This section is from the book "Telephotography: An Elementary Treatise On The Construction And Application Of The Telephotographic Lens", by Thomas Rudolphus Dallmeyer. Also available from Amazon: Telephotography and Telephotographic Lens.
It is important for following our subsequent reasoning that the reader should always refer the distance of the object from the lens to a multiple of the focal length of the lens, but bear in mind that the multiple he has to deal with to find the magnification is one (focal length) less. To give another example: suppose the lens to be one of 10 inches focal length as before, and the object he intends to reproduce is ioo inches from the lens. It is distant 10 times the focal length of the lens, but only 9 times from the front focal point, so that he knows the "conjugate" will be one-ninth of the focal length beyond the back focal point, and also that the "magnification" will be one-ninth.
Conversely (and this it should be remarked is the usual manner in which we shall have to use this law in practice), if we wish to make a certain " magnification" (say one-tenth) with a lens of given focal length (say 10 inches), the distance of the lens from the object must be one focal length more, 10+ 1, or 11 times the focal length of the lens.
In general, for a reduction of n times (i.e., a magnification of 1/n): Distance of object from lens=(n+1) times the focal length of lens.
Distance of image from lens =n+1/n times the focal length of the lens.
The reader will readily see from the foregoing that as the focal points are constant, or fixed, for any given positive lens, this lens can only give larger or smaller images of a given object by placing the lens nearer to it or further from it, in a similar manner to that employed by the pinhole camera with the screen at a fixed distance from the pinhole (Fig. 3). If we wish to produce a larger direct image of an object at a given distance with an ordinary lens the only way will be to employ another lens in which the focal points are situated at a greater distance from the centre of the lens, or one with a greater (or longer) focal length.
Let Fig. 21 represent two lenses which differ in their focal lengths, that is to say the distances between their centres and focal points differ. In the upper figure the distant object a b will form an image in the focal plane through f, at a b, of a certain size. In the lower figure the same object will form an image b' a' in the focal plane through f" at b' a'. It is easy to see that the respective sizes of ab, a' b' are directly dependent on the distances of the focal points from the centres of the lenses, or as c f : c' f". In other words, the size of an image given by a lens depends upon its focal length; and if one lens has a greater (or longer) focal length than another, the sizes of the images given by them respectively will be directly proportional to their focal lengths.
In general, if one lens is n times the focal length of another, it will give an image n times the size of the other.
We must impress upon the reader that the focal length of a lens is a measurement and definite characteristic of a lens, but he must always bear in mind that this measurement is carried out under definite conditions - viz., that the focal length of a lens is the distance between the centre of the lens and either of the focal points for parallel incident rays. This is most important, as confusion might otherwise arise as to the respective sizes of images for objects that are comparatively close.
If we compare the sizes of the image of a very distant object (say a building) given by two lenses, one of which has double the focal length of the other, one image will be found to give an image exactly double the size of the other. If, however, we compare the sizes of the image given by the two lenses of a near object, this relation no longer holds absolutely.
For example, let us compare the sizes of the images given by two lenses of 10 and 20 inches focal length respectively, of an object 100 inches from the centre of the lens : with the 10-inch lens: 100/10 = 10; 10- 1=9 ; magnification =1/9 ; " " 20- " "100/20 = 5; 5- 1=4; = 1/4 so that the sizes of the images are not now as 1 : 2 exactly. The smaller the multiple of the focal length the object be distant, the more will the relation of the sizes of the images differ from that of their true focal length ; if the object be 40 inches distant from either lens, the 10-inch lens gives a magnification of-, and the 20- „ „ „ „ „ 1, the proportion being as 1 : 3 here.
The usual statement that the right standpoint for viewing a photograph is a distance equal to the focal length of a lens only holds good when all the objects included in the photograph are very distant; it is not then true of all lenses irrespective of their optical construction. For ordinary positive lenses or lens-systems, images of near objects should be viewed as shown above, the correct viewing distance being practically identical with the distance of the conjugate focus of the nearest foreground object in the image from the centre of the lens. But, as we have asked before (p. 13), how are we to know this from the photograph itself?
The only way to evade the necessity of asking the question is to avoid including a foreground that is too near, and to employ a lens whose focal length is considerably longer than the longest side of the trimmed photograph. This practice is both cumbersome and expensive when ordinary positive lenses are employed. The advantages of the Telephotographic lens will be compared presently.
We have, as already stated, endeavoured in this chapter to simplify the study of the formation of images by considering the two principal points of a lens as combined in one, which we have termed the "centre."
If we still keep to this convention, it must be understood nevertheless that this "centre" is not necessarily the mechanical centre of the glass lens in the case of a single lens, nor midway between the two lenses forming a combination of positive lenses, as might be supposed.