64 



KNOWLEDGE. 



February, 1913. 



difference between them and the apochromatic, so incon- 

 spicuous is the secondary spectrum. For photographic work, 

 with the monochromatic screen, they are equally as useful as 

 the much higher-priced lenses. Their spherical correction is 

 excellent, at any rate up to nearly -90 N.A. We, however, 

 begin to note a falling-off in the quality of the image under 

 the higher eyepieces, though nearly all stand an eyepiece of 

 X 10 with little or no deterioration. It may be said that 

 photographs show that such powers will give good definition 

 up to nearly fifteen times their initial power. But in photo- 

 graphy there are no muscae to disturb the sight, and the effect 

 is. above all, cumulative. Even for photography this limit is 

 never exceeded, except for well-marked objects like such 

 diatoms as are well within the resolving power of the objective. 



We may, therefore, take it that one hundred and twenty-five 

 lines to the inch is about the average power of resolution for 

 the ordinary eyesight, and that an eyepiece magnification of 

 ten can be used with any objective of the present day con- 

 sistently with perfect definition and sufficiency of light. 



We must now endeavour to find out the initial power of the 

 objective suitable for each numerical aperture. This is fairly 

 obvious. 



An objective of • 13 N.A. will resolve twelve thousand five hun- 

 dred lines to the inch according to the well-known tables of the 

 Royal Microscopical Society, the accuracy of which have never 

 been impugned. To accomplish this the objective must have an 

 initial power of ten which, multipled by ten (the power of the 

 eyepiece), and then by one hundred and twenty-five (the limit 

 of resolution of ordinary eyesight), gives us the twelve thousand 

 five hundred required. 



Now an objective with an initial power of ten must be of 

 sixteen millimetres focus, i.e., two-thirds of an inch. Thus, a 

 sixteen-millimetre objective should possess an aperture of -13. 

 The lowest usually made at the present day is -20 N.A., so 

 that it has a large surplus of aperture, and would resolve 

 nineteen thousand two hundred and eighty-two lines to the 

 inch with an eyepiece magnifying 15-4 times. As a matter of 

 fact, it is easier to make a sixteen-millimetre objective of the 

 higher N.A. than as low as -13 N.A., which latter could only 

 be done easily by using a single achromatic triplet which would 

 not bear an eyepiece of more than X 7. 



We may now construct a table of magnifications of the 

 corresponding N.A., and of the focus of the objective required 

 to obtain that magnification with an eyepiece amplification of 



X10. 



Focal length of 

 Magnification. N.A. Objective. 



100 -13 16-0 mm. 



200 -26 8-0 „ 



300 -39 5-33 „ 



400 -52 4-0 „ 



500 -65 3-2 „ 



600 -78 2-67 „ 



700 -91 2-3 „ 



The point to be observed is that the above are the minimum 

 figures really required. As we have said, it is not easy to make 

 objectives of as low a numerical aperture as the first five, 

 owing to optical difficulties. Nor would an addition of 

 reasonable amount be objectionable, so long as proper working 

 distance be kept in view. It is, however, useless to give us a 

 four-millimetre (one-sixth) objective of N.A. • 88, as its aperture 

 could not be fully utilized except by employing such high 

 power eyepieces or lengthening the tube as would utterly break 

 down the critical character of the image. Anything above 

 • 52 N.A. for such an objective is of little or no value, and if 

 working distance is sacrificed to obtain a higher aperture it is 

 worse than useless. As a matter of fact • 65 N.A. is the lowest 

 aperture in which such a lens is made. Fortunately, opticians 

 are beginning to see this, and we have now objectives of this 

 focus beautifully corrected with a working distance of one 

 millimetre, thus allowing them to be used with Thoma-Zeiss 

 Haemocytometer, or with thick covers. 



It will be said that greater aperture means more light, but 

 in these days of efficient condensers this is no desideratum. 

 As a general rule, the light nowadays is oftener too much than 

 too little. 



Besides, we sacrifice depth of focus, which is too often 

 confounded with working distance, but which really varies 

 inversely with the N.A. Thus, in two six-millimetre objectives 

 of -62 and -92 N.A. respectively, the former would possess a 

 depth of focus of 1-6, the latter that of 1 • 1 only, or one and a 

 half times the latter — an important point, especially in micro- 

 photography. 



It may be added that the one-twelfth homogeneous 

 immersion usually sold cannot be improved upon either in 

 aperture or power, as will be seen if the reader will carry on 

 the figures in the above. It possesses a good proportion of 

 aperture to power and reasonable working distance, consider- 

 ing the aperture. 



It will be asked what size would the disc of confusion be 

 with such low apertures ? The usual formula being employed, 

 100 1 



we have 



inch which is at any rate a safe 



95000 X -13 123-5 

 limit above the conventional one-hundreth part of an inch. 



With Mr. Conrad Beck's formula rigidly applied, we have the 

 following table : — 



Focus of Objective 

 Magnification. N.A. with Eyepiece 10. 



100 -20 16-0 mm. 



200 -40 80 „ 



300 -60 5-33 „ 



400 -80 4-00 „ 



500 1-00 3-20 „ 



A power of four hundred and fifty would thus be the limit 

 for dry objectives of -90 N.A. The circle of confusion with 

 this series would be the one hundred and ninetieth part of an 

 inch. 



With Mr. Nelson's formula we get : — 



Focus with Eye- 

 Magnification. N.A. piece 10. 



100 -26 16-0 mm. 



200 -52 8-0 „ 



300 -78 5-33 „ 



400 1-04 4-00 „ 



His circle of confusion would thus be the two hundred and 

 forty-seventh part of an inch. 



Comparison of the above with Professor Abbe's figures is 

 not easy, since, as we have said, he decreases the power of the 

 eyepiece as that of the objective increases, but no formula is 

 given for the relationship between the two. They are simply 

 the results of experiments made in 1 874 on several objectives. 

 Professor Abbe's paper was read on June 14th, 1882. 

 The following extract only from his table will suffice. 



Total Magnification Magnification Focus of 



N.A. magnification, of eyepiece. of objective. objective. 



•10 ... 53 ... 10-0 ... 5-3 ... 47-2 mm. 



•20 ... 106 ... 8-2 ... 12-8 ... 19-4 „ 



•30 ... 159 ... 6-7 ... 23-7 ... 10-5 „ 



•40 ... 212 ... 5-6 ... 37-9 ... 6-6 „ 



•50 ... 265 ... 5-0 . ... 53-0 ... 4-7 „ 



•60 ... 317 ... 4-6 ... 68-9 ... 3-6 „ 



•70 ... 370 ... 4-3 ... 86-0 ... 2-9 „ 



•80 ... 423 ... 4-1 ... 103-2 ... 2-42 „ 



•90 ... 476 ... 4-0 ... 119-0 ... 2-10 „ 



That this table does not agree with modern practice will 

 probably be admitted. That modern objectives will bear 

 higher eyepieces than the last six or seven of the table will 

 probably also be allowed. This, however, is no fault in the 

 table itself, which I cannot doubt was correct for most objec- 

 tives of 1874. It is due to Professor Abbe himself that we 

 are able to use new formulae for new objectives. The Abbe- 

 Schott glass it is that enables the modern optician to make a 

 dry one-tenth of an inch objective almost as perfect as the 

 inch or the two-thirds of an inch. Professor Abbe's table 

 shows us only the better the result of his investigations in 

 both the theory and practice of the construction of objectives. 

 Thus, we reject most of his tables as too low. We must 

 reject Mr. Nelson's as too high. There remain Mr. Beck's 

 and my own. The latter are, as I have explained, the 

 minimum that can be adopted with safety, and some may 

 prefer for a magnification of one hundred an aperture of • 15, 



