The cause and nature of vision are properly the subject of that part of natural philosophy which is called Optics: but as light is the cause of vision, the word Optics is commonly used in a more extensive sense; and everything is looked upon as a part of Optics which relates to the nature and qualities of light. If we use the word Optics, in the stricter sense of it, for the theory of vision, the science of Optics is divided into two parts, viz. Dioptrics and Catoptrics. The laws of refraction, and the effects which the refraction of light has in vision, are the subject of Dioptrics: The laws of reflection, and the effects which the reflection of light has in vision, are the subject of Catoptrics. But this division of Optics is of no use; for there are many propositions in Optics where both parts are mixed, and many that cannot be properly reduced to either; and therefore we shall not make any use of that distinction in the following Treatise.

Of Light.

Light consists of an inconceivably great number of particles flowing from a luminous body in all manner of directions; and these particles are so small, as to surpass all human comprehension.

That the number of particles of light is inconceivably great great, appears from the light of a candle; which, if there be no obstacle in the way to obstruct the passage of its rays, will fill all the space within two miles of the candle every way with luminous particles, before it has lost the least sensible part of its substance.

A ray of light is a continued stream of these particles, flowing from any visible body in a straight line; and that these particles themselves are incomprehensibly small, is manifest from the following experiment. Make a small pin-hole in a piece of black paper, and hold the paper upright on a table facing a row of candles standing by one another; then place a sheet of pasteboard at a little distance behind the paper, and some of the rays which flow from all the candles through the hole in the paper, will form as many specks of light on the pasteboard, as there are candles on the table before the plate; each speck being as distinct and clear, as if there was only one speck from one single candle; which shews, that the particles of light are exceedingly small, otherwise they could not pass through the hole from so many different candles without confusion.—Dr Nieuwenyt has computed, that there flows more than $6,000,000,000,000$ times as many particles of light from a candle in one second of time, as there are grains of sand in the whole earth, supposing each cubic inch of it to contain $1,000,000$.

These particles, by falling directly upon our eyes, excite in our minds the idea of light. And when they fall upon bodies, and are thereby reflected to our eyes, they excite in us the ideas of these bodies. And as every point of a visible body reflects the rays of light in all manner of directions, every point will be visible in every part to which the light is reflected from it. Thus the object $ABC$ (Optical Plates, fig. no. 1.) is visible to an eye in any part where the rays $Au$, $Ab$, $Ac$, $Ad$, $Ae$, $Ba$, $Bb$, $Bc$, $Bd$, $Be$, and $Ca$, $Cb$, $Cc$, $Cd$, $Ce$, come. Here we have shewn the rays as if they were only reflected from the ends $A$ and $B$, and from the middle point $C$ of the object; every other point being supposed to reflect rays in the same manner. So that, wherever a spectator is placed with regard to the body, every point of that part of the surface which is towards him will be visible, when no intervening object stops the passage of the light.

Since no object can be seen through the bore of a bend pipe, it is evident that the rays of light move in straight lines, whilst there is nothing to refract or turn them out of their rectilineal course.

Whilst the rays of light continue in any medium of an uniform density, they are straight; but when they pass obliquely out of one medium into another which is either more dense or more rare, they are refracted towards the denser medium; and this refraction is more or less, as the rays fall more or less obliquely on the refracting surface which divides the mediums.

To prove this by experiment, set the empty vessel $ABCD$ (No. 2.) into any place where the sun shines obliquely, and observe the part where the shadow of the edge $BC$ falls on the bottom of the vessel at $E$; then fill the vessel with water, and the shadow will reach no farther than $e$; which shews, that the ray $aBE$, which came straight in the open air, just over the edge of the vessel at $B$ to its bottom at $E$, is refracted by falling obliquely on the surface of the water at $B$; and instead of going on in the rectilineal direction $aBE$, it is bent downward in the water from $B$ to $e$, the whole bend being at the surface of the water; and so of all other rays $abc$.

If a stick be laid over the vessel, and the sun's rays be reflected from a glass perpendicularly into the vessel, the shadow of the stick will fall upon the same part of the bottom, whether the vessel be empty or full; which shows that the rays of light are not refracted when they fall perpendicularly on the surface of any medium.

The rays of light are as much refracted by passing out of water into air, as by passing out of air into water. Thus, if a ray of light flows from the point $e$, under water, in the direction $eB$; when it comes to the surface of the water at $B$, it will not go on thence in the rectilineal course $Bd$, but will be refracted into the line $Ba$. Therefore,

To an eye at $e$ looking through a plane glass in the bottom of the empty vessel, the point $a$ cannot be seen, because the side $Bc$ of the vessel interposes; and the point $d$ will just be seen over the edge of the vessel at $B$. But if the vessel be filled with water, the point $a$ will be seen from $e$; and will appear as at $d$, elevated in the direction of the ray $eB$. Hence a piece of money lying at $e$, in the bottom of an empty vessel, cannot be seen by an eye at $a$, because the edge of the vessel intervenes; but let the vessel be filled with water, and the ray $ea$ being then refracted at $B$, will strike the eye at $a$, and so render the money visible, which will appear as if it were raised up to $f$ in the line $aBf$.

The time of sun-rising or setting, supposing its rays suffered no refraction, is easily found by calculation. But observation proves, that the sun rises sooner and sets later every day than the calculated time; the reason of which is plain, from what was said immediately above. For, though the sun's rays do not come part of the way to us through water, yet they do through the air or atmosphere, which being a grosser medium than the free space between the sun and the top of the atmosphere, the rays, by entering obliquely into the atmosphere, are there refracted, and thence bent down to the earth. And although there are many places of the earth to which the sun is vertical at noon, and consequently his rays can suffer no refraction at that time, because they come perpendicularly through the atmosphere; yet there is no place to which the sun's rays do not fall obliquely on the top of the atmosphere, at his rising and setting; and consequently, no clear day in which the sun will not be visible before he rises in the horizon, and after he sets in it; and the longer or shorter, as the atmosphere is more or less replete with vapours. For, let $ABC$, (No. 5.) be part of the earth's surface, $DEF$ the atmosphere that covers it, and $EBGH$ the sensible horizon of an observer at $B$. As every point of the sun's surface tends out rays of light in all manner of directions, some of his rays will constantly fall upon, and enlighten, some half of our atmosphere; and

* Anything through which the rays of light can pass, is called a medium; as air, water, glass, diamond, or even vacuum. and therefore, when the sun is at $I$, below the horizon $H$, those rays which go on in the free space $IkK$ prefer a rectilineal course until they fall upon the top of the atmosphere; and those which fall to about $K$, are refracted at their entrance into the atmosphere, and bent down in the line $KmB$, to the observer's place at $B$: and therefore, to him the sun will appear at $L$, in the direction of the ray $BmK$, above the horizon $BGH$, when he is really below it at $I$.

The angle contained between a ray of light, and a perpendicular to the refracting surface, is called the angle of incidence; and the angle contained between the same perpendicular, and the same ray after refraction, is called the angle of refraction. Thus (No. 4.) let $LBM$ be the refracting surface of a medium (suppose water,) and $ABC$ a perpendicular to that surface; let $DB$ be a ray of light, going out of air into water at $B$, and therein refracted in the line $BH$; the angle $ABD$, is the angle of incidence, of which $DF$ is the fine; and the angle $KBH$ is the angle of refraction, whose fine is $KI$.

When the refracting medium is water, the fine of the angle of incidence is to the fine of the angle of refraction as 4 to 3; which is confirmed by the following experiment, taken from Doctor Smith's Optics.

Describe the circle $DAEC$ on a plane square board, and cross it at right angles with the straight lines $ABC$, and $LBM$; then, from the intersection $A$, with any opening of the compasses, set off the equal arcs $AD$ and $AE$, and draw the right line $DFE$: then, taking $Fa$, which is three quarters of the length $FE$, from the point $a$, draw $aI$ parallel to $ABK$, and join $KI$ parallel to $BM$: so $KI$ will be equal to three quarters of $FE$ or of $DF$. This done, fix the board upright upon the leaden pedestal $O$, and stick three pins perpendicularly into the board, at the points $D$, $B$, and $I$: then set the board upright into the vessel $TUV$, and fill up the vessel with water to the line $LBM$. When the water has settled, look along the line $DB$, so as you may see the head of the pin $B$ over the head of the pin $D$; and the pin $I$ will appear in the same right line produced to $G$, for its head will be seen just over the head of the pin at $B$: which shews that the ray $IB$, coming from the pin at $I$, is so refracted at $B$, as to proceed from thence in the line $BD$ to the eye of the observer; the same as it would do from any point $G$ in the right line $DBG$, if there were no water in the vessel: and also shews, that $KI$, the fine of refraction in water, is to $DF$, the fine of incidence in air, as 3 to 4.

Hence, if $DBH$ were a crooked stick put obliquely into the water, it would appear a straight one at $DBG$. Therefore, as the line $BH$ appears at $BG$, so the line $BG$ will appear at $Bg$; and consequently, a straight stick $DBG$ put obliquely into water, will seem bent at the surface of the water in $B$, and crooked, as $DBg$.

When a ray of light passes out of air into glass, the fine of incidence is to the fine of refraction as 3 to 2; and when out of air into a diamond, as 5 to 2.

Of Glasses.

Glass may be ground into eight different shapes at least, for optical purposes, viz,

1. A plane glass, (No. 5.) which is flat on both sides, and of equal thickness in all parts, as $A$. 2. A plano-convex, which is flat on one side, and convex on the other, as $B$. 3. A double-convex, which is convex on both sides, as $C$. 4. A plano-concave, which is flat on one side, and concave on the other, as $D$. 5. A double concave, which is concave on both sides, as $E$. 6. A meniscus, which is concave on one side, and convex on the other, as $F$. 7. A flat plano-convex, whose convex side is ground into several little flat surfaces, as $G$. 8. A prism, which has three flat sides; and when viewed endwise, appears like an equilateral triangle, as $H$.

Glasses ground into any of the shapes $B$, $C$, $D$, $E$, $F$, are generally called lens.

A right line $LIK$, (No. 6.) going perpendicularly through the middle of a lens, is called the axis of the lens.

A ray of light $Gh$, falling perpendicularly on a plane glass $EF$, will pass through the glass in the same direction $hi$, and go out of it into the air in the same right course $iH$.

A ray of light $AB$, falling obliquely on a plane glass, will go out of the glass in the same direction, but not in the same right line: for in touching the glass, it will be refracted in the line $BC$; and in leaving the glass, it will be refracted in the line $CD$.

A ray of light $CD$, (No. 7.) falling obliquely on the middle of a convex glass, will go forward in the same direction $DE$, as if it had fallen with the same degree of obliquity on a plane glass; and will go out of the glass in the same direction with which it entered: for it will be equally refracted at the points $D$ and $E$, as if it had passed through a plane surface. But the rays $CG$ and $CI$ will be so refracted, as to meet again at the point $F$. Therefore, all the rays which flow from the point $C$, so as to go through the glass, will meet again at $F$; and if they go further onward, as to $L$, they cross at $F$, and go forward on the opposite sides of the middle ray $CDF$, to what they were in approaching it in the directions $HF$ and $KF$.

When parallel rays, as $ABC$, (No. 8.) fall directly upon a plano-convex glass $DE$, and pass through it, they will be so refracted, as to unite in a point $f$ behind it; and this point is called the principal focus; the distance of which, from the middle of the glass, is called the focal distance, which is equal to twice the radius of the sphere of the glass's convexity. And,

When parallel rays, as $ABC$, (No. 9.) fall directly upon a glass $DE$, which is equally convex on both sides, and pass through it; they will be so refracted, as to meet in a point or principal focus $f$, whose distance is equal to the radius or semidiameter of the sphere of the glass's convexity. But if a glass be more convex on one side than on the other, the rule for finding the focal distance is this: As the sum of the semidiameters of both convexities is to the semidiameter of either, so is double the semidiameter semidiameter of the other to the distance of the focus. Or, divide the double product of the radii by their sum, and the quotient will be the distance sought.

Since all those rays of the sun which pass through a convex glass are collected together in its focus, the force of all their heat is collected into that part; and is in proportion to the common heat of the sun, as the area of the glass to the area of the focus. Hence we see the reason why a convex glass causes the sun's rays to burn after passing through it.

All these rays cross the middle ray in the focus \( f \), and then diverge from it, to the contrary sides, in the same manner \( F/G \), as they converged in the space \( D/E \) in coming to it.

If another glass \( FG \), of the same convexity as \( DE \), be placed in the rays at the same distance from the focus, it will refract them so, as that, after going out of it, they will be all parallel, as \( abc \); and go on in the same manner as they came to the first glass \( DE \), through the space \( ABC \); but on the contrary sides of the middle ray \( B/f \); for the ray \( AD/f \) will go on from \( f \) in the direction \( f/G \), and the ray \( CE/f \) in the direction \( F/c \); and so of the rest.

The rays diverge from any radiant point, as from a principal focus: therefore if a candle be placed at \( f \), in the focus of the convex glass \( FG \), the diverging rays in the space \( E/G \) will be so refracted by the glass, as that, after going out of it, they will become parallel, as shown in the space \( cba \).

If the candle be placed nearer the glass than its focal distance, the rays will converge after passing through the glass more or less as the candle is more or less distant from the focus.

If the candle be placed farther from the glass than its focal distance, the rays will converge after passing through the glass more or less as the candle is nearer to or farther from its focus: and where the rays meet, they will form an inverted image of the flame of the candle; which may be seen on a paper placed in the meeting of the rays.

Hence, if any object \( ABC \) (No. 10.) be placed beyond the focus \( F \) of the convex glass \( def \), some of the rays which flow from every point of the object, on the side next the glass, will fall upon it; and after passing through it, they will be converged into as many points on the opposite side of the glass, where the image of every point will be formed, and consequently the image of the whole object, which will be inverted. Thus, the rays \( Ad, Ae, Af \), flowing from the point \( A \), will converge in the space \( daf \), and by meeting at \( a \) will there form the image of the point \( A \). The rays \( Bb, Be, Bf \), flowing from the point \( B \), will be united at \( b \) by the refraction of the glass, and will there form the image of the point \( B \). And the rays \( Cd, Ce, Cf \), flowing from the point \( C \), will be united at \( c \), where they will form the image of the point \( C \). And so of all the other intermediate points between \( A \) and \( C \). The rays which flow from every particular point of the object, and are united again by the glass, are called pencils of rays.

If the object \( ABC \) be brought nearer to the glass, the picture \( abc \) will be removed to a greater distance. For then more rays flowing from every single point, will all more diverging upon the glass; and therefore cannot be so soon collected into the corresponding points behind it. Consequently, if the distance of the object \( ABC \) (No. 11.) be equal to the distance \( eB \) of the focus of the glass, the rays of each pencil will be so refracted by passing through the glass, that they will go out of it parallel to each other; as \( dI, eH, fb \), from the point \( G \); \( dG, eK, fD \), from the point \( B \); and \( dK, eE, fL \), from the point \( A \); and therefore, there will be no picture formed behind the glass.

If the focal distance of the glass, and the distance of the object from the glass, be known, the distance of the picture from the glass may be found by this rule, viz., Multiply the distance of the focus by the distance of the object, and divide the product by the difference; the quotient will be the distance of the picture.

The picture will be as much bigger or less than the object, as its distance from the glass is greater or less than the distance of the object. For, as \( Be \) (No. 10.) is to \( eB \), so is \( AC \) to \( ca \). So that if \( ABC \) be the object, \( cba \) will be the picture; or if \( cba \) be the object, \( ABC \) will be the picture.

Having described how the rays of light, flowing from objects and passing through convex glasses, are collected into points, and form the images of the objects; it will be easy to understand how the rays are affected by passing through the humours of the eye, and are thereby collected into innumerable points on the bottom of the eye, and thereon form the images of the objects which they flow from. For, the different humours of the eye, and particularly the crystalline humour, are to be considered as a convex glass; and the rays in passing through them to be affected in the same manner as in passing through a convex glass.—For a description of the coats and humours of the eye, see Anatomy, p. 289.

As every point of an object \( ABC \) (No. 12.) sends out rays in all directions, some rays, from every point on the side next the glass, will fall upon the cornea between \( E \) and \( F \); and by passing on through the humours and pupil of the eye, they will be converged to as many points on the retina or bottom of the eye, and will thereon form a distinct inverted picture \( cba \) of the object. Thus, the pencil of rays \( grs \) that flows from the point \( A \) of the object, will be converged to the point \( a \) on the retina; those from the point \( B \) will be converged to the point \( b \); those from the point \( C \) will be converged to the point \( c \); and so of all the intermediate points; by which means the whole image \( abc \) is formed, and the object made visible: although it must be owned, that the method by which this sensation is carried from the eye by the optic nerve to the common sensory in the brain, and there discerned, is above the reach of our comprehension.

But that vision is effected in this manner, may be demonstrated experimentally. Take a bullock's eye whilst it is fresh, and having cut off the three coats from the back part, quite to the vitreous humour, put a piece of white paper over that part, and hold the eye towards any bright object, and you will see an inverted picture of the object upon the paper.

Since the image is inverted, many have wondered why the object appears upright. But we are to consider,

1. That inverted is only a relative term; and, 2. That there there is a very great difference between the real object and the means or image by which we perceive it. When all the parts of a distant prospect are painted upon the retina, they are all right with respect to one another, as well as the parts of the prospect itself; and we can only judge of an object's being inverted, when it is turned reverse to its natural position with respect to other objects which we see and compare it with.—If we lay hold of an upright stick in the dark, we can tell which is the upper or lower part of it, by moving our hand downward or upward; and know very well that we cannot feel the upper end by moving our hand downward. Just so we find by experience, that upon directing our eyes towards a tall object, we cannot see its top by turning our eyes downward, nor its foot by turning our eyes upward; but must trace the object the same way by the eye to see it from head to foot, as we do by the hand to feel it; and as the judgment is informed by the motion of the hand in one case, so it is also by the motion of the eye in the other.

In (No. 12.) is exhibited the manner of seeing the same object \(ABC\), by both the eyes \(D\) and \(E\) at once.

When any part of the image \(cba\) falls upon the optic nerve \(L\), the corresponding part of the object becomes invisible. On which account, nature has wisely placed the optic nerve of each eye, not in the middle of the bottom of the eye, but towards the side next the nose; so that whatever part of the image falls upon the optic nerve of one eye, may not fall upon the optic nerve of the other. Thus the point \(a\) of the image \(cba\) falls upon the optic nerve of the eye \(D\), but not of the eye \(E\); and the point \(c\) falls upon the optic nerve of the eye \(E\), but not of the eye \(D\); and therefore, to both eyes taken together, the whole object \(ABC\) is visible.

The nearer that any object is to the eye, the larger is the angle under which it is seen, and the magnitude under which it appears. Thus to the eye \(D\), (No. 14.) the object \(ABC\) is seen under the angle \(APC\); and its image \(cba\) is very large upon the retina; but to the eye \(E\), at a double distance, the same object is seen under the angle \(APC\), which is equal only to half the angle \(APC\), as is evident by the figure. The image \(cba\) is likewise twice as large in the eye \(D\), as the other image \(cba\) is in the eye \(E\). In both these representations, a part of the image falls on the optic nerve, and the object in the corresponding part is invisible.

As the sense of seeing is allowed to be occasioned by the impulse of the rays from the visible object upon the retina of the eye, and forming the image of the object thereon, and that the retina is only the expansion of the optic nerve all over the choroides; it should seem surprising, that the part of the image which falls on the optic nerve should render the like part of the object invisible; especially as that nerve is allowed to be the instrument by which the impulse and image are conveyed to the common sensory in the brain. But this difficulty vanishes, when we consider that there is an artery within the trunk of the optic nerve, which entirely obscures the image in that part, and conveys no sensation to the brain.

That the part of the image which falls upon the middle of the optic nerve is lost, and consequently the corresponding part of the object is rendered invisible, is plain by experiment. For, if a person fixes three patches, \(A, B, C\), (No. 15.) upon a white wall, at the height of the eye, and the distance of about a foot from each other, and places himself before them, shutting the right eye, and directing the left towards the patch \(C\), he will see the patches \(A\) and \(C\), but the middle patch \(B\) will disappear. Or, if he shuts his left eye, and directs the right towards \(A\), he will see both \(A\) and \(C\), but \(B\) will disappear; and if he directs his eye towards \(B\), he will see both \(B\) and \(A\), but not \(C\). For whatever patch is directly opposite to the optic nerve \(N\), vanishes. This requires a little practice, after which he will find it easy to direct his eye, so as to lose the sight of whichever patch he pleases.

We are not commonly sensible of this disappearance, because the motions of the eye are so quick and instantaneous, that we no sooner lose the sight of any part of an object, than we recover it again; much the same as in the twinkling of our eyes; for at each twinkling we are blinded; but it is so soon over, that we are scarce ever sensible of it.

Some eyes require the assistance of convex glasses to make them see objects distinctly, and others of concave. If either the cornea \(abc\), (No. 16.) or crystalline humour \(e\), or both of them, be too flat, as in the eye \(A\), their focus will not be on the retina, as at \(d\), where it ought to be, in order to render vision distinct; but beyond the eye, as at \(f\). And therefore, those rays which flow from the object \(C\), and pass through the humours of the eye, are not converged enough to unite at \(d\); and therefore the observer can have but a very indistinct view of the object. This is remedied by placing a convex glass before the eye, which makes the rays converge sooner, and imprint the image duly on the retina at \(d\).

If either the cornea, or crystalline humour, or both of them, be too convex, as in the eye \(B\), the rays that enter it from the object \(C\), will be converged to a focus in the vitreous humour, as at \(f\), and by diverging from thence to the retina, will form a very confused image thereon: and so, of course, the observer will have as confused a view of the object, as if his eye had been too flat. This inconvenience is remedied by placing a concave glass before the eye; which glass, by causing the rays to diverge between it and the eye, lengthens the focal distance so; that if the glass be properly chosen, the rays will unite at the retina, and form a distinct picture of the object upon it.

Such eyes as have their humours of a due convexity, cannot see any object distinctly at a less distance than five inches; and there are numberless objects too small to be seen at that distance, because they cannot appear under any sensible angle. The method of viewing such minute objects is by a microscope; of which there are three sorts, viz. the single, the double, and the solar.

Of Microscopes.

The single microscope is only a small convex glass, as \(cd\), (No. 17.) having the object \(ab\) placed in its focus, and the eye at the same distance on the other side; so that the rays of each pencil, flowing from every point of the object on the side next the glass, may go on parallel parallel to the eye after passing through the glass; and then, by entering the eye at C, they will be converged to as many different points on the retina, and form a large inverted picture AB upon it, as in the figure.

To find how much this glass magnifies, divide the least distance (which is about six inches) at which an object can be seen distinctly with the bare eye, by the focal distance of the glass; and the quotient will shew how much the glass magnifies the diameter of the object.

The double or compound microscope, (No. 18.) consists of an object-glass cd, and an eye-glass ef. The small object ab is placed at a little greater distance from the glass cd, than its principal focus, so that the pencils of rays flowing from the different points of the object, and passing through the glass, may be made to converge and unite in as many points between g and h, where the image of the object will be formed: which image is viewed by the eye through the eye-glass ef. For the eye-glass being so placed, that the image gh may be in its focus, and the eye much about the same distance on the other side, the rays of each pencil will be parallel, after going out of the eye-glass, as at e and f, till they come to the eye at k, where they will begin to converge by the refractive power of the humours; and after having crossed each other in the pupil, and passed through the crystalline and vitreous humours, they will be collected into points on the retina, and form the large inverted image AB thereon.

The magnifying power of this microscope is as follows. Suppose the image gh to be six times the distance of the object ab from the object-glass cd; then will the image be six times the length of the object: but since the image could not be seen distinctly by the bare eye at a less distance than six inches, if it be viewed by an eye-glass ef, of one inch focus, it will thereby be brought six times nearer the eye; and consequently viewed under an angle six times as large as before; so that it will be again magnified six times; that is, six times by the object-glass, and six times by the eye-glass; which multiplied into one another, makes 36 times; and so much is the object magnified in diameter more than what it appears to the bare eye; and consequently 36 times 36, or 1296 times, in surface.

But, because the extent or field of view is very small in this microscope, there are generally two eye-glasses placed sometimes close together, and sometimes an inch asunder; by which means, although the object appears less magnified, yet the visible area is much enlarged by the interposition of a second eye-glass, and consequently a much pleasanter view is obtained.

The solar microscope, (No. 19.) invented by Dr Lieberkun, is constructed in the following manner. Having procured a very dark room, let a round hole be made in the window-shutter, about three inches diameter, through which the sun may cast a cylinder of rays AA' into the room. In this hole, place the end of a tube, containing two convex glasses and an object, viz. 1. A convex glass aa', of about two inches diameter, and three inches focal distance, is to be placed in that end of the tube which is put into the hole. 2. The object bb', being put between two glasses (which must be concave to hold it at liberty) is placed about two inches and a half from the glass aa'.

A little more than a quarter of an inch from the object is placed the small convex glass cc', whose focal distance is a quarter of an inch.

The tube may be so placed, when the sun is low, that his rays AA may enter directly into it: but when he is high, his rays BB must be reflected into the tube by the plane mirror or looking glass CC.

Things being thus prepared, the rays that enter the tube will be conveyed by the glass aa towards the object bb', by which means it will be strongly illuminated; and the rays d which flow from it through the convex glass cc', will make a large inverted picture of the object at DD', which, being received on a white paper, will represent the object magnified in length, in proportion of the distance of the picture from the glass cc', to the distance of the object from the same glass. Thus, suppose the distance of the object from the glass to be 7/8 parts of an inch, and the distance of the distinct picture to be 12 feet or 144 inches, in which there are 1440 tenths of an inch; and this number divided by 3 tenths, gives 480; which is the number of times the picture is longer or broader than the object; and the length multiplied by the breadth, shews how much the whole surface is magnified.

Of Telescopes.

Before we enter upon the description of telescopes, it will be proper to shew how the rays of light are affected by passing through concave glasses, and also by falling upon concave mirrors.

When parallel rays, as abcdefgh, (No. 20.) pass directly through a glass AB, which is equally concave on both sides, they will diverge after passing through the glass, as if they had come from a radiant point C, in the centre of the glass's concavity; which point is called the negative or virtual focus of the glass. Thus the ray a, after passing through the glass AB, will go on in the direction kl, as if it had proceeded from the point C, and no glass been in the way. The ray b will go on in the direction mn; the ray c in the direction op, &c.—The ray G, that falls directly upon the middle of the glass, suffers no refraction in passing through it; but goes on in the same rectilineal direction, as if no glass had been in its way.

If the glass had been concave only on one side, and the other side quite plane, the rays would have diverged, after passing through it, as if they had come from a radiant point at double the distance of C from the glass; that is, as if the radiant had been at the distance of a whole diameter of the glass's concavity.

If rays come more converging to such a glass, than parallel rays diverge after passing through it, they will continue to converge after passing through it; but will not meet so soon as if no glass had been in the way, and will incline towards the same side to which they would have diverged if they had come parallel to the glass. Thus the rays f and h, going in a converging state towards the edge of the glass at B, and converging more in their way to it than the parallel rays diverge after passing through it, they will go on converging after they pass through it, though... though in a less degree than they did before, and will meet at \( i \); but if no glass has been in their way, they would have met at \( i \).

When parallel rays, (No. 21.) as \( dfa, Cmb, \) etc., fall upon a concave mirror \( AB \) (which is not transparent, but has only the surface \( Abb \) of a clear polish,) they will be reflected back from that mirror, and meet in a point \( m \), at half the distance of the surface of the mirror from \( C \) the centre of its concavity; for they will be reflected at as great an angle from a perpendicular to the surface of the mirror, as they fell upon it with regard to that perpendicular, but on the other side thereof.

Thus, let \( C \) be the centre of concavity of the mirror \( Abb \); and let the parallel rays \( dfa, Cmb, \) etc., fall upon it at the points \( a, b, \) and \( c \). Draw the lines \( Gia, Gmb, \) and \( Gce, \) from the centre \( C \) to these points; and all these lines will be perpendicular to the surface of the mirror, because they proceed thereto like so many radii or spokes from its centre. Make the angle \( Cab \) equal to the angle \( daC, \) and draw the line \( amb, \) which will be the direction of the ray \( dfa, \) after it is reflected from the point \( a \) of the mirror; so that the angle of incidence \( daC, \) is equal to the angle of reflection \( Cab; \) the rays making equal angles with the perpendicular \( Gia \) on its opposite sides.

Draw also the perpendicular \( Gce \) to the point \( c, \) where the ray \( elc \) touches the mirror; and, having made the angle \( Cei \) equal to the angle \( Cce, \) draw the line \( emi, \) which will be the course of the ray \( elc, \) after it is reflected from the mirror.

The ray \( Cmb \) passing through the centre of concavity of the mirror, and falling upon it at \( b, \) is perpendicular to it; and is therefore reflected back from it in the same line \( bmG. \)

All these reflected rays meet in the point \( m; \) and in that point the image of the body which emits the parallel rays \( da, Cb, \) and \( cc, \) will be formed; which point is distant from the mirror equal to half the radius \( bmG \) of its concavity.

The rays which proceed from any celestial object may be deemed parallel at the earth; and therefore, the images of that object will be formed at \( m, \) when the reflecting surface of the concave mirror is turned directly towards the object. Hence, the focus \( m \) of parallel rays is not in the centre of the mirror's concavity, but halfway between the mirror and that centre.

The rays which proceed from any remote terrestrial object, are nearly parallel at the mirror; not strictly so, but come diverging to it, in separate pencils, or, as it were, bundles of rays, from each point of the side of the object next the mirror; and therefore they will not be converged to a point at the distance of half the radius of the mirror's concavity from its reflecting surface, but into separate points at a little greater distance from the mirror. And the nearer the object is to the mirror, the farther these points will be from it; and an inverted image of the object will be formed in them, which will seem to hang pendant in the air; and will be seen by an eye placed beyond it (with regard to the mirror) in all respects like the object, and as distinct as the object itself.

Let \( AcB, \) (No. 22.) be the reflecting surface of a mirror, whose centre of concavity is at \( C; \) and let the upright object \( DE \) be placed beyond the centre \( C, \) and send out a conical pencil of diverging rays from its upper extremity \( D, \) to every point of the concave surface of the mirror \( AcB. \) But to avoid confusion, we only draw three rays of that pencil, as \( DA, De, DB. \)

From the centre of concavity \( C, \) draw the three right lines \( CA, Ce, CB, \) touching the mirror in the same points where the foresaid rays touch it; and all these lines will be perpendicular to the surface of the mirror. Make the angle \( CAD \) equal to the angle \( DAC, \) and draw the right line \( Ad \) for the course of the reflected ray \( DA; \) make the angle \( Ced \) equal to the angle \( DeC, \) and draw the right line \( cd \) for the course of the reflected ray \( Dd; \) make also the angle \( CBd \) equal to the angle \( DBc, \) and draw the right line \( Bb \) for the course of the reflected ray \( DB. \) All these reflected rays will meet in the point \( d, \) where they will form the extremity \( d \) of the inverted image \( ed, \) similar to the extremity \( D \) of the upright object \( DE. \)

If the pencil of rays \( Eg, Eh, \) be also continued to the mirror, and their angles of reflection from it be made equal to their angles of incidence upon it, as in the former pencil from \( D, \) they will all meet at the point \( e \) by reflection, and form the extremity \( e \) of the image \( ed, \) similar to the extremity \( E \) of the object \( DE. \)

And as each intermediate point of the object, between \( D \) and \( E, \) sends out a pencil of rays in like manner to every part of the mirror, the rays of each pencil will be reflected back from it, and meet in all the intermediate points between the extremities \( e \) and \( d \) of the image; and so the whole image will be formed, not at \( i, \) half the distance of the mirror from its centre of concavity \( C; \) but at a greater distance, between \( i \) and the object \( DE; \) and the image will be inverted with respect to the object.

This being well understood, the reader will easily see how the image is formed by the large concave mirror of the reflecting telescope, when he comes to the description of that instrument.

When the object is more remote from the mirror than its centre of concavity \( C, \) the image will be less than the object, and between the object and mirror; when the object is nearer than the centre of concavity, the image will be more remote and bigger than the object; thus, if \( DE \) be the object, \( ed \) will be its image; for, as the object recedes from the mirror, the image approaches nearer to it; and as the object approaches nearer to the mirror, the image recedes farther from it; on account of the lesser or greater divergency of the pencils of rays which proceed from the object; for, the less they diverge, the sooner they are converged to points by reflection; and the more they diverge, the farther they must be reflected before they meet.

If the radius of the mirror's concavity, and the distance of the object from it, be known, the distance of the image from the mirror is found by this rule; Divide the product of the distance and radius by double the distance made less by the radius, and the quotient is the distance required.

If the object be in the centre of the mirror's concavity, If a man places himself directly before a large concave mirror, but farther from it than its centre of concavity, he will see an inverted image of himself in the air, between him and the mirror, of a less size than himself. And if he holds out his hand towards the mirror, the hand of the image will come out towards his hand, and coincide with it, of an equal bulk, when his hand is in the centre of concavity; and he will imagine he may shake hands with his image. If he reaches his hand farther, the hand of the image will pass by his hand, and come between his hand and his body: and if he moves his hand towards either side, the hand of the image will move towards the other; so that whatever way the object moves, the image will move the contrary.

All the while a bystander will see nothing of the image, because none of the reflected rays that form it enter his eyes.

If a fire be made in a large room, and a smooth mahogany table be placed at a good distance near the wall, before a large concave mirror, so placed, that the light of the fire may be reflected from the mirror to its focus upon the table; if a person stands by the table, he will see nothing upon it but a longish beam of light: but if he stands at a distance towards the fire, not directly between the fire and mirror, he will see an image of the fire upon the table, large and erect. And if another person, who knows nothing of this matter before-hand, should chance to come into the room, and should look from the fire towards the table, he would be startled at the appearance; for the table would seem to be on fire, and, by being near the wainscot, to endanger the whole house. In this experiment, there should be no light in the room but what proceeds from the fire; and the mirror ought to be at least fifteen inches in diameter.

If the fire be darkened by a screen, and a large candle be placed at the back of the screen; a person standing by the candle will see the appearance of a fine large star, or rather planet, upon the table, as bright as Venus or Jupiter. And if a small wax taper (whose flame is much less than the flame of the candle) be placed near the candle, a satellite to the planet will appear on the table: and if the taper be moved round the candle, the satellite will go round the planet.

In a refracting telescope, the glass which is nearest the object in viewing it is called the object-glass, and that which is nearest the eye is called the eye-glass. The object-glass must be convex, but the eye-glass may be either convex or concave: and generally, in looking through a telescope, the eye is in the focus of the eye-glass; though that is not very material: for the distance of the eye, as to distinct vision, is indifferent, provided the rays of the pencils fall upon it parallel: only, the nearer the eye is to the end of the telescope, the larger is the scope or area of the field of view.

Let cd (No. 23.) be a convex glass fixed in a long tube, and have its focus at E. Then, a pencil of rays ghi, flowing from the upper extremity A of the remote object AB, will be so refracted by passing through the glass, as to converge and meet in the point f; whilst the pencil of rays klm, flowing from the lower extremity B of the same object AB, and passing through the glass, will converge and meet in the point e: and the images of the points A and B will be formed in the points f and e. And as all the intermediate points of the object, between A and B, send out pencils of rays in the same manner, a sufficient number of these pencils will pass through the object-glass cd, and converge to as many intermediate points between e and f; and so will form the whole inverted image ef of the distant object. But because this image is small, a concave glass no is so placed in the end of the tube next the eye, that its virtual focus may be at F. And as the pencils of rays pass through the concave glass, but converge less after passing through it than before, they go on further, as to b and a, before they meet; and the pencils themselves being made to diverge by passing through the concave glass, they enter the eye; and form the large picture ab upon the retina, whereon it is magnified under the angle bFa.

But this telescope has one inconvenience which renders it unfit for most purposes, which is, that the pencils of rays being made to diverge by passing through the concave glass no; very few of them can enter the pupil of the eye; and therefore the field of view is but very small, as is evident by the figure. For none of the pencils which flow either from the top or bottom of the object AB can enter the pupil of the eye at C, but are all stopped by falling upon the iris above and below the pupil: and therefore, only the middle part of the object can be seen when the telescope lies directly towards it, by means of those rays which proceed from the middle of the object. So that to see the whole of it, the telescope must be moved upwards and downwards, unless the object be very remote; and then it is never seen distinctly.

This inconvenience is remedied by substituting a convex eye-glass, as gh, (No. 24.) in place of the concave one; and fixing it so in the tube, that its focus may be coincident with the focus of the object-glass cd, as at E. For then, the rays of the pencils flowing from the object AB, and passing through the object-glass cd, will meet in its focus, and form the inverted image nEp: and as the image is formed in the focus of the eye-glass gh, the rays of each pencil will be parallel, after passing through that glass; but the pencils themselves will cross in its focus on the other side, as at e: and the pupil of the eye being in this focus, the image will be viewed through the glass, under the angle geb; and being at E, it will appear magnified, so as to fill the whole space CmepD.

But, as this telescope inverts the image with respect to the object, it gives an unpleasant view of terrestrial objects; and is only fit for viewing the heavenly bodies, in which we regard not their position, because their being inverted does not appear on account of their being round. But whatever way the object seems to move, this telescope must be moved the contrary way, in order to keep sight of it; for, since the object is inverted, its motion will be so too.

The magnifying power of this telescope is as the focal distance of the object-glass to the focal distance of the eye-glass. Therefore, if the former be divided by the latter, the quotient will express the magnifying power. When we speak of the magnifying of a telescope or micro- scope, it is only meant with regard to the diameter, not to the area nor solidity of the object. But as the instrument magnifies the vertical diameter, as much as it does the horizontal, it is easy to find how much the whole visible area or surface is magnified: for, if the diameters be multiplied into one another, the product will express the magnification of the whole visible area. Thus, suppose the focal distance of the object-glass be ten times as great as the focal distance of the eye-glass; then, the object will be magnified ten times, both in length and breadth: and 10 multiplied by 10, produces 100; which shews, that the area of the object will appear 100 times as big when seen through such a telescope, as it does to the bare eye.

Hence it appears, that if the focal distance of the eye-glass were equal to the focal distance of the object-glass, the magnifying power of the telescope would be nothing.

This telescope may be made to magnify in any given degree, provided it be of a sufficient length. For, the greater the focal distance of the object-glass, the less may be the focal distance of the eye-glass; though not directly in proportion. Thus, an object-glass, of 10 feet focal distance, will admit of an eye-glass whose focal distance is little more than 2½ inches; which will magnify near 48 times: but an object-glass, of 100 feet focus, will require an eye-glass somewhat more than 6 inches; and will therefore magnify almost 200 times.

A telescope for viewing terrestrial objects, should be so constructed, as to shew them in their natural posture. And this is done by one object-glass cd, (No. 25.) and three eye-glasses ef, gh, ik, so placed, that the distance between any two, which are nearest to each other, may be equal to the sum of their focal distances; as in the figure, where the focus of the glasses cd and ef meet at F, those of the glasses ef and gh meet at l, and of gh and ik at m; the eye being at n, in or near the focus of the eye-glass ik, on the other side. Then, it is plain, that these pencils of rays, which flow from the object AB, and pass through the object-glass cd, will meet and form an inverted image CFD in the focus of that glass; and the image being also in the focus of the glass ef, the rays of the pencils will become parallel, after passing through that glass, and cross at l, in the focus of the glass ef; from whence they pass on to the next glass gh, and by going through it they are converged to points in its other focus, where they form an erect image EmF of the object AB: and as this image is also in the focus of the eye-glass ik, and the eye on the opposite side of the same glass; the image is viewed through the eye-glass in this telescope, in the same manner as through the eye-glass in the former one; only in a contrary position, that is, in the same position with the object.

The three glasses next the eye have all their focal distances equal: and the magnifying power of this telescope is found the same way as that of the last above; viz. by dividing the focal distance of the object-glass cd, by the focal distance of the eye-glass ik, or gh, or ef, since all these three are equal.

When the rays of light are separated by refraction, they become coloured; and if they be united again, they will be a perfect white. But those rays which pass through a convex glass near its edges are more unequally refracted than those which are nearer the middle of the glass. And when the rays of any pencil are unequally refracted by the glass, they do not all meet again in one and the same point, but in separate points; which makes the image indistinct, and coloured, about its edges. The remedy is, to have a plate with a small round hole in its middle, fixed in the tube at m, parallel to the glasses. For, the wandering rays about the edges of the glasses will be stopped by the plate, from coming to the eye; and none admitted but those which come through the middle of the glass, or at least at a good distance from its edges, and pass through the hole in the middle of the plate. But this circumscribes the image, and lessens the field of view, which would be much larger if the plate could be dispensed with.

The great inconvenience attending the management of long telescopes of this kind, has brought them much into disuse ever since the reflecting telescope was invented. For one of this sort, six feet in length, magnifies as much as one of the other an hundred. It was invented by Sir Isaac Newton, but has received considerable improvements since his time; and is now generally constructed in the following manner, which was first proposed by Dr Gregory.

At the bottom of the great tube TT'TT', (No. 26.) is placed the large concave mirror DUVF, whose principal focus is at m; and in its middle is a round hole P, opposite to which is placed the small mirror L, concave toward the great one; and so fixed to a strong wire N, that it may be moved farther from the great mirror, or nearer to it, by means of a long screw on the outside of the tube, keeping its axis still in the same line Pmn with that of the great one.—Now, since in viewing a very remote object, we can scarce see a point of it but what is as large as broad as the great mirror, we may consider the rays of each pencil, which flow from every point of the object, to be parallel to each other, and to cover the whole reflecting surface DUVF. But to avoid confusion in the figure, we shall only draw two rays of a pencil flowing from each extremity of the object into the great tube, and trace their progress, through all their reflections and refractions, to the eye f, at the end of the small tube tt, which is joined to the great one.

Let us then suppose the object AB to be at such a distance, that the rays B may flow from its lower extremity B, and the rays E from its upper extremity A. Then the rays G falling parallel upon the great mirror at D, will be thence reflected converging, in the direction DG; and by crossing at I in the principal focus of the mirror, they will form the upper extremity I of the inverted image IK, similar to the lower extremity B of the object AB; and passing on to the concave mirror L (whose focus is at n) they will fall upon it at g, and be thence reflected converging, in the direction gN, because gm is longer than gn; and passing through the hole P in the large mirror, they would meet somewhere about r, and form the lower extremity B of the erect image AB, similar to the lower extremity B of the object AB. But by passing through the plano-convex glass R in their way, they form that extremity of the image at b. In like manner, the rays E, which come from the top of the object OPTICS

AB, and fall parallel upon the great mirror at F, are thence reflected converging to its focus, where they form the lower extremity K of the inverted image IK, similar to the upper extremity A of the object AB; and thence passing on to the small mirror L, and falling upon it at b, they are thence reflected in the converging state hO; and going on through the hole P of the great mirror, they would meet somewhere about q, and form there the upper extremity a of the erect image ab, similar to the upper extremity A of the object AB: but by passing through the convex glass R in their way, they meet and cross sooner, as at a, where that point of the erect image is formed.—The like being understood of all those rays which flow from the intermediate points of the object between A and B, and enter the tube TT; all the intermediate points of the image between a and b will be formed; and the rays passing on from the image, through the eye-glass S, and through a small hole e in the end of the lesser tube tt, they enter the eye f, (which sees the image ab by means of the eye-glass) under the large angle ced, and magnified in length under that angle from c to d.

In the best reflecting telescopes, the focus of the small mirror is never coincident with the focus m of the great one, where the first image IK is formed, but a little beyond it (with respect to the eye) as at n: the consequence of which is, that the rays of the pencils will not be parallel after reflection from the small mirror, but converge so as to meet in points about g, e, r; where they would form a larger upright image than ab, if the glass R was not in their way; and this image might be viewed by means of a single eye-glass properly placed between the image and the eye; but then the field of view would be less, and consequently not so pleasant; for which reason, the glass R is still retained, to enlarge the scope or area of the field.

To find the magnifying power of this telescope, multiply the focal distance of the great mirror by the distance of the small mirror from the image next the eye, and multiply the focal distance of the small mirror by the focal distance of the eye-glass; then, divide the product of the former multiplication by the product of the latter, and the quotient will express the magnifying power.

We shall here set down the dimensions of one of Mr. Short's reflecting telescopes, as described in Dr. Smith's optics.

The focal distance of the great mirror 9.6 inches, its breadth 2.3; the focal distance of the small mirror 1.5, its breadth 0.6; the breadth of the hole in the great mirror 0.5; the distance between the small mirror and the next eye-glass 14.2; the distance between the two eyeglasses 2.4; the focal distance of the eye-glass next the metals 3.8; and the focal distance of the eye-glass next the eye 1.1.

One great advantage of the reflecting telescope is, that it will admit of an eye-glass of a much shorter focal distance than a refracting telescope will; and, consequently, it will magnify so much the more: for the rays are not coloured by reflection from a concave mirror, if it be ground to a true figure, as they are by passing through a convex glass, let it be ground ever so true.

The adjusting screw on the outside of the great tube fits these telescopes to all sorts of eyes, by bringing the small mirror either nearer to the eye, or removing it farther; by which means, the rays are made to diverge a little for short-sighted eyes, or to converge for those of a long sight.

The nearer an object is to the telescope, the more its pencils of rays will diverge before they fall upon the great mirror, and therefore they will be the longer of meeting in points after reflection; so that the first image IK will be formed at a greater distance from the large mirror, when the object is near the telescope, than when it is very remote. But as this image must be formed farther from the small mirror than its principal focus n, this mirror must be always set at a greater distance from the large one, in viewing near objects, than in viewing remote ones. And this is done by turning the screw on the outside of the tube, until the small mirror be so adjusted, that the object (or rather its image) appears perfect.

In looking through any telescope towards an object, we never see the object itself, but only that image of it which is formed next the eye in the telescope. For if a man holds his finger or a stick between his bare eye and an object, it will hide part (if not the whole) of the object from his view. But if he ties a stick across the mouth of a telescope before the object-glass, it will hide no part of the imaginary object he saw through the telescope before, unless it covers the whole mouth of the tube: for, all the effect will be, to make the object appear dimmer, because it intercepts part of the rays. Whereas, if he puts only a piece of wire across the inside of the tube, between the eye-glass and his eye, it will hide part of the object which he thinks he sees: which proves, that he sees not the real object, but its image. This is also confirmed by means of the small mirror L, in the reflecting telescope, which is made of opake metal, and stands directly between the eye and the object towards which the telescope is turned; and will hide the whole object from the eye at e, if the two glasses R and S are taken out of the tube.

Of the Multiplying Glass.

The multiplying glass is made by grinding down the round side bit (No. 27.) of a convex glass AB, into several flat surfaces, as bb, bld, dk. An object C will not appear magnified when seen through this glass by the eye at H; but it will appear multiplied into as many different objects as the glass contains plane surfaces. For, since rays will flow from the object C to all parts of the glass, and each plane surface will refract these rays to the eye, the same object will appear to the eye in the direction of the rays which enter it through each surface. Thus, a ray ghH, falling perpendicularly on the middle surface, will go through the glass to the eye without suffering any refraction; and will therefore show the object in its true place at C: whilst a ray ab flowing from the same object, and falling obliquely on the plane surface bb, will be refracted in the direction be, by passing thro' the glass; and upon leaving it, will go on to the eye in the direction eH; which will cause the same object C to appear also at E, in the direction of the ray He, produced in the right line Hen. And the ray cd, flowing from the the object \( AB \), and passing through the convex glass \( CD \), to the plane mirror \( EF \), will be reflected from the mirror, and meet at \( I \), where they will form the extremity \( I \) of the image \( IK \), similar to the extremity \( A \) of the object \( AB \). The like is to be understood of the pencil \( qrs \), flowing from the lower extremity of the object \( AB \), and meeting at \( K \) (after reflection from the plane mirror) the rays form the extremity \( K \) of the image, similar to the extremity \( B \) of the object; and so of all the pencils that flow from the intermediate points of the object to the mirror, through the convex glass.

**Of the Opera-Glass.**

If a convex glass, of a short focal distance, be placed near the plane mirror in the end of a short tube, and a convex glass be placed in a hole in the side of the tube, so as the image may be formed between the last mentioned convex glass and the plane mirror; the image being viewed through this glass, will appear magnified.—In this manner, the opera-glasses are constructed; with which a gentleman may look at any lady at a distance in the company, and the lady know nothing of it.

**Of the Common Looking-Glass.**

The image of any object that is placed before a plane mirror appears as big to the eye as the object itself; and is erect, distinct, and seemingly as far behind the mirror, as the object is before it: and that part of the mirror, which reflects the image of the object to the eye (the eye being supposed equally distant from the glass with the object) is just half as long and half as broad as the object itself. Let \( AB \) (No. 29.) be an object placed before the reflecting surface \( ghi \) of the plain mirror \( CD \); and let the eye be at \( o \). Let \( Ab \) be a ray of light flowing from the top \( A \) of the object and falling upon the mirror at \( h \), and \( hm \) be a perpendicular to the surface of the mirror at \( h \); the ray \( Ab \) will be reflected from the mirror to the eye at \( o \), making an angle \( mbo \) equal to the angle \( Ahm \): then will the top of the image \( E \) appear to the eye in the direction of the reflected ray \( ob \) produced to \( E \), where the right line \( ApE \), from the top of the object, cuts the right line \( chE \), at \( E \). Let \( Bi \) be a ray of light proceeding from the foot of the object at \( B \) to the mirror at \( i \); and \( ni \) a perpendicular to the mirror from the point \( i \), where the ray \( Bi \) falls upon it: this ray will be reflected in the line \( io \), making an angle \( nio \), equal the angle \( Bin \), with that perpendicular, and entering the eye at \( o \); then will the foot \( F \) of the image appear in the direction of the reflected ray \( oi \), produced to \( F \), where the right line \( BF \) cuts the reflected ray produced to \( F \). All the other rays that flow from the intermediate points of the object \( AB \), and fall upon the mirror between \( h \) and \( i \), will be reflected to the eye at \( o \); and all the intermediate points of the image \( EF \) will appear to the eye in the direction of these reflected rays produced. But all the rays that flow from the object, and fall upon the mirror above \( h \), will be reflected back above the eye at \( o \); and all the rays that flow from the object, and fall upon the mirror below \( i \), will be reflected back below the the eye at o: so that none of the rays that fall above h, or below i, can be reflected to the eye at o; and the distance between h and i is equal to half the length of the object AB.

Hence it appears, that if a man sees his whole image in a plane looking-glass, the part of the glass that reflects his image must be just half as long and half as broad as himself, let him stand at any distance from it whatever; and that his image must appear just as far behind the glass as he is before it. Thus, the man AB (No. 30.) viewing himself in the plane mirror CD, which is just half as long as himself, sees his whole image as at EF, behind the glass, exactly equal to his own size. For, a ray AC, proceeding from his eye at A, and falling perpendicularly upon the surface of the glass at C, is reflected back to his eye, in the same line CA; and the eye of his image will appear at E, in the same line produced to E, beyond the glass. And a ray BD, flowing from his foot, and falling obliquely on the other side of the perpendicular abD, in the direction DA; and the foot of his image will appear at F, in the direction of the reflected ray AD, produced to F, where it is cut by the right line BGF, drawn parallel to the right line ACE. Just the same as if the glass were taken away, and a real man stood at F, equal in size to the man standing at B: for to his eye at A, the eye of the other man at E would be seen in the direction of the line ACE; and the foot of the man at F would be seen by the eye A, in the direction of the line ADF.

If the glass be brought nearer the man AB, as suppose to cb, he will see his image as at CDG: for the reflected ray CA (being perpendicular to the glass) will shew the eye of the image as at C; and the incident ray BB, being reflected in the line bA, will shew the foot of his image as at G; the angle of reflection abA being always equal to the angle of incidence Bba: and so of all the intermediate rays from A to B. Hence, if the man AB advances towards the glass CD, his image will approach towards it; and if he recedes from the glass, his image will also recede from it.

Of the Magic Lantern.

ABCD (No. 31.) is a tin lantern, with a tube nklm fixed in the side of it. This tube consists of two joints, one of which slips into the other; and by drawing this joint out, or pushing it in, the tube may be made longer or shorter. At kl in the end of the moveable joint of the tube a convex lens is fixed, and an object painted with transparent colours upon a piece of thin glass is placed at de somewhere in the immovable joint of the tube; so that as the tube is lengthened or shortened, the lens will be either at a greater or a less distance from this transparent object. In the side of the lantern there is a very convex lens bhc, which serves to cast a very strong light from the candle within the lantern upon the object de. Now when the rays, which shine through the object de, diverge from the several points as d, e, &c. in the object, and fall upon the lens kl, they will be made to converge to as many points f, g, &c. on the other side of the lens, and will paint an inverted picture of the object at fg upon a white wall, a sheet or a screen of white paper, provided the object is farther from the lens than its principal focus. To make this picture appear distinct and bright, it must have no other light fall upon it but what comes through the lens kl; and for this reason the whole apparatus is to be placed in a dark room EFGH. The lens kl must be very convex, so that the object de may be very near to it, and yet not be nearer than its principal focus: for by this means, as the object is near to the lens, the picture fg will be at a great distance from it, and consequently the picture will be much bigger than the object. Since the picture is inverted in respect of the object, in order to make the picture appear with the right end upwards, it is necessary that the object de should be placed with the wrong end upwards.

Of the Different Refrangibility of Light.

We have hitherto supposed that a particle of light, as it comes from the sun, is the least particle into which light can be separated. But we must now correct this supposition, by shewing, that a particle of light, as it comes from the sun, is, properly speaking, a bundle of rays, which may be separated from one another. Therefore, for the future, by a ray of light we must be understood to mean, not that collection of particles which we have hitherto called by this name, but the least particles into which light can be separated.

Rays of light are said to be differently refrangible, when at the same or equal angles of incidence some are more turned out of the way than others.

Rays are said to be differently reflexible, if some are more easily reflected than others.

Light is called homogeneous, when all the rays are equally refrangible: it is called heterogeneous, when some rays are more refrangible than others.

The colours of homogeneous light are called primary or simple colours; and those of heterogeneous light are called secondary or mixed.

The rays of the sun are not all equally refrangible: and those rays, which have a different degree of refrangibility, have likewise a different colour.

If a beam of light SF, (No. 32.) that comes from the sun, passes into a dark room through F a round hole in a window shutter EG; this beam proceeding straight forwards, and falling upon a paper at Y, would there make a round picture of the sun. This picture would be a confused one indeed, if the hole is a large one and there is no lens in the hole. However, as this round spot of light is a picture of the sun, we shall hereafter, notwithstanding its confusion, call it by this name. Now if a glass prism ABC is placed between the hole in the window-shutter and the paper at Y, the rays of this beam, by the refraction which they suffer in the prism, will be bent from their straight course, and instead of going on so as to fall upon the paper at Y, they will be turned upwards, and the picture of the sun produced by them will fall upon a paper MN that is placed above Y. If all the rays were were equally bent upwards, the picture would be a round one upon the paper MN, after the rays have been reflected, as well as when they passed straight forwards and fell upon a paper at Y. But this refracted picture PT is found to be oblong. The horizontal diameter or breadth of this oblong picture is equal to the diameter of the circular one Y; but the perpendicular diameter or height of the picture PT is much greater than its breadth. The refraction is made upwards, and not in a horizontal direction; therefore no alteration ought to be made in the breadth or horizontal diameter of the picture; because no refraction, that is not in the direction of that diameter, can make the picture either broader or narrower. And if all the rays were equally refracted upwards, such a refraction would not change the length of the picture; as it is round when it falls at Y, so it would be round when it is refracted upwards by the prism and falls at PT.

This oblong picture consists therefore of rays, which are differently refrangible: they all fall at equal angles of obliquity upon BC the first side of the prism, but in the refraction some are more turned out of the way than others: those rays which go to P, the upper part of the picture, are the most refrangible; and those which go to T, the lower part of it, are least refrangible: the rest, which fall between P and T, have intermediate degrees of refrangibility.

This oblong picture is of different colours in different parts of it. The most refrangible rays at P are violet, the least refrangible at T are red; the rays of intermediate refrangibility from the violet downwards to the red are indigo-coloured, blue, green, yellow, and orange. So that the whole picture is made up of rays of these seven different colours. We may from hence see the reason why the coloured picture consisting of differently refrangible rays should be oblong, in such a manner that the two sides of it are right lines, and the two ends semicircles. For it consists (as in No. 33.) of seven circles, the highest of which PAGQ is violet, the lowest STN is red, the five intermediate ones, BH, CI, DK, EL, OM, are indigo coloured, blue, green, yellow, and orange. The white round picture Y, (No. 32.) is formed by heterogeneous rays, that are of seven different sorts, distinguished from one another by their different degrees of refrangibility and different colours. The refraction of the prism separates these rays from one another by refracting some of them more and others less. And consequently the refracted picture will consist of seven round pictures one below another. These round pictures are so near to each other, that the highest of them APGR will mix itself with some of those below it, as with BH and CQ. This nearness of these several round pictures to each other will prevent their colours from being distinctly seen; it will likewise make the sides AS, GN, which are composed of small arcs of circles very close to one another, appear like right lines; but the two ends P and T will be semicircles.

If the centres of these circles continue at the same distance from one another, and the circles themselves are made less, as ag, bb, ci, dk, el, om, sn, they will then be distinct or will not mix with each other; and as the colours of the several parts will by this means be kept separate, so the refracted rays, instead of forming one continued oblong picture, will form seven small circular ones placed in a line perpendicular to the horizon. This separation of the several parts in the refracted picture from each other is brought about, (as in No. 34.) by making the hole F in the window-shutter very small, and by collecting the rays that come through it with a convex lens MN. For this will make a very small white picture of the sun at L, if there is no prism abc; but the refraction of this prism, if it is placed a little beyond the lens, will separate the heterogeneous rays by refracting them upwards; and instead of one small round and white picture at L, there will be seven small round pictures at PT, of which r will be violet, s indigo, u blue, x green l yellow, y orange, and z red.

That the prism ABC (No. 32.) does not make the rays diverge, so as to spread over the space PT, upon any other account but their different refrangibility, will be evident, if (as in No. 35.) a second prism DH is placed beyond the first abc. For if the rays that come from S and pass through the hole F of the window-shutter EG, were by the first prism abc made to diverge and form the oblong picture PT upon any other account besides their different refrangibility; then, supposing a second prism DH to be placed at right angles to the former, the effect must be this; the first prism abc makes the rays diverge from one another in a line PT perpendicular to the horizon, and consequently the second prism DH must make them diverge from one another in a line parallel to the horizon; so that the second prism would increase the breadth of the picture, as much as the first increased its length; and as one prism alone makes the picture a long one, both of them together would make it square, as prst. But this second refraction does not alter the figure of the picture, but only the position of it: the second prism refracts the picture sideways; and those rays, which fell the highest at P after the first refraction, are refracted sideways the most by the second prism; those which fell the lowest at T are refracted sideways the least; by which means the picture, though it continues oblong, will not be perpendicular to the horizon as PT was, but will be inclined so as to lie in the position prt. This makes it evident, that the spreading of the rays by the first refraction was owing to their different refrangibility, and to no other cause. It must be owing to their different refrangibility, because those which were most refracted upwards by the first prism are most refracted sideways by the second. It cannot be owing to any other cause, because if it was, the second prism would spread the rays in breadth as much as the first prism spreads them in length, and both prisms would make the picture square.

Those rays of light, which are most refrangible, are likewise most reflexible.

When a beam of light is admitted into a dark chamber through the hole F in the window-shutter EG, (No. 36.) and this beam falls upon a prism ACB, the sides of which AC and AB are equal, and the angle at A a right one; when the obliquity of these rays, as they are to pass out of the prism at its base BC, is less than 40 degrees, the greatest part of the beam will pass out, but some few rays will will be reflected at the surface BC. The rays, which pass through the base, form an oblong coloured picture HK, where MH is a more refrangible ray, and MK a less refrangible one. If the few rays of the beam, which are reflected from M in the direction MN, are made to pass through another prism XYV, they will likewise form an oblong coloured picture pt, where p is the most refrangible and t the least refrangible ray. This picture will be a very faint one, because there are but few rays reflected from M.

Now if the prism ACB is turned slowly round upon its axis in the direction ACB, the obliquity of the rays EM to the base BC will keep increasing, till at last this obliquity may become so great, that no rays will pass out at M, but all of them will be reflected. When this total reflection is made, the oblong picture pt, which was faint before, will become much brighter, because then not only a few rays, but all the beam, will be reflected thither. This total reflection will not be made all at once; but as the prism is turned slowly round upon its axis, the most refrangible rays MH will be first reflected, for the violet colour will disappear in the oblong picture HK, whilst all the other colours continue as bright as they were before; and when this colour disappears at HK, the same colour at p will become bright, and all the other colours at pt will continue as faint as they were before. When the prism is turned a little farther upon its axis, the indigo colour, which consists of rays that have the next greatest degree of refrangibility, will be reflected, so that this colour will disappear at HK and will become bright at pt. The same thing will happen to all the rays in their order; as the prism is turned round, each different sort of rays will be reflected sooner as the rays have a greater degree of refrangibility, or later as they have a less degree. The red rays at K, which are the least refrangible of all, will be reflected last of all. From hence therefore it appears, that the rays of the sun are differently reflexible, and that those which are most refrangible are likewise most reflexible.

Homogeneous light is refracted regularly without any dilatation or scattering of the rays.

When the rays of any one particular colour in the oblong picture of the sun, as the green rays, for instance, are separated from one another; if some of these green rays which are homogeneous, or are all equally refrangible, are transmitted through a very small round hole in a stiff pasteboard, and are refracted by a prism on the other side of the hole, the picture formed by these green rays after refraction upon a white paper held beyond the prism will not be oblong, but circular, as the hole is through which they passed. Therefore this homogeneous light is not dilated, nor are the rays of it scattered by this refraction.

The confused appearance of objects, when they are seen through refracting bodies, is owing to the different refrangibility of light.

If flies, or the letters of a small print, or any other minute objects, are placed in heterogeneous light, such as a direct beam of the sun's, which has never been separated by any refraction into its homogeneous parts; these objects being viewed through a glass-prism will be seen confusedly, their edges will appear so misty that the smaller parts of minute animals cannot easily be distinguished from one another, and the letters of the small print cannot be read. But if the same objects are placed in a beam of homogeneous light, which is separated from all other rays of a different refrangibility in the manner already described, they will appear as distinct through a prism as if they were viewed with the naked eye. Therefore we may conclude, that this confusion is owing to the different refrangibility of those rays which come from the objects; since objects never appear confused when they are seen through refracting bodies, unless they are enlightened with several sorts of rays which have different degrees of refrangibility.

It is probable that any single ray of the least refrangible sort contains a greater quantity of matter than any single ray of the most refrangible sort.

We have already seen, that at the same angles of incidence violet rays will be more refracted or more turned out of the way than red rays. And we have likewise seen, that rays are refracted when they pass out of one medium into another, by being either more or less attracted in one medium than they are in the other. Now since, when all other circumstances are equal, when red rays and violet rays fall at equal obliquities, and are to pass out of glass into air, so that the mediums, and consequently the attractive force or cause of refraction, is given; if the same cause can turn the violet rays more out of the way, or refract them more, than it does the red rays, these rays must have different moments; the most refrangible rays, or those which are most easily turned out of the way, have the least moment; and the least refrangible rays, or those which are most difficult to turn out of the way, have the greatest moment. But if all sorts of rays have the same velocity, their respective quantities of matter will be as their moments; and consequently any single ray of the most refrangible sort contains a less quantity of matter than any single ray of the least refrangible sort.

It may be upon this account that a red colour, or a pale purple, is less pleasant to the eye than a blue, green, or a yellow. The red rays strike the eye with so great a force as to be offensive to it; and the small force of the pale purple ones will produce too faint a sensation to be agreeable. The intermediate colours are therefore more pleasant to the eye, as the force of the rays is neither too great to be offensive, nor too small to produce a quick and lively sensation.

The colours of homogeneous light are so invariable, that neither any refraction nor any reflection can alter them.

If a beam of homogeneous light passes through a round hole in a pasteboard, and then is refracted by a prism on the other side of the hole, this refraction will make no alteration in the colour of the rays; if they were red, or whatever was their colour, before they entered the prism, their colour will still be the same, when they have passed through it, and fall upon a white paper held beyond the prism. This proves the first part of the proposition, that the colours of homogeneous light are to be changed by any refraction.

Red lead, when it is viewed in open day-light, or when heterogeneous rays fall upon it, will likewise be red, if it is placed in homogeneous red light; but red lead, when it is placed in any other sort of homogeneous light, will have the same colour with the rays that fall upon it and are reflected from it: if it is placed in yellow homogeneous light, it will be yellow; if in green light, it will be green; or if in blue light, it will be blue. Consequently the reflection of the rays from the red lead make no alteration in their colour; for if it did, rays of any sort reflected from the lead would be of the same colour, so that it would appear red in whatever sort of light it was placed. The same that is here said of red lead, is true of any other substance of any other colour. Grafts, which is green either in open day-light or in homogeneous green light, will not change the colour of any homogeneous rays by reflecting them, but will itself have the same colour with the rays in which it is placed; it will be red in red light, or blue in blue light, or yellow in yellow light.

From hence we may conclude, by the way, that a body is of any particular colour, not because it reflects no other rays but those of that particular colour, but because it reflects those more copiously and others more sparingly. Red lead, as it appears red in red light, so in green light it appears green, or in blue light it appears blue: consequently it reflects rays of these sorts, and in the same manner it might be shewn to reflect all other sorts of rays. But then the red colour of red lead, when it is placed in red light, is much brighter than any other colour will be that it puts on by being placed in another sort of light: consequently it reflects red rays more copiously than any other sort of rays; and for this reason, when it is placed in open day-light, where it reflects all sorts of rays at once, the red rays are so much more numerous than the rest, as to make the whole mixture of their own colour.

Colours may be produced by composition, which shall in appearance be like the colour of homogeneous light: but then these compound colours will be altered by refraction.

When, by means of two holes in the window-shutter of a dark room and of two prisms, two oblong coloured pictures are produced; if a circular piece of white paper is so placed that the red light of one picture and the yellow light of the other may fall upon it, this mixture will produce an orange colour, that in appearance will be like the primary orange colour. But between the simple and compound colour, though they are alike in appearance, there will be this difference; if the circular piece of paper, when it is enlightened with compound orange, is viewed through a prism, the rays will be found to be differently refrangible, and they will, by the refraction of the prism, be so separated from one another, that the paper seen through it will appear as two circles, one of which will be red and the other yellow; whereas, if the same paper, when it is enlightened with simple or primary orange, is viewed in like manner through a prism, the rays will be found to be equally refrangible, and the paper will appear through the prism, as it does to the naked eye, to be one orange-coloured circle distinctly terminated all round. After the same manner other homogeneous colours, as blue and yellow, when mixed together, will produce a new compound colour like the intermediate homogeneous green colour in appearance. But then the rays of this compound green will not be all of them equally refrangible, as the rays of the simple or primary green colour are.

The whiteness of the sun's light is compounded of all the primary colours mixed in a due proportion.

Let the oblong coloured picture (No. 37.) fall upon the convex lens MN; and then all the rays which are separated from one another at PT will be collected together by passing through the lens, and will meet at its focus G, in such a manner as to form around picture of the sun upon a white paper DE. This round picture, which consists of rays of all sorts, of red, orange, yellow, blue, green, indigo, and violet, is white. And this whiteness is compounded of all the primary colours mixed together. None of the rays change their colour by being mixed with the rest; each fort retains the same colour after it is mixed with the rest that it had before; neither the red rays, nor the orange, nor the yellow, nor the blue, nor the green, nor the indigo, nor the violet, are made white by being mixed with the rest at the focus; but though none of the parts are white, yet the whole mixture is white.

That the whiteness at the focus G arises from a mixture of all the primary colours, is evident. For if any of the colours are intercepted at the lens, the focus loses its whiteness, and becomes of that colour which arises from a mixture of those which are not intercepted. Thus if all the rays at PT are intercepted except the yellow, the orange, and the red, the focus will not be white, but will be orange-coloured. If all the rays are intercepted at PT, except the blue, the green, and the yellow, the focus will then be green. The orange in one case, and the green in the other case, is the compound colour arising from a mixture of those rays which are not intercepted. And in either case, if the rays that were intercepted are again suffered to pass through the lens, the focus will recover its whiteness.

It may be more difficult to shew that the rays, when they are all of them mixed at the focus, retain their proper colours, and are none of them white though the compound mixture is white. To make this out, let the paper be removed from DE, where all the rays are mixed upon it at G, to de, where it will receive the rays, after they have crossed one another at the focus, and having got beyond it diverge again. In this position of the paper, because the rays that were mixed at the focus have diverged from thence, and are again separated from one another, the oblong coloured picture will appear again at tp, so that the red colour T, which was the lowest at the lens, will be the highest at the paper de. But though the colours are thus inverted by passing the focus, yet all of them appear at tp; which would have been impossible, if each fort of rays, by being mixed with the rest at the focus, had lost their colour, and had been made white. Nor indeed is the colour of any fort of rays at all changed by being mixed with the rest at the focus; but increasing the light, and the paper by thus diminishing it, may be made to appear equally white.

The colours of all bodies are either the simple colours of homogeneous light, or such compound colours as arise from a mixture of homogeneous light.

Each sort of light has a peculiar colour of its own, which no refraction or reflection can change. Therefore the colour of no natural body can be any other than either the colour of some sort of homogeneous light, or a compound colour arising from a mixture of the several sorts. For bodies appear coloured only by reflecting light; and no reflection can give any other colours to the rays but what they had before.

Of the Colours of thin transparent Plates.

Water, air, glass, or any other transparent substance, when drawn out into thin plates, become coloured.

Water, when it is made tenacious by having soap mixed with it, may be blown up into a bubble A, (No. 38.) such as children play with. If this bubble is set under a glass, so that the motion of the air may not affect it, then as the water glides down the sides of it, and the top of it at A grows thinner, several colours will successively appear at A, and will spread themselves from thence in rings surrounding A, and descending farther and farther down the sides of the bubble, till they vanish at BC in the same order in which they appeared. Thus, for instance, the first colour that appears at A, the top of the bubble, is red: this red spot spreads itself into a circular ring round A, and then the top of the bubble A becomes blue: this blue spot spreads itself in the same manner round A, and then A becomes red a second time. Before we go on to consider what other colours arise at A, we will observe what becomes of those which arise first. The red, which first appeared at A, spreads itself into a circular ring round A: this ring grows larger, as the water glides down the sides of the bubble; so that the coloured ring glides down the bubble along with the water, till it sinks at last to BC, and there encompasses the bubble. In like manner the blue, which arises at A after the red, spreads itself and descends down the bubble, as the red ring did. The colour which arises next at A, is red a second time; this spreads itself in the same manner, and is succeeded by blue a second time. These are followed by a great variety of colours, which appear successively at A, and spread themselves from thence in this order: Red, yellow, green, blue, purple; then again red, yellow, green, blue, violet; and lastly, red, yellow, white, blue. This last blue colour is succeeded at A by a black spot, which reflects scarce any light: this spot dilates itself, but not into a circular ring as the colours had done; it becomes broader and broader, till the bubble breaks.

A thin plate of water of the same sort with this bubble, but more lasting, may be otherwise procured. If a piece of plane polished glass is placed upon the object-glass of a long telescope, as in (No. 39.) the plane surface of one glass, and the convex one of the other, will touch one another only at a single point; and if the interval between them is filled with water, as the glasses are pressed together, the same colours arise at the point of contact. contact, and spread themselves in circular rings round it in the same order as in the soap-bubble. If BC (No. 40.) is a section of the plane glaiss, and DAE a section of the convex one; when they are pressed close together, the thin plate of water that fills the interval between them will have a black spot at A; and this spot will be encompassed with rings of colours, in the same order that they stand in that figure upon the line BC, on each side of A. If the colours are reckoned in the order in which they stand on the plate of water after the black spot appears at A, and we reckon them from the spot A towards the edges of the plate at B and C; then we must call blue the first colour. But if we reckon them in the order in which they arose at A, and spread themselves; then we must begin from B or C, the edges of the plate, and go on towards A, and in this reckoning we must call red the first colour.

If there is no water between the two glasses, then the interval will be filled with air, and this thin plate of air will have the same colours that the plate of water had; with this difference only, that each of the coloured rings is larger in the plate of air than in the plate of water.

When glass is blown very thin at a lamp-furnace, thin plates of it thus formed will exhibit colours; and so likewise will thin plates of Mulcovy-glass. Metals, when they are heated, send out their surfaces scoria or vitrified parts, which cover the metals in form of a thin skin; and these scoria or thin plates cause colours upon the surface of the metal, such as are made to appear on polished steel by heating it, or on bell-metal by melting it first and then pouring it on the ground to cool in the air.

When the thin plate is denser than the medium that surrounds it, the colours are more vivid than they are when the plate is rarer than that medium.

A thin bubble is a plate of water encompassed with air; where the substance of the plate, which is water, is denser than the air, which is the medium that surrounds it. On the contrary, the plate of air between the two glasses BAC, DAE, (No. 40.) is encompassed with glass; and here the substance of the plate is rarer than that of the circumambient medium; and the colours on the bubble of water are more vivid than those on the thin plate of air.

When thin transparent plates reflect one sort of rays, they transmit the rest.

If the plate of air between the two glasses BAC, DAE, (No. 40.) is viewed by reflected light, the colours of it are those expressed on the upper part of the figure from B to C; but if we look through it, that is, if we view it by transmitted light, or if the transmitted light falls upon a white paper, the colours that we see through the plate, or that fall on the paper, are those expressed on the lower part of the figure. Now, any of the transmitted colours are what would arise from a mixture of all the remaining rays, after those of the reflected colour are separated from the sun's heterogeneous light. Thus, for instance, the fourth reflected colour from the black spot A inclusively is yellow, the transmitted colour is violet. The yellow rays, and some of the orange and green, are reflected here, so that the mixture of the reflected light will be yellow. The mixture of the transmitted light therefore will be violet, or rather such a purple as is not exactly like any of the primary colours; for we observed, that from red rays, violet, and blue, new purples may be produced.

This is the case in some natural bodies, as well as in these transparent artificial plates; for if leaf-gold, which is made thin enough to transmit light, is held against the strong light of the sun's rays, the gold, which is yellow when seen by the reflected light, will be blue when thus seen by transmitted light.

The seventh reflected colour inclusively from the black spot is blue, the seventh transmitted colour is yellow. When the rays which make the blue colour are taken out of the sun's heterogeneous light, the remaining rays will be yellow. Thus it happens likewise in some natural bodies; for an infusion of lignum nephriticum, which is blue when seen by reflected light, is yellow when seen by transmitted light.

The black spot A reflects scarce any light; and as rays of all colours are transmitted there, the transmitted colour is white; the third reflected colour from the black spot inclusively is white. Therefore, since all the rays are reflected there, no colour ought to be seen there, when we look through the plate; and accordingly that part of the plate is black.

Hence we see the reason why, if there be two liquors of full colours in two different glass vessels, suppose red and blue; though each is transparent when we look thro' it separately, yet we should not be able to see through both of them together, if one was held behind the other. For if the blue liquor, for instance, is held towards the light, and the red towards the eye; since only blue rays pass through the first liquor, and come to the second; and since the second liquor will transmit no blue rays, but only red ones; it follows, that no rays at all can come to the eye.

Indeed some transparent bodies appear of the same colour, whether we see them by reflected or transmitted light. Of this sort is most painted glass. But when this is the case, the coloured rays are reflected from the second surface of the body. Thus, if a piece of painted glass is yellow either when seen by reflected light or when seen by transmitted light, all the rays but the yellow ones are suppressed as they pass through the glass: of the yellow rays, most are transmitted at the second surface; the few which are reflected from thence will be sufficient to tinge all the light yellow, which is reflected from the first surface. This will be evident from making the body thick, and pitching it on the backside: for by this means the reflected colour will be lost; whereas, if it had been reflected from the first surface, the pitch at the second surface could not have altered it.

The more dense the substance is out of which a thin plate is made, the less is the thickness of the plate where it reflects any certain colour.

The colours are the same whether there is air or water between the two glasses BAC, DAE, (No. 40.) only the coloured circles are smaller in the plate of water than in the plate of air. Thus the yellow, for instance, which is the fourth coloured circle from the black spot, is a less circle, or is nearer to the black spot, when there is a plate of water between the glasses, than when there is a plate of air between them. But the less the distance from the point of contact A, the closer the glasses are to one another, and consequently the thinner will be the plate that lies between them; consequently that part of a plate of water where this yellow appears, is thinner than that part of a plate of water where the same colour appears. And the same holds good in any other colour. But water is more dense than air; therefore, the more dense the substance is out of which a thin plate is made, the less is the thickness of the plate where it reflects any certain colour.

The sort of colour, which is reflected from any part of a thin plate, depends only upon the thickness of the plate itself in that part: but the same colour will be made less vivid by increasing the density of the medium with which the plate is encompassed.

The colours upon any part of a thin plate of Muscovy glass are the same in sort, whether the plate is dry or wetted with water. Therefore the sort of colour in any part depends not upon the medium that encompasses the plate, but upon the thickness of the plate itself; since the colours are the same when the plate is dry and encompassed with air, or wet and so encompassed with water. But the same colours are more faint when the plate is wet, than when it is dry; and consequently, the brightness of the colours does depend upon the medium that encompasses the plate; and the denser that medium is, the fainter will be the colours; they are more faint when the plate is covered with water than when it is dry and so surrounded with air.

The rays of light have alternate fits of easy reflection and easy transmission, which return at equal intervals.

Let GF, (No. 41.) be a beam of homogeneous light consisting all of one sort of rays, as suppose all the rays that compose the beam were red ones. Then, if these rays fall upon a thin plate of air between the two glasses BAC, DAE, at A there will be a dark spot, and all the rays will be transmitted; round this spot there will be a red ring, where all the rays are reflected; round this red ring there will be a dark ring, where all the rays are transmitted. And if the thickness of the plate where all the rays are reflected in the ring nearest to A is called 1, the thickness where the dark ring appears and all the rays are transmitted will be 2. Again, at that part of the plate where the thickness is 3, all the rays will be transmitted; at the thickness 4, they will be all reflected. And thus alternately, as expressed by the lines in the figure, the rays will be reflected in all parts of the plate where the thickness is expressed by any of the uneven numbers 1, 3, 5, 7, 9, &c., and will be transmitted where the thickness is expressed by any of the even numbers 2, 4, 6, 8, 10, &c.

Now as the plate is the same in all parts, the cause of this alternate reflection and transmission must be in the rays themselves; and their dispositions to be thus alternately reflected and transmitted, are what we call fits of easy reflection and easy transmission.

The rays that are in a fit of easy reflection penetrate as far as the second surface of the plate. For if the second surface of a thin plate of Muscovy glass is wetted, the colours caused by the alternate reflection grow fainter; whereas if the reflection was made at the first surface, wetting the second could not affect the colours. But since those rays which have passed from the first surface of the plate to the second where the thickness of it is 1, are reflected, and those which have passed from the first surface to the second where the thickness of it is 2, are transmitted; and then again those which have thus passed from one surface to the other, where the thickness is 3, are reflected; and those which have passed in the same manner, where the thickness is 4, are transmitted; it follows, that the fits of easy reflection and transmission return at equal intervals. So that, if a ray was set out from A in the line AB, (No. 42.) and was to be in a fit of easy reflection when it had moved from A to c, it would be in a fit of easy transmission when it had moved to twice that distance from A, or when it was got to d: at e, or the distance 3 from A, it will be in a fit of easy reflection; at f, or the distance 4, in a fit of easy transmission; at g, or 5, in a fit of easy reflection; at B, or 6, in a fit of easy transmission: and thus, in the farther progress of the ray, the same fits will return at equal intervals.

Thus if the thickness of the plate of air, where the rays of any homogeneous colour are all reflected, is equal to Ac or 1, and the rays are in a fit of easy reflection when they come to the second surface of the plate; then, where the thickness of the plate is Ad or 2, the rays will be in a fit of easy transmission when they come to the second surface, and consequently will all pass through that surface. Again, where the thickness is Ae or 3, the rays, when they come to the second surface, will be in a fit of easy reflection, and will all be reflected; where the thickness is Af or 4, the fit of easy transmission will be returned when the rays come to the second surface, so that all of them will be transmitted. And in like manner, by such fits returning at equal intervals, the rays will be reflected where the thickness is expressed by the number 1, 3, 5, 7, 9, &c. and will be transmitted where it is expressed by 2, 4, 6, 8, 10, &c.

When a thin coloured plate is viewed obliquely, the colours of every part in the plate will be altered.

When a bubble of water or a plate of air between two glasses BAC, DAE, (No. 40.) is viewed obliquely, the coloured rings dilate themselves: and consequently a ring of any one colour, by being dilated, gets into that part of the plate where a ring of some other colour appeared when the plate was viewed directly.

If the plate is denser than the medium that encompasses it, the colours of it, when viewed obliquely, change less than they would if the plate was rarer than the medium that encompasses it.

A bubble of water is a thin plate denser than the air that encompasses it; and a plate of air between the two glasses BAC, DAE, (No. 48.) is rarer than the glass that encompasses it. Upon viewing each of these thin plates obliquely, the coloured rings on the plate of water dilate less than those on the plate of air. Therefore, since it is by this dilation of the rings that a ring of One colour gets into a part of the plate where a ring of some other colour appeared when the plate was viewed directly; that is, since it is by this dilatation of the rings that the several parts of the plates change their colours; it follows, that any part of a plate of water encompassed with air changes colour less upon being viewed obliquely, than any part of a plate of air encompassed with glass.

When the medium which encompasses a coloured transparent plate is given, the colours change less upon altering the situation of the eye, as the substance is more dense out of which that plate is made.

The matter out of which a bubble of water is made is not so dense as that out of which a bubble of glass is made, and glass is not so dense as the scoria or glairy skin thrown out by metals when they are heated. Now, any of these plates either of water, or glass, or metallic substance, when they are encompassed with the same medium air, will change their colour a little upon being viewed obliquely; but the plate of water changes the most, the plate of glass least than that, and the scoria of metals least of all.

Of the Opakeness, Transparency, and Colours of Natural Bodies.

The opakeness of bodies is owing to the many reflections and refractions which the rays of light suffer within those bodies.

The smallest parts of almost all natural bodies are transparent, as will readily be granted by those who have been used to look through microscopes. A piece of leaf-gold is transparent if it is held up against the hole of a window-shutter in a dark room; and any other substance, however opake it may seem in the open air, will appear transparent by the same means, when it is made of a sufficient thinness. Even metals become transparent, if they are dissolved in a proper menstruum, as gold in aqua regia, or silver in aqua fortis; and by being thus dissolved, are reduced to very small particles. But since even in opake bodies every single particle transmits light, or is transparent, the whole would likewise transmit light, unless the rays, when they are to pass through all the particles which make up the whole, were so turned out of the way by innumerable refractions and reflections, as to be stopped and suppressed in their passage. That this is the reason why bodies that consist of transparent particles should be opake, is evident; since opake bodies, when they are reduced to a sufficient thinness, become transparent: for then there will be but few particles lying beyond one another for the light to pass through; and as the rays will suffer fewer refractions and reflections, some of them may get through a thin plate, though all of them would be suppressed in a thicker mass of the same substance.

The medium, with which the pores of opake bodies are filled, is not of the same density with the particles of those bodies.

Bodies consist of transparent particles, and their opakeness is owing to the many reflections and refractions which the light suffers within them. Now, if the interfaces between the particles of any body were filled with a medium of the same density with the particles, the light would neither be refracted nor reflected as it passed out of the particles into the interfaces and out of the interfaces into the pores, but would pass through the body, and the body would be transparent. Consequently, in an opake body, where the light is suppressed by the refractions and reflections which it suffers, the particles that compose the body, and the medium that fills the pores or interfaces between the particles, must be of different densities.

Hence we may see the reason why paper, when it has been dipped in water or oil, is more transparent than when it is dry. For when the paper is thoroughly wetted with water or oil, the pores of it are filled with a medium that is nearly of the same density with its particles. On the contrary, though oil of turpentine and water are both of them transparent when they are separate; yet if they are shaken together so as to mix but imperfectly, the mixture becomes much less transparent, because the parts of each fluid are separated from one another, and those of the other fluid, which are of a different density, get in between them.

The parts of bodies, and their interfaces, must not be less than of a certain definite bigness to render them opake and coloured.

The most opake bodies become transparent when their particles are subtilly divided; as metals, such as gold or silver, which are opake in large masses, become transparent when the former is dissolved in aqua regia, and the latter in aqua fortis. And we observed, that at the top of a bubble of water, where the water is extremely thin, there is a black spot, which reflects scarce any light at all; though the water is encompassed with air, which is a medium of a different density. Consequently, if the diameter of the particles of which any natural substance consists was no greater than the thickness of the bubble, where it reflects no light, but transmits all, such a body would be transparent, notwithstanding the interfaces that are between its particles were filled with a medium the density of which is different from theirs.

In like manner, we observed, that when a thin plate of air lies between two pieces of glass BAC, DAE, (No.40.) there is a dark spot, which reflects no light, and transmits all, not only at the point A where the glasses touch one another, but also round that point to some distance where the glasses are very near to one another. From hence we may conclude, that though the particles of any natural substance were as dense as glass, and the medium which fills their interfaces was as rare as air; yet if these interfaces were no bigger than the interval between the two glasses BAC, DAE, at that place where all light is transmitted, such a body would be transparent.

The transparency of water seems to be owing to the causes here mentioned, to the smallness of its parts, or of its pores, or of both. For we are sure that the pores of water are filled with air, because the air may be drawn out from the water in an air-pump; and consequently, as the pores are filled with a medium of a different density from the parts, the mixture ought to be opake, like such a mixture of water and oil of turpentine as was mentioned above. But the smallness either of the parts, or or of the interstices, or of both, will prevent the mixture from being opake.

Since therefore all bodies will be transparent, if either their parts or their interstices are too small, it follows that the parts, and likewise the pores, of such bodies as are not transparent but opake and coloured, must not be less than of a certain and determinate bigness.

The colours of natural bodies depend upon the size of their particles.

Different parts of thin transparent plates, according to the different thicknesses of them, are of different colours. Now if any part of such a thin plate of glass, for instance, where it appears of one uniform colour, should be split into threads, or broken into small particles, all these particles would make a heap of powder of the same colour. And the small particles of natural bodies, since they are transparent, like so many fragments of a thin plate, must exhibit colours in the same manner.

The parts of bodies, on which their colours depend, are much denser than the medium which fills their pores.

For where the transparent plate or particle consists of a rarer substance than the medium that encompasses it, the colours are less vivid than those of natural bodies commonly are. For this reason it is that the colours of silks or cloths, when they are wetted with oil or water, become more faint; because these liquors are more nearly of the same density with the particles, than the medium is which fills the interstices when they are dry. Besides, the colours upon a transparent plate change very sensibly, unless the plate consists of a substance much denser than the medium that encompasses it; but most natural bodies are of the same colour in whatever position of the eye they are viewed. Therefore their transparent particles, upon which their colours depend, are much denser than the medium which encompasses those particles or fills the interstices between them.

Nor is the case otherwise even in those bodies which do change colour upon being viewed obliquely, such as changeable silks, or the feathers of a peacock's tail or of a pigeon's neck. For this change of colour, upon the situation of the eye being changed, is no reason for concluding that the medium which fills the interstices or pores is more nearly of the same density with the particles, upon which the colours depend, in these bodies than in others; since the change of colour is plainly owing to our seeing a different part of the body in different positions of the eye. Thus, in changeable silks, the warp is of one colour, and the woof of another; and in one position of the eye more of the warp is seen, and in another position of it more of the woof is seen. In like manner, if a pigeon's neck appears blue in one position of the eye, and crimson in another, it is because in these different positions we see different parts of the same feathers.

We cannot from the colour of a body make any conjecture about the size of the particles upon which its colours depend.

Suppose, from the appearance of the colour in any yellow body, that we had determined its yellow to be of the same sort with that which is next to the black spot in a plate of air, or water, or glass. The thickness of a plate, where it appears of this colour, is different according to the different density of the substance out of which that plate is made; the thickness of a plate of air where it appears of this colour, is greater than that of a plate of water were it appears of the same colour, and much greater still than that of a plate of glass. Suppose therefore farther, that we were able to determine exactly what is the thickness of a plate of air or water or glass, where each of them is tinged with the same yellow colour that any natural body exhibits; yet we cannot determine whether the diameter of the particles, upon which this body's colour depends, is equal to the thickness of the plate of air, or of water, or of glass, unless we could first determine whether the density of those particles is equal to the density of air, or to that of water, or to that of glass: since the particles must be larger, if their density is equal to the density of air, than if it is equal to the density of water; and larger, if it is equal to that of water than if it is equal to that of glass. And indeed we have good reason to conclude, that the density of the parts, upon which the colours of natural bodies depend, is greater even than that of glass; and consequently that the diameter of those parts is much less than the thickness of a plate of glass, where it appears of the same colour with the body. For, upon being viewed obliquely, thin plates of glass change colour, whereas natural bodies do not; and the colour of natural bodies is made more unchangeable than that of thin plates of glass, by their particles being more dense than glass.

Of the Rainbow.

When the rays of the sun fall upon a drop of rain and enter into it, some of them, after one reflection and two refractions, may come to the eye of a spectator who has his back towards the sun and his face toward the drop.

If XY (No. 43.) is a drop of rain, and the sun shines upon it in any lines sf, sd, sa, &c. most of the rays will enter into the drop; some few of them only will be reflected from the first surface; those rays, which are reflected from thence, do not come under our present consideration, because they are never refracted at all. The greatest part of the rays then enter the drop, and those passing on to the second surface will most of them be transmitted through the drop; but neither do those rays which are thus transmitted fall under our present consideration, since they are not reflected. For the rays, which are described in the proposition, are such as are twice refracted and once reflected. However, at the second surface, or hinder part of the drop, at pg some few rays will be reflected, whilst the rays are transmitted; those rays proceed in some such lines as nr, ng; and coming out of the drop in the lines rv, qt, may fall upon the eye of a spectator, who is placed anywhere in those lines, with his face towards the drop, and consequently with his back towards the sun, which is supposed to shine upon the drop in the lines sf, sd, sa, &c. These rays are twice refracted, and once reflected; they are refracted, when they pass out of the air into the drop; they are reflected from the second surface, and are refracted again, when they pass out of the drop into the air. When rays of light reflected from a drop of rain come to the eye, those are called effectual which are able to excite a sensation.

When rays of light come out of a drop of rain, they will not be effectual, unless they are parallel and contiguous.

There are but few rays that can come to the eye at all; for the greatest part of those rays which enter the drop xy (No. 43.) between x and a, pass out of the drop through the hinder surface fg; only few are reflected from thence and come out through the nearer surface between a and y. Now such rays as emerge, or come out of the drop, between a and y, will be ineffectual, unless they are parallel to one another, as rv and gt are; because such rays as come out diverging from one another, will be so far asunder when they come to the eye, that all of them cannot enter the pupil; and the very few that can enter it will not be sufficient to excite any sensation. But even rays, which are parallel, as rv, gt, will not be effectual, unless there are several of them contiguous or very near to one another. The two rays rv and gt alone will not be perceived, though both of them enter the eye; for so very few rays are not sufficient to excite a sensation.

When rays of light come out of a drop of rain after one reflection, those will be effectual which are reflected from the same point, and which entered the drop near to one another.

Any rays, as sb and cd, (No. 44.) when they have passed out of the air into a drop of water, will be refracted towards the perpendiculars bl, dl; and as the ray sb falls farther from the axis av than the ray cd, sb will be more refracted than cd; so that these rays, though parallel to one another at their incidence, may describe the lines be and de after refraction, and be both of them reflected from one and the same point e. Now all rays which are thus reflected from one and the same point, when they have described the lines ef, eg, and after reflection emerge at f and g, will be so refracted, when they pass out of the drop into the air, as to describe the lines fb, gi, parallel to one another. If these rays were to return from e in the lines eb, ed, and were to emerge at b and d, they would be refracted into the lines of their incidence bs, ds. But if these rays, instead of being returned in the lines eb, ed, are reflected from the same point e in the lines eg, ef, the lines of reflection eg and ef will be inclined both to one another and to the surface of the drop: just as much as the lines eb and ed are. First sb and eg make just the same angle with the surface of the drop: for the angle bex, which eb makes with the surface of the drop, is the complement of incidence; and the angle gey, which eg makes with the surface, is the complement of reflection; and these two are equal to one another. In the same manner we might prove that ed and ef make equal angles with the surface of the drop. Secondly, the angle bed is equal to the angle feg, or the reflected rays eg, ef, and the incident rays be, de, are equally inclined to each other. For the angle of incidence bel is equal to the angle of reflection gel, and the angle of incidence del is equal to the angle of reflection fel; consequently the difference between the angles of incidence is equal to the difference between the angles of reflection, or bel—del=gel—fel, or bed=gef. Since therefore either the lines eg ef, or the lines eb ed, are equally inclined both to one another and to the surface of the drop; the rays will be refracted in the same manner, whether they were to return in the lines eb, ed, or are reflected in the lines eg, ef. But if they were to return in the lines eb, ed, the refraction, when they emerge at b and d, would make them parallel. Therefore, if they are reflected from one and the same point e in the lines eg, ef, the refraction, when they emerge at g and f, will likewise make them parallel.

But though such rays, as are reflected from the same point in the hinder part of a drop of rain, are parallel to one another, when they emerge, and to have one condition that is requisite towards making them effectual; yet there is another condition necessary; for rays, that are effectual, must be contiguous, as well as parallel. And though rays, which enter the drop in different places, may be parallel when they emerge, those only will be contiguous which enter it nearly at the same place.

Let xy, (No. 43.) be a drop of rain, ag the axis or diameter of the drop, and ra a ray of light that comes from the sun and enters the drop at the point a. This ray sa, because it is perpendicular to both the surfaces, will pass straight through the drop in the line agb without being refracted; but any collateral rays that fall about sb, as they pass through the drop, will be made to converge to their axis, and passing out at n will meet the axis at b: rays which fall farther from the axis than sb, such as, those which fall about sc, will likewise be made to converge; but then their focus will be nearer to the drop than b. Suppose therefore i to be the focus to which the rays that fall about sc will converge, any ray sc, when it has described the line eo within the drop, and is tending to the focus i, will pass out of the drop at the point o. The rays that fall upon the drop about sd, more remote still from the axis, will converge to a focus still nearer than i, as suppose at k. These rays therefore go out of the drop at p. The rays, that fall still more remote from the axis, as ss, will converge to a focus nearer than k, as suppose at l; and the ray se, when it has described the line eo within the drop, and is tending to l, will pass out at the point o. The rays, that fall still more remote from the axis, will converge to a focus still nearer. Thus the ray sf will after refraction converge to a focus at m, which is nearer than l; and having described the line fn within the drop, it will pass out at the point n. Now here we may observe, that as any rays sb or sc, fall farther above the axis sa, the points n, or o, where they pass out behind the drop, will be farther above g; or that, as the incident ray rises from the axis sa, the arc gno increases, till we come to some ray sd, which passes out of the drop at p; and this is the highest point where any ray that falls upon the quadrant or quarter ax can pass out: for any rays se, or sf, that fall higher than sd, will not pass out in any point above p, but at the points s, or n, which are below it. Consequently, though the arc gnop increases, whilst the distance of the incident ray from the axis sa increased; till we come to the ray sd; yet afterwards, the higher the ray

ray falls above the axis \( sa \), this arc \( p \) will decrease.

We have hitherto spoken of the points on the hinder part of the drop, where the rays pass out of it; but this was for the sake of determining the points from whence those rays are reflected, which do not pass out behind the drop. For, in explaining the rainbow, we have no farther reason to consider those rays which go through the drop; since they can never come to the eye of a spectator placed anywhere in the lines \( rv \) or \( qt \) with his face towards the drop. Now, as there are many rays which pass out of the drop between \( g \) and \( p \), so some few rays will be reflected from thence; and consequently the several points between \( g \) and \( p \), which are the points where some of the rays pass out of the drop, are likewise the points of reflection for the rest which do not pass out. Therefore, in respect of those rays which are reflected, we may call \( gp \) the arc of reflection; and may say, that this arc of reflection increases, as the distance of the incident ray from the axis \( sa \) increases, till we come to the ray \( sd \); the arc of reflection is \( gn \) for the ray \( sb \), it is \( go \) for the ray \( se \), and \( gp \) for the ray \( sd \). But after this, as the distance of the incident ray from the axis \( sa \) increases, the arc of reflection decreases; for \( og \) less than \( pg \) is the arc of reflection for the ray \( se \), and \( rg \) is the arc of reflection for the ray \( sf \).

From hence it is obvious, that some one ray, which falls above \( sd \), may be reflected from the same point with some other ray which falls below \( sd \). Thus, for instance, the ray \( sb \) will be reflected from the point \( n \), and the ray \( sf \) will be reflected from the same point; and consequently, when the reflected rays \( nr \), \( nq \), are refracted as they pass out of the drop at \( r \) and \( q \), they will be parallel, by what has been shewn in the former part of this proposition. But since the intermediate rays, which enter the drop between \( sf \) and \( sb \), are not reflected from the same point \( n \), these two rays alone will be parallel to one another when they come out of the drop, and the intermediate rays will not be parallel to them. And consequently these rays \( rv \), \( qt \), though they are parallel after they emerge at \( r \) and \( q \), will not be contiguous, and for that reason will not be effectual; the ray \( sd \) is reflected from \( p \), which has been shewn to be the limit of the arc of reflection; such rays, as fall just above \( sd \), and just below \( sd \), will be reflected from nearly the same point \( p \), as appears from what has been already shewn. These rays therefore will be parallel, because they are reflected from the same point \( p \); and they will likewise be contiguous, because they all of them enter the drop at one and the same place very near to \( d \). Consequently, such rays as enter the drop at \( d \), and are reflected from \( p \) the limit of the arc of reflection, will be effectual; since, when they emerge at the fore part of the drop between \( a \) and \( y \), they will be both parallel and contiguous.

If we can make out hereafter that the rainbow is produced by the rays of the sun which are thus reflected from drops of rain as they fall whilst the sun shines upon them, this proposition may serve to show us, that this appearance is not produced by any rays that fall upon any part and are reflected from any part of those drops; since this appearance cannot be produced by any rays but those which are effectual; and effectual rays must always enter each drop at one certain place in the fore-part of it, and must likewise be reflected from one certain place in the hinder surface.

When rays that are effectual emerge from a drop of rain after one reflection and two refractions, those which are most refrangible will, at their emergence, make a less angle with the incident rays than those do which are least refrangible; and by this means the rays of different colours will be separated from one another.

Let \( fb \) and \( gi \), (No. 44.) be effectual violet rays emerging from the drop at \( fg \); and \( fn \), \( gp \), effectual red rays emerging from the same drop at the same place. Now, though all the violet rays are parallel to one another, because they are supposed effectual; and though all the red rays are likewise parallel to one another for the same reason; yet the violet rays will not be parallel to the red rays. These rays, as they have different colours, and different degrees of refrangibility, will diverge from one another; any violent ray \( gi \), which emerges at \( g \), will diverge from any red ray \( gp \), which emerges at the same place. Now, both the violet ray \( gi \), and the red ray \( gp \), as they pass out of the drop of water into the air, will be refracted from the perpendicular \( lo \). But the violet ray is more refrangible than the red one, and for that reason \( gi \), or the refracted violet ray, will make a greater angle with the perpendicular than \( gp \) the refracted red ray; or the angle \( igo \) will be greater than the angle \( pgo \). Suppose the incident ray \( sb \) to be continued in the direction \( sk \), and the violet ray \( ig \) to be continued backward in the direction \( ik \), till it meets the incident ray at \( k \). Suppose likewise the red ray \( pg \) to be continued backwards in the same manner, till it meets the incident ray at \( w \). The angle \( ikk \) is that which the violet ray, or most refrangible ray at its emergence, makes with the incident ray; and the angle \( pww \) is that which the red ray, or least refrangible ray at its emergence, makes with the incident ray. The angle \( iks \) is less than the angle \( pws \). For, in the triangle \( gawk \), \( gws \) or \( pws \) is the external angle at the base, and \( gkw \) or \( iki \) is one of the internal opposite angles; and either internal opposite angle is less than the external angle at the base. Euc. b. I. prop. 16. What has been shewn to be true of the rays \( gi \) and \( gp \) might be shewn in the same manner of the rays \( fb \) and \( fn \), or of any other rays that emerge respectively parallel to \( gi \) and \( gp \). But all the effectual violet rays are parallel to \( gi \), and all the effectual red rays are parallel to \( gp \). Therefore the effectual violet rays at their emergence make a less angle with the incident ones than the effectual red ones. And for the same reason, in all the other sorts of rays, those which are most refrangible, at their emergence from a drop of rain after one reflection, will make a less angle with the incident rays, than those do which are least refrangible.

Or otherwise: When the rays \( gi \) and \( gp \) emerge at the same point \( g \), as they both come out of water into air, and consequently are refracted from a perpendicular, instead of going straight forwards in the line \( eg \) continued, they will both be turned round upon the point \( g \) from the perpendicular \( go \). Now it is easy to conceive, that either of these lines might be turned in this manner upon the point \( g \). VEX is violet; the intermediate parts, reckoning from the red to the violet, are orange, yellow, green, blue, and indigo. Suppose the spectator's eye to be at O, and let LOP be an imaginary line drawn from the centre of the sun through the eye of the spectator: If a beam of light S coming from the sun falls upon any drop F; and the rays that emerge at F in the line FO, so as to be effectual, make an angle FOP of 42 degrees 2 minutes with the line LP; then these effectual rays make an angle of 42 degrees 2 minutes with the incident rays, by the preceding proposition, and consequently these rays will be red, so that the drop F will appear red. All the other rays, which emerge at F, and would be effectual if they fell upon the eye, are refracted more than the red ones, and consequently will pass above the eye. If a beam of light S falls upon the drop E; and the rays that emerge at E in the line EO, so as to be effectual, make an angle EOP of 40 degrees 17 minutes with the line LP; then these effectual rays make likewise an angle of 40 degrees 17 minutes with the incident rays, and the drop E will appear of a violet colour. All the other rays, which emerge at E, and would be effectual if they came to the eye, are refracted less than the violet ones, and therefore pass below the eye. The intermediate drops between F and E will for the same reasons be of the intermediate colours.

Thus we have shewn why a set of drops from F to E, as they are falling, should appear of the primary colours, red, orange, yellow, green, blue, indigo, and violet. It is not necessary that the several drops, which produce these colours, should all of them fall at exactly the same distance from the eye. The angle FOP, for instance, is the same whether the distance of the drop from the eye is OF, or whether it is in any other part of the line OF something nearer to the eye. And whilst the angle FOP is the same, the angle made by the emerging and incident rays, and consequently the colour of the drop, will be the same. This is equally true of any other drop. So that although in the figure the drops F and E are represented as falling perpendicularly one under the other, yet this is not necessary in order to produce the bow.

But the coloured line FE, which we have already accounted for, is only the breadth of the bow. It still remains to be shewn, why not only the drop F should appear red, but why all the other drops quite from A to B in the arc ATFYB should appear of the same colour. Now it is evident, that where-ever a drop of rain is placed, if the angle, which the effectual rays make with the line LP is equal to the angle FOP, that is, if the angle which the effectual rays make with the incident rays is 42 degrees 2 minutes, any of those drops will be red, for the same reason that the drop F is of this colour.

If FOP was to turn round upon the line OP, so that one end of this line should always be at the eye, and the other be at P opposite to the sun; such a motion of this figure would be like that of a pair of compasses turning round upon one of the legs OP with the opening FOP. In this revolution the drop F would describe a circle, P would be the centre, and ATFYB would be an arc in this circle. Now since, in this motion of the line and drop OF, the angle made by FO with OP, that is, the angle FOP, continues the same; if the sun was to shine upon this drop as it revolves, the effectual rays would make the same angle... angle with the incident rays; in whatever part of the arc \( \text{ATFYB} \) the drop was to be. Therefore, whether the drop is at \( A \), or at \( T \), or at \( Y \), or at \( B \), or where-ever else it is in this whole arc, it would appear red, as it does at \( F \). The drops of rain, as they fall, are not indeed turned round in this manner: but then, as innumerable of them are falling at once in right lines from the cloud, whilst one drop is at \( F \), there will be others at \( Y \), at \( T \), at \( B \), at \( A \), and in every other part of the arc \( \text{ATFYB} \): and all these drops will be red for the same reason that the drop \( F \) would have been red, if it had been in the same place. Therefore, when the sun shines upon the rain as it falls, there will be a red arc \( \text{ATFYB} \) opposite to the sun. In the same manner, because the drop \( E \) is violet, we might prove that any other drop, which, whilst it is falling, is in any part of the arc \( \text{AVEXB} \), will be violet, and consequently, at the same time that the red arc \( \text{ATFYB} \) appears, there will likewise be a violet arc \( \text{AVEXB} \) below or within it. \( FE \) is the distance between these two coloured arcs; and from what has been said it follows, that the intermediate space between these two arcs will be filled up with arcs of the intermediate colours, orange, yellow, blue, green, and indigo. All these coloured arcs together make up the primary rainbow.

The primary rainbow is never a greater arc than a semi-circle.

Since the line \( \text{LOP} \) is drawn from the sun through the eye of the spectator, and since \( P \) (No. 46.) is the centre of the rainbow; it follows, that the centre of the rainbow is always opposite to the sun. The angle \( \text{FOP} \) is an angle of 42 degrees 2 minutes, as was observed, or \( F \) the highest part of the bow is 42 degrees 2 minutes from \( P \) the centre of it. If the sun is more than 42 degrees 2 minutes high, \( P \) the centre of the rainbow, which is opposite to the sun, will be more than 42 degrees 2 minutes below the horizon; and consequently \( F \) the top of the bow, which is only 42 degrees 2 minutes from \( P \), will be below the horizon; that is, when the sun is more than 42 degrees 2 minutes high, no primary rainbow will be seen. If the sun is something less than 42 degrees 2 minutes high, then \( P \) will be something less than 42 degrees 2 minutes below the horizon; and consequently \( F \), which is only 42 degrees 2 minutes from \( P \), will be just above the horizon; that is, a small part of the bow at this height of the sun will appear close to the ground opposite to the sun. If the sun is 20 degrees high, then \( P \) will be 20 degrees below the horizon; and \( F \) the top of the bow, being 42 degrees 2 minutes from \( P \), will be 22 degrees 2 minutes above the horizon; therefore, at this height of the sun, the bow will be an arc of a circle whose centre is below the horizon; and consequently that arc of the circle, which is above the horizon, or the bow, will be less than a semicircle. If the sun is in the horizon, then \( P \), the centre of the bow, will be in the opposite part of the horizon; \( F \), the top of the bow, will be 42 degrees 2 minutes above the horizon; and the bow itself, because the horizon passes through the centre of it, will be a semicircle. More than a semicircle can never appear; because if the bow was more than semicircle, \( P \) the centre of it must be above the horizon; but \( P \) is always opposite to the sun, therefore \( P \) cannot be above the horizon, unless the sun is below it; and when the sun is set, or is below the horizon, it cannot shine upon the drops of rain, as they fall; and consequently, when the sun is below the horizon, no bow at all can be seen.

When the rays of the sun fall upon a drop of rain, some of them, after two reflections and two refractions, may come to the eye of a spectator, who has his back towards the sun and his face towards the drop.

If \( hgw \) (No. 45.) is a drop of rain, and parallel rays coming from the sun, as \( xv \), \( yw \), fall upon the lower part of it, they will be refracted towards the perpendiculars \( v \), \( w \), as they enter into it, and will describe some such lines as \( vh \), \( wi \). At \( b \) and \( i \) great part of these rays will pass out of the drop; but some of them will be reflected from thence in the lines \( hf \), \( ig \). At \( f \) and \( g \) again, great part of the rays, that were reflected thither, will pass out of the drop. But these rays will not come to the eye of a spectator at \( o \). However, here again all the rays will not pass out; but some few will be reflected from \( f \) and \( g \), in some such lines as \( fd \), \( gb \); and these, when they emerge out of the drop of water into the air at \( d \) and \( b \), will be refracted from the perpendiculars, and, describing the lines \( dt \), \( bo \), may come to the eye of a spectator who has his back towards the sun and his face towards the drop.

Those rays, which are parallel to one another after they have been once refracted and once reflected in a drop of rain, will be effectual when they emerge after two refractions and two reflections.

No rays can be effectual, unless they are contiguous, and parallel. From what was said, it appears, that when rays come out of a drop of rain contiguous to one another, either after one or after two reflections, they must enter the drop nearly at one and the same place. And if such rays as are contiguous are parallel after the first reflection, they will emerge parallel, and therefore will be effectual. Let \( xv \) and \( yw \) be contiguous rays which come from the sun, and are parallel to one another when they fall upon the lower part of the drop \( hgw \) (No. 45.). Suppose these rays to be refracted at \( v \) and \( w \), and to be reflected at \( h \) and \( i \); if they are parallel to one another, as \( hf \), \( gi \), after this first reflection, then, after they are reflected a second time from \( f \) and \( g \), and refracted a second time as they emerge at \( d \) and \( b \), they will go out of the drop parallel to one another in the lines \( dt \) and \( bo \), and will therefore be effectual.

The rays \( xv \), \( yw \), are refracted towards the perpendiculars \( v \), \( w \), when they enter the drop, and will be made to converge. As these rays are very oblique, their focus will not be far from the surface \( vw \). If this focus is at \( k \), the rays, after they have passed the focus, will diverge from thence in the directions \( kh \), \( ki \); and if \( ki \) is the principal focal distance of the concave reflecting surface \( hi \), the reflected rays \( hf \), \( ig \), will be parallel. These rays \( hf \), \( ig \), are reflected again from the concave surface \( fg \), and will meet in a focus at \( e \), so that \( ge \) will be the principal focal distance of this reflecting surface \( fg \). And because \( hi \) and \( fg \) are parts of the same sphere, the principal focal distances \( ge \) and \( ki \) will be equal to one another. When the rays have passed the focus \( e \), they will diverge from thence in the lines \( cd \), \( eb \); and we are to shew, that, when they emerge at \( d \) and \( b \), and are refracted there, they will become parallel.

Now if the rays \( vk \), \( wk \), when they have met at \( k \), were to be turned back again in the directions \( kv \), \( kw \), and were to emerge at \( v \) and \( w \), they would be refracted into the lines of their incidence \( xv \), \( yw \), and therefore would be parallel. But since \( ge \) is equal to \( ik \), as has already been shewn, the rays ed, eb, that diverge from e, fall in the same manner upon the drop at d and b, as the rays kv, kw, would fall upon it at v and w; and ed, eb, are just as much inclined to the refracting surface db, as kv, kw, would be to the surface vw. From hence it follows, that the rays ed, eb, emerging at d and b, will be refracted in the same manner, and will have the same direction in respect of one another; as kv, kw, would have. But kv and kw would be parallel after refraction. Therefore ed and eb will emerge in lines dp, bo, so as to be parallel to one another, and consequently so as to be effectual.

When rays that are effectual emerge from a drop of rain after two reflections and two refractions, those which are most refrangible will at their emergence make a greater angle with the incident rays than those do which are least refrangible; and by this means the rays of different colours will be separated from one another.

If rays of different colours, which are differently refrangible, emerge at any point b, (No. 45.) these rays will not be all of them equally refracted from the perpendicular. Thus, if bo is a red ray, which is of all others the least refrangible, and bm is a violet ray, which is of all others the most refrangible; when these two rays emerge at b, the violet ray will be refracted more from the perpendicular bx than the red ray, and the refracted angle xbm will be greater than the refracted angle xbo. From hence it follows, that these two rays, after emergence, will diverge from one another. In like manner, the rays that emerge at d will diverge from one another; a red ray will emerge in the line dp, a violet ray in the line dt. So that though all the effectual red rays of the beam bdop are parallel to one another, and all the effectual red rays of the beam bdop are likewise parallel to one another, yet the violet rays will not be parallel to the red ones, but the violet beam will diverge from the red beam. Thus the rays of different colours will be separated from one another.

This will appear farther, if we consider what the proposition affirms, That any violet or most refrangible ray will make a greater angle with the incident rays, than any red or least refrangible ray makes with the same incident rays. Thus if yw is an incident ray, bm a violet ray emerging from the point b, and bo a red ray emerging from the same point; the angle which the violet ray makes with the incident one is yrm, and that which the red ray makes with it is yso. Now yrm is a greater angle than yso. For in the triangle brs the internal angle brs is less than by the external angle at the base. Euc. b. I. prop. 16. But yrm is the complement of brs or of bry to two right ones, and yso is the complement of by to two right ones. Therefore, since bry is less than by, the complement of bry to two right angles will be greater than the complement of by to two right angles; or yrm will be greater than yso.

Or otherwise: Both the rays bo and bm, when they are refracted in passing out of the drop at b, are turned round upon the point b from the perpendicular bx. Now either of these lines bo or bm might be turned round in this manner, till it made a right angle with yw. Consequently, that ray which is most turned round upon b, or which is most refracted, will make an angle with yw that will be nearer to a right one than that ray makes with it which is least turned round upon b, or which is least refracted. Therefore that ray which is most refracted will make a greater angle with the incident ray than that which is least refracted.

But since the emerging rays, as they are differently refrangible, make different angles with the same incident ray yw, the refraction which they suffer at emergence will separate them from one another.

The angle yrm, which the most refrangible or violet rays make with the incident ones, is found by calculation to be 54 degrees 7 minutes; and the angle yso, which the least refrangible or red rays make with the incident ones, is found to be 50 degrees 57 minutes: the angles, which the rays of the intermediate colours, indigo, blue, green, yellow, and orange, make with the incident rays, are intermediate angles between 54 degrees 7 minutes and 50 degrees 57 minutes.

If a line is supposed to be drawn from the centre of the sun through the eye of the spectator; the angle, which, after two refractions and two reflections, any effectual ray makes with the incident ray, will be equal to the angle which it makes with that line.

If yw, (No. 45.) is an incident ray, bo an effectual ray, and qn a line drawn from the centre of the sun through the eye of the spectator; the angle yso, which the effectual ray makes with the incident ray, is equal to son the angle which the same effectual ray makes with the line qn. For yw and qn, considered as drawn from the centre of the sun, are parallel; bo crosses them, and consequently makes the alternate angles yro, son, equal to one another. Euc. b.I. prop. 29.

When the sun shines upon the drops of rain as they are falling; the rays that come from those drops to the eye of a spectator, after two reflections and two refractions, produce the secondary rainbow.

The secondary rainbow is the outermost CHD, No. 46. When the sun shines upon a drop of rain H; and the rays HO, which emerge at H so as to be effectual, make an angle HOP of 54 degrees 7 minutes with LOP a line drawn from the sun through the eye of the spectator; the same effectual rays will make likewise an angle of 54 degrees 7 minutes with the incident rays S, and the rays which emerge at this angle are violet ones, by what was observed above. Therefore, if the spectator's eye is at O, none but violet rays will enter it: for as all the other rays make a less angle with OP, they will fall above the spectator's eye. In like manner, if the effectual rays that emerge from the drop G make an angle of 50 degrees 57 minutes with the line OP, they will likewise make the same angle with the incident rays S; and consequently, from the drop G to the spectator's eye at O, no rays will come but red ones; for all the otherways, making a greater angle with the line OP, will fall below the eye at O. For the same reason, the rays emerging from the intermediate drops between H and G, and coming to the spectator's eye at O, will emerge at intermediate angles, and therefore will have the intermediate colours. Thus, if there are seven drops from H to G inclusively, their colours will be violet, indigo, blue, green, yellow, orange and red. This coloured line is the breadth of the secondary rainbow.

Now, if HOP was to turn round upon the line OP, like a pair of compasses upon one of the legs OP with the opening HOP, it is plain from the supposition, that, in such a revolution of the drop H, the angle HOP would be the same, same, and consequently the emerging rays would make the same angle with the incident ones. But in such a revolution the drop would describe a circle of which P would be the centre and CNHRD an arc. Consequently, since, when the drop is at N, or at R, or any where else in that arc, the emerging rays make the same angle with the incident ones as when the drop is at H, the colour of the drop will be the same to an eye placed at O, whether the drop is at N, or at H, or at R, or any where else in that arc.

Now, though the drop does not thus turn round as it falls, and does not pass through the several parts of this arc, yet, since there are drops of rain falling every where at the same time, when one drop is at H, there will be another at R, another at N, and others in all parts of the arc; and these drops will all of them be violet-coloured, for the same reason that the drop H would have been of this colour if it had been in any of those places. In like manner, as the drop G is red when it is at G, it would likewise be red in any part of the arc CWGQD; and so will any other drop, when, as it is falling, it comes to any part of that arc. Thus as the sun shines upon the rain, whilst it falls, there will be two arcs produced, a violet coloured one CNHRD, and a red one CWGQD; and for the same reasons the intermediate space between these two arcs will be filled up with arcs of the intermediate colours. All these arcs together make up the secondary rainbow.

The colours of the secondary rainbow are fainter than those of the primary rainbow; and are ranged in the contrary order.

The primary rainbow is produced by such rays as have been only once reflected; the secondary rainbow is produced by such rays as have been twice reflected. But at every reflection some rays pass out of the drop of rain without being reflected; so that the fewer the rays are reflected, the fewer of them are left. Therefore the colours of the secondary bow are produced by fewer rays, and consequently will be fainter, than the colours of the primary bow.

In the primary bow, reckoning from the outside of it, the colours are ranged in this order; red, orange, yellow, green, blue, indigo, violet. In the secondary bow, reckoning from the outside, the colours are violet, indigo, blue, green yellow, orange, red. So that the red, which is the outermost or highest colour in the primary bow, is the innermost or lowest colour in the secondary one.

Now the violet rays, when they emerge so as to be effectual after one reflection, make a less angle with the incident rays than the red ones; consequently the violet rays make a less angle with the lines OP, (No. 46.) than the red ones. But in the primary rainbow the rays are only once reflected, and the angle which the effectual rays make with OP is the distance of the coloured drop from P the centre of the bow. Therefore the violet drops or violet arc in the primary bow will be nearer to the centre of the bow, than the red drops or red arc; that is, the innermost colour in the primary bow will be violet, and the outermost colour will be red. And, for the same reason, through the whole primary bow, every colour will be nearer to the centre P, as the rays of that colour are more refrangible.

But the violet rays, when they emerge so as to be effectual after two reflections, make a greater angle with the incident rays than the red ones; consequently the violet rays will make a greater angle with the line OP, than the red ones. But in the secondary rainbow the rays are twice reflected, and the angle which effectual rays make with OP is the distance of the coloured drop from P the centre of the bow. Therefore the violet drops or violet arc in the secondary bow will be farther from the centre of the bow than the red drops or red arc; that is, the outermost colour in the secondary bow will be violet, and the innermost colour will be red. And, for the same reason, thro' the whole secondary bow, every colour will be further from the centre P, as the rays of that colour are more refrangible.