square on a line equal to the distance of the visible horizon, in which case, of course, Prof. Lodge's description of the symbols would be accurately true; but I do not think that the formula with this interpretation really illustrates his meaning.