Pagina:Scientia - Vol. VII.djvu/109

on the use of the differential calculus etc. 101

great circle) from the celestial equator varies more rapidly when the distance is near its maximum, about the times of solstice, than at other times. Thus the variation in the Sun’s Declination1 per hour at the time of the summer solstice (at noon of the day on which the solstice occurs) is 0.36 seconds. The variation at the time of the spring equinox per hour is 59.25 seconds. The change is more than 160 times greater at the equinox than what it is at the solstice. If we consider a degree (or small finite difference) of time larger than an hour, say a day, there will still exist the same kind of contrast, but less in quantity — as theory leads to expect. The change in Declination per day at the Spring Equinox is very much greater than what it is at the Summer Solstice. Like propositions are true of the Winter Solstice, and all similar cases. The principle may be extended from variations of quantities to variations of form (the special object of the Calculus of variations).

This principle seems to be of wide application in Political Economy, more especially in the larger sense of the term to which we are coming in our second Section. It has been employed as an argument in favour of the diffusion of taxation over a great number of commodities2. It may be employed to justify small interferences with the natural course of industry for the sake of large ulterior advrantages. A small readjustment of the entrepreneur’s plan of production and «margins of profitableness», recommended for some considerable non-economic advantage, becomes additionally advisable by the theorem that the economic loss is likely to be not merely small, but very small3. No doubt it is a dangerous doctrine to enunciate; very liable to be abused by Protectionists and other unscrupulous controversialists, who if you give them a differential will take an integral.

  1. The Apparent Declination; «Nautical Almanack», 1909. The variation per hour, at noon of the day on which the hourly variation is a minimum is larger or smaller according as the actual moment of solstice is further from or near to noon.
  2. «Economic Journal», Volume XVIII., p. 555 referring to Vol. VII n. 568. The argument is no doubt overborne by pratical considerations on the other side.
  3. See on the principle of incipient taxes «Economic Journal», Vol. XVIII p. 400.