268 MICROSCOPE happens when divergent pencils of rays pass from one medium into another of higher refractive index. For such divergent pencils, proceeding from air into water or oil, will be closed together or compressed ; so that the rays which, when an object is mounted in air, spread out over the whole hemisphere then form comparatively narrow pencils, and can thus be utilized by an immersion objective of smaller aperture than is required in a dry objective to admit the most diverging rays of air-pencils. It follows, therefore, that a given angle in a water or oil immersion objective represents a much larger aperture than does the same angle in an air-objective ; and thus it comes to pass that by opening out the angle of immersion objectives they may be made to receive and utilize rays of much greater divergence than can possibly enter dry objectives of even maximum aperture. The following table, abridged from that given by Professor Abbe for every - 02 of numerical aperture from 50 up to the maximum of 1 52, brings this contrast into clear view : Numerical Aperture TaWc. Angle of Aperture (=2u). Illumi Theoretical Resolving Pene- Numerical Aperture Dry Water- Homogeneous- nating I ower Power, in Lines to Power (nsinu = a). Objec tives Immersion Objectives Immersion Objectives (a 2 ). an Inch (A=0-5269/n (;) (=1). (=l-33). (n=l 52). =line E). 1-52 . /
180 2-310 146,528 658 1-42 138 12 2-016 136,888 704 1-33 180 122 6 1-770 128,212 752 1-26 142 39 111 59 1-588 121,464 794 1-18 125 3 101 50 1-302 113,752 847 1-12 114 44 94 56 1-254 107,968 893 1-06 105 42 88 26 1-124 102,184 943 1-00 18o" 97 31 82 17 1-000 96,400 1-000 0-94 140 6 89 56 76 24 884 90,616 1064 0-86 118 38 80 34 68 54 740 82,904 1-103 0-80 106 16 73 58 63 31 640 77,120 1-250 0-76 98 56 69 42 60 578 73,264 l-31fi 0-70 88 51 63 31 54 50 490 67,480 1-429 0-62 76 38 55 34 48 9 384 59,768 1-613 0-56 68 6 49 48 43 14 314 53,984 1-786 0-50 60 44 10 38 24 "->50 48,200 2-000 Thus, taking as a standard of comparison a dry objective of the maximum theoretical angle of 180, whose numerical aper ture is the sine of 90, or I OO, we find this standard equalled by a water-immersion objective whose angle of aperture is no more than 97^, and by an oil or homogeneous immersion objective of only 82, the numerical apertures of these, obtained by multi plying the sines of their respective semiangles by the refractive index of water or of oil, being I OO in each case. Each, there fore, will have as great a power of receiving and utilizing divergent rays as any dry objective can even theoretically possess. But, as the actual angle of either a water or an oil immersion objective can be opened out to the same extent as that of an air or dry objective, it follows that the aperture of the former can be augmented far beyond even the theoretical maximum of the latter. Thus the numerical aperture of a water-immersion lens of the maximum angle of 180 is 1 33, or one-third greater than that of an air-lens of the same angle ; and this aperture would be given by an oil-immersion objective of only 122. Again, the numerical aperture of an oil-immersion objective having the theo retical maximum angle of 180 would be l 52, or more than one- half greater than that of an air-lens of the same angle. And the numerical apertures corresponding to angles of 170, which have been actually attained in both cases, fall very little short of the proportions just given. So, again, an oil-immersion objective whose angle of aperture is only 60 has as high a numerical aperture (076) as a water- immersion objective of 69^, or as a dry objective of 99; and a dry objective of 140 has no greater a numerical aperture (0 94) than a water-immersion of 90 or an oil-immersion of 76. This important doctrine may be best made practically intelligible by a comparison of the relative diameters of the back lenses of dry with those of water and oil immersion objectives of the same power, from an "air-angle" of 60 to an "oil-angle" of 180, these diameters expressing, in each case, the opening between the extreme pencil-forming rays at their issue from the posterior surface of the combination, to meet in its conjugate focus for the formation of the image, the relation of which opening in each case to the focal length of the combination is the real measure of its aperture (fig. 16). Thus the dry objective of 60 angle (5 in fig. 16) has its air-angle represented by sini = = 50 numerical aperture. The dry objective of 97 (4) has its air-angle represented by sinw = | = 075 numerical aperture. And the dry objective having the (theoretical) angle of 180 (3) has its air-angle represented by sinw. = 1 00 numerical aperture, this corresponding to 96 water- angle and 82 oil-angle. But the water-immersion lens having the (theoretical) angle of 180 (2) has its water-angle represented by resinw 1 33 numerical aperture. And the oil-immersion lens having the (theoretical) angle of 180 (1) has its oil -angle represented by re sin it 1 52 "numerical apertiire. " x These theoretical apertures for water and oil immersion lenses having been found as nearly attainable in practice as the theoretical maximum for dry objectives, such lenses can utilize rays from objects mounted in balsam or other dense media, which are entirely lost for the image (since they do not exist physically) when the sam< object is in air or is observed through a film of air. And this loss cannot be compensated by an increase of illumination ; because the rays which are lost are different rays physi cally from those obtained by any illumi nation, however intense, through an aeriform medium. It is by increasing the number of diffrac tion-spectra that the additional rays thus received by objectives of great numerical aperture impart to them an increased resolv ing power for lined and dotted objects, the truth of the image formed by the recom bination of these spectra being (as already shown) essentially dependent on the number of them that the objective may be capable of receiving. But whilst the resolving power of micro scopic objectives increases in the ratio of their respective numerical apertures, and whilst their illuminating power (dependent upon the quantity of light that passes through them) increases with the square of the numerical aperture, the case is reversed with another most important quality, that of penetration or focal depth ; for this diminishes as the numerical aperture in creases, until nothing but what is precisely in the focal plane can be even discerned with objectives possessed of the highest resolving power. Thus, the penetrating power of an FIG. 16. Relative Diu- objective of 60 air-angle being expressed meters of Back Lenses as 2 000, an extension of that angle to 76 of Air, Water, and reduces it to 1 613, an extension to 89 9 11 Jmmersion Objec- reduces it to 1 429, and an extension to 99 tives. reduces it to 1 316 ; further extension to 118J reduces it to 1 163, while an objective whose air-angle is 140 has a penetrating power of only 1 064. So, again, the oil-immersion objective which has tl e numerical aperture of I OO corresponding to the theoretical air- angle of 180 has a penetrating power of I OOO; this is brought down to 752 when its angle is so increased as to make its numerical aperture 1 33, equalling the theoretical maximum of a water- immersion objective, and is 658 at the theoretical maximum (1 52J of an oil-objective. Hence it is clear that, as some of the qualities to be sought in microscopic objectives are absolutely incompatible, a preference is to be accorded to objectives of greatest resolving power but very little penetration, or to those of moderate resolving power and great penetration, according to the uses to which they are to be applied; and some general principles will now be laid down in regard to this matter, based alike on science and experience. In the first place, a marked distinction is to be drawn between those objectives of low or moderate power which are to be worked dioptrically and those of high power which are to be worked dif- fractively. The objects on which the former are to be for the most part used are either minute transparent bodies having solid forms which the observer should be able to take in as wholes (as in the case of Polycystina, the larger diatoms, Infusoria, &c. ); or trans parent sections, dissections, or injections, whose parts lie in different planes, the general relations of which he desires to study, while reserving their details for more special scrutiny ; or opaque objects, whose structure can only be apprehended from the examination of their surfaces, when the inequalities of those sur faces are seen in their relations to each other. In all these cases it is desirable that microscopic vision should resemble ordinary vision as much as possible. If the eye were so constructed as to enable us to discern only those parts of an object that lie pre cisely in the plane to which we focus it, our visual conceptions of the forms and relations of these parts, and consequently of the object as a whole, would in general be very inadequate, and often erroneous. It is because, while focussing our eye successively on the several planes of the object, we can see the relation of each to what is nearer and more remote that we can readily acquire a visual conception of its shape as a whole, and that unmistakable perception of solid form which is given by the combination of the two dissimilar perspectives of near objects in binocular vision 1 The dotted circles in the interior of 1 and 2, of the same diameter as 3, show the excess in the diameters of the back lenses of the water and oil objectives
over that of tlu- dry at their re-pective theoretical limits.