Plate CLX.

adjusted state: to perform which, you try the levels of the horizontal plates \( A_a \), by first turning the screws \( BRRR \) till the bubbles of air on the glass tubes of the spirit-levels (levels are at right angles to each other) which are central or in the middle, and remain so when you turn the upper plate \( A \) half round its centre; but if they should not keep so, there are small screws at the end of each level, which admit of being turned one way or the other as may be requisite till they are so. The plates \( Aa \) being thus made horizontal, set the latitude arch or meridian \( C \) steadily between the two-grooved sides that hold it (one of which is seen at \( D \)), by the screw behind. On this side \( D \) is divided the nonius or vernier, corresponding with the divisions on the latitude arch \( C \), and which may be subdivided into 5 minutes of a degree, and even less if required. The latitude arch \( C \) is to be so placed in \( D \), that the pole \( M \) may be in a vertical position; which is done by making 90° on the arch at bottom coincide with the 0 of the nonius. The arch is then fixed by the tightening screw at the back of \( D \). Hang a silken plumb-line on the hook at \( G \); which line is to coincide with a mark at the bottom of the latitude arch at \( H \), all the while you move the upper plate \( A \) round its centre. If it does not so, there are four screws to regulate this adjustment; two of which pass through the base \( I \) into the plate \( A \); the other two screws fasten the nonius piece \( D \) together; which when unscrewed a thread or two, the nonius piece may be easily moved to the right or left of 90° as may be found requisite.

Prob. 1. To find the latitude of the place. Fasten the latitude and hour circles together, by placing the pin \( K \) into the holes; slide the nonius piece \( E \) on the hour-circle to the sun’s declination for the given day: the sun’s declination you may know in the ephemeris by White, or other almanacs, for every day in the year. The nonius piece \( E \) must be set on that portion of the hour-circle marked \( ND \) or \( SD \), according as the sun has north or south declination. About 20 minutes or a quarter of an hour before noon, observe the sun’s shadow or spot that passes through the hole at the axis \( O \), and gently move the latitude arch \( C \) down in its groove at \( D \) till you observe the spot exactly fall on the cross line on the centre of the nonius piece at \( L \); and by the falling of this spot, so long as you observe the sun to increase in altitude, you depress the arch \( C \): but at the instant of its stationary appearance the spot will appear to go no lower; then fix the arch by the screw at the back of \( D \), and the degrees thereby cut by the nonius on the arch will be the latitude of the place required: if great exactness is wanted, allowance should be made for the refraction of the atmosphere, taken from some nautical or astronomical treatise.

Prob. 2. The latitude of the place being given, to find the time by the sun or stars. From an ephemeris as before, you find the sun’s declination for the day north or south, and set the nonius piece \( E \) on the arch accordingly. Set the latitude arch \( C \), by the nonius at \( D \), to the latitude of the place; and place the magnifying glass at \( M \), by which you will very correctly set the index carrying a nonius to the upper XII at \( S \). Take out the pin \( K \), slacken the horizontal screw \( N \), and gently move, either to the right or left as you see necessary, the hour-circle \( F \), at the same time with the other hand moving the horizontal plate \( A \) round its axis to the right and left, till the latitude-arch \( C \) falls into the meridian; which you will know by the sun’s spot falling exactly in the centre of the nonius piece, or where the lines intersect each other. The time may be now read off exactly to a minute by the nonius on the dial-plate at top, and which will be the time required. The horizontal line drawn on the nonius piece \( L \), not seen in the figure, being the parallel of declination or path that the sun-dial makes, it therefore can fall on the centre of that line at no other time but when the latitude arch \( C \) is in the meridian or due north and south. Hence the hour-circle, on moving round with the pole, must give the true time on the dial-plate at top. There is a hole to the right, and cross hairs to the left, of the centre axis hole \( O \), where the sun’s rays pass through; whence the sun’s shadow or spot will also appear on the right and left of the centre on the nonius piece \( L \), the holes of which are occasionally used as sights to observe through. If the sun’s rays are too weak for a shadow, a dark glass to screen the eye is occasionally placed over the hole. The most proper time to find a true meridian is three or four hours before or after noon; and take the difference of the sun’s declination from noon at the time you observe. If it be the morning, the difference is that and the preceding day; if afternoon, that and the following day: and the meridian being once found exact, the hour-circle \( E \) is to be brought into this meridian, a fixed place made for the dial, and an object to observe by it also fixed for it at a great distance. The sights \( LO \) must at all times be directed against this fixed object, to place the dial truly in the meridian, proper for observing the planets, moon, or bright stars by night.

Prob. 3. To find the sun’s azimuth and altitude. The latitude-arch \( C \) being in the meridian, bring the pole \( M \) into the zenith, by setting the latitude-arch to 90°. Fasten the hour-circle \( E \) in the meridian, by putting in the pin \( K \); fix the horizontal plates by the screw \( N \); and set the index of the dial-plate to XII. which is the fourth point: Now take out the pin \( K \), and gently move the hour-circle \( E \); leaving the latitude arch fixed, till the sun’s rays or spot passing through the centre-hole in the axis \( O \) fall on the centre line of the hour-circle \( E \), made for that purpose. The azimuth in time may be then read off on the dial-plate at top by the magnifying glass. This time may be converted into degrees, by allowing at the rate of 15 for every hour. By sliding the nonius piece \( E \), so that the spot shall fall on the cross line thereon, the altitude may be taken at the same time if it does not exceed 45 degrees. Or the altitude may be taken more universally, by fixing the nonius piece \( E \) to the 0 on the divisions, and sliding down the latitude arch in such a manner in the groove at \( D \), till the spot falls exactly on the centre of the nonius \( E \). The degrees and minutes then shown by the nonius at \( D \), taken from 90°, will be the altitude required. By looking through the sight holes \( L, O \), the altitude of the moon, planets, and stars, may be easily taken. Upon this principle it is somewhat adapted for levelling also: by lowering the nonius piece \( E \), equal altitudes of the sun may be had; and by raising it higher, equal depressions.

More completely to answer the purposes of a good theodolite, of levelling, and the performance of problems blems in practical astronomy, trigonometry, &c. Mr W. Jones of Holborn divides the horizontal plate D into 368°, and an opposite nonius on the upper plate A, subdividing the degrees into 5 or more minutes. A telescope and spirit-level applies on the latitude arch at H G by two screws, making the latitude arch a vertical arch; and the whole is adapted to triangular staffs with parallel plates, similar to those used with the best theodolites.

A dial more universal for the performance of problems than the above, though in some particulars not so convenient and accurate, is made by Mr Jones and other instrument-makers in London. It consists of the common equatorial circles reduced to a portable size, and instead of a telescope carries a plain sight. Its principal parts consist of the sight-piece O P, moveable over the declination's semicircle D. It has a nonius Q to the semicircle. A dark glass to screen the eye applies occasionally over either of the holes at O: these holes on the inner side of the piece are intersected by cross lines, as seen in the figure below; and to the sight P two pieces are screwed, the lower having a small hole for the sun's rays or shadow, and the upper two cross hairs or wires.

The declination circle or arch D is divided into two, 90° each; and is fixed perpendicularly on a circle with a chamfered edge, containing a nonius division that subdivides into single minutes under the equatorial circle MN, which in all cases represents the equator, and is divided into twice 12 hours, and each hour into five minutes. At right angles below this equatorial circle is fixed the semicircle of altitude AB, divided into two quadrants of 90° each. This arch serves principally to measure angles of altitude and depression; and it moves centrally on an upright pillar fixed in the horizontal circle EF. This circle EF is divided into four quadrants of 90° each, and against it there is fixed a small nonius plate at N. The horizontal circle may be turned round its centre or axis; and two spirit levels LL are fixed on it at right angles to one another.

We have not room to detail the great variety of astronomical and trigonometrical problems that may be solved by this general instrument, as described in Jones's Instrumental Dialing. One example connected with our present purpose may here suffice, viz. To find the time when the latitude is given. Supposing the instrument to be well adjusted by the directions hereafter given. The meridian of the place should be first obtained to place the instrument in, which is settled by a distant mark, or particular cavities to receive the screws at IGH, made in the base it stands on. The meridian is best found by equal altitudes of the sun. In order to take these, you set the middle mark of the nonius on the declination arch D at o, and fix it by the screw behind; then set the horary or hour circle to XII. The circle EF being next made horizontal, you direct the sights to the sun, by moving the horizontal circle EF and altitude semicircle AB: the degrees and minutes marked by the nonius on the latter will be the altitude required. To take equal altitudes, you observe the sun's altitude in the morning two or three hours before noon by the semicircle AB: leave the instrument in the same situation perfectly unaltered till the afternoon, when by moving the horizontal circle EF, only find the direction of the sight or the sun's spot to be just the same, which will be an equal altitude with the morning. The place of the horizontal circle EF against the nonius at each time of observation is to be carefully noted; and the middle degree or part between each will be the place where the semicircle AB, and sight OP, will stand or coincide with, when directed to the south or north, according to the sun's situation north or south at noon at the place of observation. Set the index or sight-piece OP very accurately to this middle point, by directing the sight to some distant object; or against it, let one be placed up: this object will be the meridian mark, and will always serve at any future time. To find the time, the meridian being thus previously known by equal altitudes of the sun (or star), and determined by the meridian mark made at a distance, or by the cavities in the base to set the screw in: Place the equatorial accordingly, and level the horizontal circle EF by the spirit-levels thereon. Set the semicircle AB to the latitude of the place, and the index of the sights O P to the declination of the sun, found by the ephemeris, as before directed. Turn the semicircle D till the sight-holes are accurately directed to the sun, when the nonius on the hour circle MN will show the time. It may easily be known when the sun's rays are directed through, by the spot falling on the lower intersections of the marks across the hole at O. See the figure S adjoining.

The adjustments of this equatorial dial are to be made from the following trials. 1st. To adjust the levels LL on EF: Place the o of any of the divisions on EF to the middle mark or stroke on the nonius at N; bring the air-bubbles in the levels in the centres of each scale, by turning the several screws at IGH: this being exactly done, turn the circle EF two 90° or half round: if the bubble of air then remains in the centre, they are right, and properly adjusted for use; but if they are not, you make them so by turning the necessary screws placed for that purpose at the ends of the level-scales by means of a turncrew, until you bring them to that fixed position, that they will return when the plate EF is turned half round. 2ndly. To adjust the line of sight OP: Set the nonius to o on the declination arch D, the nonius on the hour-circle to VI, and the nonius on the semicircle AB to 90°. Direct to some part of the horizon where there may be a variety of fixed objects. Level the horizontal circle EF by the levels LL, and observe any object that may appear on the centre of the cross wires. Reverse the semicircle AB, viz. so that the opposite 90° of it be applied to the nonius, observing particularly that the other nonius preserve their situation. If then the remote object formerly viewed still continues in the centre of the cross wires, the line of sight OP is truly adjusted; but if not, unscrew the two screws of the frame carrying the cross wires, and move the frame till the intersection appears against another or new object, which is half way between the first and that which the wires were against on the reverberation. Return the semicircle AB to its former position: when, if the intersection of the wires be found to be against the half-way-object, or that to which they were last divided, the line of sight is adjusted; if not, the operation of observing the interval of the two objects, and applying half way, must be repeated.

It is necessary to observe, that one of the wires should be in the plane of the declination circle, and the other wire at right angles; the frame containing the wires is made to shift for that purpose.

The hole at P which forms the sun's spot is also to be adjusted by directing the sight to the sun, that the centre of the shadow of the cross hairs may fall ex- actly on the upper hole; the lower frame with the hole is then to be moved till the spot falls exactly on the lower sight-hole.

Lastly, it is generally necessary to find the correc- tion always to be applied to the observations by the semicircle of altitude AB. Set the nonius to O on the declination arch D, and the nonius to XII on the equator or hour-circle: Turn the sight to any fixed and distinct object, by moving the arch AB and circle EF only: Note the degree and minute of the angle of alti- tude or depression: Reverse the declination semi- circle by placing the nonius on the hour-circle to the opposite XII: Direct the sight to the same object again as before. If the altitude or depression now gi- ven be the same as was observed in the former position, no correction is wanted; but if not the same, half the difference of the two angles is the correction to be added to all observations or rectifications made with that quadrant by which the least angle was taken, or to be subtracted from all observations made with the other quadrant. These several adjustments are abso- lutely necessary previous to the use of the instrument; and when once well done, will keep so, with care, a considerable time.

The Universal or Astronomical Equinoctial Ring-Dial, is an instrument of an old construction, that also serves to find the hour of the day in any latitude of the earth (see fig. 3.). It consists of two flat rings or circles, usually from 4 to 12 inches diameter, and of a moderate thickness; the outward ring AE representing the meridian of the place it is used at, contains two di- visions of 90° each opposite to one another, serving to let the sliding piece H, and ring G (by which the dial is usually suspended), be placed on one side from the equator to the north pole, and on the other side to the south, according to the latitude of the place. The inner ring B represents the equator, and turns dia- metrically within the outer by means of two pivots inserted in each end of the ring at the hours XII.

Across the two circles is screwed to the meridian a thin pierced plate or bridge, with a cursor C, that slides along the middle of the bridge: this cursor has a small hole for the sun to shine through. The middle of this bridge is conceived as the axis of the world, and its extremities as the poles; on the one side are delineated the 12 signs of the zodiac, and some- times opposite the degrees of the sun's declination; and on the other side the days of the month through- out the year. On the other side of the outer ring A are the divisions of 90°, or a quadrant of altitude: It serves, by the placing of a common pin P in the hole b (see fig. 4.), to take the sun's altitude or height, and from which the latitude of the place may easily be found.

Use of the dial. Place the line a in the middle of the sliding piece H over the degree of latitude of the place. Suppose, for example, 51° for London; put the line which crosses the hole of the cursor C to the day of the month or the degree of the sign. Open the instrument till the two rings be at right angles to each other, and suspend it by the ring G; that the axis of the dial represented by the middle of the bridge be parallel to the axis of the earth, viz. the north pole to the north, and vice versa. Then turn the flat side of the bridge towards the sun, so that his rays falling through the small hole in the cursor may fall exactly in a line drawn through the middle of the concave sur- face of the inner ring or hour-circle, the bright spot by which shows the hour of the day in the laid con- cave surface of the dial. Note, The hour XII cannot be shown by this dial, because the outer ring being then in the plane of the meridian, excludes the sun's rays from the inner; nor can this dial show the hour when the sun is in the equinoctial, because his rays then falling parallel to the plane of the inner circle or equinoctial, are excluded by it.

To take the altitude of the sun by this dial, and with the declination thereby to find the latitude of the place: Place a common pin p in the hole b projecting in the side of the meridian where the quadrant of altitude is: then bring the centre mark of the sliding piece H to the O or middle of the two divisions of latitude on the other side, and turn the pin towards the sun till it cuts a shadow over the degree of the quadrant of alti- tude; then what degree the shadow cuts is the altitude. Thus, in fig. 4., the shadow bg appears to cut 35°, the altitude of the sun.

The sun's declination is found by moving the cursor in the sliding piece till the mark across the hole stands just against the day of the month; then, by turning to the other side of the bridge, the mark will stand against the sun's declination.

In order to find the latitude of the place, observe that the latitude and declination be the same, viz. both north or south; subtract the declination from the meridian or greatest daily altitude of the sun, and the remainder is the complement of the latitude; which subtracted from 90°, leaves the latitude. Ex- ample:

| Deg. min. | |-----------| | The meridian altitude may be | 57° 48' | | The sun's declination for the day | 19° 18' | | Complement of latitude | 38° 30' |

The latitude = 51° 30'

But if the latitude and declination be contrary, add them together, and the sum is the complement of the latitude. This dial is sometimes mounted on a stand, with a compass, two spirit-levels, and ad- justing screws, &c. &c. (see fig. 5.), by which it is rendered more useful and convenient for finding the sun's azimuth, altitudes, variation of the needle, de- clinations of planes, &c. &c.

An Universal Dial on a plain cross, is described by Mr Ferguson. It is moveable on a joint C, for ele- vating it to any given latitude on the quadrant C o 90°, Fig. 6, 7, 8, as it stands upon the horizontal board A. The arms of the cross stand at right angles to the middle part; and the top of it, from a to n, is of equal length with either of the arms ne or mk. See fig. 6.

This dial is rectified by setting the middle line tu to the latitude of the place on the quadrant, the board A level, and the point N northward by the needle; thus, the plane of the cross will be parallel to the plane of the equator. Then, from III o'clock in the morning till VI, the upper edge k l of the arm io will cast a shadow on the time of the day on the side of the arm cm; from VI till IX, the lower edge i of the arm io will cast a shadow on the hours on the side pq. From IX in the morning to XII at noon, the edge ab of the top part an will cast a shadow on the hours on the arm nef; from XII to III in the afternoon, the edge cd of the top part will cast a shadow on the hours on the arm klm; from III to VI in the evening, the edge gh will cast a shadow on the hours on the part pq; and from VI till IX, the shadow of the edge ef will show the time on the top part ar.

The breadth of each part, ab, ef, &c., must be so great, as never to let the shadow fall quite without the part or arm on which the hours are marked, when the sun is at his greatest declination from the equator.

To determine the breadth of the sides of the arms which contain the hours, so as to be in just proportion to their length; make an angle ABC (fig. 7.) of $23\frac{1}{2}$ degrees, which is equal to the sun's greatest declination; and suppose the length of each arm, from the side of the long middle part, and also the length of the top part above the arms, to be equal to Bd. Then, as the edges of the shadow, from each of the arms, will be parallel to Be, making an angle of $23\frac{1}{2}$ degrees with the side Bd of the arm, when the sun's declination is $23\frac{1}{2}$°; it is plain, that if the length of the arm be Bd, the least breadth that it can have, to keep the edge Be of the shadow Be gd from going off the side of the arm de before it comes to the end of it ed, must be equal to ed or dB. But in order to keep the shadow within the quarter divisions of the hours, when it comes near the end of the arm, the breadth of it should be still greater, so as to be almost doubled, on account of the distance between the tips of the arms.

The hours may be placed on the arms, by laying down the cross abcd (fig. 8.) on a sheet of paper; and with a black-lead pencil held close to it, drawing its shape and size on the paper. Then take the length ae in the compasses, and with one foot in the corner a, describe with the other the quadrant ef. Divide this arc into five equal parts, and through the points of division draw light lines ag, ah, &c., containing three of them to the arm ce, which are all that can fall upon it; and they will meet the arm in those points through which the lines that divide the hours from each other, as in fig. 6, are to be drawn right across it. Divide each arm, for the three hours contained in it, in the same manner; and set the hours to their proper places, on the sides of the arms, as they are marked in fig. 33. Each of the hour spaces should be divided into four equal parts, for the half hours and quarters, in the quadrant ef; and right lines should be drawn through these division-marks in the quadrant, to the arms of the cross, in order to determine the places thereon where the subdivisions of the hours must be marked.

This is a very simple kind of universal dial; it is easily made and has a pretty, uncommon appearance in a garden.

Fig. 9. is called a Universal Mechanical Dial, as by its equinoctial circle an easy method is had of describing a dial on any kind of plane. For example: Suppose a dial is required on an horizontal plane. If the plane be immoveable, as ABCD, find drawing a meridian line as GF; or if moveable, assume the meridian at pleasure: then by means of the triangle mechanical KEF, whose base is applied on the meridian line cal dial, raise the equinoctial dial H till the index GI becomes parallel to the axis of the earth, (which is so, if the angle KEF be equal to the elevation of the pole), and the 12 o'clock line on the dial hang over the meridian line of the plane or the base of the triangle. If then, in the night-time or a darkened place, a lighted candle be successively applied to the axis GI, so as the shadow of the index or style GI fall upon one hour-line after another, the same shadow will mark out the several hour-lines on the plane ABCD. Noting the points therefore on the shadow, draw lines through them to G; then an index being fixed on G, according to the angle IGF, its shadow will point out the several hours by the light of the fire.

If a dial were required on a vertical plane, having raised the equinoctial circle as directed, push forward the index GI till the tip thereof I touch the plane. If the plane be inclined to the horizon, the elevation of the pole should be found on the same; and the angle of the triangle KEF should be made equal thereto.

Mr Fergusson describes a method of making three Dials on dials on three different planes, so that they may all show the three planes of the day by one gnomon. On the flat board ABC by one gnomon describe an horizontal dial, with its gnomon FGH; mon. the edge of the shadow of which shows the time of the day. To this horizontal board join the upright board EDC, touching the edge GH of the gnomon; then making the top of the gnomon at G the centre of the vertical fourth dial, describe it on the board EDC. Besides, on a circular plate IK describe an equinoctial dial, and, by a slit cd in the XII o'clock line from the edge to the centre, put it on the gnomon EG as far as the slit will admit. The same gnomon will show the same hour on each of these dials.

An Universal Dial, showing the hours of the day by a terrestrial globe, and by the shadows of several gnomons, at the same time: together with all the places of the earth, which are then enlightened by the sun; and those to which the sun is then rising, or on the meridian, or setting. This dial is made of a thick square piece of wood, or hollow metal. The sides are cut into semicircular hollows, in which the hours are placed; the stile of each hollow coming out from the bottom thereof, as far as the ends of the hollows project. The corners are cut out into angles, in the insides of which the hours are also marked; and the edge of the end of each side of the angle serves as a stile for casting a shadow on the hours marked on the other side.

In the middle of the uppermost side, or plane, there is an equinoctial dial; in the centre whereof an upright wire is fixed, for casting a shadow on the hours of that dial, and supporting a small terrestrial globe on its top.

The whole dial stands on a pillar, in the middle of a round horizontal board, in which there is a compass and magnetic needle, for placing the meridian stile toward ward the south. The pillar has a joint with a quadrant upon it, divided into 90 degrees (supposed to be hid from sight under the dial in the figure) for setting it to the latitude of any given place.

The equator of the globe is divided into 24 equal parts, and the hours are laid down upon it at these parts. The time of the day may be known by these hours, when the sun shines upon the globe.

To rectify and use this dial, set it on a level table, or sole of a window, where the sun shines, placing the meridian file due south, by means of the needle; which will be, when the needle points as far from the north fleur-de-lis toward the west, as it declines westward, at your place. Then bend the pillar in the joint, till the black line on the pillar comes to the latitude of your place in the quadrant.

The machine being thus rectified, the plane of its dial part will be parallel to the equator, the wire or axis that supports the globe will be parallel to the earth's axis, and the north pole of the globe will point toward the north pole of the heavens.

The same hour will then be shown in several of the hollows, by the ends of the shadows of their respective files: the axis of the globe will cast a shadow on the same hour of the day, in the equinoctial dial, in the centre of which it is placed, from the 20th of March to the 23rd of September; and, if the meridian of your place on the globe be set even with the meridian file, all the parts of the globe that the sun shines upon, will answer to those places of the real earth which are then enlightened by the sun. The places where the shade is just coming upon the globe, answer to all those places of the earth to which the sun is then setting; as the places where it is going off, and the light coming on, answer to all the places of the earth where the sun is then rising. And lastly, if the hour of VI be marked on the equator in the meridian of your place (as it is marked on the meridian of London in the figure) the division of the light and shade on the globe will show the time of the day.

The northern file of the dial (opposite to the southern or meridian one) is hid from the sight in the figure, by the axis of the globe. The hours in the hollow to which that file belongs, are also supposed to be hid by the oblique view of the figure: but they are the same as the hours in the front-hollow. Those also in the right and left hand semicircular hollows are mostly hid from sight; and so also are all those on the sides next the eye of the four acute angles.

The construction of this dial is as follows:

On a thick square piece of wood, or metal, draw the lines \(a c\) and \(b d\), as far from each other as you intend for the thickness of the file \(abc d\); and in the same manner, draw the like thickness of the other three files, \(efgh\), \(iklm\), and \(nopq\), all standing upright as from the centre.

With any convenient opening of the compasses, as \(a A\) (so as to leave proper strength of stuff when \(K I\) is equal to \(a A\)), set one foot in \(a\), as a centre, and with the other foot describe the quadrant arc \(Ac\). Then, without altering the compasses, set one foot in \(b\) as a centre, and with the other foot describe the quadrant \(dB\). All the other quadrants in the figure must be described in the same manner, and with the same opening of the compasses, on their centres \(efik\), and \(no\); and each quadrant divided into six equal parts, for as many hours, as in the figure; each of which parts must be subdivided into 4, for the half-hours and quarters.

At equal distances from each corner, draw the right lines \(Ip\) and \(Kp\), \(Lq\) and \(Mq\), \(Nr\) and \(Or\), \(Ps\) and \(Qs\); to form the four angular hollows \(IpK\), \(LqM\), \(NrO\), and \(PsQ\); making the distances between the tips of these hollows, as \(IK\), \(LM\), \(NO\), and \(PQ\), each equal to the radius of the quadrants; and leaving sufficient room within the angular points \(pqrs\), for the equinoctial in the middle.

To divide the insides of these angles properly for the hour-spaces thereon, take the following method.

Set one foot of the compasses in the point \(I\) as a centre, and open the other to \(K\); and with that opening describe the arc \(Kt\); then, without altering the compasses, set one foot in \(K\), and with the other foot describe the arc \(It\). Divide each of these arcs, from \(I\) and \(K\) to their intersection at \(t\), into four equal parts; and from their centres \(I\) and \(K\), through the points of division, draw the right lines \(I3\), \(I4\), \(I5\), \(I6\), \(I7\), and \(K2\), \(K1\), \(K0\), \(K11\); and they will meet the sides \(Kp\) and \(IpK\) of the angle \(IpK\) where the hours thereon must be placed. And these hour-spaces in the arcs must be subdivided into four equal parts, for the half hours and quarters.—Do the like for the other three angles, and draw the dotted lines, and set the hours in the insides where those lines meet them, as in the figure; and the like hour-lines will be parallel to each other in all the quadrants and in all the angles.

Mark points for all these hours on the upper side; and cut out all the angular hollows, and the quadrant ones quite through the places where their four gnomons must stand; and lay down the hours on their insides, (as in fig. 10.), and set in their gnomons, which must be as broad as the dial is thick; and this breadth and thickness must be large enough to keep the shadows of the gnomons from ever falling quite out at the sides of the hollows, even when the sun's declination is at the greatest.

Lastly, draw the equinoctial dial in the middle, all the hours of which are equidistant from each other; and the dial will be finished.

As the sun goes round, the broad end of the shadow of the file \(abcd\) will show the hours in the quadrant \(Ac\), from sun-rise till VI in the morning; the shadow from the end \(M\) will show the hours on the side \(Lq\) from V to IX in the morning; the shadow of the file \(efgh\) in the quadrant \(Dg\) (in the long days) will show the hours from sun-rise till VI in the morning; and the shadow of the end \(N\) will show the morning-hours, on the side \(Or\), from III to VII.

Just as the shadow of the northern file \(abcd\) goes off the quadrant \(Ac\), the shadow of the southern file \(iklm\) begins to fall within the quadrant \(Ft\), at VI in the morning; and shows the time, in that quadrant, from VI till XII at noon; and from noon till VI in the evening in the quadrant \(mE\). And the shadow of the end \(O\) shows the time from XI in the forenoon till III in the afternoon, on the side \(rN\); as the shadow of the end \(P\) shows the time... from IX in the morning till 1 o'clock in the afternoon, on the side Qs.

At noon, when the shadow of the eastern stile ef gb goes off the quadrant bC (in which it showed the time from VI in the morning till noon, as it did in the quadrant gD from sun-rise till VI in the morning), the shadow of the western stile n opq begins to enter the quadrant Hp; and shows the hours thereon from XII at noon till VI in the evening; and after that till sunset, in the quadrant q G: and the end Q casts a shadow on the side Ps from V in the evening till IX at night, if the sun be not set before that time.

The shadow of the end I shows the time on the side Kp from III till VII in the afternoon; and the shadow of the stile ab cd shows the time from VI in the evening till the sun sets.

The shadow of the upright central wire, that supports the globe at top, shows the time of the day, in the middle or equinoctial dial, all the summer half-year, when the sun is on the north side of the equator.

Having shown how to make sun-dials by the assistance of a good globe, or of a dialing scale, we shall now proceed to the method of constructing dials arithmetically; which will be more agreeable to those who have learned the elements of trigonometry, because globes and scales can never be so accurate as the logarithms in finding the angular distances of the hours. Yet as a globe may be found exact enough for some other requisites in dialing, we shall take it in occasionally.

The construction of sun-dials on all planes whatever may be included in one general rule: intelligible, if that of a horizontal-dial for any given latitude be well understood. For there is no plane, however obliquely situated with respect to any given place, but what is parallel to the horizon of some other place; and therefore if we can find that other place by a problem on the terrestrial globe, or by a trigonometrical calculation, and construct a horizontal dial for it; that dial applied to the plane where it is to serve will be a true dial for that place.—Thus, an erect direct south dial in 51½ degrees north latitude, would be a horizontal-dial on the same meridian, 90 degrees southward of 51½ degrees north latitude: which falls in with 38½ degrees of south latitude. But if the upright plane declines from facing the south at the given place, it would still be a horizontal plane 90 degrees from that place, but for a different longitude, which would alter the reckoning of the hours accordingly.

CASE I. 1. Let us suppose that an upright plane at London declines 36 degrees westward from facing the south, and that it is required to find a place on the globe to whose horizon the said plane is parallel; and also the difference of longitude between London and that place.

Rectify the globe to the latitude of London, and bring London to the zenith under the brafs meridian; then that point of the globe which lies in the horizon at the given degree of declination (counted westward from the south point of the horizon) is the place at which the above-mentioned plane would be horizontal.

—Now, to find the latitude and longitude of that place, keep your eye upon the place, and turn the globe eastward until it comes under the graduated edge of the brafs meridian; then the degree of the brafs meridian that stands directly over the place is its latitude; and the number of degrees in the equator, which are intercepted between the meridian of London and the brafs meridian, is the place's difference of longitude.

Thus, as the latitude of London is 51½ degrees north, and the declination of the place is 36 degrees west; elevate the north pole 51½ degrees above the horizon, and turn the globe until London comes to the zenith, or under the graduated edge of the meridian; then count 36 degrees on the horizon westward from the south point, and make a mark on that place of the globe over which the reckoning ends, and bringing the mark under the graduated edge of the brafs meridian, it will be found to be under 30½ degrees in south latitude: keeping it there, count in the equator the number of degrees between the meridian of London and the brafs meridian (which now becomes the meridian of the required place), and you will find it to be 42½. Therefore an upright plane at London, declining 36 degrees westward from the south, would be a horizontal plane at that place, whose latitude is 30½ degrees south of the equator, and longitude 42½ degrees west of the meridian of London.

Which difference of longitude being converted into time, is 2 hours 51 minutes.

The vertical-dial declining westward 36 degrees at London, is therefore to be drawn in all respects as a horizontal-dial for south latitude 30½ degrees; save only that the reckoning of the hours is to anticipate the reckoning on the horizontal-dial by 2 hours 51 minutes: for so much sooner will the sun come to the meridian of London, than to the meridian of any place whose longitude is 42½ degrees west from London.

2. But to be more exact than the globe will show us, we shall use a little trigonometry.

Let NESW be the horizon of London, whose zenith is Z, and P the north pole of the sphere; and let Zb be the position of a vertical plane at Z, declining westward from S (the south) by an angle of 36 degrees; on which plane an erect-dial for London at Z is to be described. Make the semidiameter ZD perpendicular to Zb; and it will cut the horizon in D, 36 degrees west of the south S. Then a plane, in the tangent HD, touching the sphere in D, will be parallel to the plane Zb; and the axis of the sphere will be equally inclined to both these planes.

Let WQER be the equinoctial, whose elevation above the horizon of Z (London) is 38½ degrees; and PRD be the meridian of the place D, cutting the equinoctial in R. Then it is evident, that the arc RD is the latitude of the place D (where the plane Zb would be horizontal) and the arc RQ is the difference of longitude of the planes Zb and DII.

In the spherical triangle WDR, the arc WD is given, for it is the complement of the plane's declination from S to south; which complement is 54° (viz. 90° - 36°): the angle at R, in which the meridian of the place D cuts the equator, is a right angle; and the angle RWD measures the elevation of the equinoctial above the horizon of Z, namely 38½ degrees. Say therefore, As radius is to the co-line of the plane's declination from the south, so is the co-line of the latitude of Z to the line of RD the latitude of D: which is of a different denomination. denomination from the latitude of Z, because Z and D are on different sides of the equator.

As radius - - - - - - 10.00000 To co-sine 36° 0' = RQ = 9.90796 So co-sine 51° 30' = QZ = 9.79415

To fine 30° 14' = DR = (9.70211) = the lat. of D, whose horizon is parallel to the vertical plane Zb at Z.

N.B. When radius is made the first term, it may be omitted; and then by subtracting it mentally from the sum of the other two, the operation will be shortened. Thus, in the present case,

To the logarithmic fine of WR = * 54° 0' = 9.90796 Add the logarithmic fine of RD = † 38° 30' = 9.79415

Their sum—radius - - - - - - 9.70211 gives the same solution as above. And we shall keep to this method in the following part of this article.

To find the difference of longitude of the places D and Z, say, As radius is to the co-sine of 38° 30' degrees, the height of the equinoctial at Z, so is the co-tangent of 36° degrees, the plane's declination, to the co-tangent of the difference of longitudes. Thus,

To the logarithmic fine of ‡ 51° 30' = 9.89354 Add the logarithmic tang. of § 54° 0' = 10.13874

Their sum—radius - - - - - - 10.03228 is the nearest tangent of 47° 8' = WR; which is the co-tangent of 42° 52' = RQ, the difference of longitude sought. Which difference, being reduced to time, is 2 hours 51 minutes.

3. And thus having found the exact latitude and longitude of the place D, whose horizon the vertical plane at Z is parallel, we shall proceed to the construction of a horizontal dial for the place D, whose latitude is 30° 14' south; but anticipating the time at D by 2 hours 51 minutes (neglecting the ½ min. in practice), because D is so far westward in longitude from the meridian of London; and this will be a true vertical dial at London, declining westward 36 degrees.

Assume any right line CSL for the subtiline of the dial, and make the angle KCP equal to the latitude of the place (viz. 30° 14'), to whose horizon the plane of the dial is parallel; then CRP will be the axis of the subtiline, or edge that casts the shadow on the hours of the day, in the dial. This done, draw the contingent line EQ, cutting the subtiline line at right angles in K; and from K make KR perpendicular to the axis CRP. Then KG (= KR) being made radius, that is, equal to the chord of 60° or tangent of 45° on a good sector, take 42° 52' (the difference of longitude of the places Z and D) from the tangents, and having set it from K to M, draw GM for the hour-line of XII. Take KN, equal to the tangent of an angle less by 15 degrees than KM; that is, the tangent of 27° 52': and through the point N draw CN for the hour-line of I. The tangent of 12° 52' (which is 15° less than 27° 52'), set off the same way, will give a point between K and N, through which the hour-line of II is to be drawn. The tangent of 2° 8' (the difference between 45° and 52° 52') placed on the other side of CL, will determine the point through which the hour-line of III is to be drawn: to which 2° 8', if the tangent of 15° be added, it will make 17° 8'; and this set off from K towards Q on the line EQ, will give the point for the hour-line of IV: and so of the rest.—The forenoon hour-lines are drawn the same way, by the continual addition of the tangents 15°, 30°, 45°, &c. to 42° 52' (= the tangent of KM) for the hours of XI, X, IX, &c. as far as necessary; that is, until there be five hours on each side of the subtiline. The sixth hour, accounted from that hour or part of the hour on which the subtiline falls, will be always in a line perpendicular to the subtiline, and drawn through the centre C.

4. In all erect dials, CM, the hour-line of XII, is perpendicular to the horizon of the place for which the dial is to serve; for that line is the intersection of a vertical plane with the plane of the meridian of the place, both which are perpendicular to the plane of the horizon; and any line HO, or ho, perpendicular to CM, will be a horizontal line on the plane of the dial, along which the hours may be numbered; and CM being set perpendicular to the horizon, the dial will have its true position.

5. If the plane of the dial had declined by an equal angle toward the east, its description would have differed only in this, that the hour-line of XII would have fallen on the other side of the subtiline CL, and the line HO would have a subcontrary position to what it has in this figure.

6. And these two dials, with the upper points of their titles turned toward the north pole, will serve for other two planes parallel to them; the one declining from the north toward the east, and the other from the north toward the west, by the same quantity of angle. The like holds true of all dials in general, whatever be their declination and obliquity of their planes to the horizon.

CASE II. 7. If the plane of the dial not only declines, but also reclines, or inclines. Suppose its declination from fronting the south be equal to the arc SD on the horizon; and its inclination be equal to the arc Dd of the vertical circle DZ: then it is plain, that if the quadrant of altitude ZdD on the globe cuts the point D in the horizon, and the reclination is counted up on the quadrant from D to d; the intersection of the hour circle PRd, with the equinoctial WQE, will determine Rd, the latitude of the place d; whose horizon is parallel to the given plane Zb at Z; and RQ will be the difference in longitude of the places at d and Z.

Trigonometrically thus: Let a great circle pass thro' the three points, W, d, E; and in the triangle WdD, right-angled at D, the sides WD and Dd are given; and thence the angle DWd is found, and so is the hypothenuse Wd. Again, the difference, or the sum, of DWd and DIWR, the elevation of the equinoctial above the horizon of Z, gives the angle dWd; and the hypothenuse of the triangle WRd was just now found; whence the sides Rd and WR are found, the former being the latitude of the place d, and the latter the complement of RQ, the difference of longitude sought.

Thus,

* The co-sine of 36°, or of RQ, † The co-sine of 51° 30', or of QZ. ‡ The co-sine of 38° 30', or of WDR. § The co-tangent of 36°, or of DW. Thus, if the latitude of the place \( Z \) be 52° 10' north; the declination \( SD \) of the plane \( Zb \) (which would be horizontal at \( d \)) be 36°, and the relevation be 15°, or equal to the arc \( Dd \); the fourth latitude of the place \( d \), that is, the arc \( Rd \), will be 15° 9'; and \( RQ \), the difference of the longitude, 36° 2'. From these data, therefore, let the dial (fig. 7.) be described, as in the former example.

8. There are several other things requisite in the practice of dialing; the chief of which shall be given in the form of arithmetical rules, simple and easy to those who have learned the elements of trigonometry. For in practical arts of this kind, arithmetic should be used as far as it can go; and scales never trusted to, except in the final conclusion, where they are absolutely necessary in laying down the calculated hour-distances on the plane of the dial.

Rule I. To find the angles which the hour-lines on any dial make with the subfile. To the logarithmic fine of the given latitude, or of the file's elevation above the plane of the dial, add the logarithmic tangent of the hour (*) distance from the meridian, or from the (+) subfile; and the sum minus radius will be the logarithmic tangent of the angle sought.

For \( KC \) is to \( KM \) in the ratio compounded of the ratio of \( KC \) to \( KG (=KR) \) and of \( KG \) to \( KM \); which making \( CK \) the radius 10,000,000, or 10,000, or 10, or 1, are the ratio of 10,000,000, or of 10,000,000, or of 10, or of 1, to \( KG \times KM \).

Thus, in a horizontal dial, for latitude 51° 30', to find the angular distance of XI in the forenoon, or I in the afternoon, from XII.

To the logarithmic fine of 51° 30' 9.89354†

Add the logarithmic tang. of 51° 0' 9.42805

The sum—radius is 9.32159 = the logarithmic tangent of 11° 50', or of the angle which the hour-line of XI or I makes with the hour of XII.

And by computing in this manner, with the fine of the latitude, and the tangents of 30°, 45°, 60°, and 75°, for the hours of II, III, IIII, and V in the afternoon; or of X, IX, VIII, and VII in the forenoon; you will find their angular distances from XII to be 24° 18', 38° 3', 53° 35', and 71° 6'; which are all that there is occasion to compute for. — And these distances may be set off from XII by a line of chords; or rather, by taking 1000 from a scale of equal parts, and setting that extent as a radius from C to XII; and then, taking 209 of the same parts (which are the natural tangent of 11° 50'), and setting them from XII to XI and I, on the line bo, which is perpendicular to C XII: and so for the rest of the hour-lines, which in the table of natural tangents, against the above distances, are 451, 782, 1355, and 2920, of such equal parts from XII, as the radius C XII contains 1000.

N° 100.

(*) That is, of 15°, 30°, 45°, 60°, 75°, for the hours of I, II, III, IIII, V, in the afternoon; and XI, X, IX, VIII, VII, in the afternoon.

(+) In all horizontal dials, and erect north or south dials, the subfile and meridian are the same; but in all declining dials, the subfile line makes an angle with the meridian.

(†) In which case, the radius \( CK \) is supposed to be divided into 10,000 equal parts.

** Here we consider the radius as unity, and not 10,000,000; by which, instead of the index 9, we have —1 as above; which is of no farther use than making the work a little easier.

†† The distance of one hour from VI.

§§ The co-declination of the fun.

And, lastly, set off 1257 (the natural tangent of 51° 30') for the angle of the file's height, which is equal to the latitude of the place.

Rule II. The latitude of the place, the sun's declination, and his hour distance from the meridian, being given, to find (1.) his altitude, (2.) his azimuth. (1.) Let \( d \) be the sun's place, \( dR \) his declination; and, in the triangle \( PZd \), \( PD \) the fun, or the difference, of \( dR \), and the quadrant \( PR \), being given by the supposition, as also the complement of the latitude \( PZ \), and the angle \( dPZ \), which measures the horary distance of \( d \) from the meridian; we shall (by Cafe 4. of Keill's oblique spheric Trigonometry) find the base \( Zd \), which is the fun's distance from the zenith, or the complement of his altitude.

And (2.) as fine \( Zd \): fine \( PD \) :: fine \( dPZ \): \( dZP \), or of its supplement \( DZS \), the azimuthal distance from the south.

Or the practical rule may be as follows.

Write \( A \) for the sign of the sun's altitude, \( L \) and \( l \) for the fine and co-fine of the latitude, \( D \) and \( d \) for the fine and co-fine of the sun's declination, and \( H \) for the fine of the horary distance from VI.

Then the relation of \( H \) to \( A \) will have three varieties.

1. When the declination is toward the elevated pole, and the hour of the day is between XII and VI; it is \( A = LD + Hld \), and \( H = \frac{A - LD}{ld} \).

2. When the hour is after VI, it is \( A = LD - Hld \), and \( H = \frac{LD + A}{ld} \).

3. When the declination is toward the depressed pole, we have \( A = Hld - LD \), and \( H = \frac{A + LD}{ld} \).

Which theorems will be found useful, and expeditious enough for solving those problems in geography and dialing which depend on the relation of the sun's altitude to the hour of the day.

Example I. Suppose the latitude of the place to be 51° degrees north; the time five hours distant from XII, that is, an hour after VI in the morning, or before VI in the evening; and the sun's declination 28° north. Required the sun's altitude?

Then to log. \( L = \log. \sin. 51° 30' = 1.89854 ** \)

add log. \( D = \log. \sin. 20° 0' = 1.53405 \)

Their sum 1.42759 gives \( LD = \logarithm of 0.267664 \), in the natural fines.

And, to log. \( H = \log. \sin. 15° 0' = 1.41300 \)

add \( \log. l = \log. \sin. 38° 0' = 1.79414 \)

\( \log. d = \log. \sin. 70° 0' = 1.97300 \)

Their sum 1.18014 gives \( Hld = \logarithm of 0.151408 \), in the natural fines.

And And these two numbers (0.267664 and 0.151408) make 0.419072 = A; which, in the table, is the nearest natural sine of 24° 47', the sun's altitude sought.

The same hour-distance being assumed on the other side of VI, then LD—Hld is 0.116256, the sine of 6° 40' 3"; which is the sun's altitude at V in the morning, or VII in the evening, when his north declination is 20°.

But when the declination is 20° south (or towards the depressed pole) the difference Hld—LD becomes negative; and thereby shows, that an hour before VI in the morning, or past VI in the evening, the sun's centre is 6° 40' 3" below the horizon.

Example 2. From the same data, to find the sun's azimuth. If H, L, and D, are given, then (by par. 2. of Rule II.) from H having found the altitude and its complement Zd; and the arc Pd (the distance from the pole) being given; say, as the co-sine of the altitude is to the sine of the distance from the pole, so is the sine of the hour-distance from the meridian to the sine of the azimuth distance from the meridian.

Let the latitude be 51° 30' north, the declination 15° 9' south, and the time 11 h. 24 m. in the afternoon, when the sun begins to illuminate a vertical wall, and it is required to find the position of the wall.

Then, by the foregoing theorems, the complement of the altitude will be 81° 32' 3", and Pd the distance from the pole being 109° 5', and the horary distance from the meridian, or the angle dPZ, 36°.

To log. sin. 74° 51' - 1.98464 Add log. sin. 36° 3" - 1.76022

And from the sum - 1.75386 Take the log. sin. 81° 32' 3" - 1.99525

Remains 1.75861 = log. sin. 35°, the azimuth distance sought.

When the altitude is given, find from thence the hour, and proceed as above.

This praxis is of singular use on many occasions; in finding the declination of vertical planes more exactly than in the common way, especially if the transits of the sun's centre are observed by applying a ruler with sights, either plain or telescopical, to the wall or plane whose declination is required. In drawing a meridian line, and finding the magnetic variation. In finding the bearings of places in terrestrial surveys; the transits of the sun over any place, or his horizontal distance from it, being observed, together with the altitude and hour. And thence determining small differences of longitude. In observing the variations at sea, &c.

The declination, inclination, and reclinatior, of planes, are frequently taken with a sufficient degree of accuracy by an instrument called a declinator or declinator.

The construction of this instrument, as somewhat improved by Mr. Jones, is as follows: On a mahogany board ABIK, is inserted a semicircular arch AGEB of ivory or box-wood, divided into two quadrants of 90° each, beginning from the middle G. On the centre C turns a vertical quadrant DFE, divided into 90°, beginning from the base E; on which is a moveable index CF, with a small hole at F for the sun's rays to pass through, and form a spot on a mark at C. The lower extremity of the quadrant at E is pointed, to mark the linear direction of the quadrant when applied to any other plane; as this quadrant takes off occasionally, and a plumb-line P hangs at the centre on C, for taking the inclinations and reclinations of planes. At H, on the plane of the board, is inserted a compass of points and degrees, with a magnetic needle turning on a pivot over it. The addition of the moveable quadrant and index considerably extend the utility of the declinator, by rendering it convenient for taking equal altitudes of the sun, the sun's altitude, and bearing, at the same time, &c.

To apply this instrument in taking the declination To take by of a wall or plane: Place the side ACB in an horizontal direction to the plane proposed, and observe what degree or point of the compass the N part of the needle stands over from the north or south, and it will be the declination of the plane from the north or south accordingly. In this case, allowance must be made for the variation of the needle (if any) at the place; and which, if not previously known, will render this operation very inaccurate. At London it is now 22° 30' to the west.

Another way more exact may be used, when the sun shines out half an hour before noon. The side ACB being placed against the plane, the quadrant must be so moved on the semicircle AGB, and the index CF on DE, till the sun's rays passing through the hole at F fall exactly on the mark at G, and continued so till the sun requires the index to be raised no higher: you will then have the meridian or greatest altitude of the sun; and the angle contained between G and E will be the declination required. The position of CE is the meridian or 12 o'clock line. But the most exact way for taking the declination of a plane, or finding a meridian line, by this instrument, is, in the forenoon, about two or three hours before 12 o'clock, to observe two or three heights or altitudes EF of the sun; and at the same time the respective angular polar distances GE from G: write them down; and in the afternoon watch for the same, or one of the same altitudes, and mark the angular distances or distance on the quadrant AG: Now, the division or degree exactly between the two noted angular distances will be the true meridian, and the distance at which it may fall from the G of the divisions at G will be the declination of the plane. The reason for observing two or three altitudes and angles in the morning is, that in case there should be clouds in the afternoon, you may have the chance of one corresponding altitude.

The quadrant occasionally takes off at C, in order to place it on the surface of a pedestal or plane intended for an horizontal dial; and thereby from equal altitudes of the sun, as above, draw a meridian or 12 o'clock line to set the dial by.

The base ABIK serves to take the inclination and reclinatior of planes. In this case, the quadrant is taken off, and the plummet P is fitted on a pin at the centre C: then the side IGK being applied to the plane proposed, as QL (fig. 7.), of the plumb-line cuts the semicircle in the point G, the plane is horizontal; or if it cut the quadrant in any point at S, then will GCS be the angle of inclination. Lastly, if applying the side ACB (fig. 7.) to the plane, the plummet cuts G, the plane is vertical; or if it cuts either of the quadrants, it is accordingly the angle of inclination. Hence, if the quantity of the angle of inclination be compared with the elevation of the pole and equator, it is easily known whether the plane be inclined or reclined.

Of the double Horizontal Dial, and the Babylonian and Italian Dials.

To the gnomonic projection, there is sometimes added a stereographic projection of the hour-circles, and the parallels of the sun's declination, on the same horizontal plane; the upright side of the gnomon being sloped into an edge, standing perpendicularly over the centre of the projection: so that the dial, being in its due position, the shadow of that perpendicular edge is a vertical circle passing through the sun, in the stereographic projection.

The months being duly marked on this dial, the sun's declination, and the length of the day at any time, are had by inspection (as also his altitude, by means of a scale of tangents). But its chief property is, that it may be placed true, whenever the sun shines, without the help of any other instrument.

Let \( d \) be the sun's place in the stereographic projection, \( x \) the parallel of the sun's declination, \( Zd \) a vertical circle through the sun's centre, \( Pd \) the hour-circle; and it is evident, that the diameter NS of this projection being placed duly north and south, these three circles will pass through the point \( d \). And therefore, to give the dial its due position, we have only to turn its gnomon toward the sun, on a horizontal plane, until the hour on the common gnomonic projection coincides with that marked by the hour-circle \( Pd \), which passes through the intersection of the shadow \( Zd \) with the circle of the sun's present declination.

The Babylonian and Italian dials reckon the hours, not from the meridian as with us, but from the sun's rising and setting. Thus, in Italy, an hour before sun-set is reckoned the 2nd hour; two hours before sun-set the 2nd hour; and so of the rest. And the shadow that marks them on the hour-lines, is that of the point of a slant. This occasions a perpetual variation between their dials and clocks, which they must correct from time to time, before it arises to any sensible quantity, by setting their clocks so much faster or slower. And in Italy, they begin their day, and regulate their clocks, not from sun-set, but from about mid-twilight, when the Ave Maria is said; which corrects the difference that would otherwise be between the clock and the dial.

The improvements which have been made in all sorts of instruments and machines for measuring time, have rendered such dials of little account. Yet, as the theory of them is ingenious, and they are really, in some respects, the best contrived of any for vulgar use, a general idea of their description may not be unacceptable.

Let fig. 8. represent an erect direct south wall, on which a Babylonian dial is to be drawn, showing the hours from sun-rising; the latitude of the place, whose horizon is parallel to the wall, being equal to the angle KCR. Make, as for a common dial, KG = KR (which is perpendicular to CR) the radius of the equinoctial AEQ, and draw RS perpendicular to CK for the slant of the dial; the shadow of whose point R is to mark the hours, when SR is set upright on the plane of the dial.

Then it is evident, that, in the contingent line AEQ, the spaces K1, K2, K3, &c. being taken equal to the tangents of the hour-distances from the meridian, to the radius KG, one, two, three, &c. hours after sun rising, on the equinoctial day; the shadow of the point R will be found, at these times, respectively in the points 1, 2, 3, &c.

Draw, for the like hours after sun-rising, when the sun is in the tropic of Capricorn \( V \), the like common lines CD, CE, CF, &c. and at these hours the shadow of the point R will be found in those lines respectively. Find the sun's altitudes above the plane of the dial at these hours; and with their co-tangents Sa, Sb, Sc, &c. to radius SR, describe arcs intersecting the hour-lines in the points d1, e1, f1, &c. so that the right lines 1d1, 2e1, 3f1, &c. be the lines of I, II, III, &c. hours after sun-rising.

The construction is the same in every other case; due regard being had to the difference of longitude of the place at which the dial would be horizontal, and the place for which it is to serve; and likewise, taking care to draw no lines but what are necessary; which may be done partly by the rules already given for determining the time that the sun shines on any plane; and partly from this, that on the tropical days, the hyperbola described by the shadow of the point R limits the extent of all the hour-lines.

Of the right placing of Dials, and having a true Meridian Line for the regulating of Clocks and Watches.

The plane on which the dial is to rest being duly prepared, and every thing necessary for fixing it, you may find the hour tolerably exact by a large equinoctial ring-dial, and set your watch to it. And then the dial may be fixed by the watch at your leisure.

If you would be more exact, take the sun's altitude by a good quadrant, noting the precise time of observation by a clock or watch. Then compute the time for the altitude observed; and set the watch to agree with that time, according to the sun. A Hadley's quadrant is very convenient for this purpose: for by it you may take the angle between the sun and his image reflected from a basin of water; the half of which angle, subtracting the refraction, is the altitude required. This is best done in summer; and the nearer the sun is to the prime vertical (the east or west azimuth) when the observation is made, so much the better.

Or, in summer, take two equal altitudes of the sun in the same day; one any time between 7 and 10 in the morning, the other between 2 and 5 in the afternoon; noting the moments of these two observations by a clock or watch: and if the watch shows the observations to be at equal distances from noon, it agrees exactly with the sun; if not, the watch must be corrected by half the difference of the forenoon and afternoon intervals; and then the dial may be set true by the watch.

Thus, for example, suppose you had taken the sun's altitude when it was 20 minutes past VIII in the morning. A good meridian line, for regulating clocks or watches, may be had by the following method.

Make a round hole, almost a quarter of an inch diameter, in a thin plate of metal; and fix the plate in the top of a south window, in such a manner, that it may recline from the zenith at an angle equal to the co-latitude of your place, as nearly as you can guess; for then the plate will face the sun directly at noon on the equinoctial days. Let the sun shine freely thro' the hole into the room; and hang a plumb-line to the ceiling of the room, at least five or six feet from the window, in such a place as that the sun's rays transmitted through the hole, may fall upon the line when it is noon by the clock; and having marked the said place on the ceiling, take away the line.

Having adjusted a sliding bar to a dove-tail groove, in a piece of wood about 18 inches long, and fixed a hook into the middle of the bar, nail the wood to the above-mentioned place on the ceiling, parallel to the side of the room in which the window is; the groove and bar being towards the floor: Then hang the plumb-line upon the hook in the bar, the weight or plummet reaching almost to the floor; and the whole will be prepared for farther and proper adjustment.

This done, find the true solar time by either of the two last methods, and thereby regulate your clock. Then, at the moment of next noon by the clock, when the sun shines, move the sliding-bar in the groove, until the shadow of the plumb-line bisects the image of the sun (made by his rays transmitted thro' the hole) on the floor, wall, or on a white screen placed on the north-side of the line; the plummet or weight at the end of the line hanging freely in a pail of water placed below it on the floor.—But because this may not be quite correct for the first time, on account that the plummet will not settle immediately, even in water; it may be farther corrected on the following days, by the above method, with the sun and clock; and so brought to a very great exactness.

N.B. The rays transmitted through the hole will cast but a faint image of the sun, even on a white screen, unless the room be so darkened that no sunshine may be allowed to enter but what comes thro' the small hole in the plate. And always, for some time before the observation is made, the plummet ought to be immersed in a jar of water, where it may hang freely; by which means the line will soon become steady, which otherwise would be apt to continue swinging.

Description of two New Instruments for facilitating the practice of Dialing.

I. The Dialing Sector, contrived by the late Mr Benjamin Martin, is an instrument by which dials are drawn in a more easy, expeditious, and accurate manner. It is represented on the plate as now made by Mr Jones of Holborn. The principal lines on it are the line of latitudes and the line of hours. They are found on most of the common plane scales and sectors; but in a manner that greatly confines and diminishes their use: for, first, they are of a fixed length; and, secondly, too small for any degree of accuracy. But in this new sector, the line of latitudes is laid down, as it is called, sector-wise, viz. one line of latitudes upon each leg of the sector, beginning in the centre of the joint, and diverging to the end (as upon other sectors), where the extremes of the two lines at 90° and 90° are nearly one inch apart, and their length 11½ inches: which length admits of great exactness; for at the 70th degree of latitude, the divisions are to quarters of a degree or 15 minutes. This accuracy of the divisions admits of a peculiar advantage, namely, that it may be equally communicated to any length from 1 to 23 inches, by taking the parallel distances (see fig. 5.), viz. from 10 to 10, 20 to 20, 30 to 30, and so on as is done in like cases on the lines of lines, tangents, &c. Hence its universal use for drawing dials of any prepared size.

The line of hours for this end is adapted and placed contiguous to it on the sector, and of a size large enough for the very minutes to be distinctly on the part where they are smallest, which is on each side of the hour of III.

From the construction of the line of hours before shown, the divisions on each side of the hour III are the same to each end, so that the hour-line properly is only a double line of three hours. Hence a line of 3 hours answers all the purposes of a line of 6, by taking the double extent of 3, which is the reason why upon the sector the line of hours extends only to 4½.

To make use of the line of latitude and line of hours on the sector: As single scales only, they will be found more accurate than those placed on the common scales and sectors, in which the hours are usually subdivided, but into 5 minutes, and the line of latitudes into whole degrees. But it is shown above how much more accurately these lines are divided on the dialing sector. As an example of the great exactness with which horizontal and other dials may be drawn by it, on account of this new sectoral disposition of these scales, and how all the advantages of their great length are preserved in any lesser length of the VI o'clock line c e and a f: Apply either of the distances of c e or a f to the line of latitude at the given latitude of London, suppose 51° 32' on one line to 51° 32' on the other, in the manner shown in fig. 5. and then taking all the hours, quarters, &c. from the hour-scale by similar parallel extents, you apply them upon the lines c d and f b as before described.

As the hour-lines on the sector extend to but 4½, the double distance of the hour 3, when used either singly or sectorially, must be taken, to be first applied from 51° 32' on the latitudes, to its contact on the XII o'clock line, before the several hours are laid off. The method of drawing a vertical north or south dial is perfectly the same as for the above horizontal one; only reverting the hours as in fig. 4. and making the angle of the fluke's height equal to the complement of the latitude 38° 28'.

The method of drawing a vertical declining dial by the sector, is almost evident from what has been already said in dialing. But more fully to comprehend the matter, it must be considered there will be a variation of particulars as follow: 1. Of the subtitle or line over which the file is to be placed; 2. The height of the file above the plane; 3. The difference between the meridian of the place and that of the plane, or their difference of longitude. From the given latitude of the place, and declination of the plane, you calculate the three requisites just mentioned, as in the following example. Let it be required to make an erect south dial, declining from the meridian westward 28° 43', in the latitude of London 51° 32'. The first thing to be found is the distance of the subtillar line GB (fig. 3.) from the meridian of the plane G XII. The analogy from this is: As radius is to the sine of the declination, so is the co-tangent of the latitude to the tangent of the distance sought, viz. As radius : 28° 43' :: tang. 38° 28' : tangent 20° 55'. This and the following analogy may be as accurately worked on the Gunter's line of lines, tangents, &c. properly placed on the sector, as by the common way from logarithms. Next, To find the plane's difference of longitude. As the sine of the latitude is to radius, so is the tangent of the declination to the tangent of the difference of longitude, viz. As s 51° 32': radius :: tang. 28° 43': tang. 35° 0'. Lastly, to find the height of the file: As radius is to the cosine of the latitude, so is the cosine of the declination to the sine of the file's height, viz. Radius : s 38° 28' :: s 61° 17' : s 33° 5'.

The three requisites thus obtained, the dial is drawn in the following manner: Upon the meridian line G XII, with any radius GC describe the arch of a circle, upon which set off 20° 55' from C to B, and draw GB, which will be the subtillar line, over which the file of the dial must be placed.

At right angles to this line GB, draw AQ indefinitely through the point G; then from the scale of latitudes take the height of the file 33° 5', and set it each way from G to A and Q. Lastly, take the double length of 3 on the hour-line in your compasses, and setting one foot in A or Q, with the other foot mark the line GB in D, and join AD QD, and then the triangle AD Q is completed upon the subtillar line GB.

To lay off the hours, the plane's difference of longitude being 35°, equal to 2h. 20 min. in time, allowing 15° to an hour, so that there will be 2h. 20' between the point D and the meridian G XII, in the line AD. Therefore, take the first 20' of the hour-scale in your compasses, and set off from D to 2; then take 1h. 20', and set off from D to 1; 2h. 20', and set off from D to 12; 3h. 20', from D to 11; 4h. 20' from D to 10; and 5h. 20' from D to 9, which will be 40' from A.

Then, on the other side of the subtillar line GB, you take 40' from the beginning of the scale, and set off from D to 3; then take 1h. 40', and set off from D to 4; also 2h. 40', and set off from D to 5; and so on to 8, which will be 20' from Q. Then from G the centre, through the several points 2, 1, 12, 11, 10, 9, on one side, and 3, 4, 5, 6, 7, 8, on the other, you draw the hour-lines, as in the figure they appear. The hour of VIII need only be drawn for the morning; for the sun goes off from this west decliner 20' before VIII in the evening.—The quarters, &c. are all set off in the same manner from the hour-scale as the above hours were.

The next thing is fixing the file or gnomon, which is always placed in the subtillar line GB, and which is already draw. The file above the plane has been found to be 33° 5'; therefore with any radius GB describe an obscure arch, upon which set off 33° 5' from B to S, and drawn GS, and the angle SGB will be the true height of the gnomon above the subtillar line GB.

II. The Dialing Trigon is another new instrument of great utility in the practice of dialing; and was also contrived by the late Mr Martin. It is composed of two graduated scales and a plane one. On the scale AB is graduated the line of latitudes; and on the scale AC, the line of hours; these properly joined with the plane scale BD, as shown in the figure, truly represent the gnomonical triangle, and is properly called a dialing trigon. The hour-scale AC is here of its full length; so that the hours, halves, quarters, &c. and every minute (if required) may be immediately set off by a teet point; and from what has before been observed in regard to the sector, it must appear that this method by the trigon is the most expeditious way of drawing dials that any mechanism of this sort can afford. As an example of the application of this trigon in the construction of an horizontal dial for the latitude of London 51° 32', you must proceed as follows: Apply the trigon to the 6 o'clock line af (fig. 1.) on the morning side, so that the line of latitudes may coincide with the 6 o'clock line, and the beginning of the divisions coincide with the centre a; and at 5° 32' of the line of latitudes place the 6 o'clock edge of the line of hours, and the other end or beginning of the scale close against the plane scale cd, as by the figure at d; and fastening these bars down by the several pins placed in them to the paper and board, then the hours, quarters, &c. are all marked off with a teet point instantly, and the hour-lines drawn through them as before, and as shown in the figure. When this is done for the side af or morning hours, you move the scale of latitudes and hours to the other side ef, or afternoon side, and place the hour-scale to 5° 32' as before, and push down the hours, quarters, &c. and draw the lines through them for the afternoon hours, which is clearly represented in the figure.

In like manner is an erect north or south dial drawn (see fig 2.), the operation being just the same, only reverting the hours as in the figure, and marking the angles of the file's height equal to the complement of the latitude.

This trigon may be likewise used for drawing vertical declining dials (fig. 3.), as it is with the same facility applied to the lines AQ, GB, and the hours and quarters marked off as before directed.

Mr Jones graduates on the scale BD of the trigon a line of chords, which is found useful for laying off the necessary angles of the file's height. The scales of this trigon, when not in use, lie very close together, and pack up into a portable case for the pocket. DIALING Lines, or Scales, are graduated lines, placed on rules, or the edges of quadrants, and other instruments, to expedite the construction of dials. See Plate CLVIII.

DIALING-Sector. See Dialing, p. 803, and Plate CLXI.

DIALING-Sphere, is an instrument made of brass, with several semicircles sliding over one another, on a moving horizon, to demonstrate the nature of the doctrine of spherical triangles, and to give a true idea of the drawing of dials on all manner of planes.

DIALING-Trigon. See Dialing, p. 804, and Plate CLXI.

a mine, called also Pluming, is the using of a compass (which they call dial), and a long line, to know which way the load or vein of ore inclines, or where to shift an air-shaft, or bring an adit to a desired place.