THE art of drawing dials on the surface of any given body or plane. The Greeks and the Latins call this art gnomenica et scintherica, by reason it distinguishes the hours by the shadow of the gnomon. Some call it photo-scintherica, because the hours are sometimes known by the light of the sun. Lastly, others call it horolography.

Dialling is a most necessary art: for notwithstanding we are provided with moving machines, such as clocks and watches, to show time; yet these are apt to be out of order, go wrong, and stop; consequently they stand frequently in need of regulation by some invariable instrument, as a dial; which being rightly constructed and duly placed, will always, by means of the sun, inform us of the true solar time; which time being corrected by the equation table published annually in the ephemerides, almanacks, and other books, will be the mean time to which clocks and watches are to be set.

The antiquity of dials is beyond doubt. Some attribute their invention to Anaximenes Milefus; and others to Thales. Vitruvius mentions one made by the ancient Chaldee historian Beroos, on a reclining plane, almost parallel to the equinoctial. Aristarchus Samius invented the hemispherical dial. And there were some spherical ones, with a needle for a gnomon. The discus of Aristarchus was a horizontal dial, with its limb raised up all around, to prevent the shadow stretching too far.

It was late ere the Romans became acquainted with dials. The first sun-dial at Rome was set up by Papius Cursor, about the year of the city 460; before which time, lays Pliny, there is no mention of any account of time but by the sun's rising and setting: it was set up at or near the temple of Quirinus, but went ill. About 30 years after, M. Valerius Messala being consul, brought out of Sicily another dial, which he set up on a pillar near the rostrum; but for want of its being made for that latitude, it could not go true. They made use of it 99 years; till Martius Philippus set up another more exact.

But there seem to have been dials among the Jews much earlier than any of these. Witnes the dial of Ahaz; who began to reign 400 years before Alexander, and within 12 years of the building of Rome: mentioned by Isaiah, chap. xxxviii, ver. 8.

The first professed writer on dialling is Clavius; who demonstrates all, both the theory and the operations, after the rigid manner of the ancient mathematicians; but so intricately, that few, we dare say, ever read them all. Dechales and Ozanam gave much easier demonstrations in their Courses, and Wolfius in his Elements. M. Picard has given a new method of making large dials, by calculating the hour-lines; and M. de la Hire, in his Dialling, printed in 1683, a geometrical method of drawing hour-lines from certain points determined by observation. Eberhardus Welperus, in 1625, published his Dialling, wherein he lays down a method of drawing the primary dials on a very easy foundation. The same foundation is described at length by Sebastian Munster, in his Rudimenta Mathematica, published in 1551. Sturmius, in 1672, published a new edition of Welperus's Dialling, with the addition of a whole second part, about inclining and declining dials, &c. In 1709, the same work, with Sturmius's additions, was republished, with the addition of a fourth part, containing Picard's and de la Hire's methods of drawing large dials. Paterson, Michael, and Muller, have each wrote on dialling in the German tongue; Coetius in his Horologographia Plana, printed in 1689; Gaupennius, in his Gnomonica Mechanica; Bion, in his Use of Mathematical Instruments; the late ingenious Mr Ferguson, in his Select Lectures; Mr Emmerson, in his Dialling; and Mr W. Jones, in his Instrumental Dialling.

A Dial, accurately defined, is a plane, upon which lines are described in such a manner, that the shadow of a wire, or of the upper edge of another plane, erected perpendicularly on the former, may shew the true time of the day.

The edge of the plane by which the time of the day is found, is called the file of the dial, which must be parallel to the earth's axis; and the line on which the said plane is erected, is called the subtitle.

The angle included between the subtitle and file, is called the elevation or height of the file.

Those dials whose planes are parallel to the plane of the horizon, are called horizontal dials; and those dials whose planes are perpendicular to the plane of the horizon, are called vertical or erect dials.

Those erect dials, whose planes directly front the north or south, are called direct north or south dials; and all other erect dials are called decliners; because their planes are turned away from the north or south.

Those dials whose planes are neither parallel nor perpendicular to the plane of the horizon, are called inclining or reclining dials, according as their planes make acute or obtuse angles with the horizon; and if their planes are also turned aside from facing the south or north, they are called declining-inclining or declining-reclining dials.

The intersection of the plane of the dial, with that of the meridian, passing through the file, is called the meridian of the dial, or the hour-line of XII.

Those meridians, whose planes pass through the file, and make angles of 15, 30, 45, 60, 75, and 90 degrees with the meridian of the place (which marks the hour-line of XII) are called hour-circles; and their intersections with the plane of the dial are called hour-lines.

In all declining dials, the subtitle makes an angle with the hour-line of XII, and this angle is called the distance of the subtitle from the meridian.

The declining plane's difference of longitude, is the angle formed at the intersection of the file and plane of the dial, by two meridians; one of which passes through the hour-line of XII, and the other through the subtitle.

Thus much being premised concerning dials in general, we shall now proceed to explain the different methods of their construction.

If the whole earth A P e p were transparent, and hollow, like a sphere of glass, and had its equator divided into 24 equal parts by so many meridian The universal fericircles, a, b, c, d, e, f, g, &c. one of which is the sal principle geographical meridian of any given place, as London on which (which is supposed to be at the point a); and if the hour of XII were marked at the equator, both upon that meridian and the opposite one, and all the rest of the hours in order on the rest of the meridians, those meridians would be the hour-circles of London; then, if the sphere had an opaque axis, as PE, p, terminating in the poles P and p, the shadow of the axis would fall upon every particular meridian and hour, when the sun came to the plane of the opposite meridian, and would consequently shew the time at London, and at all other places on the meridian of London.

If this sphere were cut through the middle by a solid Horizontal plane ABCD, in the rational horizon of London, one half of the axis EP would be above the plane, and the other half below it; and if straight lines were drawn from the centre of the plane to those points where its circumference is cut by the hour-circles of the sphere, those lines would be the hour-lines of a horizontal dial for London: for the shadow of the axis would fall upon each particular hour-line of the dial, when it fell upon the like hour-circle of the sphere.

If the plane which cuts the sphere be upright, as Fig. 2, AFCG, touching the given place (London) at F, and directly facing the meridian of London, it will then become the plane of an erect direct south dial: and if right lines be drawn from its centre E to those points of its circumference where the hour-circles of the sphere cut it, these will be the hour-lines of a vertical or direct south-dial for London, to which the hours are to be set as in the figure (contrary to those on a horizontal dial), and the lower half E p of the axis will cast a shadow on the hour of the day in this dial, at the same time that it would fall upon the like hour-circle of the sphere, if the dial plane was not in the way.

If the plane (still facing the meridian) be made to incline or recline any given number of degrees, the hour-circles of the sphere will still cut the edge of the plane in those points to which the hour-lines must be drawn straight from the centre; and the axis of the sphere will cast a shadow on these lines at the respective hours. The like will still hold, if the plane be made Inclining, to decline by any given number of degrees from the reclining meridian towards the east or west: provided the declination be less than 90 degrees, or the inclination be less than the co-latitude of the place: and the axis of the sphere will be a gnomon or file for the dial. But it cannot be a gnomon, when the declination is quite 90 degrees, nor when the declination is equal to the co-latitude; because in these two cases, the axis has no elevation above the plane of the dial.

And thus it appears, that the plane of every dial represents the plane of some great circle upon the earth; and the gnomon of the earth's axis, whether it be a small small wire as in the above figures, or the edge of a thin plate, as in the common horizontal dials.

The whole earth, as to its bulk, is but a point, if compared to its distance from the sun; and therefore, if a small sphere of glass be placed upon any part of the earth's surface, so that its axis be parallel to the axis of the earth, and the sphere have such lines upon it, and such planes within it, as above described; it will shew the hours of the day as truly as if it were placed at the earth's centre, and the shell of the earth were as transparent as glass.

But because it is impossible to have a hollow sphere of glass perfectly true, blown round a solid plane; or if it was, we could not get at the plane within the glass to set it in any given position; we make use of a wire-sphere to explain the principles of dialling, by joining 25 semicircles together at the poles, and putting a thin flat plate of brass within it.

A common globe of 12 inches diameter has generally 24 meridian semicircles drawn upon it. If such a globe be elevated to the latitude of any given place, and turned about until one of these meridians cut the horizon in the north point, where the hour of XII is supposed to be marked, the rest of the meridians will cut the horizon at the respective distances of all the other hours from XII. Then if these points of distance be marked on the horizon, and the globe be taken out of the horizon, and a flat board or plate be put into its place, even with the surface of the horizon; and if straight lines be drawn from the centre of the board to those points of distance on the horizon which were cut by the 24 meridian semicircles; these lines will be the hour-lines of a horizontal dial for that latitude, the edge of whose gnomon must be in the very same situation that the axis of the globe was, before it was taken out of the horizon; that is, the gnomon must make an angle with the plane of the dial equal to the latitude of the place for which the dial is made.

If the pole of the globe be elevated to the co-latitude of the given place, and any meridian be brought to the north point of the horizon, the rest of the meridians will cut the horizon in the respective distances of all the hours from X., for a direct south dial, whose gnomon must be an angle with the plane of the dial, equal to the co-latitude of the place; and the hours must be set the contrary way on this dial to what they are on the horizontal.

But if your globe have more than 24 meridian semi-circles upon it, you must take the following method for making horizontal and south dials.

Elevate the pole to the latitude of your place, and turn the globe until any particular meridian (suppose the first) comes to the north point of the horizon, and the opposite meridian will cut the horizon in the south. Then set the hour-index to the uppermost XII on its circle; which done, turn the globe westward until 15 degrees of the equator pass under the brazen meridian, and then the hour index will be at I (for the sun moves 15 degrees every hour), and the first meridian will cut the horizon in the number of degrees from the north point that I is distant from XII. Turn on until other 15 degrees of the equator pass under the brazen meridian, and the hour index will then be at II, and the first meridian will cut the horizon in the number of degrees that II is distant from XII: and so by making 15 degrees of the equator pass under the brazen meridian for every hour, the first meridian of the globe will cut the horizon in the distances of all the hours from XII to VI, which is just 90 degrees; and then you need go no farther, for the distances of XI, X, IX, VIII, VII, and VI, in the forenoon, are the same from XII as the distances of I, II, III, IV, V, and VI, in the afternoon: and these hour-lines continued through the centre, will give the opposite hour-lines on the other half of the dial.

Thus, to make a horizontal dial for the latitude of London, which is 51\( \frac{1}{2} \) degrees north, elevate the north pole of the globe 51\( \frac{1}{2} \) degrees above the north point of the horizon; and then turn the globe until the first meridian (which is that of London on the English terrestrial globe) cuts the north point of the horizon, and set the hour-index to XII at noon.

Then turning the globe westward until the index points successively to I, II, III, IV, V, and VI, in the afternoon, or until 15, 30, 45, 60, 75, and 90 degrees of the equator pass under the brazen meridian, you will find that the first meridian of the globe cuts the horizon in the following number of degrees from the north towards the east, viz. 11\( \frac{1}{2} \), 24\( \frac{1}{2} \), 38\( \frac{1}{2} \), 53\( \frac{1}{2} \), 71\( \frac{1}{2} \), and 90; which are the respective distances of the above hours from XII upon the plane of the horizon.

To transfer these, and the rest of the hours, to a Fig. 3, horizontal plane, draw the parallel right lines a c and d b, upon the plane, as far from each other as is equal to the intended thickness of the gnomon or stile of the dial, and the space included between them will be the meridian or twelve-o'clock line on the dial. Cross this meridian at right angles with the fix-o'clock line g h, and setting one foot of your compasses in the intersection a, as a centre, describe the quadrant g e with any convenient radius or opening of the compasses: then, letting one foot in the intersection b, as a centre, with the same radius describe the quadrant f h, and divide each quadrant into 90 equal parts or degrees, as in the figure.

Because the hour-lines are less distant from each other about noon, than in any other part of the dial, it is best to have the centres of these quadrants at a little distance from the centre of the dial plane, on the side opposite to XII, in order to enlarge the hour-distances thereabouts, under the same angles on the plane. Thus the centre of the plane is at C, but the centres of the quadrants are at a and b.

Lay a ruler over the point b (and keeping it there for the centre of all the afternoon hours in the quadrant f h) draw the hour-line of I through 11\( \frac{1}{2} \) degrees in the quadrant; the hour-line of II, through 24\( \frac{1}{2} \) degrees; of III, through 38\( \frac{1}{2} \) degrees; III, through 53\( \frac{1}{2} \); and V, through 71\( \frac{1}{2} \): and because the sun rises about Fig. 3, four in the morning, on the longest days at London, continue the hour-lines of III and V in the afternoon through the centre b to the opposite side of the dial.—This done, lay the ruler to the centre a of the quadrant eg; and through the like divisions or degrees of that quadrant, viz. 11\( \frac{1}{2} \), 24\( \frac{1}{2} \), 38\( \frac{1}{2} \), 53\( \frac{1}{2} \), and 71\( \frac{1}{2} \), draw the forenoon hour-lines of XI, X, IX, VIII, and VII, and because the sun sets not before eight in the evening on the longest day, continue the hour-lines of VII and VIII in the forenoon through the centre a, to VII and VIII in the afternoon; and all the hour-lines will be finished on this dial; to which the hours may be set, as in the figure.

Lastly, though 51½ degrees of either quadrant, and from its centre, draw the right line \( ag \) for the hypothenuse or axis of the gnomon \( agi \); and from \( g \), let fall the perpendicular \( gi \), upon the meridian line \( ai \), and there will be a triangle made, whose sides are \( ag \), \( gi \), and \( ia \). If a plate similar to this triangle be made as thick as the distance between the lines \( ac \) and \( bd \), and set upright between them, touching at \( a \) and \( b \), its hypothenuse \( ag \) will be parallel to the axis of the world, when the dial is truly set; and will cast a shadow on the hour of the day.

N. B. The trouble of dividing the two quadrants may be saved if you have a scale with a line of chords upon it (as represented in the plate); for if you extend the compasses from 0 to 60 degrees of the line of chords, and with that extent, as a radius, describe the two quadrants upon their respective centres, the above distances may be taken with the compasses upon the lines, and set off upon the quadrants.

To make an erect direct south dial. Elevate the pole to the co-latitude of your place, and proceed in all respects as above taught for the horizontal dial, from VI in the morning to VI in the afternoon; only the hours must be reversed as in the figure; and the hypothenuse \( ag \) of the gnomon \( agf \), must make an angle with the dial-plane equal to the co-latitude of the place. As the sun can shine no longer on this dial than from fix in the morning until fix in the evening, there is no occasion for having any more than 12 hours upon it.

To make an erect dial, declining from the south towards the east or west. Elevate the pole to the latitude of your place, and screw the quadrant of altitude to the zenith. Then, if your dial declines towards the east (which we shall suppose it to do at present), count in the horizon the degrees of declination, from the east point towards the north, and bring the lower end of the quadrant to that degree of declination at which the reckoning ends. This done, bring any particular meridian of your globe (as suppose the first meridian) directly under the graduated edge of the upper part of the brazen meridian, and set the hour to XII at noon. Then, keeping the quadrant of altitude at the degree of declination in the horizon, turn the globe eastward on its axis, and observe the degrees cut by the first meridian in the quadrant of altitude (counted from the zenith) as the hour-index comes to XI, X, IX, &c. in the forenoon, or as 15, 30, 45, &c. degrees of the equator pass under the brazen meridian at these hours respectively; and the degrees then cut in the quadrant by the first meridian, are the respective distances of the forenoon hours from XII on the plane of the dial.—Then, for the afternoon hours, turn the quadrant of altitude round the zenith, until it comes to the degree in the horizon opposite to that where it was placed before; namely, as far from the west point of the horizon towards the south, as it was set at first from the east point towards the north; and turn the globe westward on its axis, until the first meridian comes to the brazen meridian again, and the hour-index to XII: then, continue to turn the globe westward; and as the index points to the afternoon hours I, II, III, &c. or as 15, 30, 45, &c. degrees of the equator pass under the brazen meridian, the first meridian will cut the quadrant of altitude in the respective number of degrees from the zenith that each of these hours is from XII on the dial.—And note, that when the first meridian goes off the quadrant at the horizon in the forenoon, the hour-index shows the time when the sun will come upon this dial; and when it goes off the quadrant in the afternoon, the index will point to the time when the sun goes off the dial.

Having thus found all the hour-distances from XII, lay them down upon your dial-plane, either by dividing a semicircle into two quadrants of 90 degrees each (beginning at the hour-line of XII), or by the line of chords, as above directed.

In all declining dials, the line on which the file or gnomon stands (commonly called the subfile-line) makes an angle with the twelve o'clock line, and falls among the forenoon hour-lines, if the dial declines towards the east; and among the afternoon hour-lines, when the dial declines towards the west; that is, to the left hand from the twelve o'clock line in the former case, and to the right hand from it in the latter.

To find the distance of the subfile from the twelve o'clock line; if your dial declines from the south toward the east, count the degrees of that declination in the horizon from the east point towards the north, and bring the lower end of the quadrant of altitude to that degree of declination where the reckoning ends: then turn the globe until the first meridian cuts the horizon in the like number of degrees, counted from the south points towards the east; and the quadrant and first meridian will then cross one another at right angles; and the number of degrees of the quadrant, which are intercepted between the first meridian and the zenith, is equal to the distance of the subfile line from the twelve o'clock line; and the number of degrees of the first meridian, which are intercepted between the quadrant and the north pole, is equal to the elevation of the file above the plane of the dial.

If the dial declines westward from the south, count that declination from the east point of the horizon towards the south, and bring the quadrant of altitude to the degree in the horizon at which the reckoning ends; both for finding the forenoon hours and distance of the subfile from the meridian: and for the afternoon hours, bring the quadrant to the opposite degree in the horizon, namely, as far from the west towards the north, and then proceed in all respects as above.

Thus we have finished our declining dial; and in so doing we made four dials, viz.

1. A north dial, declining eastward by the same number of degrees. 2. A north dial, declining the same number west. 3. A south dial, declining east. And, 4. A south dial, declining west. Only, placing the proper number of hours, and the file or gnomon respectively, upon each plane. For (as above-mentioned) in the south-west plane, the subfile line falls among the afternoon hours; and in the south-east, of the same declination, among the forenoon hours, at equal distances from XII. And so all the morning hours on the west decliner will be like the afternoon hours on the east decliner: the south-east decliner will produce the north-west decliner; and the south-west decliner the north-east decliner, by only extending the hour-lines, hour-lines, file and subfile, quite through the centre; the axis of the file (or edge that casts the shadow on the hour of the day) being in all dials whatever parallel to the axis of the world, and consequently pointing towards the north pole of the heaven in north latitudes, and towards the south pole in south latitudes.

But because every one who would like to make a dial, may perhaps not be provided with a globe to assist him, and may probably not understand the method of doing it by logarithmic calculation; we shall show how to perform it by the plain dialling lines, or scale of latitudes and hours (as represented on the Plate), and which may be had on scales commonly sold by the mathematical-instrument-makers.

This is the easiest of all mechanical methods, and by much the best, when the lines are truly divided: and not only the half-hours and quarters may be laid down by all of them, but every fifth minute by most, and every single minute by those where the line of hours is a foot in length.

Having drawn your double meridian line \( a b, c d \), fig. 5, on the plane intended for a horizontal dial, and crooked it at right angles by the fix o'clock line \( f e \) (as in fig. 3.), take the latitude of the place with your compasses, in the scale of latitudes, and set that extent from \( c \) to \( e \), and from \( a \) to \( f \), on the fix o'clock line: then, taking the whole fix hours between the points of the compasses in the scale of hours, with that extent let one foot in the point \( c \), and let the other foot fall where it will upon the meridian line \( c d \), as at \( d \). Do the same from \( f \) to \( b \), and draw the right lines \( e d \) and \( f b \), each of which will be equal in length to the whole scale of hours. This done, setting one foot of the compasses in the beginning of the scale at \( XII \), and extending the other to each hour of the scale, lay off these extents from \( d \) to \( e \) for the afternoon hours, and from \( b \) to \( f \) for those of the afternoon: this will divide the lines \( d e \) and \( b f \) in the same manner as the hour-scale is divided at 1, 2, 3, 4, and 6; on which the quarters may also be laid down, if required. Then, laying a ruler on the point \( c \), draw the first five hours in the afternoon, from that point, through the dots at the numeral figures, 1, 2, 3, 4, 5, on the line \( d e \); and continue the lines of \( III \) and \( V \) through the centre \( c \) to the other side of the dial, for the like hours of the morning: which done, lay the ruler on the point \( a \), and draw the last five hours in the afternoon through the dots, 5, 4, 3, 2, 1, on the line \( f b \); continuing the hour-lines of \( VII \) and \( VIII \) through the centre \( a \) to the other side of the dial, for the like hours of the evening; and set the hours to their respective lines, as in the figure. Lastly, make the gnomon the same way as taught above for the horizontal dial, and the whole will be finished.

To make an erect fourth dial; take the co-latitude of your place from the scale of latitudes, and then proceed in all respects for the hour-line as in the horizontal dial; only reversing the hours, as in fig. 4. and making the angle of the file's height equal to the co-latitude.

But, lest the young diallist should have neither globe nor wooden scale, we shall now show him how he may make a dial without any of these helps. Only, if he has not a line of chords, he must divide a quadrant into 92 equal parts or degrees for taking the proper angle of the file's elevation; which is easily done.

With any opening of the compasses, as \( ZL \), fig. 6. Fig. 6. describe the two semicircles \( LF \) and \( LQ \), upon the centres \( Z \) and \( x \), where the fix o'clock line crosses the double meridian line, and divide each semicircle into 12 equal parts, beginning at \( L \) (though, strictly speaking, only the quadrants from \( L \) to the fix o'clock line Horizontal need be divided); then connect the divisions which dial, are equidistant from \( L \), by the parallel lines \( KM, IN, HO, GP, \) and \( FQ \). Draw \( VZ \) for the hypothenuse of the file, making the angle \( VZE \) equal to the latitude of your place; and continue the line \( VZ \) to \( R \). Draw the line \( R r \) parallel to the fix o'clock line; and set off the distance \( a K \) from \( Z \) to \( Y \), the distance \( b I \) from \( Z \) to \( X \), \( c H \) from \( Z \) to \( W \), \( d G \) from \( Z \) to \( T t \), and \( e F \) from \( Z \) to \( S \). Then draw the lines \( S s, T t, W w, X x, \) and \( Y y \), each parallel to \( R r \). Set off the distance \( y Y \) from \( a \) to \( 11 \), and from \( f \) to \( 1 \); the distance \( x X \) from \( b \) to \( 10 \), and from \( g \) to \( 2 \); \( w W \) from \( c \) to \( 9 \), and from \( h \) to \( 3 \); \( t T \) from \( d \) to \( 8 \), and from \( i \) to \( 4 \); \( s S \) from \( e \) to \( 7 \), and from \( n \) to \( 5 \). Then laying a ruler to the centre \( Z \), draw the forenoon hour lines through the points \( 11, 10, 9, 8, 7 \); and laying it to the centre \( x \), draw the afternoon lines through the points \( 1, 2, 3, 4, 5 \); continuing the forenoon lines of \( VII \) and \( VIII \) through the centre \( Z \), to the opposite side of the dial, for the like afternoon hours; and the afternoon lines \( III \) and \( V \) through the centre \( x \), to the opposite side, for the like morning hours. Set the hours to these lines as in the figure, and then erect the file or gnomon, and the horizontal dial will be finished.

To construct a fourth dial, draw the line \( VZ \), making an angle with the meridian \( ZL \), equal to the co-latitude of your place: and proceed in all respects as in the above horizontal dial for the same latitude, reversing the hours as in fig. 4. and making the elevation of the gnomon equal to the co-latitude.

Perhaps it may not be unacceptable to explain the method of constructing the dialling lines, and some others; which is as follows:

With any opening of the compasses, as \( EA \), fig. 7. Dialling according to the intended length of the scale, describe lines, how the circle \( ADCB \), and cross it at right angles by the constructed diameters \( CEA \) and \( DEB \). Divide the quadrant \( AB \) Fig. 7. first into 9 equal parts, and then each part into 10; so shall the quadrant be divided into 90 equal parts or degrees. Draw the right line \( AFB \) for the chord of this quadrant; and setting one foot of the compasses in the point \( A \), extend the other to the several divisions of the quadrant, and transfer these divisions to the line \( AFB \) by the arcs \( 10, 10, 20, 20, \) &c. and this will be a line of chords, divided into go unequal parts; which, if transferred from the line back again to the quadrant, will divide it equally. It is plain by the figure that the distance from \( A \) to 60 in the line of chords, is just equal to \( AE \), the radius of the circle from which that line is made; for if the arc 60, 60, be continued, of which \( A \) is the centre, it goes exactly through the centre \( D \) of the arc \( AB \).

And therefore, in laying down any number of degrees on a circle, by the line of chords, you must first open the compasses so as to take in just 60 degrees upon that line as from A to 60: and then, with that extent, as a radius, describe a circle, which will be exactly of the same size with that from which the line was divided: which done, set one foot of the compasses in the beginning of the chord line, as at A, and extend the other to the number of degrees you want upon the line; which extent, applied to the circle, will include the like number of degrees upon it.

Divide the quadrant CD into 90 equal parts, and from each point of division draw right lines, as i k l, &c. to the line CE, all perpendicular to that line, and parallel to DE, which will divide EC into a line of lines: and although these are seldom put among the dialling lines on a scale, yet they assist in drawing the line of latitudes. For if a ruler be laid upon the point D, and over each division in the line of lines, it will divide the quadrant CB into 90 unequal parts, as B a, B b, &c. shown by the right lines 10 a, 20 b, 30 c, &c. drawn along the edge of the ruler. If the right line BC be drawn, subtending this quadrant, and the nearest distances B a, B b, B c, &c. be taken in the compasses from B, and set upon this line in the same manner as directed for the line of chords, it will make a line of latitudes BC, equal in length to the line of chords AB, and of an equal number of divisions, but very unequal as to their lengths.

Draw the right line DGA, subtending the quadrant DA; and parallel to it, draw the right line r s, touching the quadrant DA at the numeral figure 3. Divide this quadrant into fix equal parts, as 1, 2, 3, &c. and through these points of division draw right lines from the centre E to the line r s, which will divide it at the points where the fix hours are to be placed, as in the figure. If every fixth part of the quadrant be subdivided into four equal parts, right lines drawn from the centre through these points of division, and continued to the line r s, will divide each hour upon it into quarters.

In fig. 8, we have the representation of a portable dial, which may be easily drawn on a card, and carried in a pocket-book. The lines a d, a b, and b c of the gnomon, must be cut quite through the card; and as the end a b of the gnomon is raised occasionally above the plane of the dial, it turns upon the uncut line c d as on a hinge. The dotted line A B must be slit quite through the card, and the thread C must be put through the slit, and have a knot tied behind, to keep it from being easily drawn out. On the other end of this thread is a small plummet D, and on the middle of it a small bead for showing the hour of the day.

To rectify this dial, set the thread in the slit right against the day of the month, and stretch the thread from the day of the month over the angular point where the curve lines meet at XII; then shift the bead to that point on the thread, and the dial will be rectified.

To find the hour of the day, raise the gnomon (no matter how much or how little) and hold the edge of the dial next the gnomon towards the sun, so as the uppermost edge of the shadow of the gnomon may just cover the shadow line; and the bead then playing freely on the face of the dial, by the weight of the plummet, will show the time of the day among the hour lines, as it is forenoon or afternoon.

To find the time of sun-rising and setting, move the thread among the hour-lines, until it either covers some one of them, or lies parallel betwixt any two; and then it will cut the time of sun-rising among the forenoon hours, and of sun-setting among the afternoon hours, for that day of the year to which the thread is set in the scale of months.

To find the sun's declination, stretch the thread from the day of the month over the angular point at XII, and it will cut the sun's declination, as it is north or south, for that day, in the proper scale.

To find on what days the sun enters the signs, when the bead, as above rectified, moves along any of the curve-lines which have the signs of the zodiac marked upon them, the sun enters those signs on the days pointed out by the thread in the scale of months.

The construction of this dial is very easy, especially if the reader compares it all along with fig. 9. Plate CLXXII. as he reads the following explanation of that figure.

Draw the occult line AB (fig. 9.) parallel to the top of the card, and cross it at right angles with the fix o'clock line ECD; then upon C, as a centre, with the radius CA, describe the semicircle AEL, and divide it into 12 equal parts (beginning at A), as A r, A s, &c. and from these points of division draw the hour-lines r, s, t, u, v, E, w, x, all parallel to the fix o'clock line EC. If each part of the semicircle be subdivided into four equal parts, they will give the half-hour lines and quarters, as in fig. 2. Draw the right line ASD o, making the angle SAB equal to the latitude of your place. Upon the centre A describe the arch RST, and set off upon it the arcs SR and ST, each equal to 23½ degrees, for the sun's greatest declination; and divide them into 23½ equal parts, as in fig. 2. Through the intersection D of the lines ECD and AD o, draw the right line FDG at right angles to AD o. Lay a ruler to the points A and R, and draw the line ARF through 23½ degrees of south declination in the arc SR; and then laying the ruler to the points A and T, draw the line ATG through 23½ degrees of north declination in the arc ST: so shall the lines ARF and ATG cut the line FDG in the proper lengths for the scale of months. Upon the centre D, with the radius DF, describe the semicircle F O G; which divide into fix equal parts, F m, m n, n o, &c. and from these points of division draw the right lines m h, n i, p k, and q l, each parallel to o D. Then setting one foot of the compasses in the point F, extend the other to A, and describe the arc AZH for the tropic of γβ; with the same extent, setting one foot in G, describe the arc AEQ for the tropic of ως. Next setting one foot in the point h, and extending the other to A, describe the arc ACI for the beginnings of the signs Ξ and Ψ; Fig. 9. and with the same extent, setting one foot in the point par. with l, describe the arc AN for the beginnings of the signs Π and ΣL. Set one foot in the point i, and having extended the other to A, describe the arc AK for the beginnings of the signs Ξ and η; and with the same extent, set one foot in k, and describe the arc AM for the beginnings of the signs ξ and ηγ. Then setting one foot in the point D, and extending the other to A, describe the curve AL for the beginnings of γν and ια; and the signs will be finished. This done, lay a ruler from the point A over the sun's declination in the arch RST; and where the ruler cuts the line EDG, make marks: marks: and place the days of the months right against these marks, in the manner shown by fig. 2. Lastly, draw the shadow line PQ parallel to the occult line AB; make the gnomon, and set the hours to their respective lines, as in fig. 2, and the dial will be finished.

There are several kinds of dials called universal, because they serve for all latitudes. One, of Mr Pardie's construction, was formerly considered as the best. It consists of three principal parts; the first whereof is called the horizontal plane, A fig. 10. because in practice it must be parallel to the horizon. In this plane is fixed an upright pin, which enters into the edge of the second part BD, called the meridional plane; which is made of two pieces, the lowest whereof B is called the quadrant, because it contains a quarter of a circle, divided into 90 degrees; and it is only into this part, near B, that the pin enters. The other piece is a semicircle D adjusted to the quadrant, and turning in it by a groove, for raising or depressing the diameter EF of the semicircle, which diameter is called the axis of the instrument. The third piece is a circle G, divided on both sides into 24 equal parts, which are the hours. This circle is put upon the meridional plane so that the axis EF may be perpendicular to the circle, and the point C be the common centre of the circle, semicircle, and quadrant. The straight edge of the semicircle is chamfered on both sides to a sharp edge, which passes through the centre of the circle. On one side of the chamfered part, the first six months of the year are laid down, according to the sun's declination for their respective days, and on the other side the last six months. And against the days on which the sun enters the signs, there are straight lines drawn upon the semicircle, with the characters of the signs marked upon them. There is a black line drawn along the middle of the upright edge of the quadrant, over which hangs a thread H, with its plummet I, for levelling the instrument. N. B. From the 23d of September to the 20th of March, the upper surface of the circle must touch both the centre C of the semicircle, and the line of γ and α; and from the 20th of March to the 23d of September, the lower surface of the circle must touch that centre and line.

To find the time of the day by this dial. Having set it on a level place in sunshine, and adjusted it by the levelling screws k and l, until the plumb-line hangs over the back line upon the edge of the quadrant, and parallel to the said edge; move the semicircle in the quadrant, until the line of γ and α (where the circle touches) comes to the latitude of your place in the quadrant: then turn the whole meridional plane BD, with its circle G, upon the horizontal plane A, until the edge of the shadow of the circle fall precisely on the day of the month in the semicircle; and then the meridional plane will be due north and south, the axis EF will be parallel to the axis of the world, and will cast a shadow upon the true time of the day among the hours on the circle.

N. B. As, when the instrument is thus rectified, the quadrant and semicircle are in the plane of the meridian, so the circle is then in the plane of the equinoctial. Therefore, as the sun is above the equinoctial in summer (in northern latitudes), and below it in winter; the axis of the semicircle will cast a shadow on the hour of the day, on the upper surface of the circle, from the 20th of March till the 23d of September; and from the 23d of September to the 20th of March, the hour of the day will be determined by the shadow of the semicircle upon the lower surface of the circle. In the former case, the shadow of the circle falls upon the day of the month, on the lower part of the diameter of the semicircle; and in the latter case, on the upper part.

The method of laying down the months and signs upon the semicircle is as follows: Draw the right line ACB, fig. 11, equal to the diameter of the semicircle Fig. 11. ADB, and cross it in the middle at right angles with the line ECD, equal in length to ADB; then EC will be the radius of the circle FCG, which is the same as that of the semicircle. Upon E, as a centre, describe the circle FCG, on which set off the arcs C h and C i, each equal to 23\( \frac{1}{2} \) degrees, and divide them accordingly into that number for the sun's declination. Then laying the edge of a ruler over the centre E, and also over the sun's declination for every fifth day of each month (as in the card-dial), mark the points on the diameter AB of the semicircle from a to g, which are cut by the ruler; and there place the days of the months accordingly, answering to the sun's declination. This done, setting one foot of the compasses in C, and extending the other to a or g, describe the semicircle a b c d e f g; which divide into fix equal parts, and through the points of division draw right lines parallel to CD, for the beginning of the lines (of which one half are on one side of the semicircle, and the other half on the other), and set the characters of the lines to their proper lines, as in the figure.

An universal dial, of a very ingenious construction, has lately been invented by Mr G. Wright of London, by Mr G. Wright. The hour-circle or arch E, (fig. 19.), and latitude arch C, are the portions of two meridian circles; one fixed, and the other moveable. The hour or dial-plate SEN at top is fixed to the arch C, and has an index that moves with the hour-circle E; therefore the construction of this dial is perfectly similar to the construction of the meridians and hour-circle upon a common globe. The peculiar problems to be performed by this instrument are, 1. To find the latitude of any place. 2. The latitude of the place being known, to find the time by the sun and stars. 3. To find the sun or star's azimuth and altitude.

Previous to use, this instrument should be in a well-adjusted state: to perform which, you try the levels of the horizontal plates A a, by first turning the screws BBBBB till the bubbles of air in the glass tubes of the spirit-levels (which levels are at right angles to each other) 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 A a 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 o of the nonius. The arch is then fixed by the tightening screw at the back of D. Hang a filken 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 fit, 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 unforewd 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 almanacks, 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 E, 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 fall 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, 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 south 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 o 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 LO, 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 in practical astronomy, trigonometry, &c. the horizontal plate D is divided into 360°, 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 HG 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, has been invented by some instrument-makers. 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 OP, fig. 20. moveable over the declination's semicircle D. It has a nonius Q to the semicircle. femicircle. 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 right 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 the under 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 femicircle 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, which is described in Jones's "Instrumental Dialling." 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 let 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 femicircle 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 femicircle 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 femicircle 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 femicircle AB to the latitude of the place, and the index of the sights OP to the declination of the sun, found by the ephemeris, as before directed. Turn the femicircle 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 direct through, by the spot falling on the lower interceptors 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 LE 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 case, by turning the several screws at IGH; this being exactly done, turn the circle EF to 90° or half round: If the bubble of the 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-cases by means of a turn-screw, until you bring them to that fixed position, that they will return when the plate EF is turned half round. 2dly, To adjust the line of sight OP: Set the nonius to o on the declination arch D, the nonius on the hour circle to VII, and the nonius on the femicircle AB to 95°. 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 femicircle 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 reverse. Return the femicircle 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 exactly 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 correction always to be applied to the observations by the femicircle 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 altitude or depression: Reverse the declination femi 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 given 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 absolutely 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. 21.) 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 divisions 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 diametrically within the outer by means of two pivots inserted in each end of the ring at the hour 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 sometimes opposite the degrees of the sun's declination; and on the other side the days of the month throughout 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 h (see fig. 22.), 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° 48' 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 passing through the small hole in the cursor may fall exactly in a line drawn through the middle of the concave surface of the inner ring or hour circle, the bright spot by which shows the hour of the day in the said concave surface of the dial. Note, The hour XII cannot be shewn 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 h, 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 altitude; then what degree the shadow cuts is the altitude. Thus, in fig. 22, the shadow h g 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 the 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.

Example.

<table> <tr> <th></th> <th>Deg. Min.</th> </tr> <tr> <td>The meridian altitude may be</td> <td>57 48</td> </tr> <tr> <td>The sun's declination for the day</td> <td>19 18</td> </tr> <tr> <td>Complement of latitude</td> <td>38 30</td> </tr> <tr> <td></td> <td>90 0</td> </tr> <tr> <td>The latitude</td> <td>51 30</td> </tr> </table>

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 adjusting screws, &c. &c. (see fig. 23.), by which it is rendered more useful and convenient for finding the sun's azimuth, altitudes, variation of the needle, declinations of planes, &c. &c.

An Universal Dial on a plain cross, is described by Universal Mr Fergusson. It is moveable on a joint C, for elevating it to any given latitude on the quadrant C o 90, 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 n e or m k. See fig. 24.

This dial is rectified by setting the middle line t u 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 i o will cast a shadow on the time of the day on the side of the arm c m; from VI till IX; the lower edge i of the arm i o will cast a shadow on the hours on the side o q; from IX in the morning to XII at noon, the edge a b of the top part a n will cast a shadow on the hours on the arm n e f; from XII to III in the afternoon, the edge e d of the top part will cast a shadow on the hours on the arm k l m; from III to VI in the evening, the edge g h will cast a shadow on the hours on the part p q; and from VI till IX, the shadow of the edge ef will show the time on the top part an. 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. 25.) of 23 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 B d. Then, as the edges of the shadow, from each of the arms, will be parallel to Be, making an angle of 23 1/2 degrees with the side B d of the arm, when the sun's declination is 23 1/2°, it is plain, that if the length of the arm be B d, the least breadth that it can have, to keep the edge Be of the shadow B e g d from going off the side of the arm d before it comes to the end of it e d, must be equal to e d or d B. 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 crofs a b c d (fig. 26.) 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 a c in the compasses, and with one foot in the corner a, describe with the other the quadrant ef. Divide this arc into six equal parts, and through the points of division draw right lines a g, a h, &c. continuing three of them to the arm c e, 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. 24. 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 crofs, 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. 27. is called an 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 immovable, as ABCD (fig. 27.), find a meridian line as GF; or if moveable, assume the meridian at pleasure: then by means of the triangle KEF, whose base is applied on the meridian line, 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 hand 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 stile GI falls 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 JGF, its shadow will point out the several hours by the light of the sun. 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 Ferguson describes a method of making three Dials on dials on three different planes, so that they may all show the time of the day by one gnomon. On the flat board ABC (fig. 28.), describe an horizontal dial, with its mon. gnomon FGH, 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 south dial, describe it on the board EDC. Besides, on a circular plate IK describe an equinoctial dial, and by a slit c d 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 indies 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 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 stile due south, by means of the needle; which will be, when the needle points as far from the north fleur-de-lis towards 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 shewn 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 23d of September: and if the meridian of your place on the globe be set even with the meridian stile, 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 shew 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, fig. 17. as far from each other as you intend for the thickness of the file a b c d; and in the same manner draw the like thickness of the other three files e f g h, i k l m, and n o p q, all standing upright as from the centre.

With any convenient opening of the compasses, as a A, (so as to have proper strength of stuff when KI is equal to a A), set one foot on a as a centre, and with the other foot describe the quadrantal arc A c. Then, without altering the compasses, set one foot on b as a centre, and with the other foot describe the quadrant d B. 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 e f i k, and n o; 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 I p and K p, I q and M q, N r and O r, P s and Q s; to form the four angular hollows I p K, L q M, N r O, and P s Q; making the distances between the tips of these hollows, as I K, L M, N O, and P Q, each equal to the radius of the quadrants; and leaving sufficient room within the angular points p q r and s, for the equinoctial in the middle.

To divide the inside 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 K t; then, without altering the compasses, set one foot in K, and with the other foot describe the arc I t. 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 I 3, I 4, I 5, I 6, I 7; and K 2, K 1, K 12, K 11; and they will meet the sides K p and I p of the angle I p K where the hours thereon must be placed. And these hour spaces in the arc 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 quadrantal ones quite through the places where their four gnomons must stand; and lay down the hours on their insides (as in fig. 18.), 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 at 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 a c b d will show the hours in the quadrant A c from sunrise till VI in the morning: the shadow from the end M will show the hours on the side L q from V to IX in the morning; the shadow of the file e f g h in the quadrant D g (in the long days) will show the hours from sunrise till VII 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 a b c d goes off the quadrant A c, the shadow of the southern file i k l m begins to fall within the quadrant F l, 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 m E. And the shadow of the end O shows the time from XI in the forenoon till III in the afternoon, on the side r N; as the shadow on the end P shows the time from IX in the morning till 1 o'clock in the afternoon, on the side Q s.

At noon, when the shadow of the eastern file e f g h goes off the quadrant h C (in which it showed the time from VI in the morning till noon, as it did in the quadrant g D from sunrise till VI in the morning), the shadow of the western file n o p q begins to enter the quadrant H p, 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 P s 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 K p from III till VII in the afternoon; and the shadow of the file a b c d shows the time from VI in the evening till the sun sets.

The shadow of the upright central wire that sup- ports 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 shewn how to make sun-dials by the assistance of a good globe, or of a dialling 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 distance of the hours. Yet as a globe may be found exact enough for some other requisites in dialling, we shall take it occasionally.

The construction of fun-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 51 1/2 degrees north latitude, would be a horizontal dial on the same meridian, 90 degrees southward of 51 1/2 degrees of north 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 in 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 51 1/2 degrees north, and the declination of the place is 36 degrees west; elevate the north pole 51 1/2 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 30 1/2 degrees in south latitude; keeping it there, count in the equator the number of degrees between the meridian of London and the brazen meridian (which now becomes the meridian of the required place), and you will find it to be 42 1/2. Therefore an upright plane at London, declining 36 degrees westward from the south, would be a horizontal plane at that place, whose latitude is 30 1/2 degrees south of the equator, and longitude 42 1/2 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 30 1/2 degrees; save only that the reckoning on 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 42 1/2 degrees west from London.

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

Let NESW (fig. 12.) be the horizon of London, whose zenith is Z, and P the north pole of the sphere; and let Zh 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 femidiameter ZD perpendicular to Zh; and it will cut the horizon in D, 36 degrees west of the south S. Then a plane, on the tangent HD, touching the sphere in D, will be parallel to the plane Zh; and the axis of the sphere will be equally inclined to both these planes.

Let WQE be the equinoctial, whose elevation above the horizon of Z (London) is 38 1/2 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 K h would be horizontal) and the arc RQ is the difference of longitude of the planes Zh and DH.

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 RWQ measures the elevation of the equinoctial above the horizon of Z, namely 38 1/2 degrees. Say therefore, as radius is to the co-fine of the plane's declination from the south, so is the co-fine of the latitude of Z to the fine of RD the latitude of D; which is of a different denomination from the latitude of Z, because Z and D are on different sides of the equator.

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

To fine 30° 14' = DR (9.70211) = the latitude of D, whose horizon is parallel to the vertical plane Zh 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 sine of WR = * 54° 9.92796 Add the logarithmic sine 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-fine of 38 1/2 degrees, the height of the equinoctial at Z, fo is the co-tangent of 36 degrees, the plane's declination, to the co-tangent of the difference of longitudes. Thus,

To the logarithmic sine 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 1/2 minutes.

3. And thus having found the exact latitude and longitude of the place D, to 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 38° 14' south; but anticipating the time at D by 2 hours 51 minutes (neglecting the 1/2 minute 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 (fig. 13.) for the subfile of the dial, and make the angle KCP equal to the latitude of the place (viz. 38° 14'), to whose horizon the plane of the dial is parallel; then CRP will be the axis of the file, or edge that casts the shadow on the hours of the day, in the dial. This done, draw the contingent line EQ, cutting the subtilar 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 CM 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° 42') 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 let 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 tangents 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 subfile. The sixth hour, counted from that hour or part of the hour on which the subfile falls, will be always in a line perpendicular to the subfile, 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 k o, perpendicular to CM, will be a horizontal line on the plane of the dial, along which line 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 towards 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 subfile 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 files 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 S (fig. 14.) be equal to the arc SD on the horizon ; and its reclinian be equal to the arc D d of the vertical circle DZ : then it is plain, that if the quadrant of altitude Z d D on the globe cuts the point D in the horizon, and the reclinian is counted upon the quadrant from D to d ; the intersection of the hour-circle PR d, with the equinoctial WQE, will determine R d, the latitude of the place d, whose horizon is parallel to the given plane Z h at Z ; and RQ will be the difference in longitude of the places at d and Z.

Trigonometrically thus: Let a great circle pass through the three points, W, d, E ; and in the triangle WD d, right-angled at D, the sides WD and D d are given ; and thence the angle DW d is found, and so is the hypothenuse W d. Again, the difference, or the sum, of DW d and DWR, the elevation of the equinoctial above the horizon of Z, gives the angle d'WR ; and the hypothenuse of the triangle WR d was just now found ; whence the sides R d 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, if the latitude of the place Z be 52° 10' north ; the declination SD of the plane Z h (which would be horizontal at d) be 36°, and the reclinian be 15°, or equal to the arch D d ; the south latitude of the place d, that is, the arc R d, will be 15° 9' ; and RQ the difference

* The co-fine of 36.0, or of RQ. † The co-fine of 51.30, or of QZ. ‡ The co-fine of 38.30, or of WDR. § The co-tangent of 36.0, or of DW. difference of the longitude, 36° 2'. From these data, therefore, let the dial (fig. 15.) be described, as in the former example.

8. There are several other things requisite in the practice of dialling; 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 construction, 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 sine 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 (fig. 13.) is to KM in the ratio compounded of the ratio of KC to KG (=KR) and of KG to KN; which making CK the radius 10,000000, or 10,0000, or 10, or 1, are the ratio of 10,000000, or of 10,0000, or of 10, or of 1, to KG×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' 9.89334 ‡ Add the logarithmic tang. of 51° 0' 9.42805 The sum—radius is - - 8.32159 = 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', 55° 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 A, 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. And lastly, set off 1257 (the natural tangent of 51°

Vol. VII. Part I.

(*) That is, of 15, 30, 45, 60, 75°, for the hours of I, II, III, IIII, and 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.

(f) In which case the radius CK is supposed to be divided into 10,0000 equal parts.

** Here we consider the radius as unity, and not 10,0000: but which, instead of the index 9, we have —1 as above; which is of no farther use than making the work a little easier.

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 (fig. 14.) be the sun's place, dR his declination; and in the triangle PZd, Pd, the sum, or the difference, of d R, 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 sun'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 fine 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 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

\[ A = LD + H ld, \quad \text{and} \quad H = \frac{A - LD}{ld}. \]

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

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

Which theorems will be found useful, and expeditious enough for solving those problems in geography and dialling which depend on the relations 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 5 hours distant from XII, that is, an hour after VI in the morning, or before VI in the evening; and the sun's declination 20° north. Required the sun's altitude?

Then to log. L=log. fin. 51° 30' 1.89354** add log. D=log. fin. 20° 0' 1.53405 Their sum 1.42759 gives

LD=logarithm. of 0.267664, in the natural fines. D d And, And, to log. H=log. fin. ‡‡ 15° 0' 1.41300 add {log. l=log. fin. ‡‡ 38° 0' 1.79414 {log. d=log. fin. §§ 70° 0' 1.97300

Their sum 1.18014 gives

H / d=—logarithm of 0.151408, in the natural fines.

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

The same hour distance being aflumed on the other side of VI, then LD—H / d = 0.116266, the fine of 6° 40' 4"; which is the sun's altitude at VI 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 H / d—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' 4" below the horizon.

Exampl. 2. From the fame 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 Z d; and the arc Pd (the distance from the pole) being given; say, As the co-fine of the altitude is to the fine of the distance from the pole, so is the fine of the hour-distance from the meridian to the fine of the azimuth distance from the meridian.

Let the latitude be 51° 35' north, the declination 15° 0' south, and the time 11h. 24m. 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' 4", and P d the distance from the meridian, or the angle d PZ, 36",

To log. fin. 74° 51' - 1.98464. Add log. fin. 36° 0' - 1.76922

And from the sum - 1.75386 Take the log. fin. 81° 32' 1.99525

Remains 1.75861=log. fin.

35°, the azimuth distance fought.

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 telecopical, 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 reclinuation of planes, are frequently taken with a sufficient degree of accuracy by an instrument called a declinator or declinatory.

The construction of this instrument is as follows: On a mahogany board ABIK, (fig. 34.) is inserted a femicircular 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 forms 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 magnetical 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 a horizontal direction to the plane proposed, and observe what declination, degree or point of the compass the N part of the needle stands over from the north or the 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 femicircle 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 CD 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 fame 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 C of the divisions at G will be the declination of the plane. The reason for observing.

†† The distance of one hour from VI. §§ The co-declination of the sun. two or three altitudes and angles in the morning is, that in cafe 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 fun, as above, draw a meridian or 12 o'clock line to fet the dial by.

The base ABIK serves to take the inclination and reclinatian of planes. In this cafe, the quadrant is taken off, and the plummet P is fitted on a pin at the centre C; then the fide IGK being applied to the plane proposed, as QL (fig. 35); if the plumb-line cuts the femicircle 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 fide ACB 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 fun's declination, on the same horizontal plane; the upright side of the gnomon being floped 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 fun's declination, and the length of the day at any time, are had by inspection (as also his latitude, by means of a scale of tangents). But its chief property is, that it may be placed true, whenever the fun shines, without the help of any other instrument.

Let d (fig. 14.) be the fun's place in the stereographic projection, x dy &c the parallel of the fun's declination, Z d a vertical circle through the fun's centre, P d 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 P d, which passes through the intersection of the shadow Z d with the circle of the fun's present declination.

The Babylonian and Italian dials reckon the hours not from the meridian as with us, but from the fun's rising and setting. Thus, in Italy, an hour before sunset is reckoned the 23d hour; two hours before sunset the 22d hour; and so of the rest. And the shadow that marks them on the hour-lines, is that of the point of a fide. 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 fetting their clocks so much faster or slower. And in Italy, they begin their day, and regulate their clocks, not from sunset, 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. 16. 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 fide 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 K 1, K 2, K 3, &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 sunrising, when the sun is in the tropic of Capricorn &c 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 fun's altitudes above the plane of the dial at these hours; and with their co-tangents S d, S e, S f, &c. to radius SR, describe arcs intersecting the hour lines in the points d, e, f, &c. so that the right lines 1 d, 2 e, 3 f, &c. be the lines of I, II, III, &c. hours after sunrising.

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 fun 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 fun'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 fet the watch to agree with that time, according to the fun. A Hadley's quadrant is very convenient for this purpuse: for by it you may take the angle between the fun and his image reflected from a bason of water; the half of which angle, subtracting the refraction, is the altitude required. 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 observation 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 25 minutes past VIII in the morning by the watch; and found, by observing in the afternoon, that the sun had the same altitude 10 minutes before III; then it is plain that the watch was 5 minutes too fast for the sun: for 5 minutes after XII is the middle time between VIII. 20 m. in the morning, and IIIh. 50 m. in the afternoon; and therefore, to make the watch agree with the sun, it must be set back five minutes.

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 colatitude 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 through 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 whole, 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 the 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 further 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 the 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 through 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 sunlight may be allowed to enter but what comes through 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 Dialling.

I. The Dialling Sector, contrived by the late Mr Benjamin Martin, is an instrument by which dials are drawn in a more easy, expeditious, and accurate manner. The principal lines on it are the line of latitudes and the line of hours (fig. 32.). They are found on most of the common plane scales and sectors; but in a manner that greatly confines and diminishes their use; for the first, they are of a fixed length; and secondly, too final 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 11 1/2 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 division 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. 33.), viz. from 10 to 10, 20 to 20, 30 to 40, and so on, as is done in like cases on the lines of sines, tangents, &c. Hence its universal use for drawing dials of any proposed 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 distinct 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 4 1/2.

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 dialling 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 af, (fig. 32.) : Apply either of the distances of c e or af 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, quarters, &c. from the hour scale by similar parallel extents, you apply them upon the lines e d and f b as before described.

As the hour-lines on the sector extend to but 4', the double distance of the hour 3, when used either singly or sectorally, 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. 1, and making the angle of the file'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 dialling. But more fully to comprehend the matter, it must be considered there will be a variation of particulars as follows: 1. Of the subfile 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 subfile line GB (fig. 31.) 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 south, viz. As radius : 28° 43' :: tang. 38° 28' : tangent 29° 55'. This and the following analogy may be as accurately worked on the Gunter's line of sines, tangents, &c. properly placed on the sector, as by the common way for 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° 5'. Lastly, to find the height of the file: As radius is to the co-sine of the latitude, so is the co-sine of the declination to the sine of the file's height, viz. Radius : s 38° 28' :: s 65° 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 28° 55' from C to B, and draw GB, which will be the subfile 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 ADQD, and then the triangle ADQ is completed upon the subfile GB.

To lay off the hours, the plane's difference of longitude being 35°, equal to 2 h. 23 min. in time, allowing 15° to an hour, so that there will be 2 h. 23' 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 1 h. 20', and set off from D to 1 ; 2 h. 20', and set off from D to 12 ; 3 h. 20', from D to 11 ; 4 h. 20' from D to 10 ; and 5 h. 20' from D to 9, which will be 40' from A.

Then, on the other side of the subfile line GB, you take 40' from the beginning of the scale, and set off from D to 3 ; then take 1 h. 40', and set off from D to 4 ; also 2 h. 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 fun 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 subfile line GB, and which is already drawn. 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 draw GS, and the angle SGB will be the true height of the gnomon above the subfile GB.

II. The Dialling Trigon is another new instrument of great utility in the practice of dialling; and was also contrived by the late Mr Martin. It is composed of two graduated scales and a plane one. On the scale AB (fig. 36.) is graduated the line of latitudes; and on the scale AC, the line of hours: these properly conjoined with the plane scale BD, as shown in the figure, truly represent the gnomonical triangle, and is properly called a dialling trigon. The hour-scale AC is here of its full length; so that the hours, halves, quarters, &c. and every single minute (if required) may be immediately set off by a steel 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 the 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 a f (fig. 29.) 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 51° 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 c d, as by the figure at d, and falling the 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 steel point instantly, and the hour lines drawn through them as before, and as shown in the figure. When this is done for the side a f or morning hours, you move the scale of latitudes and hours to the other side, c e, or afternoon side, and place the hour-scale to 51° 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. 30.), the operation being just the same, only reversing the hours as in the figure, and marking the angles 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. 31.) as it is with the same facility applied to the lines AQ, GB, and the hours and quarters marked off as before directed.

On the scale BD of the trigon is graduated 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.

DIA DIALLING 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 CLXXI.

DIALLING Sector. See DIALLING, p. 212. and Plate CLXXIV.

DIALLING Sphere, is an instrument made of brafs, 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.

DIALLING Trigon. See DIALLING, p. 213, and Plate CLXXIV.

a mine, called also Plummimg, 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.

DIALOGISM, in Rhetoric, is used for the soliloquy of persons deliberating with themselves. See SOLOQUY.

DIALOGUE, in matters of literature, a conversation between two or more persons either by writing or by word of mouth.

Composition and Style of written DIALOGUE. As the end of speech is conversation, no kind of writing can be more natural than dialogue, which represents this. And accordingly we find it was introduced very early, for there are several instances of it in the Mosaic history. The ancient Greek writers also fell very much into it, especially the philosophers, as the most convenient and agreeable method of communicating their sentiments and instructions to mankind. And indeed it seems to be attended with very considerable advantages, if well and judiciously managed. For it is capable to make the driest subjects entertaining and pleasant, by its variety, and the different characters of the speakers. Besides, things may be canvassed more minutely, and many lesser matters, which serve to clear up a subject, may be introduced with a better grace, by questions and answers, objections and replies, than can be conveniently done in a continued discourse. There is likewise a further advantage in this way of writing, that the author is at liberty to choose his speakers: and therefore, as Cicero has well observed, when we imagine that we have persons of an established reputation for wisdom and knowledge talking together, it necessarily adds a weight and authority to the discourse, and more closely engages the attention. The subject matter of it is very extensive; for whatever is a proper argument of discourse, public or private, serious or jocose; whatever is fit for wise and ingenious men to talk upon, either for improvement or diversion, Dialogue is suitable for a dialogue.

From this general account of the nature of dialogue, it is easy to perceive what kind of style best suits it. Its affinity with EPISTLES, shows there ought to be no great difference between them in this respect. Indeed, some have been of opinion, that it ought rather to sink below that of an epistle, because dialogues should in all respects represent the freedom of conversation; whereas epistles ought sometimes to be composed with care and accuracy, especially when written to superiors. But there seems to be little weight in this argument, since the design of an epistle is to say the same things, and in the same manner, as the writer judges would be most fit and proper for him to speak, if present. And the very same thing is designed in a dialogue, with respect to the several persons concerned, in it. Upon the whole, therefore, the like plain, easy, and simple style, suited to the nature of the subject, and the particular characters of the persons concerned, seems to agree to both.

But as greater skill is required in writing dialogues than letters, we shall give a more particular account of the principal things necessary to be regarded in their composition, and illustrate them chiefly from Cicero's excellent dialogues concerning an orator.—A dialogue then consists of two parts; an introduction, and the body of the discourse.

1. The introduction acquaints us with the place, time, persons, and occasion of the conversation. Thus Cicero places the scene of his dialogues at Crassus's country seat; a very proper recess, both for such a debate and the parties engaged in it. And as they were persons of the first rank, and employed in the greatest affairs of state, and the discourse held them for two days; he represents it to have happened at the time of a festival, when there was no business done at Rome, which gave them an opportunity to be absent.

And because the greatest regard is to be had in the choice of the persons, who ought to be such as are well acquainted with the subject upon which they discourse; in these dialogues of Cicero, the two principal disputants are Crassus and Antony, the greatest orators of that age, and therefore the most proper persons to dispute upon the qualifications necessary for their art. One would think it scarce necessary to observe, that the conference should be held by persons who lived at the same time, and so were capable to converse together. But yet some good writers have run into the impropriety of feigning dialogues between persons who lived at distant times. Plato took this method, in which he has been followed by Macrobius. But others,

PLATE CLXXI.

Fig. 1. Fig. 2. Fig. 3. Fig. 5. Fig. 4. Fig. 6. Fig. 7. Fig. 8.

Line of Chords Scale of Latitudes Scale of Hours

The Line of Chords The Line of Latitudes The Line of Hours

PLATE CLXXII.

Fig. 9. Fig. 10. Fig. 11. Fig. 12. Fig. 13. Fig. 14. Fig. 15. Fig. 16. Fig. 17. Fig. 18.

Engraved by W. Archibald Edin.

PLATE CLXXIII.

Fig. 19. Fig. 20. Fig. 21. Fig. 22. Fig. 23. Fig. 24. Fig. 25. Fig. 26. Fig. 27. Fig. 28.

Engraved by W. Archibald Edin.

PLATE CLXXIV.

Fig. 29. Fig. 30. Fig. 31. Fig. 32. Fig. 33. Fig. 34. Fig. 35. Fig. 36.

 Dialogue, who have been willing to bring persons to discourse together, who lived in different ages, without such incongruity, have wrote dialogues of the dead. Lucian has made himself most remarkable in this way. As to the number of persons in a dialogue, they may be more or less; so many as can conveniently carry on a conversation without disorder or confusion may be admitted. Some of Cicero's dialogues have only two, others, three or more, and those concerning an orator seven. And it is convenient they should all, in some respects, be persons of different characters and abilities; which contributes both to the variety and beauty of the discourse, like the different attitudes of figures in a picture. Thus, in Cicero's dialogues last-mentioned, Crassus excelled in art, Antony principally for the force of his genius, Catullus for the purity of his style, Scevola for his skill in the law, Caesar for wit and humour; and though Sulpitius and Cotta, who were young men, were both excellent orators, yet they differed in their manner. But there should be always one chief person, who is to have the main part of the conversation; like the hero in an epic poem or a tragedy, who excels the rest in action; or the principal figure in a picture, which is most conspicuous. In Plato's dialogues, this is Socrates; and Crassus in those of Cicero above-mentioned.

It is usual likewise, in the introductions, to acquaint us with the occasion of the discourse. Indeed this is not always mentioned; as in Cicero's dialogue of the parts of oratory, where the son begins immediately with desiring his father to instruct him in the art. But it is generally taken notice of, and most commonly represented as accidental. The reason of which may be, that such discourses appear most natural; and may likewise afford some kind of apology for the writer in managing his different characters, since the greatest men may be supposed not always to speak with the utmost exactness in an accidental conversation. Thus Cicero, in his dialogues concerning an orator, makes Crassus occasionally fall upon the subject of oratory, to divert the company from the melancholy thoughts of what they had been discoursing of before, with relation to the public disorders, and the dangers which threatened their country. But the introduction ought not to be too long and tedious. Mr Addison complains of this fault in some authors of this kind. "For though (as he says) some of the finest treatises of the most polite Latin and Greek writers are in dialogue, as many very valuable pieces of French, Italian, and English, appear in the same dress; yet in some of them there is so much time taken up in ceremony, that, before they enter on their subject, the dialogue is half over."

2. We come now to the body of the discourse, in which some things relating to the persons, and others to the subject, are proper to be remarked.

And as to the persons, the principal thing to be attended to is to keep up a justness of character through the whole. And the distinct characters ought to be so perfectly observed, that from the very words themselves, it may be always known who is the speaker. This makes dialogue more difficult than single description by reason of the number and variety of characters which are to be drawn at the same time, and each of them managed with the greatest propriety. The principal speaker should appear to be a person of great fame and wisdom, and best acquainted with the subject. No question ought to be asked him, or objection started to what he says, but what he should fairly answer. And what is said by the rest should principally tend to promote his discourse, and carry it through in the most artful and agreeable manner. When the argument is attended with difficulties, one other person or more may be introduced, of equal reputation or near it, but of different sentiments, to oppose him, and maintain the contrary side of the question. This gives opportunity for a thorough examination of the point on both sides and answering all objections. But if the combatants are not pretty equally matched, and masters of the subject, they will treat it but superficially. And through the whole debate there ought not to be the least wrangling, peevishness, or obstinacy; nothing but the appearance of good humour and good breeding, the gentleman and the friend, with a readiness to submit to conviction and the force of truth, as the evidence shall appear on one side or the other. In Cicero, these two characters are Crassus and Antony. And from them Mr Addison seems to have taken his Philander and Cynthia in his Dialogues upon the usefulness of ancient medals, which are formed pretty much on Cicero's plan. When younger persons are present, or such who are not equally acquainted with the subject, they should be rather upon the inquiry than dispute: And the questions they ask should be neither too long nor too frequent, that they may not too much interrupt the debate, or appear over talkative before wiser and more experienced persons. Sulpitius and Cotta sustain this character in Cicero, and Eugenius in Mr Addison. And it is very convenient there should be one person of a witty and jocose humour, to enliven the discourse at proper seasons, and make it the more entertaining, especially when the dialogue is drawn out to any considerable length. Caesar has this part in Cicero. And in Mr Addison, Cynthia is a person of this turn, and opposes Philander in a merry way. Mr Addison's subject admitted of this: but the seriousness and gravity of Cicero's argument required a different speaker for the jocose part. Many persons ought not to speak immediately after one another. Horace's rule for plays is:

To crowd the stage is odious and absurd. Let no fourth actor strive to speak a word.

Though Scaliger and others think a fourth person may sometimes be permitted to speak in the same scene without confusion. However, if this is not commonly to be allowed upon the stage, where the actors are present, and may be distinguished by their voice and habit; much less in a dialogue, where you have only their names to distinguish them.

With regard to the subject, all the arguments should appear probable at least, and nothing be advanced which may seem weak or trivial. There ought also to be an union in dialogue, that the discourse may not ramble but keep up the main design. Indeed, short and pleasant digressions are sometimes allowable for the ease and entertainment of the reader. But every thing should be so managed, that he may still be able to carry on the thread of the discourse in his mind, and keep the main argument in view, till the whole is finished. The writers of dialogue have not confined their discourses to any certain space of time; but either concluded them with the day, or broke off when their speakers have been tired, and reassembled them again the next day. Thus Cicero allows two days for his three dialogues concerning an orator; but Mr Addison extends his to three days, allowing a day for each. Nor has the same method always been observed in composing dialogues. For sometimes the writer, by way of narrative, relates a discourse which passed between other persons. Such are the dialogues of Cicero and Mr Addison last mentioned, and many others both of the ancients and moderns. But, at other times, the speakers are introduced in person, as talking to each other. This, as Cicero observes, prevents the frequent repetition of those words, he said, and he replied; and by placing the hearer, as it were, in the conversation, gives him a more lively representation of the discourse, which makes it the more affecting. And therefore Cicero, who wrote his dialogue of old age in this manner, in which Cato, who was then in years, largely recounts the satisfaction of life which may be enjoyed in old age, tells his friend Atticus, he was himself so affected with that discourse, that when he reviewed it sometimes, he fancied they were not his own words, but Cato's. There are some other dialogues of Cicero, written in the same way; as that Of friendship and Of the parts of Oratory. And both Plato and Lucian generally chose this method.