Therefore, it follows, that, since the shadow of a square inch of any surface occupies at twice the distance of the surface from the luminous point the space of four square inches, the intensity of the light diminishes as the square of the distance increases. If, consequently, we remove the two sources of light to such distances from an object that they may illuminate it in equal degrees, we are authorized to conclude that their original intensities are inversely as the squares of the distances.
Hence, if two lights of unequal illuminating powers shine upon the same surface at equal obliquities, and an opaque body be interposed between them and the illuminated surface, the two shadows produced must differ in blackness or intensity in the same degree. For the shadow formed by intercepting the greater light, will be illuminated by the smaller light only; and reversely, the other shadow will be illuminated by the greater light; that is to say, the stronger light will be attended with the deeper shadow.
Now it is easy by removing the stronger light to a greater distance, to make the shadow which it produces equal to that afforded by the less light. Experiments of this kind may be made in the following manner:
Fasten a sheet of white paper against the wall of a room, and place the two lights intended to be compared, so that the rays of light from each fall with nearly the same angle of incidence upon the middle of the paper. In this situation, if a book or other object be held to intercept part of the light, which would have fallen on the paper, the shadows may be made to appear as in this figure:
where A represents the surface illuminated by one of the lights only; B, the surface illuminated by the other light; C, the perfect shadow from which both lights are excluded. It will easily be understood that the lights about D and E, near the angle F, will fall with equal incidences when the double shadow is made to occupy the middle of the paper; and consequently, if one or both of the lights be removed directly towards or from the paper, as the appearances may require, until the two shadows at E and D have the same intensity, the quantities of light emitted by each, will be as the squares of the distances from the paper.
By experiments of this kind, many useful particulars may be shewn; for, since the cost and duration of candles, and the consumption of coal gas, or oil in lamps, are easily ascertainable, it may be shewn whether more or less light is obtained at the same expense during a given time, by burning a number of small lights, instead of one or more of greater intensities. And thus we may compare the power of different kinds of lamps or candles, with gas lights of different intensities, so as to determine the relative cost of each particular kind of the combustible substance employed for furnishing light. For example; if a candle and a gas burner supplying coal gas, adjusted by a stop-cock, produce the same darkness of shadow, at the same distance from the wall, the strength or intensity of light is the same.
An uniform degree of intensity of the gas light may readily be produced, by opening or shutting the stop-cock, if more or less light be required, and the candle kept carefully snuffed to produce the most regular and greatest quantity of light. The size of the flame, in experiments of this kind, of course becomes unnecessary, and will vary very much with the quality or chemical constitution of the coal gas. The bulk of the gas consumed, and the weight of tallow or oil used by weighing the candle or oil before and after the experiment furnish the data for calculating the relative cost of tallow, or oil and gas, when compared with each other.
The following statement exhibits the quantity of coal gas consumed in a given time, by different kinds of argand lamps. An argand burner which measures in the upper rim half an inch in diameter, between the holes from which the gas issues, when furnished with five apertures 1⁄25 part of an inch in diameter, consumes two cubic feet of gas in an hour, when the gas flame is one and a half inch high. The illuminating power produced by this burner is equal to three tallow candles eight in the pound.
An argand burner three quarters of an inch in diameter between the holes in the upper rim, and perforated with holes, 1⁄30 of an inch in diameter, consumes three cubic feet of gas in an hour, when the flame is two and a quarter inches high, and produces a light equal in intensity to four tallow candles, eight in a pound.