This section is from the book "Complete Self-Instructing Library Of Practical Photography", by J. B. Schriever. Also available from Amazon: Complete Self-Instructing Library Of Practical Photography.
627. If the focal length - the distance of the plate from the lens - is different, the volume of light alone does not determine the speed, for the intensity, or concentration, of the light varies with the distance between lens and plate (the focal length of the lens). As the light travels farther away from the lens, its intensity is diminished. Fig. 21 will help us understand this. A cone of light emanating from a source (a) - in this case the diaphragm opening in a lens - forms a disc of light of a certain size on a screen placed at (b) and a larger disc on a more distant screen (c). The areas of these discs, as we already know, compare as the squares of their diameters. Their diameters are proportionate to their distances from the source of light; consequently, the areas of the two discs must be proportionate to the squares of those distances.
628. The total amount of light falling on the two screens is the same, but on the more distant screen it is distributed over a larger area, and its concentration, its intensity, is therefore necessarily less. In Fig. 21 the distance of the screen (c) from the source of light is twice the distance of the screen (b). Area being proportionate to square of distance, the area of the larger disc is consequently four times the area of the smaller one. Each corresponding part (each one-fourth) of the larger disc can therefore only receive one-fourth of the total amount of light, and the intensity can be only one-fourth of the total intensity. Thus we find that the intensity of the light is inversely proportionate to the area - the smaller the area covered by the same quantity of light, the greater is the intensity, and vice versa. As the area is proportionate to the square of the distance, and as the distance here means the focal length of the lens, we conclude that the intensity of the light - and the speed of the lens in so far as it depends on that intensity - is inversely proportionate to the square of the fonts.
629. If the volume of light remains constant, variation in speed is determined by variation in focal length only. In two or more lenses of the same aperture, the volume of light transmitted is the same, and speed is inversely proportionate to square of focus. The speed of a 3 inch and a 6 inch focus lens of the same aperture will compare inversely as 9 and 36, that is, directly as 36 and 9; in other words, the 3 inch focus lens will be four times as quick as the 6 inch focus lens.
630. So far we have presumed that when one of the factors, either the diameter of aperture or the focal length, varied, the other remained constant, and the variation of the one factor alone then governed the variation of speed. We should bear in mind, however, that with a variation in distance of the object there is also a corresponding variation in distance between lens and plate, with a consequent variation in intensity of light, even though the aperture remains constant. This variation is of little importance when the lens is used at or near its equivalent focal distance from the plate, as for general landscape work, for reasons which we understand from what was pointed out in connection with conjugate foci. But when we photograph near-by objects requiring the plate at a greater distance than the equivalent focus, the variation requires consideration. Using a 12 inch focus lens, take a distant landscape requiring the plate at the equivalent focal distance, and a near object requiring a distance of 15 inches between lens and plate. With the same aperture giving the same volume of light, the intensity of the light - the speed of the lens - would be vastly different in the two instances. Being in inverse proportion to square of distance, it would be inversely as 12x12 and 15x15, that is, as 15x15 and 12x12, or as 225 and 144, or as 25 and 16. Exposure time being in inverse proportion to intensity and speed, it would be directly proportionate to square of distance between lens and plate, that is, as 12x12 and 15x15, or as 16 and 25. The near object would therefore require fully one-half again the exposure required for the landscape.
631. In our previous comparison of the speed of two or more lenses we considered only the variation of one factor at a time, assuming the other factor, either the aperture or the focal length, to be the same in both or all of them. If both aperture and focal length are different, the problem of determining the relative speed becomes somewhat less simple. We consider the variation in volume and intensity of light together - in connection with each other - instead of separately. A certain diaphragm opening admits a certain volume of light. With any variation in diameter of this opening, the volume of light varies directly as the square of the diameter. The light possesses a certain intensity at the diaphragm opening. This intensity is gradually diminished, and the diminution is proportionate to the square of the distance traveled. As the relative speed ultimately depends on the relative intensity of light action on the plate, and as we cannot directly compare volume and intensity - quantity and quality - it becomes necessary to have volume represented by intensity, or to estimate the relative intensity of different volumes of the same light at the same point. A certain volume of light represents a certain amount of intensity - initial intensity we may term it - at the point where that volume is estimated. An increase in volume of the same light quite naturally means a corresponding increase in the amount of intensity and vice versa. The initial intensity therefore varies as the volume of light varies, or as the square of the diameter of aperture. As this intensity is diminished in proportion to the square of the distance, we can without further preliminaries conclude that the ratio of the square of aperture to square of focus denotes the relative ultimate intensity, the relative speed.