An Elementary Treatise on Optics/Chapter 15

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Chap XIV.: The Eye

An Elementary Treatise on Optics
by Henry Coddington

Chapter 15

Chap. .: Optical Phænomena

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2920520An Elementary Treatise on Optics — Chapter 15Henry Coddington

CHAP. XV.

OPTICAL INSTRUMENTS.

The common Looking-glass.

149. This instrument is not quite so simple as it seems at first sight. It consists of a plate of glass more or less thick, with a sheet of metal at the back of it. This metal, which is an amalgam of block-tin and mercury, fits tightly to the glass, and its surface being smooth and bright, reflects any light thrown on it. However the rays have to pass through the glass on their passage to and from the speculum, and therefore suffer two refractions, which do not indeed affect their general directions, if the surfaces of the glass be accurately parallel, but which have a tendency to cause irregularity in them.

This will easily be seen from Fig. 154, which is purposely drawn on an exaggerated scale. $ABCD$ represents a portion of the glass, $CD$ of the mirror (sections, of course, as usual), $QR$ , $QR'$ are two rays of an oblique pencil, of which the former is incident on the glass at $R$ , refracted to $S$ , reflected to $T$ , and refracted towards $V$ ; the other in like manner. Now let $QR$ , $TV$ meet in $s$ : the refractions at $R$ and $T$ being equal, the effect on the whole is the same as if a reflexion took place at a mirror parallel to $CD$ , placed at $s$ ; in like manner $QR'$ , is as it were, reflected at $s'$ . So far all is regular enough, but are $s$ , $s'$ equidistant from the mirror $CD$ ? because if not, the case is not the simple one of a reflexion at $s$ , $s'$ , instead of one at $S$ , $S'$ . Now strictly speaking $Ss$ , $S's'$ are not equal, for it will be seen by referring to p. 47, that there is an aberration in these refractions, of the amount of $m 2 -1 / 2 n \cdot∆\cdottan θ 2$ , $∆$ being the thickness of the glass, and $m$ the ratio of refraction out of glass into air, which being less than unity, the aberration is negative, and as it increases with $θ$ , the angle of incidence, $s'$ must be nearer the surface than $s$ , and the rays will not diverge from $q'$ , as if reflected at a plane mirror at $s$ , but will form a caustic to which $q'T$ , $q″T'$ will be tangents. Hence, the image of any object situated obliquely, with respect to the mirror and the eye, will be more or less confused, besides being situated in a different place from that which it would occupy, in the case of a plain metallic speculum without glass.

There is another irregularity attending these looking-glasses, which is easily perceived, when the incidence of the rays is oblique; it is an additional reflexion at the first surface of the glass. Indeed, when the obliquity is great, there may be observed a series of reflexions, at the two surfaces of the glass, each of the alternate ones attended with an emergence of part of the rays, as shown in Fig. 155, where $QR$ represents a pencil of rays, partly reflected, and partly refracted at $R$ ; the refracted rays are reflected at $S$ , and in part emerge at $T$ , in part are reflected at $T$ , and at $V$ , and meeting the upper surface again at $X$ are divided, some escaping in the direction $Xx$ , some suffering another reflexion, and so on.

When the image of a candle is seen in a common looking-glass, the following phænomena may be observed, as the obliquity of the incidence is increased: at first one image is seen, then two, then several in a row, apparently decreasing in magnitude and brightness as their distance increases. At a certain point the distances between these images attain their maximum, diminish, and vanish, so that the images are all in a line with the eye, and there appears only one. Past this point the attendant images emerge on the opposite side of the principal one, and become more and more distinctly separated, till on the candle being placed in the plane of the glass, they disappear altogether.

The aberration in the refraction appears to be the cause of the latter phænomena: its effect is the same as if the different pencils which convey to the eye the appearances of the different images, were reflected by plane mirrors, at different distances.

The Concave Mirror.

150. This usually consists of a plano-convex lens of glass, silvered on the convex side, as represented in Fig. 156, where $BAb$ is, in fact, the mirror, the glass lens serving only to give it the required form. The first effect of this glass, is evidently to throw an object at $Q$ to $q$ , making $Aq = m \cdot AQ$ (that is, to increase its apparent distance from the mirror about one half): then in order to find the place of the image of $q$ , given by the metallic mirror, we have the equation

$1 / ∆ + 1 / ∆' = 2 / r$ , or $∆'= ∆ r / 2∆- r$ .

But this is modified again by the glass which alters $∆'$ to $∆' / m$ , so that upon the whole, if we put $δ$ for the original distance of the object from the mirror, (neglecting the thickness of the lens,) that of the ultimate image is

$1 / m \cdot mδr / 2 mδ - r$ , or $δr / 2 mδ - r$ .

When the reflexion takes place obliquely, there is of course a good deal of irregularity, but we cannot enter into the discussion of it at present.

The Multiplying Glass.

151. This is a plano-convex lens, of which the convex side is ground into several plane faces, so as to form a collection of prisms. Fig. 157 will serve to illustrate this, though not exactly in the same manner as those commonly referred to in this Treatise, because the figure of the instrument in question is not one of revolution, though something of the same general appearance.

$AB$ , $BC$ , $CD$ , $DE$ , $EF$ represent plane surfaces, which refract the rays falling on them and produce several images $q$ , $q'$ , $q″$ , $q'$ , $q ″$ …. thus to appearance $multiplying$ the object $Q$ , whence the name.

The Reflecting Goniometer. Fig. 158.

152. The purpose of this instrument, is to measure the inclination of two planes of a chrystal, by bringing them successively into the same position, which is known by their reflecting the light from a given object in the same direction, and observing on the graduated rim of the instrument, the number of degrees through which the chrystal has been turned.

Hadley's Sextant.

153. The principle of this instrument is, that when a ray or pencil of rays is reflected successively by two plane mirrors, inclined to each other, the angle between the first and last directions of the light, is double the inclination of the mirrors. The manner in which this is applied is as follows, $ABC$ (Fig. 159) is an instrument in form of a sector of a circle; $AD$ a moveable radius, carrying a mirror at the centre $A$ . This is placed in such a position, that a pencil of light from $S$ , may be reflected at $A$ , and again at $E$ , where there is a mirror fixed on the limb of the instrument, so that it may pass along the axis of the tube $F$ , which is directed to some determined object, or to the horizon. This is managed in practice, by having the glass at $E$ silvered on the part next to the instrument, and the other part transparent, so that the one object may be seen through the plain glass, and the reflexion of the other in the mirror.

When the two mirrors are parallel, of course the first and last rays are parallel, and the image of a distant object appears just where the object itself does. The position of the moveable radius, with regard to the arc or limb of the instrument, is marked in that particular case; and when this radius is moved so as to bring the image of another object to touch the former, their angular distance is found, from its being twice the angle of the mirror, that is, twice the angle through which the radius has moved, which angle is measured by the portion of the limb it has swept over.

The limb is graduated, so that the zero point answers to that position of the radius, which makes the image of an object cover it, and half degrees of the arc of the instrument are counted as degrees; so that the angle between two objects is found at once, by observing the number of degrees marked on the limb, when they are brought into apparent contact.

The plain mirrors on this instrument are sometimes supplied with advantage by two prisms, which are placed so as to produce a total reflexion of the light. Fig. 160, represents a sextant invented by Professor Amici of Modena, which is described in the Correspondance Astronomique du Baron de Zach.

The Kaleidoscope.

154. The curious images seen in this amusing toy, are produced by two glass mirrors placed so as to touch along one edge, and to form, by their inclination, an angle which is some submultiple of two right angles. The eye is placed as in Fig. 161, and the object is generally a collection of bright glass beads and fragments, inclosed between two disks of glass, at the other end of the instrument. The glass mirrors are smoked, or covered with some rough black substance on the back, to prevent reflexion at the second surface.

The formation of the images will be easily understood from Fig. 162, where $AO$ , $BO$ , are the mirrors, inclined at an angle of 60 degrees. The figure $a$ , placed between them, is reflected at each so as to occupy, in inverted positions, the two sections $AOF$ , $BOC$ . These images are again reflected at $BO$ , $AO$ , and from other images in $COD$ , $FOE$ , which, in this case, conspire to form by another pair of reflexions another image in $EOD$ which completes the circle.

The Camera Lucida. Fig. 163.

155. This pretty little instrument invented by Dr. Wollaston, and which is of great assistance in drawing from nature, consists mainly of a glass prism, the section of which, represented in Fig. 164, is a trapezium, having one right angle ( $B$ ), and an angle of 135° opposite to it at $D$ . It will easily be seen that any ray of light entering the prism perpendicularly to the face $CD$ , and therefore unrefracted, will be reflected by the faces $CB$ , $BA$ , so as to emerge perpendicularly through $AD$ , and therefore appear to come from under the prism. The consequence of this is, that if the instrument be turned towards a distant object, the rays proceeding from it, emerging after the reflexion near $A$ , will appear to form an image of the object on a paper placed under the prism, where it may be traced out with a pencil.

There are sometimes attached to this instrument, a concave lens, which may be placed in front of $CD$ , and a convex one which is placed under the point $A$ , by means of which the apparent distances of the paper and the image may be equalized, so that both may be seen distinctly at the same time, the concave lens bringing the image to the paper, or the convex throwing the paper to a distance, according as the sight of the person, using the instrument, is short or long.

The Camera Obscura.

156. In this instrument, or toy, or whatever it may be called, an image of a distant object, either stationary, or moving, is produced by means of a convex lens, plane mirrors being added either to reflect the rays to the lens, or to throw the image in any required direction, so as to fall on a sheet, or other object placed for its reception.

In Fig. 165, which represents the camera obscura shown by the Plumian Professor, in his Experimental Lectures, $A$ is the lens; $BD$ the plane mirror placed at half a right angle to its axis, so as to throw the refracted rays from the horizontal direction into the vertical one; $CD$ is a plate of ground glass, to receive the image on its under side; $EF$ is a lid with a curtain attached to it, which the spectator puts over his head, to exclude all extraneous light.

Fig. 166, represents another form of this instrument. Here the lens $A$ is horizontal, and the external rays are directed through it by the mirror $B$ placed as before, at an angle of 45 degrees to its axis. The image is received on a surface $C$ , which is curved so as to fit it, and viewed through an aperture at $D$ .

The effect of the camera obscura may be produced without a lens, merely by admitting light through a small aperture in a shutter into a darkened room, for the rays from each point of an external object being allowed to illuminate only a very small surface, will produce a tolerably distinct image without being made to converge, provided this image be received on a white surface, placed near the hole in the shutter. (See Fig. 167.)

The Magic Lantern. Fig. 168.

157. Here a convex lens produces a distant, and therefore magnified image of a near transparent object, which is strongly enlightened by a lamp placed behind it; a concave mirror is generally put behind the lamp to throw as much of the light as possible on the object, which is some group of figures painted on a glass slide, and is inverted so that the image may be cast in an erect position, on a wall or a white sheet, at a proper distance.

In making this instrument it is to be observed, that the place of the slide must be farther from the lens than its principal focus, else the image will be thrown to an infinite distance, or become imaginary.

The Phantasmagoria. Fig. 169.

158. This is merely a magic lantern, in which the figures on the slide instead of being painted on the glass, are left transparent, or slightly tinted, all the rest of the glass being darkened. There is also a contrivance by which the distance of the lens from the slide is altered, when the place of the machine is changed, so as to keep the image on the fixed screen, which in this case is placed between the lantern and the spectator, and made in some degree transparent. The variation in the distance of the image from the lens, and therefore in its magnitude, is meant to give it the appearance of advancing and retiring. The deception in this, however, is incomplete, unless the brightness of the image be made to increase, instead of diminishing, as it increases in size: this may be effected by modifying the quantity of light thrown on the slide.

The Simple Microscope.

159. When a very small object is to be examined, the first expedient that occurs, is to put it as near one's eye as is conveniently possible, because the angle it subtends at the center of the eye being thus increased, it so far appears larger. There is, however, a limit to this, though some persons, particularly those who are short-sighted, can avail themselves of it to a greater extent than others. The cause of the limitation is, as the student will remember, that when an object is placed too near the eye, the extreme rays of any pencil admitted by the pupil are not refracted to the focus on the retina, and the vision becomes indistinct.

This evil may be remedied, in the first place, by supplying the fault of the pupil, that is, by stopping the extreme rays, and admitting the others through a small aperture in a plate of some opaque substance. This accordingly is found to answer pretty well in some cases, but only where there is a good deal of light proceeding from the object.

In order to explain this more completely and correctly, we will suppose $PQR$ , Fig. 170, to be a small object, $1 / 20$ of an inch long, for instance, which is to be examined by a person who cannot see distinctly any thing nearer to his eye than 6 inches. It is plain that the greatest angle the object can subtend at his eye, is that of which the tangent is $1 / 6\times20$ , or $1 / 120$ .

Now let $A$ be a convex lens of half an inch principal focal length, placed at a quarter of an inch from the eye $O$ , and let $AQ$ be $12 / 25$ , and consequently, $Aq$ the distance of the image, 12 inches, its length $pr = Aq / AQ$ . $PR$ , that is, $12\cdot 25 / 12 \cdot 1 / 20$ , or $25 / 20$ ; which is an inch and a quarter.

The angle it subtends at the eye has for its tangent this length divided by the distance $Oq$ , which is $12 1 / 4$ , or $49 / 4$ ; it is, therefore, $25 / 20 \cdot 4 / 49$ , or $1 / 8$ nearly.

The apparent lengths with the naked eye, and with the lens, are therefore, as $1 / 120$ and $1 / 8$ , or as 8 and 120, of which numbers the latter being fifteen times the former, the object is said to be magnified in length fifteen times.

In order to generalize this, let $c$ be the nearest distance for correct vision,

and let	$k = OA$ ,	the distance of the eye from the lens,
	$∆= AQ$ ,
	$F = AF$ ;
	$δ = Aq$ .

Then since $δ = ∆ F / F -∆$ , the linear magnitudes of the object and image are as $∆: δ$ , that is, as $F -∆: F$ .

The angular magnitudes, that is, the angles subtended by the object and image at $O$ , are as $F -∆ / ∆+ k : F / δ + k$ , but the fairer way of stating the matter is to compare the angular magnitudes of the object at the distance $c$ and the image at the distance $δ + k$ : these are as

$F -∆ / c : F / δ + k$ ,

and the magnifying power of the lens is

$F / δ + k \cdot c / F -∆$ .

This of course is increased by diminishing $k$ . If we make this $=0$ by placing the eye close to the lens, the magnifying power becomes

$F / δ \cdot c / F -∆$ , which is equal to $c / ∆$ ,

and is inversely as the distance $AQ$ , which may consequently be diminished with advantage, as long as $Aq$ is left greater than $c$ . The limiting value of $AQ$ is easily found to be $cF / c + F$ , which gives for the extreme magnifying power

$c + F / F$ , or $c / F +1$ .^[1]

The greatest angle under which the image of a line of the length $α$ can be viewed is

$c + F / cF α$ , or $α {1 / F + 1 / c$ }.

It appears from the two last observations, that a long-sighted person derives most advantage from a simple microscope, but that a short sight enables one to view a minute object more closely, and to use a greater magnifying power with a given microscope.

It need hardly be said, that the shorter the focal length of a lens, or the greater its power, (see p. 68.) the more it will magnify.

When a very great power is required, it is not uncommon to use a minute spherule of glass, of water, or of colourless varnish, stuck in a needle-hole in a plate of metal, which should be ground hollow on both sides, so as to be as thin as possible, where the aperture is made. The distance of the principal focus of a sphere from its surface being only half the radius, the magnifying power of such an apparatus is very great.^[2]

The following Table, abridged in part from the Encyclopædia Britannica, gives the magnifying power of small convex lenses or spherules, supposing the least distance of distinct vision to be 7 inches, and the object to be placed (l) at the focus, (2) at the least distance for distinct vision.

Focal length
in terms of
an inch.

No. of times that the
length of the object
is magnified.

No. of times that
the surface is
magnified.

——————

.10

007.00

008.00

000049

000064

.75

009.33

010.33

000087

000107

.50

014.00

015.00

000196

000225

.40

017.50

018.50

000306

000342

.30

023.33

024.33

000544

000592

.20

035.00

036.00

001225

001296

.15

046.66

047.66

002181

002276

.10

070.00

071.00

004900

005401

.05

140.00

141.00

019600

019881

.03

233.33

234.33

054289

054912

.02

350.00

351.00

122500

123201

.01

700.00

701.00

490000

491401

The next Table is for a person whose least distance of correct vision is 5 inches.

Focal length
in terms of
an inch.

No. of times that
the length
is magnified.

No. of times that
the surface is
magnified.

——————

.10

005.00

006.00

000025

000036

.75

006.67

007.67

000044

000058

.50

010.00

011.00

000100

000121

.40

012.50

013.50

000156

000195

.30

016.67

017.67

000278

000312

.20

025.60

026.00

000625

000676

.15

033.33

034.33

001111

001179

.10

050.00

051.00

002500

002601

.05

100.00

101.00

010000

010201

.03

166.67

167.67

027779

028110

.02

250.00

251.00

062500

062901

.01

500.00

501.00

250000

251001

Sometimes instead of one single strong lens, two or more are used, having together the same power. The advantage gained by this is, that the aberration arising from the spherical form of the glasses is lessened.

As for the rest, whether these lenses be placed close together, or at some distance, they may be equally considered as constituting one single refracting instrument.

The Compound Microscope. Fig. 171.

160. In this instrument, a double magnification takes place, for an enlarged real image of the object is produced by means of a convex lens, (called an object glass,) and this image is viewed through another lens, the eye glass, in the same manner as an object is with a simple microscope.

In order to find the magnifying power, we will suppose the first image $q$ to be at the least possible distance from the second lens $B$ , and we will represent the focal length of the first lens by $F$ , that of the second by $F'$ , and the distance between them by $b$ .

Then the image $q$ is greater than the object $Q$ in the ratio of $Aq : AQ$ , that is, of $b - F' + F : F$ .^[3]

The final image is greater than this would be, when viewed at the least distance of correct vision $c$ , in the ratio of $c / F' +1:1$ , supposing $q$ to be as near as possible.

On the whole then, the magnifying power is

$b - F' + F / F (c / F' +1)$ .

The image seen is inverted with respect to the object, as may be seen by the figure.

Amici's reflecting Microscope. Fig. 172.

161. In the 18^th volume of the Transactions of the Italian Society, there is a Memoir giving a detailed account of a catoptrical microscope, invented by Professor Amici, of Modena.^[4]

At one end of a tube $12$ inches long, and $1 1 / 10$ in diameter, is placed a concave metallic speculum of a spheroidal form, having its foci at the distances $2 6 / 10$ and $12$ inches, respectively. The object to be examined is placed on a little shelf projecting from the stand, half an inch below the tube, and reduced to the nearer focus by a plain mirror, placed at half a right angle to the axis of the speculum, at the distance of $1 1 / 2$ inch from it. The image, formed in the farther focus, is viewed through a lens, which may be changed at pleasure, so as to increase or diminish the magnifying power, the object remaining unmoved, which gives this instrument a great advantage, in point of convenience, over the common compound refracting microscope.

In order to make the object sufficiently bright, there are attached to the instrument two concave mirrors, one of $3$ inches diameter, and $2 1 / 2$ focus at the foot of the stand, and the other directly over the object, having an aperture in the center like that in the tube, to admit the rays to the plain mirror.

This instrument has a magnifying power of near a million, and is found extremely convenient from the horizontal position of the tube, which enables the observer to examine an object more at his ease, and for a longer time, than when stooping over a microscope of the common construction.

The Professor has contrived, by a very ingenious arrangement, to convert his instrument into a species of camera lucida, which enables him to draw any object on a magnified scale.

Dr. Smith's Reflecting Microscope. Fig. 173.

162. Here the office of the object-glass is executed by a pair of mirrors $A$ , $B$ , the former concave, the latter convex, having apertures pierced through their centers to give a passage to the light. By this means, a small object being placed at $Q$ , a first reflexion at $A$ would produce an image at $q$ , which being farther from the center of the surface, would be larger than the object, but a second reflexion at the surface $B$ sends back the light which is proceeding to form this, and thus throws the image to $q 2$ still farther magnified; there an eye glass receives the rays, and transmits them with the proper divergence for distinct vision.

A small screen is placed at $c$ to prevent the rays from coming directly from the object to the lens.

The Solar Microscope. Fig. 174.

163. This is merely a sort of magic lantern, in which the light of the Sun, reflected by plane mirrors, and condensed by lenses, is thrown on a minute transparent object, of which a magnified image is formed by means of a lens.

In the figure, the object to be exhibited is placed near the focus of the first combination of lenses, so as to be entirely enlightened by the rays coming through them, and not to be burnt, which would be the case, were it exactly at the focus.

The Heliostat. Fig. 175.

164. This instrument consists of a plane mirror, which is made to revolve by clock-work about an axis parallel to that of the heavens, so as to reflect the Sun's light constantly in one same direction, during the course of the day.

It is found very useful in many optical experiments where a small pencil of solar light admitted into a darkened room, is to be subjected to reflexion, refraction, dispersion, &c.

It is a very convenient appendage to the solar microscope, which is the reason of its description being inserted here.

The Astronomical Telescope. Fig. 176.

165. Generally speaking, the Telescope is in construction analogous to the compound microscope: the only difference is, that the latter instrument is used to examine an object placed near it, which is not distinctly visible, on account of its minuteness, whereas the former is calculated to assist the eye in viewing objects, which look small or indistinct, only on account of their distance.

The telescope substitutes for a distant object an image at any distance required, which subtends at the eye a much larger angle than the object, and from which more light proceeds, than the eye could receive from that object. The manner in which this is effected in this instance, is as follows.

By means of a convex lens $A$ , called the object glass, there is produced an inverted image of a distant object, which image is, of course, at the principal focus of the glass, and has collected on it nearly all the light which, proceeding from the object, falls on the surface of the object glass.

This image is viewed and magnified by means of an eye glass, another convex lens, which is placed so that the image is at its focus, or a little within it, in which latter case, a second image (though not a real one,) is produced at the most convenient distance for correct vision.

It is evident then, that if the eye glass be a powerful lens, which it always is in practice, the final image subtends a much larger angle at the eye which is placed close to the eye glass, than the object does, and it is brighter, for the eye receives in the one case all the light which falls on the object glass, in the other, only as much of that coming from the object as can pass through the pupil of the eye.

166. If the foci of the two lenses coincide, the magnifying power of this instrument is expressed by $AF / Bf$ , or $F / F'$ , if $F$ , $F'$ represent the focal lengths of the object and eye glass. If the final image be brought to the nearest distance of distinct vision ( $c$ ), it is greater in the proportion of $c / F' +1$ to $c / F'$ , (see compound microscope,) and therefore its value is

$F / c (c / F' +1)$ .

It appears from this, that to make a telescope of great magnifying power, the object glass should be of considerable focal length or small power, and the eye glass, on the contrary, very powerful.^[5]

In consequence, before the discovery of reflecting telescopes, and achromatic combinations of lenses, astronomers were obliged to use instruments of so great a length as to be hardly manageable, till an ingenious mechanic, named Hartsocker, divided the object glass from the eye piece, fixing the former with its frame on the roof of a house, or on a high pole. Huyghens, improving upon Hartsocker's plans, made use of the apparatus represented in Fig. 177.

When merely common lenses are used in this telescope, the eye glass is limited in power by the necessary confusion in the first image from the aberrations, which become very sensible when that image is much magnified, particularly those arising from the unequal refraction of the light.

For this reason, all good telescopes of this kind are made with an achromatic compound lens for an object glass.

The spherical aberration may be lessened, preserving the magnifying power, by using a very weak object glass, assisted by another, called a field-glass, as in Fig. 178.

167. The field of view of a telescope is the lateral extent of prospect it affords in one position, the greater or less portion of the heavens it makes visible at once. To determine this we have only to find the extreme direction in which a ray can pass from one lens through the other: for this we must join the corresponding extremities of the object and eye glass, Fig. 179; the lines $Mm$ , $Nn$ will thus bound the visible image $rqp$ , and the field of view is measured by the angle $rAp$ .^[6]

In point of fact, the field of view is not so large as represented here, for the point $p$ receives but one single ray that can fall on the eye glass, and therefore will not be visible.

If the opposite extremities of the glasses be joined by lines $mn$ , $Nm$ , they will bound a part of the image $tqs$ , which transmits all its light.

It will be seen from the figure, that the eye must be placed at a little distance from the eye glass to receive the rays proceeding from the extreme verge of the field of view.

168. The image is inverted as in the compound microscope. It may be set upright by an additional pair of lenses $C$ , $D$ , (Fig. 180), which are placed so as to have a common focus, and, usually, have no effect on the magnifying power.

This construction is used in what are called Day Telescopes, which are chiefly employed for viewing distant terrestrial objects; for observations on the heavenly bodies, the additional glasses are dispensed with, in order to save the light that is lost by the two additional refractions.

The lenses in the figures are drawn of their full diameter, but in practice it is usual to limit their apertures in order to diminish the aberrations, (see Fig. 181). In this case we must consider the lenses as extending no farther than these apertures.

169. In telescopes to be used for astronomical observations it is usual to put a net-work of fine wire or sometimes of spider's web at the focus of the object glass, in order to determine the precise position of a star as it passes by them. This apparatus is called a Micrometer, and its simplest form is represented in Fig. 182, having five parallel wires dividing the diameter of a circular diaphragm into equal parts, and a sixth bisecting them all perpendicularly.

Another kind of wire micrometer consists of two parallel wires, the one fixed, and the other moveable by means of a fine screw, with a third perpendicular to them. This is represented in Fig. 183; it is used for measuring the apparent diameters of the heavenly bodies.

169. There is another kind of micrometer used, which may as well be described here while we are on the subject: it is called the divided object glass micrometer, and is, in fact, an object glass divided into two by a plane passing through its axis, and of which the two parts, when placed with their centres not coinciding, act as two separate lenses, (Figs. 184, and 185.) The use of this construction is to make the images of two stars, not far distant from each other, coincide on the axis of the telescope, and to determine the angular distance by observing how much the centers of the half lenses have been displaced, by graduated scales on the edges.

Sometimes the eye glass of a telescope is divided in this manner instead of the object glass, and this is thought by many persons a preferable arrangement.

Galileo's Telescope. Fig. 186.

170. In this the rays refracted through the object glass, and proceeding to form an image at its focus, as in the Astronomical Telescope, are stopped by a concave lens, which, if it have a common focus with the object glass, makes them emerge parallel, or if placed nearer, gives them any degree of divergency suited to the eye.

By this means the first image is made imaginary, and the second, likewise imaginary, is thrown on the opposite side of the eye glass, and is therefore erect.

This is the principle of the opera glass, which has the advantage of being much shorter than the astronomical telescope in which the eye glass is beyond the first image, besides representing objects in their natural erect position.

This was the first telescope invented.

The magnifying power, which in this instrument is commonly very low, is represented as before by the fraction $F / F'$ .

This telescope might be used as a microscope, but it would require to be lengthened, as the first image would be thrown farther from the object glass than its principal focus; the magnifying power would in this case be increased.

The field of view is here found by joining the opposite extremities of the glasses by the lines $Mn,\ Nm,$ (Fig. 187.) which mark on the first image the extreme points $p,\ r,$ to which a ray belonging can fall on the eye glass. The angle $rAp [errata 1]$ thus measures the field of view which is much larger than in the astronomical telescope. The lines $Mm,\ Nm$ define the bright part of the field.

Dr. Herschel's Telescope. Fig. 188.

171.The construction of this instrument is better seen in Fig. 189, where $A$ represents a concave metallic speculum giving an image of a distant object at its focus $q,$ where it is viewed through an eye glass $B.$

In practice, the image is thrown to one side as in Fig. 190, as otherwise the head of the observer would intercept the best part of the incident light.

The magnifying powder is plainly ${\frac {f}{F'}},\ f$ being the focal length of the speculum, and $F'$ that of the eye glass. Dr. Herschel has constructed telescopes of this kind that magnify several thousand times,^[7] but he generally used powers of only 500 or 600 which gave more brightness to the image.

The visible part of the image is bounded by the lines $Mm,\ Nn,$ determining the points $r,\ p,$ from which, single rays are sent to the eye glass. The field of view is measured by the angle $rEp,$ which is nearly the apparent magnitude of the eye glass seen from the speculum, and is of course very small, for which reason there is often attached to telescopes of this kind a small refracting one, of low magnifying power but considerable field, which has its axis parallel to that of the other, and is called a finder, as it serves to direct the large telescope to any desired point, as a particular star.

172.Reflecting telescopes in general have these advantages over refractors:

They are free from chromatic aberration, being subject only to that of the eye glass, which is never considered.

They are shorter, cæteris paribus, for the focal length of a concave speculum is only half its radius, whereas that of a glass lens with equal surfaces is the whole radius.

They give brighter images, for there is less light lost in their reflexions, than in the refraction through an object glass.

Notwithstanding this, they are not so much used, because they are less manageable from their weight, more expensive, more apt to get out of order, and more troublesome to use with any nicety, as the least shaking of the instrument or its stand causes great confusion in the image, which is not the case in refracting telescopes.

Sir Isaac Newton's Telescope. Fig. 191.

173. This differs from Dr. Herschel's only in having a plane mirror placed at an angle of 45° to the axis, which throws the image to the side of the instrument, where the eye glass is placed. Newton sometimes used a rectangular prism of glass for a plane reflector.

The magnifying power is of course the same as in Herschel's telescope, as is likewise the field of view, provided the plane reflector be large enough to convey all the rays to the eye glass.

The Gregorian Telescope. Fig. 192.

174. In this the image formed, as in the last two instruments, at the focus $q$ of a concave speculum $A$ , is reflected by a second small concave mirror $B$ having its focus $f$ a little beyond $q$ , so that there is a second image at $q'$ , which is erect, and is viewed through an eye-piece fixed in an aperture in the center of the principal speculum.

This appears at first sight to be a very disadvantageous construction, as the central rays are all stopped by the smaller mirror, and the best part of the great speculum is lost. It is, however, to be observed, that the small reflector is usually of very confined dimensions, and when the object speculum is well ground, it is found that the lateral rays converge quite sufficiently well to make a distinct image.

To find the magnifying power, we must compare the angle subtended by the second image at the eye glass with that of the object or first image at the center of the great mirror: but since the focal length of the mirror is half its radius, this is the same thing as the angle of the first image seen from the mirror. The magnifying power may then be measured by

$p'q' / pq \cdot Aq / Aq'$ , that is, by $eq' / eq \cdot Aq / Cq'$ .

In order to exhibit this more conveniently, we will put $f$ , $f'$ , $F$ for the focal lengths of the great and small mirror and the lens respectively, $l$ for the distance of the mirrors, which is about the length of the telescope.

Then is $eq = q$ , $eq' = q'$ , since

$1 / q$	$= 1 / q' + 1 / f'$ ,
( $eq' / eq$ or) $q' / q$	$= q' + f' / f'$ .

$Cq'$ is the focal length of the lens, $F$ . $Aq$ is nearly equal to $Af$ or $l - f'$ , and $eq'$ about the same thing as $eA$ or $l -2 f'$ , so that the magnifying power is nearly

$l - f' / f' \cdot l - f' / F$ , or $(l - f') 2 / f' \cdot F$ .

To determine the field of view in this telescope we must join the corresponding extremities of the eye glass and the small reflector; this will mark the extreme points of the second image, and the angle subtended by it at the eye glass, divided by the magnifying power, will show the extent of view taken in at once by the instrument.

This telescope, which was the first reflecting one invented, is found in practice very preferable to Newton's, and in general to Dr. Herschel's, whose construction is fit only for a very large instrument. In the first place it is more convenient than either, as the observer has the object in view before him, and can easily direct the instrument to it; but it has one more solid advantage which is this: the metallic specula never can be worked perfectly true, so that the images formed by them are necessarily a little imperfect: now in Gregory's telescope, the two mirrors correct each other if they are properly matched. For this reason a careful optician always tries several small mirrors and chuses the best.

Cassegrain's Telescope. Fig. 193.

175. This bears the same relation to Gregory's that Galileo's does to the astronomical telescope; the small speculum is placed between the large one and its focus, and is convex, so that the second image is thrown near the eye glass as before: this image is, however, inverted.

This instrument is of course shorter than Gregory's, and it appears, in theory at least, to have a considerable advantage over it in that the spherical aberrations of the two reflectors tend to correct each other,^[8] and the second image should therefore be more perfect; this is, however, not discernible in practice, and for some reason or other the construction is seldom used.

The magnifying power is here $(l + F') 2 / f'F$ .

176. Dr. Brewster, in the Edinburgh Quarterly Journal for October, 1822, suggests a construction for a reflecting telescope something like that of Sir Isaac Newton. He proposes to substitute for his plane mirror, or reflecting prism, an achromatic pair of prisms which should divert the image in an oblique direction towards one side of the instrument where the eye glass should be placed so as to view it directly, (Fig. 194.).

He also suggests the idea of dividing the converging light among three or four such prisms which should convey images to different parts of the instrument where they might be viewed at once by several different observers.

Mr. Airey of Trinity College has invented a new reflecting telescope, in which lenses, silvered on one side, are substituted for metallic mirrors so as to combine reflexion and refraction. We understand he has been able to give great perfection to his instrument by calculating the forms of the lenses so as to correct all the aberrations.

↑ When the object is at the focus, and the image infinitely distant, the magnifying power is $c / F$ , one less than this.
↑ The smallest, and therefore the most powerfully magnifying spherules ever made, were some that Di Torre, of Naples, sent to the Royal Society. One of them was only $1 / 144$ th of an inch in diameter, and was said to magnify 2560 times; but Mr. Baker, to whom they were given for examination, could not make any use of them, though he very nearly destroyed his eye-sight in the attempt.
↑ From the known equation $1 / ∆″ = 1 / ∆ - 1 / F$ , we deduce $∆= ∆″ F / ∆″+ F$ , and therefore $∆″ / ∆ = ∆″+ F / F$ .
↑ This account is borrowed from the Edinburgh Philosophical Journal, Vol. 2.
↑ The brightness of the image varies as the surface or aperture of the object glass directly, and the magnitude of the image, (that is, as the focal length), inversely. On this account, the lens should be large in proportion to its focal length, which it may safely, to a certain extent, which is determined by the aberrations becoming sensible.
↑ This is nearly equal to the angle subtended by the object glass at the eye-glass, which is the common measure of the field of view.
↑ His great telescope is 39 feet 4 inches in length, 4 feet in diameter, and magnifies 6000 times.
↑ The aberration in the first reflection is from the center, in the second towards it, and both the centers lie on the same side of the foci.

Errata

↑ Original: MvN was amended to rAp: detail

[1] When the object is at the focus, and the image infinitely distant, the magnifying power is $c / F$ , one less than this.

[2] The smallest, and therefore the most powerfully magnifying spherules ever made, were some that Di Torre, of Naples, sent to the Royal Society. One of them was only $1 / 144$ th of an inch in diameter, and was said to magnify 2560 times; but Mr. Baker, to whom they were given for examination, could not make any use of them, though he very nearly destroyed his eye-sight in the attempt.

[3] From the known equation $1 / ∆″ = 1 / ∆ - 1 / F$ , we deduce $∆= ∆″ F / ∆″+ F$ , and therefore $∆″ / ∆ = ∆″+ F / F$ .

[4] This account is borrowed from the Edinburgh Philosophical Journal, Vol. 2.

[5] The brightness of the image varies as the surface or aperture of the object glass directly, and the magnitude of the image, (that is, as the focal length), inversely. On this account, the lens should be large in proportion to its focal length, which it may safely, to a certain extent, which is determined by the aberrations becoming sensible.

[6] This is nearly equal to the angle subtended by the object glass at the eye-glass, which is the common measure of the field of view.

[8] His great telescope is 39 feet 4 inches in length, 4 feet in diameter, and magnifies 6000 times.

[9] The aberration in the first reflection is from the center, in the second towards it, and both the centers lie on the same side of the foci.

[7] Original: MvN was amended to rAp: detail

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