MICROMETER

MICROMETER, an instrument, by the help of which the apparent magnitudes of objects viewed thro' telescopes or microscopes are measured with great exactness.

I. The first TELESCOPIC micrometers were only mechanical contrivances for measuring the image of an object in the focus of the object-glass. Before these contrivances were thought of, astronomers were accustomed to measure the field of view in each of their telescopes, by observing how much of the moon they could see through it, the semidiameter being reckoned at 15 or 16 minutes; and other distances were estimated by the eye, comparing them with the field of view. Mr Gascoigne, an English gentleman, however, fell upon a much more exact method, and had a Treatise on Optics prepared for the press; but he was killed during the civil wars in the service of Charles I. and his manuscript was never found. His instrument, however, fell into the hands of Mr R. Townly, who says, that by the help of it he could mark above 40,000 divisions in a foot.

Mr Gascoigne's instrument being shown to Dr Hooke, he gave a drawing and description of it, and proposed several improvements in it, which may be seen in Phil. Trans. abr. Vol. I. p. 217. Mr Gascoigne divided the image of an object, in the focus of the object-glass, by the approach of two pieces of metal ground to a very fine edge, in the place of which Dr Hooke would substitute two fine hairs stretched parallel to one another. Two other methods of Dr Hooke's, different from this, are described in his Posthumous Works, p. 497, 498. An account of several curious observations that Mr Gascoigne made by the help of his micrometer, particularly in the mensuration of the diameters of the moon and other planets, may be seen in the Phil. Trans. Vol. XLVIII. p. 190.

Mr Huygens, as appears by his System of Saturn, published in 1659, used to measure the apparent diameters of the planets, or any small angles, by first measuring the quantity of the field of view in his telescope; which, he says, is best done by observing the time which a star takes up in passing over it, and then preparing two or three long and slender brass plates, of various breadths, the sides of which were very straight, and converging to a small angle. In making use of these pieces of brass, he made them slide in two slits, that were made in the sides of the tube, opposite to the place of the image, and observed in what place it just covered the diameter of any planet, or any small distance that he wanted to measure. It was observed, however, by Sir Isaac Newton, that the diameters of planets, measured in this manner, will be larger than they should be, as all lucid objects appear to be when they are viewed upon dark ones.

In the Ephemerides of the Marquis of Malvasia, published in 1662, it appears that he had a method of measuring small distances between fixed stars and the diameters of the planets, and also of taking accurate draughts of the spots of the moon; and this was by a net of silver wire, fixed in the common focus of the object and eye-glass. He also contrived to make one of two stars to pass along the threads of this net, by turning it, or the telescope, as much as was necessary for

that purpose; and he counted, by a pendulum-clock, beating seconds, the time that elapsed in its passage from one wire to another, which gave him the number of the minutes and seconds of a degree contained between the intervals of the wires of his net, with respect to the focal length of his telescope.

In 1666, Melfis Azout and Picard published a description of a micrometer, which was nearly the same with that of the Marquis of Malvasia, excepting the method of dividing it, which they performed with more exactness by a screw. In some cases they used threads of silk, as being finer than silver wires. Dechales also recommends a micrometer consisting of fine wires, or filken threads, the distances of which were exactly known, disposed in the form of a net, as peculiarly convenient for taking a map of the moon.

M. de la Hire says, that there is no method more simple or commodious for observing the digits of an eclipse than a net in the focus of the telescope. These, he says, were generally made of filken threads; and that for this particular purpose six concentric circles had also been made use of, drawn upon oiled paper; but he advises to draw the circles on very thin pieces of glass with the point of a diamond. He also gives several particular directions to assist persons in the use of them. In another memoir he shows a method of making use of the same net for all eclipses, by using a telescope with two object-glasses, and placing them at different distances from one another.

Different Constructions of Micrometers. The first we shall describe is that by Mr Huygens. Let ABCD be a section of the telescope at the principal focus of the object-glass, or where the wires are situated, which are placed in a short tube containing the eye-glass, and may be turned into any position by turning that tube; $mn$ is a fine wire extended over its centre; $vw$, $xy$, are two straight plates whose edges are parallel and well defined, and perpendicular to $mn$; $vw$ is fixed, and $xy$ moves parallel to it by means of a screw, which carries two indexes over a graduated plate, to show the number of revolutions and parts of a revolution which it makes. Now to measure any angle, we must first ascertain the number of revolutions and parts of a revolution corresponding to some known angle, which may be thus done: 1st, Bring the inner edges of the plates exactly to coincide, and set each index to 0; turn the screw, and separate the plates to any distance; and observe the time a star $m$ is in passing along the wire $mn$ from one plate to the other: for that time, turned into minutes and seconds of a degree, will be the angle answering to the number of revolutions, or the angle corresponding to the distance. Thus, if $d\; =\; \backslash cos.$ of the star's declination, we have $15\text{'}\; dm$, the angle corresponding to this distance; and hence, by proportion, we find the angle answering to any other. 2dly, Set up an object of a known diameter, or two objects at a given distance, and turn the screw till the edges of the plates become tangents to the object, or till their opening just takes in the distance of the two objects upon the wire $mn$; then from the diameter, or distance of the two objects from each other, and their distance from the glass, calculate the angle, and observe the number of revolutions and parts corresponding. 3dly, Take the diameter of the sun on any day, by making the edges of the plates tangents to the opposite limbs, and find,

Micrometer. from the nautical almanac, what is his diameter on that day. Here it will be best to take the upper and lower limbs of the sun when on the meridian, as he has then no motion perpendicular to the horizon. If the edges do not coincide when the indexes stand at 0, we must allow for the error. Instead of making a proportion, it is better to have a table calculated to show the angle corresponding to every revolution and parts of a revolution. But the observer must remember, that when the micrometer is fixed to telescopes of different focal lengths, a new table must be made. The whole system of wires is turned about in its own plane, by turning the eye-tube round with a hand, and by that means the wire mn can be thrown into any position, and consequently angles in any position may be measured. Dr Bradley added a small motion by a rack and pinion to set the wires more accurately in any position.

Instead of two plates, two wires were afterwards put; and Sir Isaac Newton observed, that the diameters of the planets measured by the plates were somewhat bigger than they ought, as appeared by comparing Mr Huygens's measures with others taken with the wires; and also by comparing the diameter of mercury observed in and out of the sun's disk, the latter being the greatest. Dark objects on bright ones appear less, and light objects on dark ones appear greater, than if they were equally bright; owing, perhaps, to the brighter image on the retina diffusing itself into the darker; and the bright image of the planet being intercepted by the plates, the faint diffused light becomes more sensible, and is mistaken for the edge of the planet.

But the micrometer, as now contrived, is of use, not only to find the angular distance of bodies in the field of view at the same time, but also of those which, when the telescope is fixed, pass through the field of view successively; by which means we can find the difference of their right ascensions and declinations. Let Aa, Bb, Cc, be three parallel and equidistant wires, the middle one bisecting the field of view; HOR a fixed wire perpendicular to them passing through the centre of the field; and Ff, Gg, two wires parallel to it, each moveable by a micrometer screw, as before, so that they can be brought up to HOR, or a little beyond. Then to find the angular distance of two objects, bring them very near to Bb, and in a line parallel to it, by turning about the wires, and bring one upon HOR, and by the micrometer screw make Ff or Gg pass through the other; then turn the screw till that wire coincides with HOR, and the arc which the index has passed over shows their angular distance. If the objects be further remote than you can carry the distance of one of the wires Ff, Gg, from HOR, then bring one object to Ff and the other to Gg; and turn each micrometer screw till they meet, and the sum of the arcs passed over by each index gives their angular distance. If the objects be two stars, and one of them be made to run along HOR, or either of the moveable wires as occasion may require, the motion of the other will be parallel to these wires, and their difference of declinations may be observed with great exactness; but in taking any other distances, the motion

of the stars being oblique to them, it is not quite so easy to get them parallel to Bb; because if one star be brought near, and the eye be applied to the other to adjust the wires to it, the former star will have gotten a little away from the wire. Dr Bradley, in his account of the use of this micrometer, published by Dr Maskelyne in the Philosophical Transactions for 1772, thinks the best way is to move the eye backwards and forwards as quick as possible; but it seems to me to be best to fix the eye at some point between, by which means it takes in both at once sufficiently well defined to compare them with Bb. In finding the difference of declinations, if both bodies do not come into the field of view at the same time, make one run along the wire HOR, as before, and fix the telescope and wait till the other comes in, and then adjust one of the moveable wires to it, and bring it up to HOR, and the index gives the difference of their declinations. The difference of time between the passage of the star at either of the cross moveable wires, and the transit of the other star over the cross fixed wire (which represents a meridian), turned into degrees and minutes, will give the difference of right ascension. The star has been here supposed to be bisected by the wire; but if the wire be a tangent to it, allowance must be made for the breadth of the wire, provided the adjustment be made for the coincidence of the wires. In observing the diameters of the sun, moon, or planets, it may perhaps be most convenient to make use of the outer edges of the wires, because they appear most distinct when quite within the limb; but if there should be any sensible inflection of the rays of light in passing by the wires, it will be best avoided by using the inner edge of one wire and the outward edge of the other; for by that means the inflection at both limbs will be the same way, and therefore there will be no alteration of the relative position of the rays passing by each wire. And it will be convenient in the micrometer to note at what division the index stands when the moveable wire coincides with HOR; for then you need not bring the wire when a star is upon it up to HOR, only reckon from the division at which the index then stands to the above division.

With a micrometer therefore thus adapted to a telescope, Mr Servington Savery of Exeter proposed, a new way of measuring the difference between the greatest and least apparent diameters of the sun, although the whole of the sun was not visible in the field of view at once. The method we shall briefly describe. Place two object-glasses instead of one, so as to form two images whose limbs shall be at a small distance from each other; or instead of two perfect lenses, he proposed to cut a single lens into four parts of equal breadths by parallel lines, and to place the two segments with their straight sides against each other, or the two middle frustums with their opposite edges together; in either case, the two parts which before had a common centre and axis, have now their centres and axes separated, and consequently two images will be formed as before by two perfect lenses. Another method in reflectors was to cut the large concave reflector through the centre, and by a contrivance to turn up the outer edges whilst the straight ones remained

mained fixed; by which means the axis of the two parts became inclined, and formed two images. Two images being formed in this manner, he proposed to measure the distance between the limbs when the diameters of the sun were the greatest and least, the difference of which would be the difference of the diameters required. Thus far we are indebted to Mr Savery for the idea of forming two images; and the admirable uses to which it was afterwards applied, we shall next proceed to describe.

The divided object-glass micrometer, as now made, was contrived by the late Mr John Dollond, and by him adapted to the object-end of a reflecting telescope, and has been since by the present Mr P. Dollond his son applied with equal advantage to the end of an achromatic telescope. The principle is this: The object-glass is divided into two segments in a line drawn through the centre; each segment is fixed in a separate frame of brass, which is moveable, so that the centres of the two segments may be brought together by a handle for that purpose, and thereby form one image of an object; but when separated they will form two images, lying in a line passing through the centre of each segment; and consequently the motion of each image will be parallel to that line, which can be thrown into any position by the contrivance of another handle to turn the glass about in its own plane. The brass-work carries a vernier to measure the distance of the centres of the two segments. Now let E and H be the centres of the two segments, F their principal focus, and P Q two distant objects in FE, FH, produced, or the opposite limbs of the same object PBQD; then the images of P and Q, formed by each segment, or the images of the opposite limbs of the object PBQD, coincide at F: hence two images $EF$, $HF$ of that object are formed, whose limbs are in contact; therefore the angular distance of the points P and Q is the same as the angle which the distance EH subtends at F, which, as the angles supposed to be measured are very small, will vary as EH extremely nearly; and consequently if the angle corresponding to one interval of the centres of the segments be known, the angle corresponding to any other will be found by proportion. Now to find the interval for some one angle, N° 218.

take the horizontal diameter of the sun on any day, by separating the images till the contrary limbs coincide, and read off by the vernier the interval of their centres, and look into the nautical almanac for the diameter of the sun on that day, and you have the corresponding angle. Or if greater exactness be required than from taking the angle in proportion to the distances of their centres, we may proceed thus:—Draw FG perpendicular to EH, which therefore bisects it; then one half EH, or EG, is the tangent of half the angle EFH; hence, half the distance of their centres: tangent of half the angle corresponding to that distance :: half any other distance of the centres: tangent of half the corresponding angle (A).

Hence the method of measuring small angles is manifest; for we consider P, Q, either as two objects whose images are brought together by separating the two segments, or as the opposite limbs of one object PBQD, whose images, formed by the two segments E, H, touch at F: in the former case, EFH gives the angular distance of the two objects; and in the latter, it gives the angle under which the diameter of the object appears. Hence, to find the angular distance of two objects, separate the segments till the two images which approach (a) each other coincide; and to find the diameter of an object, separate the segments till the contrary limbs of the images touch each other, and read off the distance of the centres of the segment from the vernier (c), and find the angle as directed in the last article. From hence appears one great superiority in this above the wire micrometer; as, with this, any diameter of an object may be measured with the same ease and accuracy; whereas with that we cannot with accuracy measure any diameter, except that which is at right angles to its apparent motion.

But, besides these two uses to which the instrument seems so well adapted, Dr Maskelyne has shown, in the Philosophical Transactions for the year 1771, how it may be applied to find the difference of right ascensions and declinations. For this purpose, two wires at right angles to each other, bisecting the field of view, must be placed in the principal focus of the eye-glass, and moveable about in their own plane.—Let

(a) If the object be not a distant one, let $f$ be the principal focus; then $Ff\; :\; FG\; ::\; FG\; :\; FK$ ($FG$ being produced to meet a line joining the apparent places of the two objects P, Q), $\backslash therefore$ dividendo, $fG\; :\; FG\; ::\; GK\; :\; FK$, and alternando, $fG\; :\; GK\; ::\; FG\; :\; FK\; ::$ (by similar triangles) $EH\; :\; PQ$, hence $\backslash frac\{EH\}\{fG\}\; =\; \backslash frac\{PQ\}\{GK\}$, therefore the angle subtended by EH at $f\; =$ the angle subtended by PQ at G; and consequently, as $fG$ is constant, the angle measured at G is, in this case also, in proportion to EH. The instrument is not adapted to measure the angular distance of bodies, one of which is near and the other at a distance, because their images would not be formed together.

(b) Besides these two images, there will be two others receding from each other, for each segment gives an image of each object.

(c) To determine whether there be any error of adjustment of the micrometer scale, measure the diameter of any small well defined object, as Jupiter's equatorial diameter, or the longest axis of Saturn's ring, both ways, that is, with $o$ on the vernier to the right and left of $o$ on the scale, and half the difference is the error required; which must be added to or subtracted from all observations, according as the diameter measured with $o$ on the vernier, when advanced on the scale, is less or greater than the diameter measured the other way. And it is also evident, that half the sum of the diameters thus measured gives the true diameter of the object.

Micrometer.
Fig. 4.
Let $HR$ be the field of view, $HR$ and $Cc$ the two wires; turn the wires till the westernmost star (which is the bed, having further to move) run along $ROH$; then separate the two segments and turn about the micrometer till the two images of the same star lie in the wire $Cc$; and then, partly by separating the segments, and partly by raising or depressing the telescope, bring the two innermost images of the two stars to appear and run along $ROH$, as $a,\; b$, and the vernier will give the difference of their declinations; because, as the two images of one of the stars coincided with $Cc$, the image of each star was brought perpendicularly upon $HR$, or to $HR$ in their proper meridian. And, for the same reason, the difference of their times of passing the wire $Cc$ will give their difference of right ascensions. These operations will be facilitated, if the telescope be mounted on a polar axis. If two other wires $KL,\; MN$, parallel to $Cc$, be placed near $H$ and $R$, the observation may be made on two stars whose difference of meridians is nearly equal to $HR$ the diameter of the field of view, by bringing the two images of one of the stars to coincide with one of these wires. If two stars be observed whose difference of declinations is well settled, the scale of the micrometer will be known.

It has hitherto been supposed, that the images of the two stars can be both brought into the field of view at once upon the wire $ROH$; but if they cannot, set the micrometer to the difference of their declinations as nearly as you can, and make the image which comes first run along the wire $ROH$, by elevating or depressing the telescope; and when the other star comes in, if it do not also run along $ROH$, alter the micrometer till it does, and half the sum of the numbers shown by the micrometer at the two separate observations of the two stars on the wire $ROH$ will be the difference of their declinations. That this should be true, it is manifestly necessary that the two segments should recede equally in opposite directions; and this is effected by Mr Dollond in his new improvement of the object-glass micrometer.

The difference of right ascensions and declinations of Venus or Mercury in the sun's disk and the sun's limb may be thus found. Turn the wires so that the north limb $n$ of the sun's image $AB$, or the north limb of the image $V$ of the planet, may run along the wire $ROH$, which therefore will then be parallel to the equator, and consequently $Cc$ a secondary to it; then separate the segments, and turn about the micrometer till the two images $V,\; v$ of the planet pass $Cc$ at the same time, and then by separating the segments, bring the north limb of the northernmost image $V$ of the planet to touch $HR$, at the time the northernmost limb $n$ of the southernmost image $AB$ of the sun touches it, and the micrometer shows the difference of declinations of the northernmost limbs of the planet and sun, for the reason formerly given $\backslash dagger$, we having brought the northernmost limbs of the two innermost images $V$ and $AB$ to $HR$, these two being manifestly interior to $v$ and the northernmost limb $N$ of the image $PQ$. In the same manner we take the difference of declinations of their southernmost limbs; and
Vol. XI. Part II.

half the difference of the two measures, (taking immediately one after another) is equal to the difference of the declinations of their centres, without any regard to the sun's or planet's diameters, or error of adjustment of the micrometer; for as it affects both equally, the difference is the same as if there were no error: and the difference of the times of the transits of the eastern or western limbs of the sun and planet over $Cc$ gives the difference of their right ascensions.

Instead of the difference of right ascensions, the distance of the planet from the sun's limb, in lines parallel to the equator, may be more accurately observed thus: Separate the segments, and turn about the wires and micrometer, so as to make both images $V,\; v$, run along $HR$, or so that the two intersections $I,\; T$, of the sun's image may pass $Cc$ at the same time. Then bring the planet's and sun's limbs into contact, as at $V$, and do the same for the other limb of the sun, and half the difference gives the distance of the centre of the planet from the middle of the chord on the sun's disk parallel to the equator, or the difference of the right ascensions of their centres, allowing for the motion of the planet in the interval of the observations, without any regard to the error of adjustment, for the same reason as before. For if you take any point in the chord of a circle, half the difference of the two segments is manifestly the distance of the point from the middle of the chord; and as the planet runs along $HR$, the chord is parallel to the equator.

In like manner, the distances of their limbs may be Fig. 7. measured in lines perpendicular to the equator, by bringing the micrometer into the position already described*, and instead of bringing $V$ to $HR$, separate the segments till the northernmost limbs coincide as at $V$; and in the same manner make their southernmost images to coincide, and half the difference of the two measures, allowing for the planet's motion, gives the difference of the declinations of their centres.

Hence the true place of a planet in the sun's disk may at any time of its transit be found; and consequently the nearest approach to the centre and the time of ecliptic conjunction may be deduced, although the middle should not be observed.

But however valuable the object-glass micrometer undoubtedly is, difficulties sometimes have been found in its use, owing to the alteration of the focus of the eye, which will cause it to give different measures of the same angle at different times. For instance, in measuring the sun's diameter, the axis of the pencil coming through the two segments from the contrary limbs of the sun, as $PF,\; QF$, fig. 3. crossing one another in the focus $F$ under an angle equal to the sun's semidiameter, the union of the limbs cannot appear perfect, unless the eye be disposed to see objects distinctly at the place where the images are formed; for if the eye be disposed to see objects nearer to or further off than that place, in the latter case the limbs will appear separated, and in the former they will appear to lap over ($b$). This imperfection led Dr Mafkelyne

Micrometer. It is likely to inquire, whether some method might not be found of producing two distinct images of the sun, or any other object, by bringing the axis of each pencil to coincide, or very nearly so, before the formation of the images, by which means the limbs when brought together would not be liable to appear separated from any alteration of the eye; and this he found would be effected by the refraction of two prisms, placed either without or within the telescope; and on this principle, placing the prisms within, he constructed a new micrometer, and had one executed by Mr Dollond, which upon trial answered as he expected. The construction is as follows.

Let AB be the object-glass; ab the image, suppose of the sun, which would have been formed in the principal focus Q; but let the prisms PR, SR be placed to intercept the rays, and let EF, WG, be two rays proceeding from the eastern and western limbs of the sun, converging, after refraction at the lens, to a and b; and suppose the refraction of the prisms to be such, that in fig. 8. the ray EFR, after refraction at R by the prism PR, may proceed in the direction RQ; and as all the rays which were proceeding to a suffer the same refraction at the prism, they will all be refracted to Q; and therefore, instead of an image ab, which would have been formed by the lens alone, an image Qc is formed by those rays which fall on the prism PR; and for the same reason, the rays falling on the prism SR will form an image Qd; and in fig. 9. the image of the point b is brought to Q, by the prism PR, and consequently an image Qd is formed by those rays which fall on PR; and for the same reason, an image Qc is formed by the rays falling on SR. Now in both cases, as the rays EFR, WGR, coming from the two opposite limbs of the sun, and forming the point of contact of the two limbs, proceed in the same direction RQ, they must thus accompany each other through the eye-glass and also through the eye, whatever refractive power it has, and therefore to every eye the images must appear to touch. Now the angle aRb is twice the refraction of the prism, and the angle aCb is the diameter of the sun; and as these angles are very small, and have the same subtense ab, we have the angle aRb : angle aCb :: CQ : RQ. — Now as CQ is constant, and also the angle aRb, being twice the refraction of the prism, the angle aCb varies as RQ. Hence the extent of the scale for measuring angles becomes the focal length of the object glass, and the angle measured is in proportion to the distance of the prisms from the principal focus of the object glass; and the micrometer can measure all angles (very small ones excepted, for the reason afterwards given*) which do not exceed the sum of the refractions of the prisms; for the angle aCb, the diameter of the object to be measured, is always less than the angle aRb, the sum of the refractions of the prisms, except when the prisms touch the object glass, and then they become equal. The scale can never be out of adjustment, as the point o, where the measurement begins, answers to the focus of the object glass, which is a fixed point for all distant objects, and we have only to find the

value of the scale answering to some known angle: Micrometer. for instance, bring the two limbs of the sun's images into contact, and measure the distance of the prisms from the focus, and look in the nautical almanac for the sun's diameter, and you get the value of the scale.

In fig. 8. the limb Q, of the image Qc, is illuminated by the rays falling on the object glass between A and F, and of the image Qd by those falling between B and G; but in fig. 9. the same limbs are illuminated by the rays falling between B and F, A and G respectively, and therefore will be more illuminated than in the other case; but the difference is not considerable in achromatic telescopes, on account of the great aperture of the object-glass compared with the distance FG.

It might be convenient to have two sets of prisms, one for measuring angles not exceeding 36°, and therefore fit for measuring the diameters of the sun and moon, and the lucid parts and distances of the eclipses in their eclipses; and another for measuring angles not much greater than 1°, for the convenience of measuring the diameters of the planets. For as QC : QR :: sum of the refractions of the prisms : angle aCb, the apparent diameter of the object, it is evident that if you diminish the third term, you must increase the second in the same ratio, in order to measure the same angle; and thus by diminishing the refractive angle of the prisms, you throw them further from Q, and consequently avoid the inconvenience of bringing them near to Q, for the reason in the next paragraph; and at the same time you will increase the illumination in a small degree. The prisms must be achromatic, each composed of two prisms of flint and crown glass, placed with their refracting angles contrariways, otherwise the images will be coloured.

In the construction here described, the angle measured becomes evanescent when the prisms come to the principal focus of the object glass, and therefore o on the scale then begins: but if the prisms be placed in the principal focus they can have no effect, because the pencil of rays at the junction of the prisms would then vanish, and therefore it is not practicable to bring the two images together to get o on the scale. Dr Maskelyne, therefore, thought of placing another pair of prisms within, to refract the rays before they came to the other prisms, by which means the two images would be formed into one before they came to the principal focus, and therefore o on the scale could be determined. But to avoid the error arising from the multiplication of mediums, he, instead of adding another pair of prisms, divided the object glass through its centre, and sliding the segments a little it separated the images, and then by the prisms he could form one image very distinctly, and consequently could determine o on the scale; for by separating the two segments you form two images, and you will separate the two pencils so that you may move up the two prisms, and the two pencils will fall on each respectively, and the two images may be formed into one. In the instrument which Dr Maskelyne had made, o on the scale was chosen to be about $\backslash frac\{1\}\{3\}$ of the focal length of the object-glass.

appear to coincide: but if we increase the refractive power of the eye, then each pencil is brought to a focus, and they cross each other before the rays come to the retina, consequently the two limbs on the retina will lap over; and if we diminish the refractive power of the eye, then each pencil being brought to a focus beyond the retina, and not crossing till after they have passed it, the two limbs on the retina must be separated.

Micrometer. glass, and each prism refracted 27°. By this means all angles are measured down to 0.

In the Philosophical Transactions for 1779, Mr Ramsden has described two new micrometers, which he contrived with a view of remedying the defects of the object-glass micrometer.

1. One of these is a catastrophic micrometer, which, beside the advantage it derives from the principle of reflection, of not being disturbed by the heterogeneity of light, avoids every defect of other micrometers, and can have no aberration, nor any defect arising from the imperfection of materials or of execution; as the extreme simplicity of its construction requires no additional mirrors or glasses to those required for the telescope; and the separation of the image being effected by the inclination of the two specula, and not depending on the focus of any lens or mirror, any alteration in the eye of an observer cannot affect the angle measured. It has peculiar to itself the advantages of an adjustment, to make the images coincide in a direction perpendicular to that of their motion; and also of measuring the diameter of a planet on both sides of the zero, which will appear no inconsiderable advantage to observers who know how much easier it is to ascertain the contact of the external edges of two images than their perfect coincidence.

Fig. 10. A represents the small speculum divided into two equal parts; one of which is fixed on the end of the arm B; the other end of the arm is fixed on a steel axis X, which crosses the end of the telescope C. The other half of the mirror A is fixed on the arm D, which arm at the other end terminates in a socket y, that turns on the axis X; both arms are prevented from bending by the braces a a. G represents a double screw, having one part e cut into double the number of threads in an inch to that of the part g; the part e having 100 threads in one inch, and the part g 50 only. The screw e works in a nut F in the side of the telescope, while the part g turns in a nut H, which is attached to the arm B; the ends of the arms B and D, to which the mirrors are fixed, are separated from each other by the point of the double screw pressing against the stud h, fixed to the arm D, and turning in the nut H on the arm B. The two arms B and D are pressed against the direction of the double screw e g by a spiral spring within the part n, by which means all shake or play in the nut H, on which the measure depends, is entirely prevented.

From the difference of the threads on the screw at e and g, it is evident, that the progressive motion of the screw through the nut will be half the distance of the separation of the two halves of the mirror; and consequently the half mirrors will be moved equally in contrary directions from the axis of the telescope C.

The wheel V fixed on the end of the double screw has its circumference divided into 100 equal parts, and numbered at every fifth division with 5, 10, &c. to 100, and the index I shows the motion of the screw with the wheel round its axis, while the number of revolutions of the screw is shown by the divisions on the same index. The steel screw at R may be turned by the key S, and serves to incline the small mirror at right angles to the direction of its motion. By turning the finger-head T (fig. 11.), the eye-tube P is

adjust nearer or farther from the small mirror, to adjust the telescope to distinct vision; and the telescope itself hath a motion round its axis for the convenience of measuring the diameter of a planet in any direction. The inclination of the diameter measured with the horizon is shown in degrees and minutes by a level and vernier on a graduated circle, at the breech of the telescope.

“ It is necessary to observe (says Mr Ramsden), that, besides the table for reducing the revolutions and parts of the screw to minutes, seconds, &c. it may require a table for correcting a very small error which arises from the eccentric motion of the half-mirrors. By this motion their centres of curvature will (when the angle to be measured is large) approach a little towards the large mirror: the equation for this purpose in small angles is insensible; but when angles to be measured exceed ten minutes, it should not be neglected. Or, the angle measured may be corrected by diminishing it in the proportion the versed sine of the angle measured, supposing the eccentricity radius, bears to the focal length of the small mirror.”

Mr Ramsden preferred Cassigraian's construction of the reflecting telescope to either the Gregorian or Newtonian; because in the former, errors caused by one speculum are diminished by those in the other. From a property of the reflecting telescope (which, he observes, has not been attended to), that the apertures of the two specula are to each other very nearly in the proportion of their focal lengths, it follows, that their aberrations will be to each other in the same proportion; and these aberrations are in the same direction, if the two specula are both concave; or in contrary directions, if one speculum is concave and the other convex. In the Gregorian construction, both specula being concave, the aberration at the second image will be the sum of the aberrations of the two mirrors; but in the Cassigraian construction, one mirror being concave and the other convex, the aberration at the second image will be the difference between their aberrations. By assuming such proportions for the foci of the specula as are generally used in the reflecting telescope, which is about as 1 to 4, the aberration in the Cassigraian construction will be to that in the Gregorian as 3 to 5.

2. The other is a dioptric micrometer, or one suited to the principle of refraction. This micrometer is applied to the erect eye-tube of a refracting telescope, and is placed in the conjugate focus of the first eye-glass: in which position, the image being considerably magnified before it comes to the micrometer, any imperfection in its glass will be magnified only by the remaining eye-glasses, which in any telescope seldom exceeds five or six times. By this position also the size of the micrometer glass will not be the $\backslash frac\{1\}\{100\}$ part of the area which would be required if it was placed in the object-glass; and, notwithstanding this great disproportion of size, which is of great moment to the practical optician, the same extent of scale is preserved, and the images are uniformly bright in every part of the field of the telescope.

Fig. 12. represents the glasses of a refracting telescope; xy, the principal pencil of rays from the object-glass O; tt and uu, the axis of two oblique pencils; a, the first eye-glass; m, its conjugate focus, or the

Micrometer-
te.
place of the micrometer; $b$ the second eye-glass; $c$ the third; and $d$ the fourth, or that which is nearest the eye. Let $a$ be the diameter of the object-glass, $e$ the diameter of a pencil at $m$, and $f$ the diameter of the pencil at the eye; it is evident, that the axis of the pencils from every part of the image will cross each other at the point $m$; and $e$, the width of the micrometer-glass, is to $a$ the diameter of the object-glass as $m\; a$ is to $g\; o$, which is the proportion of the magnifying power at the point $m$; and the error caused by an imperfection in the micrometer glass placed at $m$ will be to the error, had the micrometer been at $O$, as $m$ is to $a$.

Fig. 13. represents the micrometer; $A$, a convex or concave lens divided into two equal parts by a plane across its centre; one of these semi-lenses is fixed in a frame $B$, and the other in the frame $E$; which two frames slide on a plate $H$, and are pressed against it by thin plates $a\; a$: the frames $B$ and $E$ are moved in contrary directions by turning the button $D$; $L$ is a scale of equal parts on the frame $B$; it is numbered from each end towards the middle with 10, 20, &c. There are two verniers on the frame $E$, one at $M$ and the other at $N$, for the convenience of measuring the diameter of a planet, &c. on both sides the zero. The first division on both these verniers coincides at the same time with the two zeros on the scale $L$; and, if the frame is moved towards the right, the relative motion of the two frames is shown on the scale $L$ by the vernier $M$; but if the frame $B$ be moved towards the left, the relative motion is shown by the vernier $N$.—This micrometer has a motion round the axis of vision, for the convenience of measuring the diameter of a planet, &c. in any direction, by turning an endless screw $F$; and the inclination of the diameter measured with the horizon is shown on the circle $g$ by a vernier on the plate $V$. The telescope may be adjusted to distinct vision by means of an adjusting screw, which moves the whole eye-tube with the micrometer nearer or farther from the object-glass, as telescopes are generally made; or the same effect may be produced in a better manner, without moving the micrometer, by sliding the part of the eye tube $m$ on the part $a$, by help of a screw or pinion. The micrometer is made to take off occasionally from the eye-tube, that the telescope may be used without it.

Still, however, micrometers remained in several respects imperfect. In particular, the imperfections of the parallel-wire micrometer in taking the distance of very close double stars, are the following.

When two stars are taken between the parallels, the diameters must be included. Mr Herschel informs us, he has in vain attempted to find lines sufficiently thin to extend them across the centres of the stars so that their thickness might be neglected. The single threads of the silk-worm, with such lenses as he uses, are so much magnified that their diameter is more than that of many of the stars. Besides, if they were much less than they are, the power of deflection of light would make the attempt to measure the distance of the centres this way fruitless: for he has always found the light of the stars to play upon those lines and separate their apparent diameters into two parts. Now since the spurious diameters of the stars thus included, as Mr. Herschel assures us, are continually changing according to the state of the air,

and the length of time we look at them, we are, in some respect, left at an uncertainty, and our measures taken at different times and with different degrees of attention, will vary on that account. Nor can we come at the true distance of the centres of any two stars, one from another, unless we could tell what to allow for the semidiameters of the stars themselves; for different stars have different apparent diameters, which, with a power of 227, may differ from each other as far as two seconds.

The next imperfection is that which arises from a deflection of light upon the wires when they approach very near to each other; for if this be owing to a power of repulsion lodged at the surface, it is easy to understand, that such powers must interfere with each other, and give the measures larger in proportion than they would have been if the repulsive power of one wire had not been opposed by a contrary power of the other wire.

Another very considerable imperfection of these micrometers is a continual uncertainty of the real zero. Mr Herschel has found, that the least alteration in the situation and quantity of light will affect the zero, and that a change in the position of the wires, when the light and other circumstances remain unaltered, will also produce a difference. To obviate this difficulty, whenever he took a measure that required the utmost accuracy, his zero was always taken immediately after, while the apparatus remained in the same situation it was in when the measure was taken; but this enhances the difficulty, because it introduces an additional observation.

The next imperfection, which is none of the smallest, is that every micrometer that has hitherto been in use requires either a screw, or a divided bar and pinion, to measure the distance of the wires or divided image. Those who are acquainted with works of this kind are but too sensible how difficult it is to have screws that shall be perfectly equal in every thread or revolution of each thread; or pinions and bars that shall be so evenly divided as perfectly to be depended upon in every leaf and tooth to perhaps the two, three, or four thousandth part of an inch: and yet, on account of the small scale of those micrometers, these quantities are of the greatest consequence; an error of a single thousandth part inducing in most instruments a mistake of several seconds.

The last and greatest imperfection of all is, that these wire micrometers require a pretty strong light in the field of view; and when Mr Herschel had double stars to measure, one of which was very obscure, he was obliged to be content with less light than is necessary to make the wires perfectly distinct; and several stars on this account could not be measured at all, though otherwise not too close for the micrometer.

Mr Herschel, therefore, having long had much occasion for micrometers that would measure exceeding small distances exactly, was led to bend his attention to the improvement of these instruments; and the result of his endeavours has been a very ingenious instrument called a lamp-micrometer, which is not only free from the imperfections above specified, but also possesses the advantages of a very large scale. This instrument is described in the Philosophical Transactions for 1782; and the construction of it is as follows: