William Benjamin Carpenter.

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angles, those nearer the perpendicular are refracted less than those more-
inclined to the refracting surface. When a pencil of rays, however, im-
pinges on the surface of a denser medium (as when rays passing throughi
Air fall upon "Water or Glass), some of the incident rays are reflected from,
that surface, instead of entering it and undergoing refraction; and the pro-
portion of these rays increases with the increase of their obliquity. Hence-
there is a loss of light in every case in which pencils of rays are made to
pass through lenses or prisms: and this diminution in the brightness of
the image formed by refraction will bear a proportion, on the one hand,
to the number of surfaces through which the rays have had to pass; and,
on the other, to the degree of obliquity of the incident rays, and to the-
difference of the refractive powers of the two media. Hence, in the
assage of a pencil of rays out of Glass into Air, and then from Air into
again, the loss of light is much greater than it is when some
medium of higher refractive power than air is interposed between the two-
glass surfaces; and advantage is taken of this principle in the construc-
tion of Achromatic objectives for the Microscope, the component lenses
of each pair or triplet ( 14) being cemented together by Canada Balsam ^
as also in the interposition of Water or some other liquid between the
covering-glass of the object and the front lens of the objective, in the
' immersion lenses' now coming into general use ( 19). On the other
hand, advantage is taken of the partial reflection of rays passing from air
into glass at an oblique angle to the surface of the latter, in the construc-
tion of the ingenious (non-stereoscopic) Binoculars of Messrs. Powell and
Lealand and of Mr. Wenham ( 81).

2. When, on the other hand, a ray, w o, emerges from a dense
medium into a rare one, instead of following the straight course, it is
bent from the perpendicular according to the same ratio; and to find the
course of the emergent ray, the sine of the angle of incidence must be
multiplied by the ' index of refraction/ which will give the sine of the


angle of refraction. And thus, when an emergent ray falls very obliquely
upon the surface of the denser medium, the refraction which it would
sustain in passing forth into the rarer medium, tending as it does to
deflect it still farther from the perpendicular, becomes so great that the ray
cannot pass out at all, and is reflected back from the plane which separates
the two media, into the one from which it was emerging. This internal

beyond which an oblique ray suffers internal reflection, varies for differ-
ent substances in proportion to their respective indices of refraction.
Thus, the index of refraction of Water being 1.336, no ray can pass out
of it into a vacuum, 1 if its angle of incidence exceed 48 28', since the
sine h li' of that angle, H o c', multiplied by 1.336 equals the radius;
and, in like manner, the 'limiting angle' for Flint-glass, its index of
refraction being 1.60, is 38 41'. This fact imposes certain limits upon
the performance of microscopic Lenses, since of the rays which would
otherwise pass out from glass into air all the more oblique are kept back;
whilst, on the other hand, it enables the Optician to make most advan-
tageous use of glass Prisms for the purpose of reflection, the proportion
of the light which they throw back being much larger than that returned
from the best polished metallic surfaces, and the brilliancy of the reflected
image being consequently greater. Such prisms are of great value to the
Microscopist for particular purposes, as will hereafter appear. ( 33-

3. The Lenses employed in the construction of Microscopes are chiefly
convex ; those of the opposite kind, or concave, being only used to make
certain modifications in the course of the rays passing through convex
lenses, whereby their performance is rendered more exact ( 11, 13). It
is easily shown to be in accordance with the laws of refraction already
cited, that when a bundle of parallel rays, passing through air, impinges
upon a spherical surface of glass, these rays will be made to converge.
For the perpendicular to every point of that surface is the radius drawn
from the centre of the sphere to that point, and prolonged through it; so
that, whilst any ray which coincides with the radial perpendicular will go
on without change* in its course towards the centre of the sphere; every
ray which falls upon the spherical surface at an inclination to its pro-
longed radius undergoes refraction in a degree proportionate (as already
explained) to that inclination. And the effect upon the whole bundle-
will be such, that its rays will be caused to meet at a point, called the
focus, some distance beyond the centre of curvature. This effect will be;
somewhat modified by the passage of the rays into air again through a,
plane surface of glass, perpendicular to the axial ray (Fig. 2); and a lens
of this description, called & plano-convex lens, will hereafter be shown to
possess properties which render it very useful in the construction of
Microscopes. But if, instead of passing through a plane surface, the
rays re-enter the air through a second convex surface, turned in the oppo-
site direction, as in a double-convex lens, they will be made to converge

1 The reader may easily make evident to himself the internal reflection of
Water, by nearly filling a wine-glass with -water, and holding it at a higher level
than his eye, so that he sees the surface of the fluid obliquely from beneath : no
object held above the water will then be visible through it, if the eye be placed
beyond the limiting angle; whilst the surface itself will appear as if silvered,
through its reflecting back to the eye the light which falls upon it from beneath.


Parallel rays, falling on a plano-convex lens of
glass, brought to a focus at the distance of the
diameter of its sphere of curvature; and con-
versely, rays diverging from that point, ren-
dered parallel.

still more. This will be readily comprehended when it is borne in mind
that the contrary direction of the second surface, and the contrary direc-
tion of its refraction (this being from the denser medium instead of into

i fc )> antagonize each other; so
that the second convex surface
exerts an influence on the course
of the rays passing through it,
which is almost exactly equiva-
lent to that of the first. Hence
the focus of SL double-convex lens
will be at just half the distance,
or (as commonly expressed) will
be half the length of the focus
of a plano-convex, lens having
the same curvature on one side
(Fig. 3).

4. The distance of the Focus
from the spherical surface will
depend not merely upon its de-
gree of curvature, out also upon the refracting power of the substance
of which it may be formed; since the lower the index of refraction, the
FIG. 3. less will the oblique rays be de-

flected towards the axial ray, and
the more remote will be their point
of meeting; and conversely, the
greater the refractive index, the
more will the oblique rays be de-
flected towards the axial ray, and
the nearer will be their point of
convergence. A lens made of any
substance whose index of refrac-
tion is 1.5, will bring parallel rays
to a focus at the distance of its
diameter of curvature, after they
have passed through one convex
surface (Fig. 2), and at the dis-
tance of its radius of curvature,
after they have passed through
two convex surfaces (Fig. 3); and as this ratio almost exactly expresses
the refractive power of ordinary crown or plate Glass, we may for all
practical purposes consider the ' principal focus ' (as the focus for parallel
rays is termed) of a double-convex lens to be at the distance of its radius,
that is, in the centre of curvature, and that of & plano-convex, lens to be
at the distance of twice its radius, that is, at the other end of the diame-
ter of its sphere of curvature.

5. It is evident from what has preceded, that as a Double-convex lens
brings parallel rays to a focus in its centre of curvature, it will on the
other hand cause those rays which are diverging from that centre before
they impinge upon it, to assume a parallel direction (Fig. 3); so that, if
a luminous body be placed in the principal focus of a double-convex lens,
its divergent rays, falling on one surface of the lens, as a cone, will pass
forth from its other side as a cylinder. If, however, the rays which fall
upon a double-convex lens be diverging from the farther extremity of the
diameter of its sphere of curvature, they will be brought to. a focus at an

Parallel rays, falling on a double-convex lens,
brought to a focus in the centre of its sphere of
curvature: conversely, rays diverging from that
point rendered parallel.


Rays diverging from the farther extremity of
one diameter of curvature of a double-convex lens,
brought to a focus at the same distance on the
other side.

equal distance on the other side of the lens (Fig. 4); but the more the
point of divergence is approximated to the centre or principal focus, the
farther removed from the other
side will be the point of conver-
gence (Fig. 5), until, the point
of divergence being at the cen-
tre, there is no convergence at
all, the rays being merely render-
ed parallel (Fig. 3); whilst if
the point of divergence be be-
yond the diameter of the sphere
of curvature, the point of con-
vergence will be within it (Fig.
5). The farther removed the
point of divergence, the more
nearly will the rays approach
tne paralJel direction i until, at
length, when the object is very
distant, its rays in effect become
parallel, and are brought together in the principal focus (Fig. 3). If, on
the other hand, the point of divergence be within the principal focus,
they will neither be brought to converge, nor be rendered parallel, but
will diverge in a diminished de-
gree (Fig. 6). And conversely,
if rays already converging fall
upon a double-convex lens, they
will be brought together at a
point nearer to it than its centre
of curvature (Fig. G). The same
principles apply equally to a
plano-convex lens; allowance be-
ing made for the double distance
of its principal focus. They also
apply to a lens whose surfaces
have different curvatures; the
principal focus of such a lens be-
ing found by multiplying the
radius of one surface by the rad-
ius of the other, and dividing this and vice versa.
product by half the sum of the
same radii. The rules by which
the foci of convex lenses may be
found, for rays of different de-
grees of convergence and diver-
gence, will be found in works on

6. The refracting influence of
concave lenses will evidently be
precisely the opposite of that of
convex. Rays which fall upon
them in a parallel direction, will
be made to diverge as if from the
principal focus, which is here

Called the negative foCUS. This in a dimiriisheU'degree.

diverging from points more distant than


will be for a plano-concave lens, at the distance of the diameter or
the sphere of curvature; and for a double-concave, in the centre of that
sphere. In the same manner, rays which are converging to such a
degree, that, if uninterrupted, they would have met in the principal
focus, will be rendered parallel; if converging more they will still meet,
but at a greater distance; and if converging less, they will diverge as
from a negative focus at a greater distance than that for parallel rays.
If already diverging, they will diverge still more, as from a negative
focus nearer than the principal focus; but this negative focus will ap-
proach the principal focus, in proportion as the distance of the point
of divergence is such that the direction of the rays approaches the

7. If a lens be convex on one side and concave on the other, forming
what is called a meniscus, its effect will depend upon the proportion be-
tween the two curvatures. If they are equal, as in a watch-glass, scarcely
any perceptible effect will be produced; if the convex curvature be the
greater, the effect will be that of a less powerful convex lens; and if the
concave curvature be the more considerable, it will be that of a less
powerful concave lens. The focus of convergence for parallel rays in the
first case, and of divergence in the second, may be found by dividing the
product of the two radii by half their difference.

8. Hitherto we have considered only the effects of lenses either on a
' bundle ' of parallel rays, or on a ' pencil ' of rays issuing from a single
luminous point, and that point situated in the line of its axis. If the
point be situated above the line of its axis, the focus will be below it, and
vice versa. The surface of every luminous body may be regarded as com-
prehending an infinite number of such points, from every one of which a
pencils of rays proceeds, to be refracted in its passage through a lens
according to the laws already specified; so that a complete but inverted
image or picture of the object is formed upon any surface placed in the
focus and adapted to receive the rays. It will be evident from what has
gone before, that if the object be placed at twice the distance of the princi-
pal focus, the image, being formed at an equal distance on the other side
of the lens ( 5), will be of the same dimensions with the object : whilst,
on the other hand, if the object (Fig. 7, a b) be nearer the lens, the

j,, IO 7 image A B will be farther from

it, and of larger dimensions;
but if the object A B be farther
from the lens, the image a b
will be nearer to it, and smaller
than itself. Further, it is to
be remarked that the larger
the image in proportion to
the object, the less bright will
it be, beca'use the same amount
of light has to be spread over
a greater surface; whilst an

Formation of Images by Convex Lenses. imag? that ig smaller than the

object will be more brilliant in the same proportion.

9. A knowledge of these general facts will enable the learner to un-
derstand the ordinary action of the Microscope; but the instrument is
subject to certain optical imperfections, the mode of remedying which
cannot be comprehended without an acquaintance with their nature. One
of these imperfections results from the unequal refraction of the rays


-\vhich pass through lenses whose curvatures are equal over their whole
surfaces. If the course of the rays passing through an ordinary convex
lens be carefully laid down (Fig. 8), it will be found that they do not all
meet exactly in the foci already
stated; but that the focus F of
the rays AB, AB, which have
passed through the marginal
portion of the lens, is much
closer to it than that of the rays
a by a b, which are nearer the
line of its axis. This may
be shown experimentally, by
* stopping out ' either the cen-
tral or the marginal portion of
the lens; for it will then be Diagram illustratin s s * herical ^*"-
found that the rays which are allowed to pass through the latter alone
form a distinct image at F; whilst those which pass through the former
alone form a distinct image at /. Hence, if the whole aperture be in
use, and a screen be held in the focus F of the marginal portion of the
lens, the rays which have passed through its central portion will be
stopped by it before they have come to a focus; whilst, if the screen be
carried back into the focus f of the latter, the rays which were most dis-
tant from the axis will have previously met and crossed, so that they will
come to it in a state of divergence, and will pass to c and d. In either
case, therefore, the image will have a certain degree of indistinctness; and
there is no one point to which all the rays can be brought by a single lens
of spherical curvature. The distance F/, between the focal points of the
central and of the peripheral rays of any lens, is termed its Spherical
Aberration. It is obvious that the desired effect could be produced by
such an increase of the curvature round the centre of the lens, and such
a diminution of the curvature towards its circumference, as would make
the two foci coincident. And the requisite conditions may be theoreti-
cally fulfilled by a single lens, one of whose surfaces, instead of being
spherical, is a portion of an ellipsoid or hyperboloid of certain proportions.
But the difficulties in the way of the mechanical execution of lenses of
this description are such, that for practical purposes this plan of construc-
tion is altogether unavailable; besides which, their performance would
only be perfectly accurate for parallel rays.

10. Various means have been devised for reducing the aberration of
lenses of spherical curvature. In the first place, it may be kept down by
using ordinary lenses in the most advantageous manner. Thus the aber-
ration of a Plano-convex lens whose convex side is turned towards paral-
lel rays, is only ly^ths of its thickness; whilst, if its plane side be turned
towards them, the aberration is 4J- times the thickness of the lens.
Hence, when a plano-convex lens is used to form an image by bringing
to a focus parallel or slightly-diverging rays from a distant object, its
convex surface should be turned towards the object; but, when it is used
to render parallel the rays which are diverging from a very near object,
its plane surface should be turned towards the object. The single lens
liaving the least spherical aberration, is a Double-convex whose radii are
as one to six: when the flattest face of this is turned towards parallel
rays, the aberration is nearly 3-J times its thickness; but when its most
convex side receives or transmits them, the aberration is only lyj^ths of
its thickness. Spherical Aberration is further diminished by reducing


the aperture or working-surface of the lens, so as to employ only the rays
that pass through its central part, which, if sufficiently small in propor-
tion to the whole sphere, will bring them all to nearly the same focus.
Such a reduction is made in the Object-glasses of common (non-achro-
matic) Microscopes; in which, whatever be the size of the lens itself, the
greater portion of its surface is rendered inoperative by a stop, which is a
plate with a circular aperture interposed between the lens and the rest of
the instrument. If this aperture be gradually enlarged, it will be seen
that, although the image becomes more and more illuminated, it is at the
same time becoming more and more indistinct; and that, in order to gain
defining power, the aperture must be reduced again. Now, this reduction
is attended with two great inconveniences: in the first place, the loss of
intensity of light, the degree of which will depend upon the quantity
transmitted by the lens, and will vary therefore with its aperture; and,
secondly, the diminution of the Angle of Aperture, that is, of the angle
a b c (Fig. 10) made by the most diverging of the rays of the pencil issu-
ing from any point of an object, that can enter the lens and take part in
the formation of an image of it; on the extent of which angle (as will be
shown hereafter) depend some of the most important qualities of a Micro-
scope. "

11. The Spherical Aberration may'be approximately corrected, how-
ever, by making use of combinations of lenses, so disposed that their op-
posite aberrations shall correct each other, whilst magnifying power is
still gained. For it is easily seen that, as the aberration'of a concave
lens is just the opposite of that of a convex lens, the aberration of a con-
vex lens placed in its most favorable position may be corrected by that of
a concave lens of much less power in its most unfavorable position; so
that, although the power of the convex lens is weakened, all the rays
which pass through this combination will be brought to one focus. It is
thus that the Optician aims to correct the Spherical Aberration, in the
construction of those combinations of lenses which are now employed as
Object-glasses in all Compound Microscopes that are of any real value as
instruments of observation. But this correction is not always perfectly
made: and the want of it becomes evident in ihefog by which the dis-
tinctness of the image, and especially the sharpness of its outlines, is im-
paired; and in the eidola, or false images, on each side of the best focal
point, which impair the perfection of the principal image, and can be
themselves brought into view when proper means are used for their de-
tection. 1 The skill of the best constructors of Microscopic objectives has
been of late years successfully exerted in the removal of the * residual
errors' to which these eidola were due; so that objectives of the largest
angular aperture are now made truly aplanatic, the corrections for Sphe-
rical Aberration being applied with/a perfection which was formerly sup-
posed to be attainable only in the case of Objectives of small or moderate
aperture. Still, the difficulty (and the consequent cost) of producing
such objectives, constitutes one out of many reasons for the preference
of objectives ef moderate aperture, in which the correction for Spherical
Aberration can be easily made complete, for all the ordinary purposes of
scientific investigation ( 17).

12. But spherical aberration is not the only difficulty with which the
Optician has to contend in the construction of Microscopes; for one

1 See Dr. Royston Pigott's description of his " Searcher for Aplanatic Images,""
and its uses, in the "Philos. Transact." for 1870, p. 59.


equally serious arises from the unequal ref rang Utility of the several Col-
ored rays which together make up White or colorless light, 1 so that they
are not all brought to the same focus, even by a lens free from spherical
aberration. It is this difference in their refrangibility, which causes their
complete separation or ' dispersion ' by the Prism into a spectrum; and it
manifests itself, though in a less degree, in the image formed by a convex
lens. For if parallel rays of white light fall upon a convex surface, the
most refrangible of its component rays, namely, the violet, will be brought
to a focus at a point somewhat nearer to the lens than the principal focus,
which is the mean of the whole; and the converse will be true of the red
rays, which are the least refrangible, and whose focus will therefore be
more distant. Thus in Fig. 9, the rays of white light, A B, A" B", which
fall on the peripheral portion of the lens, are so far decomposed, that the
violet rays are brought to a focus a c, and crossing there, diverge again,
and pass on towards F F, whilst
the red rays are not brought to
a focus until D, crossing the di-
vergent violet rays at E E. The
foci of the intermediate rays of
the spectrum (indigo, blue, green,
yellow, and orange) are interme-
diate between these two extremes.
The distance c D between the foci
of the violet and of the red rays
respectively, is termed Chromatic

Aberration. If the image be re- Diagram illustrating Chromatic Aberration.

ceived upon a screen placed at c

the focus of the violet rays violet will predominate in its own color, and it
will be surrounded by a prismatic fringe in which blue, green, yellow,
orange, and red may be successively distinguished. If, on the other
hand, the screen be placed at D the focus of the red rays the image
will have a predominantly red tint, and will be surrounded by a series of
colored fringes in inverted order, formed by the other rays of the spec-
trum which have met and crossed. 3 The line E E, which joins the points
of intersection between the red and the violet rays, marks the ' mean
focus/ that is, the situation in which the colored fringes will be narrow-
est, the ' dispersion ' of the colored rays being the least. As the axial
ray A' B' undergoes no refraction, neither does it sustain any dispersion;
and the nearer the rays are to the axial ray, the less dispersion do they
suffer. Again, the more oblique the direction of the rays, whether they
pass through the central or the peripheral portion of the lens, the greater
will be the refraction they undergo, and the greater also will be their
dispersion; and thus it happens that when, by using only the central part
of a lens ( 13), the chromatic aberration is reduced to its minimum, the
central part of a picture may be tolerably free from false colors, whilst.

Online LibraryWilliam Benjamin CarpenterThe microscope and its revelations (Volume 1) → online text (page 2 of 51)