PART III.

Sect. I. The Application of the foregoing Theory to several natural Phenomena.

§ 1. Of the Rainbow.

This beautiful phenomenon hath engaged the attention of all ages. By some nations it hath been deified; though the more sensible part always looked upon it as a natural appearance, and endeavoured, however imperfectly, to account for it. The observations of the ancients and philosophers of the middle ages concerning the rainbow were such as could not have escaped the notice of the most illiterate husbandmen who gazed at the sky; and their hypotheses were such as deserve no notice. It was a considerable time even after the dawn of true philosophy in this western part of the world, before we find any discovery of importance on this subject. Maurolycus was the first who pretended to have measured the diameters of the two rainbows with much exactness; and he reports, that he found that of the inner bow to be 45 degrees, and that of the outer bow 56; from which Des Cartes takes occasion to observe, how little we can depend upon the observations of those who were not acquainted with the causes of appearances.

One Clichtovius (the same, it is probable, who distinguished himself by his opposition to Luther, and who died in 1543) had maintained, that the second bow is the image of the first, as he thought was evident from the inverted order of the colours. For, said he, when we look into the water, all the images that we see reflected by it are inverted with respect to the objects themselves; the tops of the trees, for instance, that stand near the brink, appearing lower than the roots.

That the rainbow is opposite to the sun, had always been observed. It was, therefore, natural to imagine, that the colours of it were produced by some kind of reflection of the rays of light from drops of rain, or vapour. The regular order of the colours was another circumstance that could not have escaped the notice of any person. But, notwithstanding mere reflection had in no other case been observed to produce colours, and it could not but have been observed that refraction is frequently attended with that phenomenon, yet no person seems to have thought of having recourse to a proper refraction in this case, before one Fletcher of Breslau, who, in a treatise which he published in 1571, endeavoured to account for the colours of the rainbow by means of a double refraction and one reflection. But he imagined that a ray of light, after entering a drop of rain, and suffering a refraction both at its entrance and exit, was afterwards reflected from another drop, before it reached the eye of the spectator. He seems to have overlooked the reflection at the farther side of the drop, or to have imagined that all the bendings of the light within the drop would not make a sufficient curvature to bring the ray of the sun to the eye of the spectator. That he should think of two refractions, was the necessary consequence of his supposing that the ray entered the drop at all. This supposition, therefore, was all the light that he threw upon the subject. B. Porta supposed that the rainbow is produced by the refraction of light in the whole body of rain or vapour, but not in the separate drops.

After all, it was a man whom no writers allow to have had any pretensions to philosophy, that hit upon this curious discovery. This was Antonio De Dominis, bishop of Spalatro, whose treatise De Radiis Visus et Lucis, was published by J. Bartolus in 1611. He first advanced, that the double refraction of Fletcher, with an intervening reflection, was sufficient to produce the colours of the bow, and also to bring the rays that formed them to the eye of the spectator, without any subsequent reflection. He distinctly describes the progress of a ray of light entering the upper part of the drop, where it suffers one refraction, and after being thereby thrown upon the back part of the inner surface, is from thence reflected to the lower part of the drop; at which place undergoing a second refraction, it is thereby bent, so as to come directly to the eye. To verify this hypothesis, this person (no philosopher as he was) proceeded in a very sensible and philosophical manner. For he procured a small globe of solid glass, and viewing it when it was exposed to the rays of the sun, in the same manner in which he had supposed that the drops of rain were situated with respect to them, he actually observed the same colours which he had seen in the true rainbow, and in the same order.

Thus the circumstances in which the colours of the rainbow were formed, and the progress of a ray of light through a drop of water, were clearly understood; but philosophers were a long time at a loss when they endeavoured to assign reasons for all the particular colours, and for the order of them. Indeed nothing but the doctrine of the different refrangibility of the rays of light, which was a discovery reserved for the great Sir Isaac Newton, could furnish a complete solution of this difficulty. De Dominis supposed that the red rays were those which had traversed the least space in the inside of a drop of water, and therefore retained more of their native force, and consequently, striking the eye more briskly, gave it a stronger sensation; that the green and blue colours were produced by those rays, the force of which had been, in some measure, obtunded in passing through a greater body of water; and that all the intermediate colours were composed (according to the hypothesis which generally prevailed at that time) of a mixture of these three primary ones. That the different colours were caused by some difference in the impulse of light upon the eye, and the greater or less impression that was thereby made upon it, was an opinion which had been adopted by many persons, who had ventured to depart from the authority of Aristotle.

Afterwards the same De Dominis observed, that all the rays of the same colour must leave the drop of water in a part similarly situated with respect to the eye, in order that each of the colours may appear in a circle, the centre of which is a point of the heavens, in a line drawn from the sun through the eye of the spectator. The red rays, he observed, must issue from the drop nearest to the bottom of it, in order that the circle of red may be the outermost, and therefore the most elevated in the bow.

Notwithstanding De Dominis conceived so justly of the manner in which the inner rainbow is formed, he was far from having as just an idea of the cause of the exterior bow. This he endeavoured to explain in the very same manner in which he had done the interior, viz. by one reflection of the light within the drop, preceded and followed by a refraction; supposing only that the rays which formed the exterior bow, were returned to the eye by a part of the drop lower than that which transmitted the red of the interior bow. He also supposed that the rays which formed one of the bows came from the superior part of the sun's disk, and those which formed the other from the inferior part of it. He did not consider, that upon these principles, the two bows ought to have been contiguous; or rather, that an indefinite number of bows would have had their colours all intermixed; which would have been no bow at all.

When Sir Isaac Newton discovered the different refrangibility of the rays of light, he immediately applied his new theory of light and colours to the phenomena of the rainbow, taking this remarkable object of philosophical inquiry where De Dominis and Defearde, for want of this knowledge, were obliged to leave their investigations imperfect. For they could give no good reason why the bow should be coloured, and much less could they give any satisfactory account of the order in which the colours appear.

If different particles of light had not different degrees of refrangibility, on which the colours depend, the rainbow, besides being much narrower than it is, would be colourless; but the different refrangibility of differently coloured rays being admitted, the reason is obvious, both why the bow should be coloured, and also why the colours should appear in the order in which they are observed. Let A, (fig. 2.) be a plate drop of water, and S a pencil of light; which, on its leaving the drop of water, reaches the eye of the spectator. This ray, at its entrance into the drop, begins to be decomposed into its proper colours; and upon leaving the drop, after one reflection and a second refraction, it is farther decomposed into as many small differently-coloured pencils, as there are primitive colours in the light. Three of them only are drawn in this figure, of which the blue is the most, and the red the least refracted.

The doctrine of the different refrangibility of light enables us to give a reason for the size of a bow of each particular colour. Newton, having found that the fines of refraction of the most refrangible and least refrangible rays, in passing from rain-water into air, are in the proportion of 185 to 182, when the fine of incidence is 138, calculated the size of the bow; and he found, that if the sun was only a physical point, without sensible magnitude, the breadth of the inner bow would be 2 degrees; and if to this 30' was added, for the apparent diameter of the sun, the whole breadth would be $2\frac{1}{2}$ degrees. But as the outermost colours, especially the violet, are extremely faint, the breadth of the bow will not, in reality, appear to exceed two degrees. He finds, by the same principles, that the breadth of the exterior bow, if it was everywhere equally vivid, would be $4^\circ 20'$. But in this case there is a greater deduction to be made, on account of the faintness of the light of the exterior bow; so that, in fact, it will not appear to be more than 3 degrees broad.

The principal phenomena of the rainbow are all explained on Sir Isaac Newton's principles in the following propositions.

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 (fig. 6.) is a drop of rain, and the sun shines upon it in any lines s f, s d, s a, &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 p g, some few rays will be reflected, whilst the rays are transmitted; those rays proceed in some such lines as \(nr, nq\); 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 \(sf, sd, sa, etc.\). 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, these 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 (fig. 6.) between X and a, pass out of the drop thro' the hinder surface \(pg\); 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 inef- fectual, unless they are parallel to one another, as \(rv\) and \(qt\) 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 \(rv, qt\), will not be effectual, unless there are several of them contiguous or very near to one another. The two rays \(rv\) and \(qt\) alone will not be perceived, though both of them enter the eye; for 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 \(cd\), (fig. 7.) 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 \(fh, gt\), 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 \(eb\) and \(eg\) make just the same angle with the surface of the drop: for the angle \(be\), which \(eb\) makes with the surface of the drop, is the complement of incidence, and the angle \(eg\), 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 so 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, (fig. 6.) be a drop of rain, \(ag\) the axis or diameter of the drop, and \(sa\) 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 \(agh\) 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 \(co\) 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 \(se\), 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

Of any rays \( s_b \) or \( s_c \), fall farther above the axis \( s_a \), 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 \( s_a \), the arc \( g_n \) increases, till we come to some ray \( s_d \), 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 \( s_e \) or \( s_f \), that fall higher than \( s_d \), will not pass out in any point above \( p \), but at the points \( o \) or \( n \), which are below it. Consequently, tho' the arc \( g_n \) increases, whilst the distance of the incident ray from the axis \( s_a \) increases, till we come to the ray \( s_d \); yet afterwards, the higher the ray falls above the axis \( s_a \), this arc \( p_o \) 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 \( r_v \) or \( q_t \) 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 \( g_p \) the arc of reflection; and many say, that this arc of reflection increases, as the distance of the incident ray from the axis \( s_a \) increases, till we come to the ray \( s_d \); the arc of reflection is \( g_n \) for the ray \( s_b \), it is \( g_o \) for the ray \( s_c \), and \( g_p \) for the ray \( s_d \). But after this, as the distance of the incident ray from the axis \( s_a \) increases, the arc of reflection decreases; for \( o_g \) less than \( p_g \) is the arc of reflection for the ray \( s_e \), and \( n_g \) is the arc of reflection for the ray \( s_f \).

From hence it is obvious, that some one ray, which falls above \( s_d \), may be reflected from the same point with some other ray which falls below \( s_d \). Thus, for instance, the ray \( s_b \) will be reflected from the point \( n \), and the ray \( s_f \) will be reflected from the same point; and consequently, when the reflected rays \( n_r \), \( n_g \), are refracted as they pass out of the drop at \( r \) and \( g \), 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 \( s_f \) and \( s_b \), are not reflected from the same point \( n \), these two rays alone will be the 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 \( r_o \), \( q_t \), 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 \( s_d \) is reflected from \( p \), which has been shewn to be the limit of the arc of reflection; such rays as fall just above \( s_d \), and just below \( s_d \), 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 \( a \). 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 shew 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 \( f_b \) and \( g_i \), (fig. 7.) be effectual violet rays Plate emerging from the drop at \( f_g \); and \( f_n \), \( g_p \), effectual CCXIII. 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 \( g_i \), which emerges at \( g \), will diverge from any red ray \( g_p \), which emerges at the same place. Now, both the violet ray \( g_i \), and the red ray \( g_p \), as they pass out of the drop of water into the air, will be refracted from the perpendicular \( l_o \). But the violet ray is more refrangible than the red one; and for that reason \( g_i \), or the refracted violet ray, will make a greater angle with the perpendicular than \( g_p \) the refracted red ray; or the angle \( i_g_o \) will be greater than the angle \( p_g_o \). Suppose the incident ray \( s_b \) to be continued in the direction \( s_k \), and the violet ray \( i_g \) to be continued backwards in the direction \( i_k \), till it meets the incident ray at \( k \). Suppose likewise the red ray \( p_g \) to be continued backwards in the same manner, till it meets the incident ray at \( w \). The angle \( i_k_s \) is that which the violet ray, or most refrangible ray at its emersion, makes with the incident ray; and the angle \( p_w_s \) is that which the red ray, or least refrangible ray at its emersion, makes with the incident ray. The angle \( i_k_s \) is less than the angle \( p_w_s \). For, in the triangle, \( g_w_k \), \( g_w_s \), or \( p_w_s \), is the external angle at the base, and \( g_k_w \) or \( i_k_s \) is one of the internal opposite angles; and either internal opposite angle is less than the external angle at the base. (Eucl. b. I. prop. 16.) What has been shewn to be true of the rays \( g_i \) and \( g_p \) might be shewn in the same manner of the rays \( f_b \) and \( f_n \), or of any other rays that emerge respectively parallel to \( g_i \) and \( g_p \). But all the effectual violet rays are parallel to \( g_i \), and all the effectual red rays are parallel to \( g_p \). Therefore the effectual violet rays at their emersion make a less angle 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 emersion from a drop of rain after one reflection, will make a less angle with the incident rays, than those do which are less refrangible.

Or otherwise: When the rays $g i$ and $g p$ 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 $e g$ continued, they will both be turned round upon the point $g$ from the perpendicular $g o$. Now it is easy to conceive, that either of these lines might be turned in this manner upon the point $g$ as upon a centre, till they become parallel parallel to $s b$ the incident ray. But if either of these lines or rays were refracted so much from $g o$ as to become parallel to $s b$, the ray so much refracted, would, after emersion, make no angle with $s k$, because it would be parallel to it. And consequently that ray which is most turned round upon the point $g$, or that ray which is most refrangible, will after emersion be nearest parallel to the incident ray, or will make the least angle with it. The same may be proved of all other rays emerging parallel to $g i$ and $g p$ respectively, or of all effectual rays; those which are most refrangible will after emersion make a less angle with the incident rays, than those do which are least refrangible.

But since the effectual rays of different colours make different angles with $s k$ at their emersion, they will be separated from one another: so that if the eye was placed in the beam $f g h i$, it would receive only rays of one colour from the drop $x a g y$; and if it was placed in the beam $f g n p$, it would receive only rays of some other colour.

The angle $s u p$, which the least refrangible or red rays make with the incident ones when they emerge so as to be effectual, is found by calculation to be 42 degrees 2 minutes. And the angle $s k i$, which the most refrangible rays make with the incident ones when they emerge so as to be effectual, is found to be 40 degrees 17 minutes. The rays which have the intermediate degrees of refrangibility, make with the incident ones intermediate angles between 42 degrees 2 minutes, and 40 degrees 17 minutes.

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

Let the eye of the spectator be at $i$, (fig. 7.) and let $q t$ be the line supposed to be drawn from the centre of the sun through the eye of the spectator; the angle $g i t$, which any effectual ray makes with this line, will be equal to the angle $i k s$, which the same ray makes with the incident ray $s b$ or $s k$. If $s b$ is a ray coming from the centre of the sun, then since $q t$ is supposed to be drawn from the same point, these two lines, upon account of the remoteness of the point from whence they are drawn, may be looked upon as parallel to one another. But the right line $k i$ crossing these two parallel lines will make the alternate angles equal. Euc. b. I. prop. 29. Therefore $k i t$ or $g i t$ is equal to $s k i$.

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 one reflection and two refractions, produce the primary rainbow.

If the sun shines upon the rain as it falls, there are commonly seen two bows, as AFB, CHD, (fig. 9.); or if the cloud and rain does not reach over that whole side of the sky where the bows appear, then only a part of one or of both bows is seen in that place where the rain falls. Of these two bows, the innermost AFB is the more vivid of the two, and this is called the primary bow. The outer part TFY of the primary bow is red, the inner part 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° 2' with the line LP; then these effectual rays make an angle of 42° 2' 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° 17' with the line LP; then these effectual rays make likewise an angle of 40° 17' 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 wherever 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 cident rays is $42^\circ 2'$, 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 with the incident rays, in whatever part of the arc ATFYB the drop was to be. Therefore, whether the drop is at A, or at T, or at Y, or at B, or wherever 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 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 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 AVEXB, will be violet; and consequently, at the same time that the red arc ATFYB appears, there will likewise be a violet arc 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 semicircle.

Since the line LOP is drawn from the sun through the eye of the spectator, and since P (fig. 9.) is the centre of the rainbow; it follows, that the centre of the rainbow is always opposite to the sun. The angle FOL is an angle of $42^\circ 2'$, as was observed, or F the highest part of the bow is $42^\circ 2'$ from P the centre of it. If the sun is more than $42^\circ 2'$ high, P the centre of the rainbow, which is opposite to the sun, will be more than $42^\circ 2'$ minutes below the horizon; and consequently F the top of the bow, which is only $42^\circ 2'$ from P, will be below the horizon; that is, when the sun is more than $42^\circ 2'$ minutes high, no primary rainbow will be seen. If the sun is something less than $42^\circ 2'$ high, then P will be something less than $42^\circ 2'$ below the horizon; and consequently F, which is only $42^\circ 2'$ from P, will be just above the horizon; that is, a small part of the bow at this height of the fun will appear close to the ground opposite to the sun. If the sun is $20^\circ$ high, then P will be $20^\circ$ below the horizon; and F the top of the bow, being $42^\circ 2'$ from P, will be $22^\circ 2'$ 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 $42^\circ 2'$ 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 a 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 (fig. 8.) is a drop of rain, and parallel rays coming from the sun, as $v_1$, $w_1$, fall upon the lower part of it, they will be refracted towards the perpendiculars $v_1$, $w_1$, as they enter into it, and will describe some such lines as $v_1$, $w_1$. 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 $b_f$, $i_g$. 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 $f_d$, $g_b$; and these, when they emerge out of the drop of water into the air at $b$ and $d$, will be refracted from the perpendiculars, and, describing the lines $d_t$, $b_o$, may come to the eye of the spectator who has his back towards the sun and his face towards the drop.

These 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 $v_1$ and $w_1$ be contiguous rays which come from the sun, and are parallel to one another when they fall upon the lower part of the drop, suppose these rays to be refracted at $v$ and $w$, and to be reflected at $b$ and $i$; if they are parallel to one another, as $b_f$, $g_i$, 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 $d_t$ and $b_o$, and will therefore be effectual.

The rays $v_1$, $w_1$, are refracted towards the perpendiculars $v_1$, $w_1$, 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 $v_1w_1$. If this focus is at $k$, the rays, after they have passed the focus, will will diverge from thence in the directions $k_b$, $k_i$; and if $k_i$ is the principal focal distance of the concave reflecting surface $h_i$, the reflected rays $h_f$, $i_g$, will be parallel. These rays $e_f$, $i_g$, are reflected again from the concave surface $f_g$, and will meet in a focus at $e$, so that $g_e$ will be the principal focal distance of this reflecting surface $f_g$. And because $h_i$ and $f_g$ are parts of the same sphere, the principal focal distances $g_e$ and $k_i$ will be equal to one another. When the rays have passed the focus $e$, they will diverge from thence in the lines $e_d$, $e_b$; 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 $v_k$, $w_k$, when they have met at $k$, were to be turned back again in the directions $k_v$, $k_w$, and were to emerge at $v$ and $w$, they would be refracted into the lines of their incidence, $v_z$, $w_y$, and therefore would be parallel. But since $g_e$ is equal to $i_k$, as has already been shewn, the rays $e_d$, $e_b$, that diverge from $e$, fall in the same manner upon the drop at $d$ and $b$, as the rays $k_v$, $k_w$, would fall upon it at $v$ and $w$; and $e_d$, $e_b$, are just as much inclined to the refracting surface $d_h$, as $k_v$, $k_w$, would be to the surface $v_w$. From hence it follows, that the rays $e_d$, $e_b$, emerging at $d$ and $b$, will be refracted in the same manner, and will have the same direction in respect of one another, as $k_v$, $k_w$, would have. But $k_v$ and $k_w$ would be parallel after refraction. Therefore $e_d$ and $e_b$ will emerge in lines $d_p$, $b_o$, 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 $b$, (fig. 8.) these rays will not be all of them equally refracted from the perpendicular. Thus, if $b_o$ is a red ray, which is of all others the least refrangible, and $b_m$ 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 $b_x$ than the red ray, and the refracted angle $x_b_m$ will be greater than the refracted angle $x_b_o$. 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 $d_p$, a violet ray in the line $d_z$. So that though all the effectual red rays of the beam $bdmt$ 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 $y_w$ is an incident ray, $b_m$ a violet ray emerging from the point $b$, and $b_o$ a red ray emerging from the same point; the angle which the violet ray makes with the incident one is $y_r_m$, and that which the red ray makes with it is $y_s_o$. Now $y_r_m$ is a greater angle than $y_s_o$. For in the triangle $b_r_s$ the internal angle $b_r_i$ is less than $b_r_y$ the external angle at the base. Euc. B. I. prop. 16. But $y_r_m$ is the complement of $b_r_s$ or of $b_r_y$ to two right ones, and $y_s_o$ is the complement of $b_r_y$ to two right ones. Therefore, since $b_r_y$ is less than $b_r_s$, the complement of $b_r_y$ to two right angles will be greater than the complement of $b_r_s$ to two right angles; or $y_r_m$ will be greater than $y_s_o$.

Or otherwise: Both the rays $b_o$ and $b_m$, when they are refracted in passing out of the drop at $b$, are turned round upon the point $b$ from the perpendicular $b_x$. Now either of these lines $b_o$ or $b_m$ might be turned round in this manner, till it made a right angle with $y_w$. Consequently, that ray which is most turned round upon $b$, or which is most refracted, will make an angle with $y_w$ 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 $y_w$, the refraction which they suffer at emergence will separate them from one another.

The angle $y_r_m$, which the most refrangible or violet rays make with the incident ones, is found by calculation to be $54^\circ 7'$; and the angle $y_s_o$, which the least refrangible or red rays make with the incident ones, is found to be $50^\circ 57'$: the angles, which the rays of the intermediate colours, indigo, blue, green, yellow, and orange, make with the incident rays, are intermediate angles between $54^\circ 7'$ and $50^\circ 57'$.

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 $y_w$ (fig. 8.) is an incident ray, $b_o$ an effectual ray, and $q_n$ a line drawn from the centre of the sun through $o$ the eye of the spectator; the angle $y_s_o$, which the effectual ray makes with the incident ray, is equal to $s_o_n$ the angle which the same effectual ray makes with the line $q_n$. For $y_w$ and $q_n$, considered as drawn from the centre of the sun, are parallel; $b_o$ crosses them, and consequently makes the alternate angles $y_s_o$, $s_o_n$, 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 these drops to the eye of a spectator, after two reflections and two refractions, produce the secondary rainbow.

The secondary rainbow is the outermost CHD, fig. 9. When the sun shines upon a drop of rain $H$; and the rays $H_O$, which emerge at $H$ so as to be effectual, make an angle $H_O_P$ of $54^\circ 7'$ with $L_O_P$ a line drawn from the sun through the eye of the spectator; the same effectual rays will make likewise an angle of $54^\circ 7'$ 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

Of the Rainbow.

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 $50^\circ 57'$ 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 other rays, 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, 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 anywhere 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 anywhere 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 everywhere 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 (fig. 9.) 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, through the whole secondary bow, every colour will be further from the centre P, as the rays of that colour are more refrangible.

§ 2. Of Coronas, Parhelia, &c.

Under the articles CORONA and PARHELION a pretty full account is given of the different hypotheses concerning these phenomena, and likewise of the method by which these hypotheses are supported, from the known laws of refraction and reflection; to which therefore, in order to avoid repetition, we must refer.

§ 3. Of the Apparent Place, Distance, Magnitude, and Motion of Objects.

Philosophers in general had taken for granted, that the place to which the eye refers any visible object seen by reflection or refraction, is that in which the visual ray meets a perpendicular from the object upon the reflecting or refracting plane. But this method of judging of the place of objects was called in question by Dr Barrow, who contended that the arguments brought in favour of the opinion were not conclusive. These arguments are, that the images of objects appear straight in a plane mirror, but curved in a convex or concave one; that a straight thread, when partly immersed perpendicularly in water, does not appear crooked as when it is obliquely plunged into the fluid; but that which is within the water seems to be a continuation of that which is without. With respect to the reflected image, however, of a perpendicular right line from a convex, or concave mirror, he Part III.

Apparent he says, that it is not easy for the eye to distinguish place, &c., the curve that it really makes; and that, if the appearance of a perpendicular thread, part of which is plunged in water, be closely attended to, it will not favour the common hypothesis. If the thread is of any shining metal, as silver, and viewed obliquely, the image of the part immersed will appear to detach itself sensibly from that part which is without the water, so that it cannot be true that every object appears to be in the same place where the refracted ray meets the perpendicular; and the same observation he thinks may be extended to the case of reflection. According to this writer, we refer every point of an object to the place from which the pencils of light, that give us the image of it, issue, or from which they would have issued if no reflecting or refracting substance intervened. Pursuing this principle, he proceeds to investigate the place, in which the rays issuing from each of the points of an object, and which reach the eye after one reflection or refraction, meet; and he found, that, if the refracting surface was plane, and the refraction was made from a denser medium into a rarer, those rays would always meet in a place between the eye and a perpendicular to the point of incidence. If a convex mirror be used, the case will be the same; but if the mirror be plane, the rays will meet in the perpendicular, and beyond it if it be concave. He also determined, according to these principles, what form the image of a right line will take, when it is presented in different manners to a spherical mirror, or when it is seen through a refracting medium.

Probable as Dr Barrow thought the maxim which he endeavoured to establish, concerning the supposed place of visible objects, he has the candour to mention an objection to it, and to acknowledge that he was not able to give a satisfactory solution of it. It is this. Let an object be placed beyond the focus of a convex lens; and if the eye be close to the lens, it will appear confused, but very near to its true place. If the eye be a little withdrawn, the confusion will increase, and the object will seem to come nearer; and when the eye is very near the focus, the confusion will be exceedingly great, and the object will seem to be close to the eye. But in this experiment the eye receives no rays but those that are converging; and the point from which they issue is so far from being nearer than the object, that it is beyond it; notwithstanding which, the object is conceived to be much nearer than it is, though no very distinct idea can be formed of its precise distance. It may be observed, that, in reality, the rays falling upon the eye in this case in a manner quite different from that in which they fall upon it in other circumstances, we can form no judgment about the place from which they issue. This subject was afterwards taken up by Berkley, Smith, Montucla, and others.

M. De la Hire made several valuable observations concerning the distance of visible objects, and various other phenomena of vision, which are well worth our notice. He also took particular pains to ascertain the manner in which the eye conforms itself to the view of objects placed at different distances. He enumerates five circumstances, which assist us in judging of the distance of objects, namely, their apparent magnitude, the strength of the colouring, the direction of the two eyes, the parallax of the objects, and the distinctness place, &c., of their small parts. Painters, he says, can only take advantage of the two first mentioned circumstances, and therefore pictures can never perfectly deceive the eye; but in the decorations of theatres, they, in some measure, make use of them all. The size of objects, and the strength of their colouring, are diminished in proportion to the distance at which they are intended to appear. Parts of the same object which are to appear at different distances, as columns in an order of architecture, are drawn upon different planes, a little removed from one another, that the two eyes may be obliged to change their direction, in order to distinguish the parts of the nearer plane from those of the more remote. The small distance of the planes serves to make a small parallax, by changing the position of the eye; and as we do not preserve a distinct idea of the quantity of parallax, corresponding to the different distances of objects, it is sufficient that we perceive there is a parallax, to be convinced that these planes are distant from one another, without determining what that distance is; and as to the last circumstance, viz., the distinctness of the small parts of objects, it is of no use in discovering the deception, on account of the false light that is thrown upon these decorations.

To these observations concerning deceptions of sight, we shall add a similar one of M. Le Cat, who took notice that the reason why we imagine objects to be larger when they are seen through a mist, is the dimness or obscurity with which they are then seen; this circumstance being associated with the idea of great distance. This he says is confirmed by our being surprised to find, upon approaching such objects, that they are so much nearer to us, as well as so much smaller, than we had imagined.

Among other cases concerning vision, which fell under the consideration of M. De la Hire, he mentions one which is of difficult solution. It is when a candle, in a dark place, and situated beyond the limits of distinct vision, is viewed through a very narrow chink in a card; in which case a considerable number of candles, sometimes so many as six, will be seen along the chink. This appearance he attributes to small irregularities in the surface of the humours of the eye, the effect of which is not sensible when rays are admitted into the eye through the whole extent of the pupil, and consequently one principal image effaces a number of small ones; whereas, in this case, each of them is formed separately, and no one of them is so considerable as to prevent the others from being perceived at the same time.

There are few persons, M. De la Hire observes, who have both their eyes perfectly equal, not only with respect to the limits of distinct vision, but also with respect to the colour with which objects appear tinged when they are viewed by them, especially if one of the eyes has been exposed to the impression of a strong light. To compare them together in this respect, he directs us to take two thin cards, and to make in each of them a round hole of a third or a fourth of a line in diameter, and, applying one of them to each of the eyes, to look through the holes on a white paper, equally illuminated; when a circle of the paper will appear to each of the eyes, and, placing Apparent the cards properly, these two circles may be made to place, &c., touch one another, and thereby the appearance of the same object to each of the eyes may be compared to the greatest advantage. To make this experiment with the greatest exactness, it is necessary, he says, that the eyes be kept shut some time before the cards be applied to them.

M. De la Hire first endeavoured to explain the cause of those dark spots which seem to float before the eyes, especially those of old people. They are most visible when the eyes are turned towards an uniform white object, as the snow in the open fields. If they be fixed when the eye is so, this philosopher supposed that they were occasioned by extravasated blood upon the retina. But he thought that the movable spots were occasioned by opaque matter floating in the aqueous humour of the eye. He thought the vitreous humour was not sufficiently limpid for this purpose.

By the following calculation M. De la Hire gives us an idea of the extreme sensibility of the optic nerves. One may see very easily, at the distance of 4000 toises, the sail of a wind-mill, 6 feet in diameter; and the eye being supposed to be an inch in diameter, the picture of this sail, at the bottom of the eye, will be $\frac{1}{500}$ of an inch, which is less than the 666th part of a line, and is about the 66th part of a common hair, or the 8th part of a single thread of silk. So small, therefore, must one of the fibres of the optic nerve be, which he says is almost inconceivable, since each of these fibres is a tube that contains spirits. If birds perceive distant objects as well as men, which he thought very probable, he observes that the fibres of their optic nerves must be much finer than ours.

The person who first took much notice of Dr Barrow's hypothesis was the ingenious Dr Berkley, bishop of Cloyne, who distinguished himself so much by the objections which he started to the reality of a material world, and by his opposition to the Newtonian doctrine of fluxions. In his essay on a new theory of vision, he observes, that the circle formed upon the retina, by the rays which do not come to a focus, produce the same confusion in the eye, whether they cross one another before they reach the retina, or tend to do it afterwards; and therefore that the judgment concerning distance will be the same in both the cases, without any regard to the place from which the rays originally issued; so that in this case, as, by receding from the lens, the confusion, which always accompanies the nearness of an object, increases, the mind will judge that the object comes nearer.

But, say Dr Smith, if this be true, the object ought always to appear at a less distance from the eye than that at which objects are seen distinctly, which is not the case: and to explain this appearance, as well as every other in which a judgment is formed concerning distance, he maintains, that we judge of it by the apparent magnitude of objects only, or chiefly; so that, since the image grows larger as we recede from the lens through which it is viewed, we conceive the object to come nearer. He also endeavours to shew, that, in all cases in which glasses are used, we judge of distance by the same simple rule; from which he concludes universally, that the apparent distance of an object seen in a glass, is to its apparent distance seen by the naked eye, as the apparent magnitude to the naked eye is to its apparent magnitude in the glass. Apparent

But that we do not judge of distance merely by the place, &c., angle under which objects are seen, is an observation as old as Alhazen, who mentions several instances, in which, though the angles under which objects appear be different, the magnitudes are universally and instantaneously deemed not to be so. And Mr Robins clearly shews the hypothesis of Dr Smith to be contrary to fact in the most common and simple cases. In microscopes, he says, it is impossible that the eye should judge the object to be nearer than the distance at which it has viewed the object itself, in proportion to the degree of magnifying. For when the microscope magnifies much, this rule would place the image at a distance, of which the sight cannot possibly form any opinion, as being an interval from the eye at which no object can be seen. In general, he says, he believes, that whoever looks at an object through a convex glass, and then at the object itself, without the glass, will find it to appear nearer in the latter case, though it be magnified in the glass; and in the same trial with the concave glass, though by the glass the object be diminished, it will appear nearer through the glass than without it.

But the most convincing proof that the apparent distance of the image is not determined by its apparent magnitude is the following experiment. If a double convex glass be held upright before some luminous object, as a candle, there will be seen two images, one erect, and the other inverted. The first is made simply by reflection from the nearest surface, the second by reflection from the farther surface, the rays undergoing a refraction from the first surface both before and after the reflection. If this glass has not too short a focal distance, when it is held near the object, the inverted image will appear larger than the other, and also nearer; but if the glass be carried off from the object, though the eye remain as near to it as before, the inverted image will diminish so much faster than the other, that, at length, it will appear very much less than it, but still nearer. Here, says Mr Robins, two images of the same object are seen under one view, and their apparent distances immediately compared; and here it is evident, that those distances have no necessary connection with the apparent magnitude. He also shews how this experiment may be made still more convincing, by sticking a piece of paper on the middle of the lens, and viewing it thro' a short tube.

M. Bouguer adopts the general maxim of Dr Barrow, in supposing that we refer objects to the place from which the pencils of rays seemingly converge at their entrance into the pupil. But when rays issue from below the surface of a vessel of water, or any other refracting medium, he finds that there are always two different places of this seeming convergence; one of them of the rays that issue from it in the same vertical circle, and therefore fall with different degrees of obliquity upon the surface of the refracting medium; and another, of those that fall upon the surface with the same degree of obliquity, entering the eye laterally with respect to one another. Sometimes, he says, one of these images is attended to by the mind, and sometimes the other, and different images may be observed by different persons. An object plunged in water Apparent water affords an example, he says, of this duplicity of place, &c., images.

If BA b, fig. 1. be part of the surface of water, and the object be at O, there will be two images of it, in two different places; one at G, on the caustic by refraction, and the other at E, in the perpendicular AO, which is as much a caustic as the other line. The former image is visible by the rays ODM, O dm, which are one higher than the other, in their progress to the eye; whereas the image at E is made by the rays ODM, O ef, which enter the eye laterally. This, says he, may serve to explain the difficulty of Father Tacquet, Barrow, Smith, and many other authors, and which Newton himself considered as a very difficult problem, though it might not be absolutely insoluble.

G. W. Krafft has ably supported the opinion of Dr Barrow, that the place of any point, seen by reflection from the surface of any medium, is that in which rays issuing from it, infinitely near to one another, would meet; and considering the case of a distant object, viewed in a concave mirror, by an eye very near to it, when the image, according to Euclid and other writers, would be between the eye and the object, and the rule of Dr Barrow cannot be applied, he says that in this case the speculum may be considered as a plane, the effect being the same, only the image is more obscure.

Dr Porterfield gives a distinct and comprehensive view of the natural methods of judging concerning the distance of objects.

The conformation of the eye, he observes, can be of no use to us with respect to objects that are placed without the limits of distinct vision. As the object, however, does then appear more or less confused, according as it is more or less removed from those limits, this confusion assists the mind in judging of the distance of the object; it being always esteemed so much the nearer, or the farther off, by how much the confusion is greater. But this confusion hath its limits also, beyond which it can never extend; for when an object is placed at a certain distance from the eye, to which the breadth of the pupil bears no sensible proportion, the rays of light that come from a point in the object, and pass the pupil, are so little diverging, that they may be considered as parallel. For a picture on the retina will not be sensibly more confused, tho' the object be removed to a much greater distance.

The most universal, and frequently the most sure means of judging of the distance of objects is, he says, the angle made by the optic axis. For our two eyes are like two different stations, by the assistance of which distances are taken; and this is the reason why those persons who are blind of one eye, so frequently miss their mark in pouring liquor into a glass, snuffing a candle, and such other actions as require that the distance be exactly distinguished. To convince ourselves of the usefulness of this method of judging of the distance of objects, he directs us to suspend a ring in a thread, so that its side may be towards us, and the hole in it to the right and left hand; and taking a small rod, crooked at the end, retire from the ring two or three paces, and having with one hand covered one of our eyes, to endeavour with the other to pass the crooked end of the rod thro' the ring. This, says he, appears very easy; and yet, upon trial, perhaps once in 100 times we shall not succeed, especially if we move the rod a little quickly.

Our author observes, that by persons recollecting the time when they began to be subject to the misfortunes above-mentioned, they may tell when it was that they lost the use of one of their eyes; which many persons are long ignorant of, and which may be a circumstance of some consequence to a physician.† See the use of this second method of judging of distances Medicine, De Chales limited to 120 feet; beyond which, he says, we are not sensible of any difference in the angle of the optic axis.

A third method of judging of the distance of objects, consists in their apparent magnitudes, on which so much stress was laid by Dr Smith. From this change in the magnitude of the image upon the retina, we easily judge of the distance of objects, as often as we are otherwise acquainted with the magnitude of the objects themselves; but as often as we are ignorant of the real magnitude of bodies, we can never, from their apparent magnitude, form any judgment of their distance.

From this we may see why we are so frequently deceived in our estimates of distance, by any extraordinary magnitudes of objects seen at the end of it; as, in travelling towards a large city, or a castle, or a cathedral church, or a mountain larger than ordinary, we fancy them to be nearer than we find them to be. This also is the reason why animals, and all small objects, seen in valleys, contiguous to large mountains, appear exceedingly small. For we think the mountain nearer to us than if it were smaller; and we should not be surprised at the smallness of the neighbouring animals, if we thought them farther off. For the same reason, we think them exceedingly small when they are placed upon the top of a mountain, or a large building; which appear nearer to us than they really are, on account of their extraordinary size.

Dr Jurin clearly accounts for our imagining objects, when seen from a high building, to be smaller than they are, and smaller than we fancy them to be when we view them at the same distance on level ground. It is, says he, because we have no distinct idea of distance in that direction, and therefore judge of things by their pictures upon the eye only; but custom will enable us to judge rightly even in this case.

Let a boy, says he, who has never been upon any high building, go to the top of the monument, and look down into the streets; the objects seen there, as men and horses, will appear so small as greatly to surprise him. But 10 or 20 years after, if in the mean time he has used himself now and then to look down from that and other great heights, he will no longer find the same objects to appear so small. And if he was to view the same objects from such heights as frequently as he sees them upon the same level with himself in the streets, he supposes that they would appear to him just of the same magnitude from the top of the monument, as they do from a window one story high. For this reason it is, that statues placed upon very high buildings ought to be made of a larger size than those which are seen at a nearer distance. Apparent stance; because all persons, except architects, are apt place, &c., to imagine the height of such buildings to be, much less than it really is.

The fourth method by which Dr Porterfield says that we judge of the distance of objects, is, the force with which their colour strikes upon our eyes. For if we be assured that two objects are of a similar and like colour, and that one appears more bright and lively than the other, we judge that the brighter object is the nearer of the two.

The fifth method consists in the different appearance of the small parts of objects. When these parts appear distinct, we judge that the object is near; but when they appear confused, or when they do not appear at all, we judge that it is at a greater distance. For the image of any object, or part of an object, diminishes as the distance of it increases.

The sixth and last method by which we judge of the distance of objects is, that the eye does not represent to our mind one object alone, but at the same time all those that are placed betwixt us and the principal object, whose distance we are considering; and the more this distance is divided into separate and distinct parts, the greater it appears to be. For this reason, distances upon uneven surfaces appear less than upon a plane; for the inequalities of the surfaces, such as hills, and holes, and rivers, that lie low and out of sight, either do not appear, or hinder the parts that lie behind them from appearing; and so the whole apparent distance is diminished by the parts that do not appear in it. This is the reason that the banks of a river appear contiguous to a distant eye, when the river is low and not seen.

Dr Porterfield very well explains several fallacies in vision depending upon our mistaking the distances of objects. Of this kind, he says, is the appearance of parallel lines, and long vistas consisting of parallel rows of trees; for they seem to converge more and more, as they are farther extended from the eye. The reason of this, he says, is because the apparent magnitudes of their perpendicular intervals are perpetually diminishing, while, at the same time, we mistake their distance. Hence we may see why, when the two parallel rows of trees stand upon an ascent, whereby the more remote parts appear farther off than they really are, because the line that measures the length of the vistas now appears under a greater angle than when it was horizontal, the trees, in such a case, will seem to converge less, and sometimes, instead of converging, they will be thought to diverge.

For the same reason that a long vista appears to converge more and more the farther it is extended from the eye, the remoter parts of a horizontal walk or a long floor will appear to ascend gradually; and objects placed upon it, the more remote they are, the higher they will appear, till the last be seen on a level with the eye; whereas the ceiling of a long gallery appears to descend towards a horizontal line, drawn from the eye of the spectator. For this reason, also, the surface of the sea, seen from an eminence, seems to rise higher and higher the farther we look; and the upper parts of high buildings seem to stoop, or incline forwards over the eye below, because they seem to approach towards a vertical line proceeding from the spectator's eye; so that statues on the top of such buildings, in order to appear upright, must recline, or bend backwards.

Our author also shews the reason why a windmill, seen from a great distance, is sometimes imagined to move the contrary way from what it really does, by our taking the nearer end of the sail for the more remote. The uncertainty we sometimes find in the course of the motion of a branch of lighted candles, turned round at a distance, is owing, he says, to the same cause; as also our sometimes mistaking a convex for a concave surface, more especially in viewing seas, and impressions, with a convex glass or a double microscope; and lastly, that, upon coming in a dark night into a street, in which there is but one row of lamps, we often mistake the side of the street they are on.

Far more light was thrown upon this curious subject by M. Bouguer.

The proper method of drawing the appearance of two rows of trees that shall appear parallel to the eye, is a problem which has exercised the ingenuity of several philosophers and mathematicians. That the apparent magnitude of objects decreases with the angle under which they are seen, has always been acknowledged. It is also acknowledged, that it is only by custom and experience that we learn to form a judgment both of magnitudes and distances. But in the application of these maxims to the above-mentioned problem, all persons, before M. Bouguer, made use of the real distance instead of the apparent one; by which only the mind can form its judgment. And it is manifest, that, if any circumstances contribute to make the distance appear otherwise than it is in reality, the apparent magnitude of the object will be affected by it; for the same reason, that, if the magnitude be misapprehended, the idea of the distance will vary.

For want of attending to this distinction, Tacquet pretended to demonstrate, that nothing can give the idea of two parallel lines (rows of trees for instance) to an eye situated at one of their extremities, but two hyperbolic curves, turned the contrary way; and M. Varignon maintained, that in order to make a vista appear of the same width, it must be made narrower, instead of wider, as it recedes from the eye.

M. Bouguer observes, that very great distances, and those that are considerably less than them, make nearly the same impression upon the eye. We, therefore, always imagine great distances to be less than they are, and for this reason the ground-plan of a long vista always appears to rise. The visual rays come in a determinate direction; but as we imagine that they terminate sooner than they do, we necessarily conceive that the place from which they issue is elevated. Every large plan, therefore, as AB, fig. 4, viewed Plate CCXII, by an eye at O, will seem to lie in such a direction as A b; and consequently lines, in order to appear truly parallel on the plane AB, must be drawn so as that they would appear parallel on the plane A b, and be from thence projected to the plane AB.

To determine the inclination of the apparent ground-plane A b to the true ground-plane AB, our ingenious author directs us to draw upon a piece of level ground, two straight lines of a sufficient length, (for which purpose lines fastened to small sticks are very con- Part III.

Apparent convenient), making an angle of 3 or 4 degrees with place, &c., one another. Then a person, placing himself within of objects.

the angle, with his back towards the angular point, must walk backwards and forwards till he can fancy the lines to be parallel. In this situation, a line drawn from the point of the angle thro' the place of his eye, will contain the same angle with the true ground-plane which this does with the apparent one.

M. Bouguer then shews other more geometrical methods of determining this inclination; and says, that by these means he has often found it to be 4 or 5 degrees, though sometimes only 2 or 2½ degrees. The determination of this angle, he observes, is variable; depending upon the manner in which the ground is illuminated, and the intensity of the light. The colour of the soil is also not without its influence, as well as the particular conformation of the eye, by which it is more or less affected by the same degree of light, and also the part of the eye on which the object is painted. When, by a slight motion of his head, he contrived, that certain parts of the soil, the image of which fell towards the bottom of his eye, should fall towards the top of the retina, he always thought that this apparent inclination became a little greater.

But what is very remarkable, and what he says he can assure his reader may be depended upon, is, that, if he look towards a rising ground, the difference between the apparent ground-plan and the true one, will be much more considerable, so that they will sometimes make an angle of 25 or 30 degrees. Of this he had made frequent observations. Mountains, he says, begin to be inaccessible when their sides make an angle from 35 to 37 degrees with the horizon, as then it is not possible to climb them but by means of stones or shrubs, to serve as steps to fix the feet on. In these cases, both he and his companions always agreed that the apparent inclination of the side of the mountain was 60 or 70 degrees.

These deceptions are represented in fig. 3, in which, when the ground plan, AM, or AN, are much inclined, the apparent ground plan Am, or An, makes a very large angle with it. On the contrary, if the ground dips below the level, the inclination of the apparent to the true ground-plan diminishes, till, at a certain degree of the slope, it becomes nothing at all; the two plans AP and Ap being the same, so that parallel lines drawn upon them would always appear fo. If the inclination below the horizon is carried beyond the situation AP, the error will increase; and what is very remarkable, it will be on the contrary side; the apparent plan Ar being always below the true plan AR, so that if a person would draw upon the plan AR lines that shall appear parallel to the eye, they must be drawn converging, and not diverging, as is usual on the level ground, because they must be the projections of two lines imagined to be parallel, on the plan Ar, which is more inclined to the horizon than AR.

These remarks, he observes, are applicable to different planes exposed to the eye at the same time. For if BH, fig. 2, be the front of a building, at the distance of AB from the eye, it will be reduced in appearance to the distance Ab; and the front of the building will be bb, rather inclined towards the spectator, unless the distance be inconsiderable.

After making a great number of observations upon this subject, our author concludes, that when a man stands upon a level plane, it does not seem to rise sensibly but at some distance from him. The apparent plane, therefore, has a curvature in it, at that distance, the form of which is not very easy to determine; so that a man standing upon a level plane, of infinite extent, will imagine that he stands in the centre of a basin. This is also, in some measure, the case with a person standing upon the level of the sea.

He concludes with observing that there is no difficulty in drawing lines according to these rules, so as to have any given effect upon the eye, except when some parts of the prospect are very near the spectator, and others very distant from him; because, in this case, regard must be had to the conical or conoidal figure of a surface. A right line passing at a small distance from the observer, and below the level of his eye, in that case almost always appears sensibly curved at a certain distance from the eye; and almost all figures, in this case, are subject to some complicated optical alteration to which the rules of perspective have not as yet been extended. If a circle be drawn near our feet, and within that part of the ground which always appears level to us, it will always appear to be a circle, and at a very considerable distance it will appear an ellipse; but between those two situations, it will not appear to be either the one or the other, but will be like one of those ovals of Descartes, which is more curved on one of its sides than the other.

On these principles a parterre, which appears distorted when it is seen in a low situation, appears perfectly regular when it is viewed from a balcony or any other eminence. Still, however, the apparent irregularity takes place at a greater distance, while the part that is near the spectator is exempt from it. If AB, fig. 5, be the ground-plane, and AA be a perpendicular, under the eye, the higher it is situated, at O, to the greater distance will T, the place at which the plane begins to have an apparent ascent along Tb, be removed.

All the varieties that can occur with respect to the visible motion of objects, are succinctly summed up by Dr Porterfield under 11 heads, with which we shall present our readers.

1. An object moving very swiftly is not seen, unless it be very luminous. Thus a cannon-ball is not seen if it is viewed transversely; but if it be viewed according to the line it describes, it may be seen, because its picture continues long on the same place of the retina; which, therefore, receives a more sensible impression from the object.

2. A live coal swung briskly round in a circle appears a continued circle of fire, because the impressions made on the retina by light, being of a vibrating, and consequently of a lasting nature, do not presently perish, but continue till the coal performs its whole circuit, and returns again to its former place.

3. If two objects, unequally distant from the eye, move with equal velocity, the more remote one will appear the slower; or, if their velocities be proportional to their distances, they will appear equally swift.

4. If two objects, unequally distant from the eye, move with unequal velocities in the same direction, their apparent velocities are in a ratio compounded of the direct ratio of their true velocities, and the reciprocal procal one of their distances from the eye.

5. A visible object moving with any velocity appears to be at rest, if the space described in the interval of one second be imperceptible at the distance of the eye. Hence it is that a near object moving very slowly, as the index of a clock, or a remote one very swiftly, as a planet, seems to be at rest.

6. An object moving with any degree of velocity will appear at rest, if the space it runs over in a second of time be to its distance from the eye as 1 to 1400.

7. The eye proceeding straight from one place to another, a lateral object, not too far off, whether on the right or left, will seem to move the contrary way.

8. The eye proceeding straight from one place to another, and being sensible of its motion, distant objects will seem to move the same way, and with the same velocity. Thus, to a person running eastwards, the moon on his right hand appears to move the same way, and with equal swiftness; for, by reason of its distance, its image continues fixed upon the same place of the retina, from whence we imagine that the object moves along with the eye.

9. If the eye and the object move both the same way, only the eye much swifter than the object, the last will appear to go backwards.

10. If two or more objects move with the same velocity, and a third remain at rest, the moveable ones will appear fixed, and the quiescent in motion, the contrary way. Thus clouds moving very swiftly, their parts seem to preserve their situation, and the moon to move the contrary way.

11. If the eye be moved with great velocity, lateral objects at rest appear to move the contrary way. Thus to a person sitting in a coach, and riding briskly through a wood, the trees seem to retire the contrary way; and to people in a ship, &c., the shores seem to recede.

At the conclusion of these observations, our author endeavours to explain another phenomenon of motion, which, though very common and well known, had not, as far as he knew, been explained in a satisfactory manner. It is this: If a person turns swiftly round, without changing his place, all objects about will seem to move round in a circle the contrary way; and this deception continues not only while the person himself moves round, but, which is more surprising, it also continues for some time after he ceases to move, when the eye, as well as the object, is at absolute rest.

The reason why objects appear to move round the contrary way, when the eye turns round, is not so difficult to explain: for though, properly speaking, motion is not seen, as not being in itself the immediate object of sight; yet by the sight we easily know when the image changes its place on the retina, and thence conclude that either the object, the eye, or both, are moved. But by the sight alone we can never determine how far this motion belongs to the object, how far to the eye, or how far to both. If we imagine the eye at rest, we ascribe the whole motion to the object, though it be truly at rest. If we imagine the object at rest, we ascribe the whole motion to the eye, though it belongs entirely to the object; and when the eye is in motion, though we are sensible of its motion, yet, if we do not imagine that it moves so swiftly as it really does, we ascribe only a part of the place, &c., motion to the eye, and the rest of it we ascribe to the object, though it be truly at rest. This last, he says, is what happens in the present case, when the eye turns round; for though we are sensible of the motion of the eye, yet we do not apprehend that it moves so fast as it really does; and therefore the bodies about appear to move the contrary way, as is agreeable to experience.

But the great difficulty still remains, viz. Why, after the eye ceases to move, objects should, for some time, still appear to continue in motion, though their pictures on the retina be truly at rest, and do not at all change their place. This, he imagined, proceeds from a mistake we are in with respect to the eye, which, though it be absolutely at rest, we nevertheless conceive it as moving the contrary way to that in which it moved before; from which mistake, with respect to the motion of the eye, the objects at rest will appear to move the same way which the eye is imagined to move; and, consequently, will seem to continue their motion for some time after the eye is at rest.

M. Le Cat well explains a remarkable deception, by which a person shall imagine an object to be on the opposite side of a board, when it is not so, and also inverted, and magnified. It is illustrated by fig. 3, in which D represents the eye, and CB a large black board, pierced with a small hole. E is a large white board, placed beyond it, and strongly illuminated; and d a pin, or other small object, held betwixt the eye and the first board. In these circumstances, the pin shall be imagined to be at F, on the other side of the board, where it will appear inverted, and magnified; because what is in fact perceived, is the shadow of the pin upon the retina; and the light that is stopped by the upper part of the pin coming from the lower part of the enlightened board, and that which is stopped by the lower part coming from the upper part of the board, the shadow must necessarily be inverted with respect to the object.

There is a curious phenomenon relating to vision, which some persons have ascribed to the inflection of light, but which Mr Melville explains in a very different and very simple manner.

When any opaque body is held at the distance of three or four inches from the eye, so that a part of some more distant luminous object, such as the window, or the flame of a candle, may be seen by rays passing near its edge, if another opaque body, nearer to the eye, be brought across from the opposite side, the edge of the first body will seem to swell outwards, and meet the latter; and in doing so will intercept a portion of the luminous object that was seen before.

This appearance he explains in the following manner: Let AB, fig. 1, represent the luminous object, to which the sight is directed, CD the more distant opaque body, GH the nearer, and EF the diameter of the pupil. Join ED, FD, EG, FG, and produce them till they meet AB in K, N, M, and L. It is plain that the parts AN, MB, of the luminous object cannot be seen. But taking any point a between N and K, and drawing aDd, since the portion dF of the Part III.

Concavity the pupil is filled with light flowing from that point, of the Sky, it must be visible. Any point b, between a and K, must fill fF, a greater portion of the pupil, and therefore must appear brighter. Again, any point c, between b and K, must appear brighter than b, because it fills a greater portion gF with light. The point K itself, and every other point in the space KL, must appear very luminous, since they send entire pencils of rays EKF, ELF, to the eye; and the visible brightnesses of every point from L towards M, must decrease gradually, as from K to N; that is, the spaces KN, LM, will appear as dim shadowy borders, or fringes, adjacent to the edges of the opaque bodies.

When the edge G is brought to touch the right line KF, the penumbras unite; and as soon as it reaches NDF, the above phenomenon begins; for it cannot pass that right line without meeting some line aDd, drawn from a point between N and K, and, by intercepting all the rays that fall upon the pupil, render it invisible. In advancing gradually to the line KDE, it will meet other lines bDf, cDg, &c. and therefore render the points b, c, &c. from N to K, successively invisible; and therefore the edge of the fixed opaque body CD must seem to swell outwards, and cover the whole space NK; while GH, by its motion, covers MK. When GH is placed at a greater distance from the eye, CD continuing fixed, the space OP to be passed over in order to intercept NK is less; and therefore, with an equal motion of GH, the apparent swelling of CD must be quicker; which is found true by experience.

If ML represent a luminous object, and REFQ any plane exposed to its light, the space FQ will be entirely shaded from the rays, and the space FE will be occupied by a penumbra, gradually darker, from E to F. Let now GH continue fixed, and CD move parallel to the plane EF; and as soon as it passes the line LF, it is evident that the shadow QF will seem to swell outwards; and when CD reaches ME, so as to cover with its shadow the space RE, QF, by its extension, will cover FE. This is found to hold true likewise by experiment.

§ 4. Of the Concave Figure of the Sky.

This apparent concavity is only an optical deception founded on the incapacity of our organs of vision to take in very large distances.—Dr Smith, in his Complete System of Optics, hath demonstrated, that, if the surface of the earth was perfectly plane, the distance of the visible horizon from the eye would scarce exceed the distance of 5000 times the height of the eye above the ground, supposing the height of the eye between five and six feet: beyond this distance, all objects would appear in the visible horizon. For, let OP be the height of the eye above the line PA drawn upon the ground; and if an object AB, equal in height to PO, be removed to a distance PA equal to 5000 times that height, it will hardly be visible by reason of the smallness of the angle AOB. Consequently any distance AC, how great forever, beyond A, will be invisible. For since AC and BO are parallel, the ray CO will always cut AB in some point D between A and B; and therefore the angle AOC, or AOD, will always be less than AOB, and therefore AD or AC will be invisible. Consequently Concavity all objects and clouds, as CE and FG, placed at all of the Sky-distances beyond A, if they be high enough to be visible, or to subtend a bigger angle at the eye than AOB, will appear at the horizon AB; because the distance AC is invisible.

Hence, if we suppose a vast long row of objects, or why a very vast long wall ABZY, built upon this plane, and long row of objects its perpendicular distance OA from the eye at O to must appear be equal to or greater than the distance OA of the circular visible horizon, it will not appear straight, but circular, Fig. 5, as if it was built upon the circumference of the horizon accgy: and if the wall be continued to an immense distance, its extreme parts YZ, will appear in the horizon at yz, where it is cut by a line Oy parallel to the wall. For, supposing a ray YO, the angle YOy will become insensibly small. Imagine this infinite plane OAYy, with the wall upon it, to be turned about the horizontal line O like the lid of a box, till it becomes perpendicular to the other half of the horizontal plane LMy, and the wall parallel to it, like a vast ceiling over head; and then the wall will appear like the concave figure of the clouds over-head. But though the wall in the horizon appear in the figure of a semicircle, yet the ceiling will not, but much flatter. Because the horizontal plane was a visible surface, which suggested the idea of the same distances quite round the eye; but in the vertical plane extended between the eye and the ceiling, there is nothing that affects the sense with an idea of its parts but the common line Oy; consequently the apparent distances of the higher parts of the ceiling will be gradually diminished in ascending from that line. Now when the sky is quite overcast with clouds of equal gravities, they will all float in the air at equal heights above the earth, and consequently will compose a surface resembling a large ceiling, as flat as the visible surface of the earth. Its concavity therefore is not real, but apparent: and when the heights of the clouds are unequal, since their real shapes and magnitudes are all unknown, the eye can seldom distinguish the unequal distances of those clouds that appear in the same directions, unless when they are very near us, or are driven by contrary currents of the air. So that the visible shape of the whole surface remains alike in both cases. And when the sky is either partly overcast, or perfectly free from clouds, it is matter of fact that we retain much the same idea of its concavity as when it was quite overcast.

The concavity of the heavens appears to the eye, which is the only judge of an apparent figure, to be a less portion of a spherical surface than a hemisphere. Dr Smith says, that the centre of the concavity is much below the eye; and by taking a medium among several observations, he found the apparent distance of its parts at the horizon to be generally between three and four times greater than the apparent distance of its parts overhead. For let the arch ABCD represent Fig. 6, the apparent concavity of the sky, O the place of the eye, OA and OC the horizontal and vertical apparent distances, whose proportion is required. First observe when the sun or the moon, or any cloud or star, is in such a position at B, that the apparent arches BA, BC, extended on each side of this object towards the horizon... Blue colour horizon and zenith, seem equal to the eye; then taking the altitude of the object B with a quadrant, or a cross-staff, or finding it by astronomy from the given time of observation, the angle AOB is known. Drawing therefore the line OB in the position thus determined, and taking in it any point B at pleasure in the vertical line CO produced downwards, seek the centre E of a circle ABC, whose arches BA, BC, intercepted between B and the legs of the right angle AOC, shall be equal to each other; then will this arch ABCD represent the apparent figure of the sky. For by the eye we estimate the distance between any two objects in the heavens by the quantity of sky that appears to lie between them; as upon earth we estimate it by the quantity of ground that lies between them. The centre E may be found geometrically by constructing a cubic equation, or as quick and sufficiently exact by trying whether the chords BA, BC, of the arch ABC drawn by conjecture are equal, and by altering its radius BE till they are so. Now in making several observations upon the sun, and some others upon the moon and stars, they seemed to our author to bisect the vertical arch ABC at B, when their apparent altitudes or the angle AOB was about 23 degrees; which gives the proportion of OC to OA as 3 to 10 or as 1 to 3 nearly. When the sun was but 30 degrees high, the upper arch seemed always less than the under one; and, in our author's opinion, always greater when the sun was about 18 or 20 degrees high.

§ 5. Of the Blue Colour of the Sky, and of Blue and Green Shadows.

The opinions of ancient writers concerning the colour of the sky merit no notice. The first who gave any rational explanation was Fromondus. By him it was supposed, that the blueness of the sky proceeded from a mixture of the white light of the sun with the black space beyond the atmosphere, where there is neither refraction nor reflection. This opinion prevailed very generally even in modern times, and was maintained by Otto Guerick and all his contemporaries, who asserted that white and black may be mixed in such a manner as to make a blue. Mr Bouguer had recourse to the vapours diffused through the atmosphere, to account for the reflection of the blue rays rather than any other. He seems however to suppose, that it arises from the constitution of the air itself, whereby the fainter coloured rays are incapable of making their way through any considerable tract of it. Hence he is of opinion, that the colour of the air is properly blue; to which opinion Dr Smith seems also to have inclined.

To this blue colour of the sky is owing the appearance of blue and green shadows in the mornings and evenings.—These were first taken notice of by M. Buffon in the month of July 1742, when he observed that the shadows of trees which fell upon a white wall were green. He was at that time standing upon an eminence, and the sun was setting in the cleft of a mountain, so that he appeared considerably lower than the horizon. The sky was clear, excepting in the west, which, though free from clouds, was lightly shaded with vapours, of a yellow colour, inclining to red. Then the sun itself was exceedingly red, and was seemingly, at least, four times as large as he appears to be at mid-day. In these circumstances, he saw distinctly the shadows of the trees, which were 30 or 40 feet from the white wall, coloured with a light green, inclining to blue. The shadow of an arbour, which was three feet from the wall, was exactly drawn upon it, and looked as if it had been newly painted with verdigris. This appearance lasted near five minutes; after which it grew fainter, and vanished at the same time with the light of the sun.

The next morning, at sun-rise, he went to observe other shadows, upon another white wall; but instead of finding them green, as he expected, he observed that they were blue, or rather of the colour of lively indigo. The sky was serene, except a slight covering of yellowish vapours in the east; and the sun arose behind a hill, so that it was elevated above his horizon. In these circumstances, the blue shadows were only visible three minutes; after which they appeared black, and in the evening of same day he observed the green shadows exactly as before. Six days passed without his being able to repeat his observations, on account of the clouds; but the 7th day, at sun-set, the shadows were not green, but of a beautiful sky-blue. He also observed, that the sky was, in a great measure, free from vapours at that time; and that the sun felt behind a rock, so that it disappeared before it came to his horizon. Afterwards he often observed the shadows both at sun-rise and sun-set; but always observed them to be blue, though with a great variety of shades of that colour. He shewed this phenomenon to many of his friends, who were as much surprized at it as he himself had been; but he says that any person may see a blue shadow, if he will only hold his finger before a piece of white paper at sun-rise or sun-set.

The first person who attempted to explain this phenomenon was the Abbé Mazeas, in a memoir of the society in Berlin for the year 1752. He observed, that when an opaque body was illuminated by the moon and a candle at the same time, and the two shadows were cast upon the same white wall, that which was enlightened by the candle was reddish, and that which was enlightened by the moon was blue. But, without attending to any other circumstances, he supposed the change of colour to be occasioned by the diminution of the light; but M. Melville, and M. Bouguer, both independent of one another, seem to have hit upon the true cause of this curious appearance, and which hath been already hinted at. The former of these gentlemen, in his attempts to explain the blue colour of the sky, observes, that since it is certain that no body assumes any particular colour, but because it reflects one sort of rays more abundantly than the rest; and since it cannot be supposed that the constituent parts of pure air are gross enough to separate any colours of themselves; we must conclude with Sir Isaac Newton, that the violet and blue making rays are reflected more copiously than the rest, by the finer vapours diffused through the atmosphere, whose parts are not big enough to give them the appearance of visible opaque clouds. And he shews, that in proper circumstances, the bluish colour of the sky-light may be actually seen on bodies illuminated by it, as, he says, it is objected should always happen upon this hypothesis. For that, if on a clear cloudless day a sheet of Blue colour of white paper be exposed to the sun's beams, when of the sky any opaque body is placed upon it, the shadow which is illuminated by the sky only will appear remarkably bluish compared with the rest of the paper, which receives the sun's direct rays.

M. Beguelin, who has taken the most pains with this subject, observes, that as M. Buffon mentions the shadows appearing green only twice, and that at all other times they are blue, this is the colour which they regularly have, and that the blue was changed into green by some accidental circumstance. Green, he says, is only a composition of blue and yellow, so that this accidental change may have arisen from the mixture of some yellow rays in the blue shadow; and that perhaps the wall might have had that tinge, so that the blue is the only colour for which a general reason is required. And this, he says, must be derived from the colour of pure air, which always appears blue, and which always reflects that colour upon all objects without distinction; but which is too faint to be perceived when our eyes are strongly affected by the light of the sun, reflected from other objects around us.

To confirm this hypothesis, he adds some curious observations of his own, in which this appearance is agreeably diversified. Being at the village of Boucholtz in July 1764, he observed the shadows projected on the white paper of his pocket-book, when the sky was clear. At half an hour past 6 in the evening, when the sun was about four degrees high, he observed that the shadow of his finger was of a dark grey, while he held the paper opposite to the sun; but when he inclined it almost horizontally, the paper had a bluish cast, and the shadow upon it was of a beautiful bright blue.

When his eye was placed between the sun and the paper laid horizontally, it always appeared of a bluish cast; but when he held the paper, thus inclined, between his eye and the sun, he could distinguish, upon every little eminence occasioned by the inequality of the surface of the paper, the principal of the prismatic colours. He also perceived them upon his nails, and upon the skin of his hand. This multitude of coloured points, red, yellow, green, and blue, almost effaced the natural colour of the objects.

At three quarters past six, the shadows began to be blue, even when the rays of the sun fell perpendicularly. The colour was the most lively when the rays fell upon it at an angle of 45° degrees; but with a less inclination of the paper, he could distinctly perceive, that the blue shadow had a border of a stronger blue, on that side which looked towards the sky, and a red border on that side which was turned towards the earth. To see these borders, the body that made the shadow was obliged to be placed very near the paper; and the nearer it was, the more sensible was the red border. At the distance of three inches, the whole shadow was blue. At every observation, after having held the paper towards the sky, he turned it towards the earth, which was covered with verdure, holding it in such a manner, that the sun might shine upon it while it received the shadows of various bodies, but, in this position, he could never perceive the shadow to be blue or green at any inclination with respect to the sun's rays.

At seven o'clock, the fun being still about two degrees high, the shadows were of a bright blue, even blue colour when the rays fell perpendicularly upon the paper, but of the sky were the brightest when it was inclined at an angle of 45°. At this time he was surprised to observe, that a large tract of sky was not favourable to this blue colour, and that the shadow falling upon the paper placed horizontally was not coloured, or at least the blue was very faint. This singularity, he concluded, arose from the small difference between the light of that part of the paper which received the rays of the sun, and that which was in the shade in this situation. In a situation precisely horizontal, the difference would vanish, and there could be no shadow. Thus too much or too little of the sun's light produced, but for different reasons, the same effect; for they both made the blue light reflected from the sky to become insensible. This gentleman never saw any green shadows, but when he made them fall on yellow paper. But he does not absolutely say, that green shadows cannot be produced in any other manner; and supposes, that if it was on the same wall that M. Buffon saw the blue shadows, seven days after having seen the green ones, he thinks that the cause of it might be the mixture of yellow rays, reflected from the vapours, which he observes were of that colour.

These blue shadows, our author observes, are not confined to the times of the sun-rising and sun-setting; but on the 19th of July, when the sun has the greatest force, he observed them at three o'clock in the afternoon, but the sun shone through a mist at that evening time.

If the sky is clear, the shadows begin to be blue; when, if they be projected horizontally, they are eight times as long as the height of the body that produces them, that is, when the centre of the sun is 7° 8' above the horizon. This observation, he says, was made in the beginning of August.

Besides these coloured shadows, which are produced by the interception of the direct rays of the sun, our author observed others similar to them at ever hour of the day, in rooms into which the light of the sun was reflected from some white body, if any part of the clear sky could be seen from the place, and all unnecessary light was excluded as much as possible. Observing these precautions, he says that the blue shadows may be seen at any hour of the day, even with the direct light of the sun; and that this colour will disappear in all those places of the shadow from which the blue sky cannot be seen.

All the observations that our author made upon the yellow or reddish borders of shadows above-mentioned, led him to conclude, that they were occasioned by the interception of the sky-light, whereby part of the shadow was illuminated either by the red rays reflected from the clouds, when the sun is near the horizon, or from some terrestrial bodies in the neighbourhood. This conjecture is favoured by the necessity he was under of placing any body near the paper, in order to produce this bordered shadow, as he says it is easily demonstrated, that the interception of the sky-light can only take place when the breadth of the opaque body is to its distance from the white ground on which the shadow falls, as twice the sine of half the amplitude of the sky to its cosine.

At the conclusion of his observations on these blue shadows, Irradiations shadows, he gives a short account of another kind of of the Sun's them, which, he does not doubt, have the same ori- gin. These he often saw early in the spring when he was reading by the light of a candle in the morning, and consequently the twilight mixed with that of his candle. In these circumstances, the shadow that was made by intercepting the light of his candle, at the di- stance of about six feet, was of a beautiful and clear blue, which became deeper as the opaque body which made the shadow was brought nearer to the wall, and was exceedingly deep at the distance of a few inches only. But wherever the day-light did not come, the shadows were all black without the least mixture of blue.

§ 6. Of the Irradiations of the Sun's Light appearing through the interstices of the Clouds.

This is an appearance which every one must have observed when the sky was pretty much overcast with clouds at some distance from each other. At that time several large beams of light, something like the appear- ance of the light of the sun admitted into a smoky room, will be seen, generally with a very considerable degree of divergence, as if the radiant point was sit- uated at no great distance above the clouds. Dr Smith observes, that this appearance is one of those which serve to demonstrate that very high and remote objects in the heavens do not appear to us in their real shapes and positions, but according to their perspective projections on the apparent concavity of the sky. He acquaints us, that though these beams are generally seen diverging, as represented in fig. 7, it is not al- ways the case. He himself, in particular, once saw them converging towards a point diametrically oppo- site to the sun; for, as near as he could conjecture, the point to which they converged was situated as much below the horizon, as the sun was then elevated above the opposite part of it. This part is represented by the line \(D_1\), and the point below it in opposition to the sun is \(E_2\); towards which all the beams \(v_1, v_2, \ldots\) appeared to converge.

"Observing," (says our author,) "that the point of convergence was opposite to the sun, I began to suspect that this unusual phenomenon was but a case of the usual apparent divergence of the beams of the sun from his apparent place among the clouds, as represented in fig. 7. I say, an apparent divergence; for though noth- ing is more common than for rays to diverge from a luminous body, yet the divergence of these beams in such large angles is not real but apparent. Because it is impossible for the direct rays of the sun to cross one another at any point of the apparent concavity of the sky, in a greater angle than about half a degree. For the diameter of the earth being so extremely small, in comparison to the distance of the sun, as to subtend an angle at any point of his body of but 20 or 22 seconds at most; and the diameter of our visible horizon being extremely smaller than that of the earth; it is plain, that all the rays which fall upon the horizon, from any given point of the sun, must be inclined to each other in the smallest angles imaginable; the greatest of them being as much smaller than that angle of 22 seconds, as the diameter of the visible horizon is smaller than that of the earth. All the rays that come to us from any given point of the sun may therefore be consid- ered as parallel to each other; as the rays \(eB_g\) from the irradiations point \(e\), or \(fB_b\) from the opposite point \(f\); and con- sequently the rays of these two pencils that come from opposite points of the sun's real diameter, and cross each other in the sun's apparent place \(B\) among the clouds, Fig. 9. can constitute no greater an angle with each other than about half a degree; this angle of their intersec- tion \(eB_f\) being the same as the sun would appear un- der to an eye placed among the clouds at \(B\), or (which is much the same), to an eye at \(O\) upon the ground. Because the sun's real distance \(OS\) is inconceivably greater than his apparent distance \(OB\). Therefore the rays of the sun, as \(B_g, B_b\), do really diverge from his apparent place \(B\) in no greater angles \(gB_b\) than about half a degree. Nevertheless they appear to diverge from the place \(B\) in all possible angles, and even in op- posite directions. Let us proceed then to an explana- tion of this apparent divergence, which is not self-evid- ent by any means; though at first sight we are apt to think it is, by not distinguishing the vast difference between the true and apparent distances of the sun.

"What I am going to demonstrate is this. Sup- posing all the rays of the sun to fall accurately paral- lel to each other upon the visible horizon, as they do very nearly, yet in both cases they must appear to di- verge in all possible angles. Let us imagine the hea- vens to be partly overcast with a spacious bed of bro- ken clouds, \(v_1, v_2, \ldots\) lying parallel to the plane of the visible horizon, here represented by the line \(AOD\). And when the sun's rays fall upon these clouds in the parallel lines \(v_1, v_2, \ldots\) let some of them pass through their intervals in the lines \(v_1, v_2, \ldots\) and fall upon the plane of the horizon at the places \(t_1, t_2, \ldots\). And since the rest of the incident rays \(v_1, v_2, \ldots\) are supposed to be intercepted from the place of the spectator at \(O\) by the cloud \(x\), and from the intervals between the transmitted rays \(v_1, v_2, \ldots\) by the clouds \(v_1, v_2, \ldots\), a small part of these latter rays \(v_1, v_2, \ldots\) when reflected every way from some certain kind of thin vapours float- ing in the air, may undoubtedly be sufficient to affect the eye with an appearance of lights and shades, in the form of bright beams in the places \(v_1, v_2, \ldots\) and of dark ones in the intervals between them: just as the like beams of light and shade appear in a room by reflections of the sun's rays from a smoky or dusty air within it: the lights and shades being here occa- sioned by the transmission of the rays through some parts of the window, and by their interruption at other parts.

"Now if the apparent concavity of this bed of clouds \(v_1, v_2, \ldots\) to the eye at \(O\), be represented by the arch \(ABCD\), and be cut in the point \(B\) by the line \(OB_x\) drawn parallel to the beams \(v_1, v_2, \ldots\); it will be evident by the rules of perspective, that these long beams will not appear in their real places, but upon the concave \(AB\) \(CD\) diverging every way from the place \(B\), where the sun himself appears, or the cloud \(x\) that covers his body, as represented separately in full view in fig. 7.

"And for the same reason, if the line \(BO\) be pro- duced towards \(E\), below the plane of the horizon \(AOD\), and the eye be directed towards the region of the sky directly above \(E\), the lower ends of the same real beams \(v_1, v_2, \ldots\) will now appear upon the part \(DF\) of this concave; and will seem to converge towards the point \(F_1\), situated just as much below the horizon as the op- posite Part III.

Irradiations posit point B is above it: which is separately represented in full view in fig. 8.

"For if the beams vt, vt, be supposed to be visible throughout their whole lengths, and the eye be directed in a plane perpendicular to them, here represented by the line OF; they and their intervals will appear broader in and about this plane, because these parts of them are the nearest to the eye; and therefore their remoter parts and intervals will appear gradually narrower towards the opposite ends of the line BE. As a farther illustration of this matter, we may conceive the spectator at O to be situated upon the top of so large a descent OHI towards a remote valley IK, and the sun to be so very low, that the point E, opposite to him, may be seen above the horizon of this shady valley. In this case it is manifest, that the spectators at O would now see these beams converging so far as to meet each other at the point E in the sky itself.

"I do not remember to have ever seen any phenomenon of this kind by moon-light; not so much as of beams diverging from her apparent place. Probably her light is too weak after reflections from any kind of vapours, to cause a sensible appearance of lights and shades so as to form these beams. And in the unusual phenomenon I well remember, that the converging sun-beams towards the point below the horizon were not quite so bright and strong as those usually are that diverge from him; and that the sky beyond them appeared very black (several showers having passed that way), which certainly contributed to the evidence of this appearance. Hence it is probable that the thinness and weakness of the reflected rays from the vapours opposite to the sun, is the chief cause that this appearance is so very uncommon in comparison to that other of diverging beams. For as the region of the sky round about the sun, is always brighter than the opposite one; so the light of the diverging beams ought also to be brighter than that of the converging ones. For, though rays are reflected from rough unpolished bodies in all possible directions, yet it is a general observation, that more of them are reflected forwards obliquely, than are reflected more directly backwards. Besides, in the present case, the incident rays upon the opposite region to the sun, are more diminished by continual reflections from a longer tract of the atmosphere, than the incident rays upon the region next the sun.

"The common phenomenon of diverging beams, I think, is more frequent in summer than in winter, and also when the sun is lower than when higher up; probably because the lower vapours are denser and therefore more strongly reflective than the higher; because the lower sky-light is not so bright as the upper; because the air is generally quieter in the mornings and evenings than about noon-day; and lastly, because many sorts of vapours are exhaled in greater plenty in summer than in winter, from many kinds of volatile vegetables; which vapours, when the air is cooled and condensed in the mornings and evenings, may become dense enough to reflect a sensible light."

§ 7. Of the Illumination of the Shadow of the Earth by the refraction of the Atmosphere.

The ancient philosophers, who knew nothing of the refractive power of the atmosphere, were very much perplexed to account for the body of the moon being visible when totally eclipsed. At such times she generally appears of a dull red colour, like tarnished copper, or of iron almost red hot. This, they thought, was the moon's native light, by which she became visible when hid from the brighter light of the sun. Plutarch indeed, in his discourse upon the face of the moon, attributes this appearance to the light of the fixed stars reflected to us by the moon; but this must be by far too weak to produce that effect. The true cause of it is the scattered beams of the sun bent into the earth's shadow by refractions through the atmosphere in the following manner.

"Let the body of the sun, says Dr Smith, be represented by the greater circle ab, and that of the earth by ccxv. the lesser one cd; and let the lines ace and bde touch them both on their opposite sides, and meet in e beyond the earth; then the angular space ced will represent the conic figure of the earth's shadow, which would be totally deprived of the sun's rays, were none of them bent into it by the refractive power of the atmosphere. Let this power just vanish at the circle hi, concentric to the earth, so that the rays ab and bi, which touch its opposite sides, may proceed unrefracted, and meet each other at k. Then the two nearest rays to these that flow within them, from the same points a and b, being refracted inwards through the margin of the atmosphere, will cross each other at a point l, somewhat nearer to the earth than k; and in like manner, two opposite rays next within the two last will cross each other at a point m, somewhat nearer to the earth than l, having suffered greater refractions, by passing through longer and denser tracts of air lying somewhat nearer to the earth. The like approach of the successive intersections k, l, m, is to be understood of innumerable couples of rays, till you come to the intersection n of the two innermost; which we may suppose just to touch the earth at the points o and p. It is plain then, that the space bounded by these rays on, np, will be the only part of the earth's shadow wholly deprived of the sun's rays. Let fmg represent part of the moon's orbit when it is nearest to the earth, at a time when the earth's dark shadow on p is the longest: in this case I will shew that the ratio of tm to tn is about 4 to 3; and consequently that the moon, tho' centrally eclipsed at m, may yet be visible by means of those scattered rays above-mentioned, first transmitted to the moon by refraction through the atmosphere, and from thence reflected to the earth.

"For let the incident and emergent parts ag, rn, Fig. 2, of the ray agorn, that just touches the earth at o, be produced till they meet at u, and let agu produced meet the axis st produced in x; and joining ui and um, since the refractions of an horizontal ray passing from o to r, or from o to q, would be alike and equal, the external angle nux is double the quantity of the usual refraction of an horizontal ray; and the angle aux is the apparent measure of the sun's semidiameter seen from the earth; and the angle uxt is that of the earth's semidiameter tu seen from the sun (called his horizontal parallax); and lastly, the angle umt is that of the earth's semidiameter seen from the moon, (called her horizontal parallax); because the elevation of the point u above the earth, is too small to make a sensible error in the quantity of these angles; whose measures by astronomical tables are as follow:

The The sun's least apparent semidiameter = ang. aus = 15°-50 The sun's horizontal parallax = ang. ust = 30°-10 Their difference is = ang. txu = 15°-40 Double the horizontal refraction = ang. nux = 67°-30 Their sum is = ang. tnu = 83°-10

The moon's greatest horizontal parallax = ang. tmu = 62°-10

Therefore we have \( tm : tn :: (ang. tnu : ang. tmu :: 83°-10' : 62°-10') \cdot 4 : 3 \) in round numbers; which was to be proved. It is easy to collect from the moon's greatest horizontal parallax of 62°-10', that her least distance \( tm \) is about 55½ semidiameters of the earth; and therefore the greatest length \( tn \) of the dark shadow, being three quarters of \( tm \), is about 41½ semidiameters.

"The difference of the last mentioned angles \( tnu, tmu \), is \( man=21' \), that is, about two thirds of 31°-40', the angle which the whole diameter of the sun subtends at \( u \). Whence it follows, that the middle point \( m \) of the moon centrally eclipsed, is illuminated by rays which come from two thirds of every diameter of the sun's disk, and pass by one side of the earth; and also by rays that come from the opposite two thirds of every one of the said diameters, and pass by the other side of the earth. This will appear by conceiving the ray \( ag \) or \( rn \) to be inflexible, and its middle point \( o \) to abide upon the earth, while the part \( rn \) is approaching to touch the point \( m \); for then the opposite part \( ga \) will trace over two thirds of the sun's diameter. The true proportion of the angles \( num, aus \), could not be preserved in the scheme, by reason of the sun's immense distance and magnitude with respect to the earth.

"Having drawn the line \( ata \), is is observable, that all the incident rays, as \( ag, ax \), flowing from any one point of the sun to the circumference of the earth, will be collected to a focus \( a \), whose distance \( ta \) is less than \( tm \) in the ratio of 62 to 67 nearly; and thus an image of the sun will be formed at \( a \), whose rays will diverge upon the moon. For the angle \( tua \) is the difference of the angles \( xua, ust \) found above; and \( ta : tm :: ang. tmu : ang. txu :: 62°-10' : 67°-30' \).

"The rays that flow next above \( ag \) and \( ax \), by passing through a thinner part of the atmosphere, will be united at a point in the axis \( ta \) somewhat farther from the earth than the last focus \( a \); and the same may be said of the rays that pass next above these, and so on; whereby an infinite series of images of the sun will be formed, whose diameters and degrees of brightness will increase with their distances from the earth.

"Hence it is manifest why the moon eclipsed in her perigee is observed to appear always duller and darker than in her apogee. The reason why her colour is always of the copper kind between a dull red and orange, I take to be this. The blue colour of a clear sky shews manifestly that the blue-making rays are more copiously reflected from pure air than those of any other colour; consequently they are less copiously transmitted through it among the rest that come from the sun, and so much the less as the tract of air through which they pass is the longer. Hence the common colour of the sun and moon is whitish in the meridian, and grows gradually more inclined to diluted yellow, orange, and red, as they descend lower, that is, as the rays are transmitted through a longer tract of air; which tract being still lengthened in passing to the moon and back again, causes a still greater loss of the blue-making rays in proportion to the rest; and so the resulting colour of the transmitted rays must lie between a dark orange and red, according to Sir Isaac Newton's rule for finding the result of a mixture of colours. We have an instance of the reverse of this case in leaf-gold, which appears yellow by reflected, and blue by transmitted rays. The circular edge of the shadow in a partial eclipse appears red; because the red-making rays are the least refracted of all others, and consequently are left alone in the conical surface of the shadow, all the rest being refracted into it."

§ 8. Of the Measures of Light.

That some luminous bodies give a stronger, and others a weaker light, and that some reflect more light than others, was always obvious to mankind; but no person, before Mr Bouguer, hit upon a tolerable method of ascertaining the proportion that two or more lights bear to one another. The methods he most commonly used were the following:

He took two pieces of wood or pasteboard EC and CD, fig. 4, in which he made two equal holes P and Q, over which he drew pieces of oiled or white paper. Upon these holes he contrived that the light of the different bodies he was comparing should fall; while he placed a third piece of pasteboard FC, to prevent the two lights from mixing with one another. Then placing himself sometimes on one side, and sometimes on the other, but generally on the opposite side of this instrument, with respect to the light, he altered their position till the papers in the two holes appeared to equally enlightened. This being done, he computed the proportion of their light by the squares of the distances at which the luminous bodies were placed from the objects. If, for instance, the distances were as three and nine, he concluded that the light they gave were as nine and eighty-one. Where any light was very faint, he sometimes made use of lenses, in order to condense it; and he inclosed them in tubes or not, as his particular application of them required.

To measure the intensity of light proceeding from the heavenly bodies, or reflected from any part of the sky, he contrived an instrument which resembles a kind of portable camera obscura. He had two tubes, of which the inner was black, fastened at their lower extremities by a hinge C, fig. 5. At the bottom of these tubes were two holes, R and S, three or four lines in diameter, covered with two pieces of fine white paper. The two other extremities had each of them a circular aperture, an inch in diameter; and one of the tubes consisted of two, one of them sliding into the Part III.

Optics.

Measures of the other, which produced the same effect as varying Light. the aperture at the end. When this instrument is used, the observer has his head, and the end of the instrument C, so covered, that no light can fall upon his eye, besides that which comes thro' the two holes S and R, while an assistant manages the instrument, and draws out or shortens the tube DE, as the observer directs. When the two holes appear equally illuminated, the intensity of the lights is judged to be inversely as the squares of the tubes.

In using this instrument, it is necessary that the object should subtend an angle larger than the aperture A or D, seen from the other end of the tube; for, otherwise, the lengthening of the tube has no effect. To avoid, in this case, making the instrument of an inconvenient length, or making the aperture D too narrow, he has recourse to another expedient. He constructs an instrument, represented fig. 6, consisting of two object-glasses, AE and DF, exactly equal, fixed in the ends of two tubes six or seven feet, or, in some cases, ten or twelve feet long, and having their foci at the other ends. At the bottom of these tubes B, are two holes, three or four lines in diameter, covered with a piece of white paper; and this instrument is used exactly like the former.

If the two objects to be observed by this instrument be not equally luminous, the light that issues from them must be reduced to an equality, by diminishing the aperture of one of the object-glasses; and then the remaining surface of the two glasses will give the proportion of their lights. But for this purpose, the central parts of the glass must be covered in the same proportion with the parts near the circumference, leaving the aperture such as is represented fig. 7, because the middle part of the glass is thicker and less transparent than the rest.

If all the objects to be observed lie nearly in the same direction, our author observes, that these two long tubes may be reduced into one, the two object-glasses being placed close together, and one eye-glass sufficing for them both. The instrument will then be the same with that of which he published an account in 1748, and which he called a heliometer, or astrometer.

Our author observes, that it is not the absolute quantity, but only the intensity of the light, that is measured by these two instruments, or the number of rays, in proportion to the surface of the luminous body; and it is of great importance that these two things be distinguished. The intensity of light may be very great, when the quantity, and its power of illuminating other bodies, may be very small, on account of the smallness of its surface; or the contrary may be the case, when the surface is large.

Having explained these methods which M. Bouguer took to measure the different proportions of light, we shall subjoin in this place a few miscellaneous examples of his application of them.

It is observable, that when a person stands in a place where there is a strong light, he cannot distinguish objects that are placed in the shade; nor can he see anything upon going immediately into a place where there is very little light. It is plain, therefore, that the action of a strong light upon the eye, and also the impression which it leaves upon it, makes it insensible to the effect of a weaker light. M. Bouguer had the curiosity to endeavour to ascertain the proportion between the intensities of the two lights in this case; and by throwing the light of two equal candles upon a board, he found that the shadow made by intercepting the light of one of them, could not be perceived by his eye, upon the place enlightened by the other, at little more than eight times the distance; from whence he concluded, that when one light is eight times eight, or sixty-four times less than another, its presence or absence will not be perceived. He allows, however, that the effect may be different on different eyes; and supposes that the boundaries in this case, with respect to different persons, may lie between 60 and 80.

Applying the two tubes of his instrument, mentioned above, to measure the intensity of the light reflected from different parts of the sky; he found, that when the sun was 25 degrees high, the light was four times stronger at the distance of eight or nine degrees from his body, than it was at 31 or 32 degrees. But what struck him most was to find, that when the sun is 15 or 20 degrees high, the light decreases on the same parallel to the horizon to 110 or 120 degrees, and then increases again to the place exactly opposite to the sun.

The light of the sun, our author observes, is too strong, and that of the stars too weak, to determine the variation of their light at different altitudes: but as, in both cases, it must be in the same proportion with the diminution of the light of the moon in the same circumstances, he made his observations on that luminary, and found, that its light at 19° 16', is to its light at 66° 11', as 1681 to 2500; that is, the one is nearly two thirds of the other. He chose those particular altitudes, because they are those of the sun Great variation at the two solstices at Crofie, where he then resided. When one limb of the moon touched the horizon of the sea, its light was 2000 times less than at the altitude of 66° 11'. But this proportion he acknowledges must be subject to many variations, the atmosphere near the earth varying so much in its density. From this observation he concludes, that at a medium light is diminished in the proportion of about 2500 to 1681, in traversing 7469 toises of dense air.

Lastly, our accurate philosopher applied his instrument to the different parts of the sun's disk, and found that the centre is considerably more luminous than the extremities of it. As near as he could make the observation, it was more luminous than a part of the disk planets, 4/5ths of the semi-diameter from it, in the proportion of 28 to 35; which, as he observes, is more than in the proportion of the sines of the angles of obliquity. On the other hand, he observes, that both the primary and secondary planets are more luminous at their edges than near their centres.

The comparison of the light of the sun and moon is a subject that has frequently exercised the thoughts of philosophers; but we find nothing but random conjectures, before our author applied his accurate measures in this case. In general, the light of the moon is imagined to bear a much greater proportion to that of the sun than it really does; and not only are the imaginations of the vulgar, but those of philosophers also, imposed upon with respect to it. It was a great fur- Measures of surprise to M. De la Hire to find that he could not, by the help of any burning mirror, collect the beams of the moon in a sufficient quantity to produce the least sensible heat. Other philosophers have since made the like attempts with mirrors of greater power, tho' without any greater success; but this will not surprise us, when we see the result of M. Bouguer's observations on this subject.

In order to solve this curious problem concerning the comparison of the light of the sun and moon, he compared each of them to that of a candle in a dark room, one in the day-time, and the other in the night following, when the moon was at her mean distance from the earth; and, after many trials, he concluded that the light of the sun is about 300,000 times greater than that of the moon; which is such a disproportion, that, as he observes, it can be no wonder that philosophers have had so little success in their attempts to collect the light of the moon with burning glasses. For the largest of them will not increase the light 1000 times; which will still leave the light of the moon, in the focus of the mirror, 300 times less than the intensity of the common light of the sun.

To this account of the proportion of light which we actually receive from the moon, it cannot be displeasing to the reader, if we compare it with the quantity which would have been transmitted to us from that opaque body, if it reflected all the light it receives. Dr Smith thought that he had proved, from two different considerations, that the light of the full moon would be to our day-light as one to about 90,900, if no rays were lost at the moon.

Dr Smith's calculation. In the first place, he supposes that the moon, enlightened by the sun, is as luminous as the clouds are at a medium. He therefore supposed the light of the sun to be equal to that of a whole hemisphere of clouds, or as many moons as would cover the surface of the heavens. But on this Dr Priestley observes, that it is true, the light of the sun shining perpendicularly upon any surface would be equal to the light reflected from the whole hemisphere, if every part reflected all the light that fell upon it; but the light that would in fact be received from the whole hemisphere (part of it being received obliquely) would be only one-half as much as would be received from the whole hemisphere if every part of it shone directly upon the surface to be illuminated.

In his Remarks, par. 97, Dr Smith demonstrates his method of calculation in the following manner:

"Let the little circle \( cfdg \) represent the moon's body half enlightened by the sun, and the great circle \( ae \), a spherical shell concentric to the moon, and touching the earth; \( ab \), any diameter of that shell perpendicular to a great circle of the moon's body, represented by its diameter \( cd \); \( e \) the place of the shell receiving full moon-light from the bright hemisphere \( fdg \). Now, because the surface of the moon is rough like that of the earth, we may allow that the sun's rays, incident upon any small part of it, with any obliquity, are reflected from it every way alike, as if they were emitted. And therefore, if the segment \( df \) shone alone, the points \( a, e \), would be equally illuminated by it; and likewise if the remaining bright segment \( dg \) shone alone, the points \( b, e \) would be equally illuminated by it. Consequently, if the light at the point \( a \) was increased by the light at \( b \), it would become equal to the full moon-light at \( e \). And conceiving the same transfer to be made from every point of the hemispherical surface \( bbik \) to their opposite points in the hemisphere \( kaeb \), the former hemisphere would be left quite dark, and the latter would be uniformly illustrated with full moon-light; arising from a quantity of the sun's light, which, immediately before its incidence on the moon, would uniformly illustrate a circular plane equal to a great circle of her body, called her disk. Therefore the quantities of light being the same upon both surfaces, the density of the sun's incident light, is to the density of full moon-light, as that hemispherical surface \( bok \) is to the said disk; that is, as any other hemispherical surface whose centre is at the eye, to that part of it which the moon's disk appears to possess very nearly, because it subtends but a small angle at the eye: that is, as radius of the hemisphere to the verified sine of the moon's apparent semidiameter, or as 10,000,000 to 1106\(\frac{2}{3}\) or as 90,400 to 1; taking the moon's mean horizontal diameter to be 16° 7".

"Strictly speaking, this rule compares moon-light at the earth with day-light at the moon; the medium of which, at her quadratures, is the same as our daylight; but is less at her full in the duplicate ratio of 365 to 366, or thereabout; that is, of the sun's distances from the earth and full moon: and therefore full-moon light would be to our day-light, as about 1 to 90,900, if no rays were lost at the moon.

"Secondly, I say that full-moon light is to any other moon-light as the whole disk of the moon to the part that appears enlightened, considered upon a plane surface. For now let the earth be at \( b \), and let Fig. 9, \( dl \) be perpendicular to \( fg \), and \( gm \) to \( cd \): then it is plain, that \( gl \) is equal to \( dm \); and that \( gl \) is equal to a perpendicular section of the sun's rays incident upon the arch \( dg \), which at \( b \) appears equal to \( dm \); the eye being unable to distinguish the unequal distances of its parts. In like manner, conceiving the moon's surface to consist of innumerable physical circles parallel to \( cfdg \), as represented at A, the same reason holds for every one of these circles as for \( cfdg \). It follows then, that the bright part of the surface visible at \( b \), when reduced to a flat as represented at B, by the crescent \( pdqwp \), will be equal and similar to a perpendicular section of all the rays incident on that part, represented at C by the crescent \( pgqlp \). Now the whole disk being in proportion to this crescent, as the quantities of light incident upon them; and the light falling upon every rough particle, being equally rarified in diverging to the eye at \( b \), considered as equidistant from them all; it follows, that full moon-light is to this moon-light as the whole disk \( pdqc \) to the crescent \( pdqwp \).

"Therefore, by compounding this ratio with that in the former remark, day light is to moon-light as the surface of an hemisphere whose centre is at the eye, to the part of that surface which appears to be possessed by the enlightened part of the moon.

Mr Michell made his computation in a much more simple and easy manner, and in which there is much less danger of falling into any mistake. Considering the distance of the moon from the sun, and that the density... Part III.

Of optical density of the light must decrease in the proportion of the square of that distance, he calculated the density of the sun's light, at that distance, in proportion to its density at the surface of the sun; and in this manner he found, that if the moon reflected all the light it receives from the sun, it would only be the 45,000th part of the light we receive from the greater luminary. Admitting, therefore, that moon light is only a 300,000th part of the light of the sun, Mr Michell concludes, that it reflects no more than between the 6th and 7th part of what falls upon it.

Sect. IV. Of Optical Instruments.

§ 1. The Multiplying-glas.

The multiplying-glas is made by grinding down the round side b i k (fig. 1.) of a convex glas A B, into several flat surfaces, as b b, b l d, d k. An object C will not appear magnified when seen through this glas by the eye at H; but it will appear multiplied into as many different objects as the glas contains plane surfaces. For, since rays will flow from the object C to all parts of the glas, 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 g i H, falling perpendicularly on the middle surface, will go through the glas to the eye without suffering any refraction; and will therefore shew the object in its true place at C; whilst a ray a b flowing from the same object, and falling obliquely on the plane surface b b, will be refracted in the direction b e, by passing through the glas; and, upon leaving it, will go on to the eye in the direction e H; which will cause the same object C to appear also at E, in the direction of the ray H e, produced in the right line H e n. And the ray c d, flowing from the object C, and falling obliquely on the plane surface d k, will be refracted (by passing through the glas,) and leaving it at f) to the eye at H; which will cause the same object to appear at D, in the direction H f m.—If the glas be turned round the line g / H, as an axis, the object C will keep its place, because the surface b l d is not removed; but all the other objects will seem to go round C, because the oblique planes, on which the rays a b c d fall, will go round by the turning of the glas.

§ 2. Mirrors.

1. The Plane Mirror, or common Looking-glas. 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 glas with the object,) is just half as long and half as broad as the object itself. Let A B (fig. 3.) be an object placed before the reflecting surface g b i of the plane mirror C D; and let the eye be at o. Let A b be a ray of light flowing from the top A of the object and falling upon the mirror at b, and b m be a perpendicular to the surface of the mirror at b; the ray A b will be reflected from the mirror to the eye at o, making an angle m b o, equal to the angle A b m: then will the top of the image E appear to the eye in the direction of the reflected ray o b produced to E, where the right line A p E, from the top of the object, cuts the right line o b E, at E. Let B i be a ray of light proceeding from the foot of the object at B to the mirror at i; and n i a perpendicular to the mirror from the point i, where the ray B i falls upon it: this ray will be reflected in the line i o, making an angle n i o, equal the angle B i n, 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 o i, produced to F, where the right line B F cuts the reflected ray produced to F. All the other rays that flow from the intermediate points of the object A B, and fall upon the mirror between b and i, will be reflected to the eye at o; and all the intermediate points of the image E F 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 b, 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 eye at o; so that none of the rays that fall above b, or below i, can be reflected to the eye at o; and the distance between b and i is equal to half the length of the object A B.

Hence it appears, that if a man sees his whole size of a image in a plane looking-glas, the part of the glas looking that reflects his image must be just half as long and glas in half as broad as himself; let him stand at any distance man will from it whatever; and that his image must appear just see his as far behind the glas as he is before it. Thus, the man whole i-AB (fig. 4.) viewing himself in the plane mirror CD, mage, which is just half as long as himself, sees his whole image as at E F, behind the glas, exactly equal to his own size. For, a ray AC proceeding from his eye at A, and falling perpendicularly upon the surface of the glas 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 glas. And a ray BD, flowing from his foot, and falling obliquely on the glas at D, will be reflected as obliquely on the other side of the perpendicular a b D, 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 B G F, drawn parallel to the right line ACE. Just the same as if the glas 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 A D F.

If the glas be brought nearer the man A B, as suppose to c b, he will see his image as at C D G: for the reflected ray C A (being perpendicular to the glas) will shew the eye of the image as at C; and the incident ray B b, being reflected in the line b A, will shew the foot of his image as at G; the angle of reflection a b A being always equal to the angle of incidence B b a: and so of all the intermediate rays from A to B. Hence, if the man A B advances towards the glas C D, his image will approach towards it; and if he recedes from the glas, his image will also recede from it. If the object be placed before a common looking-glass, and viewed obliquely, three, four, or more images of it will appear behind the glass.

To explain this, let ABCD (fig. 11.) represent the glass; and let EF be the axis of a pencil of rays flowing from E, a point in an object situated there. The rays of this pencil will in part be reflected at F, supposing into the line FG. What remains will (after refraction at F, which we do not consider here) pass on to H; from whence (on account of the quicksilver which is spread over the second surface of glasses of this kind to prevent any of the rays from being transmitted there) they will be strongly reflected to K, where part of them will emerge and enter an eye at L. By this means one representation of the said point will be formed in the line LK produced, suppose in M: Again, another pencil, whose axis is EN, first reflected at N, then at O, and afterwards at P, will form a second representation of the same point at Q: And thirdly, another pencil, whose axis is ER, after reflection at the several points R, S, H, T, V, successively, will exhibit a third representation of the same point at X; and so on in infinitum. The same being true of each point in the object, the whole will be represented in the like manner; but the representations will be faint, in proportion to the number of reflections the rays suffer, and the length of their progress within the glass. We may add to these another representation of the same object in the line LO produced, made by such of the rays as fall upon O, and are from thence reflected to the eye at L.

This experiment may be tried by placing a candle before the glass as at E, and viewing it obliquely, as from L.

2. Of Concave and Convex Mirrors. The effects of these in magnifying and diminishing objects have been already in general explained; but for the better understanding the nature of reflecting telescopes, it will still be proper to subjoin the following particular description of the effects of concave ones.

When parallel rays, (fig. 2.) as df, Cmb, etc., fall upon a concave mirror AB (which is not transparent, but has only the surface A B 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 A B; and let the parallel rays df, Cmb, etc., fall upon it at the points a, b, c. Draw the lines Ci a, Cmb, and Cbc, 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 da C, and draw the line am b, which will be the direction of the ray df, after it is reflected from the point a of the mirror: so that the angle of incidence da C, is equal to the angle of reflection Cab; the rays making equal angles with the perpendicular Ci a on its opposite sides.

Draw also the perpendicular Cbc to the point c, where the ray etc. touches the mirror; and having made the angle Cce equal to the angle Cce, draw the line cm i, which will be the course of the ray etc., after it is reflected from the mirror.

The ray Cmb passing thro' 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 b m C.

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 etc., will be formed; which point is distant from the mirror equal to half the radius b m C of its concavity.

The rays which proceed from any celestial object may be esteemed 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 half way 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; images and will be seen by an eye placed beyond it (with regard to the mirror) in all respects like the object, and concave mirrors, as distinct as the object itself.

Let AcB (fig. 3.) 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, DC, DB.

From the centre of concavity C, draw the three right lines CA, CC, 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 Ccd equal to the angle DcC, 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 Bd 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 Ef, 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, Of Optical 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, the image and object will be coincident, and equal in bulk.

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.

§ 3. Microscopes.

1. The Single Microscope is only a very small globule of glass, or a convex lens, whose focal distance is very short. It is represented by cd, fig. 6. The object ab is 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 to the eye after passing through the glass; and then, by entering the eye at C, they will be converged to as many points on the retina, and form a large inverted picture AB upon it. The magnifying power of this microscope is thus explained by Dr Smith. "A minute object pq, seen distinctly through a small glass AE by the eye put close to it appears so much greater than it would to the naked eye, placed at the least distance qL from whence it appears sufficiently distinct, as this latter distance qL is greater than the former qE. For having put your eye close to the glass EA, in order to see as much of the object as possible at one view, remove the object pq to and fro till it appear most distinctly, suppose at the distance qL. Then conceiving the glass AE to be removed, and a thin plate, with a pin-hole in it, to be put in its place, the object will appear distinct, and as large as before, when seen through the glass, only no so bright. And in this latter case it appears so much greater than it does to the naked eye at the distance qL, either with the pin-hole or without it, as the angle pEq is greater than the angle pLq, or as the latter distance qL is greater than the former qE. Since the interpolation of the glass has no other effect than to render the appearance distinct, by helping the eye to increase the refraction of the rays in each pencil, it is plain that the greater apparent magnitude is entirely owing to a nearer view than could be taken by the naked eye. If the eye be so perfect as to see distinctly by pencils of parallel rays falling upon it, the distance Eq, of the object from the glass, is then the focal distance of the glass. Now, if the glass be a small round globule, of about \( \frac{1}{2} \) th of an inch diameter, its focal distance Eq, being three quarters of its diameter, is \( \frac{3}{4} \) th of an inch; and if qL be eight inches, the distance at which we usually view minute objects, this globule will magnify in the proportion of 8 to \( \frac{3}{4} \), or of 160 to 1.

2. The Double or Compound Microscope (fig. 7.) consists of an object-glass cd, and an eye-glass ef. The final 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 eyeglass 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 eyeglass, 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.

By this combination of lenses, the aberration of the light from the figure of the glass, which in a globule of the kind abovementioned is very considerable, is in some measure corrected. This appeared so feasible to be the case, even to former opticians, that they very soon began to make the addition of another lens. The instrument, however, receives a considerable improvement by the addition of a third lens. For, says Mr Martin, it is not only evident from the theory of this aberration, that the image of any point is rendered less confused by refraction through two lenses, Of Optical Instruments

than by an equal refraction through one; but it also follows, from the same principle, that the same point has its image still less confused when formed by rays refracted through three lenses, than by an equal refraction through two; and therefore a third lens added to the other two, will contribute to make the image more distinct, and consequently the instrument more complete. At the same time the field of view is amplified, and the use of the microscope rendered more agreeable, by the addition of the other lens. Thus also we may allow a somewhat larger aperture to the object-lens, and thereby increase the brightness of objects, and greatly heighten the pleasure of viewing them. For the same reason, Mr Martin has proposed a four-glass microscope, which answers the purposes of magnifying and of distinct vision still more perfectly.

The magnifying power of double microscopes is easily understood, thus: The glass L next the object PQ is very small, and very much convex, and consequently its focal distance LF is very short; the distance LQ of the small object PQ is but a little greater than LF; so that the image pq may be formed at a great distance from the glass, and consequently may be much greater than the object itself. This picture pq being viewed through a convex glass AE, whose focal distance is qE, appears distinct as in a telescope. Now the object appears magnified upon two accounts; firstly, because, if we viewed its picture pq with the naked eye, it would appear as much greater than the object, at the same distance, as it really is greater than the object, or as much as Lq is greater than LQ; and, secondly, because this picture appears magnified through the eye-glass as much as the least distance at which it can be seen distinctly with the naked eye, is greater than qE, the focal distance of the eye-glass. For example, if this latter ratio be five to one, and the former ratio of Lq to LQ be 20 to 1; then, upon both accounts, the object will appear 5 times 20, or 100 times greater than to the naked eye.

Fig. 2 represents a compound microscope with three lenses. By the middle one GK the pencils of rays coming from the object-glass are refracted so as to tend to a focus at O; but being intercepted by the proper eye-glass DF, they are brought together at I, which is nearer to that lens than its proper focus at L; so that the angle DIF, under which the object now appears, is larger than DLF, under which it would have appeared without this additional glass; and consequently the object is more magnified in the same proportion. Dr Hooke tells us, that, in most of his observations, he made use of a double microscope with this broad middle-glass when he wanted to see much of an object at one view, and taking it out when he would examine the small parts of an object more accurately; for the fewer refractions there are, the more bright and clear the object appears.

In microscopic lenses whose focal distances are not much shorter than half an inch, there is no need to contract their apertures for procuring distinct vision; the pupil itself being small enough to exclude the extraneous magnetic straggling rays. But in smaller lenses, where apertures are necessary, Dr Smith has demonstrated, that, to preserve the same degree of distinctness, their apertures must be as their focal distances, and then the apparent brightness will decrease in a duplicate ratio of their focal distances: so that, by using smaller glasses, the apparent magnitude and the obscurity of the object will both increase in the same ratio. For the ratio of PD to PF being invariable, the angle Fig. 3. PFD is also invariable; and consequently the quantity of light received from the point F is also invariable; because the apertures of the lenses, whether smaller or larger, must all be situated at such distances from F as just to receive all the rays contained in a cone described by turning the angle PFD about the axis PF, neither more nor less. But the apparent magnitude of the object, or the surface of its picture upon the retina, is reciprocally as PF square; and consequently, the light being the same, its brightness is directly as PF square. By this theory it appears, that a minute object cannot be magnified to infinity by a single lens, though it were possible to make it as small as we please, without some method of increasing its light. Nevertheless, this imperfection in single microscopes is not so great as at first sight one would take it to be, or as in fact we find it; the reason may be, because the eye is capable of discerning objects tolerably well by above 20,000 different degrees of light. But though the brightness of the object were increased by throwing new light upon it, yet Huygens observes, that the power of the microscope will still be limited by the breadth of the pencils which enter the pupil; which is equal to the breadth of the aperture. For, if this breadth be less than \( \frac{1}{7} \) or \( \frac{1}{8} \) of a line, he affirms that the edges of the object will begin to appear indistinct. But by double microscopes this author has made it appear, that we may magnify objects at pleasure, provided it was possible to form their object-glasses sufficiently small.

3. The Solar Microscope (fig. 4.), invented by Dr Lieberkühn, 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. A convex glass a, 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 a. 3. 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 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 it by the mirror 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 cc, which flow from it through the convex lens 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 the 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 GH = 5; then \(2, 25 : 25 :: 1 : 1\) nearly. So that by this construction you may increase the light upon objects, or their images, at least seven times with ease, or ten times with very little trouble or expense.

§ 4. Telescopes.

I. The Refracting Telescope.

After what has been said concerning the structure of the compound microscope, and the manner in which the rays pass through it to the eye, the nature of the common astronomical telescope will easily be understood: for it differs from the microscope only in that the object is placed at so great a distance from it, that the rays of the same pencil, flowing from thence, may be considered as falling parallel to one another upon the object-glass; and therefore the image made by that glass is looked upon as coincident with its focus of parallel rays.

The 6th figure will render this very plain; in which AB is the object emitting the several pencils of rays A d c, B c d, &c., but supposed to be at so great a distance from the object-glass c d, that the rays of the same pencil may be considered as parallel to each other; they are therefore supposed to be collected into their respective foci at the point m and p, situated at the focal distance of the object-glass c d. Here they form an image E, and crossing each other proceed diverging to the eye-glass b g; which being placed at its own focal distance from the points m and p, the rays of each pencil, after passing through that glass, will become parallel among themselves; but the pencils themselves will converge considerably with respect to one another, even so as to cross at e, very little farther from the glass g b than its focus; because, when they entered the glass, their axes were almost parallel, as coming through the object-glass at the point k, to whose distance the breadth of the eye-glass in a long telescope bears very small proportion. So that the place of the eye will be nearly at the focal distance of the eye-glass, and the rays of each respective pencil being parallel among themselves, and their axes crossing each other in a larger angle than they would do if the object were to be seen by the naked eye, vision will be distinct, and the object will appear magnified.

The power of magnifying in this telescope is as the focal length of the object-glass to the focal length of the eye-glass.

Dem. In order to prove this, we may consider the angle A k B as that under which the object would be seen by the naked eye; for in considering the distance of the object, the length of the telescope may be omitted, as bearing no proportion to it. Now the angle under which the object is seen by means of the telescope, is g e h, which is to the other A k B, or its equal g k b, as the distance from the centre of the object-glass to that of the eye-glass. The angle, therefore, under which an object appears to an eye affixed by a telescope of this kind, is to that under which it would be seen without it, as the focal length of the object-glass to the focal length of the eye-glass.

It is evident from the figure, that the visible area, or space which can be seen at one view when we look through this telescope, depends on the breadth of the eye-glass, and not of the object-glass; for if the eye-glass be too small to receive the rays g m, p b, the extremities... Of Optical tremities of the object could not have been seen at all; Instruments a larger breadth of the object-glass conduces only to the rendering each point of the image more luminous by receiving a larger pencil of rays from each point of the object.

It is in this telescope as in the compound microscope, where we see, when we look through it, not the object itself, but only an image of it at CED: now that image being inverted with respect to the object, as it is, because the axes of the pencils that flow from the object cross each other at k, objects seen through a telescope of this kind necessarily appear inverted.

This is a circumstance not at all regarded by astronomers: but for viewing objects upon the earth, it is convenient that the telescope should represent them in their natural posture; to which use the telescope with three eye-glasses, as represented fig. 7, is peculiarly adapted, and the progress of the rays through it from the object to the eye is as follows:

AB is the object sending out the several pencils Acd, Bcd, &c. which passing through the object-glass cd, are collected into their respective foci in CD, where they form an inverted image. From hence they proceed to the first eye-glass ef, whose focus being at l, the rays of each pencil are rendered parallel among themselves, and their axes, which were nearly parallel before, are made to converge and cross each other: the second eye-glass gh, being so placed that its focus shall fall upon m, renders the axis of the pencils which diverge from thence parallel, and causes the rays of each which were parallel among themselves to meet again at its focus EF on the other side, where they form a second image inverted with respect to the former, but erect with respect to the object. Now this image being seen by the eye at ab through the eyeglass ik, affords a direct representation of the object, and under the same angle that the first image CD would have appeared, had the eye been placed at l, supposing the eye-glasses to be of equal convexity; and therefore the object is seen equally magnified in this as in the former telescope, that is, as the focal distance of the object-glass to that of any one of the eye-glasses, and appears erect.

If a telescope exceeds 20 feet, it is of no use in viewing objects upon the surface of the earth; for if it magnifies above 90 or 100 times, as those of that length usually do, the vapours, which continually float near the earth in great plenty, will be so magnified as to render vision obscure.

2. The Galilean Telescope with the concave eye-glass is constructed as follows:

AB (fig. 5.) is an object sending forth the pencils of rays ghi, klm, &c. which, after passing through the object-glass cd, tend towards eEf (where we will suppose the focus of it to be), in order to form an inverted image there as before; but in their way to it are made to pass through the concave glass no, so placed that its focus may fall upon E, and consequently the rays of the several pencils which were converging towards those respective focal points e, E, f, will be rendered parallel among themselves; but the axes of those pencils crossing each other at F, and diverging from thence, will be rendered more diverging, as represented in the figure. Now these rays entering the pupil of an eye, will form a large and distinct image ab upon the retina, which will be inverted with respect to the object, because the axes of the pencils cross in F; and the angle the object will appear under will be equal to that which the lines aF, bF, produced back through the eye-glass, form at F.

It is evident, that the least pupil of the eye is, the less is the visible area seen through a telescope of this kind; for a least pupil would exclude such pencils as proceed from the extremities of the object AB, as is evident from the figure. This is an inconvenience that renders this telescope unfit for many uses; and is only to be remedied by the telescope with the convex eye-glasses, where the rays which form the extreme parts of the image are brought together in order to enter the pupil of the eye, as explained above.

It is apparent also, that the nearer the eye is placed to the eye-glass of this telescope, the larger is the area seen through it; for, being placed close to the glass, as in the figure, it admits rays that come from A and B, the extremities of the object, which it could not if it was placed farther off.

The degree of magnifying in this telescope is in the same proportion with that in the other, viz. as the focal distance of the object-glass is to the focal-distance of the eye-glass.

For there is no other difference but this, viz. that as the extreme pencils in that telescope were made to converge and form the angle geb (fig. 6.), or i n k (fig. 7.), these are now made to diverge and form the angle aFb (fig. 5.) which angles, if the concave glass in one has an equal refractive power with the convex one in the other, will be equal, and therefore each kind will exhibit the object magnified in the same degree.

There is a defect in all these kinds of telescopes, not to be remedied in a single lens by any means whatever, which was thought only to arise from hence, viz. that spherical glasses do not collect rays to one and the same point. But it was happily discovered by Sir Isaac Newton, that the imperfection of this sort of telescope, so far as it arises from the spherical form of the glasses, bears almost no proportion to that which is owing to the different refrangibility of light. This diversity in the refraction of rays is about a 28th part of the whole; so that the object-glass of a telescope cannot collect the rays which imperfectly flow from any one point in the object into a least room than the circular space whose diameter is about the 56th dioptrical part of the breadth of the glass (a). Therefore, since telecopes.

(a) To shew this, let AB, fig. 1. represent a convex lens, and let CDF be a pencil of rays flowing from the point D: let H be the point at which the least refrangible rays are collected to a focus; and I, that where the most refrangible concur. Then, if IH be the 28th part of EH, IK will be a proportional part of FC (the triangles IIJK and HEC being similar): consequently LK will be the 28th part of FC. But MN will be the least space into which the rays will be collected, as appears by their progress represented in the figure. Now MN is but about half of KL; and therefore it is about the 56th part of CF: so that the diameter of the space, into which the rays are collected, will be about the 56th part of the breadth of that part of the glass thro' which the rays pass. Which was to be shewn. Of Optical Instruments each point of the object will be represented in so large a space, and the centres of those spaces will be contiguous, because the points in the object the rays flow from are so; it is evident, that the image of an object made by such a glass must be a most confused representation, though it does not appear so when viewed through an eye-glass that magnifies in a moderate degree; consequently the degree of magnifying in the eye-glass must not be too great with respect to that of the object-glass, lest the confusion become sensible.

Notwithstanding this imperfection, a dioptrical telescope may be made to magnify in any given degree, provided it be of sufficient length; for the greater the focal distance of the object-glass is, the lens may be the proportion which the focal distance of the eye-glass may bear to that of the object-glass, without rendering the image obscure. Thus, an object-glass whose focal distance is about four feet, will admit of an eye-glass whose focal distance shall be little more than an inch, and consequently will magnify almost 48 times; but an object-glass of 40 foot focus will admit of an eye-glass of only four-inch focus, and will therefore magnify 120 times; and an object-glass of 100 foot focus will admit of an eye-glass of little more than six-inch focus, and will therefore magnify almost 200 times.

The reason of this disproportion in their several degrees of magnifying is to be explained in the following manner. Since the diameter of the spaces, into which rays flowing from the several points of an object are collected, are as the breadth of the object-glass, it is evident that the degree of confusion in the image is as the breadth of that glass; for the degree of confusion will only be as the diameters or breadths of those spaces, and not as the spaces themselves. Now the focal length of the eye-glass, that is, its power of magnifying, must be as that degree; for, if it exceeds it, it will render the confusion sensible; and therefore it must be as the breadth or diameter of the object-glass. The diameter of the object-glass, which is as the square root of its aperture or magnitude, must be as the square-root of the power of magnifying in the telescope; for unless the aperture itself be as the power of magnifying, the image will want light: the square root of the power of magnifying will be as the square root of the focal distance of the object-glass; and therefore the focal distance of the eye-glass must be only as the square root of that of the object-glass. So that in making use of an object-glass of a longer focus, suppose, than one that is given, you are not obliged to apply an eye-glass of a proportionably longer focus than what would suit the given object-glass, but such an one only whose focal distance shall be to the focal distance of that which will suit the given object-glass, as the square root of the focal length of the object-glass you make use of, is to the square root of the focal length of the given one. And this is the reason that longer telescopes are capable of magnifying in a greater degree than shorter ones, without rendering the object confused or coloured.

3. Dollond's Telescopes.—The general principle on which this artist's celebrated improvement of the refracting telescope depends, hath been already mentioned; namely, that by the different powers of refraction in two kinds of glasses, and by their different powers of dispersing the rays, the errors arising from the different refrangibility of the light are in a great measure, if not totally, corrected.—For this purpose the object-glasses of his telescopes are composed of three distinct lenses, Plate two convex and one concave; of which the concave one is placed in the middle, as is represented in fig. 6. where a and c shew the two convex lenses, and b the concave one, which is by the British artists placed in the middle. The two convex ones are made of green glass, and the middle one of white flint glass, and are all ground to spheres of the same radius. When put together, they refract the rays in the following manner. Let ab, ab, be two red rays of the sun's light falling parallel on the first green convex lens c. Supposing there was no other lens present but that one, they would then be converged into the lines b e, b e, and at last meet in the focus q. Let the lines g h, g h, represent two violet rays falling on the surface of the lens. These are also refracted, and will meet in a focus; but as they have a greater degree of refrangibility than the red rays, they must, of consequence, converge more by the same power of refraction in the glass, and meet sooner in a focus, suppose at r.—Let now the concave lens d d be placed in such a manner as to intercept all the rays before they come to their focus. As this lens is ground to the same radius with the convex one, it must have the same power to cause the rays diverge that the former had to make them converge; that is, supposing them both to be made of the same materials. In this case, the red rays would become parallel, and move on in the line oo, oo; But the concave lens, being made of white glass, has a greater refractive power, and therefore they diverge a little after they come out of it; and if no third lens was interposed, they would proceed diverging in the lines o p t, o p t; but, by the interposition of the third lens a v o, they are again made to converge, and meet in a focus somewhat more distant than the former, as at x. By the concave lens the violet rays are also refracted, and made to diverge: but having a greater degree of refrangibility, the same power of refraction makes them diverge somewhat more than the red ones; and thus, if no third lens was interposed, they would proceed in such lines as l m n, l m n. Now as the differently coloured rays fall upon the third lens with different degrees of divergence, it is plain, that the same power of refraction in that lens will operate upon them in such a manner as to bring them all together to a focus very nearly at the same point. The red rays, it is true, require the greatest power of refraction to bring them to a focus; but they fall upon the lens with the least degree of divergence. The violet rays, though they require the least power of refraction, yet have the greatest degree of divergence; and thus all meet together at the point x, or very nearly so.

But, though we have hitherto supposed the refraction of the concave lens to be greater than that of the convex ones, it is easy to see how the errors occasioned by the first lens may be corrected by it, though it should have even a less power of refraction than the convex one. Thus, let a b, a b, fig. 8, be two rays of red light falling upon the convex lens c, and refracted into the focus q; let also g h, g h, be two violet rays Of Optical Instruments

converged into a focus at \( r \); it is not necessary, in order to their convergence into a common focus at \( x \), that the concave lens should make them diverge; it is sufficient if the glass has a power of dispersing the violet rays somewhat more than the red ones; and many kinds of glasses have this power of dispersing some kinds of rays, without a very great power of refraction. It is better, however, to have the object-glass composed of three lenses; because there is then another correction of the aberration by means of the third lens; and it might be impossible to find two lenses, the errors of which would exactly correct each other. It is also easy to see, that the effect may be the same whether the concave glass is a portion of the same sphere with the others or not; the effect depending upon a combination of certain circumstances, of which there is an infinite variety.

By means of this correction of the errors arising from the different refrangibility of the rays of light, it is possible to shorten dioptric telescopes considerably, and yet leave them equal-magnifying powers. The reason of this is, that the errors arising from the object-glass being removed, those which are occasioned by the eye-glass are inconsiderable: for the error is always in proportion to the length of the focus in any glass; and in very long telescopes it becomes exceedingly great, being no less than \( \frac{1}{2} \) of the whole; but in glasses of a few inches focus it becomes trifling. Refracting telescopes which go by the name of Dollond's, are therefore now constructed in the following manner. Let \( AB \) (fig. 1.) represent an object-glass composed of three lenses as above described, and converging the rays \( 1, 2, 3, 4, \) &c. to a very distant focus as at \( x \). By means of the interposed lens \( CD \), however, they are converged to one much nearer, as at \( y \), where an image of the object is formed. The rays diverging from thence fall upon another lens \( EF \), where the pencils are rendered parallel, and an eye placed near that lens would see the object magnified and very distinct. To enlarge the magnifying power still more, however, the pencils thus become parallel are made to fall upon another at \( GH \); by which they are again made to converge to a distant focus; but, being intercepted by the lens \( IK \), they are made to meet at the nearer one \( z \); whence diverging to \( LM \), they are again rendered parallel, and the eye at \( N \) sees the object very distinctly.

From an inspection of the figure it is evident, that Dollond's telescope thus constructed is in fact two telescopes combined together; the first ending with the lens \( EF \), and the second with \( LM \). In the first we do not perceive the object itself, but the image of it formed at \( y \); and in the second we perceive only the image of that image formed at \( z \). Nevertheless such telescopes are exceedingly distinct, and represent objects so clearly as to be preferred, in viewing terrestrial things, even to reflectors themselves. The latter indeed have greatly the advantage in their powers of magnifying, but they are much deficient in point of light. Much more light is lost by reflection than by refraction: and as in these telescopes the light must unavoidably suffer two reflections, a great deal of it is lost; nor is this loss counterbalanced by the greater aperture which these telescopes will bear, which enables them to receive a greater quantity of light than the refracting ones. The metals of reflecting telescopes also are very much subject to tarnish, and require much more dexterity to clean them than the glasses of refractors; which makes them more troublesome and expensive, though for making discoveries in the celestial regions they are undoubtedly the only proper instruments.

II. The Reflecting Telescope.

1. Of Sir Isaac Newton's Reflecting Telescope. The inconveniences arising from the great length of refracting telescopes are sufficiently obvious; and these, together with the difficulties arising from the different refrangibility of light, induced Sir Isaac Newton to give attention to the subject of reflection, and endeavour to realize the ideas of himself and others concerning the possibility of constructing telescopes upon this principle. The instrument he contrived is represented fig. 9, where \( ABCD \) is the tube, \( BC \) a concave reflecting metal, \( EF \) a plain reflecting metal fixed to the tube by means of the stem \( HI \). \( MN \) represents a distant object emitting pencils of rays from each point, two only of which are here represented, and those cut off before they reach the metal, to prevent confusion in the figure. Now it is evident from what has been explained above, that these rays, were they not intercepted in their way, would return after reflection at the concave surface \( BC \), and form an inverted image at \( OP \), supposing that to be the place of the focus of reflected rays. But in this case the reflected rays are intercepted in their return to that place by the plain metal, and are thereby thrown sidewise; and instead of forming the image \( OP \), are made to form the image \( QR \): which, because the rays have as yet suffered no refraction, is not liable to the imperfection which arises from the different refrangibility of the rays of light, nor to any other except what may arise from an imperfect polish, or the want of the form of one of the conic sections in the reflector \( BC \); and therefore may be viewed by an eye at \( T \) with a very small lens or eye-glass \( KL \), without appearing either coloured or confused.

2. The Gregorian telescope. This remedies the inconvenience of the Newtonian one, by which objects are found with difficulty. This defect, indeed, was in some measure removed by having a small refracting telescope with two hairs, or wires, running thro' the tube in the common focus of the two glasses, and crossing each other at right angles; and the object being first viewed through this small telescope was afterwards easily found by the reflector. But the inconvenience is more effectually remedied by the following construction.

At the bottom of the great tube \( TTTT \) (fig. 8.) 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 \( M \), 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 at least as broad as the great mirror, we may consider the rays of each pencil, which flow... Optical 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 \( t \), which is joined to the great one.

Let us then suppose the object AB to be at such a distance, that the rays C may flow from its lower extremity B, and the rays E from its upper extremity A. Then the rays C 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 D of the erect image ad, 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 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 bO; 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 ad, 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 the 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 lever tube \( t \), 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 q, 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.

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 this telescope 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.

§ 5. Camera Obscura.

The camera obscura is made by a convex glass CD (fig. 2.) placed in a hole of a window-shutter. Plate Then if the room be darkened, so as no light can enter but what comes thro' the glass, the pictures of all the objects (as fields, trees, buildings, men, cattle, cattle, &c.) on the outside, will be shewn in an inverted order, on a white paper placed at GH in the focus of the glass; and will afford a most beautiful and perfect piece of perspective or landscape of whatever is before the glass, especially if the sun shines upon the objects.

If the convex-glass CD be placed in a tube in the side of a square box, within which is the plane mirror EF, reclining backwards in an angle of 45 degrees from the perpendicular k q, the pencils of rays flowing from the outward objects, and passing thro' the convex glass to the plane mirror, will be reflected upwards from it, and meet in points, as I and K (at the same distance that they would have met at H and G, if the mirror had not been in the way,) and will form the aforesaid images on an oiled paper stretched horizontally in the direction IK: on which paper the outlines of the images may be easily drawn with a black-lead pencil; and then copied on a clean sheet, and coloured by art, as the objects themselves are by nature.—In this machine, it is usual to place a plane glass, unpolished, in the horizontal situation IK, which glass receives the images of the outward objects; and their outlines may be traced upon it by a black-lead pencil.

N.B. The tube in which the convex-glass CD is fixed, must be made to draw out, or push in, so as to adjust the distance of that glass from the plane mirror, in proportion to the distance of the outward objects; which the operator does, until he sees their images distinctly painted on the horizontal glass at IK.

The forming a horizontal image, as IK, of an upright object AB, depends upon the angles of incidence of the rays upon the plane mirror EF, being equal to their angles of reflection from it. For, if a perpendicular be supposed to be drawn to the surface of the plane mirror at e, where the ray A a C e falls upon it, that ray will be reflected upwards in an equal angle with the other side of the perpendicular, in the line e d l. Again, if a perpendicular be drawn to the mirror from the point f, where the ray A b f falls upon it, that ray will be reflected in an equal angle from the other side of the perpendicular, in the line f h I. And if a perpendicular be drawn from the point g, where the ray A c g falls upon the mirror, that ray will be reflected in an equal angle from the other side of the perpendicular, in the line g i I. So that all the rays of the pencil a b c, flowing from the upper extremity of the object AB, and passing thro' 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 q r s, 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, thro' the convex 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 thro' 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.

§ 6. Magic Lantern.

ABCD (fig. 5,) is a tin lantern, with a tube n k l m 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 k l, 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 d e, somewhere in the immoveable 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 b b c, which serves to cast a very strong light from the candle within the lantern upon the object d e. Now when the rays, which shine through the object d e, diverge from the several points as d, e, &c. in the object, and fall upon the lens k l, 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 f g 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 k l; and for this reason the whole apparatus is to be placed in a dark room EFGH. The lens k l must be very convex, so that the object d e 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 f g 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 d e should be placed with the wrong end upwards.

Sect. V. A Description of the above and other Optical Instruments, fitted with their Apparatus; with an account of the methods of applying them to the purposes for which they are intended.

§ 1. Camera Obscura and Magic Lantern.

See Dioptics, p. 2477 to p. 2482.

§ 2. The Graphical Perspective.

This instrument consists of two lenses A B and C D, fig. 1, which are placed at twice their focal distance plate from one another; and in their common focus is ano-CCXIX, ther glass EF, divided into equal parts with the point of a diamond. Though this instrument does not magnify objects, yet the angle under which any object is seen is easily known by it; and since this angle varies with the distance of objects, it is easily applied to the purpose of measuring inaccessible heights and distances; Part III.

Optical distances; and since the field of view is divided into equal squares, it is useful in drawing the perspective appearance of objects. As all foreign light is excluded by the tube in which these lenses are inclosed, pictures seen through this machine have a fine relief; on which account, as also because objects appear inverted through it, the images of a camera obscura are viewed to particular advantage by its means. If a lens of a greater focal length be fixed at a proper distance from the centre of the tube, this instrument will be a telescope, and will magnify the prints which are looked at through it; and if a small lens be used, it will be a microscope, and the same micrometer will serve for both.

§ 3. Of the Single Microscope.

The famous microscopes made use of by Mr Leeuwenhoek, were all, as Mr Baker assures us, of the single kind, and the construction of them the most simple possible, each consisting only of a single lens set between two plates of silver, perforated with a small hole, with a moveable pin before it to place the object on, and adjust it to the eye of the beholder. He informs us also, that lenses only, and not globules, were used in every one of these microscopes.

The single microscope now most generally known and used is that called Wilson's Pocket Microscope. The body is made of brass, ivory, or silver, and is represented by AA, BB. CC is a long fine-threaded male-screw that turns into the body of the microscope. D a convex glass at the end of the screw. *, Two concave round pieces of thin brass, with holes of different diameters in the middle of them, to cover the above-mentioned glass, and thereby diminish the aperture when the greatest magnifiers are employed. EE, three thin plates of brass within the body of the microscope; one of which is bent semicircularly in the middle, so as to form an arched cavity for the reception of a tube of glass, the use of the other two being to receive and hold the sliders between them. F, a piece of wood or ivory, arched in the manner of the semicircular plate, and cemented thereto. G, the other end of the body of the microscope, where a hollow female screw is adapted to receive the different magnifiers. H, is a spiral spring of steel, between the end G and the plates of brass, intended to keep the plates in a right position, and counteract the long screw CC. I, is a small turned handle, for the better holding of the instrument, to screw on or off at pleasure.

To this microscope belong six or seven magnifying glasses: six of them are set in silver, brass, or ivory, as in the figure K, and marked 1, 2, 3, 4, 5, 6; the lowest numbers being the greatest magnifiers. L, is the seventh magnifier, set in the manner of a little barrel, to be held in the hand for the viewing of any larger object. M, is a flat slip of ivory, called a slider, with four round holes thro' it, wherein to place objects between two pieces of glass, or mucov talc, as they appear dddd. Eight such sliders, and one of brass, are usually sold with this microscope; some with objects placed in them, and others empty for viewing any thing that may offer: but whoever pleases to make a collection, may have as many as he desires. The brass slider is to confine any small object, that it may be viewed without crushing or destroying it.

N, is a forceps, or pair of pliers, for the taking up of insects or other objects, and adjusting them to the glasses. O, is a little hair-brush or pencil, wherewith to wipe any dust from off the glasses, or to take up any small drop of liquid, which one would want to examine, and put it upon the talc, or ifinglas. P is a tube of glass contrived to confine living objects, such as frogs, fishes, &c., in order to discover the circulation of the blood. All these are contained in a little neat box, very convenient for carrying in the pocket.

When an object is to be viewed, thrust the ivory slider, in which the said object is placed, between the two flat brass plates EE: observing always to put that side of the slider where the brass rings are, fartherst from the eye. Then screw on the magnifying glass you intend to use, at the end of the instrument G; and looking thro' it against the light, turn the long screw CC, till your object be brought to fit your eye; which will be known by its appearing perfectly distinct and clear. It is most proper to look at it first through a magnifier that can show the whole at once, and afterwards to inspect the several parts more particularly with one of the greatest magnifiers; for thus you will gain a true idea of the whole, and of all its parts. And tho' the greatest magnifiers can show but a minute portion of any object at once, such as the claw of a flea, the horn of a louse, or the like, yet by gently moving the slider which contains the object, the eye will gradually overlook it all.

As objects must be brought very near the glasses when the greatest magnifiers are made use of, be careful not to fetch them by rubbing the slider against them as you move it in or out. A few turns of the screw CC will easily prevent this mischief, by giving them room enough. You may change the objects in your sliders for what others you think proper, by taking out the brass rings with the point of a pen-knife; the ifinglas will then fall out, if you but turn the sliders; and after putting what you please between them, by replacing the brass rings you will fasten them as they were before. It is proper to have some sliders furnished with talc, but without any object between them, to be always in readiness for the examination of fluids, salts, sands, powders, the farina of flowers, or any other casual objects of such sort as need only be applied to the outside of the talc.

The circulation of the blood may be easiest seen in the tails or fins of fishes, in the fine membranes between a frog's toes, or best of all in the tail of a water-newt. If your object be a small fish, place it within the tube, and spread its tail or fin along the side thereof; if a frog, choose such an one as can but just be got into your tube; and, with a pen, or small stick, expand the transparent membrane between the toes of the frog's hind foot as much as you can. When your object is so adjusted, that no part of it can intercept the light from the place you intend to view, unscrew the long screw CC; and thrust your tube into the arched cavity, quite thro' the body of the microscope; then screw it to the true focal distance, and you will see the blood passing along its vessels with a rapid motion, and in a most surprising manner. The third or fourth magnifiers may be used for frogs or fishes; but for the tails of water-newts, the fifth or sixth will do; because the globules of their blood are twice as large as those of frogs or fish. The first or second magnifier cannot well be employed for this purpose; because the thickness of the tube in which the object lies, will scarce admit its being brought so near as the focal distance of the magnifier.

§ 5. The Single Microscope with Reflection.

In fig. 2. A is a scroll of brass fixed upright on a round pedestal of wood B, so as to stand perfectly firm and steady. C is a brass screw, that passes through a hole in the upper limb of the scroll, into the side of the microscope D, and screws it fast to the said scroll. E, is a concave speculum set in a box of brass, which hangs in the arch G by two small screws f f, that screw into the opposite sides thereof. At the bottom of this arch is a pin of the same metal, exactly fitted to a hole b in the wooden pedestal, made for the reception of the pin. As the arch turns on this pin, and the speculum turns on the ends of the arch, it may, by this twofold motion, be easily adjusted in such a manner as to reflect the light of the sun, of the sky, or of a candle, directly upwards through the microscope that is fixed perpendicularly over it; and by so doing, may be made to answer almost all the purposes of the large double reflecting microscope. The body of the microscope may also be fixed horizontally, and objects viewed in that position by any light you choose, which is an advantage the double reflecting microscope has not. It may also be rendered further useful by means of a slip of glass, one end of which being thrust through between the plates where the sliders go, and the other extending to some distance, such objects may be placed thereon as cannot be applied in the sliders; and then, having a limb of brass that may fasten to the body of the microscope, and extend over the projecting glass a hollow ring wherein to screw the magnifiers, all sorts of subjects may be examined with great convenience, if a hole be made in the pedestal, to place the speculum exactly underneath, and thereby throw up the rays of light.

The pocket-microscope, thus mounted, says Mr Baker, "is as easy and pleasant in its use; as fit for the most curious examination of the animalcules and salts in fluids, of the farines in vegetables, and of the circulation in small animals; in short, is as likely to make considerable discoveries in objects that have some degree of transparency, as any microscope I have ever seen or heard of."

§ 5. Of the Double Refracting and Reflecting Microscopes.

Double microscopes are so called as being a combination of two or more lenses.

The only advantage which the double refracting microscope hath over the single one is, that it takes in a larger field of view; and therefore hath yielded to the double reflecting microscope, which gives a clearer view of objects, with a greater power of magnifying at the same time.

The body of this microscope AAAA is a large tube, supported by three brass pillars b b b, rising from a wooden pedestal C; in which pedestal is a drawer D, to hold the object-glasses and other parts of the apparatus. A lesser tube e e slides into the greater, and sends from its bottom another tube f, much smaller than itself, with a male screw g at the end thereof, whereon to screw the object-glass or magnifier. There are five of these magnifiers, numbered 1, 2, 3, 4, 5; which numbers are also marked on the inner tube, to direct whereabout to place it according to the magnifier made use of; but if then it fits not the eye exactly, slide the inner tube gently higher or lower, or turn the screw of the magnifier gradually, till the object appears distinct. The greatest magnifiers have the smallest apertures and the lowest numbers.

L, is a circular plate of brass fixed horizontally between the three brass pillars, and in the centre thereof a round hole M is adapted to receive a proper contrivance N for holding ivory sliders wherein objects are placed; this contrivance consists of a spiral steel wire confined between three brass circles, one whereof is moveable for the admission of a slider. O is a round brass plate with several holes for placing objects in, some of which are usually furnished with them at the shops; but two holes are commonly reserved for small concave glasses, whereon to place a drop of any liquid, in order to view the animalcules, &c. There is also a piece of white ivory, and a piece of black ebony, of the same size and shape as the holes for objects; the ivory is for holding such opaque objects as are black, and the ebony such as are white, by which contrariety of colours they will be seen more distinctly. At the bottom of this object-plate is a button to slip into a slit P, that fits it, on the circular plate of brass; and by turning it round on this, all the objects may be examined successively with very little trouble.

Q, is a concave speculum set in a box of brass, and turning in an arch R, upon two small screws s s. From the bottom of the arch comes a pin, which being let down into a hole t, in the centre of the pedestal, enables the speculum to turn either vertically or horizontally, and to reflect the light directly upwards on the object to be viewed. V, is a plano-convex lens, which by turning on two screws **, when the pin at the bottom of it is placed in the hole W, for its reception in the circular plate L, will transmit the light of a candle to illuminate any opaque object that is put on the round piece of ivory or on the ebony for examination; and it may be moved higher or lower, as the light requires. This glass is of service to point the sunshine or the light of a candle upon any opake object, but in plain day-light is of no great use. X, is a cone of black ivory, to fasten on a shank underneath the brass circular plate L, principally when the first or second magnifier is made use of, and the object very transparent; for objects are rendered much more distinctly visible, by intercepting some part of the oblique rays which come from the speculum. The brass fish-pan Y is to fasten any small fish upon, to see the circulation of the blood in its tail. For this purpose, the tail of the fish must be expanded across the oblong hole at the smallest end of the pan; then by slipping the button on the backside of the pan into the slit P thro' the circular plate L, Part III.

Optical Instruments

L, the spring that comes from the button will make it steady, and present it well to view. But if it be a frog, a newt, or an eel, in which the circulation is desired to be shewn, a glass tube is fitted for the purpose. The tail of a newt or eel, or, in a frog, the web between the toes of the hind-feet, are the parts where it may be seen best. When the object is well expanded on the inside of the tube, slide the tube along under the circular brass plate L, (where there are two springs and a cavity made in the shank to hold it), and bring your object directly under the magnifier.

There are three of these glass tubes smaller one than another, and the size of the object must direct which of them is to be used; but, in general, the less room the creature has to move about in, the easier will it be managed.

The cell 2, with a concave and a plane glass in it, is intended to confine fleas, lice, mites, or any small living objects, during pleasure; and by placing it over the hole M, in the middle of the circular brass plate, they may be viewed with much convenience. Three loose glasses, viz. one plane, and two concave, belong also to this microscope; and are designed to confine objects, or to place them upon occasionally. The long steel wire 3, with a pair of pliers at one end, and a point on the other, to hold fast or stick objects upon, slips backward or forward in a short brass tube where-to a button is fastened, which fits into the little hole x, near the edge of the brass plate L; and then the object may be readily brought to a right position, and a light be cast upon it either by the speculum underneath, or, if it be opaque, by the plano-convex lens V. 4, Is a flat piece of ivory called a slider, with four round holes thro' it, and objects placed in them, between mucovory tales or ifinglases, kept in by brass wires. It is proper to have a number of these sliders, filled with curious objects, always ready, as well as some empty ones for anything new that offers. When made use of, thrust them between the brass rings of the contrivance on purpose for them, which shoots into the round hole M, in the centre of the brass plate L. This keeps them steady, and at the same time permits them to be moved to and fro for a thorough examination. 5, Is a little round ivory box to hold pieces of ifinglases for the sliders; 6, a small hair-brush to wipe off any dust from the glasses, or to apply a drop of any liquid; 7, a pair of nippers to take up any object to be examined.

§ 6. The Microscope for Opaque Objects.

This microscope remedies the inconvenience of having the dark side of an object next the eye, which formerly was an unsurmountable objection to the making observations on opaque objects with any considerable degree of exactness or satisfaction: for, in all other contrivances commonly known, the narrowness of the instrument to the object (when glasses that magnify much are used) unavoidably overshadows it so much, that its appearance is rendered obscure and indistinct. And notwithstanding ways have been tried to point light upon an object, from the sun or a candle, by a convex glass placed on the side thereof, the rays from either can be thrown upon it in such an acute angle only, that they serve to give a confused glare, but are insufficient to afford a clear and perfect view of the object. But in this microscope, by means of a concave speculum of silver highly polished, in whose centre a magnifying lens is placed, such a strong and direct light is reflected upon the object, that it may be examined with all imaginable ease and pleasure. The several parts of this instrument, made either of brass or silver, are as follow.

Thro' the first side A, passes a fine screw B, the plate other end of which is fastened to the moveable side C. CCXIX. D is a nut applied to this screw, by the turning of fig. 4, which the two sides A and C are gradually brought together. E, is a spring of steel that separates the two sides when the nut is uncrewed. F a piece of brass, turning round in a socket, whence proceeds a small spring-tube moving upon a rivet, thro' which tube there runs a steel wire, one end whereof terminates in a sharp point G, and the other hath a pair of pliers H fastened to it. The point and pliers are to thrust into, or take up and hold, any insect or object; and either of them may be turned upwards, as best suits the purpose. I, is a ring of brass, with a female screw within it, mounted on an upright piece of the same metal; which turns round on a rivet, that it may be set at a due distance when the least magnifiers are employed. This ring receives the screws of all the magnifiers. K, is a concave speculum of silver, polished as bright as possible; in the centre of which is placed a double convex lens, with a proper aperture to lock thro' it. On the back of this speculum, a male screw, L, is made to fit the brass ring I, to screw into it at pleasure. There are four of these concave specula of different depths, adapted to four glasses of different magnifying powers, to be used as the objects to be examined may require. The greatest magnifiers have the least apertures. M, is a round object-plate, one side of which is white and the other black: The intention of this is to render objects the more visible, by placing them, if black, on the white side, or, if white, on the black side. A steel ring, N, turns down on each side to make any object fast; and issuing from the object-plate is a hollow pipe to screw it on the needle's point G. O is a small box of brass, with a glass on each side, contrived to confine any living object, in order to examine it: this also has a pipe to screw upon the end of the needle G. P, is a turned handle of wood, to screw into the instrument when it is made use of. Q, a pair of brass pliers to take up any object, or manage it with conveniency. R is a soft hair-brush for cleaning the glasses, &c. S, is a small ivory box for ifinglases, to be placed, when wanted, in the small brass box O.

When you would view any object with this microscope, screw the speculum, with the magnifier you think proper to use, into the brass ring L. Place your object, either on the needle G in the pliers H, on the object-plate M, or in the hollow brass box O, as may be most convenient: then holding up your instrument by the handle P, look against the light thro' the magnifying lens; and by means of the nut D, together with the motion of the needle, by managing its lower end, the object may be turned about, raised, or depressed, brought nearer the glass, or removed farther from it, till you hit the true focal distance, and the light be seen strongly reflected from the speculum upon the object, by which means it will be shewn in a manner. Optical manner surprisingly distinct and clear; and for this purpose the light of the sky or of a candle will answer very well. Transparent objects may also be viewed by this microscope: only observing, that when such come under examination, it will not always be proper to throw on them the light reflected from the speculum; for the light transmitted thro' them, meeting the reflected light, may together produce too great a glare. A little practice, however, will show how to regulate both lights in a proper manner.

§ 7, The Solar Microscope.

This instrument is composed of a tube, a looking-glass, a convex lens, and Wilson's single pocket microscope before described. The sun's rays being directed thro' the tube, by means of the looking-glass, upon the object, the image or picture of the object is thrown distinctly and beautifully upon a screen of white paper, or a white linen sheet, placed at a proper distance to receive the same; and may be magnified to a size not to be conceived by those who have not seen it: for the farther the screen is removed, the larger will the object appear; inasmuch, that a house may thus be magnified to the length of five or six feet, or even a great deal more; though it is more distinct, when not enlarged to above half that size.—The apparatus for this purpose is as follows.

A, a square wooden frame, thro' which pass two long screws affixed by a couple of nuts i, i. Fasten it firmly to a window-shutter, wherein a hole is made for its reception; the two nuts being let into the shutter, and made fast thereto. A circular hole is made in the middle of this frame to receive a piece of wood, B, of a circular figure; whose edge, that projects a little beyond the frame, composes a shallow groove 2, wherein runs a catgut 3; which, by twisting round, and then crossing over a brass pulley 4, (the handle whereof, 5, passes thro' the frame) affords an easy motion for turning round the circular piece of wood B, with all the parts affixed to it. C is a brass tube covered with seal-skin; which, screwing into the middle of the circular piece of wood, becomes a case for the uncovered brass tube D to be drawn backwards or forwards in. E is a smaller tube, of about one inch in length cemented to the end of the larger tube D. F is another brass tube, made to slide over the above described tube E; and to the end of this the microscope must be screwed, when we come to use it. G, a convex lens, whose focus is about 12 inches, designed to collect the sun's rays, and throw them more strongly upon the object. H is a looking-glass of an oblong figure, set in a wooden frame, fastened by hinges in the circular piece of wood B, and turning about therewith by means of the abovementioned cat-gut. I is a jointed wire, partly brass, and partly iron; the brass part whereof, 6, which is flat, being fastened to the looking-glass, and the iron part, 7, which is round, passing thro' the wooden frame, enable the observer, by putting it backwards or forwards, to elevate or depress the glass according to the sun's altitude. There is a brass ring at the end of the jointed wire, whereby to manage it with the greater ease. The extremities of the cat-gut are fastened to a brass pin, by turning of which it may be braced up, if at any time it becomes too slack.

When this microscope is employed, the room must be rendered as dark as possible; for on the darkness of the room, and the brightness of the sunshine, depend the sharpness and perfection of your image. Then putting the looking-glass G thro' the hole in your window-shutter, fasten the square frame A to the shutter by its two screws and nuts i, i. This done, adjust your looking-glass to the elevation and situation of the sun, by means of the jointed wire H, together with the cat-gut and pulley, 3, 4. For the first of these raising or lowering the glass, and the other inclining it to either side, there results a twofold motion, which may easily be so managed as to bring the glass to a right position, that is, to make it reflect the sun's rays directly thro' the lens, 5, upon the paper screen, and form thereon a spot of light exactly round. But tho' the obtaining a perfect circular spot of light upon the screen before you apply the microscope, is a certain proof that your looking-glass is adjusted right, that proof must not always be expected: for the sun is so low in winter, that if it shines in a direct line against the window, it cannot then afford a spot of light exactly round; but if it be on either side, a round spot may be obtained, even in December. As soon as this appears, screw the tube C into the brass collar provided for it in the middle of your wood-work, taking care not to alter your looking-glass: then screwing the magnifier you choose to employ to the end of your microscope, in the usual manner, take away the lens at the other end thereof, and place a slider, containing the object to be examined, between the thin brass plates, as in the other ways of using the microscope.

Things being thus prepared, screw the body of your microscope to the short brass tube F; which slip over the small end E of the tube D, and pull out the said tube D less or more, as your object is capable of enduring the sun's heat. Dead objects may be brought within about an inch of the focus of the convex lens, 5; but the distance must be shortened for living creatures, or they will soon be killed.

If the light falls not exactly right, you may easily, by a gentle motion of the jointed wire and pulley, direct it thro' the axis of the microscopic lens. The short tube F, to which the microscope is screwed, renders it easy, by sliding it backwards or forwards on the other tube E, to bring the objects to their focal distance; which will be known by the sharpness and clearness of their appearance: they may also be turned round by the same means, without being in the least disordered.

The magnifiers most useful in the solar microscope are in general, the fourth, fifth, or sixth. The screen on which the representations of the objects are thrown, is usually composed of a sheet of the largest elephant paper, strained on a frame which slides up or down, or turns about at pleasure on a round wooden pillar, after the manner of some fire-screens. Larger screens may also be made of several sheets of the same paper paited together on cloth, and let down from the ceiling with a roller like a large map.

"This microscope (says Mr Baker) is the most entertaining of any; and perhaps the most capable of making discoveries in objects that are not too opaque: as it shows them much larger than can be done any any other way. There are also several conveniences attending it, which no other microscope can have: for the weakest eyes may use it without the least straining or fatigue; numbers of people together may view any object at the same time, and, by pointing to the particular parts thereof, and discoursing on what lies before them, may be able better to understand one another, and more likely to find out the truth, than in other microscopes, where they must peep one after another, and perhaps see the object neither in the same light nor in the same position. Those also who have no skill in drawing, may, by this contrivance, easily sketch out the exact figure of any object they have a mind to preserve a picture of; since they need only fasten a paper on the screen, and trace it out thereon either with a pen or pencil, as it appears before them. It is worth the while of those who are desirous of taking many draughts in this way, to get a frame, wherein a sheet of paper may be put in or taken out at pleasure; for if the paper be fingle, the image of an object will be seen almost as plainly on the back as on the fore side, and, by standing behind the screen, the shade of the hand will not obstruct the light in drawing, as it must in some degree when one stands before it.

§ 8. Universal Microscope.

ABC, is the body of the microscope. D, is a joint, by which it is moveable vertically. E, is a hollow square socket, with F, a screw, by which it is fixed to the part at D. DQR is a strong brass pillar or stand. S, T, V, the tripod, or three feet, on which it stands. GHI is a stage on which objects of different sorts are placed to be viewed. K, is a strong screw by which the stage is rendered moveable horizontally. MN, are two brass sockets, connected by an adjusting screw, and moveable up and down upon the square part of the stand. O is a screw for fixing the socket M. P is a long adjusting screw by which the socket N is moveable, and the objects upon the stage adjusted to the view. W is a concave mirror, or speculum, fixed at X, just under the central part of the stage, for illuminating transparent objects. Y is a concave lens moveable at Z, in a spring socket; by this lens opaque bodies are sufficiently enlightened for the view.

This compound microscope is in the best manner adapted to view transparent objects; for if they are such as can be put into the concave glass in the middle of the stage at H, then they will be sufficiently enlightened by the reflector W below, in one side of which is a plane speculum, and in the other a concave one, as both sorts are occasionally necessary.

If the transparent objects are such as may be included between tales in the ivory sliders, then there is a part ABCD, called the slider-holder, which is fitted to the hole at H in the middle of the stage, and in which all the variety of objects in sliders may be viewed to great perfection by reflected light from the speculum W below.

As some curious experiments with transparent objects require the light to be very pure, and adjusted to a proper degree, there is an inverted cone of brass, ABC, to be placed in the under part of the hole at Fig. 3. H by its broad end or base AC; and by the narrow end B only the interior and purer light contained in the upper and denser part of the large cone of rays reflected by the speculum W, can illuminate the objects to be viewed. This cone is indeed of more immediate use in the single microscope, to be mentioned by and by. Thus, it is plain, in one or other of these ways all kinds of transparent objects are to be viewed in the utmost perfection.

In this construction it is also as evident, that every opaque object may be shewn as well in this as in that which is usually called the opaque microscope: because here is all the same apparatus for that purpose, and much more; for this is both a single and compound opaque microscope.

Thus, if any opaque object be laid upon the glass at H, it may be very strongly illuminated by the lens Y, moveable higher or lower in the socket Z, to make the light upon the object greater or less, as occasion requires. In this case you have the advantage of a large and delightful field of view, and objects of all shapes and sizes are immediately viewed upon the stage GHI, as it is so easily moved up and down by the sliding sockets M and N, fastened in any position by the screw O, and adjusted for the most accurate inspection by the screw P.

In many cases it may be requisite to view light-coloured objects upon a dark ground, and the contrary: therefore, to answer such purposes, there is provided a round flat piece of ivory, with one side white and the other black, fitted into the hole at G, and to be taken in and out at pleasure.

To answer these purposes still more generally, there is a pair of pincers, AB, moveable in a brass spring socket at C, and by the shank at D it is fitted in the hole of the stage at I, where it has a horizontal motion, and also a vertical motion (up and down) by its joint at E. By the pincers at the end A any object may be very readily adjusted to the view, and illuminated by the lens Y. Also at the other end of the pincers B, there is a small cylindric piece of ivory F screwed on, with one end black, the other white, for the above-mentioned purposes.

But oftentimes objects will be found which require the use of the pincers AB, and are seen to the greatest advantage by light reflected upon them; for which purpose a small concave metallic speculum AB is screwed into the end CD of a tube CDEF, which is made to go on upon the pipe of the microscope; and the large cone of reflected light from the mirror W will pass through the hole H to this small concave AB, by which it will be reflected upon the object in the pincers at A, but more especially upon the ivory at F, where exceeding small objects require the greatest degree of light they can bear.

Further, to make this microscope answer all the ends of a single opaque microscope, there is a brass piece AB, with a square hole or socket, to go on upon the shank at D (fig. 1.) when the body of the microscope ABC is taken off, and is there made fast by the screw C. To this is annexed a strong brass ring DE, into which are screwed the same Liberthuns, as they are called, or concaves with a small lens in the middle of each, as are used in the single opaque microscope; and being applied to objects on the stage GHI, or in the players, they are viewed in the same manner here as they are there; with this additional advantage, that in the present microscope, the light is much more intense from the speculum W, than the common light but once reflected in the usual form of this instrument.

Lastly, every thing shewn in the aquatic microscope is to be seen equally in this; because, since the hole at H in the stage is very large, it will admit of a concavity sufficiently capacious for any purposes of viewing objects in water, of any fort whatsoever. And not only the magnifiers, but the mode of applying them, is nearly the same here as in those of the common form. But in this construction, you have both the single and compound aquatic microscope: for the stage GHI being moveable horizontally, and the magnifier at A moving vertically on the joint at D, it is plain, every part of the water in the concave glass at H may be brought under it, and the most minute objects well enlightened by the concave AB, and shewn with great distinctness.

The other parts of the apparatus are common to all microscopes. Every body knows the use of the ivory sliders, for holding and applying transparent objects as above directed, (fig. 7.) But in fig. 8. you have the form of a brass slider, with several small glasses concaves fixed in one side, and over them a flip of clear plain glass is made to slide in the frame, and thereby to confine in the hollow of the glasses very small living objects, as a flea, louse, mite, &c. and prevent their crawling out of the field of view.

There is also what is called a bug-box, consisting of two parts: the lowest contains a large concave; and the upper part contains a plane glass, which, being screwed upon the concave, will confine any larger animal, as a bug, an ant, a spider, a small fly, &c.

As the circulation of the blood is one of the noblest experiments of the microscope, to ample provision is made for it by a set of glass tubes of different sizes for applying the transparent parts of proper objects for that purpose, such as small fish, tadpoles, and water-newts, the best subject of all; such a tube is AB (fig. 9.) It being necessary to stop the open end B, when the animal is in, with a cork, there is a small hole at the other end A to give air to the animal. These tubes are applied to the hole H in the stage by two steel springs on the under part, bent to receive them.

In case it be required to view the circulation in the tail of a large fish, as a gudgeon, loach, &c. there is an instrument of brass called the fish-pan, (fig. 10.) contrived of a proper form to hold and confine it; where ABD is the incurved plate to receive the body of the fish; CFG is a ribbon to tie the fish to the said plate or pan, and is kept tight by a spring behind at H. At the end AD, is a long transverse hole or slit, over which the transparent tail of the fish is placed; and then by the shank at E, on the underside, it is put upon the stage thro' a hole at I, and there easily adapted to the magnifier A, by moving it every way under the same.

It is often required to see what many small objects are, and how they are best disposed in sliders, glasses, instrument tubes, &c. for which purpose there is a hand-magnifier ABC, (fig. 11.) containing a lens of about one inch focal distance, to be used upon all such occasions.

Besides the above particulars, there is a pair of nippers (or forceps) to take up small objects, in order to place them on the stage, between the talcs, &c. also a camel-hair brush, for cleansing talcs, glasses, &c. A small wire, with a spiral screw at the end, for holding cotton, &c. for cleansing the glass tubes. A little ivory box with spare talcs, and wires to fasten them in the sliders. A piece of shagreen leather is useful upon all occasions for wiping the glasses of every fort, as it will cleanse them well without hurting their surfaces.

§ 9. Clark's Improved Pocket Microscope.

This is represented in Plate CCXX. where ABC DEF (fig. 2. no 1.) is a box three inches broad, four inches long, and one inch deep, covered with shagreen, and having the lid open, which when shut is fastened by clasps as in the figure. This box serves for the pedestal as well as case of the instrument. abcd, Is a solid piece of wood, fixed in the middle of the box; on which is screwed a brass plate ef, having in the middle a female screw for holding the other parts now to be described.

αβγδεζ. Is a piece of brass of the shape represented in the plate, having a male screw at α, answering to the female one abovementioned at γ, and by which this part is firmly fixed upright in the middle of the box. On the lower part of this piece of brass is fastened, but in such a manner as to be moveable at the joint s, a semicircle of brass ef, in which is a concave speculum iii, ground to a focus of about eight inches: it is moveable in the ring, by means of two pivots; and as the ring itself is also moveable, it is plain that the speculum may be moved to a proper distance from the standard. The face of it is placed next the standard when the instrument is put into the case, in order to prevent the polished surface from injury. γ Is a piece of solid brass, which goes into a dovetail slit in the part next to be described, and which Mr Clark calls the stage. This consists of two pieces, no 3. One of these, x x x x, is a parallelogram of brass, in which the other part yzyz slides up or down by means of the screw mm. From the upper part of this, proceeds at right angles another piece yzrs. This is formed of two pieces of brass riveted at y and z; and joined to each other at their extremities by the cross piece rs. Upon this slides another piece 1, 2, 3, 4, having in it a round hole s, and which can be made to approach either to yz or to rs as occasion requires. This part, by means of the dovetail slit at xx, may be put on the solid piece of brass at γ of the former, and secured in a perpendicular position on the top of it.

The last part of this microscope is represented no 4. It consists of a solid piece of brass opqrs, to which is fixed at right angles the piece o p v; v is a small plane speculum, with the reflecting side downwards, and, by means of a joint, capable of being raised up or let down as the observer finds necessary. To the under side Part III.

Optical side of the fore-part of this is placed a small brass circle, the edges of which appear at \( \alpha \) and \( \lambda \), and which is fully represented in no. 5. Round this circle the magnifiers are disposed, and over against each of them is engraved its magnifying power, expressed by 1, 2, 3, 4, 5; the highest numbers magnifying most. This circle is movable; and so disposed, that, as it turns round, the magnifiers appear successively through the hole at \( \alpha \), at the same time that the power of each is shown by the cipher which appears through the little square hole at 1, no. 4. This whole part of the machine slides up and down on the back of the other, by means of a dovetail; and thus, though the part \( \rho \) \( \lambda \) \( v \) is always above and parallel to the stage, yet it may be brought nearer to it or removed farther off at pleasure.

On the solid back plate are marked the numbers 1, 2, 3, 4, 5, to show the foci of the different magnifiers. Concerning the proper method of using this microscope, Mr. Clark gives the following directions.

When the case is opened, take out the microscope, which consists of two separate parts; screw the under part (on which the speculum is) into the brass plate in the inside of the case, which is the base for the instrument while in use.—Put the other end into the dovetail slit behind the handle of the adjusting screw \( m \), no. 3. place the microscope so as the speculum may front the light; then gently move up the back part by the button for that purpose, till the figure 1, on the plate \( \rho \) \( q \) \( t \) (no. 4.) appears just above the stage; then turn round the circular plate which contains the magnifiers, being five in number, till 1 appear in the square hole atop. Put the slider with the objects into the stage; give the concave speculum such an inclination as to throw the rays through the object immediately under the magnifier: thereabout distinct vision will be had; if not entirely so, a turn or two of the adjusting screw will either raise or depress the stage, as the eye or object requires: and so on with each magnifier and corresponding figure, always taking care that the speculum be in such a situation as to throw the light properly up.

The slider for opaque objects consists of three divisions; first, ebony, for laying all white or light-colored objects on, such as seeds, sands, mineral, &c. The second division ivory, for all dark and black bodies. The third division glass, which opens and shuts; when open, for the circulation of the blood in tadpoles, &c. when shut, for containing any live object to be examined, and for all kinds of animalcules in fluids, solutions of salts, &c. Likewise there are on the side of the above slider, a pair of small forceps, that turn out at pleasure, to hold any opaque or transparent object, such as a fly, spider, &c. which may be viewed with the aid of one or both speculums to great advantage. See no. 6.

When this microscope is employed for examining an opaque object, the upper speculum must be bent down to such an angle as to throw the rays reflected from the under speculum upon the opaque object in view: with the sun or candle-light, those two speculums have a most delightful effect. The rays from the under speculum, passing through the square opening \( y \) \( z \), 1, 2, (no. 3.) behind the stage, fall on the small upper plane speculum, which, moving on an axis, may be placed in a direction so as to illuminate the opaque object with the whole light proceeding from the large concave speculum. In this operation all the magnifiers but no. 5, may be used with success and satisfaction.

§ 10. Of Extempor Microscopes.

For those who cannot conveniently procure the apparatus of any of the above-mentioned microscopes, it may afford some entertainment to try the magnifying power of small globules of water, which in some cases is very considerable. The inventor of this method of viewing objects was Mr. Stephen Gray, who gives an account of it in the Philosophical Transactions No. 221, 223. "Having observed, says he, some irregular particles within the glass globules (for microscopes), and finding that they appeared distinct, and prodigiously magnified when held close to my eye; I concluded, that if I conveyed a small globule of water close to my eye, in which there were any opacous or less transparent particles than water, I might see them distinctly. I therefore took on a pin a small portion of water, which I knew to have in it some minute animals, and laid it on the end of a small piece of brass wire that then lay by me, about \( \frac{1}{10} \) of an inch in diameter, till there was formed somewhat more than an hemisphere of water. Then keeping the wire erect, I applied it to my eye, and standing at a proper distance from the light, I saw them and some irregular particles, as I had predicted; but most enormously magnified. For, whereas they were scarce discernible by my glass microscope, they appeared within the globule not much different in form, nor less in magnitude, than ordinary peas. They cannot be well seen by day-light, unless the room be darkened; but most distinctly by candle-light. They may also be very well seen by the light of the full moon."

But Mr. Gray tells us, that these little animals will appear more distinctly, if drops of water be conveyed by a pin's point into a round hole made in a brass plate whose thickness is about one tenth of an inch, and the diameter of the cylindrical hole a little less than half a tenth; observing to fill it till near an hemisphere of water be extant on each side of it. Now, supposing the axis of this cylinder of water to be terminated by equal spherical surfaces, and to be exactly equal to three diameters of the spheres of those surfaces; in that case the little animals seen by reflection from the farther surface, will appear just twice as big in diameter, as if they were placed in the focus of one of those spheres of water, and were seen thro' it as in common microscopes. His description of the animalcules thus observed is curious. "They are (says he) of a globular form, and but little less transparent than the water they swim in. They have sometimes two dark spots diametrically opposite; but these are rarely seen. There are sometimes two of these globular insects sticking together, and the place of junction is opacous: possibly they may be in the act of generation. They have a twofold motion; a swift progressive regular one, and at the same time a rotation about their axes, at right angles to the diameter that joins their dark spots; but this is only seen when they move slowly. They are almost of an incredible minuteness." Mr. Leeuwenhoek is moderate enough Optical enough in his computation, when he tells us that he could scarcely equal a grain of coarse sand. But I believe it will seem a paradox to him when he is told, that he may see them by only applying his eye to a portion of water wherein they are contained. I have examined many transparent fluids, as water, wine, brandy, vinegar, beer, spittle, urine, &c. and do not remember to have found any liquors without these insects. But I have not seen many in motion, except in common water that has stood, for sometimes a longer, at others a shorter time. In the rivers, after the water has been thickened by rain, there are such infinite numbers of them, that the water seems in great part to owe its opacity and whiteness to those globules. Rain-water, as soon as it falls, has many, and snow-water has more of them. The dew that stands on glass windows has many of them; and for as much as rains and dews are continually ascending and descending, I believe we may say the air is full of them. They seem to be of the same specific gravity with the water they swim in; the dead remaining in all parts of the water. Of the many thousands that I have seen, I could discern no sensible difference in their diameters; they appear of equal bignesses in water that has been boiled: they retain their shapes, and will sometimes revive."

The same ingenious author describes another microscope of his own invention, as follows. "A.B. I call the frame of the microscope; it may be about \( \frac{1}{4} \) of an inch in thickness. At A there is a small hole near \( \frac{1}{8} \) of an inch in diameter, in the middle of a spherical cavity about \( \frac{1}{6} \) of an inch in diameter, and in depth somewhat more than half the thickness of the brass. Opposite to this, at the other side of the brass, there is another spherical cavity, half as broad as the former; and so deep as to reduce the circumference of the small hole above-mentioned, almost to a sharp edge. In these cavities the water is to be placed, being taken upon a pin or a large needle, and conveyed into them till there be formed a double convex lens of water; which, by the concavities being of different diameters, will be equivalent to a double convex lens of unequal convexities. By this means I find the object is rendered more distinct than by a plano-convex of water, or by a double convex formed on the plane surfaces of a piece of metal. CDE is the supporter whereon to place the object; if it be water, in the hole C; if a solid, on the point F. This is fixed to the frame of the microscope by the screw E, where it is bent upwards, that its upper part CF may stand at a distance from the frame A.B. It is moveable about the screw E as a centre, in order that either the hole C, or the point F, may be exposed before the microscope A, and that the object may be brought to, and fixed in its focus. There is another screw, about half an inch in length, which goes through a round plate in the frame of the microscope AE, the screw and plate taking hold of the supporter about D, where there is a slit somewhat larger than the diameter of the screw. This is requisite for the admission of the hole C, or point F, according to the nature of the object, into the focus of the watery lens at A. For, by turning the screw G, the supporter is carried to or from the same; which may be sooner done, if, while one turns the screw with one hand, the other holds the microscope by the end Instrument B; and one be looking through the water, till the object be seen most distinctly. The supporter must be made of a thin piece of brass, well hammered, that, by its spring, it may better follow the motion of the screw. I choose rather to fix the supporter by the screw E, than by a rivet; because it may now, by the help of a knife, be unscrewed, and, by the other screw G, be brought close to the frame of the microscope, without weakening its spring, and so become more conveniently portable. If the hole C in the supporter be filled with water, but not so as to be spherical, all objects that will bear it are seen therein more distinctly. The hole at B is made for seeing animals in water by reflection from its farther surface as above described."

§ II. To find the Magnifying Power of Glasses employed in Single Microscopes.

The apparent magnitude of any object, as must appear from what hath been already delivered, is measured by the angle under which it is seen; and this angle is greater or smaller, according as the object is near to or far off from the eye; and of consequence the less the distance at which it can be viewed, the larger it will appear. The naked eye is unable to distinguish any object brought exceedingly near it; but looking through a convex lens, however near the focus of that lens be, there an object may be distinctly seen; and the smaller the lens is, the nearer will be its focus, and in the same proportion the greater will be its magnifying power. From these principles it is easy to find the reason why the first or greatest magnifiers are so extremely minute; and also to calculate the magnifying power of any convex lens employed in a single microscope: For as the proportion of the natural light is to the focus, such will be its power of magnifying. If the focus of a convex lens, for instance, be at one inch, and the natural light at eight inches, which is the common standard, an object may be seen through that lens at one inch distance from the eye, and will appear in its diameter eight times larger than it does to the naked eye; but as the object is magnified every way, in length as well as in breadth, we must square this diameter to know how much it really is enlarged; and we then find that its superficies is magnified 64 times.

Again, suppose a convex lens whose focus is only one-tenth of an inch distant from its centre; as in eight inches, the common distance of distinct vision with the naked eye, there are 80 such tenths, an object may be seen through this glass 80 times nearer than with the naked eye. It will, of consequence, appear 80 times longer, and as much broader, than it does to common sight; and therefore is 6400 times magnified. If a convex glass be so small that its focus is only \( \frac{1}{20} \) of an inch distant, we find that eight inches contains 160 of these twentieth-parts; and of consequence, the length and breadth of any object seen through such a lens will be magnified 160 times, and the whole surface 25,600 times. As it is an easy matter to melt a drop or globe of a much smaller diameter than a lens can be ground, and as the focus of a globe is no farther off than a quarter of its own diameter, able discoveries were to be made with such glasses as, magnifying but moderately, exhibited the object with the greatest brightness and distinction."

In a single microscope, if you want to learn the magnifying power of any glass, no more is necessary than to bring it to its true focus, the exact place whereof will be known by an object's appearing perfectly distinct and sharp when placed there. Then, with a pair of small compasses, measure, as nearly as you can, the distance from the centre of the glass to the object you was viewing, and afterwards applying the compasses to any ruler, with a diagonal scale of the parts of an inch marked on it, you will easily find how many parts of an inch the said distance is. When that is known, compute how many times those parts of an inch are contained in eight inches, the common standard of light, and that will give you the number of times the diameter is magnified: squaring the diameter will give the superficies; and, if you would learn the solid contents, it will be shewn by multiplying the superficies by the diameter.

The superficies of one side of an object only can be seen at one view; and to compute how much that is magnified, is most commonly sufficient: but sometimes it is satisfactory to know how many minute objects are contained in a larger; as suppose we desire to know how many animalcules are contained in the bulk of a grain of sand; and to answer this, the cube, as well as the surface, must be taken into the account. For the greater satisfaction of those who are not much versed in these matters, we shall here subjoin the following

**TABLE of the MAGNIFYING POWERS of CONVEX GLASSES, employed in Single Microscopes, according to the distance of their focus:** Calculated by the scale of an inch divided into 100 parts.

Shewing how many times the DIAMETER, the SUPERFICIES, and the CUBE of an OBJECT, is magnified, when viewed through such glasses, to an eye whose natural sight is at eight inches, or 800 of the 100th-parts of an inch.

| Magnifies the Diameter | Magnifies the Superficies | Magnifies the Cube of an Object | |------------------------|--------------------------|-------------------------------| | 1/2, or 50 | 16 | 4,096 | | 3/10, or 40 | 20 | 8,000 | | 3/10, or 30 | 26 | 17,576 | | 1/10, or 20 | 40 | 64,000 | | 1/5 | 53 | 148,877 | | 1/4 | 57 | 185,193 | | 1/3 | 61 | 226,981 | | 1/2 | 66 | 287,496 | | 1/1 | 72 | 373,248 | | The focus of a glass at 1/10, or 10 | 80 | 512,000 | | 9/10 | 88 | 681,472 | | 8/10 | 100 | 1,000,000 | | 7/10 | 114 | 1,481,544 | | 6/10 | 133 | 2,352,637 | | 5/10 | 160 | 4,096,000 | | 4/10 | 200 | 8,000,000 | | 3/10 | 266 | 18,821,006 | | 2/10 | 400 | 64,000,000 |

The greatest magnifier in Mr Leeuwenhoek's cabinet of microscopes, presented to the Royal Society, has its focus, as nearly as can well be measured, at one-twentieth of an inch distance from its centre; and consequently magnifies the diameter of an object 160 times, and the superficies 25,600. But the greatest magnifier in Mr Wilson's single microscopes, as they are now made, has usually its focus at no farther distance than about the 50th part of an inch; whereby it has a power of enlarging the diameter of an object 400, and its superficies 160,000 times. The magnifying power of the solar microscope must be calculated in a different manner; for here the difference between the focus of the magnifier and the distance of the screen or sheet whereon the image of the object is cast, is the proportion of its being magnified. Suppose, for instance, the lens made use of has its focus at half an inch, and the screen is placed at the distance of five feet, the object will then appear magnified in the proportion of five feet to half an inch: and as in five feet there are 120 half-inches, the diameter will be magnified 120 times, and the surfaces 14,400 times; and, by putting the screen at farther distances, you may magnify the object almost as much as you please; but Mr Baker advises to regard distinctness more than bigness, and to place the screen just at that distance where the object is seen most distinct and clear.

With regard to the double reflecting microscope, Mr Baker observes, that the power of the object-lens is indeed greatly increased by the addition of two eyeglasses; but as no object lens can be used with them so minute a diameter, or which magnifies itself near so much as those that can be used alone, the glasses of this microscope, upon the whole, magnify little or nothing more than those of Mr Wilson's single one; the chief advantage arising from a combination of lenses being the sight of a larger field or portion of an object magnified in the same degree.

§ 12. To find out the real Size of Objects seen by Microscopes.

Though, by the directions already given, the magnifying powers of microscopes may be easily calculated; yet if we examine extremely minute objects, the real size of them will still remain uncertain. For, though we may know that they are magnified so many thousand times, we can by that make but a very imperfect computation of their natural and true size; nor indeed can we come to any certain conclusion as to that, but by the mediation of some larger object whose dimensions we really know. For as bulk itself is merely comparative, the only way we can judge of the bigness of any thing is by comparing it with something else, and finding out how many times the lesser is contained in the larger body. The simplest and most practicable methods of doing this in microscopical objects are the following:

1. Mr Leeuwenhoek's method of computing the size of salts in fluids, of the animalcules in feminine majulino, in pepper-water, &c. was by comparing them with a grain of sand. By this, however, we must understand the coarse sea-sand, usually called scouring-sand, which is equal in bigness to several grains of writing sand. But to make our calculations still more certain, we must suppose them to be of such a size, that 100 of them placed in a row shall extend an inch in length. Mr Leeuwenhoek then made his calculations in the following manner.

He viewed thro' his microscope a single grain of sand, which we will suppose to be magnified as the round figure ABCD. Then, observing an animalcule swimming or running across it, (which suppose to be of the size 1,) considering and measuring this by his eye, he concludes, that the diameter of this animalcule is only \( \frac{1}{12} \) of the diameter of the grain of sand; consequently, according to the common rules, the superficies of the grain of sand is 144 times, and the whole contents 1728 times, larger than the animalcule.

Suppose again, that he sees among these another and smaller species of animalcules; one of which, 2, he likewise measures by his eye, and computes its diameter to be four times less than the former: then, according to the foregoing rules, the surface of this second animalcule will be 16, and the whole bulk 64, times less than the animalcule 1.

If farther, upon a nicer view, he discovers a third kind of animalcule, 3, so exceedingly minute, that examining it in the former manner, he concludes the diameter to be 10 times smaller than the second sort; it will then follow, that 1000 of them are only equal in bigness to one of that sort. Hence, of the first sort, 1728 would be contained in a grain of sand; of the second, 110,592; and of the third, 110,592,000.

In this manner may the comparative size of small objects be judged of with tolerable certainty: particularly in the solar microscope; since the image of the object and of the grain of sand, or whatever else is thought proper to compare with it, may be really measured by a ruler or a pair of compasses, and the difference of their diameters most exactly found.

2. Mr Hooke describes his method in the following words. "Having rectified the microscope to see the desired object thro' it very distinctly; at the same time that I look upon the object thro' the glass with one eye, I look upon other objects at the same distance with my other bare eye: by which means I am able, by the help of a ruler divided into inches and small parts, and laid on the pedestal of the microscope, to cast as it were the magnified appearance of the object upon the ruler, and thereby exactly to measure the diameter it appears of thro' the glass; which being compared with the diameter it appears of to the naked eye, will easily afford the quantity of its being magnified." This method is recommended by Mr Baker as very good for multitude of objects; and he declares from his own experience, that a little practice will render it exceedingly easy and pleasant.

3. Another very curious method for this purpose is described by Dr Jurin in his Physico-Mathematical Dissertations. Wind a piece of the finest silver-wire you can get a great many times about a pin, or some other such slender body, so closely as to leave no interval between the wire-threads; to be certain of which, they must be carefully examined with a glass. Then, with a small pair of compasses, measure what length of pin the wire covers; and applying the compasses with that measure to a diagonal scale of inches, you will find how much it is; after which, by counting the number of wire-rounds contained in that length, you will easily discover the real thickness of the single wire. This being known, cut it into very small pieces; and, when you examine the object, if it be opaque, strew some of these wires upon it; if transparent, under it; and by your eye compare the parts of the object with the thickness of such bits of wire as lie fairest to the view. By this method, Dr Jurin observed, that four globules of human blood would generally cover the breadth of a wire which he had found to be \( \frac{1}{48} \)th part of an inch; and consequently that the diameter of a single globule was \( \frac{1}{35} \)th of an inch; which was also confirmed by Leeuwenhoek, from observations made on the blood with a piece of the same wire.

4. Mr Martin in his Optics gives another method sufficiently easy. On a circular piece of glass, let a number of parallel lines be carefully drawn, with the fine point of a diamond, at the distance of \( \frac{1}{36} \)th of an inch from each other. If this be placed in the focus of the eye-glass of a microscope, the image of the object will be seen upon these lines, and the parts thereof may be compared with the intervals whereby its true magnitude or dimensions may be very nearly known; for the intervals of these lines, tho' scarce discernible to the naked eye, appear very large thro' the microscope. A contrivance of this kind may also be invented for such microscopes as glasses cannot be applied to in the above manner, by placing it under or behind the object, which will answer the same purpose. Here it will be easy to find what proportion an object, or any part thereof, bears to an interval between two lines, and then determine it in parts of an inch: for if the width of an object appears just one interval, we shall know it to be just one fourtieth part of an inch; if half an interval, the 80th; if a quarter of an interval, the 160th; if one fifth, only the 200th part of an inch.

5. Dr Smith has an invention similar to this for taking exact draughts of objects viewed in double microscopes: for he advises to get a lattice made with small silver wires, or small squares drawn upon a plane glass by the strokes of a diamond, and to put into the place of the image formed by the object-glass. Then, by transferring the parts of the object seen in the squares of the glass or lattice upon similar corresponding squares drawn upon paper, the picture thereof may be exactly taken. A micrometer may also be applied to microscopes of the same form with those applied to telescopes; for by opening the hairs of the micrometer till they exactly correspond to a certain length, suppose \( \frac{1}{36} \)th of an inch, and by observing the number of revolutions in this opening, the diameter of any other object, answering to a known number of revolutions, may be found by the golden rule.

§ 13. Of the Field of View in Microscopes.

This is always in proportion to the diameter of the lens made use of, and its power of magnifying, by which it may be determined: since, if the lens is extremely small, it magnifies a great deal, and consequently a very minute portion of an object only can be distinguished thro' it; for which reason the greatest magnifiers never should be employed but for the most minute objects. This consideration will direct to the use of such magnifiers as are most proper to be employed, which is of the utmost consequence in microscopic observations. On this subject Mr Baker gives the following short rule, viz. that the field of view differs not greatly from the size of the lens; and that the whole of any object much beyond that size, cannot be conveniently viewed thro' it. There is some difference, as to the visible area of an object, as seen thro' single or double microscopes; for the double shew a larger portion of it than the single, tho' magnified as much.

§ 14. Of Microscopic Objects, and the Method of preparing them for being examined.

Mr Hooke gives a general account of microscopic objects under the following denominations, viz. "exceeding small bodies, exceeding small pores, and exceeding small motions." The first must either be the parts of larger bodies; or things, the whole of which is exceedingly minute, such as small seeds, insects, fangs, salts, &c. The second are the interstices between the solid parts of bodies, as in stones, minerals, shells, &c. or the mouths of minute vessels in vegetables, the pores in the skins, bones, &c. of animals.—Exceeding small motions are the movements of the several parts or members of minute animals, or the motion of the fluids contained either in animal or vegetable bodies.

Many, as Mr Baker observes, even of those who have purchased microscopes, are so little acquainted with their general and extensive usefulness, and so much at a loss for objects to examine by them, that, after diverting themselves and their friends some few times with what they find in the slides bought with them, or two or three more common things, the microscopes are laid aside as of little farther value; and a supposition that this must be the case, prevents many others from buying them: whereas, among all the inventions that ever appeared in the world, none perhaps can be found so constantly capable of entertaining, improving, and satisfying the mind of man.

An examination of objects, in order to discover truth, requires a great deal of attention, care, and patience, together with some considerable skill and dexterity, (to be acquired by practice chiefly), in the preparing, managing, and applying them to the microscope. When any object comes to be examined, the size, contexture, and nature of it, should be duly considered, in order to apply it to such glasses and in such a manner as may shew it best. The first step towards this should constantly be, to view it thro' a magnifier that can take in the whole at once: for, by observing how the parts lie as to one another, we shall find it much easier to examine and judge of them separately if there be occasion. After having made ourselves acquainted with the form of the whole, we may divide it as we please; and the smaller the parts into which it is divided, the greater must be the magnifiers with which we view them.

The transparency or opacity of an object must also be regarded, and the glasses made use of must be suited to it accordingly: for a transparent object will bear a much greater magnifier than one that is opaque; since the nearness required in a large magnifier unavoidably darkens an opaque object, and prevents its being seen, unless by the microscope contrived on purpose for such objects. Most objects, however, become transparent by being divided into extremely thin or minute parts. Contrivance therefore is requisite to reduce them into such thinness or smallness as may render them most fit for examination.

The nature of the object, whether it be alive or dead, a solid or a fluid, an animal, a vegetable, or a mineral substance, must likewise be considered, and all the circumstances attended to, that we may apply it in the most convenient manner. If it be a living animal, mal care must be taken to squeeze, hurt, or discompose it as little as possible, that its right form, posture, and temper, may be discovered. If a fluid, and too thick, it must be thinned with water; if too thin, we must let some of its watery parts evaporate. Some substances are fittest for observation when dry, others again when moistened; some when fresh, and some after being kept a while.

Light is a thing next to be taken care of; for on this the truth of all our examination depends, and a very little experience will show how differently objects appear in one position and kind of it, from what they do in another. So that we should turn them every way, and view them in every degree of light, from brightness even to obscurity; and in all positions to each degree; till we are certain of their true form, and that we are not deceived. For, as Mr Hooke says, it is very difficult, in many objects, to distinguish between a promiscuity and a depression, between a shadow and a black stain; and, in colour, between a reflection and a whiteness. The eye of a fly, for instance, in one kind of light, appears like a lattice drilled through with abundance of holes; in the sunshine, like a surface covered with golden nails; in one position like a surface covered with pyramids, in another with cones, and in other positions of quite other shapes.

The degree of light must be duly suited to the object: which if dark, will be best seen in a full and strong light; but, if very transparent, the light should be proportionally weak; for which reason there is a contrivance both in the single and double microscope to cut off abundance of its rays when such transparent objects are viewed by the greatest magnifiers.

The light of a candle, for many objects, and especially such as are exceedingly minute and transparent, is preferable to day-light. For others, daylight is best; that is, the light of a bright cloud. As for sunshine, it is reflected from objects with so much glare, and exhibits such gaudy colours, that nothing can be determined by it with certainty; and therefore it is to be accounted the worst light that can be had.

This opinion of sunshine, however, must not be extended to the solar microscope, which cannot be used to advantage without its brightest light: for in that way we see not the object itself, whereon the sunshine is cast, but only the image or shadow of it exhibited upon a screen; and therefore no confusion can arise from the glaring reflection of the sun-beams from the object to the eye, which is the case in other microscopes: but then, in this way, we must rest contented with viewing the true form and shape of an object without expecting to find its natural colour, since no shadow can possibly wear the colours of the body it represents.

Most objects require some management in order to bring them properly before the glasses. If they are flat and transparent, and such as will not be injured by pressure, the best method is to inclose them in sliders, between two Mulcovy talcs or ifinglafs. This way, the feathers of butterflies, the scales of fishes, the farinae of flowers, &c. the several parts and even whole bodies of minute insects, and a thousand other things, may very conveniently be preserved. Every curious observer, therefore, will have them always ready to receive any accidental object, and secure it for future examination: and a dozen or two of these sliders properly furnished are a fine natural history.

In making a collection of objects, the sliders should not be filled promiscuously, but care taken to sort the objects according to their size and transparency; in such a manner that none may be put together in the same slider but what may be properly examined by the same magnifier; and then the slider should be marked with the number of the magnifier its objects are fitted for: that is, the most transparent, or minutest objects of all, which require the first magnifier to view them by, should be placed in a slider or sliders marked with number I, those of the next degree in sliders marked with number II, and so of the rest. This method will save abundance of time and trouble in shifting the magnifiers, which, without such sorting, must perhaps be done two or three times, in overlooking a single slider. The numbers marked out upon the sliders will likewise prevent our being at any loss what glass to apply to each. In placing your objects in sliders, a convex glass of about an inch focus, to hold in the hand, and thereby adjust them properly between the tales, before you fasten them down with the brass rings, will be found very convenient.

Small living objects, such as lice, fleas, gnats, small bugs, minute spiders, mites, &c. may be placed between these tales, without killing or hurting them, if care be taken not to press down the brass rings that keep in the tales, and will remain alive even for weeks in this manner. But if they are larger than to be treated thus, either put them in a slider with concave glasses intended for that use, or in the cell described above, or else examine them stuck on the pin or held between the piers; either of which ways they may be viewed at pleasure.

If fluids come under examination, to discover the animalcules that may be in them, take up a small drop with your pen or hair-pencil, and place it on a single ifinglaf, which you should have in a slider ready, or else in one of the little concave glasses, and so apply it. But in case, upon viewing it, you find, as often happens, the animalcules swarming together, and so exceedingly numerous, that running continually over one another, their kinds and real form cannot be known; some part of the drop must be taken off the glass, and then a little fair water added to the rest, will make them separate, and shew them distinct and well. And this mixture of water is particularly necessary in viewing the femen masculinum of all creatures; for the animalcules therein contained are inconceivably minute, and yet crowded together in such infinite numbers, that, unless it be diluted a great deal, they cannot be sufficiently separated to distinguish their true shape.

But if we view a fluid, to find what salts it may have in it, a method quite contrary to the foregoing must be employed: for then the fluid must be suffered to evaporate, that the salts, being left behind upon the glass, may the more easily be examined.

Another, and indeed the most curious way of examining amining fluids, is by applying them to the microscope in exceedingly small capillary tubes made of the thinnest glass possible. This was Mr Leeuwenhoek's method of discovering the shapes of salts floating in vinegar, wine, and several other liquors; and such tubes should be always ready to use as occasion requires.

For the circulation of the blood, frogs, newts, or fishes, are commonly made use of; and there are glass tubes in the single microscope, and a fish-pan as well as tubes in the double one, on purpose to confine these creatures, and bring the proper parts of them to view: these parts, in newts and fishes, are the tails, and in frogs the fine filmy membrane between the toes of the hinder legs. Though, if we can contrive to fallen down the creature, and bring our object to the magnifier, the circulation cannot possibly be seen so plainly anywhere as in the mesentery, or thin transparent membrane which joins the guts together; and this part, by pulling out the gut a little, may easily be adjusted to the magnifier.

To dissect minute insects, as fleas, lice, gnats, mites, &c. and view their internal parts, requires a great deal of patience and dexterity; yet this may be done in a very satisfactory manner, by means of a fine lancet and needle, if they are placed in a drop of water: for their parts will then be separated with ease, and lie fair before the microscope, so that the stomach and other bowels may be plainly distinguished and examined.

We should always have ready for this purpose, little slips of glass, about the size of a slider, to place objects on occasionally; some of which slips should be made of green, blue, and other coloured glasses, many objects being much more distinguishable when placed on one colour than on another. We should likewise be provided with glass tubes of all sizes, from the finest capillaries that can be blown, to a bore of half an inch diameter.

There is, perhaps, no better way of preserving transparent objects in general, than by placing them between clear sliding glasses in sliders: but opaque bodies, such as lands, seeds, woods, &c. require different management, and a collection of them should be prepared in the following manner.

Cut cards into small slips, about half an inch in length, and one tenth of an inch in breadth: wet them half their length with a strong but very transparent gum-water, and with that stick on your object. As the spots of cards are red and black, by making your slips of such spots, you will obtain a contrast to objects of almost any colour; and by fixing black things on the white, white on the black, blue or green on the red or white, and all other coloured objects on slips most contrary to themselves, they will be shewn to the best advantage. These slips are intended chiefly for the microscope for opaque objects, to be applied between the nippers: but they will also be proper for any other microscope that can show opaque bodies. A little square box should be contrived to keep these slips in, with a number of very shallow holes in it just big enough to hold them. If such holes were cut through that pasteboard of which the covers of books are made, exactly fitted to the box, and a paper pasted on one side of it to serve for a bottom to it, three or four such pasteboards stored with objects might lie upon one another in same box, and contain 100 or more slips; with objects fastened on them, always ready for examination. It will not be found amiss to provide some slips larger than others, for the reception of different sized objects. But this will, perhaps, be better understood by an inspection of fig. 3. The box should likewise be furnished with a pair of pliers, to take up and adjust the slips, and therefore a convenient place is contrived therein to hold them, as is shown in the figure.

There is no advantage in examining any object with a greater magnifier than what shows the same distinctly; and therefore, if you can see it well with the third or fourth glass, never use the first or second; for the less a glass magnifies, the better light you will have, the easier you can manage the object, and the clearer it will appear. It is much to be doubted, whether the true colours of objects can be judged of when seen through the greatest magnifiers: for as the pores or interstices of an object must be enlarged according to the magnifying power of the glass made use of, and the component particles of matter must by the same means appear separated many thousands of times farther than they do to the naked eye, their reflections of the rays of light will probably be different, and exhibit different colours. And indeed the variety of colouring which some objects appear dressed in, may serve as a proof of this.

The motions of living creatures themselves, or of the fluids contained within them, as seen through the microscope, are likewise not to be determined without due consideration: for, as the moving body, and the space wherein it moves, are magnified, the motion must probably be so too; and therefore that rapidity wherewith the blood seems to pass along through the vessels of small animals must be judged of accordingly.

§ 15. Dollond's Achromatic Telescope.

Mr Dollond's telescopes are of two kinds. 1. Those in which only the eye-piece slides, so as to be drawn out as far as is necessary for procuring distinct vision. Of this form are all the larger instruments; which are therefore generally fixed upon a stand, for viewing objects with greater steadiness.—2. Those which are composed of several sliding tubes, for the convenience of being put into the pocket.

The usual method of making the sliding tubes of telescopes has been with paper covered with vellum; but as such tubes have been found liable to several inconveniences from being affected by the moisture of the air, they are now contrived to be made exceedingly thin of brass, and the outside of mahogany.

The sliding tubes are all made to stop, when drawn out to the proper length; so that, by applying one hand to the outside tube A, fig. 4, and the other hand to the end of the smallest tube B, the telescope may be, at one pull, drawn out to its whole length, as is represented by fig. 5.; then any of the tubes may be slipped in a little while you look through, and the object rendered distinct to any sight.

To make the tubes slide properly, they all pass through short springs or tubes, which are screwed in at a, b; and c, fig. 5. These springs may be unscrewed from the ends of the sliding tubes by means of the milled edges which project above the tubes, and the tubes taken from one another when required. There are four convex eye-glasses to these telescopes, whose surfaces and focal-lengths are so proportioned as to render the field of view very large. These eye-glasses are all contained in the smallest sliding tube; three of them may be seen by unscrewing the tube at eee; and the fourth, which is at the end of the tube, may be come at by unscrewing the spring at c.

These telescopes are of three different lengths and sizes, usually called 1 foot, 2 feet, and 3 feet.

| Length when in Use | Length when shut up | Aperture of the Achromatic Object-glas. | Weight | |--------------------|---------------------|--------------------------------------|--------| | 14 Inches. | 5 Inches. | 1,1 Inches. | 6 Ounces. | | 28 ditto. | 9 ditto. | 1,6 ditto. | 16 ditto. | | 40 ditto. | 10 ditto. | 2,0 ditto. | 30 ditto. |

The best achromatic telescopes which Mr Dollond has yet made, are those with a triple object-glass of about 45 inches focal distance, with an aperture of 3½ inches. Some of these magnify the diameters of objects 150 times, with great distinctness, and light sufficient for most astronomical purposes. When fitted (for terrestrial objects) with an eye-piece magnifying about 70 or 80 times, they give most agreeable vision.

The object-glass of one of these telescopes was found to have the following radii (in inches) of curvature for its different surfaces, beginning with that next the object: $26\frac{1}{2}$, $37\frac{1}{2}$, $19\frac{1}{2}$, $26\frac{1}{2}$, $26\frac{1}{2}$. But it does not appear that Mr Dollond and the best artists abide by a fixed rule in their constructions; for telescopes of the same length and magnifying power, and made by the same artist, have different constructions of the object-glass. This may be expected, when we consider the variable nature of the flint-glass. It is probable, that these very fine object-glasses have been produced by trials pro re nata of different curvatures.

Till some method can be discovered of making flint glass free from veins, which differ in their refracting power, it is not probable that larger telescopes than those now mentioned will be produced.

§ 16. Of the Newtonian Reflecting Telescope.

Fig. 1. shews one of those telescopes made by the Hon. Samuel Molyneux, and presented by him to king John V. of Portugal. ABC represents a triangular board or table supported by the globe D, and by the annexed carvings and marks, and which serves for the basis or pedestal of the instrument. Upon occasion this board may be taken off by unscrewing three iron screws, the heads of which lie near the three volutes at the three corners. At E is represented a small key or handle which turns some wheel-work, concealed under the board of the table, and which serves to give an horizontal circular motion to the pillar F placed in the middle, and to the superincumbent tube HIKL. If this should ever be out of order by taking off the upper board, it may be rectified. At G is represented another handle which gives the tube its perpendicular motion; so that while the observer sits with his right side applied to the side of the table AC at the end C, by turning the two handles E and G, he can give the tube any required elevation or azimuth, and thereby follow the motion of the heavenly bodies very commodiously.

The telescope itself consists of two metallic specula optical and an eye-glass, which are to be duly placed in the instrument tube HIKL left open at the end HL. The large concave spherical speculum k is to be placed within the tube at IK; in which are fixed three stops, or bits of wood, against which the polished surface of the speculum being applied, the axis of reflection will fall exactly in the axis of the tube. In the brass plate which closes this end of the tube, there are three screws intended for holding the metal in this situation; but many cautions are requisite with regard to this metal and the placing of it. In the first place, it is never to be touched, but by screwing into the backside of it a handle l, which fits the hole therein. In the next place, great care must be taken not to breathe on it, or to expose it to damp air. If anything of that kind happens, it must be wiped thoroughly dry with a linen cloth before a fire; and it may be sometimes in like manner cleaned with a rag wetted in spirit of wine; provided the spirit be not left to evaporate, for that would leave an humid sediment which would hurt the polish. In the third place, unless when in use, it should be constantly kept with its face downwards on a piece of plane glass made on purpose.

The speculum is a portion of a concave sphere whose diameter is about eight feet eight inches, and which of consequence collects the rays into a focus about 26 inches distant from its surface. The laws of reflection are such, that any error in the figure of this speculum will produce about five times as great an irregularity in the picture formed in its focus, as the like irregularity would cause in a common refracting telescope. It hath been found by experiment, that an error of less than 1000th part of an inch is capable of vitiating its figure; so that great care must be taken in placing the metal in the tube for use, against the three stops above-mentioned; and that the three screws at IK be gently screwed, only just sufficient to bear the metal truly against the stops; for the smallest excess of stress in the screws against the back of the metal may distort and very much damage its figure. There is also a piece of wood m, having a round hole p in it, and carrying a small brass arm n, which holds the other smaller speculum o, which is plane. This speculum must be always preserved from the air when out of use. When the telescope is to be used, the cover of the small speculum must first be taken off; then place it in the tube at the hole M, which exactly fits the above-said square piece. Press it in pretty tight and true; which, if duly performed, the centre of the small speculum will be placed in the axis of the large concave one, and will reflect the parallel rays which enter at the open end HL, to the round hole p in the said square piece, in which hole one of the two eye-glasses in its cell q is to be placed; and then the instrument is prepared for use. The observer is therefore to place himself at the side of the tube, and to look in at M, where he will see the images of the objects which lie at his left-hand. In taking out or putting in the little speculum at o, great care must be used to avoid shaking or bending the arm; for the smallest accident of that kind will certainly disorder its situation. There are three screws at the back, the middlemost of which fixes it to the arm mn; the other two only press upon the back, and serve to adjust... Part III.

Optical just its situation to an exact angle of 45° with the axis of the great speculum. There are two eye-glasses, whereof the one that hath the largest aperture being made use of, the instrument will magnify as much as a common refracting telescope of about 20 or 22 feet long; and with the eye-glass that hath the smallest apertures, it will magnify as much and as distinctly as one of 35 or 40 feet.

At P stands a round button of ivory; and at Q is represented a small pin of ivory, which may be seen with a small white thread fixed to it, at the end of the tube H. This thread at the other end is fixed in the inside of the tube; and towards the middle of it, it is wound once round the inward end of the ivory button P. From this disposition, by turning round the ivory button P, the whole slider of black ebony wood NO, with the small speculum and the eye-glass applied at M, may be made to approach to or recede from the large speculum at the other end IK; and by this means its true distance, and the distinct appearance of the object, must be found, for various distances of the same object, and for the various eyes of different observers; which variety in different persons, from the great magnifying power of the eye-glass in this instrument, will be considerably more sensible than in a refracting telescope. But the true distance of the specula will immediately be found in all cases, by turning this ivory pin P backwards and forwards very slowly and gently; and, for celestial objects, the true distance being once found for the observer's eye, a small mark may be made across the slider, and upon the edge of the tube, to bring it speedily, and without any difficulty, to its proper place at another time. By the little ivory pin at Q, the string may be tightened or relaxed to make the slider NO move most easily as occasion requires. Either of the eye-glasses being applied in the cylindrical hole p, in the square piece m p, may also be made to approach to or recede from the focus, by turning round the small tube g in which they are inserted, the outside whereof is wrought into a fine screw for that purpose. Distinct vision may also by that means be obtained for different eyes, without moving the whole slider NO.

RS represents a small refracting telescope, whose axis is parallel to the axis of the reflector. In its focus there are placed two cross-hairs, and its only use is to find out any object more readily by the reflector. The eye being applied at S, turn the two handles at E and G, till the point of the object to be viewed in the reflector falls exactly on the cross-hairs; then the eye applied at M to the reflector, will see the same object distinctly; with this caution, that as the whole instrument with its basis can easily be moved, the most convenient situation for the observer will be to keep the tube HH nearly at right angles to the side AC, and to fit with the side AC flat against his right side near C, as hath been already mentioned. And in finding the object at first with the small refracting telescope, it is most convenient to stand at the corner of the table C. The handle G, may be inserted at either side of the pillar F, as convenience shall require. At the small pillars TV, which support and hold the small telescope RS, there are some small screws near T, which being relaxed, the direction of the tube RS way be altered horizontally by pushing the tube with the hand sidewise, either way, as occasion requires, and then tightening the screws again. And at V there are screws and a springing piece of brass, which, being relaxed or tightened, will in like manner alter its elevation, so as to restore the parallelism of the tubes in case of any accident that may have disturbed them. In making observations, it will be found convenient not to touch the table, but only to move the handles, as the motions of the star or other celestial body directs; for in an instrument that magnifies so much, the least motion or trembling is magnified proportionally.

In this telescope Dr Smith takes notice of a remarkable deception; namely, that, when the reflector is compared with a refracting telescope of equal magnifying power, the observer always imagines that the latter has the advantage. For this he does not pretend to account, but looks upon it to be an optical deception common to all mankind.

§ 17. The Gregorian Reflecting Telescope.

This is represented, fig. 16. O is a three-footed plate pedestal of wood or metal, in the middle of which CCXXIX. is fastened, by means of the large screw R, the stand AB. On the top of this stand is fastened the plate CD, having in it a socket to receive the brass ball D. This plate is composed of two parts; and by means of the screws L, M, the socket is tightened or loosened on the ball, so that it can either allow it a free motion, or keep it firm in its place. This ball is fully shewn at fig. 17.; and, with its stalk F, is soldered into the piece of brass FG, which is again fastened on the body of the telescope by means of the screws HI. The whole length of the telescope is represented by ab; the eye-piece, or the part which contains the eye-glasses, by ar; the other part, containing the specula, is represented by cb. Fig. 4. and 5. shew the proportional size of the two specula to one another. The larger is placed at c, in the same manner and with the same precautions as have been already mentioned with regard to the Newtonian telescope; the other is placed on a short arm within the tube, in such a manner, that it occupies exactly the middle of it: and by means of a rod r p q m o, having the upper end of it turned into a screw, the little speculum can be removed from the other, or brought nearer to it, as occasion requires. In the eye-piece, ar, are two eye-glasses, which receive the light reflected from the little speculum; and the eye being applied at a, the observer sees those objects which are placed directly before the mouth of the tube.

The following are the proportions of an excellent Gregorian telescope, made by Mr James Short of Edinburgh, which may serve as a model for calculating others of any given length.

| Inches | |--------| | Focal distance of the larger speculum | 9.6 | | Its breadth or aperture | 2.3 | | Focal distance of the lesser speculum | 1.5 | | Its breadth | 0.6 | | Breadth of the hole in the larger speculum | 0.5 | | Distance between the lesser speculum and the next eye-glass | 14.2 | | Distance between the two eye-glasses | 2.4 | | Focal distance of the eye-glass next the metals | 3.8 | | Focal distance of the eye-glass next the eye | 1.1 |

This telescope was found by experiment to magnify 60 times in diameter, and to take in an angle of 19° to the naked eye; and of consequence the magnified angle was equivalent to 19°.

For finding the magnifying power of a telescope by experiment, Dr Smith tells us, that the following method was pursued by Mr Hauksbee, Mr Folkes, and Dr Jurin. Having fixed a paper circle of one inch diameter upon a wall, at the distance of 2674 inches from the eye-glass of the telescope, they viewed it in the telescope with one eye, while, with the other eye naked, they viewed two parallel lines drawn upon paper, 12 inches asunder, moving them gradually to and fro, till they appeared to touch two opposite points of the circle seen in the telescope; and then the perpendicular distance of the lines was found to be 132 inches. In this position of the objects, the angle at the eye made by the rays that came from the extremities of the diameter of the one-inch circle, was equal to the angle subtended at the other eye by the 12-inch interval of the parallel lines; and therefore the ratio of this angle to that which the said circle would subtend at the naked eye, viewing it at the said distance of 2674 inches, is the magnifying power of the telescope; and is compounded of the direct ratio of the subtenses of these angles, and the inverse ratio of the distances of the subtenses from the eye; that is, of 12 to 1, and of 2674 to 142; which make the ratio of 226 to 1, very nearly.

Supposing a larger paper circle had been placed at so great a distance, that its picture might have been formed by the speculum in its principal focus; the telescope would have magnified it more than our one-inch circle, in the ratio of the distance of this latter circle from the principal focus, to its distance from the centre of the sphere of the speculum; because the diameter of the picture of the remoter circle would have been greater, in this ratio, than that of the one-inch circle, supposing these circles to subtend the same angle at the centre of the speculum. But this ratio, in the present experiment, being only 2674 to 2671, gives only an inconsiderable increase to the magnifying power already determined.

Thus we have an easy and accurate method of examining the goodness of a telescope of any kind. First, by giving it the least eye-glass that will shew the new moon, or rather Jupiter and Saturn, with sufficient light and distinctness when the air is quiet and pure; and then by finding how much it magnifies by the method abovementioned. But if several telescopes of the same kind have nearly the same length, those are the best in their kind with which you can read a print at the greatest distance. That the reader may have some notion of the powers of telescopes in this way, we shall subjoin a short account of the effects of some reflectors made by Mr James Short of Edinburgh, as related by Mr MacLaurin.

With a reflecting telescope, of which the speculum was quick-silvered glass, and focal distance 15 inches, the Philosophical Transactions could easily be read at the distance of 230 feet; by another of the same dimensions, the Transactions could be read at 280 feet distance. By a telescope of the same kind, whose focal distance was nine inches, Mr MacLaurin read in the Transactions at the distance of 138 feet; and another much smaller print at the distance of 125 feet.

It is not mentioned whether these telescopes were of the Newtonian or Gregorian form; though it is most probable that they were of the former kind.

As the light produced by these glass speculums was very faint, Mr Short next applied himself to the construction of metallic ones, and the effects of these were vastly greater; but as they were of the Gregorian form, it is doubtful whether we are to ascribe their superiority entirely to the use of metallic speculums, or to the more advantageous construction of them. These telescopes had focal distances of two inches and 6 tenths; of four inches; six, nine, and fifteen inches. By those of four inches, the satellites of Jupiter were seen very distinctly; and he could read the Philosophical Transactions at above 125 feet distance. By those of six inches focus, he read at 160 feet distance; by those of nine inches, he read at 220 feet distance; and by those of fifteen inches, he was able to read the Transactions at 500 feet distance. With these last he also several times saw the five satellites of Saturn. The effects of these 15-inch telescopes of Mr Short's therefore were equal to those of the best 17-feet refractors ever known; for it was thought wonderful that Cassini should observe all the satellites of Saturn with a 17-feet refracting telescope.

§ 18. The Solar Telescope.

This instrument is of the nature of the camera obscura, and shews the image of the sun in a darkened room, as that of an insect is shewn by the solar microscope. AB, fig. 6, represents a part of the window-plate shutter of a darkened room; CD the frame, which, by CCXXII. means of a screw, contains the scioptic ball EF, in a hole of the said shutter adapted to its size. This ball is perforated with a hole abc. Through the middle, on the side bc, is forewed into the said hole a piece of wood, and in that is screwed the end of a common refracting telescope GHIK, with its object-glass GH, and one eye-glass at IK; and the tube is drawn out to such a length, that the focus of each glass may fall near the same point. This being done, the telescope and ball are moved about in such a manner, as to receive the sunbeams perpendicularly on the lens GH, through the cylindric hole of the ball; by this glass they will be collected all in one circular spot m, which is the image of the sun. The lens IK is to be moved nearer to, or farther from, the said image m, as the distance at which the secondary image of the sun is to be formed requires, which is done by sliding the tube IKLM backward and forward in the tube LMNO. Then of the first image of the sun m, will be formed another, PQ, very large, luminous, and distinct.

In this manner the sun's face may be viewed at any time without offence to weak eyes; and whatever changes happen herein may be duly observed. The spots are here all of them conspicuous, and easy to be observed under all their circumstances of beginning to appear, increase, division of one into many, &c. By the solar telescope also we view an eclipse of the sun to the best advantage; as having it by this means in our power to represent the sun's disk as large as we please, and consequently to render the eclipse proportionally conspicuous. Also the circle of the sun's disk may be so divided by lines and circles drawn thereon, that the quantity of eclipse, estimated in digits, may this way be most exactly determined; also the moments of the beginning, middle, and end of it, for determining the longitude of the place. The transits of Mercury and Venus over the sun are thus also beautifully represented, and the planets appear very round, black, and well-defined. Their comparative diameters with that of the sun may also be thus observed, the times of ingress and egress, &c. better than by any other method hitherto invented.

By the solar telescope the clouds are most beautifully represented passing before the sun, according to their various degrees of rarity or density; but these, Mr Martin says, are best observed by the camera obscura. He takes notice of an unusual phenomenon which he once observed in looking at the image of the sun by this instrument. The window looked towards the west, and the spire of Chichester cathedral was directly before it at the distance of about 50 or 60 yards. The images of the sun and spire were very large, being made by a lens of 12 feet focal distance, and it was very agreeable to observe the manner in which the sun was for some time eclipsed by the spire. Once, as he observed the occultation of the sun behind the spire, just as the disk disappeared, he saw several small, bright, round balls, running towards the sun from the dark part of the room, even to the distance of 20 inches. Their motion was a little irregular, but rectilinear, and seemed accelerated as they approached the sun. These luminous globules also appeared on the other side of the spire, and preceded the sun, running out into the dark room, sometimes more, sometimes less together, in the same manner as they followed the sun at its occultation. "They appeared," says he, "to be in general about \( \frac{1}{8} \) of an inch in diameter; and therefore must have been very large luminous globes in some part of the heavens, whose light was extinguished by that of the sun, so that they appeared not in open day-light; but whether of the meteor kind, or what sort of bodies they might be, I could not conjecture."

§ 19. The Heliostata.

The use of this machine is to take off the inconveniences which arise from the motion of the earth, in making experiments on the solar light. By this motion it happens, that the image of the sun formed by the solar telescope can never be steady, but continually shifts its place on the screen upon which it is thrown; and the like may be said of the solar microscope. Any contrivance therefore by which this apparent motion can be prevented, and the light of the sun fixed upon any particular spot, or in one certain direction, must certainly be of the highest utility. The heliostata answers the purpose completely; and is an invention of Dr 's Gravefande, who gives the following description of it.

"This machine consists of two principal parts, each of which consists of many smaller parts. The first is a plane metallic speculum, supported by a stand; the other is a clock, which directs the speculum.

"We make use of a metallic speculum, because there is a double reflection in a glass one. The magnitude and figure of it are not material; mine is rectangular, four inches long and three broad.

"This is put into a wooden frame, which is surrounded with wooden rulers, cut in, whereby the speculum is retained.

"To sustain this, without hindering its motion, to Plate CCXXIV. the said wooden frame, behind, is applied the brass plate \( a \), whose ends, being bent, are fastened to the wood sidewise.

"This speculum \( S \) is suspended by the handle \( AA \), small screws being put through holes in the end of it, which go into the ends \( a, a \), of the said frame, and whose parts, which are in the holes of the handle, are cylindrical, so that the speculum turns freely upon its axis, which, if it were made sensible, would pass along the surface of the speculum.

"The handle is joined to the cylinder \( C \), whose axis, if it were continued, would concur with the middle point of the said axis of the revolution of the speculum.

"To the same point answers the tail \( DE \), which is joined perpendicularly to the hind part of the speculum. This tail is cylindrical; and is made of a brass wire, which is straight, firm, and whose diameter is about a sixth part of an inch.

"The cylinder \( C \) is put upon the wooden stand \( P \), whose upper part is represented by itself: whilst this is done, the iron cylinder \( c \), whose surface is smooth, goes into a cavity in the cylinder \( C \), which is of copper; by which means this turns freely about its axis, so that, by the motion of the tail \( DE \), the position of the speculum is very easily altered as you please.

"This is raised and depressed, by means of the three brass screws \( B, B, B \), which are turned with a key, and go through a plate of the same metal, which is applied to the bottom of the stand for that purpose; and which stands out in three places, to receive the screws.

"If the speculum is to be raised higher, as may easily be done, we put the speculum together with its stand, upon a small board, which has low feet, and is made for that purpose.

"The other part of the machine is a clock, as has been said above. This is represented at \( H \); the index performs its revolution in 24 hours.

"The plane of the clock is inclined to the horizon, according to the inclination of the equator in the place where the machine is made use of; that is, in this our city of Leyden is \( 37^\circ 49' \).

"But this machine may be made use of in other places, whose latitudes differ one or two degrees from this place, as will appear.

"The clock is sustained by the copper pillar \( FG \); this consists of two parts, which are joined by the screws \( d, d \), between which, as in a sheath, is moved an iron plate, in the middle of which there is a slit, through which the said screws \( d, d \), pass. This plate is joined fast to the lower plate of the clock itself, which is raised and depressed by this method, and fastened by the screws \( d, d \). It may also be raised higher by the screws \( I, I, I \), which go thro' the thick copper plate \( LLM \), upon which the pillar \( FG \) stands.

The extreme parts of this plate \( L, L \), are terminated in such manner, that \( b c \) and \( c b \) make one right line, through which we suppose a vertical plane to pass: this Optical will be perpendicular to the horizontal lines, which may be drawn on the plane of the clock; such as are \( f g, h i \).

"The machine is so ordered, that the plane of the clock may have the inclination beforementioned, when the plane \( LLM \) is horizontal; in which situation it is easily placed by means of the screws \( I, I, I \), by help of the plumb-line \( Q \), whose point should answer to the point \( o \), which is marked upon the surface \( LLM \).

"If the machine were to be used in another place, whose latitude differed from that for which the machine was constructed, another point, as \( o \), would be marked, in which case the plane \( LLM \) would be inclined to the horizon.

"The axis of the wheel, which moves the index, is pretty thick, and is perforated cylindrically; but the cavity inclines a little to a conical figure, for towards the bottom it is somewhat narrower.

"The index itself is represented at ON. This is of brass, and its tail \( p q \) exactly fills the cavity mentioned last, into which it is thrust tight, that it may stick, and that the wheel may carry the index with it as it moves; whose situation may yet be altered, and set to any hour.

"This tail has also a cylindrical hole; and through this passes the small brass wire \( ld \), which remains in any situation, whilst it is raised or depressed.

"At the end \( O \) of the index there is a small cylinder \( n \), which is perforated cylindrically.

"The length of the index is measured in the line, perpendicular to \( ld \), drawn from the axis of the cylinder \( n \) to the axis of the wire \( ld \). In my machine this length is six inches.

"The iron tail \( t \) of the piece \( T \) goes into the cavity of the cylinder \( n \); this tail exactly fills the cavity, but yet moves freely in it.

"Between the legs of the piece \( T \), the small pipe \( R \) may be suspended at different heights, thro' which the tail \( DE \) of the speculum may be moved freely, which fills the pipe very exactly. This small pipe is suspended, as was said of the speculum. The small

| 21 Mar. | 1 Mar. | 21 Fe. | 11 Fe. | |---------|--------|--------|-------| | 0 | 8 | 17 | 32 |

| 21 Sept. | 11 Oct. | 21 Oc. | 1 No. | |----------|---------|--------|------|

"On the opposite side of the ruler, there is also drawn a small line, which accurately answers to \( v x \), whose divisions are contained in this second small table.

| 21 Mar. | 11 Ap. | 21 Ap. | 1 May | |---------|--------|--------|-------| | 0 | 11 | 22 | 36 |

| 21 Sep. | 1 Sep. | 21 Au. | 11 Au. | |---------|--------|--------|-------|

"These things being thus ordered; to fix the machine, it is put upon a plane that is horizontal, or nearly so.

"First we join the placer to the stand \( P \), which we raise as much as is necessary, that the ruler \( YZ \) being reduced to a just length, which we turn at pleasure, and incline in every respect, that is with respect to the place and direction, may agree to the sun's ray, which we undertake to fix.

"We so order the other part of the machine, that the lines \( b c, bc \), may agree to a meridian line which has been drawn on the plane; and it is so disposed by means of the screws \( I, I, I \), that the plumb-line \( Q \) may answer to the point \( o \).

Screws \( r, r \), pass through the said legs, and the ends of them go into the parts \( m, m \), of the pipe, and remain there; then the pipe turns freely about the axis which passes through \( mm \); for the parts of the small screws are cylindrical, which answers to the holes in the legs of the piece \( T \).

"When the machine is to be fixed, we make use of another machine, which we shall call a placer.

"The cylinder \( C \), together with its speculum, is removed from the stand \( P \), upon which is placed the brass pillar \( VX \). This sticks tighter to \( e \) than the cylinder \( C \), that the pillar may keep its place, whilst the machine is settled.

"Upon the head \( X \) the ruler \( YZ \) moves round a centre, so that it may be inclined to the horizon as you please, and keep its position. The length of the arm \( YX \) is determined at pleasure. The arm \( XZ \) is of a peculiar construction, and a certain length.

"To the said ruler, which is not extended beyond \( y \), there are applied two others, as \( xZ \), between which the first is inclosed; these are joined at \( Z \), and also cohere by means of the screws \( z, z \), which pass thro' a slit in the first ruler. On this ruler is marked the small line \( vx \), whose length is equal to nine hundredth-parts of the length of the index, and which is divided in the manner which will be mentioned presently.

"The arm \( XZ \) is equal to the length of the index, if it be measured between the centre of motion at \( X \) and the end \( Z \), when the end \( x \) of the outward ruler agrees to \( v \), where the divisions of the small line \( vx \) begin.

"The divisions of this small line are unequal, and determine the length of the arm at different times of the year, by applying \( x \) to the division which answers to the day in which the machine is used.

"But in order to mark the divisions, we suppose the length of the arm to be divided into 1000 equal parts, that is, \( vx \) into 90 equal parts; but the distances from the point \( v \) are set down in the following small table.

| 1 Fe. | 21 Ja. | 11 Ja. | 21 Dec. | |-------|--------|--------|--------| | 47. | 64. | 77. | 90. |

| 1 No. | 21 No. | 1 Dec. | 21 Dec. | |-------|--------|--------|--------| | 1 Au. | 21 Jul.| 11 Jul.| 21 Jun.|

"The index \( NO \) is turned, that the sun's rays may pass directly thro' the pipe \( R \), which is turned and inclined, as is required. The brass wire \( ld \) is then raised or depressed, that the shadow of the end of it may pass through the middle of the pipe.

"This whole part is moved to the placer, which is ordered as has been said before. But the clock is so moved towards the placer, and raised, that the end \( l \) of the brass wire \( ld \) may agree to the end \( Z \) of the ruler \( YZ \).

"We must continually have regard to the plummet \( Q \), that it may always answer to the point \( o \); we must also take care, that after the clock is moved, the sun's rays and the shadow of the point \( l \) may pass thro' Optical the small pipe R as before, that the position with respect to the meridian may not be disturbed.

"The pillar VX with its ruler YZ is removed, the stand P being left in its place, on which the cylinder C with its speculum is put. The piece T is taken out of its place, that the tail DE of the speculum may be put thro' the pipe R; when the piece T is put in the same place again, every thing is ready.

"Then the rays reflected from the middle of the speculum, to which all the other rays, reflected from the speculum, are parallel, agree, as to place and direction, with the position which the ruler of the placer had; and whilst the tail of the speculum is moved, as the clock moves, whose index follows the sun, its situation is altered with respect to the sun; but the ray, reflected from the middle point of the speculum, remains fixed.

"If the index NO being taken away, we substitute the index K, the machine may be used as a common clock.

"The experiments concerning light must be made in the dark; for this reason the machine, when made use of in the experiments, must be shut up in a box or case.

"This case is represented at A; it stands upon feet that have rollers joined to them, that it may be easily moved. It is open at one end, which end is moved to a window, through which the sun's rays come freely to the speculum.

"But the box is every way larger than the window, that, by being applied close to the wall, the light may be hindered from entering into the chamber; to this end, the box is moved as near the wall as possible, and the screws C, C, which are fastened to the fore-feet, are turned till they touch the ground.

"The door in my machine is opposite to the window; it might have been otherwise disposed. We transmit the rays through the fore part B; we make choice of this, by reason of the make of the place in which the experiments concerning light were made. In this part there are two apertures three inches broad, and about 13 inches high, one of which is represented open at DE.

"These are closed on the outside by pieces of wood, which are moveable between wooden rulers. Each piece serves either aperture, that they may be changed. One of them F is three feet long, and has a hole in its middle. The aperture ab is five inches long, and two broad.

"This is closed by the copper plate GH, in which there are two holes, c, d; the diameter of that is two thirds of an inch, the diameter of this is less. These holes are stopped by the plates I and K, which are applied to the first plate GH, and are moveable about the centres i and k; the magnitudes of the holes may also be varied, by turning the last plates, as the figure shews.

"The board F is hollowed behind, in order to receive the object-glass of a telescope of 16, 20 or 25 feet, according to the magnitude of the place in which the experiments are made; the centre of this glass ought to answer to the centre of the hole c.

"This board F is pretty long; the holes of the small plate may answer to any part of the aperture of the box, the other part of the aperture remaining shut.

For this reason the second board is shorter; it is sufficient if the aperture be closed with this. These boards are fastened by the screws M, M.

"We have shewn how the box is to be applied to the window; but this cannot be done thus, if we would make the experiments in the hours in which the sun's rays enter the window very obliquely. In this case, that the rays may come to the speculum, the box must answer to a part of the window only; the remaining part is closed any other way: I make use of a curtain to exclude all the sun's rays."

§ 20. Equatorial Telescope, or Portable Observatory.

The Equatorial Telescope was contrived by Mr James Short; and consists of two circular planes or plates AA, supported upon four pillars; and these again supported by a cross-foot or pedestal moveable at each end by the four screws BBBB. The two circular plates AA are moveable, the one above the other, and called the horizontal plates, as representing the horizon of the place; and upon the upper one are placed two spirit-levels, to render them at all times horizontal; these levels are fixed at right angles to each other. The upper plate is moved by a handle C which is called the horizontal handle, and is divided into 360°, and has a nonius index divided into every three minutes.—Above this horizontal plate is a semicircle DD; divided into twice 90°, which is called the meridian semicircle, as representing the meridian of the place; and is moved by a handle E, called the meridian handle, and has a nonius index divided into every three minutes. Above this meridian semicircle is fastened a circular plate, upon which are placed two other circular plates FF, moveable the one upon the other, and which are called the equatorial plates; one of them, representing the plane of the equator, is divided into twice twelve hours, and these subdivided into every ten minutes of time. This plane is moved by a handle G, called the equatorial handle, and has a nonius index for shewing every minute. Above this equatorial plate there is a semicircle HH, which is called the declination semicircle, as representing the half of a circle of declination, or horary circle, and is divided into twice 90°, being moved by the handle K, which is called the declination handle. It has also a nonius index, for subdividing into every three minutes. Above this declination semicircle is fastened a reflecting telescope LL, the focal length of its great speculum being 18 inches.

In order to adjust this instrument for observation, the first thing to be done is to make the horizontal plates level by means of the two spirit-levels, and the four screws in the cross pedestal. This being done, you move the meridian semicircle, by means of the meridian handle, so as to raise the equatorial plates to the elevation of the equator in the place, which is equal to the complement of the latitude, and which, if not known, may also be found by this instrument. By this telescope the following problems may be solved.

To find the Hour of the Day, and Meridian of the Place. First find, from the astronomical tables, the sun's declination for the day, and for that particular time of the day; then set the declination semicircle to the declination of the sun, taking particular notice whether it is north or south, and set the declination semicircle... circle accordingly. You then turn about the horizon- cal handle and the equatorial handle, both at the same time, till you find the sun precisely concentrical with the field of the telescope. If you have a clock or watch at hand, mark that instant of time; and by looking upon the equatorial plate and nonius index, you will find the hour and minute of the day, which comparing with the time shewn by the clock or watch, shews how much either of them differ from the sun.

In order to find the meridian of the place, and consequently to have a mark by which you may always know your meridian again, you first move the equa- torial plate by means of the equatorial handle, till the meridian of the plate, or hour-line of 12, is in the mid- dle of the nonius index; and then by turning about the declination handle till the telescope comes down to the horizon, you observe the place or point which is then in the middle of the field of the telescope, and a supposed line drawn from the centre of this field to that point in the horizon, in your meridian line. The best time of the day for making this observation for finding your meridian, is about three hours before noon, or as much after it. The meridian of the place may be found by this method so exactly, that it will not differ from the true meridian above 10" in time; and if a proper allowance be made for refraction at the time of observation, it will be still more exact. The line thus found will be of use afterwards, as being the foundation of all astronomical observations.

To find a Star or Planet in the Day-time. The instrument remaining rectified as already directed, you set the declination semicircle to the declination of the star or planet you want to see; and then you set the equatorial plate to the right ascension of the star or planet at that time; and looking through the telescope, you will see the star or planet; and having once got it into the field of the telescope, you cannot lose it again; for as the diurnal motion of a star is parallel to the equator, by moving the equato- rial handle so as to follow it, you will at any time, while it is above the horizon, recover it, if it is gone out of the field.

The easiest method for seeing a star or planet in the day-time is this: Your instrument being adjusted as before directed, you bring the telescope down, so as to look directly at your meridian mark; and then you set it to the declination and right ascension, as before mentioned. By this instrument most of the stars of the first and second magnitude have been seen even at noon-day, when the sun was shining very bright; as also Mercury, Venus, and Jupiter. Saturn and Mars are not so easily seen, on account of the faintness of their light, except when the sun is but a few hours above the horizon.

And in the same manner, in the night-time, when you can see a star, planet, or any new phenomenon, such as a comet, you may find its declination and right ascension immediately, by turning about the equatorial handle and declination handle, till you see the object; and then, looking upon the equatorial plate, you find its right ascension in time; and you find, upon the declination semicircle, its declination in degrees and minutes.

In order to have the other uses of this instrument, you must make the equatorial plates become parallel to the horizontal plates; and then this telescope be- comes an equal altitude instrument, a transit instru- ment, a theodolite, a quadrant, an azimuth instrument, and a level. The method of applying it to these different purposes is obvious. As there is also a box with a magnetic needle fastened in the lower plate of this in- strument, by it you may adjust it nearly in the meri- dian, and by it you may also find the variation of the needle. If you set the horizontal meridian and the equatorial meridian in the middle of their nonius indexes, and direct your telescope to your meridian mark, you observe how many degrees from the meri- dian of the box the needle points at, and this distance or difference is the variation of the needle.

§ 21. The Binocular Telescope.

The binocular telescope consists of two distinct tele- scopes severally directed from each eye to the same object, and combined together in the following man- ner. In fig. 1. ab and cd are two equal telescopes laid in a long box, nearly parallel to each other; the intervals between the eye-hlops a and c being equal to the interval of the pupils, and that of the centres of the object-glasses somewhat less than the other. Both ends of the telescopes pass through oblong slits in both ends of the box; and the interval between them may be widened or contracted at either end by a long screw-pin laid over each end of both the tele- scopes; the threads of each half of the screws being wrought contrary ways, and called a right and left handed screw. For these halves being put through two nuts e, f, fixed to the upper sides of the tele- scopes, it comes to pass, that by turning the screw- pin one way, the two telescopes will accede to, and the other way they will recede from, each other; till, by one of these screw-pins, the interval between the eye-hlops a, c becomes equal to the intervals of the pupils of the observer; and by the other, the axes of the telescope become directed to the same object; which will be known exactly if there are cross hairs in the focus of each telescope, and even without them. For before this position is obtained, the objects will appear double, and afterwards single; and a much stronger and brighter appearance of the object will be obtained than by a single telescope.—There are other contrivances, besides that of a two-handed screw, by which the telescopes may be made to approach to or recede from each other. To exclude all useless and hurtful light from the eyes, the eye-hlops are made hollow and very broad, to cover some part of the temples; and their inner parts are cut away, to admit the upper part of the nose between them. Two re- flecting telescopes, as well as two refracting ones, might be combined into a binocular telescope; and, for the purpose of celestial discoveries, promises to be a very useful instrument.

§ 22. Of making Celestial Observations.

In the day-time there is little difficulty in finding the exact time of the transit of such stars as are ca- pable of being discovered by the telescope over the middle of the field, because the cross-hairs placed in the focus of the object-glass receive a sufficient quan- tity of light to render them visible. But, in the night- time, these hairs are not visible, and therefore the ob- server Optical Instruments

Part III.

Optical sever hath not any mark to direct him; hence astro- nomers are obliged to enlighten the crofs-hairs artifi- cially, in order to render them visible; and this with- out letting the luminous body interfere with the ob- ject which they intend to view; and, for doing this, two ways are proposed.

1. The object-glas of the telescope may be obliquely enlightened by placing a candle near to it in an oblique situation, so that its smoke or flame may not interfere with the object. But if the object-glas should hap- pen to be pretty deep in the tube, it cannot be suffi- ciently enlightened by this means; and besides, if the telescope is above five feet long, there will be a considerable difficulty in throwing a sufficient quanti- ty of light upon the crofs hairs.

2. By some an opening is made in the side of the tube near the focus of the object-glas, through which the crofs hairs are illuminated by means of a candle. But this method also is attended with inconveniences; for the observer is incommoded by the light being so near his eyes, and the hairs themselves become liable to accidents through their exposed situation. An error also attends this method; which is, that, according to the position of the light illuminating these hairs, they will appear in different situations. For example, when the horizontal hair is enlightened above, we per- ceive a luminous line which may be taken for the hair itself, and which appears at its upper superficies. On the other hand, when the hair is enlightened under- neath, the luminous line will appear at the lower sur- face of it; and the error will be the diameter of the hair, which often amounts to more than five seconds. M. de la Hire, however, found a remedy for this in- convenience. He often observed, that in moon-shine nights, when the weather was a little foggy, the crofs hairs were distinctly seen; but when the heavens were serene, they could scarcely be perceived: he therefore covered with a piece of gauze, or fine silk-crape, that end of the tube next the object-glas; and this method succeeded so well, that a link placed at a good distance from the telescope so enlightened the crape, that the crofs-hairs distinctly appeared, and the sight of the stars was by no means obscured.

In making solar observations, a smoked glass must be used for preserving the eyes; and which may be thus prepared. Take two equal and well polished round pieces of flat glass; upon the surface of one of which, all round its limb, glue a pasteboard ring: then put the other piece of glass into the smoke of a lamp, taking it several times out, and putting it in again, lest the heat should break it, until the smoke be so thick, that the lamp can scarcely be seen through it: but the smoke must not be all over of the same thickness, that so a place may be chosen answering to the splendour of the sun. This being done, the glass, thus blackened, must be glued to the pasteboard ring abovementioned, with its black side next to the other glass, that the smoke may not be rubbed off.

There are two kinds of observations relating to the stars: one is, when they are in the meridian; the other, when they are in vertical circles. If the position of the meridian be known, the quadrant with which the observation is made, must be placed in the plane of that circle, and then the meridian altitudes are easily observed by means of the plumb-line. The meridian altitude of a star may likewise be had by a pendulum clock, if the exact time of the star's passing by the mer- idian be known. It must be observed, however, that stars have the same altitude a minute before and after their passing the meridian, if they be not in or near the zenith; but if they be, their altitudes must be taken every minute when they are in or near the me- ridian, and then their greatest or least altitudes will be those in question. The position of a given vertical circle must be found by the following method. 1. The quadrant and telescope remaining in the same situation wherein it was when the altitude of a star, together with the time of its passage by the intersection of the crofs hairs in the focus of the object-glas, was taken, we observe the time when the sun, or some fixed star, whose longitude and latitude is known, arrives to the vertical hair in the telescope; and from thence the po- sition of the said vertical circle will be had, and also the observed star's true place. But if the sun, or some other star, does not pass by the mouth of the tube; and if a meridian line be otherwise well drawn upon a floor, or very level ground, in the place of observation; a plumb-line must be suspended from a fixed place, at about 18 or 20 feet from the quadrant; under which a mark must be made on the floor, in a right line with the plumb-line. You must next put a thin piece of brass or pasteboard very near the object-glas, in the middle of which there is a slit vertically placed, and passing through the centre of the circular figure of the object-glas. Now, by means of this slit, the before- mentioned plumb-line may be perceived through the telescope, which before could not be seen, because of its nearness. Then the plumb-line must be removed and suspended, so that it be perceived in a right line with the vertical hair in the focus of the object-glas, and a point marked on the floor directly under it. And if a right-line be drawn through this point, and that marked under the plumb-line before it was re- moved, the said line will meet the meridian drawn upon the floor; and so we shall have the position of the vertical circle in which the observed star is, with respect to the meridian, the angle whereof may be measured in af- fuming known lengths upon the two lines from the point of concourse; for if, through the extremities of these known lengths, a line or base be drawn, we shall have a triangle, whose three sides being known, the angle at the vertex may be found, which will be the angle made by the vertical circle and meridian.

Under the article Quadrant, is shown the proper method of fixing that instrument exactly in the meri- dian: but where the observer has no conveniency of this kind, it will be proper to use a portable quadrant, by means of which the altitude of a star must be ob- served a little before its passage over the meridian, every minute, if possible, until its greatest or least altitude be had; by which means, though we have not the true position of the meridian, yet we know the meridian altitude of the star. But although this meth- od is very good, yet if a star passes by the meridian near the zenith, we cannot have its meridian altitude by repeated observations every minute, unless by chance; because in every minute of an hour, the alti- tude augments 15 minutes of a degree; and in these kind of observations, the inconvenient situation of the observer, the variation of the star's azimuth several de- grees Optical Instruments must have, and the difficulty in well replacing it vertically again, hinders our making observations oftener than once in four minutes, during which time the difference in the star's altitude will be one degree. In these cases, therefore, it will be better to have the true position of the meridian, in order to place the instrument exactly in it; or to move it so that one may observe the altitude of the star the moment it passes the meridian.

The refraction may be found in the following manner. Having the meridian altitudes, and the declination of two stars of nearly equal altitudes, find also, by the directions already given, the apparent meridian altitude of some star near the pole; and if the complement of that star's declination be added thereto or taken therefrom, we shall have the apparent height of the pole. After the same manner may the apparent height of the equator be found by means of the meridian altitude of some star near it, and adding or subtracting its declination. Then these heights of the pole and equator being added together, they will always make more than 90 degrees, because both of them are raised by the refraction: but taking 90 degrees from this sum, the remainder will be double the refraction of either of the stars observed, at the same height; and therefore taking this refraction from the apparent height of the pole, or equator, we shall have their true altitude.

To illustrate this: Suppose the meridian altitude of a star observed below the north pole to be $30^\circ 15'$, and complement of its declination $59'$; whence the apparent height of the pole will be $35^\circ 15'$. Also let the apparent meridian altitude of some other star, observed near the equator, be $30^\circ 40'$, and its declination $40^\circ 9'$; whence the apparent height of the equator will be $54^\circ 49'$. Therefore the sum of the heights of the pole and equator thus found will be $90^\circ 4'$; from which subtracting $90^\circ$, there remain $4'$, which is double the refraction at $30^\circ 28'$ of altitude, which is about the middle of the heights found. Therefore at the altitude of $30^\circ 15'$, the refraction will be somewhat above $2'$, viz. $2' 1''$; and at the altitude of $30^\circ 40'$, the refraction will be $1' 59''$. Lastly, if $2' 1''$ be taken from the apparent height of the pole, $35^\circ 15''$, the remainder $33^\circ 12' 59''$ will be the true height of the pole; and so the true height of the equator will be $54^\circ 47' 1''$, as being the complement of height of the pole to $90^\circ$. The refraction and height of the pole found according to this way, will be so much the more exact as the height of the stars is greater; for if the difference of the altitudes of the stars should be even $2'$ when their altitudes are above $30^\circ$, we may by this method have the refraction and the true height of the pole; because, in this case, the difference of refraction in altitudes differing only two degrees is not perceptible.

The quantity of refraction may also be found by the observation of one star only, whose meridian altitude is $90^\circ$, or a little less: for the height of the pole or equator above the place of observation being otherwise known, we shall have the star's true declination by its meridian altitude; because refractions near the zenith are insensible. Now, if we observe by a pendulum the exact times when the said star comes to every degree of altitude, as also the time of its passage by the meridian, which may be known by the equal altitudes of the star being east and west, the refraction may be found by the solution of a case in spherical trigonometry: for here, in a spherical triangle, we have the distance between the pole and zenith, the complement of the star's declination, and the angle comprehended by the arcs abovementioned; namely, the difference of mean time between the passage of the star by the meridian and its place, converted into degrees and minutes, to which must be added the proper proportional part of the mean motion of the sun in the proportion of $59' 8''$ per day; therefore the true arc of the vertical circle between the zenith and true place of the star may be had. But the apparent arc of the altitude of the star is had by observation, and the difference of these arcs will be the quantity of refraction at the height of the star.

To find the time of the equinox and solstice by observation, we must proceed in the following manner. Having found the height of the equator, the refraction, and the sun's parallax at the same altitude, it will not be difficult afterwards to find the time in which the centre of the sun is in the equator; for if, from the apparent meridian altitude of the sun's centre the same day that it comes to the equinox, be taken the convenient refraction, and then the parallax be added thereto, the true meridian altitude of the sun's centre will then be had. Now the difference of this altitude and the height of the equinoctial will show the true time of the true equinox before or after noon; and if the sum of the seconds of that difference be divided by $59$, the quotient will show the hours and fractions which must be added to or subtracted from the true hour of noon to have the time of the true equinox. The hours of the quotient must be added to the time of noon, if the meridian altitude of the sun be less than the height of the equator about the time of the vernal equinox; but they must be subtracted, if it be found greater. We must proceed in a contrary manner when the sun is near the autumnal equinox.

The solstices are found with much more difficulty; for one observation only is not sufficient, because about this time the difference between the meridian altitudes in one day and the next succeeding day is almost insensible. The exact meridian altitude of the sun must therefore be taken for 12 or 15 days before the solstice, and as many after, that so one may find the same meridian angle by little and little; to the end that, by the proportional parts of the alteration of the sun's meridian altitude, we may the more exactly find the time when the sun's altitude is the same before and after the solstice, being in the same parallel to the equator. Now, having found the time elapsed between both situations of the sun, you must take half of it, and seek in the tables the true place of the sun at these three times. This being done, the difference of the extreme place of the sun must be added to the mean place, in order to have it with comparison to the extremes; but if the mean place found by calculation does not agree with the mean place found by comparison, you must take the difference, and add to the mean time, that which answers to that difference, if the mean time found by calculation be lesser; but, if greater, it must be subtracted, in order to have the time of the fol- Optical slice. Here it must be noticed, that an error of a few seconds in the observed altitude of the sun will make an alteration of an hour in the true time of the solstice; whence it is plain, than the true time of the solstice cannot be had but with instruments very well divided, and several very exact observations.

With regard to eclipses, the beginning, end, and total emersion may be estimated with sufficient exactness without telescopes; excepting the beginning and ending of lunar eclipses, where an error of one or two minutes may be made, because it is difficult to determine with certainty the extremity of the shadow. But the quantity of the eclipse, that is, the eclipsed portion of the sun and moon's disk, which is measured by digits, or the 12th part of the sun and moon's diameter, and minutes, or the 60th part of digits, cannot be well known without a telescope joined to some instrument. One method of observing them by the solar telescope hath been already described; but this applies only to eclipses of the sun, those of the moon not being discoverable by reason of the faintness of the light: for these therefore micrometers must be used, which are placed in the focus of the telescope, and of which various kinds are described under the article Micrometer. The eclipses of the satellites of Jupiter are to be observed in the same manner, but require a better telescope than what is necessary for observing the eclipses of our moon.

It is here proper to take notice of the method of obviating a difficulty formerly taken notice of; namely, that in a serene night we often find the light of Jupiter and its satellites observed through the telescope to diminish by degrees, so that it is impossible to determine exactly the true times of the immersion and emersion of the satellites. This proceeds from the dew which falls upon the surface of the object-glass, and intercepts the light. A very sure remedy is, to make a tube of blotting paper, about two feet long, and of a sufficient bigness to go about the end of the tube of the telescope next to the object-glass, which will very effectually drink up the dew, and hinder it from coming to the object-glass; and by this means we may make our observations with sufficient exactness.

§ 15. Telescopic Instruments for finding Time by observing when the Sun or any Star has equal Altitudes on each side of the Meridian.

One of these instruments was made by Mr Roger Cotes, and was contrived by him for the purpose of regulating a pendulum-clock presented to the Royal Society by Sir Isaac Newton, to whom he sent the following description. "AB is a strong wooden axis about six feet in length; CD and DE on one side, EF and FG on the other, are pieces framed to each other and to the axis as firmly as was possible. Into the piece CD, and at the angle F, were fixed strong wooden pins nearly parallel to each other, and perpendicular to the plane CDEFG. PQ is the cylindrical brass tube of a five-foot telescope (belonging to our sextant); this was well fastened with iron staples and screws to the piece of wood IKML, whose under plane surface is here represented as objected to view. Into this surface there was perpendicularly fixed a strong wooden pin N, which was designed to hang the upper end of the telescope upon any of the pins in CD, whilst its lower end rested upon the pin F. Now, that the telescope might be taken off, and yet afterwards be again placed accurately in the same position, I ordered the edges IK and CD, which touched each other, to be rounded like the surface of a cylinder, as also the edge EF into which the pin F was fixed, and again the cylindrical tube of the telescope fitted, so that the contact in both places might be made in a point. Upon the same account the pins in CD were made a little hollow, as is represented at R; and the pin F was a frustum of a cone, that thereby the telescope might more surely touch the edges CD and EF. Into the two ends of the wooden axis were strongly fixed two pieces of well-tempered steel: that at the upper end A was a cylinder well turned, which moved in a collar, whose cavity, represented by S, was figured like two hollow and inverted frustums of cones joined together; the lower at B was a cone moving in a conical socket of a somewhat larger angle. This socket had liberty to be moved horizontally, and to be fixed in any position by two screws, which pressed against it sideways at right angles to each other. The instrument being thus prepared, I fixed a needle V, at the lower end of the wooden axis, whose point stood out from it about an inch; then suspending a fine plumb-line TVX from the upper end of the same axis, I altered the position of the instrument by the screws, until the plumb-line came to beat against the point of the needle in the whole revolution of the instrument, and there I fixed it as prepared for use."

So far Mr Cotes. The plumb-line or fine wire TVX was suspended by a loop T upon a brass pin that screwed into the top of the axis AB; a nick being filed round the pin to stay the loop from sliding out of it. Then by screwing the pin in or out, the plumb-line was brought to the same distance from the axis AB as the point of the needle is at V; which was fixed in the end of a thick wooden pin Y, not in the axis of it, but towards one side; so that by turning the pin round itself, in a hole bored through the axis AB, the needle's point might describe a small circle, and be brought to touch the plumb-line when parallel to the axis of motion of the wood AB.

Dr Smith gives the following description of an excellent instrument of this kind, in the noble collection of the right honourable the earl of Ilay. "It is made Earl of all of brass, except a square steel axis ab 30 inches in length. To one side of the upper part of this axis instrument, there is fixed a small sextantal arch cd represented separately at E; its centre a being at the top of the axis ab. The telescope MN is also 30 inches long, and is fixed along the diameter of a semicircle of the same radius as the sextant, and concentric to it. The telescope with the semicircle being moveable about this centre upon the plane of the fixed sextant acd, may be fastened to it in any elevation by two nuts and screws c, d, fixed in the ends of the sextantal arch; a circular slit being made all along the limb of the semicircle for these screw-pins to slide in. Closer under these arches, the axis ab is surrounded by a short cylinder e, about an inch in diameter, well turned and polished. The lower end of the axis is formed into a fine conical point b. The frame in which the axis turns, is a long hollow parallelopiped wanting two sides." Optical sides. Its other sides \( f g \), are two brass plates, equal in length to the part \( b e \) of the axis, and are screwed together edgewise. It has for its bases two equal plates \( h, i \), four inches square. In the middle of the upper square there is a round hole large enough to receive the cylinder \( e \), without touching it; and over this hole is fixed a triangular hole in another plate; one of whose sides is moveable by a screw, to make all the sides of the triangle touch the cylinder. Upon the lower square there lies a smaller plate \( i \), with a fine centre-hole to receive the point \( b \) of the axis. This centre-plate is moveable sideways by two screws at right angles to each other, which, when the frame is firmly fixed into a notch of a free-stone pillar, will bring the axis \( a b \) exactly perpendicular to the horizon. This position is known by a spirit-level \( l m \) fixed at right angles to the axis above the cylinder, upon the side opposite to the semicircle. Along the top of the level there is a sliding pointer to be set to the end of the air-bubble; and when the position of the axis is so adjusted by the screws below, that the air-bubble keeps to the pointer for a whole revolution of the instrument, the axis \( a b \) is certainly perpendicular to the horizon; and then the line of sight through the telescope describes a circle of equal altitudes in the heavens. There are several of these circles described in the heavens, even when the telescope is fixed to the sextant arch. For the round hole in its focus has five wires parallel to the horizon at equal intervals from one another, as at \( k \); and they are crossed at right angles in the middle by two other upright wires at a small distance from each other. The design of so many wires is to observe when the same star is successively covered by every one of the five, both in the east and west; so that the time of its passage over the meridian may be had more accurately, by taking a medium among all the observations. The distances between the five wires need be no greater than to afford time enough to write down the several observations, which must be taken when the star is between the perpendicular wires.

Time shewn by a clock may be called mechanical, to distinguish it from solar and sidereal time. By observing when a star has equal altitudes before and after its culmination or apsidal to the meridian, we have the mechanical time of its culmination. Then by subtracting the sun's right ascension computed to this mechanical time, from the star's right ascension determined for the same time, we have the solar time of the star's culmination, and consequently the difference between the mechanical and solar times.

Thus, by finding the mechanical times, when the same star culminates any two nights, rather at a distance from each other than successive, we have the difference between a sidereal day and a mechanical day; and consequently between a mechanical day and a solar day of a mean length.

Hence any number of mechanical minutes may be converted into solar or into sidereal minutes by the rule of three.

These observations will answer the purpose the more exactly as the star is nearer to the prime vertical; because the variation of its altitude is here greater in a given time, than if it were situated in any other vertical oblique to the meridian. In the latitude of 50 degrees an error of one minute in altitude, at any point of the prime vertical, will cause an error of 63 seconds in time; and in the latitude of 55 degrees it will cause an error of near 7 seconds; as that excellent geometer Mr Cotes has shewn in his treatise concerning the Elevation and Limits of Errors in mixed mathematics, published at the end of his admirable book called Harmonia Mensurarum. It is also the safest to choose a star as high as possible, lest a different state of the atmosphere should cause a different refraction of the visual rays, and consequently an error in the times of observation.

The solar time may also be found by observing when the sun himself has equal altitudes in the morning and evening; if we correct the time of the latter observation by a just allowance for the variation of the sun's declination, as follows. Upon a celestial globe let the pole be at \( P \); the vertex of the observer's place at \( V \); the complement of its latitude \( PV \); its meridian \( PVBG \); a circle of equal altitudes \( ABCD \), described about its pole \( V \); and passing thro' the sun's centre at \( A \) at the time of the morning observation, and through it at \( D \) at the time of the evening observation; two circles of declination, \( PAF, PDI \), cutting the equator \( FGHI \), in \( F \) and \( I \); a parallel of declination \( ACE \), cutting the circle \( PDI \) in \( E \), and \( ABD \) in \( C \); three equal vertical arches \( VA, VC, VD \); and lastly, a third circle of declination \( PCH \) cutting the equator in \( H \). Now, had not the sun varied his declination from \( A \) or \( E \) to \( D \), in the evening he would have had the same altitude at \( C \) as in reality he has at \( D \). And then as the angles \( VPC, VPA \), would have been equal, so the times of the evening and morning observations would also have been equidistant from noon; being measured by those angles, or by the arches \( GF, GH \). Therefore the angle \( CPD \), or the arch \( IH \), which measures it, is also the measure of a portion of time to be subtracted from the evening observation, if the sun's declination varies northwards, otherwise to be added, to give the time sought equidistant from noon. Let another circle of declination \( PKL \) bisect the small angle \( HPI \), and consequently the small arches \( CD \) and \( HI \) in \( k \) and \( l \). Draw the vertical arch \( KV \); and in the triangle \( KPV \), we have given \( PV \) the complement of the latitude, and \( PK \) half the sum of the given complements, \( PC \) or \( PA \) and \( PD \), of the sun's declinations at the times of the two observations, and lastly, the included angle \( KPV \), by converting half the interval of time between the observations into degrees and minutes. Hence by trigonometry we have the angle \( PKV \), of an intermediate magnitude between \( PCV \) and \( PDV \), and therefore fitter to be used instead of either of them. Hence also we shall have the arch \( IH \); by taking it in proportion to \( DE \) the difference of the declinations, as the co-tangent of the angle \( PCV \), to the sine of the arch \( PC \) or \( PK \). For \( IH \) is to \( DE \) in a ratio compounded of \( IH \) to \( CE \) and of \( CE \) to \( DE \), that is, of the radius to the sine of the arch \( PC \), and of the co-tangent of the angle \( DCE \) or \( PCV \) to the radius: as appears by taking away the common angle \( DCP \) from the right angles \( ECP \) and \( DCV \); and by considering the small triangle \( DCE \), right-angled at \( E \), as if it were rectilinear. The calculation supposes the sun's centre has equal altitudes at \( A \) and \( D \); which is agreeable to the Mechanism observations that determine when his upper or his under limb has equal altitudes."

Sect. VI. Of the Mechanism of the principal Optical Instruments.

As part of the mechanism of optical instruments depends on a knowledge of the arts of casting, polishing and soldering bras and other metals, of turning wood, &c. we must necessarily confine ourselves in this section to some directions concerning the grinding and polishing of the glasses and specula for microscopes and telescopes, with the method of putting them together and adjusting them; leaving the fabrication of the other parts to be learned by the ingenuity and industry of those who choose to employ themselves in that manner, and for which scarce any rules that would be of much service can be laid down.

§ 1. Of making Glasses-globules for Microscopes.

Though these are not found to be of such use as small convex lenses, yet as they are easily made, and are still used through choice by some persons, we shall here give some of the best methods of making them. Mr Butterfield's method of making these was by the flame of a spirit-lamp, which, instead of a wick, had several folds of fine silver-wire doubled up and down like a skin of thread. Having prepared some fine glass, beaten to powder and washed very clean, he took a little of it upon the sharp point of a silver needle wetted with spittle, and held it in the flame, turning it about till it became quite round, but no longer, for fear of burning it. The art lies in giving the globules an exact roundness, which can only be learned by experience. When a great many globules were thus formed, he rubbed them clean with soft leather. Then, having several small pieces of thin brass plates, twice as long as broad, he doubled them up into the form of a square, and punched a fine hole through the middle of them; and having rubbed off the bur about the holes with a whetstone, and blacked the insides of the plates with the smoke of a candle, he placed a globule between the two holes, and tacked the plates together with two or three rivets.

Dr Hooke, for the same purpose, used to take a very clear piece of glass, and draw it out into long threads in a lamp; then he held these threads in the flame, till they ran into round globules at the end. He next fastened the globules with sealing-wax to the end of a stick, so that the threads stood upwards, and grinding off the ends of the threads upon a whetstone, polished them upon a smooth metal plate with a little putty.

Mr Stephen Gray tells us, that, for want of a spirit-lamp, he laid a small particle of glass, about the bigness of the intended globule, upon the end of a piece of charcoal, and by the help of a Blow-Pipe, with the flame of a candle, he soon melted it into a globule. By this means he made them indifferently clear, and the smallest very round; but the larger, by resting upon that side, became a little flatted, and received a roughness on that side. He was therefore wont to grind and polish them upon a brass plate till he reduced them to hemispheres; but he found, that the small round globules, besides that they magnified more, shewed objects more distinctly than the hemispheres.

§ 2. Of grinding Lenses for Telescopes and Microscopes. Mechanism

A lens is more easily made of an equal convexity on both sides than of any other figure, because the same tools will serve for grinding both its surfaces. And a glass of this figure will make as perfect an image as any other, because the aberrations of the rays occasioned by the spherical figures of the surfaces, whatever be the proportion of the semidiameters, are inconsiderable in long telescopes, in proportion to what is occasioned by the different refrangibility of the rays, which last error Mr Dollond has shewn how to correct. If it be proposed, then, to make a glass of equal convexities, that shall have a given focal distance, the radius of the spherical surface will be found by taking it in proportion to the given focal distance as 12 to 11, putting the fine of incidence to the fine of refraction out of air into glass as 17 to 18, which Sir Isaac Newton hath accurately determined it to be. The focal distance of the glass being given, its aperture may be found by the following table.

| Length of the telescope, or focal distance of the object glass. | Linear aperture of the object glass. | Focal distance of the eye-glass. | Linear amplification, or magnifying power. | |-----------------|-----------------|-----------------|-----------------| | Feet. | Inch & Dec. | Inch & Dec. | | | 1 | 0.55 | 0.61 | 20 | | 2 | 0.77 | 0.85 | 28 | | 3 | 0.95 | 1.05 | 34 | | 4 | 1.09 | 1.20 | 40 | | 5 | 1.23 | 1.35 | 44 | | 6 | 1.34 | 1.47 | 49 | | 7 | 1.45 | 1.60 | 53 | | 8 | 1.55 | 1.71 | 56 | | 9 | 1.64 | 1.80 | 60 | | 10 | 1.73 | 1.90 | 63 | | 13 | 1.97 | 2.17 | 72 | | 15 | 2.12 | 2.32 | 77 | | 20 | 2.45 | 2.70 | 89 | | 25 | 2.74 | 3.01 | 100 | | 30 | 3.00 | 3.30 | 109 | | 35 | 3.24 | 3.56 | 118 | | 40 | 3.46 | 3.81 | 126 | | 45 | 3.67 | 4.04 | 133 | | 50 | 3.87 | 4.26 | 141 | | 55 | 4.06 | 4.47 | 148 | | 60 | 4.24 | 4.66 | 154 | | 70 | 4.58 | 5.04 | 166 | | 80 | 4.90 | 5.39 | 178 | | 90 | 5.20 | 5.72 | 189 | | 100 | 5.48 | 6.03 | 199 | | 120 | 6.00 | 6.60 | 218 | | 140 | 6.48 | 7.13 | 235 | | 160 | 6.93 | 7.62 | 252 | | 180 | 7.35 | 8.09 | 267 | | 200 | 7.75 | 8.53 | 281 |

TABLE These proportions, in Huygens's table for refracting telescopes, are measured by the Rheinland foot, which is to the English foot as 139 to 135; so that, taking their lengths as many English feet, their apertures and eye-glasses and linear amplifications should be severally diminished in the subduplicate ratio of 139 to 135; that is, nearly in the ratio of 139 to 137, or about \( \frac{1}{6} \) or \( \frac{1}{7} \)th part of the whole.

Because the figure of the glass cannot be made exactly true to its very edges, the breadth of it may be about half an inch more than the diameter of its aperture, or even three quarters, or a whole inch more, if its focal distance be between 50 and 200 feet. Mr. Huygens directs in general to make the breadth of the concave tool or plate in which an object-glass must be ground, almost three times the breadth of the glass; though in another place he speaks of grinding a glass whose focal distance was 200 feet, and breadth 8½ inches, in a plate only 15 inches broad. But for eye-glasses, and others of a shorter radius, the tool must be in proportion to the breadth of these glasses, to afford sufficient room for the hand in polishing. Huygens made his tools of copper or cast brass; which, for fear they should change their figures by bending, can hardly be cast too thick: nevertheless he found by experience, that a tool 14 inches broad and half an inch thick was sufficient for grinding glasses to a sphere of 36 feet diameter; when the tool was strongly cemented upon a cylindrical stone an inch thick, with hard cement made of pitch and ashes.

In order to make moulds for casting such tools as are pretty much concave, he directs that wooden patterns should be turned in a lathe a little thicker and broader than the tools themselves. But for tools that belong to spheres above 20 or 30 feet diameter, he says it is sufficient to use flat boards turned circular to the length and breadth required. When the plates are cast, they must be turned in a lathe exactly to the concavity required. And for this purpose it is requisite to make a couple of brass gauges in the following manner.

Take a wooden pole a little longer than the radius of the spherical surface of the glass intended, and through the end of it strike two small steel points at a distance from each other, equal to the radius of the sphere intended; and by one of the points hang up the pole against a wall, so that this upper point may have a circular motion in a hole or socket made of brass or iron firmly fixed in the wall. Then take two equal plates of brass or copper well hammered and smoothed, whose length is somewhat more than the breadth of the tool of cast brass, and whose thickness may be a tenth or a twelfth part of an inch, and the breadth two or three inches. Then having fastened these plates flat against the wall in an horizontal position, with the moveable point in the pole strike a true arch upon each of them. Then file away the brass on one side exactly to the arch struck, so as to make one of the brass edges convex and the other concave; and, to make the arches correspond more exactly, fix one of the plates flat upon a table, and grind the other against it with emery. These are the gauges to be made use of in turning the brass tools exactly to the sphere required.

But if the radius of the sphere be very large, the gauges must be made in the following manner. Suppose the line AE, fig. 1., drawn upon the brass plate, to be the tangent of the required arch AFB, whose radius, for example, is 36 feet, and diameter 72. From A set off the parts AE, EE, &c. severally equal to an inch, and let them be continued a little beyond half the breadth of the tool required. Then, as 72 feet or 864 inches is to one inch, so let one inch be to a fourth number; this will be the number of decimal parts of an inch in the first line EF, reckoning from A. Multiply this fourth number successively by the numbers 4, 9, 16, 25, &c. the squares of 2, 3, 4, 5, &c. and the several products will be the numbers of decimal parts contained in the 2d, 3d, 4th, and 5th EF respectively. But because these numbers of parts are too small to be taken from a scale by a pair of compasses, subtract them severally from an inch represented by the lines EG; and the remainders being taken from a scale of an inch divided into decimal parts, and transferred by the compasses from G to F, will determine the points F, F, &c. of the arch required; after which the brass plate must be filed away exactly to the points of this arch, and polished as before.

To apply the brass tool to a turning lathe in order to turn the concave surface of it exactly spherical, let the brass, fig. 2, represent a view of some part of the lathe, taken from a point directly over it; let ab represent Fig. 2, 3, 4, a strong flat disk of brass half an inch thick at least, having a strong iron screw-pin firmly fixed in the centre of it, and standing out exactly perpendicular to one side; by which it may be screwed into the end c of the mandrel or axis of the lathe, represented by cd. This disk is represented separately in fig. 3, and must be well soldered to the backside of the tool ef, which therefore, in the middle of it, must be made plane, and exactly parallel to the circumference of its opposite surface, in order that the circumference may be carried round the axis of the lathe in a plane perpendicular to it. The mandrel or axis cd turns upon a point d in the puppet-head of the lathe, and in an iron collar represented by st.

Let ghik represent a board nailed fast on the other puppet-head; and let the concave gauge gh be laid upon this board, with its concave arch parallel to the concavity of the tool ef, and be screwed down to the board with flat-headed screws sunk into the brass. Let lmnop represent such another board lying upon the former, with the convex gauge lm screwed to the under side of it; so that, by moving this upper board, the arch of the convex gauge may be brought to touch the concave one, and to slide against it. The turning tool pq is laid upon the moveable board, and is held fast to it by a broad-headed screw at r, to be turned... Mechanism or unturned by the hand upon occasion. To know whether the concave gauge be exactly parallel to the concavity of the tool \( e \) foreward fast to the mandrel, direct the point \( p \) of the turning tool \( q \) to touch any point of the tool \( e \) near its circumference: then having fixed the turning tool \( q \) by its screw \( r \), turn the bras tool \( e \) half round, and move the upper board till the point \( p \) of the turning tool be brought over against the same mark upon the tool \( e \); and if it just touches it as before when the gauges coincide, all is right. If not, the position of the head of the lathe may be altered a little by striking it with a mallet. But the best way is, to make this examination of the situation of the concave gauge, when only one end of it is fixed to the lathe by a single tack or screw, about which it may easily be moved into its true position. And while the tool or plate \( e \) is turning, the same examination of its parallelism to the gauge must be frequently repeated; otherwise its surface will take a false figure. It is convenient that the upper board \( l \) should project over both the gauges; and to keep its surface parallel to that of the under board, two round-headed nails, or a plate of bras, as thick as the gauges, must be fixed to its under surface, towards the opposite side \( n \). Care must be taken to drill the holes in the gauges, through which they are screwed to the boards, not too near the polished arches for fear of altering their figure by the yielding of the bras. The tool and all the parts of the lathe must be fixed very firm; because any trembling motion will cause the graving tool \( q \) to indent the bras. After the tool is well turned, it must be separated from the bras \( a b \) by melting the solder with live coals laid upon it. In a similar manner may a convex tool be turned by transposing the gauges.

Mr Huygens advises first to form the plates or tools in a turning lathe; and then to grind them together with emery; that is to say, the concave and convex tool of the same sphere together. But the tools of very large spheres, he would have ground at first quite plane by a stone-cutter; and then ground hollow with a round flat stone and emery to the proper gauge. And he prescribes to use for this grinding first a stone half as broad as the tool, and after that another nearly of the whole breadth of it; and in this way of forming the tools, it will be convenient to tie a little frame of thick paper, or rather of thin palteboard, about an inch high, round the tool, in order to keep in the emery; and in grinding, the whole must be made extremely firm. When the tool is to be polished, it must still remain upon the stone pedestal; otherwise it will be in danger of bending a little in the operation.

For polishing the tools when ground, Mr Huygens directs the concave tool to be daubed with soap; after which, he takes the round stone above-mentioned, somewhat less than the tool, (or the convex tool itself), and heats it; then he pours upon it some hot melted cement (made of pitch and fine powdered and sifted ashes, as much as he can mix with it), and then he turns over the stone and cement upon the concave tool, into which also he had poured a good quantity of the same cement; having first laid three little pieces of bras, of equal thicknesses, on the circumference of it, in order to press and keep this crust of cement of an exact equal thickness in all its parts; and thus he lets them cool together. Then taking the stone from the tool, and turning it up, he fits upon the cement that sticks to it a crust of very fine emery; and with a flat iron spatula, about one third of an inch thick, gently warmed, he presses lightly the emery, to stick to and incrustate upon the cement. The whole is then gently warmed, viz. the stone, cement, and emery, and he again replaces it upon the concave tool, and leaves it again to cool; so that he has by this means a crust of emery exactly of the figure of his tool; and with this he polishes the tool dry, without the addition of any wet, pressing it hard on the surface of the tool. To press it the harder, he places upon it a long pole, a little bent, to make it spring, whose upper end is fixed to the ceiling of the room, or else is pressed downwards by a strong iron spring; and he thinks it is necessary to have two persons to rub the stone upon the tool. Here, however, it must be observed, that great care must be taken in this, and in all cases where this way of grinding by a pole is made use of, to fix the point of pressure exactly in the middle.

To bring the concave tool still nearer to perfection, take equal pieces, about an inch square, of blue hone, such as are used by engravers for polishing their copper, and place as many of them as you can upon the surface of the tool to be polished, laying the grain of them, some one way, some another; sticking them as close as you can to one another with soap and common white starch: then fill up all the interstices of the hones with clean dry sand, to about two thirds of the thickness of the hones; then having a border of paper or palteboard put round the tool as before, shake the tool gently, that the sand may equally subside, and blow it everywhere to an equal depth with a pair of bellows. Then take some hard cement, extremely hot, and pour it all over the hones; then having cleaned the stone, or convex tool, which before was incrustated with pitch and emery, place this stone (or convex tool) warmed, on the top of the cement, and let all cool together. Then rubbing the tool with this polish made with hones, by applying your pole to the top of the stone as before, you will know when the tool is brought to perfection, by wiping off the filth, in which case all parts of it will appear equally bright by looking upon it obliquely against the light. If you would use this polisher again, it must be kept in a cool cellar, leaving the hones uppermost; otherwise in warm weather they will change their situation in the cement, even by their own weight.

The cement used for fastening the glasses is made of several different compositions, according to the fancy fastening on the operator. Cherubin informs us, that it was usually made of common black pitch and fine sifted vine-ashes: but he himself made it of rosin and ochre, or rosin and Spanish white; pounding the rosin first, and mixing it with a due quantity of the powder, and then fitting the mixture upon hot melted pitch, and, while hot, well mixing and incorporating the whole. By others, the cement is made of pitch and common coal-ashes sifted fine. In all cases, it is harder or softer, as more or less of the ashes or other fine powder is put into it: and in the present case, for polishing these tools, it must be made as hard as possible, by putting in a large quantity of ashes; for otherwise, if the cement... Mechanism is not hard enough, the particles of the emery will be loosened by the heat in grinding, and then will only run round upon the tool, without working out the little inequalities thereof. If the emery should be found to grow blunt, a very little more of it may be dusted dry upon the tool, by which its sharpness and cutting quality will be a little recovered; but if the cement be sufficiently hard at first, the emery will always remain sufficiently sharp.

The best kind of glass is perfectly white; but great care must be taken in choosing it totally free from veins. To discover these veins, one should look very obliquely against a small light in a room otherwise dark. In this manner one may examine pieces of a polished looking-glass, of which object-glasses are sometimes made; but because these are seldom of a sufficient thickness for this purpose, it will be proper to take some pieces of the same sort of glass before it is polished, and get it ground to an equal thickness and polished a little by the common glass-grinders, in order to judge what pieces are fit for use. Sometimes little veins will appear like fine threads, which scarce do any harm. Sometimes their imperfections cannot be discovered by the former way of trial; and yet after the glass is well formed and polished, they will appear by reflection in the following manner. In a dark room place the glass upright upon a table, turning that surface from you which is suspected to be faulty; then holding a lighted candle in your hand, so that the middle of the broad light reflected from the first surface may fall upon your eye, recede from the glass till the rays reflected from the back surface shall just begin to invert the candle; then the whole glass will appear all over bright, and then you will discover its defects, and the imperfections of the polish. When the glass is a portion of a large sphere, we use a small perspective, three or four inches long, to magnify the defects.

The pieces of glass above-mentioned should be much broader than the intended object-glass, that there may be room enough for choosing the best part of them. For planing and smoothing these large pieces of glass, plates of cast-iron may be made use of; such as are sold at the iron-mongers shops, after they have been ground and planed on a stone-cutter's engine. Upon the plate of glass, with a diamond-pointed compass, strike a circle representing the object-glass; and also another concentric circle, with a radius about a tenth or twelfth part of an inch bigger: And also two other such circles, on the other side of the glass, directly opposite to the former; which may be done by means of the circular glass to be afterwards described. The larger parts of the glass may be separated from the outward circle by a red-hot iron, or by a strong broad vice, opened exactly to the thickness of the glass. The remaining inequalities may be taken off by a grind-stone; beginning with the largest first, and taking care that they do not splinter. Then, having warmed the glass, cement a wooden handle to it, and in a common deep tool for eye-glasses, making use of white clear sand and water, grind the circumference of the glass exactly true to the innermost circle on each side of it. Then, having made a great many small cavities with a punch upon one side of a round copper-plate, and having fixed the other side of it upon the middle of the round glass, by cement made with two parts of rosin or hard pitch, and one part of wax, place the steel-point of the springing pole above described, being 14 or 15 feet long, into that cavity of the copper plate which lies nearest the thickest part of the glass; then work the glass by the pole with sand and water upon a flat plate of cast iron, of a round figure, the plate having been planed with sand and water by a stone-cutter. Then having examined the thicknesses of the glass in several places by a hand-vice, which is better than a pair of calipers, by repeating the same operation, it will soon be reduced to an equal thickness in all its parts. Towards the end of this operation it will be convenient to make use of sifted emery, because the sand will scratch too deep; and then it will also be necessary to place the steel-point of the pole exactly over the centre of the under surface of the glass; otherwise that surface will take a cylindrical or convex figure, even though it was exactly plane before you began to grind it; and when concave glasses are to be polished, it is also absolutely necessary to place the point of pressure exactly over the centre of the under surface of the glass. To bring one of the little cavities in the copper-plate exactly over that centre, a circular glass is made use of, formed from a broken looking-glass with the quicksilver rubbed off. On this must be described, with a diamond-pointed compass, eight or ten concentric circles, about a quarter of an inch distant from each other, so that the larger circles may be somewhat bigger than the circumference of the glass to be polished. Lay this circular glass upon the surface of the glass to be polished; and move it to and fro till you perceive that the circumference of the glass to be polished is exactly parallel to the nearest circle upon the circular glass; then, having inverted both the glasses, lay the circular glass upon a table, and having laid a small live coal upon the copper-plate, to make it moveable on the cement, place one point of a pair of compasses in one of the little cavities, and move the copper till a circumference described with the other point coincides exactly with any one circle upon the circular glass, and the business is done. It is convenient to paste three slender threads of fine linen directed towards the centre of the circular glass, that the other glass may not slide too easily upon it, and that they may not scratch one another. The cavities punched in the copper-plate, and also in the point of the pole, should be triangular, to hinder the rotation of the glass; which is still more necessary in giving it the last polish. Here also we must observe whether the circumference remains exactly circular on both sides of it, which must be tried with compasses; and if it be not, it must be corrected again by grinding it exactly circular in a common tool for making eye-glasses; which will contribute very much to its taking an exact spherical surface when it comes to be ground in its proper tool. For if any part of the circumference be protuberant, it will hinder the adjoining parts of the surface from wearing so much as they should do, and of consequence will spoil its surface.

When the glass is thoroughly planed and rounded as above, take away the plate with the several cavities, and, with some of the same cement, fix on a smaller round piece of brass or rather steel truely flat, Mechanism and turned about the bigness of a farthing, but thicker, having first made in the centre thereof, with a triangular steel punch, a hole about the bigness of a goose-quill, and about the depth of $\frac{1}{4}$ of an inch; and at the very bottom of this triangular hole, a small round hole must be punched, somewhat deeper, with a very fine steel punch. A small steel point about an inch long must be truly shaped and fitted to this triangular hole, and, at the very apex, to the small, round, deeper impression. Nevertheless, it must not be fitted to exactly, but that it may have some liberty to move to and fro; the apex always continuing to press upon the surface of the round hole below. This steel triangular point must be fixed to one end of a pole; to the other end of which another round iron point must be fixed, of about five or six inches long, to play freely up and down in a round hole in a piece of brass let into a board fixed in the ceiling for that purpose, perpendicularly over the bench, and over the centre of the tool, which must be strongly and truly fixed horizontally thereon. Mr Huygens directs the brass plate to be fixed to the glass by means of cement, and takes no notice of any other method whatever; though it is plain, that it is hardly possible, in this, or any other case, to bring the cement to a fluidity sufficient to fix two plane surfaces exactly parallel to one another, without heating the glass, and the brass also, to a very great degree, and thus endangering the glass considerably. To avoid this, some have used platter of Paris; others cement an intermediate glass to the brass or wood, and then fix the glass to be ground to the outward surface of the cemented glass with common glue. It may easily be done, however, with common sizingglass or fish-glue, which will run very fluid, and will fix the glass and the brass of itself strongly together. Some common soft red wax is to be stuck on the edges of the brass, to keep wet from getting to the glue.

For grinding glasses truly plane by this method, Mr Huygens prescribes the pole to be about 15 feet long; but, in grinding upon a concave plate, the pole is most conveniently made of the same length with the radius of the sphere, though Dr Smith is of opinion that it would not be material if made considerably shorter, as the height of the room may allow. It is necessary to have, lying by, an ordinary piece of coarse glass ground in the same tool, called a bruiser; whereby, when any new emery is laid on the tool in grinding the glass, it must be first run over and smoothed, for fear that any little coarse grains should remain and scratch the glass.

Things being thus prepared, some pots of emery of various finenesses must be prepared. Take of the roughest sort a small half-pugil, wetting and daubing it pretty equably on the tool; then lay on your glass, and fix up the pole, continuing to grind for a quarter of an hour; not pressing upon the pole, but barely carrying the glass round thereby; then take the like quantity of some fine emery, and work another quarter of an hour therewith; and then take the like quantity of emery still finer; and work for the same time: after which you must work for an hour and an half with some of the finest emery you have, taking away by little and little some of the emery with a wet sponge. It must neither be kept too moist nor too dry, but about the consistence of pap. Much depends on this last circumstance. For, if it is too dry, the emery will clog and stick, and incorporate in such a manner as to cut little or none at all, unless here and there, where its body chances to be broke; and in those places it will scratch and cut the glass irregularly: or if it is too much diluted, it will, from the irregular separation of its parts, cut in some places more than in others, as in the former case.

But Mr Huygens tells us, that this method of using various sorts of fresh emery is not good; as in this way, he finds by experience, that the best glasses are often scratched. For this reason, he advises to take a large quantity of emery of the first or second sort, and work with it from first to last, taking away by little and little every half hour, or quarter of an hour, more and more of the emery with a wet sponge. By this means he could bring the glass extremely smooth and fine, so that a candle or glass window could be seen through it pretty well defined; which is a mark of its being sufficiently well ground for receiving the last polish. But, if the glass has not acquired this degree of transparency, it is certain, says Mr Huygens, that too much emery remains; and therefore it must still be diminished, and the operation continued. He found common well-water most proper in this operation of grinding; and he took care to move the glass in circles, taking an inch beyond the centre of the tool, and somewhat beyond its outside; and he found in a glass of 200 feet, whose diameter was $8\frac{3}{4}$ inches, which he ground in a tool of 15 inches diameter, that the figure of the tool in grinding would alter considerably, unless he carried the glass round an inch beyond the centre of the tool one way, and $3\frac{1}{2}$ inches beyond the skirts of it another way; but if he carried it no farther than a straw's breadth beyond the skirts of the tool, and of consequence farther beyond the centre, the glass would always grind falsely, so that he could never afterwards bring the outsides of it to a true and fine polish.

When you first begin to grind, and the emery begins to be smooth, the glass will stick a little to the tool and run stiff. Then fresh emery is to be added. When it afterwards comes to be polished, it will, if large, require a considerable strength to move it; but this inconvenience will happen less in grinding by the pole than in grinding with the hand. For the warmth of the hand makes the substance of the glass swell; and not only increases the sticking of the glass, but in some measure may spoil the figure of it, as also of the tool. When it is ground with the pole, it never sticks very strongly, unless when you take the glass off from the tool, and keep it from it for some time, and then apply it to the tool again; and this in large glasses; for by this means the glass gets from the air a greater warmth than it had on the tool; and being again applied to the tool, its lower surface is suddenly contracted by the cold, and thus sticks to the tool. Wherefore, says Mr Huygens, you must in that case wait till the glass and the tool come to be of one temperature. The like effect is observable in grinding when there is a fire in the room; and hence we may see the great nicety requisite in grinding these large glasses, and the necessity of attending even to the minutest circumstances. Instead of emery, Father Cherubin prescribes the grit of a hard grindstone, well beaten into fine powder, and fitted. The same thing hath been done by common white-land washed clean, taking away by little and little the grit as it became finer and finer. Nay, glasses have been frequently polished off in this manner without the use of any other material whatever. This method is called drying off on sand; because, as the matter grows finer and finer, they wet it less and less, till for the last quarter of an hour (the whole work lasting nearly two hours) they only wet it by breathing upon it; and at the very last, not at all. This method, however, is now entirely disused; for which Dr Smith assigns, as one reason, the violent labour requisite at the last; another and better reason, he says, may be, the great improbability of grinding or polishing true by this method, by reason of the uncertain and unequal force of the hand. But if this last is the reason, Dr Smith is of opinion, that the method might be restored, and greatly improved by adding a pole, and spring to press down the pole, or some analogous contrivance. And in all methods of grinding hitherto invented, the artist must allow time to bring his glass by grinding to the smoothest and finest surface that he possibly can, before he attempts to give the last polish. For the smoother you bring it in grinding, the less labour you will have in polishing; in which consists not only the greatest difficulty, but the greatest danger of spoiling all you have already done.

In order to give the last and finest polish to glasses, Mr Huygens directs us to proceed as follows. "Having removed the little brass plate from the glass, take a very thick flate, or rather a block of blue or grey stone; let it be half an inch thick, and let it be ground true and round at the stone-cutter's; its diameter being somewhat smaller than the diameter of your glass, leaving a hole quite through in the centre, of about an inch diameter. Then make some cement of two parts rosin or hard pitch, and one part wax; and taking a piece of thick kersey cloth, truly and equally wrought, cut this cloth round, and leave a like hole one inch diameter in the middle. Then warming the stone and also warming the glass, and spreading thinly and equably upon them some of this cement, lay on the cloth, and thereupon lay also the glass, having left in the middle a space the breadth of a shilling uncemented and blacked with a candle. Then provide an hollow conical plate of iron or steel (shaped like an high-crowned hat) having the basis of the cone 1 inch diameter, and having round the basis a flat border about 2½ inches diameter, and having the depth or altitude of the cone exactly of the thickness of the flate, cloth, and cement, to which the glass is fixed. The vertex of this cone must go down thro' the flate and cloth; so that being cemented on the flate, the said vertex may approach to the glass within a hair's breadth, and lie perpendicularly over the centre of the lower surface of the glass: and this must be adjusted by the circular glass described above. Within the vertex of this hollow cone, the lower point of the pole is to be applied in polishing; but it may be first proper to be observed, that fish-glue and a brass plate, in lieu and of the dimensions of the aforesaid flate, may perhaps be better. Mr Huygens observes also, that the angle of the cone should be 80 or 90 degrees, and that the hollow vertex of it should be solid enough to receive a small impression from a round steel punch, to put the point of the pole into, which might otherwise have too much liberty, and slip from the vertex. The design of the black spot in the middle of the glass, is to discover by the light of a candle obliquely reflected from your glass, after it has been polished some time, whether it be perfectly clear, and free from the appearance of any bluish colour like that of ashes.

Before the work of polishing is begun, it is proper to stretch an even well wrought piece of linen over the tool, dusting thereupon some very fine tripoly. Then taking the glass in your hand, run it round 40 or 50 times thereupon; and this will chiefly take off the roughness of the glass about the border of it, which otherwise might too much wear away the lower parts of the tool, in which the glass is chiefly to obtain its last polish. This cloth is then to be removed, and the glass is to be begun to be polished upon the very naked tool itself. But first there is to be prepared some very fine tripoly, and also some blue vitriol, otherwise called cyprian, English and Hungarian vitriol finely powdered; mix four parts of tripoly with one of vitriol: 6 or 8 grains of this mixture (which is about the quantity of two large peas) is sufficient for a glass 5 inches broad. This compound powder must be wetted with about 8 or 10 drops of clear vinegar in the middle of the tool; and it must be mixed and softened thoroughly with a very fine small mullet. Then with a coarse painting brush, take great care to spread it thinly and equably upon the tool, or at least upon a much larger space in the middle of it than the glass shall run over in the polishing. This coat must be laid on very thin, (but not too thin neither), otherwise it will waste away too much in the polishing, and the tool will be apt to be surrowed thereby, and to have its figure impaired; insomuch that sometimes a new daubing thereof must be laid on, which it is not easy to do so equably as at first. This daubing must be perfectly dried by holding over it a hot clean frying-pan, or a thin pan of iron, with light charcoal therein for that purpose; then leave all till the tool is perfectly cold. Then having some other very fine tripoly very well washed and ground with a mullet, and afterwards dried and finely powdered, take some of the same and throw it thinly and equably on the tool so prepared; then take your coarse glass which lay by you, and smooth all the said tripoly very equably and finely: then take your glass to be polished, and wipe it thoroughly clean from all cement, grease, or other filth which may stick to it, with a clean cloth dipped in water, a little tinged with tripoly and vitriol; then taking your glass in your hand, apply it on the tool, and move it gently twice or thrice, in a straight line, backwards and forwards; then take it off, and observe whether the marks of the tripoly, sticking to the glass, seem to be equably spread over the whole surface thereof; if not, it is a sign that either the tool or the glass is too warm; then you must wait a little and try again till you find the glass takes the tripoly everywhere alike. Then you may begin boldly to polish, and there will be no great danger of spoiling the figure of the glass; which in the other case would infallibly happen. If the tool be warmer than the glass, it will touch the glass harder. Mechanism of Optical Instruments

Mr Huygens says, that if the work of polishing were to be performed by strength of hand only, it would be a work of very great labour, and even could not be performed in glases of 5 or 6 feet focal distance: and he seems to think it absolutely necessary that an extraordinary great force or pressure should be applied upon the glass. For this purpose he has therefore contrived and described two methods for sufficiently increasing the pressure; both of which chiefly consist in applying the force of a strong spring to press down the centre of the glass upon the polisher.

This operation of polishing, as it is one of the most difficult and nice points of the whole, hath been very variously attempted and described by various authors. Sir Isaac Newton, Pere Cherubin, Mr Huygens, and the common glass-grinders, have taken different methods in this matter. Sir Isaac is the only person who seems not to insist on the necessity of a very violent and strong pressure. In the English 8vo edition of his Optics, p. 95, he hath these words: "An object-glass of a 14 foot telescope, made by an artificer at London, I once mended considerably, by grinding it on pitch with putty, and leaning very easily on it in the grinding, lest the putty should scratch it. Whether this may not do well enough for polishing these reflecting glasses, I have not yet tried. But he that shall try either this or any other way of polishing which he may think better, may do well to make his glasses ready for polishing by grinding them without that violence wherewith our London workmen press their glasses in grinding: for by such violent pressure, glasses are apt to bend a little in the grinding, and such bending will certainly spoil their figure."

As to his own method of polishing glasses, he nowhere expressly describes it; but his method of polishing reflecting metals he doth; and it was thus, in his own words, p. 92. "The polish I used was in this manner. I had two round copper plates each six inches in diameter, the one convex the other concave, ground very true to one another. On the convex I ground the object-metal or concave, which was to be polished, till it had taken the figure of the convex and was ready for a polish. Then I pitched over the convex very thinly, by dropping melted pitch upon it, and warming it to keep the pitch soft, whilst I ground it with the concave copper wetted to make it spread evenly all over the convex. Thus by working it well, I made it as thin as a groat; and after the convex was cold I ground it again, to give it as true a figure as I could. Then I took putty, which I had made very fine by washing it from all its groser particles; and laying a little of this upon the pitch, I ground it upon the pitch with the concave copper till it had done making a noise; and then upon the pitch I ground the object-metal with a brisk motion for about two or three minutes of time, leaning hard upon it. Then I put fresh putty upon the pitch, and ground it again till it had done making a noise, and afterwards ground the object-metal upon it as before. And this work I repeated till the metal was polished, grinding it the last time with all my strength for a good while together, and frequently breathing upon the pitch to keep it moist, without laying on any more fresh putty. The object-metal was 2 inches broad, and about ½ of an inch thick to keep it from bending. I had two of these metals, and when I had polished them both, I tried which was best, and ground the other again to see if I could make it better than that which I kept. And thus by many trials I learned the way of polishing, till I made those two reflecting perspectives I spoke of above. For this art of polishing will be better learned by repeated practice than by my description. Before I ground the object-metal on the pitch, I always ground the putty on it with the concave copper, till it had done making a noise; because, if the particles of the putty were not by this means made to stick fast in the pitch, they would, by rolling up and down, grate and fret the object-metal, and fill it full of little holes. It seems not improbable, that glasses may also be polished, with proper care, by the same method."

Pere Cherubin polishes with tripoly or putty; or first with tripoly, and afterwards with putty. But what he seems most to approve of is putty alone. He polishes in the same tool he grinds in, and describes various ways of doing it. He prescribes to stretch very tight a fine thin leather, fine English fustian, or fine Holland, or any fine linen, or fine silk taffety or satin, all of an equable thickness, as near as may be, upon the tool; then he daubs thinly on this surface, thus stretched, a streak of putty wetted to the consistence of thick syrup, about as broad as the glass, or a little more, passing through the centre of the tool directly from him; then smoothing the putty by running his bruiser, and pressing it backwards and forwards to him and from him, he at length lays on the glass cemented to its handle, and giving it always the same motion, strongly pressing to him and from him along the streak of putty, and by such pressure forcing the surface of the silk, already somewhat stretched, close to the surface of the tool, to which the figure of the glass was exactly adapted, he says that he could by that means obtain an excellent fine polish on any of the abovementioned substances. Before every stroke, he turned the glass a little on its axis; and its handle was on this occasion considerably heavier than usual in grinding, which he commends as very useful in this business; and if new putty was wanting, he made no difficulty in laying it on as often as necessary, always carefully smoothing it thereon with the bruiser before the glass was applied.

This method, according to Dr Smith, might be improved by moving the glass, not by hand, but by improving the pole and spring, somewhat after the manner of Huygens; especially if the pole were contrived not to move loose in a round brass hole above, but on a strong point pressed down by some spring; the length of the pole being equal to the radius of the tool, and the point where the spring presses the upper end of the pole, being truly perpendicular over the centre of the tool, and exactly in the centre of its sphere.

Another method preferred by Cherubin is as follows. He takes a sheet of very fine paper; and examining it carefully by looking upon it, and thro' it, he takes off with a sharp pen-knife all the little lumps, Mechanism hard parts, and inequalities, that he can find; then he soaks it in clean water; then he dries it between two fine linen cloths, tho' not so much as to make it quite dry, but to leave it dampish; then, with some very thin starch or paste, he daubs equably all over the surface of his tool as thin as possible, but some everywhere; then he lays on his paper very gently and slowly, letting it touch and stick first at one side, and by degrees more and more towards the middle, and at last so as to cover the whole. This is done slowly, in order to let the air get out; then, with the palm of his hand he presses the centre, and every where round about it towards the circumference, to make the paper stick everywhere; and this he does three or four times while it is drying, to get out all the air. He lets it dry of itself, then revives with his knife as before; then he hath a very coarse bruiser of glass, whose circumference is sharply ground round, and at right angles to its surface, which he had coarsely ground before in the same tool. With this, and with a very heavy handle, he smooths and polishes and rubs off all the remaining inequalities of the paper; and when this is done, he lays on a streak of tripoly, and polishes as in his other method.

At CC is represented a square beam of wood, a little longer than the diameter of the tool, and about 1½ inch thick: the two extremities of it at C and C are bent downwards, and then are again directed parallel to the whole length, and serve for handles for the workman to lay hold of. In the middle of this beam there is fixed an iron spike, so long, that when the lower surfaces of the handles, C, C, are placed upon a plane, the point of the spike shall just touch the plane. This point presses upon the apex of the hollow cone, which descends through the hole in the slate, which, by the interposition of a cloth, was cemented to the glass B lying upon the tool A. To increase this pressure, a sort of bow, DED, is shaped out of a deal-board, half an inch thick, and five feet long, being seven inches broad in the middle, and tapered narrower towards its extremities, so as almost to end in a sharp point. The middle of the bow is fixed to the floor by an iron staple at E driven crofs it; and is bent into an arch by a rope FFIF; to which two other ropes are tied at I and I; the interval II being equal to the length of the beam CC. One of these ropes ICCG goes over the back of the beam CC, passing through a hole in each handle at C and C, and then is lapped round a cylindrical peg at G, that passes through two wooden chaps, to the bottom of which the other rope is tied that comes from the other I. So that, by turning the peg G, to lap the rope about it, the bow DD may be bent as much as you please. The tool A is placed upon a strong square board fixed to the table O on one side, and supported on the other side by the post P. Then the workman sits down, and taking hold of the handles CC, he draws the glass to him and from him over the tool A, with a moderate motion; and after every 20 or 24 strokes, he turns the glass a little about its axis. This way of polishing took up two or three hours, and was very laborious as well as tedious; because the glass, being so much pressed downwards, was moved very slowly.

Instead of the bow DD, Mr Huygens afterwards invented another spring by sloping the flat ends of a couple of dealboards αβγδ; and by nailing the flat slopes together very firmly, that the boards might make an acute angle βαγδ. One of these boards so joined was laid upon the floor under the polishing table, the ends βγ being under the middle of the tool A. So that they lay quite out of the way of the workman, who before was a little incumbered by the ends of the bow DD. The boards at the end α were 8 or 10 inches broad, and from thence went tapering almost to a point at β and γ. The board αγ lying upon the floor, the end β, of the upper board, was pulled downwards by a rope βξ that passed under a pulley ξ fixed to the floor, and then was lapped round a strong peg η that turned stiff in a hole in the floor. Under the end γ the middle of a strong stick δη was fixed at right angles to the board αγ, and cords were tied to each end of this stick at δ, δ, which went over the polishing beam C, C, as in the former machine. This stick was lifted up but very little from the floor at the time of polishing; and by consequence the ropes ξC, ξC were long enough to give liberty of motion to the polishing beam CC. Two iron pins η, η, passing through the ends of the boards at α, were screwed into the floor; but the heads of the pins stood up above the boards, to give them liberty to rise up when the rope βξ was stretched.

To facilitate the labour of moving the glass backwards and forwards in the tool, Dr Smith made the following addition to the machine. At M is represented a strong hand made of wood or iron, having a square cavity cut through the bottom of it, for the polishing beam CC to pass through, not tight, but at some liberty. To one side of this hand M is annexed a long board LL, by means of an iron bolt. The breadth of the lower surface of this board LL is equal to the breadth of the hand M, being 2½ inches; its thickness is half an inch, and its length equal to three semidiameters of the tool. The board LL must be drawn backwards and forwards lengthwise over a block H firmly fixed to a table O; the thickness of the block being such, that the board LL may lie an inch higher than the surface of the tool A. The wooden hooks at π, and the pins at Σ, keep the motion of the board in the same direction, by hindering it from slipping either upwards or sideways. Over this board, at right angles to it, and over the middle of the block H, there lies a wooden roller, having a strong iron axis which turns in the holes of two iron plates fixed to the ends of the block. The thickness of the roller is about an inch and an half. Thro' two holes bored thro' this roller, and made wider at one end of them, two strong cords are made to pass with knots at one end of them, to be drawn into the wider parts of the holes, that they may neither slip through, nor stand out from the roller. Then each cord is lapped round the cylinder several times; and one end of each is pegged firmly into the board LL at the end towards M, and the other ends of them are lapped round a peg at N; which being turned round, will stretch the cords as much as you please. At one end of the axis of this roller there is a handle Q, which being turned round backwards and forwards alternately, the board LL with the glass annexed to it is moved to and fro, so far, that about a third part of its diameter shoots both ways over the margin of the tool. The spike in the middle of the beam CC presses the glass a little obliquely, because the hand M holds the beam CC, not tight, but somewhat loosely, to the end that the glass may pass over the tool without trembling. Nevertheless this inclination of the spike must be very small; and may easily be increased or diminished several ways. Two pins or stops must be fixed to the under surface of the board LL, to determine the length of the stroke. The tool A, or rather the stone to which it is cemented, is squeezed fast between the block H, and a strong stop on the opposite side of the stone, by the interpolation of a wedge. The workman fits upon a round stool; and, when one hand is tired with turning the roller, he applies the other; and therefore is not so soon tired as with the other machine, which required both hands, and also a reciprocating motion of the whole body. A longer handle Q was also made, which turned at both ends, for the convenience of using both hands at once.

After every 20 or 24 strokes, it is necessary to give the glass a small turn about its axis; which is easily done by laying hold of the plate fixed to it, with one hand, while the other hand goes on with the polishing motion. The tool must also be moved a little after every 25 or 50 strokes, by drawing it half a straw's breadth towards that part of it which the glass has left, and by drawing it back again after as many more strokes. At the beginning of the work the tripoly will be gathered into little lumps in some places of the tool, but will be dispersed again in a little time; and then the area of the tool will become perfectly smooth. If the tripoly does not appear to stick equally to the glass in all parts, and to be diffused over it in slender straight streaks, the frying-pan with coals in it must be held over the tool again, till you perceive the area or coat of tripoly is not quite so cold as the other parts of the tool. Then let tripoly be rubbed upon the tool again, and let the glass be pressed over it with your hand, to try whether it sticks equally to the glass in every place. When it does, you may proceed in the work of polishing. But when vitriol is used instead of verdigrase, all that is said about warming the tool may be omitted; because these coats always touch the glass as they should do, and stick better than before. The tool ought also, without being warmed, to be rubbed with tripoly over the coat, that the latter may be preferred more entire, and that the glass may touch it better, which must always be repeated after 200 or 400 strokes in polishing. The glass should also be taken from the tool after 200 strokes, by withdrawing the bolt L, which connects the hand M to the board LL, and by removing the beam CC. Then rub your finger upon the glass, or a clean rag, or a bit of leather, to examine how much it is polished.

To save the trouble of counting the strokes, there is a wooden wheel A X, seven or eight inches broad, placed against a board fixed to the side of a wall. It turns easily about an axis, and has 24 teeth, like those of a saw, which are pushed round by a bended wire TYX in the following manner. The wire turns about a centre Y; and while one end of it is pulled by the string TV tied to the end of the board LL, the opposite end YX pushes back a long spring RS, fixed to the board at R; which, by pressing upon the wire at S, causes the part YX to bend a little, and so the point X, in returning to the wheel (the string being relaxed) falls a little lower into the next tooth, and pushes it forward in the position represented in the figure. There is a springing catch at A, which stays the wheel after every stroke at X. Lastly, there is a pin fixed in the circumference of the wheel at Z, which, by pressing the tail of a hammer, and letting it go again, causes a bell to sound after every revolution of the wheel, and gives notice that the glass must be turned a little about its centre. It is easy to understand, that another piece of wheel-work, having three or four indexes, whose revolutions are in decimal progression, may be fixed to the block H, and impelled by the strokes of the board LL; by which means, without any trouble of counting, one may be informed how many strokes go to polish a glass. A glass five or six inches broad requires about 3000 strokes upon each surface to bring it to perfection. You must carefully examine the middle of the glass opposite to the blacking, whether any place appears darkish or of an ash-colour; or whether any small spots appears by an oblique reflection of the light of a candle, or of a small beam of light let into a dark room; for the other parts of the glass will appear perfectly fine much sooner than the middle.

After the glass has been sufficiently polished, let the stone, the cloth, and the cement, be warmed over a pan of charcoal, till the cement grows so soft that the glass may be separated from it by a side-motion. Then, whatever cement remains upon the glass must be wiped off with a hot cloth dipped in oil or tallow, and last of all with cleaner cloths. Then if it does not appear perfectly polished, (for we are often deceived in this point), the work must be repeated again, by glueing the glass to the plate as before; then it must be wiped very clean, and made a little rough, as we said before. We must also lay a new fund, or coat, upon the tool, if the old one be spoiled; provided no other glass has been polished in the tool in the mean time. The old fund may be washed off from the tool with a little vinegar. Lastly, take care always to choose the thickest and clearest pieces of glass, to avoid a great many difficulties that arise from the unequal pressure in polishing.

§ 3. To Centre an Object-glass.

A circular object-glass is said to be truly centered when the centre of its circumference is situated in the axis of the glass, and to be ill centered when the centre of the circumference lies beside the axis. Thus, let d be the centre of the circumference of an object-glass a b c; and suppose e to be the point where its axis cuts its upper surface. If the points d and e do not coincide, the glass is ill centered. Let a f g be the greatest circle that can be described about the centre e; and by grinding away all the margin without this circle, the glass will become truly centered. The best method for finding the centre e which lies in the axis of the glass, according to Dr Smith, is as follows.

Let a couple of short cylindrical tubes be turned How to know when wood or brass, and let the convexity of the narrower a glass is be so fitted to the concavity of the wider as just to turn ill-centered, round in it with ease, but without waddling; and let Fig. 2. Mechanism the planes of the bases of the tubes be exactly perpendicular to their sides. Place the base of the narrower tube upon a smooth brass plate or a wooden board of an equal thickness; and with any sharp-pointed tool describe a true circle upon the board round the outward circumference of the base; and upon the centre of this circle, to be found when the tube is removed, describe a larger circle upon the board. These two circles should be so proportioned, that the one may be somewhat greater, and the other somewhat smaller, than any of the glasses intended to be centered by them. Then, having cleared out all the wood within the inner circle, put the end of the tube into this hole, and there fasten it with glue, so that the base of the tube may lie in the surface of the board; then, having fixed the wider tube very firmly in a hole made in a window-shutter, and having darkened the room, lay the glass to be centered upon the board fixed to the narrower tube; and having placed the centre of it as nearly as you can guess over the centre of the hole, fix it to the board with two or three lumps of pitch, or soft cement, placed at its circumference. Then put the narrower tube into the wider as far as it can go, and fix up a smooth screen of white paper to receive the pictures of objects that lie before the window; and when they appear distinct upon the screen, turn the inner tube round upon its axis; and if the centre of the glass happens to be in this axis, the picture will be perfectly at rest upon the screen; if not, every point of it will describe a circle. With a pencil mark the highest and lowest places of any one circle, described by some remarkable point in that part of the picture which appears most distinct; and when this point of the picture is brought to the highest mark, stop the circular motion of the tube, and keeping it in that position depress the object-glass till the point aforeaid falls exactly in the middle between the two marks. Then turn the tube round again, and the point of the picture will either rest there, or will describe a much smaller circle than before; which must be reduced to a quiescent point by repeating the same operation. The centre of refraction of the glass will then lie in the axis of the tube, and by consequence will be equidistant from the circumference of the large circle described upon the board fixed to it. Now to describe a circle upon the glass about its centre of refraction, let a long slender plate of brass be bent square at each end, as represented in the figure, leaving a piece in the middle equal in length to the diameter of the large circle that was described upon the board; and let the square ends of the plate be filed away, so that a little round pin may be left in the middle of each. Then, having laid it over the glass, along any diameter of the large circle, make two holes in the board to receive the pins a and b; and find the centre of this circle upon the long plate. Then upon the centre c, describe a circle as large as you can, upon the glass underneath, with a diamond-pointed compass, and grind away all the margin as far as this circle in a deep tool for grinding eye-glasses; and then the glass will be truly centered. If the pitch or cement be too soft to keep the glass from slipping, while the circle is describing, it may be fixed firmer with wax or harder cement.

The chief advantage of having a glass well centered is this, that the rays coming through any given hole, of a well whose centre coincides with the axis of the glass, will form a distinct image than if that centre lay beside the axis; because the aberrations of the rays from the geometrical focus of the pencil, are as the distances of their points of incidence from the centre of refractions in the glass.

If the picture be received upon the unpolished side of a piece of plane glass, instead of the paper ST, its motion may be discerned more accurately by viewing it from behind through a convex eye-glass; as in a telescope where crooked hairs are usually strained over a hole put into the place of the rough glass. Therefore as object-glasses are commonly included in cells that screw upon the end of the tube, one may examine whether they be pretty well centered, by fixing the tube, and by observing, while the cell is unscrewed, whether the hairs keep fixed upon the same lines of an object seen through the telescope.

§ 4. Of the Composition of the Metals for the Specula of Reflecting Telescopes.

The properties required in the metal for the speculum of a reflecting telescope are, whiteness, hardness, and immunity from rust, or at least that it may be as little liable to tarnish as possible. Various compositions have been recommended; but the best is that published by Mr Mudge in the Phil. Trans. for 1777. His metal is a composition of copper and tin, in the proportion of two pounds of the former to 14½ ounces of the latter. If the proportion of tin was increased only by a single half ounce, the metal became so hard that it... it could not be polished. Nevertheless he tells us, that one Mr Jackson, a mathematical instrument-maker, used the tin in as large a proportion as one third of the whole. This indeed gives the metal its utmost whiteness, but at the same time renders it so exceedingly hard, that the finest washed emery will not cut it without breaking up its surface; and the common blue stones used in grinding specula will not touch it. With great pains, however, Mr Jackson found a stone which would work upon this metal, and was at the same time of a texture sufficiently fine not to injure its surface; but what this stone was, or where it was to be found, he would not discover.

Another very essential property in the metal for specula is its compactness; and in this every one of those formerly tried was deficient; neither was Mr Mudge able to remove this defect till after a great many experiments. Sometimes, indeed, he says that he succeeded in casting a single metal, or perhaps two or three, without this imperfection; but most frequently he was unsuccessful, without his being in any degree able to assign a reason. The pores were so very small, that they were not perceptible when the metal had received a good face and figure upon the hones, nor till the last and highest polish had been given; then it frequently appeared as if dotted over with millions of microscopic pores, which were exceedingly prejudicial in two respects; for, first, they became in time a lodgement for a moisture which tarnished the surface; and, secondly, on polishing the speculum, the putty necessarily rounded off the edges of the pores, in such a manner as to spoil a great part of the metal, by the loss of as much light and sharpness in the image as there were defective points of reflection in the metal; and, to add to the misfortune, this fault was not discovered till a great deal of pains had been taken in grinding, and even polishing the speculum; which was at once rendered useless by this mortifying discovery.

At last Mr Mudge was extricated from these difficulties by accident. Having made a great number of experiments, and entirely exhausted his copper, he recollected that he had some metal which was preserved out of curiosity, and was part of one of the bells of St Andrew's which had been recast. This he melted with a little fresh tin, and, contrary to his expectation, it turned out perfectly free from pores, and in every respect as fine a metal as could be desired. At first he could not account for this success, but afterwards discovered it by reflecting on the circumstances of his process. He had always melted the copper first, and, when it was sufficiently fused, he added the proportional quantity of tin; and as soon as the two were mixed, and the scoria taken off, the metal was poured into the moulds. He now began to consider that putty was calcined tin, and suspected that the excessive heat which copper necessarily undergoes before fusion, was sufficient to reduce part of the tin to this state of calcination, which therefore might fly off from the composition in the state of putty, at the time the metal was poured out. On this idea he furnished himself with some more Swedish copper and grain-tin. The former he melted as usual, and mixed the tin along with it, casting the mixture into an ingot. This was porous, as he had expected; but after a second fusion, it became perfectly close; nor, after this, did he ever meet with the above-mentioned imperfection in a single instance. All that is necessary to be done, therefore, in order to procure a metal with the requisite properties for a speculum, is to melt the copper and tin in the above-mentioned proportions; then, having taken off the scoria, cast it into an ingot. This metal must be a second time melted to cast the speculum; but as it will fuse with a small heat in this compound state, it should be poured off as soon as melted, giving it no more heat than is absolutely necessary. It must be observed, however, that the same metal, by frequent melting, loses something of its hardness and whiteness; when this is the case, it becomes necessary to enrich that metal by the addition of a little tin, perhaps of half an ounce to a pound. And indeed, when the metal is first made, if, instead of adding the 14½ ounces of tin to the copper all at once, about an ounce of the former is reserved, and added to it in the second melting, the composition will be more beautiful, and the grain much finer. That the metal may have a good surface, it is necessary, before it is poured off, to throw into the crucible a spoonful of charcoal-dust; immediately after which the metal must be stirred with a wooden spatula, and poured into the moulds.

§ 5. Of preparing the Moulds; Grinding, Casting, and Polishing, the Metal.

For this purpose Dr Smith prefers the following method. "Having in the first place considered of what length one would propose the instrument to be, and consequently what diameter it will be necessary to make the give to the large speculum, for which there are ample gauges, instructions by Sir Isaac Newton's table in the Philosophical Transactions aforesaid, allowing about an inch more than the aperture in the table for the false figure of the edges, which very often happens; I say, having determined these things, take a long pole of fir deal, or any wood, of a little more than double the length of the instrument intended, and strike through each end of it two small steel points, and by one of them hang up the same against a wall perpendicularly; then take two pieces of thin plate-brass well hammered, a little thicker than a fixpence; these may be about an inch and a half broad, and let their length be in respect of the diameter of the speculum as 3 to 2, viz., if the speculum be 8 inches diameter, these may be about 12. Fix each of these strongly with rivets between two thin bits of wainscot, so that a little more than a quarter of an inch in the breadth may stand out from between the boards. Then fix up these pieces horizontally against the wall under your pole; and therewith, as with a beam compass, strike an arch upon each of them: then file each of them with a smooth file to the arch struck, so as one may be a convex and the other a concave arch of the same circle. These brasses are the gauges to keep the speculum, and the tools on which it is ground, always to the same sphere. And that they may be therefore perfectly true to each other, it is necessary to grind them with fine emery one against the other, laying them on a flat table for that purpose, and fixing one of them to the table.

When the gauges are perfectly true, let a piece of wood be turned about 2 tenths of an inch broader for the speculum, and somewhat thicker, which Mechanism which it is best to cast in no case less than 2 tenths of an inch thick, and for specula of 6, 8, or 10 inches broad, this should be at least 3 or 4 tenths thick when finished. This board being turned, take some common pewter, and mix with it about $\frac{1}{10}$ of regulus of antimony; and with that wooden pattern cast one of this pewter, which will be considerably harder than common pewter. Let this pewter pattern be truly turned in a lathe, and examined by means of the gauges aforesaid, as a pattern for casting the specula themselves; and take care when it is turned that it be at least $\frac{1}{10}$ of an inch thicker, and about $\frac{1}{10}$ of an inch broader, than the speculum intended to be cast therefrom.

"The manner of making the moulds for casting is now to be explained; and will serve for a direction as well for casting this pewter pattern, as afterwards for casting thereby the speculum itself. The flasks had best be of iron, and must be at least two inches wider every way than the speculum intended. In each flask there should be the thickness at least of one inch of sand. The casting-sand which the common founders use from Highgate, will do as well as any; and any sand will do which is mixed with a small proportion of clay to make it stick. The sand should be as little wet as may be, and well beaten but not too hard. The ingates should be cut so as to let the metal flow in, in four or five streams, over the whole upper part of the mould; otherwise whatever pores happen in the metal will not be so equally dispersed as they should be over the whole face of the metal, these pores generally falling near the ingate streams. Let the flasks dry in the sun for some hours, or near a very gentle fire; otherwise they will warp, and give the speculum, when cast, a wrong figure. For besides saving the trouble in grinding, it is best on many accounts to have the speculum cast of a true figure; and it is for this reason, that it is best to cast it from a hard pewter pattern, and not from a wooden one as founders usually cast."

With regard to the proper metal, opticians have been greatly at a loss, till of late that Mr Mudge has discovered a composition which answers every purpose as well as can be expected, and of which an account hath been already given. "The metal being duly cast, the surface of it is to be ground quite bright upon a common grindstone; keeping it, by means of your convex gauge, as near the figure as may be. When all the outward surface and sand-holes, false parts, and inequalities, are ground off, then provide a good thick stone; a common small grindstone will do very well. Let its diameter be to the diameter of the speculum as 6 to 5; with another coarse stone and sharp sand or coarse emery rub this stone till it fits the concave gauge; and then with water and coarse emery at first, and afterwards with finer, rub your speculum upon this stone until it forms itself into a true portion of a sphere fitting your convex gauge. A different method of moving the metal upon the stone will incline it to form itself somewhat of a smaller or larger sphere. If it be struck round and round, after the manner of glaas-grinders, the stone will wear off at the outides, and the metal will form itself into the portion of a less sphere. If it be struck cross and cross the middle, it will flat the stone, and become somewhat of a larger sphere. There should be used but very little emery at a time, and it ought to be frequently changed; otherwise the metal will always be of a smaller sphere than the stone, and will hardly take a true figure, especially at the outside. For the better grinding the metal, it is necessary, that this stone should be placed firm upon a strong round board fixed firmly on a post to the floor, as is usual with glass-grinders; and the same table or pillar will serve for the further grinding and polishing the speculum.

"When the metal is cast and rough-figured, which should be done with taking off as little of the surface of the metal as possible, (because that crust seems generally to be harder and more solid than the inner parts) the sides and back of it should be smoothed and finished; lest the doing that afterwards should make the metal cast, and spoil the figure of the foreside.

"A round brass plate must be cast of sufficient breadth and thickness (for a speculum of six inches in diameter requisite Mr Hadley used a brass plate 8 or 9 inches broad, polishing and half an inch thick.) Let one side be turned to the concavity you design your speculum to have, on the other side let it have such an handle fastened as may make it easily manageable. This handle should be as short as conveniently it can, and be fixed to the plate's back rather by some other method than either by screwing it into a hole in the metal, or by a broad shoulder screwed against the back of it, for fear of bending the plate. Have ready a round marble of about $\frac{1}{3}$ or $\frac{1}{4}$ broader than the brass plate, and an inch or an inch and a quarter thick: let this be cut by a stone-cutter to the same convexity on one side as the concavity of the plate, and then grind it with the plate and emery till all the marks of the chisel are out. This marble is to be covered with pieces of the finest blue hone or whetstone, choosing those that are nearest of a breadth and thickness; but chiefly those that when wetted appear most even and uniform in their colour and grain. They are to be cut into square bits; and these, having each one side ground concave on the convex marble with emery or fine sand, are to be fixed close down on it with some tough and strong cement in the manner of a pavement, leaving a space of a small straw's breadth between each; their grain being likewise placed in an alternate direction, as represented in the figure. I choose rather to disperse the squares that come out of the same whetstone, than to keep them together. They must then be reduced to one common convex surface to fit the brass plate; and if the cement happen to rise anywhere between them, so as to come up even with the surface, it must be dug out; and so, from time to time, as often as the stones wear down to it. Upon these square pieces of whetstone the last figure is to be given to the speculum.

"Besides these, there will be wanted for the last polish, either a very thick round glass plate, (its diameter being about the middle size between that of the brass tool and the speculum itself,) or if that cannot be procured of near half an inch in thickness, a piece of true black marble of the evenest grain and freest from white veins or threads, may do in its stead. This glass or marble must be figured on one side to the brass tool likewise, and is to serve for finishing of the polish of..." Mechanism of the speculum, when covered with sarcenet as shall be directed.

"A smaller bras or metal plate of the same concavity with the larger will be useful, as well to help to reduce the figure of the hones whenever it appears to be too convex, as to serve for a bruiser to rub down any gritty matter happening to be amongst your putty before you put the speculum on the polisher, when you renew the powder. Any of the speculums which prove bad in eating, will serve for this purpose.

When all is thus far ready, let the marble with the blue hones be fixed in such a manner that it may be often washed during your work, by throwing upon it about half a quarter of a pint of water at a time without inconvenience. Then place the bras tool on the hone pavement; and rub it backwards and forwards with almost a direct motion; yet carrying it by turns a little to the right and left, so as to go a little over the edges of the pavement every way, regularly turning the tool on its own axis, and also changing the direction of the stroke on the hones. This continue, keeping them always very wet, till you have got out all the rings remaining in the plate from the turning, and the blackness from grinding the marble or glass in it; and, towards the latter end, often washing away the mud which comes from the whetstones. When this is done, lay the bras tool down, and in it grind again with fine emery the glass or marble designed for the last polisher, giving it as true a figure as possible.

Choose a piece of fine sarcenet as free from rows and great threads as you can. Let it be three or four inches broader than the glass or marble; and turn down the edges of the sarcenet round the sides of the glass, &c. Strain it by lacing it on the backside as tight and smooth as you can, having first cleared it of all wrinkles and folds with a smooth iron, and drawn out the knots and gouty threads. Then wet it all over as evenly as you can with a pretty strong solution of common pitch in spirit of wine; and when the spirit is dried out, repeat the same; and if any bubbles or blisters appear under the sarcenet, endeavour to let them out with the point of a needle. This must be repeated till the silk is not only stuck everywhere firmly down to the glass or marble, but is quite filled with the pitch. A large painter's pencil, made of squirrel's hair, is of use for spreading this varnish equally on the silk, especially when it begins to be full. It must then be set by for some days, for the spirit to dry well out of it, and the pitch to harden, before anything more be done to it. If you do not care to wait so long, the pitch may be melted into the silk without dissolving it in spirits. In order to this, strain a second thin silk over the first, but you need not be curious in the choice of it; and having heated all together as hot as you think the silk or glass will safely bear, pour on it a little melted pitch (first strained through a rag) so much as you judge sufficient to fill both silks; it must be kept hot for some time till the pitch seems to have spread itself evenly all over. If you cannot get it to sink all into the upper silk, but it stands above it anywhere, it is a sign that there was too much pitch laid on, which should be taken away in those places while it remains liquid, with a hot rag pressed down on it. When all is cold again, strip off the outward silk, and cut away the useless loose edges of the inward. To take off the superfluous pitch where it lies too thick, and reduce the whole to a regular surface, it must be rubbed in the bras tool with a little soap and water, till they are coloured of a pretty deep brown with the pitch; then wash them away, and repeat the same with more soap and water, till the weaving of the silk appears everywhere as equally as you can make it. As this work takes up some time, you may expedite it by putting a few drops of spirit of wine to the soap and water (which will help them to dissolve and wear away the pitch somewhat faster till it comes towards a conclusion; and if there are any places where the pitch lies very thick, you may scrape it away with a sharp knife.

This polisher must be carefully kept from all dust and grit, but particularly from emery and filings of hard metals, and therefore should not be used in the place where the others come. After they have served a good while, they are more apt to sleek the metals than at first; to prevent which, their surfaces may be taken off by rubbing them with soap and water in the tool as before, and then striking them over once or twice with the abovementioned solution of pitch with a pencil, proceeding as before; only that you must not now put any spirit to your soap and water, nor will you need to change them above once or twice.

You may now begin to give the figure to your speculum on the hones, rubbing it and the bras tool on them by turns, till both are all over equally bright; having first fixed on to the middle of the back of your speculum a small and low handle, with only pitch strained through a rag. For of all cements, that seems the least apt to bend the metals in sticking these handles, &c. on them.

The polisher being fixed likewise in a proper manner for your work, rub either the metal itself, or rather the before-mentioned bruiser, being first also figured on the hones, with a little putty, washed very fine, and fair water, till it begins to show some polish. Then if you find it takes the polish unequally, that is, more or less about the edges than in the middle, it is a sign the bras tool and metal, &c. are more or less concave than to answer the convexity of the polisher; and must be reduced to the curvature of this, rather than to attempt an alteration in the figure of the polisher, which would be a much more difficult as well as laborious work. If the speculum appears too flat, the larger bras tool must be worked on the hones for some time, keeping its centre near their circumference, with a circular motion; but concluding for four or five minutes with such a motion as was before described. Then figure the metal anew on the hones, and try it again on the polisher as before. If the metal be too concave, the surface of the hones may be flatted by rubbing the smaller bras plate, or the before-mentioned ill-cast metal, on the middle of them; with a direct but short stroke, so as but just to reach over their circumference with the edge of it. Then the larger bras is to be worked on them in the same manner; and last of all the metal to be polished. When you find the bras tool and hones, &c. answer the curvature of the polisher, you may then examine the truth of the figure of the speculum more strictly, to avoid the loss of time and labour in finishing its polish while the figure is imperfect.

Place "Place the speculum in a vertical posture on a table about 3 or 4 feet from the floor. On another table set a candle whose flame should be about the level of the middle of the speculum, and very near the centre of its concavity. About \( \frac{1}{2} \) an inch before the flame, place a flat tin, or thin brass plate about 3 inches broad, but 4 or 5 high, having several holes about the middle, of different shapes and sizes; some of them as small as examining the point of the sharpest needle will make them, the biggest about the size of a large mustard-seed: darken the room, and move this candle and plate about on the table, till the light from the brightest part of the flame, passing through some of the larger holes to the speculum, is reflected back so as to form the images of those holes close without one of the side-edges of that thin plate. Those largest images in this case will be visible, (although the speculum have no other polish than what the hones give it), when received on a thick white card held close to that edge of the plate, if the back of the card be either blacked or so shaded that the candle may not shine through it, and the eye be also screened from the candle's direct light. If any difficulty happens in discerning them, the plate may be removed, and the image of the whole flame will be easily seen. Have ready an eye-glass whose focal distance may be something greater than the double of that of the eye-glass you intend for the instrument when finished: you may try several at your discretion. Let this be supported by a small stand moveable on the table, and capable of raising and sinking it as the height of the flame requires, and of turning it into any direction. By means of this stand, bring the eye-glass into such a position, that the light from some of the holes, after its reflection from the speculum, may be received perpendicularly on its surface; and that its distance from the speculum be such, that the reflected images of the holes may be seen distinctly through it, near the edge of the thin plate, by the light coming immediately from the speculum: guide the candle and thin plate with one hand, and the stand carrying the eye-glass with the other, till you have got them into such a situation, that you see distinctly at the same time, through the eye-glass, the edge of the thin plate, and the image of one of the holes close to it. Measure the exact distance of the middle of the speculum from the thin plate directly against the flame, and also from the edge close to which you see the image of the hole. If these measures are the same, set it down as the exact radius of concavity of your speculum, and proper curvature for any that are to be polished on your polisher, though that will allow some latitude: if the measures aforesaid differ, take the mean between them.

"You will now also judge of the perfection of the spherical figure of your metal by the distinctness with which you see the representations of the holes, with their raggedness, dents, and small hairs sticking in them; and you will be able to judge of this more exactly, and likewise to discover the particular defects of your speculum, by placing the eye-glass so as to see one of the smallest holes in or near its axis; and then by shoving the eye-glass a very little forward towards the speculum, and pulling it away, by turns, letting the candle and plate stand still in the mean time. By this means you will observe in what manner the light from the metal comes to a point, to form the images, and opens again after it has past it. If the mechanism of the light, just as it comes to or parts from the point, appears not round, but oval, square, or triangular, &c. it is a sign that the sections of the specular surface, through several diameters of it, have not the same curvature. If the light, just before it comes to a point, have a brighter circle round the circumference, and a greater darkness near the centre, than after it has crossed and is parting again; the surface is more curve towards the circumference, and flatter about the centre, like that of a prolate spheroid round the extremities of its axis; and the ill effects of this figure will be more sensible when it comes to be used in the telescope. But if the light appears more hazy and undefined near the edges, and brighter in the middle before its meeting than afterwards, the metal is then more curve at its centre and less towards the circumference; and if it be in a proper degree, may probably come near the true parabolic figure. But the skill to judge well of this, must be acquired by observation.

"In performing the foregoing examination, the image must be reflected back as near the hole itself as the eye's approach to the candle will admit of; that the obliquity of the reflection may not occasion any sensible errors: in order to which, the eye should be screened from the candle; and the glaring light, which may disturb the observation, may be still more effectually shut out, by placing a plate, with a small hole in it, in that focus of the eye-glass which is next the eye. A is the speculum, B the candle and plate with the Fig. 10. small holes, C the cell with the eye-glass and plate behind it.

"Instead of the flame of the candle and plate with small holes, I sometimes make use of a piece of glass thick stuck with globules of quicksilver, strained thro' a leather, and allowed to fall on it in a dew; placing this glass near a window, and the speculum at a distance on the side of the room, being itself and every thing about it as much in the dark as can be. The light of the window reflected from the globules of mercury, appearing as so many stars, serves instead of the small holes, with this advantage, that the reflection from the metal may be very near at right angles.

"If the figure of the metal appears not satisfactory, the hones must be worked with the bras tool and the water for 2 or 3 minutes with the motion, &c. first directed; then work the metal on them with the like motion, and such length of the stroke as may carry the edge of it about \( \frac{1}{6} \) or \( \frac{1}{4} \) of its diameter beyond that of the hone pavement each way; carry it likewise by turns to the right and left, to about the same distance. Continue this about 5 minutes, not pressing the metal down to the hones with any more than its own weight, and observe that the oftener the mud is washed away, the more truly spherical the figure of the speculum will generally be: but the leaving a little more of this mud on the stones has sometimes seemed to give the metal a parabolic figure. I have likewise given it the same, by concluding with a kind of spiral motion of the centre of the metal, near the circumference of the hones, in the manner represented in fig. 11, for about half a minute.

"If after several trials the metal appears to have always the same kind of defect, and answering to the same Mechanism of Optical Instruments

Mechanism of Optical Instruments

When the figure is to your mind, you may proceed to finish the polish on the farce net with very little putty, and that diluted with a great deal of water. Before you put the putty and water on it, observe, by holding it very obliquely between your eyes and the light, if it have any lifts or stripes across it, which appear more glossy than the rest. If it be so, let the motion of the metal in polishing be directly athwart these lifts, and not along with them, nor even circular. In other respects you may observe the same directions as were before given for its motion on the hones; not forgetting, after every 15 or 20 strokes, to turn it on its axis about \( \frac{1}{4} \) or \( \frac{3}{4} \) of a revolution. As the polisher grows dry, you will find the metal stick to it more and more stiff; at which time it both polishes faster and with a better glost; only take care that it grows not so dry as for the metal to take hold of the farce net and cut it up, or for the pitch and putty to fix in little knobs here and there on it; which, if it happen, will presently spoil the figure. As fast therefore as the farce net appears to be growing dry at any of its edges, touch the place with the end of a feather dipped in clean water: you may use the same putty at least half an hour. As often as you change it, wash the old clean away, and rub the new about first with your bruiser, to see if there be any gritty or grofs particles in it, and rub them away for fear of scratching the metal; then laying down the edge of the speculum a little way on the edge of the polisher, where it is well covered with water, slide it on the middle, and then proceed. The less putty you use at a time, the slower the work will advance; but if you use too much, it will spoil a little of the figure round the edges. It will not want any considerable force to press it down; but if it be of 5 or 6 inches diameter or more, it will be very laborious to go through the polish without some kind of machine.

Mr Mudge is of opinion that all this troublesome method is entirely unnecessary, and of the same opinion is an anonymous French author who wrote on this subject in the year 1738. The latter tells us what is certainly agreeable to reason and experience, that the more complicated the machines are by which we attempt to accomplish any purpose, the more liable we are to error by reason of their perpetual tendency to go wrong, and the necessary multiplication of inaccuracy is a complicated motion. Four tools, according to Mr Mudge, are all that are necessary; viz. the rough grinder to work off the rough face of the metal; a bras convex grinder, on which the metal is to receive its spherical figure; a bed of hones, which is to perfect that figure, and to give the metal its smooth fine face; and a concave tool or bruiser, with which both the bras grinder and the hones are to be formed. A polisher may be considered as an additional tool; but as the bras grinder is used for this purpose, and its pitchy surface is expeditiously and without difficulty formed by the bruiser, the apparatus is therefore not enlarged.

The tool by which the rough surface of the metal is rendered smooth and fit for the hones, is best made of lead stiffened with about a sixth part of tin. This tool should be at least a third more in diameter than the metal which is to be ground; and for one of any size, not less than an inch thick. It may be cemented upon a block of wool, in order to raise it higher from the bench. This leaden tool being cast, it being fixed in the lathe, and turned as true as possible by the gauge to the figure of the intended speculum, making a hole or pit in the middle for a lodgement to the emery, of four inches; when this is done, deep grooves must be cut across its surface with a graver, as is represented fig. 7. These grooves will serve to lodge the emery, and by their means the tool will cut a great deal faster. There is no reason to fear any alteration in the convexity of this tool by working the metal upon it; for the emery will bed itself in the lead, and so far arm the surface of it, that it will preserve its figure, and cut the metal very fast. Any kind of low handle, fixed on the back of the metal, with soft cement, will be sufficient; but it should cover two thirds of its back, to prevent its bending.

This way of working (says Mr Mudge) will cut the metal faster, and with more truth, than the method described by Dr Smith; for should the surface and rough parts be attempted to be ground off by a common grindstone by hand, though you did it as near the gauge as possible, yet the metal would be so much out of truth when applied to the succeeding tool, that no time would be saved by it. For this purpose Mr Mudge used to employ a common labourer, who soon acquired such dexterity at working upon the tool, that in two hours time he would give a metal of four inches diameter so good a face and figure as even to fit it for the hones.

When all the sand-holes and irregularities on the face of the metal are ground off, and the whole surface is smooth and regularly figured, the speculum is then ready for the bras grinder, and must be laid aside for the present.

The bras grinding-tool is formed in the following manner. Procure a round stout piece of Hamburgh bras, at most a fifth part larger than the metal to be polished; and let it be well hammered, by the assistance of the gauge, into a degree of convexity suitable to the intended speculum. Having done this, scrape and clean the concave side so thoroughly, that it may be well tinned all over; than cast upon it, after it has been pressed a proper depth into the sand, the composition of tin and lead above-mentioned, in such quantity, that it may, for a speculum of four inches diameter, be at least an inch and half thick, and with a base considerably broader than the top, in order that it may stand firmly upon the bench hereafter to be described. This being done, it must be fixed and turned in the lathe with great care, and of such a convexity as exactly to suit the concave gauge. More care will be necessary in forming this tool than the former, especially that no rings be left in turning; nor will the succeeding hone-tool require so much exactness, as any defects in turning will, by a method hereafter mentioned, be easily removed; but any inequality or want of truth in the bras tool will occur. Mechanism of Optical Instruments

How to form the bed of hones.

Manner of forming the bruiser.

Of grinding the speculum, the bras tool, and the bruiser, together.

Mechanism of Optical Instruments

A great deal of trable before it can be ground out by the emery. This tool must have a hole, somewhat less than that in the metal to be worked upon it, in the middle, quite through to the bottom. When this tool is finished off in the lathe, its diameter should be one-eighth wider than the metal.

The hones should be of the best sort of those recommended by Dr Smith. They should be cemented in small pieces (in a kind of pavement, as hath been already mentioned) upon a thick round piece of marble, or metal made of lead and tin in the proportions above directed, in such a manner, that the lines between the stones may run straight from one end to the other; so that placing the teeth of a fine saw in each of these divisions, they may be cleared from one end to the other of the cement which riles between the stones. This bed of hones should be at least one-fourth larger than the metal which is to be ground upon it; but there is no necessity for turning the metal on which the hones are cemented to the same convexity with the gauge. As soon as the hones are cemented down, and the joints cleared by the saw, this tool must be fixed in the lathe, and turned as exactly true to the gauge as possible; which done, it must be laid aside for the present. The next tool to be made is the bruiser.

The bruiser should be made of thick stout bras like the former, perfectly sound, about a quarter of an inch thick, and hammered as near to the gauge as possible. It should then be scraped, cleaned, and tinned on the convex side, as the former tool was on the concave, and the same thicknesses of lead and tin cast upon it. The general shape of this should differ from the former; for as that increased in diameter at the bottom for the sake of standing firmly, so this should be only as broad at bottom as at top, as it is to be used occasionally in both those positions. When this tool is fixed in the lathe, and turned off concave to the convex gauge with great truth likewise, its diameter ought to be the middle size between the hones and the polisher.

Having with the lathe roughly formed the convex bras grinder, the bed of hones, and the concave polishers, the convex and concave bras tools and the metal must be wrought alternately and reciprocally upon each other with fine emery and water, so as to keep them as nearly to the same figure as possible; in order to which, some washed emery must be procured. This is best done by putting it into a vial, which must be half filled with water and well shaken up, so that, as it subsides, the coarsest may fall to the bottom first, and the finest remain at the top; and whenever fresh emery is laid upon the tools, the best method (which we should also observe with the putty in polishing) will be, to shake gently the bottle, and pour out a small quantity of the turbid mixture.

The tools being now all ready, upon a firm post in the middle of a room, you are to begin to grind the bras convex tool with the bruiser upon it, working the latter crosswise, with strokes sometimes across its diameter, at others a little to the right and left, and always so short, that the bruisers may not pass above half an inch within the surface of the bras tool either way, shifting the bruiser round its axis every half dozen strokes or thereabout. You must likewise, every now and then, shift your own position, by walking round, and working at different sides of the bras tool; at times the stroke should be carried round and round, but not much over the tool; in short, they must be directed in such a manner, and with such equability, that every part of both tools may wear equally. This habit of grinding, as well as the future one of polishing, will soon be acquired. When you have wrought in this manner about a quarter of an hour with the bruiser upon the tool, it will be then necessary to change them, and, placing the bruiser upon its bottom, to work the convex tool upon that in the same manner.

When, by working in this equable manner alternately with the bruiser and tool, and occasionally adding fresh emery, you have nearly got out all the vestiges of the turning tool, and brought them both nearly to a figure, it will then be time to give the same form to the metal. This must be done by now and then grinding it upon the bras tool with the same kind of emery; taking care, however, by working the two former tools frequently together, to keep all three exactly in the same curve. The best kind of handle for the metal is made of lead, a little more than double its thickness, and somewhat less in diameter, of about three pounds weight, with a hole in the middle, (for reasons to be afterwards shown), a little larger than that in the metal; this handle should be cemented on with pitch. The upper edge of this weight should be rounded off, that the fingers may not be hurt; and a groove about the bigness of a little finger be turned round just below it, for the more conveniently holding and taking the metal off the tools.

When the bruiser, bras tool, and metal, are all brought to the same figure, and have all a true good figuring the metal upon the hones, it is necessary to observe, however, that the hones should be placed in a vessel of water, with which they should be quite covered for at least an hour before they are used; otherwise they will be continually altering their figure when the metal comes to be ground upon them. The same precaution is also necessary if you are called off from the work while you are grinding the metal; for, if they be suffered to grow dry, the same inconvenience will ensue.

In order to give a proper figure to the hones, and exactly suitable to that of the bras tool, bruiser, and metal, when the hones are fixed down to the block, some common flour emery (unwashed), with a good deal of water, must be put upon them, and the bruiser being placed upon the hones and rubbed thereon with a few strokes and a light hand, the inequalities of the stone will be quickly worn off; but, as a great deal of mud will be suddenly generated, it must be washed off every quarter of a minute with plenty of water. By a repetition of this two or three times, the hones (being of a very soft and friable nature) will be cut down to the figure without wearing or altering the bruiser at all. Tho' this business may be quickly done, and can be continued but for a few strokes at a time, it is absolutely necessary that those strokes be carried in the same direction, and with the same care, which was observed in grinding the former tools together. As soon as the hones have received the general figure of the bruiser, and all the turning strokes are worn out from them, the emery must be carefully washed off; in order to which, it will be necessary to clear it from the joints with a brush, under a stream of water. The bruiser and metal must likewise be cleared in the same manner, and with equal care, from any lurking particles of emery.

The hones being fixed down upon the block, you now begin to work the bruiser upon them with very cautious, regular, short strokes, forward and backward, to the right and left, turning the axis of the bruiser in the hand, while you move round the hones by shifting your position, and walking round the block. The whole now depends upon a knack in working, which should be conducted nearly in the following manner. Having placed the bruiser on the centre of the hones, slide it in an equable manner forward and backward, with a stroke or two directly across the diameter, a little on one side, and so on the other. Then, shifting your position an eighth part round the block, and having turned the bruiser in your hand about as much, give it a stroke or two round and round, but not far over the edges of the hones, and then repeat the cross-strokes as before: those round strokes, which ought not to be above two or three at most, are given every time you shift your own position and that of the metals previous to the cross ones, in order to take out any stripes, either in the hones or bruiser, which may be supposed to be occasioned by the straight cross strokes. During the time of working, no mud must be suffered to collect upon the hones, so as to destroy the perfect contact between the two tools; and therefore they must every now and then be washed clean by throwing some water upon them. When, by working in this manner, all the emery strokes are ground off from the bruiser, and it has acquired a good figure and clean surface, you may then begin with the metal upon the hones, in the same cautious manner, washing off the mud as fast as it collects; though that will be much less now than when the bruiser was ground upon them. Every now and then, however, the bruiser must be rubbed gently and lightly upon the hones, which will, as it were, by sharpening them, and preventing too great smoothness, occasion them to cut the metal faster.

After having, by working in this manner, taken out all the emery strokes, and given a fine face and true figure to the metal, which will be pretty well known by the great equality there is in the feel while you are working, and by which an experienced workman will form a pretty certain judgement, you may then try your metal, and judge of its figure by the following more certain method.

Wash the hone pavement quite clean; then put the metal upon the centre of it, and give two or three strokes round and round only, not carrying, however, the edges of the metal much over the hones; this will take out the order of straight strokes: then, having again washed the hones, and placed the speculum upon their centre, with gentle pressure, slide it towards you, till its edge be brought a little over that of the hones; then carry it quite across the diameter as far on the other side, and having given the metal a light stroke or two in this direction, take it off the tool. The metal being wiped quite dry, place it up on a table at a little distance from a window; stand yourself as near the window, at some distance from the metal, and looking obliquely on its surface, turn it round its axis, and you will see at every half turn the grain given by the last cross strokes flash upon your eye at once over the whole surface of the metal. This, says Mr Mudge, is as certain a proof of a true spherical figure, as the operose and difficult method described by Dr Smith: for as there is nothing soft or elastic either in the metal, or in the hones, this glare is a certain proof of a perfect contact in every part of the two surfaces; which there could not be, if the spheres were not both perfect and precisely the same. Indeed there is one accidental circumstance which affords its aid in this and other similar cases; namely, that a concave and convex surface ground together, though ever so irregular at first, (will, if the working be uniform and proper, consisting, especially at last, of cross strokes in every possible direction across the diameter) be formed into portions of true and equal spheres. Had it not been for this lucky necessity, it would have been impossible to have produced that correctness which is essential in the speculum of a good reflecting telescope by any mechanical contrivance whatever. For when it is considered, that the errors in reflection are four times as great as in refraction, and that the least defect in figure is magnified by the powers of the instrument, anything short of perfection in the figure of the speculum would be evidently perceived by the want of distinctness in the performance.

Here, however, Mr Mudge observes, that he all along supposes, both in forming the tools, and at last in figuring the metal (and the same is to be observed in the future process of polishing), that no kind of pressure is used that may endanger the bending or irregularly grinding them: they should therefore be held with a light hand, and loosely between the fingers; and the motion given should be in a horizontal direction, with no other pressure than their own dead weight.

Having now finished the metal on the hones, and rendered it both in point of figure and surface fit for the last and most essential part of the process, viz. that of polishing, we shall now proceed to describe it as minutely as possible, though many little circumstances must necessarily be omitted, and can only be supplied by experience.

The polishing of the speculum is the most difficult and essential part of the process; and every experienced workman knows, to his vexation, that the most trifling error here will be sufficient to spoil the figure of his metal, and render all his preceding caution useless. On this occasion also Mr Mudge makes the following remarks on the method of polishing used by Hadley and Molyneux, and already described from Dr Smith's Optics. "First then, says he, the tool itself used by them in polishing the metal is formed with infinite difficulty. The first described polisher is directed to be made by covering the tool with sarsenet, which is to be saturated with a solution of pitch in spirit of wine, by successive applications of it with a brush, till it is covered, and by the evaporation of the spirit of wine filled with this extract of pitch; the surface is then to be worked down and finished with Mechanism the bruiser. This is all very easy in imagination; but whoever has used this method (which I have myself, unsuccessfully, several times) must have found it attended with infinite labour, and at last the business done in a very unsatisfactory manner; for the pitch by this process will be deprived of an essential part of its composition. The spirit of wine dissolves none but the resinous parts of its substance, which is hard and untractable; and if you use soap or spirit of wine to soften or dissolve it, it will equally affect the whole surface, the lower as well as the higher parts of it. And suppose that, with infinite labour with the bruiser, it is at last reduced to a fine uniform surface, it is nevertheless too hard ever to give a good polish with that lustre which is always seen in good metals. Nor will it give a good spherical figure: for a perfect sphere is formed, as I observed before, by that intimate accommodation arising from the wear and yielding of both tool and metal; whereas, in this method there is such a stubbornness in the polisher, that the figure of the metal, whether good or bad, must depend upon the truth of the former, which is very seldom perfect.

"If the polisher be made in the second manner proposed, viz. by straining the pitch through an outer covering, which is afterwards to be stripped off, the superficies of pitch and sarcenet is so very thin, that the putty working into them forms a surface hard and untractable, so that it is impossible to give the speculum a fine polish. Accordingly all those metals which are wrought in this way have an order of scratches instead of polish, discovering itself by a greyish visible surface. Besides, supposing this tool perfectly finished, and answering its purpose ever so well, it is impossible that it can produce in the speculum any other than a spherical figure; and indeed nothing else is expected from this method, as is evident from the experiment recommended to ascertain the truth of it.

You are directed to place a small luminous object in the centre of the sphere of which the metal is a segment; and then having adjusted an eye-glass at the distance of its own focal length from the object, and so situated that the image of the object formed by the speculum may be visible to the eye, you are to judge of the perfect figure of the metal by the sharpness and distinctness with which the image appears. From hence it is very evident, that as the object and image are both distant from the metal by exactly its radius, nothing but a true spherical figure of the speculum can produce a sharp and distinct image; and that the image could not be distinct if the figure of the speculum were parabolic. Consequently, if the same speculum used in a telescope were to receive parallel rays, there would necessarily be a considerable aberration produced, and a consequent imperfection in the image. Accordingly, there never was a good telescope made in this manner; for if the number of degrees, or the portion of the sphere of which the great metal is a part, were as considerable as it ought to be, the instrument would bear but a very low charge, unless a great part of the circumference of the metal were cut off by an aperture, and the ill effects of the aberration by that means in some measure prevented.

"If ever a finished metal turned out without this defect, and has been found perfectly sharp and distinct, it must have been owing to an accidental parabolic tendency, nowadays the natural result of the process, and therefore quite unexpected, and most probably unknown to the workman."

Our author next acquaints us, that, from observing the high polish of some of the metals made by Mr Short, and concluding that the high lustre of the polish could never have been produced in the manner above described, but by some more soft and tender substance, he was directed to make use of pitch itself, especially as Sir Isaac Newton mentions his having used that substance in his operations. Accordingly, shortening Dr Smith's process, he made a set of tools in the manner above-mentioned, except that he was obliged to make some subsequent alteration in the polisher. Having given a good spherical figure to the brass tool and the bruiser, and likewise to the metal upon the hones, and made the brass convex tool so hot as just not to hurt the finger, he tied a lump of common pitch, which should neither be too hard nor too soft, in a rag, and holding it in a pair of tongs over a still fire where there was no rising dust, till it was ready to strain through the linen, he caused it to drop on the several parts of the convex tool, till he supposed it would cover the whole surface to about twice the thickness of a shilling; then spreading the pitch as equally as he could, he suffered the polisher (the name he gives to the tool so prepared) to grow quite cold. He then made the bruiser so hot as almost to burn his fingers; and having fixed it to the bench with its face upwards, he suddenly placed the polisher upon it, and quickly slid it off; by this means rendering the surface of the pitch somewhat more equal. The pitch is then to be wiped off from the bruiser with a little tow; and by touching the surface with a tallow candle, and wiping it a second time, it will then be perfectly clean, and fit for a second process of the same sort, which must again be performed as quickly as possible; and this is ordinarily sufficient to give a general figure to the surface of the pitch. The bruiser and the polisher are then suffered to grow perfectly cold; when the pitch, considering what has been taken off, will be about the thickness of a shilling.

Here, however, it is necessary to observe, that the pitch should neither be very hard and resinous, nor too soft; if the former, it will be so untractable as not to work kindly; and if too soft, it will in working alter its figure faster than the metal, and too readily fit itself to the irregularity of its figure, if it have any. When both tools are perfectly cold, he gave the polisher a gentle warmth, and then fixed the bruiser to the block with its face upwards; and (having, with a large camel's-hair brush, spread over the face of the polisher a little water and soap to prevent sticking, with short, straight, and round strokes, he worked it upon the bruiser, every now and then adding a little more water and soap, till the pitch upon the polisher had a fine surface and the true form of the bruiser; and this he continued till they both grew perfectly cold together: in this manner the polisher was formed in about a quarter of an hour. But here a difficulty arose. For when he began to polish the metal, he found that the edge of the hole in the speculum collected the pitch towards the middle of the polisher; hence, though in this method of working he he could give an exquisite polish, as the putty lodged itself in the pitch exceedingly well, yet the figure of the metal was injured in the middle; nor indeed did the work go on with that equability which is the inseparable attendant on a good figure. In order to obviate this difficulty, he cast some metals with a continued face, the holes not going quite through, within perhaps the thickness of a sixpence. In this way he finished two or three metals, and the work went on very well; but when he came to open the holes, even though the utmost caution was used, the metals were found to be imperfect. This he attributed to an alteration of the figure from the removal of even that small portion of metal after the speculum had been finished. This he supposes to have been the cause of his spoiling a very distinct and perfect two-foot metal, which bore a charge of 200 times, only by opening the sharp part of the edge of the hole, because he thought that it bounded the field: so essentially necessary is an exquisite correctness of figure in the speculum of a perfect reflector.

This experiment not succeeding, instead of casting the metal without a hole, he made one quite through the middle of the polisher, a little less than that in the speculum. This perfectly answered the purpose; no more inconvenience arose from the gathering of the pitch, for it had now no greater tendency to collect at the centre than the sides; and thus he finished several metals successively, excellent both in figure and polish. One of these, of 2 inches diameter and 7.5 focal length, bore a charge of 60 times and upwards.

In this method of working, the polishing goes on in an agreeable, uniform, and smooth manner; and the small degree of yielding in the pitch, which is actually not more than the wearing of the metal, produces that mutual accommodation of surfaces so necessary to a true figure. In the beginning of the polish, and indeed for some time during the progress of it, (always remembering now and then to move the metal round its axis), he worked round and round, not far from, and always equally distant from, the centre; except that every time, previous to the shifting the metal on its axis, he used a cross stroke or two; and when the polish was nearly completed, he used mostly cross strokes, giving a round stroke or two likewise every time he turned the metal on its axis. In this method of working, he always observed that the metal polished fastest in the middle; insomuch that one half or two thirds of it would be completely polished, when the circumference was scarcely touched by the tool. Observing this in some of the first metals, and not considering that this way of polishing was in fact a species of grinding, and as perfect as that upon the hones, he went on reluctantly with the work, almost despairing of being able to produce a good figure. However, he was always agreeably disappointed; for when the polish was extended to the edge, or within a tenth of an inch of it, he almost constantly found the figure good, and the performance of the metal very distinct. But this same circumstance of apparent defect in the metals, was in fact that to which their perfection was owing; for they all, contrary to his expectation, turned out parabolic. On the other hand, when he chanced to find that a metal, when first applied to the polisher, took the polish equally all over, and consequently the business did not take up above 10 minutes; yet the metal constantly turned out good for nothing. From frequent observations, however, he at last found a method of giving a correct parabolic figure and an exquisite polish at the same time.

In polishing the speculum, in order to avoid the intrusion of any particles of emery, it would not be right to polish in the same room where the metal and tools were ground, nor in the same cloths which were worn in the former process; at least it would be necessary to keep the bench quite wet, to prevent any dust from rising.

Having then made the polisher, by coating the brass convex tool equally with pitch, which we suppose smoothed and finished with the brass tool in the manner before described, and which is a very easy process, the whole operation is begun and finished in the following manner.

The leaden weight, or handle, upon the back of the metal, should be divided into eight parts, by so many deep strokes of a graver upon the upper surface of the lead, marking each stroke with the numbers 1, 2, 3, 4, and so on, that the turns of the metal in the hand may be known to be uniform and regular.

To prevent any mischief from coarse particles of putty, it must be washed immediately before using. In order to this, put about half an ounce of putty into an ounce phial, and fill it two-thirds with water; then having shaken the whole, let the putty subside, and stop the bottle with a cork.

In a tea-cup with a little water, there should be a full-sized camel's-hair brush, and a piece of dry clean soap in a galley-pot: a soft piece of sponge will also be necessary. These, as well as the metal polisher and polisher, should be constantly covered from dust.

The polisher being fixed down, and the camel's-hair brush being first wetted and rubbed a little over the soap, let every part of the tool be brushed over therewith; then work the polisher with short, straight, and round strokes, lightly upon the tool, and continue to do so, now and then turning it, till the polisher have a good face, and be fit for the metal. Then having shaken up the putty in the phial, and touched the polisher in five or fix places with the cork wetted with that and the water, place the polisher upon the tool, and give a few strokes upon the putty to rub down any gritty particles; after which, having removed it, work the metal lightly upon the polisher round and round, carrying the edges of the speculum, however, not quite half an inch over the edge of the tool, and now and then with a cross stroke.

The first putty, and indeed all the succeeding applications of it, should be wrought with a considerable while; for if time be not given for the putty to bed itself in the pitch, and any quantity of it lie loose upon the polisher, it will accumulate into knobs, which will injure the figure of the metal; and therefore as often as ever such knobs arise, they must be carefully scraped off with the point of a penknife, and the loose stuff taken away with the brush. After the putty is well wrought into the pitch, some more may be added in the same manner, but never much at a time; and always remembering to work upon it first with the polisher, for fear any gritty particles may find their way upon... Mechanism upon the polisher. If the bruiser be apt to stick, and do not slide smoothly upon the pitch, the surface of either tool may be occasionally brushed over with the soap and water, but it must be remembered that the wet brush must be but lightly rubbed upon the soap.

In the beginning of this process little effect is produced, and the metal does not seem to polish fast, in some measure owing to its taking the polish in the middle, and perhaps because neither that nor the bruiser move evenly upon the polisher: but a little perseverance will bring the whole into a good temper of working; and, when the pitch is well defended by the coating of the putty, the process will advance apace, and the former acquiring possibly some little warmth, the metal moves more agreeably over it, with an uniform and regular friction. All this while the metal must have no more pressure than that which it derives from its own weight and that of the handle; and the polisher must never be suffered to grow dry, but, as often as it has any tendency to do so, the edges of it must be moistened with the hair-pencil; and now and then, even when fresh putty is not laid on, the surface of the polisher should be touched with the brush to keep it moist.

When the polish of the metal nearly reaches the edge (for it always, as we said before, begins in the middle) you must alter your method of working; for now the round strokes must be gradually altered for the short and straight ones. Supposing then you are just beginning to alter them; after having put on fresh putty, and gently rubbed it with two or three strokes of the bruiser, you place the metal on the tool, and after a stroke or two round and round, give it a few forward and backward, and from side to side, but with the edges very little over the tool; then having turned the metal one-eighth round in your hand, and having moved yourself as much round the block (which must be remembered throughout the whole process) you go on again with a stroke or two round, to lead you only to the crofs strokes, which are now to be principally used, and with more boldness. After this has been done some time, the metal will begin to move stiffly as the friction now increases, and the speculum polishes very beautifully and fast; and the whole surface of the polishing tool will be equally covered with a fine metallic bronze. The tool, even now, must not be suffered to become dry; a single round stroke in each of your stations and turnings of the metal will be sufficient, and the rest must all be crofs ones, for we are completing a circular figure. You must now be very diligent for the polisher drying, and the friction increasing very fast, the benefits of the spherical figure is nearly at an end. As the metal wears much, its surface must be now and then cleaned, with a piece of shanny leather, from the black stuff which collects upon it; and the polisher likewise from the same matter, with a soft piece of wet sponge. You will now be able to judge of the perfect spherical figure of the metal and tool, when there is a perfect correspondence between the surfaces, by the fine equable lines there is in working, which is totally free from all jerks and inequalities. Having proceeded thus far, you may put the last finishing to this figure of the metal by bold crofs strokes, only three or four in the directions of each of the eight diameters, turning the metal at the same time: this must be done quickly; for it ought, in this part of the mechanism particularly, to be remembered, that, if you permit the tool to grow quite dry, you will never be able, with all your force, to separate that and the metal, without destroying the polisher by heat.

The metal has now a beautiful polish and a true spherical figure, but will by no means make a sharp distinct image in the telescope: for the speculum (if it be tried in the manner hereafter recommended) will not be found to make parallel rays converge without great aberration; indeed the deviation will be so great, as to be very sensibly perceived by a great indistinctness in the image.

In order then to give the speculum the last and finishing figure, which is done by a few strokes, it must be particularly remarked, that by working the metal round and round, the sphere of the polisher by this means growing less, it wears fatter in the middle; and as a segment of a sphere may become parabolic, by opening the extremes gradually from within outwards, so it may be equally well done by increasing the curvature in the middle, in a certain ratio, from without inwards.

Supposing then the metal to be now truly spherical, stop the hole in the polisher, by forcing a cork into it underneath, about an inch, so that it do not reach quite to the surface; and having washed off any mud that may be on the surface of the tool with a wet soft piece of sponge, whilst the surface of it is a little moist, place the centre of the metal upon the middle of the polisher; then having, with the wet brush, lodged as much water round the edge of the metal as the projecting edge will hold, fill the hole of the metal and its handle with water, to prevent the evaporation of the moisture, and the consequent adhesion between the speculum and polisher, and let the whole rest in this state two or three hours: this will produce an intimate contact between the two, and by parting with any degree of warmth they may have acquired by the vicinity of the operator, they will grow perfectly cold together.

By this time you may push out the cork from the polisher, to discharge the water, and give the metal the parabolic figure in the following manner.

Move the metal, gently and slowly at first, a very little round the centre of the polisher (indeed after this rest it will move stiffly); then increasing by degrees the diameter of these strokes, and turning the metal frequently round its axis, give it a larger circular motion, and this without any pressure but its own weight, and holding it loosely between the fingers: this manner of working may safely be continued about two minutes, moving yourself as usual round the block, and carrying the round strokes in their increased and largest state, not more than will move the edge of the metal half an inch or five-eighths over the tool. The speculum must not all this while be taken off from the polisher; and consequently no fresh putty can be added. It will not be safe to continue this motion longer than the time above mentioned; for if the parabolic tendency be carried the least too far, it will be impossible to recover a true figure of that kind but by going through the whole process for the spherical one in the manner before described, by the crofs strokes upon the polisher, which takes a great deal of time. However, when there is occasion, it may be done; and Mr Mudge has several times recovered the circular figure when he had inadvertently gone too far with the parabolic, and ultimately finished the metal on the polisher without the use of the hones.

It will now be proper to try the figure of the speculum; and that is always best done by placing it in the telescope it is intended for. In order to this, Mr Mudge uses the instrument as a kind of microscope; placing the object, however, at such a distance that the rays may be nearly parallel. At about 20 yards, a watch-paper, or some such object, on which there are some very fine hair-strokes of a graver, is fixed up. The lead must then be taken off from the back of the speculum; which is best done by placing the edge of a knife at the junction of the lead and metal, when, by striking the back of it with a slight blow, the pitch immediately separates, and the handle drops off; the remaining pitch may be scraped off with a knife, taking care that none of the dust stick to the polished face of the metal.

Having placed the speculum in the cell of the tube, and directed the instrument to the object, make an annular kind of diaphragm with card-paper, so as to cover a circular portion of the middle part of the metal between the hole and the circumference, equal in breadth to about an eighth part of the diameter of the speculum; this paper ring should be fixed in the mouth of the telescope, and remain so during the whole experiment; for the part of the metal covered by it is supposed to be perfect, and therefore unemployable.

There must likewise be two other circular pieces of card-paper cut out, of such sizes, that one may cover the centre of the metal by completely filling the hole in the last described annular piece; and the other, such a round piece as shall exactly fit into the tube, and so broad as that the inner edge may just touch the outward circumference of the middle annular piece. It would be convenient to have these two last pieces so fixed to an axis that they may be put in their places, or removed from thence, so easily, as not to displace or shake the instrument. All these pieces therefore together will completely shut up the mouth of the telescope.

Let the round piece which covers the centre of the metal, or that which has no hole in it, be removed; and, by a nice adjustment of the screw, let the image (which is now formed by the centre of the mirror) be made as sharp and distinct as possible. This being done, every thing else remaining at rest, replace the central piece, and remove the outside annular one, by which means the circumference only of the speculum will be exposed, and the image now formed will be from the rays reflected from the outside of the metal. If there be no occasion to move the screw and the little metal, and the two images formed by these two portions of the metal be perfectly sharp and equally distinct, the speculum is perfect, and of the true parabolic curve; or at least the errors of the great and little speculum, if there be any, are corrected by each other.

If, on the contrary, under the last circumstance, the image from the outside of the metal should not be distinct, and it should become necessary, in order to make it so, that the little speculum be brought nearer, it is plain that the metal is not yet brought to the parabolic figure; but if, on the other hand, in order to procure distinctness, you be obliged to move the little speculum farther off, then the figure of the great speculum has been carried beyond the parabolic, and hath assumed an hyperbolic form. When the latter is the case, the circular figure of the metal must be recovered (after having fixed on the handle with soft pitch) by bold cross strokes upon the polisher, finishing it again in the manner above described. If the speculum be not yet brought to the parabolic form, it must cautiously have a few more round strokes upon the polisher; indeed a very few of them in the manner before described make in effect a greater difference in the speculum than would be at first imagined. If a metal of a true spherical figure were to be tried in the above-mentioned manner in the telescope (which Mr Mudge has frequently done) the difference of the foci of the two segments of the metal would be so considerable, as to require two or three turns of the screw to adjust them; so very great is the aberration of a spherical figure of the speculum, and so improper to procure that sharpness and precision so necessary to a good reflecting telescope.

This is by no means the case with the object-glasses of refractors; for besides that they are in fact never so distinct as well-finished reflectors, the apertures of them are so exceedingly small, compared to the latter, and the number of degrees employed so very small, that the inconvenience of a spherical figure is not so much perceived. Accordingly we observe in the generality of reflectors, (whose specula, unless by accident, are always spherical), that the only true rays which form the distinct image arise from the middle of the metal: and unless the defect be remedied by a considerable aperture, which destroys much light, the false reflection from the inside of the metal produces a greyish kind of haziness, which is never seen in Mr Short's, or indeed in any good telescopes.

Supposing that the two foci of the different parts of the metal perfectly coincide; and that, by the union of them when the apertures are removed, the telescope shews the objects very sharp and distinct, you are not, however, even then to conclude that the instrument is not capable of farther improvement: for you will perceive a sensible difference in the sharpness of the images, under different positions of the great speculum with respect to the little one, by turning round the great metal in its cell, and opposing different parts of it to different parts of the little metal, correcting by this means the error of one by the other. This attempt should be persevered in for some time, turning round the great speculum about one-sixteenth at a time, and carefully observing the most distinct situation each time the eyepiece is screwed on: when, by trying and turning the great metal all round, the distinctest position is discovered, the upper part of the metal should be marked with a black stroke, in order that it may always be lodged in the cell in the same position. This is the method Mr Short always used; and the caution is of so much consequence, that he thought it necessary to mention it very particularly in his printed directions for the use of the instrument.

And, farther, Mr Short frequently corrected the errors of the great by the little metal in another way. If the great speculum did not answer quite well in the Mechanism telescope, he cured that defect sometimes by trying of the effect of several metals successively, by this means correcting the errors of one by the other; for in several of his telescopes which have passed thro' our author's hands, when the sizes and powers have been the same, he has found that the great metals, tho' very distinct in their proper telescopes, yet have, when taken out and changed from one to the other, spoiled both telescopes, rendering them exceedingly indistinct, which could arise from no other circumstance. For this reason he supposes it was, that Mr Short kept, ready furnished, a great many large metals of the same focal length, so that, when he wanted to mount a telescope, he might from a great choice be able to combine those metals which suited each other best. Our author is strongly inclined to believe this was the case, not only from the above observation, but because he shewed him a box of finished metals, in which he is sure there were a dozen and a half of the same focal length.

To return: A little use in working will make the whole of the process of grinding and polishing very easy and certain; for though we have endeavoured to be as particular as possible, it is yet scarcely possible to supply a want of dexterity, arising from habit only, by the most laboured and minute description. And though the above account may appear irksome to the reader, as it lies cold before the eye, it is hoped, whoever attempts to make the instrument, will not complain of it as tediously particular.

It may, however, be farther remarked, that when the metal begins to move stiffly upon the polisher, and particularly when the figure is almost brought to the parabolic form, it will be necessary to fix the elbows against the sides, in order to give momentum and equability to the motion of the hand by that of the whole body.

The same polisher will serve for several metals, if it be somewhat warmed when you begin to use it.

There is another circumstance, and a material one too, which must not be omitted; it is this. For the very same reason that the pitch should not be too hard or soft, the work will not proceed well in the heat of summer, or the cold of winter: in the latter, it may be possible to remedy the defect by having the room warmed with a stove; and in the summer, the other inconvenience may perhaps be avoided by using a harder kind of pitch; but our author much doubts in either case whether the work will go on so kindly: he has himself always wrought in spring and autumn.

The process of polishing, and indeed grinding upon the hones, will not go on so well if it be not continued uninterruptedly from beginning to end; for if the work of either kind be left but for a quarter of an hour, and you then return to it again, it will be some time before the tool and metal can get into a kindly way of working; and till they do, you are hurting what was done before.

We have all along supposed that the metal we have been working was about four inches diameter; if it be either larger or smaller, the sizes of the hones, bruiser, and polisher, must be proportionably different. Our author says he never found any ill consequence arising from the different expansion from heat and cold in any of the tools, though they be made of different metals and substances, unless the inconvenience, occasioned by the interruption before hinted at, be thought to result from thence; for the alteration produced in the surface of the speculum, both by grinding and polishing, is so much quicker than any that can be supposed to arise from the former cause, that it is never attended with any practical consequence.

Magnifying very minute objects, and particularly test of reading at a distance, have been generally considered as the surest test of the goodness of a telescope; and indeed, when the page is placed at a great distance, so that the letters subtend but a very small angle at the eye, if then they appear with great precision and sharpness, it is most probable that the instrument is a good one. But we are, nevertheless, sometimes apt to be deceived by this method; nor is it always possible to determine upon the different merits of two instruments of equal power, by this mode of examination; for when the letters are removed to the utmost extent of the powers of the two instruments, the eye is apt to be prejudiced by the imagination. If two or three words can be here and there made out, all the rest are guessed at by the sense; insomuch that an observer, zealous for the honour of his instrument, is very apt to deceive himself in spite of his intentions. The surer test is by figures, where you can procure no aid from this sort of deception. In order to examine reflecting telescopes, our author made upon a piece of copper, and on a black ground, six lines consisting of about 12 pieces of gold figures, and each line of figures differing in magnitude; from the smallest that could be distinctly made, to those of about two-tenths of an inch long; moreover, the figures in the several lines were differently disposed, and the sum of each line also differed. It is evident that by this method all guess is precluded; and that of two instruments, of the same powers, that which can make out the least order of figures, which will be known by the sum, is the best telescope. Such a plate he caused to be fixed up for experiments against the top of a steeple, about 300 yards north of his house; and it will serve to give some idea of the distinctions with which very small figures could be made out at that distance, by saying, that in a clear state of the air, and with the sun behind him, with a telescope of 18 inches focal length, which count Bruhl did him the honour to accept, and now has in his possession, he has seen the legs of a small fly, and the shadows of them, with great precision and exactness.

"I cannot conclude, (says our author,) without indulging myself in an observation on the amazing sagacity of Sir Isaac Newton in every subject upon which he thought fit to employ his attention. It was he who first proposed, and indeed practised, the polishing with pitch: a substance, which at first sight, perhaps, every one but himself would have thought very improper, from its softness, to produce that correctness of figure so necessary upon these occasions; and yet I do believe, that it is the only substance in nature that is perfectly calculated for the purpose: for at the same time that it is soft enough to suffer the putty to lodge very freely on its surface, and for that reason to give a most tender and delicate polish; it is likewise totally inelastic, and therefore never, from that principle, suffers any alteration in the figure you give it. If the first makers of the instrument, therefore, had given proper credit to, or had simply followed the hint Mechanism hint Sir Isaac gave, it would have saved them infinite trouble, and they would have produced much better instruments; but the pretended refinement, of drawing a tincture from pitch with spirits of wine, affords you only the resinous, hard, and untractable part of the pitch, divested of all that part of its original substance, which is necessary to give it that accommodating pliability in which its excellence consists.

It is needless to swell this account with a detail of the process for polishing the little speculum, as it must be conducted in the same manner which has been already described in that of the large one: only observing, that as the little metal has an uninterrupted face, without a hole, so there is no occasion for one in the polisher; and likewise that, as a spherical figure is all that need here be practically attempted, so the difficulty in finishing is infinitely short of that of the other.

As it is always necessary to solder to the back of the little speculum a piece of brass, as a fixture for the screw to adjust its axis, Mr Mudge mentions a safe and neat method of doing it, which may be very useful to the optical or mathematical instrument-maker upon other occasions. Having cleaned the parts to be soldered very well, cut out a piece of tin-foil the exact size of them; then dip a feather into a pretty strong solution of sal ammoniac in water, and rub it over the surfaces to be soldered; after which place the tin-foil between them as fast as you can (for the air will quickly corrode their surfaces so as to prevent the folder taking), and give the whole a gradual and sufficient heat to melt the tin. If the joints to be soldered have been made very flat, they will not be thicker than a hair: though the surfaces be ever so extensive, the soldering may be conducted in the same manner; only care must be taken, by general pressure, to keep them close together. In this manner, for instance, a silver graduated plate may be soldered on to the brass limb of a quadrant, so as not to be discernible by anything but the different colour of the metals. This method was communicated to our author by the late Mr Jackson, who during his life kept it a secret, as he used it in the construction of his quadrants.

In Plate CCXXVIII. are figured the shape of the leaden tool for rough-grinding; the hones; and the apparatus to be applied in the mouth of the telescope, to ascertain the true figure of the speculum.

Fig. 7. The grinder for working off the rough face of the metal: the black strokes represent deep grooves made with a graver.

Fig. 8. The bed of hones, which is to complete the spherical figure of the speculum, and to render its surface fit for the polisher.

Fig. 9. An apparatus for examining the parabolic figure of the speculum.

AA. The mouth of the telescope, or edge of the great tube.

BB. A thin piece of wood fastened into and flush with the end of the tube; to which is permanently fixed the annular piece of pasteboard CC, intended to cover and to prevent the action of the corresponding part of the speculum.

D. Another piece of pasteboard, fixed by a pin to the piece of wood BB, on which it turns as on a centre; so that the great annular opening HH, may be shut up by the ring FF, or the aperture GG by the imperforate piece E, in such manner, that in the first instance the reflection may be from the centre, and in the latter from the circumference, of the great speculum.

§ 6. Description of the different parts of which Reflecting Telescopes are composed, and of fitting up an instrument of that kind; with the rectification of Telescopic sights of Quadrants, &c.

In Plate CCXXIX. all these are distinctly represented.

Fig. 1. Shews the form of a pair of pincers necessary on several occasions, particularly for breaking off the corners of a piece of glass, in order to make the eye-glasses.

Fig. 2. 3. Two wooden frames for confining the sand in which the metallic specula are to be cast.

Fig. 6. 7. Two iron moulds in which are to be cast two models of lead for the specula. These models are afterwards to be turned as exactly as possible to the gauges, and then used for giving the form necessary to the sand in the frames.

Having cast the specula, and polished them according to the directions already given, you must next provide a tube of plate brass, well smoothed, for the body of your telescope; and whose length must be determined by the focal length of the large speculum. This tube must be painted black on the inside, in order to reflect as little extraneous light as possible. When the telescopes are small, brass is the usual material of the tube; but when large, the expense will be lessened by making them of wood. This tube must have a slit in one of its sides, for allowing the small mirror to slide up and down.

AB is a circle of brass, to be soldered round the mouth of the tube CD, in order to keep on the cover. Of this cover EFHI shews one piece, which is another brazen circle fitting the one AB so closely that it cannot be taken off or put on without some difficulty. G is a solid plate, which being fitted to the middle vacancy of the former, completes the cover.

LM is another circle of the same materials which contains the large speculum, and to which is soldered the piece NO, having a hole in it to receive the eye-piece of the telescope. xyz Is a thin piece of copper, a little bent, on which the speculum is laid, and which by its spring keeps it stiff in its place.

The eye-piece may be composed of two tubes RPQ and YdZ, of which the latter slides upon the former. KZ is the extremity of the eye-piece; and hath a small hemisphere perforated, in order to admit the light to the eye. The various parts of this extremity are represented at b, c, and v, which will give a more perfect idea of it than any description. In this eye-piece are placed the two lenses which magnify the image from the mirrors, and which are kept in their proper places by the rings S, T, V, X.

The small mirror is now to be fixed exactly in the middle of the tube, which is best done by such a contrivance as is shewn at gbi, k and n. f is a ruler of brass, which, sliding along with the piece g, preserves, by means of its crofs branches, the small mirror from falling from side to side as the telescope happens to be turned. l is a round piece of brass, fastened on the Mechanism of immoveable piece i, and which holds the small speculum. By means of a dove-tail slit it can be moved up and down till we find the position of the speculum is right, after which it is to be firmly screwed on to the piece i. m is a small piece of brass, having a female screw, in which the rod r o is screwed, and which moves the little speculum up or down, as it shall be found necessary for procuring distinct vision.

Having now got all the parts of your telescope, it is necessary in the first place to see that the tubes are perfectly straight and round; after which you may proceed to place your mirrors in the telescope, and to prove their situation by the following method.—AB CD, fig. 8. is a circle drawn on a round piece of pasteboard, having a small hole in the centre at E. FGHI (fig. 9.) is another perforated circular piece of pasteboard, having two hairs crossing each other as in the figure. The former of these is placed just behind the large mirror; the latter in the place where the nearest eye-glass should stand. If the light passing through the small hole in the large circle falls exactly on the intersection of the hairs, it shews that the large speculum is properly placed; if not, its situation must be altered till this is accomplished.

AB(fig. 10.) shews the shape of another piece of pasteboard, likewise perforated in the centre at C. The small circle is to be of the diameter of the lesser speculum: and when the pasteboard is put exactly in its focus, the light will pass straight through the little hole and eye-piece, so as to be distinctly visible if the position of the speculum is exactly right; but if that is not the case, the light will fall either to one side or other, and the position of the speculum must be altered accordingly.

Fig. 12. Shews a telescope of the Newtonian form, in which the plane speculum is somewhat nearer the large one than in those formerly described: in consequence of which this requires a small eye-piece at the side, that the magnifying glass may be placed at a sufficient distance from it. This telescope is to be adjusted in the following manner. Let there be provided two circles of pasteboard, represented fig. 13. and 14., both of which are perforated in such a manner, that the tube of the telescope may just enter the perforation. The circle fig. 13. is divided into quadrants, at each of which is pricked a hole with a pin. That represented fig. 14. is also divided into quadrants; but, instead of pin holes, has black lines drawn upon it. The former is to be fixed on the open end of the tube, and the latter on the end where the concave speculum is placed. The telescope is then to be turned towards the sun, so that the little specks of light passing through the pin-holes of the circle fig. 13. fall upon the black circular lines of fig. 14. Have then ready another piece of pasteboard, fig. 15. perforated with pin-holes in five different places as there represented. This piece must exactly fit the opening of the telescope; and while the tube continues thus turned straight to the sun, look through the eye-piece. If all the specks of light coming through the holes in the pasteboard are seen distinctly in the plane mirror, it is a sign that the mirrors are in a proper position with regard to each other: but if not, some of them will not be seen at all, or will appear confused and indistinct; in which case, the situation of the mirrors must be altered till the light appears bright and distinct.

In the application of telescopes to astronomical instruments and many other purposes, it is absolutely necessary to fix the plane of the cross-hairs exactly upon the plane of the picture of an object; which may easily be done from a knowledge of the following properties. First, let the interval between the two convex glasses of the telescope be adjusted to show an object distinctly; and if the hairs appear confused, they will seem to dance upon the object, while the eye moves sideways; and in dancing, if they seem to move the same way as the eye does, they lie behind the picture of the object; but if they move the contrary way, they lie before it; and must be removed accordingly, till they appear distinct; and then they will also seem fixed upon the object, notwithstanding the motion of the eye. Secondly, let the interval between the hairs and the eye-glass be first adjusted, till the hairs appear distinct; then, if the object appears confused, it will also appear to dance while the eye moves sideways; and in dancing, if it moves the same way as the eye does, its picture is behind the hairs; if the contrary way, its picture is before them; and to bring it to the hairs, either the object-glass must be moved, or else the hairs and eye-glass both together. In both these cases, it is the confused object (for the hairs may also be called so) that seems to move, and the distinct one to stand still; as in vision with the naked eye. For, to a person in motion, suppose he be walking, any object appears fixed that he fixes his eyes upon and sees distinctly, while the rest that are nearer or farther off appear confused and in motion; the reason of it is too obvious to need an explanation. But to shew it in the telescope, let b be the intersection of the cross-hairs, and h k a pencil of rays flowing from it, which, after refraction through the eye-glass c a i, belong to the focus k, either at a finite or infinite distance. Draw h e, the axis of this pencil, cutting the object in Q, and its picture in q; and let the emergent rays of the pencil g a b, flowing from q, cut the emergent rays of the former pencil in the points p, and belong to the focus b, either at a finite or an infinite distance. Now, if the eye be placed at any point o in the common axis of these pencils, the points h, Q, will both appear in the same direction o; but if the eye be moved sideways from o to p, the point Q will appear in the direction p a, and the point b in the direction p i. And from hence the reason of the foregoing cases will be sufficiently manifest, by attending to the figures. Lastly, while the focuses h q are disjoined, the mutual inclination of the emergent rays in one pencil, must be different from the mutual inclination of the emergent rays in the other; and so the humours of the eye cannot be adapted to collect the rays in both pencils to two distinct points. If one be distinct, the other will be confused, and in a different part of the retina; (except when the eye is in the axis;) but when the focuses h, q, are united, the focuses k, b, of the emergent rays will also be united; and consequently the coinciding rays of both pencils will be united in the same point of the retina, wherever the pupil of the eye be placed; and therefore the corresponding points of the object and cross-hairs will appear fixed together without any parallax. When the place of the hairs is thus determined, it may be of use to measure their distance from the object-glas; which is the exactest way of finding its focal distance, if the object be very remote. And to keep this distance always the same whenever the telescope is used, it is convenient to have marks or stops at the end of each joint of the tube. For then, whatever eyeglas be applied, the object and hairs will both appear distinct at the same time, and without parallax. Instead of hairs, the finest silver wires are now made use of, but are still called hairs.

A line drawn from the intersection of the hairs through the centre of refractions in the object-glas, whether it coincides with the axis of the glas or is inclined to it, is called the line of collimation or line of sight; because this line produced, falls upon the object in that point whose image falls upon the intersection of the hairs; and therefore the straight ray that describes this line, answers to the visual ray by which we take aim at an object with plain sights. Hence, when the object-glas and hairs are firmly fixed in a strong tube, or to a straight ruler, it is manifest, that the line of sight is as immutable with respect to the tube, as if two little holes or plain sights were substituted in the places of the intersection of the hairs, and of the centre of refractions in the object-glas.

In order to set the line of sight parallel to a given line upon the plane of an instrument, the object-glas must be firmly fixed, and the ring or plate that carries the cross-hairs must have two gradual motions in its own plane by two screws at right angles to each other; for by this means the intersection of the hairs may be moved to any given point in that plane. These motions are effected by three brass plates laid over one another. The uppermost, having a circular hole in it, over which the hairs are strained, slides over the middlemost in the direction of an oblong hole cut in it, whose breadth is somewhat greater than that of the hole above it; and these two together slide sideways over the undermost plate, in which there is a larger oval hole. We shall describe these plates more particularly in a contrary order. On each side of the oval hole in the middle of the plate R last mentioned, two brass ledges m, n, are firmly riveted to receive the dovetail sides of the plate S; and the contiguous ends of both these plates are turned up square at b and c; and through a hole b, in the middle of the part turned up in the larger plate R, there works a pretty thick screw a b c, whose fore-end c being filed to a neck, goes through a hole e in the lip of the other plate S; and in the end of the neck c there is made a small screw hole to receive a screw-pin d; so that by turning the screw a b c with a kind of a watch-key, the plate S is moved backwards or forwards between the ledges m, n. The figure T represents two more ledges o, p, that are to be riveted upon the plate S; these ledges are part of the plate T turned up at right angles to them, in which part there is the like contrivance of a screw a b c d to move a third plate V between the ledges o, p, at right angles to the former motion. The silver wires are strained over the hole in the plate V by four small pegs, that fix them in four little holes. The other end of the plate R, opposite to the part b that carries the screw, is bent square the contrary way to the part b; or, which answers the same purpose, one ledge e f of the plate X bent square, is riveted to the backside of the plate R at the end opposite to the screw b; and its other ledge g h is screwed to the side of the tube of the telescope; and the necks of the screws go through long slits in this ledge, to give liberty of placing it accurately at the due distance from the object-glas; and for the purpose of letting this bras work into the tube, two large slits must be cut in two contiguous sides of it; one of which may best be covered with a thin piece of horn, to admit the light of a candle upon the hairs in observing small stars in the night time.

To make the line of sight through a moveable telescope parallel to a given line YZ upon a fixed plane; Fig. 5, let the ends of the tube of the telescope, whether square or cylindrical, be put through two holes in two square plates a b c d, and e f g h, made exactly equal to each other, and so fixed to the tube that the sides of the one may be exactly parallel to the sides of the other; which is easily done by applying their corners a, e, to the given line YZ, and by drawing two lines a i, e k, perpendicular to it, upon the given plane, and by making all the corresponding sides, as a b, e f, coincide with these perpendiculars. Then observe what point of a remote object is covered by the intersection of the hairs when the corners a, e, touch the given line YZ, and likewise what other point is covered by them when the opposite corners c, g, touch the same line in the same places, that is, when the telescope is turned upside down or half round. Then conceiving these two points of the object to be connected by a straight line, move the cross-hairs by the two screws, till you judge their intersection bisects that line; and by repeating the same practice, you may soon bring the intersection of the hairs to cover one and the same point of the object, when the opposite corners of the squares are successively applied to the line YZ; and then the line of sight will be parallel to it.

To show the reason of this practice, we may suppose the centre of refraction in the object-glas to be any point l of the square a b c d, and the intersection of the hairs to be any point m of the square e f g h. Upon the plane of the first square, and through its centre o, draw l o, and take o λ equal to o l; also upon the plane of the second square, and through its centre p, draw m p, and take p ν equal to p m. Join l m and ν μ, and supposing l n and ν parallel to the axis o p, join m n, n ν, ν μ. Then because the respective sides about the equal angles m p, n μ, p ν, are made equal, the lines m n, n ν, opposite to them, are also equal and parallel. Now the parallel lines n l, p o, ν μ, produced will fall upon a remote object in three points so close together as to appear like a single point thro' the telescope; and consequently the planes of the parallel triangles l m n, ν μ, produced, will cut the same object in two parallel lines so close together as to appear but one line through the telescope; and since the angles n l n, ν μ, are equal, the intersection of the hairs, now at m and then turned half round to ν, will cover two points in that line equidistant from the point abovementioned, and on opposite sides of it; therefore, by removing the intersection from m to ν, it will appear to bisect the interval between those two points; and then the line of sight ν l will be parallel. Mechanism to the axis \( p \), and to the sides of the parallelopiped, and also to the given line \( YZ \).

A telescope thus prepared, may be useful upon several occasions; as if it be required to rectify the hairs in a telescope fixed to any instrument, so as to set the line of sight parallel to a given line upon the plane of the instrument. Apply the corners of the squares of the telescope abovementioned to the given line, and observing what point of a remote object is covered by its cross hairs, move the cross hairs of the fixed telescope till they cover the same point of the object, and the business is done.

But the telescopic sights of quadrants and sextants, whose planes may be readily placed in any given posture, may be rectified by a plumb-line. We shall therefore transcribe an account of these rectifications from Mr Molyneux's Dioptries, p. 238. "I come now to the rectification of these sights on quadrants and sextants, for taking angles. This may be done either before or after the division into degrees, &c. are made on the limb of the quadrant. If it be done before, then we suppose the telescope TL fixed to the quadrant, which we suppose continued a little farther than the fourth part of a circle. Choosing then an object pretty near the horizon; let us look thro' the telescope, in the usual posture of observation, and observe the point in the object marked by the cross hairs; and at the same time we are to note most nicely the point \( c \), which the plumb-line \( f \), hung from the centre \( f \) of the quadrant, cuts on the limb. Then we are to invert the quadrant into the posture of fig. 19. (which is easily done by the usual contrivances for managing great quadrants, by toothed semicircles and endless screws) keeping still the telescope TL nearly upon the same height from the ground as before, unless the object we look at be so far distant, that the breadth of the quadrant subtends but an insensible angle. But yet for certainty, it is better to keep the telescope, as it is said, upon the same height from the floor; then direct the telescope TL, that the cross hairs may cover exactly the same point in the object, as before in the posture of fig. 18. And hanging now the plumb-line \( a \) on the limb of the quadrant, let us remove it to and fro, till we find out the exact point \( a \), from which the plumb-line being hung, shall most nicely hang over the centre of the quadrant \( f \). Then carefully marking the point \( a \), let us divide the arch \( ca \) into two equal parts in \( b \); and drawing \( bf \), the point \( b \) is the point from which we are to begin the divisions of the quadrant: and the line of collimation through the telescopic sight, stands exactly at right angles to the line \( bf \). So that the quadrant \( bfd \) being completed and divided, the said line of sight thro' the telescope runs exquisitely parallel to the line \( fd \).

"In the next place, supposing the quadrant \( bfd \) truly completed and divided; and that we designed to fix thereto the telescopic sight TL, so that the line of sight may run exactly at right angles to the line \( bf \), or parallel to the line \( df \); we are to do as in the foregoing praxis. And if, in dividing the arch \( ac \), we find its half exactly coincident with the point \( b \), we have our desire. But if it differs from the point \( b \), and falls between \( b \) and \( d \), then the line of collimation through the telescope stands at an obtuse angle with the line \( bf \); and the instrument errs in excess: if this half arch fall without \( b \) and \( d \), then the line of collimation makes an acute angle with the line \( bf \); and the instrument errs in defect. And by often trials, we are to remove the cross hairs within the tube, so much as is requisite to correct this error. And when we have thus rectified them to their due place, there they are to be strongly fixed. Or else, in observations taken by this instrument, we are to make allowance for this error; by subtracting from (if it be in excess), or by adding to (if it be in defect), each observation, so much as we find the error to be.

"The reason of this rectification is most plain; for it is manifest, that \( cfd \) wants of a full quadrant, as much as \( afd \) exceeds a quadrant. So the difference of the two arches in the two postures being \( ac \); half this difference \( bc \) added in fig. 18. or \( ab \) subtracted in fig. 19. makes \( bda \) a complete quadrant.

"If we find our instrument errs in taking angles, and we desire to know the error more nicely than perhaps the divisions of the instrument itself will show it, we are to do thus. Supposing the quadrant \( bfd \) already accurately divided, and that the plumb-line plays over the point \( c \); and upon the inversion of the instrument, we find that before we can get it to play exactly over the centre \( f \), we must hang it over the point \( e \), so that the arch \( eb \) exceeds \( bc \) by the arch \( ea \); it is plain that the angle \( efa \) is the error of the instrument: for had the plumb-line hung over \( a \), and over the centre \( f \); in this latter posture, the instrument had been exact; because \( a \) is as much on one side \( b \), as \( c \) is on the other side \( b \). Wherefore \( efa \) being the angle by which our instrument errs in observation, let us turn the instrument into the usual posture of observation, as in fig. 18. and hanging the plumb-line on the centre \( f \); let us bring it to play nicely on the point \( e \), and observe what distant object is covered by the cross hairs: then let us bring it to play exactly on the point \( a \), and observe likewise what distant object is pointed at by the telescope-hairs. Lastly, by a large telescope and micrometer, let us measure the angle between these two objects, and we shall have the angle of error much more nicely than it is possible the angle \( efa \) should be given by the divisions on the limb of the quadrant \( ea \). And thus much for adjusting a quadrant.

"A sextant is rectified in like manner; if we consider, that if from the centre \( f \) to the beginning of the divisions \( d \) there be drawn the radius \( fa \), and it be divided equally in \( c \), and from \( c \) there be suspended the plumb-line \( cb \): when the plumb-line hangs over the 60th degree at \( b \), then the line \( fd \) lies horizontal; and consequently, if the line of collimation thro' the tube be parallel to \( fd \), this line also lies horizontal. To try which, whilst the sextant stands in this posture, observe the object marked by the cross hairs; then invert the sextant, and over the point \( b \) hang the plumb-line; and when from the point \( b \) the plumb-line hangs over the middle point \( c \), then again is the line \( fd \) horizontal in this posture. Mark, then, whether the cross hairs cover the same object as before: if they do, then the line of collimation is parallel to \( fa \); if they do not, but the point in the object marked in this latter posture be higher than the point marked in the first posture, the instrument errs in excess; if it be lower, the instru- times more dim when we looked thro' the glass, than when we beheld it with our naked eyes; and this, even on a supposition that the glass transmitted all the light which fell upon it, which no glass can do. But if the focal distance of the glass was only four inches, tho' its diameter remained as before, the inconvenience would be vastly diminished, because the glass could then be placed twice as near the object as before, and consequently would receive four times as many rays as in the former case, and therefore we would see it much brighter than before. Going on thus, still diminishing the focal distance of the glass, and keeping its diameter as large as possible, we will perceive the object more and more magnified, and at the same time very distinct and bright. It is evident, however, that with regard to optical instruments of the microscopic kind, we must sooner or later arrive at a limit which cannot be passed. This limit is formed by the following particulars.

1. The quantity of light lost in passing through the glass. 2. The diminution of the glass itself, by which it receives only a small quantity of micropores. 3. The extreme shortness of the focal distance of great magnifiers, whereby the free access of the light to the object which we wish to view is impeded, and consequently the reflection of the light from it is weakened. 4. The aberrations of the rays, occasioned by their different refrangibility.

To understand this more fully, as well as to see how far these obstacles can be removed, let us suppose the lens made of such a dull kind of glass that it transmits only one half of the light which falls upon it. It is evident that such a glass, of four inches focal distance, and which magnifies the diameter of an object twice, still supposing its own breadth equal to that of the pupil of the eye, will show it four times magnified in surface, but only half as bright as if it was seen by the naked eye at the usual distance; for the light which falls upon the eye from the object at eight inches distance, and likewise the surface of the object in its natural size, being both represented by 1, the surface of the magnified object will be 4, and the light which makes that magnified object visible only 2; because though the glass receives four times as much light as the naked eye does at the usual distance of distinct vision, yet one half is lost in passing through the glass. The inconvenience in this respect can therefore be removed only as far as it is possible to increase the clearness of the glass, so that it shall transmit nearly all the rays which fall upon it; and how far this can be done, hath not yet been ascertained.

The second obstacle to the perfection of microscopic glasses is the small size of great magnifiers, by which, notwithstanding their near approach to the object, they receive a smaller quantity of rays than might be expected. Thus, suppose a glass of only $\frac{1}{10}$th of an inch focal distance; such a glass would increase the visible diameter 80 times, and the surface 6400 times. If the breadth of the glass could at the same time be preserved as great as that of the pupil of the eye, which we shall suppose $\frac{1}{10}$th of an inch, the object would appear magnified 6400 times, at the same time that every part of it would be as bright as it appears to the naked eye. But if we suppose that this magnifying glass is only $\frac{1}{25}$th of an inch in diameter, it will then only receive $\frac{1}{4}$th of the light which other- Mechanism otherwise would have fallen upon it; and therefore, instead of communicating to the magnified object a quantity of illumination equal to 6400, it would communicate only one equal to 1600, and the magnified object would appear four times as dim as it does to the naked eye. This inconvenience however is still capable of being removed, not indeed by increasing the diameter of the lens, because this must be in proportion to its focal distance, but by throwing a greater quantity of light on the object. Thus, in the above-mentioned example, if four times the quantity of light which naturally falls upon it could be thrown upon the object, it is plain that the reflection from it would be four times as great as in the natural way; and consequently the magnified image, at the same time that it was as many times magnified as before, would be as bright as when seen by the naked eye. In transparent objects this can be done very effectually by a concave speculum, as in the reflecting microscope already described: but in opaque objects the case is somewhat more doubtful; neither do the contrivances for viewing these objects seem entirely to make up for the deficiencies of the light from the magnifying smallness of the lens and shortness of the focus.

The third obstacle arises from the shortness of the focal distance in large magnifiers: but in transparent objects, where a sufficient quantity of light is thrown on the object from below, the inconvenience arises at last from straining the eye, which must be placed nearer the glass than it can well bear; and this entirely supercedes the use of magnifiers beyond a certain degree.

The fourth obstacle arises from the different refrangibility of the rays of light, and which frequently causes such a deviation from truth in the appearances of things, that many people have imagined themselves to have made surprising discoveries, and have even published them to the world; when in fact they have been only as many optical deceptions, owing to the unequal refractions of the rays. For this there seems to be no remedy, except the introduction of achromatic glasses into microscopes as well as telescopes. How far this is practicable, hath not yet been tried; but when these glasses shall be introduced, (if such introduction is practicable), microscopes will then undoubtedly have received their ultimate degree of perfection.

With regard to telescopes, those of the refracting kind have evidently the advantage of all others, where the aperture is equal, and the aberrations of the rays are corrected according to Mr Dollond's method; because the image is not only more perfect, but a much greater quantity of light is transmitted than what can be reflected from the best materials hitherto known. Unluckily, however, the imperfections of the glass set a limit to these telescopes, as hath already been observed, so that they cannot be made above three feet and an half long. On the whole, therefore, the reflecting telescopes are preferable in this respect, that they may be made of dimensions greatly superior; by which means they can both magnify to a greater degree, and at the same time throw much more light into the eye.

With regard to the powers of telescopes, however, they are all of them exceedingly less than what we would be apt to imagine from the number of times Instrument which they magnify the object. Thus, when we hear of a telescope which magnifies 200 times, we are apt to imagine, that, on looking at any distant object through it, we should perceive it as distinctly as we would with our naked eye at the 200th part of the distance. But this is by no means the case; neither is there any theory capable of directing us in this matter: we must therefore depend entirely on experience.

The best method of trying the goodness of any telescope is by observing how much farther off you are able to read with it than you can with the naked eye. But that all deception may be avoided, it is proper to choose something to be read where the imagination cannot give any assistance, such as a table of logarithms, or something which consists entirely of figures; and hence the truly useful power of the telescope is easily known. In this way Mr Short's large telescope, which magnifies the diameter of objects 1200 times, is yet unable to afford sufficient light for reading at more than 200 times the distance at which we can read with our naked eye.

With regard to the form of reflecting telescopes, it is now pretty generally agreed, that when the Gregorian ones are well constructed, they have the advantage of those of the Newtonian form. One advantage evident at first sight is, that with the Gregorian telescope an object is perceived by looking directly through it, and consequently is found with much greater ease than in the Newtonian telescope, where we must look into the side. The unavoidable imperfection of the specula common to both, also gives the Gregorian an advantage over the Newtonian form. Notwithstanding the utmost care and labour of the workmen, it is found impossible to give the metals either a perfectly spherical, or a perfectly parabolical form. Hence arises some indistinctness of the image formed by the great speculum, which is frequently corrected by the little one, provided they are properly matched. But if this is not done, the error will be made much worse: and hence many of the Gregorian telescopes are far inferior to the Newtonian ones; namely, when the specula have not been properly adapted to each other. There is no method by which the workman can know the specula which will fit one another, without a trial; and therefore there is a necessity for having many specula ready made of each form, that in fitting up a telescope those many be chosen which best suit each other.

The brightness of any object seen through a telescope, in comparison with its brightness when seen by the naked eye, may in all cases be easily found by the following formula. Let \( n \) represent the natural distance of a visible object, at which it can be distinctly seen; and viewed by let \( d \) represent its distance from the object-glass of the telescope-instrument. Let \( m \) be the magnifying power of the telescope-instrument; that is, let the visual angle subtended at the eye by the object when at the distance \( n \), and viewed without the instrument, be to the visual angle produced by the instrument as \( 1 \) to \( m \). Let \( a \) be the diameter of the object-glass, and \( p \) be that of the pupil. Let the instrument be so constructed, that no parts of the pencils are intercepted for want of sufficient The brightness of vision through the instrument will be expressed by the fraction \( \frac{a}{m} \), the brightness of natural vision being 1. But although this fraction may exceed unity, the vision through the instrument will not be brighter than natural vision. For, when this is the case, the pupil does not receive all the light transmitted through the instrument.

In microscopes, \( n \) is the nearest limits of distinct vision, nearly 8 inches. But a difference in this circumstance, arising from a difference in the eye, makes no change in the formula, because \( m \) changes in the same proportion with \( n \).

In telescopes, \( n \) and \( d \) may be accounted equal, and the formula becomes \( \frac{a^2}{m^2} \).

**INDEX**

**A**

- **Aerial speculums mentioned by Mr Grey**, p. 46. - **A phenomenon similar to what is exhibited by them explained by M. le Cat**, p. 61. - **Strongly reflects the rays proceeding from beneath the surface of water**, p. 36. - **Kaline salt diminishes the mean refraction, but not the dispersive power, of glasses**, p. 18. - **Lambert (M. d’), his discoveries concerning achromatic telescopes**, p. 5485. - **Lehzen’s discoveries concerning the refraction of the atmosphere**, p. 6. - **His conjectures about the cause of it**, ib. - **He gave the first hint of the magnifying power of glasses**, ib. - **Apparent place of objects seen by reflection, first discovered by Kepler**, p. 26. - **Atmosphere varies in its refractive power at different times**, p. 19. - **About (Mr), makes an object-glass of an extraordinary focal length**, p. 85.

**B**

- **Bacon (Roger), his discoveries**, p. 8. - **Bacon (Lord), his mistake concerning the possibility of making images appear in the air**, p. 25. - **Beams of light: remarkable appearance of the boundaries of two contiguous ones**, p. 5499, col. 2. - **Beaume (Mr), cannot fire inflammable liquids with burning-glasses**, p. 43. - **Berkeley (Dr), his hypothesis concerning the apparent place of objects**, p. 5570, col. 2. - **Objected to by Dr Smith**, ib. - **Binocular telescope invented by Father Rheita**, p. 83. - **Black marble, in some cases reflects very powerfully**, p. 35. - **Bouguer’s experiments concerning the quantity of light lost by reflection**, p. 32. - **His observations concerning the apparent place of objects**, p. 5572, col. 2. - **Boyle’s experiments concerning the light of differently coloured substances**, p. 27. - **Brilliant curious appearance of the shadow of one**, p. 55. - **Burning-glasses of the ancients described**, p. 24.

**C**

- **Camera obscura explained**, p. 5583. - **Campani, a celebrated maker of telescopes**, p. 84. - **On what the goodness of his telescopes depended**, ib. - **Candle, sometimes appears multiplied when seen through a chink**, p. 5569, col. 2. - **Cat (M. le), explains a singular phenomenon**, p. 61. - **Cat, experiment with one plunged under water**, p. 5535, col. 1. - **Celestial observations; how to make them**, p. 5565. - **Clouds: why they cause certain motions in the shadows of bodies**, p. 62. - **Cold, why most intense on the tops of mountains**, p. 42. - **Colours discovered to arise from refraction**, p. 15. - **Supposed by Dechales to arise from the inflection of light**, p. 49. - **Produced by a mixture of shadows**, p. 57. - **Concave glasses for short-sighted people, when first invented**, p. 67. - **Contact of bodies in many cases apparent without being real**, p. 45. - **Crystal hath some reflective properties different from other transparent substances**, p. 38.

**D**

- **Deception in vision: a remarkable one explained by M. le Cat**, p. 5574, col. 2. - **Dechales’s observations on the inflection of light**, p. 49. - **Descartes: his discoveries concerning vision**, p. 65. - **Dioptric instruments: difficulties attending the construction of them**, p. 110. - **Distance of objects not judged merely by the angle under which they are seen**, p. 5570, col. 2. - **Divini: a celebrated maker of telescopes**, p. 84. - **Microscopes made by him**, p. 98. - **Dollond (Mr), discovers a method of correcting the errors arising from refraction**, p. 17. - **He discovers a mistake in one of Sir Isaac Newton’s experiments**, p. 5483. - **Discovers the different refractive and dispersive power of glasses**, p. 5483. - **Difficulties occurring in the execution of his plan**, p. 5484. - **His improvements in the refracting telescope**, p. 90.

**E**

- **Equatorial telescope, or portable observatory**, p. 92, and p. 5603. - **Euler (Mr), first suggested the thought of improving refracting telescopes**, p. 17. - **His controversy with Clairault, &c. ib** - **Eye: the density and refractive powers of its humours first ascertained by Scheiner**, p. 64. - **Description of it**, p. 115. - **Dimensions of the intensible spot of it**, p. 120. - **Eyes are seldom both equally good**, p. 5569, col. 2.

**F**

- **Fontana claims the honour of inventing telescopes**, p. 72. - **Funk (Baron Alexander), his observation concerning the light in mines**, p. 46.

**G**

- **Galilean telescope of more difficult construction than others**, p. 79. - **Galileo makes a telescope without any pattern**, p. 73. - **Account of his discoveries with it**, p. 74. - **Why furred named Lyncaeus**, p. 75. - **Was ignorant of the true rationale of telescopes**, p. 77.

**Glas: globes: their magnifying powers known to the ancients**, p. 3. - **Different kinds of them**, ib. - **Table of the different compositions of glass for correcting the errors in reflecting telescopes**, p. 5486. - **Show various colours when split into thin laminae**, p. 30. - **Table of the quantities of light reflected from glass not quicksilvered, at different angles of incidence**, p. 5594. - **Glasses: difference in their powers of refraction and dispersion of the light**, p. 5483. - **Globes have shorter shadows than cylinders**, p. 544, 56. - **Remarkable difference between their shadows and those of metallic plates**, p. 56. - **Globules for microscopes, how made by Adams**, p. 101. - **See Microscope**.

**Graphical perspective**, p. 5584.

**Grey (Mr), his temporary microscopes**, p. 5591.

**Grimaldi first observes that colours arise from refraction**, p. 15. - **Inflection of light first discovered by him**, p. 5497, col. 2. - **His discoveries concerning this inflection**, p. 48.

**H**

- **Hairs, remarkable appearance**. INDEX.

of their shadowes, 56 Hartsoeker's microscopes, 98 Helioscopia described, p. 5601 Hire (M. de la), his reason why rays of light seem to proceed from luminous bodies, when viewed with the eyes half-shut, 50 Hooke (Dr.), his discoveries concerning the inflection of light, 47. His objection against Hevelius founded on a mistake, 125 Horizon, its extent on a plane surface, p. 5567, col. 1. Horizontal moon, Ptolemy's hypothesis concerning it, 5 Jansen (Zacharias), the first inventor of telescopes, 70. Made the first microscope, 95, 96 Images: Lord Bacon's mistake concerning the possibility of making them appear in the air, 25. Another on the same subject by Vitellio, ib. B. Porta's method of producing this appearance, ib. Kircher's method, ib. Images of objects appear double when a quantity of water is poured into a vessel containing quicksilver, p. 5494. Inflection of light, discoveries concerning it, p. 5497, col. 2. Dr Hooke's discoveries concerning it, 47. Grimaldi's observations, 48. Dechales's observations, 49. Sir Isaac Newton's discoveries, 51. Maraldi's discoveries, 52 Inversion, a curious instance of it observed by Mr Grey, 46 Irradiations of the sun's light explained, 146, 147. Not observed by moon-light, 148. More frequent in summer than in winter, 149 Jupiter's satellites discovered by Jansen, 71. by Galileo, 74 Kepler first discovered the true reason of the apparent place of objects, 26. His discoveries concerning vision, 63. Improved the construction of telescopes, 30. His method first put in practice by Scheiner, 81 Lead increases the dispersive power of glass, p. 5485 Lenses, their effects first explained by Kepler, 77. Different kinds of them, p. 5530, col. 2. How to find their foci, n° 114. Of the appearances of objects thro' them, p. 5441—5444. The uses of having several lenses in a compound microscope, p. 5577, col. 2. How to grind them for telescopes and microscopes, p. 5609—5617 Leeuwenhoek's microscopes, 99 Light, its phenomena difficult to be accounted for, 1. Quantity of it absorbed by platter of Paris, 40. By the moon, ib. Mr Melville's observations on the manner in which bodies are heated by it, 42. No heat produced by it in a transparent medium unless it falls on the surface, ib. Of its different refrangibility, p. 5554—5558. M. Bouguer's contrivances for measuring it, n° 152. Of its general properties, p. 5519, col. 2. Lignum nephriticum, remarkable properties of its infusion, 28 Lines can be seen under smaller angles than spots, and why, 125 Liquid substances cannot be fired by the solar rays concentrated, 43 Magic lanthorn, p. 5584 Magnitudes, why we are so frequently deceived concerning them, p. 5571, col. 2. Mairan (M.), his observations on the inflection of light, 58 Maraldi's discoveries concerning the inflection of light, 52 Maurolycus, his discoveries, 9, 63 Media of different kinds; appearances of objects thro' them, p. 5541 Mery (M.), strange experiment of his with a cat, p. 5535, col. 1. Melville (Mr.), a curious phenomenon explained by him, p. 5574, col. 2. Microscopes, their history, 9. Made by Jansen, 95, 96. By Divini, 97. By Hartsoeker, 98. By Leeuwenhock, 99. By Wilson, 100. Temporary ones by Mr Grey, 102. Reflecting microscopes by Dr Barker, 103. Dr Smith's reflecting microscope, 104. Solar microscope, 105. Microscope for opaque objects, ib. Reflected light introduced into the solar microscope, 106. Martin's improvements in it, 107. Microscopes with six glasses, 109. Microscopes of various kinds described, p. 5577. How to preserve the distinctions of objects in them, p. 5578. Mr Martin's method of increasing the light on any object, n° 164. The single one described, p. 5855. Single with reflection, p. 5586. Double reflecting and refracting, ib. For opaque objects, p. 5587. Solar, p. 5588. Universal, p. 5589. Clark's improved pocket-microscope, p. 5590. Extempore microscopes, p. 5591. To find the magnifying power of microscopes, p. 5592. To find the real size of objects seen by microscopes, p. 5594. Of the field of view in microscopes, p. 5595. Of microscopic objects and the method of preparing them, p. 5595. How to make glass-globules for them, p. 5609. Advantages arising from the use of microscopes, p. 5635 Mines better illuminated in cloudy than in clear weather, 46 Mirrors, plane ones described, p. 5575. Why three or four images of objects are sometimes seen in them, 16c. Concave and convex ones, p. 5576. Aerial images formed by concave ones, 161 Mist causes objects to appear larger than their natural size, p. 5569 Moon, Maraldi's mistake concerning the shadow of it, 55. Why she appears more dull when eclipsed in her perigee than in her apogee, n° 151. Variation of her light at different altitudes, 152. Calculation of her light by M. Bouguer, 156. By Dr Smith, 157. By Mr Mitchell, 158. Moon-eyed people, why so called, n° 120. Mudge's directions for grinding the specula of telescopes, p. 5623, col. 1. Multiplying glass described, Newton (Sir Isaac), his discoveries concerning colours, n° 15. Mistaken in one of his experiments, p. 54. His discoveries concerning the inflection of light, n° Nollet (Abbé), cannot fire flammable liquids by burning glasses, 43 Objects appear magnified when viewed through small holes, 78. Why seen upright, 1. Of their apparent place, magnitude, and distance, p. 5568. Why they appear so small when seen from an high building, p. 55. Why a very long row of men must appear circular, p. 5567. Object-glasses of an extraordinary focal length made by different persons, 85. How to centre an object-glass, p. 5617. Observatory (Portable). Equatorial Telescope. Optic nerve insensible of light, 117, 118. Optical instruments, discoveries concerning them, p. 5560. Different instruments described, 5575. Their mechanism, 5609. Optics, the first treatise on the science written by Claudius Ptolemaeus, 4. Account of Vitellio's treatise on optics, 7. Of a treatise on optics attributed to Euclid, 23 Painters cannot perfectly deceive the eye, p. 5569. Parallel lines, why they seem to converge when much extended, p. 5572. Phenomena explained by the theory laid down in this treatise, p. 5561. Planets, more luminous at the edges than in the middle of their disks, n° 40. Plates, Maraldi's experiments concerning their shadows, 5. Porta (Joannes Baptista), his discoveries, 10. Porterfield (Dr.), his opinion of the methods of judging of the distances of objects, p. 5571, col. 2. Prisms in some cases reflect as strongly as quicksilver, 38. Index.

R. Rainbow variously accounted for, p. 5561. Explained on the Newtonian principles, p. 5562—5568

Rays of light, why they seem to proceed from any luminous object when viewed with the eyes half shut, 50

Reflected light, table of its quantity from different substances, 39

Reflection of light, opinions of the ancients concerning it, 22. Bouguer's experiments concerning the quantity of light lost by it, 32. Method of ascertaining the quantity lost in all the varieties of reflection, ib. Buffon's experiments on the same subject, 33. Bouguer's discoveries concerning the reflection of glass, and of polished metal, 34. Great differences in the quantity of light reflected at different angles of incidence, 35. No reflection but at the surface of a medium, 42. Treatise on reflection, p. 5544 Cause of reflection, ib. Is not performed by the light impinging on the solid parts of bodies at the first surface, n° 127. Nor at the second, 128 Very great from a vacuum, ib. Supposed to arise from a repulsive power, 129. Objections to this hypothesis, 130. Attractive force supposed to be the cause, 131. Another hypothesis, 132. Sir Isaac Newton's hypothesis, 133. Laws of reflection, p. 5546—5551

Refracting telescopes, how amended by Mr Dollond, n° 17.

Refraction known to the ancients, n° 2. Its law discovered by Snellius, 11. Explained by Descartes, 12. Fallacy of his hypothesis discovered, 13. Experiments of the Royal Society for determining the refractive powers of different substances, ib. M. de la Hire's experiments on the same subject, ib. Refraction of air accurately determined, p. 5489. Mistake of the Academy of Sciences concerning the refraction of air, ib. Allowance for refraction first thought of by Dr Hooke, n° 14. Colours discovered to arise from thence, n° 15. Mr Dollond discovers how to correct the errors of reflecting telescopes arising from refraction, n° 17. Refraction defined, 3. Explained by an attractive power, 113. Sines of refraction by different substances, p. 5521. Laws of refraction particularly explained and demonstrated, p. 5521—5530.

Retina, its extreme sensibility, p. 5570.

Rhetia invents the terrestrial telescope, n° 81. and the binocular one, 83.

Saturn's ring discovered by Galileo, n° 74.

Scheiner completes the discoveries concerning vision, 64. First puts in practice Kepler's improvements in the telescope, n° 81.

Shadows of bodies; observations concerning them, 47—60. Bounded by fringes of coloured light, p. 5502, col. 2. Of green and blue ones, n° 138—145. Of the illumination of the shadow of the earth by the atmosphere, n° 150.

Sky, its concave figure explained, p. 5567, col. 1. Why the concavity of it appears less than a hemisphere, 136. Of its blue colour, 137.

Short-sightedness and long-sightedness described, 123.

Spectacles, when first invented, 66.

Specula for reflecting telescopes, the best composition for them, p. 5613. Method of preparing the moulds, casting and grinding the metals, p. 5619.

Spots of the sun first discovered by Galileo, n° 74. M. de la Hire's explanation of the spots which float before the eyes of old people, p. 5570.

Stars, their twinkling explained by Mr Michel, n° 20. By Mr Muschenbroek, 21.

By other philosophers, ib. A momentary change of colour observed in some stars, ib. Why visible at the bottom of a well. How observed in the day-time, n° 93.

Sun and planets; variation in the light of different parts of their disks, n° 155.

Surfaces of transparent bodies have the property of extinguishing light, and why, n° 37. Supposed to consist of small transparent planes, 39; 40; 41. Of the appearance of bodies seen by light reflected from plane and spherical surfaces, p. 5551.

T.

Telescopes; different accounts of the invention of them, n° 68—70. The first an exceedingly good one, 71. One made by Galileo without a pattern, 73. The rationale of them first discovered by Kepler, 76. General reason of their effects, 78. Improved by Kepler, 80. Of the different constructions of them, 81. Vision most distinct in those of the Galilean kind, 82. Terrestrial telescope invented by Father Rhetia, 81. Method of managing them without tubes, 84. Why dioptric telescopes must be made so long, 87. Of the apertures of refracting telescopes, 88. History of the reflecting telescope, 89. Mr Smith's proposal for shortening them, 91. Apelles's proposal to bend their tubes, 94. Several kinds of telescopes described, p. 5579.

Refracting telescope, ib. Imperfection in dioptric Telescopes, n° 165. Remedied by Mr Dollond, 166. Sir Isaac Newton's reflecting telescope, p. 5582—5597. Gregorian telescope, p. 5582—5599. Dollond's achromatic telescope, p. 5597. Method of determining the magnifying power of a telescope, p. 5600. Solar telescope, p. 5601. Equatorial telescope described, p. 5603. Binocular telescope described, 5604. Of the different parts of which a reflecting telescope is composed, n° 215. How to adjust the mirrors 210, 211. Comparison of the different kinds of telescopes with each other, 219. Difference between the magnifying and truly useful power of a telescope, ib.

Telescopic instruments for measuring time, p. 5607. Telescopic sights; how to rectify them, 215.

Thin plates; Mr Boyle's account of the colours observable in them, n° 29. Dr Hooke's account, 30.

Torre (T. di) his extraordinary magnifiers for microscopes, 108.

Tour (M. du) his observations on the inflection of light, n° 59. His hypothesis by which he accounted for the phenomena, 60.

V.

Vacuum; strong reflections often proceed from it, n° 120—128.

Visible motion of objects, Dr Porterfield's observations on it, p. 5573, col. 2.

Vision; its nature first discovered by Maurolycus, n° 9. Discoveries concerning it, p. 5507, col. 1. Treatise of it, p. 5532. Dispute concerning its seat, n° 119. Bright and obscure, 121. At different distances, 122. Least angle of it, 124. Why single with two eyes, 126. Curious particulars relating to deceptions, p. 5540.

Vision; why a long horizontal one seems to ascend, p. 5672, col. 1.

W.

Water in some cases reflects more powerfully than quicksilver, n° 35. Table of the quantity of light reflected from it at different angles, p. 5494. Remarkably strong reflection into it from the air, n° 36.

Windmill; why its apparent motion is sometimes contrary to the real one, p. 5572, col. 2. p. 5574, col. 1.

Ifon's microscope, n° 100.