632. For example let us take two lenses of respectively 2 and 4 inches diameter and 6 and 12 inches focus. Their relative initial intensity as based on relative volume of light is as 2x2 and 4x4, or as 4 and 16. The intensity is diminished in proportion to the square of the focus, or as 36 and 144. Therefore, the relative ultimate intensity, or relative speed, must be as 4-36 and 16-144, or as 1-9 and 1-9; that is, the two lenses have exactly the same speed with those apertures. The fraction denoting the relative ultimate intensity is termed the intensity ratio, or ratio of intensity, and when that is the same for any two or more lenses, their speed is the same.

633. We can analyze this example independently of those figures. The 12 inch focus lens has twice the diameter of the 6 inch focus lens and consequently transmits four times as much light; but the four times larger volume, after traveling twice the distance, is distributed over a four times larger area, and the ultimate intensity is therefore the same with both lenses.

634. Further analysis shows that the diameter of aperture bears the same relation to focus in both of them - or, as we generally term it, both lenses have the same relative aperture. The ratio of 2 to 6 is as 1 to 3, or 1-3, and the ratio of 4 to 12 is as 1 to 3, or 1-3; that is, the diameter is in both instances equal to one-third of the focal length of the lens -the aperture ratio (the ratio of diameter of aperture to focal length) is 1-3.

635. The intensity ratio, as we can plainly see, is the square of the aperture ratio.

We therefore conclude that

/. - Speed depends on the aperture ratio, or relative aperture.

II. - Speed is proportionate to the square of the aperture ratio.

III. - The same aperture ratio, or relative aperture, means the same speed.

636. Thus, if we know the ratio of the diameter of aperture to focal length, regardless of their actual measurements, it is an easy and simple matter to calculate the relative speed.

637. Various systems have been devised to express and mark the relative value of certain openings. Of these systems we shall concern ourselves with two only, the focal ratio system and the uniform system.

638. The focal ratio system is so called, because it is based directly on the ratio of aperture to focus. The diameter of the aperture is represented as a fraction of the focal length (the fractional diameter) - f/2, f/3, f/4, f/8, etc., meaning that the diameter of the aperture is 1-2, 1-3, 1-4, 1-8, etc., of the focal length of the lens, or that the aperture ratio is 1-2, 1-3, 1-4, 1-8, etc. The intensity ratio, or relative intensity, relative speed, as we have previously seen, is the square of the aperture ratio, or, 1-4, 1-9, 1-16, 1-64, etc. Exposure time being in inverse proportion to speed, relative exposure must be as 4, 9, 16, 64, etc.

639. Thus the fractional diameter gives directly, or indirectly, aperture ratio (relative aperture), relative speed and relative exposure. The ratio number, 2, 3, 4, 8, etc., as denominator, with one as numerator, gives the aperture ratio; and the square of the aperture ratio gives the relative intensity or relative speed - and inversely the relative exposure. The following table will help make this clear:

Fractional diameter

f/2

f/3

f/4

f/8,

etc.

Aperture ratio. . . .

1/2

1/3,

1/4.

1/8

etc.

Intensity ratio, or Relative intensity

1/4

1/9

1/16,

1/64,

etc.

Relative exposure..

4

9

16

64

etc.

640. The intensity, or speed, being in direct proportion to the square of the aperture ratio, the speed value of any two openings can thus be easily compared. The speed of f/2 and f/4 for instance, is proportionate to 1-4 and 1-16 respectively. But these fractions compare inversely as their denominators, and the denominators are the squares of the ratio numbers; consequently the speed is in inverse proportion to the squares of the ratio numbers. Thus the speed of f/2, compares with the speed of f/4 inversely as the square of 2 compares with the square of 4, that is, inversely as 4 and 16, or directly as 16 and 4, or as 4 and 1.

641. Exposure time being in inverse proportion to speed (the more speed, the less exposure, and vice versa), it follows that exposures are directly as the squares of the ratio numbers.

642. It is needless to say that the same ratio number means the same relative aperture, and consequently the same speed and the same exposure time with all lenses.

643. Depth of focus being in inverse proportion to diameter of aperture and therefore directly proportionate to the ratio number, it follows that, I - With the same lens, or with several lenses of the same focal length, depth of focus with the various apertures is directly proportionate to the ratio numbers; II - With the same ratio number (relative aperture) and the same focal length, depth of focus as well as speed is the same.

644. The apertures which are marked have been so chosen that each succeeding smaller one has one-half the area of the preceding larger one and consequently requires double the exposure.

645. Beginning with f/1, an aperture the diameter of which is equal to the focal length of the lens, the series of apertures will be as follows: f/1, f/1.41, f/2, f/2.83, f/4, f/5.6, f/8, f/ll.3, f/16, f/22.6, f/32, f/45.25, f/64, f/90.5.

646. The Uniform System (U. S.) is based on the focal ratio system as shown in the following table, and the aperture numbers give directly the relative exposures:

Relative Aperture

f/4

f/5.6

f/8

f/11.3

f/16

f/22.6

f/32

U. S. No.

1

2

4

8

16

32

64

f/45.25

f/64

f/90.5

U. S. No.

128

256

512

647. An aperture of f/4 is here taken as requiring a unit exposure and is marked 1. The next smaller opening, f/5.6, requiring double the exposure of f/4, is marked 2; the next one, f/8, requiring double the exposure of f/5.6 and four times the exposure of f/4, is marked 4, and so on. The numbers 1, 2, 4, etc., thus have reference only to the comparative exposures, and the exposures read directly as those numbers. If No. 1 requires 1 second, No. 16 requires 16 seconds. If No. 8 requires 3 seconds, No. 32 requires 12 seconds. If No. 128 requires 2 seconds, No. 16 requires 1/4 second, and so on.