Account of the Skerryvore lighthouse

by Alan Stevenson

The intensity of light decreases as the observer recedes from the luminous body, in proportion to the square of his distance. Suppose a [beam of light] to proceed from a radiant at F, and we shall have the rays which, of course, move in straight lines, gradually receding from each other, as b, b′, b″, b‴, and c, c′, c″, c‴, so that the section of the beam will increase with the distances F b, and F c; and the same number of rays, being thus spread over spaces continually increasing, will illuminate the surfaces with a less intensity. This decrease of intensity will, therefore, be in the inverse ratio of the extent of the transverse parallel sections of the luminous cones at b and c, which, we know, increase as the square of their distances from the apex of the cone at F. Hence we conclude, that the intensity of any section of a divergent beam of light decreases as the square of its distance from the radiant. This law furnishes us with a simple measure of the comparative intensity of lights. If we suppose two lights so placed that they may separately illuminate adjacent portions of a vertical screen of paper, we may, by repeatedly comparing the luminousness of those surfaces, and moving one of the lights farther from, or nearer to the screen, at length cause the separate portions of the paper to become equally luminous. This arrangement, however, has many practical difficulties, which I shall not wait to specify; but shall at once indicate a more simple and equally correct mode of obtaining the same result, by means of the shadows cast by the lights from an opaque rod, in a vertical position at O ([fig. 94]), placed between them, and a screen covered with white paper on which the shadows fall. It is obvious that the light at F would cause the object O to cast a shadow at SS, while the light at F′ would cast a shadow at S′S′. But while the shadow at S would still receive light from F′, S′ would receive light from F, so that those two shadows are, in fact, the only portions of the screen which are each illuminated only by one of the lights, while every other portion of its surface receives light from both the radiants at F and F′. If we suppose F to be the weaker light, we can bring it nearer the screen, until the shadow S′S′, shall become similar in appearance to the shadow SS; and we shall have the ratio of the intensity of the light at F to that of the light at F′, as (FS′)² is to (F′S)², which distances must be measured with the greatest exactness. Such is the mode commonly used in estimating the comparative intensities of two lights; but there are various precautions which are needful in order to prevent errors in comparing the deepness of the shadows, and to insure the greatest attainable accuracy in the estimate of the power of the lights, which I shall endeavour briefly to describe.

Fig. 94.

More accurate comparison of the intensity of Lights. The difficulties of estimating the deepness or sharpness of the shadow is very great, and many persons seem quite incapable of arriving at any right judgment in this matter. The same person also will discover such unaccountable variations in his decision after observations made at short intervals of time, as, one would think, can only arise from a sudden change of the intensity of one or both lights. M. Peclet, in his Traité de l’éclairage, gives, as the result of his experience (and I can fully confirm his result by my own), that those differences depend less frequently on any real difficulty of estimating the deepness of the shadows, than on variations in the position of the observer, or rather in the angle at which he views the shadows, and that, consequently, in proportion to the distance between the two shadows, this source of error is increased. Any thing like a glossy texture of the surface of the screen, which then, of course, becomes a reflector, also tends to aggravate this evil. Thus, if the two lights which are to be compared be placed on a table, in such situations as to spread pretty far apart on the screen the shadows of a vertical rod placed between them; and if the shadow nearer to the observer seem to be a little deeper or sharper than the other, let the observer look at them from the other side of the table, and their difference will be reversed, and that which seemed the paler, will become the deeper. Again, if the difference between the two shadows be very great when seen from the right side of the screen, it may happen that, on viewing them from the left hand, the difference may still be in favour of the same shadow, but in a much less degree.

“When I observed this effect,” says M. Peclet,[84] “I tried to view the shadows through a transparent screen, but I remarked the same variations. They were indeed even more sensible; for a variation in the distance of the eye of a few centimètres, made a prodigious change in the deepness of the shadows. I observed also that the shadow was much deeper when seen in the line of the light, and that in every other direction, it became paler in proportion as the eye receded from that direction.

[84] Traité de l’éclairage, p. 214.

“In all the cases which I have just described, the differences of the tints when the position is changed, increase in proportion as the shadows are farther separate; and they grow very minute when the shadows are almost touching each other.

Fig. 95.

“Let AB ([fig. 95]) be a white opaque surface, a, a luminous body, and m, a black opaque body, then the shadow b′ cast on AB, will appear deeper when observed from P, than as seen from Q. This is a fact which may be easily verified, and the cause of which is easily conceived. In fact, the surface AB, although it disperses the light, must still reflect more of it, in the directions in which the regular reflection takes place; and hence the rays which are reflected round about the shadow, must have a greater intensity in the direction of P than in that of Q, and, consequently, the shadow b′ must appear deeper from the point P than from Q.