LENS

, a piece of glass or other transparent substance, having its two surfaces so formed that the rays of light, in passing through it, have their direction changed, and made to converge and tend to a point beyond the lens, or to become parallel after converging or diverging, or lastly to diverge as if they had proceeded from a point before the lens. Some lenses are convex, or thicker in the middle; others concave, or thinner in the middle; while others are plano-convex, or plano-concave; and some again are convex on one side and concave on the other, which are called meniscuses, the properties of which see under that word. When the particular sigure is not considered, a lens that is thickest in the middle is called a convex lens; and that which is thinnest in the middle is called a concave le<*>s, without farther distinction.

These several forms of lenses are represented in the annexed figure: where A, B are convex lenses, and C, D, E are concave ones; also A is a plano-convex, B is convexo-convex, C is plano-concave, D is concavo-concave, and E is a meniscus.

In every lens, the right line perpendicular to the two surfaces, is called the Axis of the lens, as F G; the points where the axis cuts the surface, are called the Vertices of the l<*>ns; also the middle point between them is called the Centre; and the distance between them, the Diameter.

Some confine lenses within the diameter of half an inch; and such as exceed that thickness, they call Lenticular Glasses.

Lenses are either blown or ground.

Blown Lenses, are small globules of glass, melted in the flame of a lamp or taper. See Microscope.

Ground Lenses, are such as are ground or rubbed into the desired shape, and then polished. For a method of grinding them, and description of a machine for that purpose, see Philos. Trans. vol. xli. pa. 555, or<*>Abr. viii. 281.

Maurolycus first delivered something relative to the nature of lenses; but we are chiefly indebted to Kepler for explaining the doctrine of refraction through mediums of different forms, the chief substance of which may be comprehended in the cases following.

Let DA be a ray of light falling upon a conver dense medium, having its centre at E. When the ray arrives at A, it will not proceed in the same direction At; but it will be there bent, and thrown into a direction AT, nearer the perpendicular AE. In the same manner, another ray falling on B, at an equal distance on the other side of the vertex C, and parallel to the former ray DA, will be refracted into the same point T. And it will also be found, that all the intermediate parallel rays will converge to the same point, very nearly.

On the other hand, if the rays fall parallel on the inside of this denser medium, as in the fig. below, they will tend from the perpendicular EAf; and converge to a point T in the air, or any rarer medium. Also the ray incident on B, at the same distance from the vertex C, will converge to the same place T, together with all the intermediate parallel rays.

Since therefore rays are made to converge when they pass either from a rarer or a denser medium terminated by a convex surface, and converge again when they pass from the same medium convex towards the rarer, a lens which is convex on both sides must, on both accounts, make parallel rays converge to a point beyond it. Thus, the parallel rays between A and B, falling upon the convex surface of the glass AB, would in that dense medium have converged to T; but that medium being terminated by another convex surface, they will be made more converging, and be collected at some place F, nearer to the lens.

Again, to explain the effects of a concave glass, let AB be the concave side of a dense medium, the centre of concavity being at E, In this case, DA will be re-| fracted towards the perpendicular EA; and so likewise will the ray incident at B; in consequence of which they will diverge from one another within the dense medium. The intermediate rays will also diverge more or less, as they recede from the axis TC; which, being in the perpendicular, will go straight on.

If the rays be parallel within the dense medium, they will diverge when they pass from thence into a rarer medium, through a concave surface. For the ray DA will be refracted from the perpendicular AE, as will also the ray that is incident at B, together with all the intermediate rays, in proportion to their distance from the axis or central ray TC.

Therefore, if a dense medium, as the glass AB, be terminated by two concave surfaces, parallel rays passing through it will be made to diverge by both the sides of it. Thus the first surface AB will make them diverge as if they had come from the point T; and with the effect of the second surface added to this, they will diverge as from a nearer point, F.

It was Kepler, who by these investigations first gave a clear explanation of the effects of lenses, in making the rays of a pencil of light converge or diverge. He shewed that a plano-convex lens makes rays, that were parallel to its axis, meet at the distance of the diameter of the sphere of convexity; but that if both sides of the lens be equally convex, the rays will have their focus at the distance of the radius of the circle corresponding to that degree of convexity. But he did not investigate any rule for the foci of lenses unequally convex. He only says, in general, that they will fall somewhere in the medium, between the foci belonging to the two different degrees of convexity. It is to Cavalerius that we owe this investigation: he laid down this rule, As the sum of both the diameters is to one of them, so is the other to the distance of the focus. And it is to be noted that all these rules, concerning convex lenses, are applicable to those that are concave, with this difference, that the focus is on the contrary side of the glass. See Montucla, vol. 2, pa. 176; or Priestley's Hist. of Vision, pa. 65, 4to.

Upon this principle it was not difficult to find the foci of pencils of rays issuing from any point in the axis of the lens; since those that are parallel will meet in the focus; and if they issue from the focus, they will be parallel on the other side. If they issue from a point between the focus and the glass, they will continue to diverge after passing the lens, but less than before; while those that come from beyond the focus, will converge after passing the glass, and will meet in a place beyond the opposite focus. This philosopher particularly observed, that rays which issue from twice the distance of the focus, will meet at the same distance on the other side. The most important of these observations have been already illustrated by proper figures, and from them the rest may be easily conceived. Later optical writers have assigned the distances at which rays will meet, that issue from any other place in the axis of a lens; but Kepler was too much intent upon his astronomical and other pursuits, to give much attention to geometry. But, from the whole, Montucla gives the following rule concerning this subject: As the excess of the distance of the object from the glass, above the distance of the focus, is to the distance of the focus; so is this distance, to the place of convergency beyond the glass. And the same rule will find the point of divergency, when the rays issue from any place between the lens and the focus: for then the excess of the distance of the object from the glass, above that of the focus, is negative, which is the same distance taken the contrary way. Montucla, vol. 2, pa. 177.

And from the principle above-mentioned, it will not be difficult to understand the application of lenses, in the rationale of telescopes and microscopes. On these principles too is founded the structure of refracting burning glasses, by which the sun's light and heat are exceedingly augmented in the focus of the lens, whether convex or plano-convex; since the rays, falling parallel to the axis of the lens, are reduced into a much narrower compass; so that it is no wonder they burn some bodies, melt others, and produce other extraordinary phenomena.

In the Philos. Trans. vol. xvii. 960, or the Abr. i. 191, Dr. Halley gives an ingenious investigation of the foci of rays refracted through any lenses, nearly as follows:

Let BEL be a double convex lens, C the centre of the segment EB, and K the centre of the segment EL; BL the thickness or diameter of the lens, and D a point in the axis; it is required to find the point F, or focus, where the rays proceeding from D shall be collected, after being refracted through the lens at A and a, points very near to the axis BL. Put the distance DA or DB=d, the radius CA or CB=r, and the radius Ka or KL=R; also the thickness of the lens BL=t, and m to n the ratio of the fine of the angle of incidence DAG to the sine of the refracted angle HAG or CAM; or m to n will be the ratio of those angles themselves nearly, since very small angles are to each other in the same ratio as their sines. Hence m is as the angle DAG or DAC, n is as the angle HAG or MAC, and because in this case the sides are as their oppo-| site angles, therefore , or which is as the [angle]C; from this take n or the [angle] MAC, and there remains as the [angle]M; hence again , that is ; which shews in what point the rays would be collected after one refraction, viz, when nr is less than But when , the point would be at an infinite distance, or the rays will be parallel to the axis; and when nr is greater than , then MB is negative, or M falls on the other side of the lens beyond D, and the rays still continue to diverge after the first refraction.

The point M being now found, to or from which the rays proceed after the first refraction, and BM - BL being thus given, which call D, by a process like the former it follows that FL, or the focal distance sought, is equal to . And here, instead of D substituting MB - LB or , and putting p for , the same theorem will become , the focal distance sought, in its most general form, including the thickness of the lens; being the universal rule for the foci of double convex glasses exposed to diverging rays.

But if t the thickness of the lens be rejected, as not sensible, the rule will be much shorter, viz, .

If therefore the lens consist of glass, whose refraction is as 3 to 2, it will be . And if it be of water, whose refraction is as 4 to 3, it will be . But, if the lens could be made of diamond, whose refraction is as 5 to 2, it would be .

If the incident rays, instead of diverging, be converging, the distance DB or d will be negative, and then the theorem for a double convex glass lens will be or , in which case therefore the focus is always on the other fide of the glass.

And if the rays be parallel, as coming from an infinite distance, or nearly so, then will d be negative, as well as the terms in the theorem in which it is found; and therefore, the other term prR will be nothing in respect of those infinite terms; and by omitting it, the theorem will be , or for glass .

And here if r = R, or the two sides of the glass be of equal convexity, this last will become barely 2r2/2r or barely r = f the focus, which therefore is in the centre of the convexity of the lens.

If the lens be a meniscus of glass; then, making r negative, the theorem is or for diverging rays, or for converging rays, and or for parallel rays.

If the lens be a double concave glass, r and R will be both negative, and then the theorem becomes for diverging rays, always negative; for converging rays; and for parallel rays.

And here, if the radii of curvature r and R be equal, this last will be barely - r = f for parallel rays falling on a double concave glass of equal curvature.

Lastly, when the lens is a plano-convex glass; then, r being infinite, the theorem becomes for diverging rays, for converging rays, and for parallel rays.

The theorems for parallel rays, as coming from an infinite distance, take place in the common resracting telescopes. And those for converging rays are chiefly of use to determine the focus resulting from any sort of lens placed in a telescope, between the focus of the object-glass and the glass itself; the distance between the said focus of the object-glass and the interposed lens being made = - d; while those for diverging rays are chiefly of use in microscopes, reading glasses, and other cases in which near objects are viewed.

It is evident that the foregoing general theorem will serve to find any of the other circumstances, as well as the focus, by considering this as given. Thus, for instance, suppose it be required to find the distance at which an object being placed, it shall by a given lens be represented as large as the object itself; which is of singular use in viewing and drawing them, by transmitting the image through a glass in a dark room, as in the camera obscura, which gives not only the true figure and shades, but the colours themselves as vivid as the life. Now in this case d is = f, which makes the theorem become , and| this gives . But if the two convexities belong to equal spheres, so as that r = R, then it is d = pr, or = 2r when the lens is glass. So that if the object be placed at the diameter of the sphere distant from the lens, then the focus will be as far distant on the other side, and the image as large as the object. But if the glass were a plano-convex, the same distance would be just twice as much.

Again, recurring to the first general theorem, including t, the thickness of the lens; let the lens be a whole sphere; then t = 2r, and r = R; and hence the theorem reduces to .

And here if d be infinite, the theorem contracts to or ; or for glass : shewing that a sphere of glass collects the sun's rays at half the radius of the sphere without it. And for a sphere of water, the focus is at the distance of a whole radius.

For another example; when a hemisphere is exposed to parallel rays; then d and R being infinite, and t=r, the theorem becomes . That is, in glass it is 4/3r, and in water <*>/4r.

Several other corollaries may be deduced from the foregoing principles. As,

1st. That the thickness of the lens, being very small, the focus will remain the same, whether the one side or the other be exposed to the rays.

2d. If a luminous body be placed in a focus behind a lens, whether plano-convex, or convex on both sides; or whether equally or unequally so; the rays become parallel after refraction, as the refracted rays become what were before the incident rays. And hence, by means of a convex lens, or a little glass bubble full of water, a very intense light may be projected to a great distance. Which furnishes us with the structure of a lamp or lantern, to throw an intense light to an immense distance: for a lens, convex on both sides, being placed opposite to a concave mirror, if there be placed a lighted candle or wick in the common focus of both, the rays reflected back from the mirror to the lens will be parallel to each other; and after refraction will converge, till they concur at the distance of the radius, after which they will again diverge. But the candle being likewise in the focus of the lens, the rays it throws on the lens will be parallel; and therefore a very intense light meeting with another equally intense, at the distance of the diameter from the lens, the light will be surprising: and though it afterwards decrease, yet the parallel and diverging rays going a long way together, it will be very great at a great distance. Lanterns of this kind are of considerable service in the night time, to discover remote objects; and are used with success by fowlers and fishermen, to collect their prey together, that so it may be taken.

If it be required to have the light, at the same time, transmitted to several places, as through several streets, &c, the number of lenses and mirrors must be increased.

3d. The images of objects are shewn inverted in the focus of a convex lens: nor is the focus of the sun's rays any thing else, in effect, but the image of the sun inverted. Hence, in solar eclipses, the sun's image, eclipsed as it is, may be burnt by a large lens on a board, &c, and exhibit a very entertaining phenomenon.

4th. If a concave mirror be so placed, as that an inverted image, sormed by refraction through a lens, be found between the centre and the focus, or even beyond the centre, it will again be inverted by reslection, and so appear erect; in the first case beyond the centre, and in the latter between the centre and the focus. And on these principles the camera obscura is constructed.

5th. The image of an object, delineated beyond a convex lens, is of such a magnitude, as it would be of, were the object to shine into a dark room through a small hole, upon a wall, at the same distance from the hole, as the focus is from the lens.—When an object is less distant from a lens than the focus of parallel rays, the distance of the image is greater than that of the object; otherwise, the distance of the image is less than that of the object: in the former case, therefore, the image is larger than the object; in the latter, it is less.

When the images are less than the objects, they will appear more distinct and vivid; because then more rays are accumulated into a given space. But if the images be made greater than the objects, they will not appear distinctly; because in that case there are fewer rays which meet after refraction in the same point; whence it happens, that rays proceeding from different points of an object, terminate in the same point of an image, which is the cause of confusion. Hence it appears, that the same aperture of a lens may be admitted in every case, if we would keep off the rays which produce confusion. However, though the image be then more distinct, when no rays are admitted but those near the axis, yet for want of rays the image is apt to be dim.

6th. If the eye be placed in the focus of a convex lens, an object viewed through it, appears erect, and enlarged in the ratio of the distance of the object from the eye, to that of the eye from the lens, if it be near; but infinitely if remote.

7th. An object viewed through a concave lens, appears erect, and diminished in a ratio compounded of the ratios of the space in the axis between the point of incidence, and the point to which an oblique ray would pass without refraction, to the space in the axis between the eye and the middle of the object; and the space in the same axis between the eye and the point of incidence, to the space between the middle of the object and the point to which the oblique ray would pass without refraction.

Finally, it may be observed, that the very small magnifying glasses used in microscopes, most properly come under the denomination of lens, as they most approach to the figure of the lentil, a seed of the vetch or pea kind, from whence the name is derived; but the reading glasses, and burning glasses, and all that magnify, come under the same denomination; for their surfaces are convex, although less so. A drop of water is a lens, and it will serve as one; and many have used it by way of | lens in their microscopes. A drop of any transparent fluid, inclosed between two concave glasses, acquires the shape of a lens, and has all its properties. The crystalline humour of the eye is a lens exactly of this kind; it is a small quantity of a translucent fluid, contained between two concave and transparent membranes, called the coats of the eye; and it acts as the lens made of water would do, in an equal degree of convexity.

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ABCDEFGHKLMNOPQRSTWXYZABCEGLMN

Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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LEE
LEGS
LEIBNITZ (Godfrey-William)
LEMMA
LEMNISCATE
* LENS
LEO
LEPUS
LEUCIPPUS
LEVEL
LEVELLING