in general, to know how much the surface thereof is magnified: but when
it is necessary to know how many minute objects are contained in a
larger, as for instance, how many given animalculæ are contained in the
bulk of a grain of sand, then we must cube the first number, by which
means we shall obtain the solidity or magnified bulk.
The foregoing rule has been also applied to estimate the magnifying
power of the compound microscope. To this application, Mr. Magny, in the
“Journal d’Economie pour le mois d’Aout 1753,” has made several
objections: one or two of these I shall just mention; the first is the
difficulty of ascertaining with accuracy the precise focus of a small
lens; the second is the want of a fixed or known measure, with which to
compare the focus when ascertained. These considerations, though
apparently trifling, will be found of importance in the calculations
which are relative to deep magnifiers. To this it may be further added,
that the same standard or fixed measure cannot be assumed for a
short-sighted, that is used for a well-constituted eye. To obviate these
difficulties, and some errors in the methods which were recommended by
Mess. Baker and Needham, Mr. Magny offers the following
PROPOSITION. All convex lenses of whatsoever foci, double the apparent
diameter of an object, provided that the object be at the focus of the
glass on one side, and the eye be at the same distance, or on the focus
of the glass, at the opposite side.
EXPERIMENT. Take a double convex lens, of six or eight inches focus, and
fix it as at A, Fig. 1, Plate II. A, into the piece A, which is fixed
perpendicular to the rule F G, and may be slid along it by means of its
socket: the rule is divided into inches and parts. Paste a piece of
white paper, two or three tenths of an inch broad, and three inches
long, on the board D; draw three lines with ink on this piece of paper,
so as to divide it into four equal parts, taking care that the middle of
the paper corresponds with the center of the lens. There is also a
sliding eye-piece, which is represented at e.
Take this apparatus into the darkest part of the room, but opposite to
the window; direct the glass towards any remarkable and distant object
which is out of doors, and move the sliding piece B, until the image of
the object on the paper be sharp and clear. The distance between the
face of the paper and the lens (which is shewn on the side of the rule
by the divisions thereon) is the focus of the glass; now set the
eye-piece e E to the same distance on the other side of the glass, then
with one eye close to the sight at e, look at the magnified image of the
lines, and with the other eye at the lines themselves: the image, seen
by means of the glass, and expressed in the figure by the dotted lines,
will be double the breadth of the same object seen by the natural eye.
This will be found to be true, whatsoever is the focus of the lens with
which the experiment is made.
This experiment is rendered more simple to those who are not accustomed
to observe with both eyes at the same time, by making use of half a
lens, and placing the diameter perpendicular to the rule, as they may
then readily view the magnified image and real object with the same
glance of the eye, and thus compare them together with ease and
accuracy.
Let the angle A F B, Fig. 3. Plate II. A, represent that which is formed
at the naked eye, by the rays of light which pass from the extremities
of the object, and unite at the eye in the point F. The angle D F E is
formed of the two rays, which at first proceeded parallel to each other
from the extremities of the object, but that were afterwards so
refracted, or bent, by passing through the glass, as to unite at its
focal point F. C O is equal to the focal distance of the lens on the
side next the object, C F equal thereto on the side next the eye, F O
the distance of the eye.
From the allowed principles of optics, it is evident, that the object
would appear double the size to the eye at C, than it would to the eye
when placed at F; because the distance F O is double the distance C O.
We have only to prove then, that the angle A C B is equal to the angle I
F K, in order to establish the proposition.
The optical axis is perpendicular to the glass and the surface of the
object. The rays A I, B K, which flow from the points A B are parallel
to each other, and perpendicular to the glass, till they arrive at it;
they are then refracted and proceed to F, where they form the triangle I
F K, resting on the base I K: now as C F is equal to C O, and I K is
equal to A B, the two triangles A C B, I F K are similar, and
consequently the angle at C is equal to the angle F. If the visual rays
are continued to the surface of the object, they will form the triangle
D F E, equiangled to the triangle A B C; and therefore, as C O is to A
B, so is F D to D E; and consequently, the apparent diameter of the
object seen through the lens is double the size that it is when viewed
by the naked eye. No notice is here taken of the double refraction of
the rays, as it does not affect the demonstration.
If you advance towards M, half the focal distance, the apparent diameter
will be only increased one-third. If, on the contrary, the point of
sight is lengthened to double the distance of its focus, then the
magnified diameter will appear to be three times that of the real
object. Mr. Magny concludes from hence, that there is an impropriety in
estimating the magnifying power of the eye glass of compound
microscopes, by seeing how often its focus is contained in eight or ten
inches; and to obviate these defects, he recommends two methods to be
used, which reciprocally confirm each other.
The first and most simple method to find how much any compound
microscope magnifies an object, is the same which is described by Dr.
Hooke in his Micrographia, and is as follows: place an accurate scale,
which is divided into very minute parts of an inch, on the stage of your
microscope; adjust the microscope, till these divisions appear distinct;
then observe with the other eye how many divisions of a rule, similarly
divided and held at the stage, are included in one of the magnified
divisions: for if one division, as seen with one eye through the
microscope, extend to thirty divisions on the rule, which is seen by the
naked eye, it is evident, that the diameter of the object is increased
or magnified thirty times.
For this purpose, we often use a small black ebony rule, (see Fig. 4.
Plate II. A,) three or four tenths of an inch broad, and about seven
inches long; at each inch is fixed a piece of ivory, the first inch is
entirely of ivory, and subdivided into ten equal parts.
2. A piece of glass, Fig. 2, fixed in a brass or ivory slider; on the
diameter of this are drawn two parallel lines, about three-tenths of an
inch long; each tenth being divided, one into three, the second into
four, the third into five parts. To use this, place the glass, Fig. 2,
on the middle of the stage, and the rule, Fig. 4, on one side, but
parallel to it; then look into the microscope with one eye, keeping the
other open, and observe how many parts one-tenth of a line in the
microscope takes in upon the parts of the rule seen by the naked eye.
For instance, suppose with a fourth magnifier that one-tenth of an inch
magnified answers in length to forty-tenths or parts on the rule, when
seen by the naked eye, then this magnifier increases the diameter of the
object forty times.
This mode of actual admeasurement is, without doubt, the most simple
that can be used; by it we comprehend, as it were, at one glance, the
different effects of combined glasses; it saves the trouble, and avoids
the obscurity that attends the usual modes of calculation; but many
persons find it exceedingly difficult to adopt this method, because they
have not been accustomed to observe with both eyes at once. We shall
therefore proceed to describe another method, which has not this
inconvenience.
OF THE NEEDLE MICROMETER.
Fig. 8. Plate II. A, represents this micrometer. The first of this kind
was made by my father, and was described by him in his Micrographia
Illustrata. It consists of a screw, which has fifty threads to an inch;
this screw carries an index, which points to the divisions on a circular
plate, which is fixed at right angles to the axis of the screw. The
revolutions of the screw are counted on a scale, which is an inch
divided into fifty parts; the index to these divisions is a flower de
luce marked upon the slider, which carries the needle point across the
field of the microscope. Every revolution of the micrometer screw
measures ¹⁄₅₀ part of an inch, which is again subdivided by means of the
divisions on the circular plate, as this is divided into twenty equal
parts, over which the index passes at every revolution of the screw; by
which means, we obtain with ease the measure of one-thousandth part of
an inch; for 50, the number of threads on the screw in one inch, being
multiplied by 20, the divisions on the circular plate, are equal to
1000; so that each division on the circular plate shews that the needle
has either advanced or receded one-thousandth part of an inch.
To place this micrometer on the body of the microscope, open the
circular part F K H, Fig. 8. Plate II. A, by taking out the screw G,
throw back the semicircle F K which moves upon a joint at K, then turn
the sliding tube of the body of the microscope, so that the small holes
which are in both tubes may exactly coincide, and let the needle g of
the micrometer have a free passage through them; after this, screw it
fast upon the body by the screw G.
The needle will now traverse the field of the microscope, and measure
the length and breadth of the image of any object that is applied to it.
But further assistance must be had, in order to measure the object
itself, which is a subject of real importance; for though we have
ascertained the power of the microscope, and know that it is so many
thousand times, yet this will be of little assistance towards
ascertaining an accurate idea of its real size; for our ideas of bulk
being formed by the comparison of one object with another, we can only
judge of that of any particular body, by comparing it with another
whose size is known: the same thing is necessary, in order to form an
estimate by the microscope; therefore, to ascertain the real measure of
the object, we must make the point of the needle pass over the image of
a known part of an inch placed on the stage, and write down the
revolutions made by the screw, while the needle passed over the image of
this known measure; by which means we ascertain the number of
revolutions on the screw, which are adequate to a real and known measure
on the stage. As it requires an attentive eye to watch the motion of the
needle point, as it passes over the image of a known part of an inch on
the stage, we ought not to trust to one single measurement of the image,
but ought to repeat it at least six times; then add the six measures
thus obtained together, and divide their sum by six, or the number of
trials; the quotient will be the mean of all the trials. This result is
to be placed in a column of a table, next to that which contains the
number of the magnifiers.
By the assistance of the sectoral scale, we obtain with ease a small
part of an inch. This scale is shewn at Fig. 5, 6, 7. Plate II. A, in
which the two lines c a c b, with the side a b, form an isosceles
triangle; each of the sides is two inches long, and the base one-tenth
of an inch. The longer sides may be of any given length, and the base
still only of one-tenth of an inch. The longer lines may be considered
as the line of lines upon a sector opened to one-tenth of an inch.
Hence, whatever number of equal parts ca cb are divided into, their
transverse measure will be such a part of one-tenth as is expressed by
their divisions. Thus, if it be divided into ten equal parts, this will
divide the inch into one-hundred equal parts; the first division next c
will be equal to one-hundredth part of an inch, because it is the tenth
part of one-tenth of an inch. If these lines be divided into twenty
equal parts, the inch will be by those means divided into two hundred
equal parts. Lastly, if a b c a be made three inches long, and divided
into one-hundred equal parts, we obtain with ease the one-thousandth
part. The scale is represented as solid at Fig. 6, but as perforated at
Fig. 5 and 7; so that the light passes through the aperture, when the
sectoral part is placed on the stage.
To use this scale, first fix the micrometer, Fig. 8. Plate II. A, to the
body of the microscope; then fit the sectoral scale, Fig. 7, in the
stage, and adjust the microscope to its proper focus or distance from
the scale, which is to be moved till the base appears in the middle of
the field of view; then bring the needle point g, Fig. 8, by turning the
screw L, to touch one of the lines c a exactly at the point answering to
20 on the sectoral scale. The index a of the micrometer, Fig. 8, is to
be set to the first division, and that on the dial plate to 20, which is
both the beginning and end of its divisions; we are then prepared to
find the magnifying power of every magnifier in the compound microscope
which we are using.
EXAMPLE. Every thing being prepared agreeable to the foregoing
directions, suppose you are desirous of ascertaining the magnifying
power of the lens marked No. 4; turn the micrometer screw, until the
point of the needle has passed over the magnified image of the tenth
part of one inch; then the division, where the two indices remain, will
shew how many revolutions, and parts of a revolution, the screw has
made, while the needle point traversed the magnified image of the
one-tenth of an inch; suppose the result to be twenty-six revolutions of
the screw, and fourteen parts of another revolution, this is equal to 26
multiplied by 20, added to 14; that is, 534 thousandth parts of an
inch.
The twenty-six divisions found on the strait scale of the micrometer,
while the point of the needle passed over the magnified image of
one-tenth part of an inch, were multiplied by 20, because the circular
plate C D, Fig. 8, is divided into twenty equal parts; this produced
520; then adding the fourteen parts of the next revolution, we obtain
534 thousandth parts of an inch, or 5-tenths and 34-hundredth parts of
another tenth, which is the measure of the magnified image of 1-tenth of
an inch, at the aperture of the eye glasses, or at their foci. Now if we
suppose the focus of the two eye-glasses to be one inch, the double
thereof is two inches; or if we reckon in the thousandth part of an
inch, we have two thousand parts for the distance of the eye from the
needle point of the micrometer. Again, if we take the distance of the
image from the object at the stage at six inches, or six thousandths,
and add thereto two thousand, double the distance of the focus of the
eye glass, we shall have eight thousand parts of an inch for the
distance of the eye from the object; and as from the proposition, page
51, we gather that the glasses double the image, we must double the
number 534 found upon the micrometer, which then makes 1068: then, by
the following analogy, we shall obtain the number of times the
microscope magnifies the diameter of the object; say, as 240, the
distance of the eye from the image of the object, is to 800, the
distance of the eye from the object, so is 1068, double the measure
found on the micrometer, to 3563, or the number of times the microscope
magnifies the diameter of the object. By working in this manner, the
magnifying power of each lens used with the compound microscope may be
easily found, though the result will be different in different compound
microscopes, varying, according to the combination of the lenses, their
distance from the object, and one another, &c.
Having discovered the magnifying power of the microscope, with the
different object lenses that are used therewith, our next subject is to
find out the real size of the objects themselves, and their different
parts; this is easily effected, by finding how many revolutions of the
micrometer-screw answer to a known measure on the sectoral scale, or
other object placed on the stage; from the number thus found, a table
should be constructed, expressing the value of the different revolutions
of the micrometer with that object lens, by which the primary number was
obtained. Similar tables must be constructed for each object lens. By a
set of tables of this kind, the observer may readily find the measure of
any object he is examining; for he has only to make the needle point
traverse over this object, and observe the number of revolutions the
screw has made in its passage, and then look into his table for the real
measure which corresponds to this number of revolutions, which is the
measure required.
ACCOUNT OF GLASS, PEARL, &c. MICROMETERS, BY THE EDITOR.
Having seen some glass, &c. micrometers with exquisite fine divisions,
for the purposes of applying to microscopes and telescopes; and in
accuracy, being equivalent to the micrometer just described by our
author, I judge, some account of their application and uses here will be
very acceptable to the curious and inquisitive reader. A particular
description of these as made by the ingenious Mr. Coventry, has been
already given in the Encyclopædia Britannica, Vol. XI. p. 708.
The singular dexterity which Mr. Coventry and others now possess, of
cutting by an engine fine parallel lines upon glass, pearl, ivory, and
brass, at such minute distances as, by means of a microscope, are proved
to be from the 100th to the 5000dth part of an inch, render this sort of
micrometer the easiest and most accurate means of obtaining the exact
natural size of the object to be magnified, and how many times that
object is magnified. Mr. B. Martin, and other opticians, many years ago
applied divided slips of glass, ivory, and horn to the body, in the
focus of the eye glass of microscopes; but the thickness of the whole
medium of the glass was found to diminish the distinct view of the
object: ivory and horn, from their variable texture, were found to
expand and contract too readily to be commodious. It is therefore to Mr.
Cavallo that we are indebted for the happy thought of adapting slips of
divided pearl to telescopes, to ascertain their power, &c. which
substance the opticians now find to be the best for microscopical
micrometers. It possesses a sufficient degree of transparency, when made
about the thickness of writing paper; is a steady substance; admits very
easily of the finest graduations, and is generally made in breadth about
the 20th part of an inch.
Fig. 9. Plate II. A, is a representation of this scale, with divisions
of the 200ths of an inch, every fifth and tenth division being left
longer than the others, which only go to about the middle. If the eye
glass of the microscope or telescope, to which this micrometer is to be
applied, magnify very much, its divisions may be proportionably minute.
To measure by this micrometer the size of an object in a single
microscope, nothing more is required than to lay it on the micrometer,
and adjust it to the focus of the magnifier, noticing how many
divisions it covers or coincides with. Supposing the parallel lines to
be the 1000dths of an inch, and the object covers two divisions, its
real size is the 500th of an inch; if five, 200th of an inch, &c.
To find how much the object is magnified, is not so easily done by the
single, as by the compound microscope, as has been before explained. The
following simple method has been adopted by Mr. Coventry, and which may
be considered tolerably accurate. Adjust a micrometer under the
microscope, suppose 100th of an inch of divisions, with a small object
on it, if square, the better; notice how many divisions one side of the
object covers, suppose ten; then cut a piece of white paper something
larger than the magnified appearance of the object; fix one eye on the
object through the microscope, and the other at the same time on the
paper, lowering it down till the object and the paper appear level and
distinct: then cut the paper till it appear exactly the size of the
magnified object; the paper being then measured, suppose an inch square:
now, as the object under the magnifier, which appeared to be one inch
square, was in reality only ten hundredths, or the tenth of an inch, the
experiment proves that it is magnified ten times in length, one hundred
times in superficies, and one thousand times in cube, which is the
magnifying power of the glass; and in the same manner a table may be
made of the power of all the other glasses.
In using the compound microscope, the real size of the object is found
by the same method as in the single; but to demonstrate the magnifying
power to greater certainty, adopt the following method. Lay a two-feet
rule on the stage, and a micrometer level with its surface, (an inch
suppose, divided into 100 parts:) with one eye see how many of those
parts are contained in the field of the microscope, suppose 50; and with
the other, at the same time, look for the circle of light in the field
of the microscope, which with a little practice will soon appear
distinct; mark how much of the rule, from the center of the stage, is
intersected by the circle of light, which will be half the diameter of
the field. Suppose eight inches; consequently the whole diameter will be
sixteen. Now, as the real size of the field by the micrometers appeared
to be only 50 hundredths, or half an inch, and as half an inch is only
one 32d part of 16 inches, it shews the magnifying power to be 32 times
in length, 1024 superficies, and 32768 in cube or bulk. For accuracy, as
well as for comparative observations, the rule should always be a
certain distance from the eye; eight inches in general is a proper
distance.
Another way, and the most easy for finding the magnifying power of
compound microscopes, is by using two micrometers of the same divisions;
one adjusted under the magnifier, the other fixed in the body of the
microscope in the focus of the eye glass. Notice how many divisions of
the micrometer in the body are seen in one division of the micrometer
under the magnifier, which again must be multiplied by the power of the
eye glass. Example: Ten divisions of the micrometer in the body are
contained in one division under the magnifier; so far the power is
increased ten times: now, if the eye glass be one inch focus, such glass
will of itself magnify about eight times in length, which, with the ten
times magnified before, will be eight times ten, or 80 times in length,
6400 superficies, and 512000 cube.
Fig. 10. Plate II. A, represents the field of view of the compound
microscope, with the pearl micrometer, as applied to the aperture in the
body, called the eye stop; and a magnified micrometer that is laid on
the stage, shewing that one of the latter contains ten of the former.
A set of ivory and glass micrometers, about six in number, besides one
or two pearl ones for the eye stops, are generally packed up with the
best sort of microscopes made by Messrs. W. and S. Jones, Opticians,
Holborn. They are divided into lines and squares, from the 100th to the
1000dth parts of an inch; and, besides measuring the magnifying powers
of microscopes, are generally found useful in measuring the diameters,
proportions, &c. of opake and transparent objects, even of the minutest
kind. The smallest divisions of the glass micrometer to be useful, are
those divided into the 4000dth part of an inch; and as these may be
crossed again with an equal number of lines in the same manner, they
form squares of the SIXTEEN MILLIONTH part of an inch surface, each
square of which appearing under the microscope true and distinct. And,
even small as this is, animalculæ are found so minute as to be contained
in one of these squares!
Glass micrometers with squares, applied to the solar microscope, divide
the objects into squares on the screen in such a manner, as to render a
drawing from it very easy; and are employed with great advantage in the
lucernal microscope.
The micrometers are constructed with moveable frames or tubes, so as to
be either applied or taken away in the readiest manner.
For the uses of the pearl micrometer as applied to the telescope, see
Mr. Cavallo’s pamphlet descriptive of its use, 8vo. 1793, and the
Philosophical Transactions for 1791.
CHAP. III.