Part 6
From what has been said, it appears plainly, the advantages we gain by microscopes are derived, first, from their magnifying power, by which the eye is enabled to view more distinctly the parts of minute objects: secondly, that by their assistance, more light is thrown into the pupil of the eye, than is done without them. The advantages procured by the magnifying power, would be exceedingly circumscribed, if they were not accompanied by the latter: for if the same quantity of light be diffused over a much larger surface, its force is proportionably diminished; and therefore the object, though magnified, will be dark and obscure. Thus, suppose the diameter of the object to be enlarged ten times, and consequently the surface one-hundred times, yet, if the focal distance of the glass were eight inches, provided this were possible, and its diameter only about the size of the pupil of the eye, the object would appear one-hundred times more obscure when viewed through the glass, than when it was seen by the naked eye; and this even on the supposition that the glass transmitted all the light which fell upon it, which no glass can do. But if the glass were only four inches focal distance, and its diameter remained as before, the inconvenience would be vastly diminished, because the glass could be placed twice as near the object as before, and would consequently receive four times as many rays as in the former case, and we should, therefore, see it much brighter than before. By going on thus, diminishing the focal distance of the glass, and keeping its diameter as large as possible, we shall perceive the object proportionably magnified, and yet remain bright and distinct. Though this is the case in theory, yet there is a limit in optical instruments, which is soon arrived at, but which cannot be passed. This arises from the following circumstances.[27]
[27] Encyclopædia Britannica, last edition, vol. xiii, p. 357.
1. The quantity of light lost in passing through the glass.
2. The diminution in the diameter of the glass or lens itself, by which it receives only a small quantity of rays.
3. The extreme shortness of the focal distance of great magnifiers, whereby the free access of the light to the object we wish to view is impeded, and consequently the reflection of the light from it is weakened.
4. The aberration of the rays, occasioned by their different refrangibility.
To make this more clear, let us suppose a lens made of such dull kind of glass, that it transmits only one half the light that falls upon it. It is evident, that supposing this lens to be of four inches focus, and to magnify the diameter of the object twice, and its own breadth equal to that of the pupil of the eye, the object will be four times magnified in surface, but only half as bright as if it was seen by the naked eye at the usual distance; for the light which falls upon the eye from the object at eight inches distance, and likewise the surface of the object in its natural size, being both represented by 1, the surface of the magnified object will be 4, and the light which makes it visible only 2; because, though the glass receives four times as much light as the naked eye does at the usual distance of distinct vision, yet one half is lost in passing through the glass. The inconvenience, in this respect, can only be removed so far as it is possible to increase the transparency of the glass, that it may transmit nearly all the rays which fall upon it; and how far this can be done, has not been yet ascertained.
The second obstacle to the perfection of microscopic glasses, is the small size of great magnifiers; by which means, notwithstanding their near approach to the object, they receive a smaller quantity of light than might be expected. Thus, suppose a glass of only one-tenth of an inch focal distance, such a glass would increase the visible diameter eighty times, and the surface 6400 times. If the breadth of the glass could at the same time be preserved as great as the pupil of the eye, which we shall suppose one-tenth of an inch, the object would appear magnified 6400 times, and every part would be as bright as it appears to the naked eye. But if we suppose the lens to be only ¹⁄₂₀ of an inch diameter, it will then only receive one-fourth of the light which would otherwise have fallen upon it; therefore, instead of communicating to the magnified object a quantity of light equal to 6400, it would communicate an illumination suited only to 1600, and the magnified object would appear four times as dim as it does to the naked eye. This inconvenience can, however, in a great degree be removed, by throwing a much larger quantity of light on the object. Various methods of effecting this purpose will be pointed out in the course of this work.
The third obstacle arises from the shortness of the focal distance in large magnifiers; this inconvenience can, like the former, be remedied in some degree, by artificial means of accumulating light; but still the eye is strained, as it must be brought nearer the glass than it can well bear, which in some measure supersedes the use of very deep lenses, or such as are capable of magnifying beyond a certain degree.
The fourth obstacle arises from the different refrangibility of the rays of light, which frequently causes such deviations from truth in the appearance of things, that many have imagined themselves to have made surprising discoveries, and have communicated them as such to the world; when, in fact, they have been only so many optical deceptions, owing to the unequal refraction of the rays. In telescopes, this error has been happily corrected by the late Mr. Dollond’s valuable discovery of achromatic glasses; but how far this invention is applicable to the improvement of microscopes, has not yet been ascertained; and, indeed, from some few trials made, there is reason for supposing they cannot be successfully applied to microscopes with high powers; so that this improvement is yet a desideratum in the construction of microscopes, and they may be considered as being yet far from their ultimate degree of perfection.[28]
[28] How many useful and ingenious discoveries have arisen from accidental circumstances? To adduce one recent instance only--Aerostation, a science, which after having baffled the skill and ingenuity of philosophers for a series of years, and by many illiterate persons deemed an idea bordering on absurdity, has been of late discovered, and successfully applied to practice. EDIT.
OF THE MAGNIFYING POWERS OF THE MICROSCOPE.
We have already treated of the apparent magnitude of objects, and shewn that they are measured by the angles under which they are seen, and that this angle is greater or smaller according as the object is nearer to, or further from, the eye; and, consequently, the less the distance at which it can be viewed, the larger it will appear: but from the limits of natural vision, the naked eye cannot distinguish an object that is very near to it; yet, when assisted by a convex lens, distinct vision is obtained, however short the focus of the lens, and, consequently, how near soever the object is to the eye; and the shorter the focus of the lens is, the greater will be the magnifying power thereof. From these considerations, it will not be difficult to estimate the magnifying power of any lens used as a single microscope; for this will be in the same proportion that the limits of natural sight bear to the focus of the lens. If, for instance, the convex lens is of one inch focus, and the natural sight of eight inches, an object seen through that lens will have its diameter apparently increased eight times; but, as the object is increased in every direction, we must square this apparent diameter, to know how much the object is really magnified; and thus multiplying 8 by 8, we find the superficies is magnified 64 times.
From these principles, the following general rule for ascertaining the magnifying power of single lenses, is deduced. Place a small thin transparent object on the stage of the microscope, adjust the lens till the object appears perfectly distinct, then measure the distance accurately between the lens and the object, reduce the measure thus found to the hundredths of an inch, and calculate how many times this measure is contained in eight inches, first reducing the eight inches into hundredths, which will give you the number of times the diameter of the object is magnified; which number multiplied into itself, or squared, gives the apparent superficial magnitude of the object.
As only one side of an object can be viewed at a time, it is sufficient, 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.