or SUN-DIAL, an instrument serving to measure time, by means of the shadow of the sun. The word is formed from the Latin dies, "days," because indicating the hour of the day.

The ancients also called it sciaterricum, from its doing it by the shadow.

Definitions. Dial is more accurately defined, 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 show 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 subfile.

The angle included between the subfile 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 subfile makes an angle with the hour-line of XII; and this angle is called the distance of the subfile 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 thro' the hour-line of XII, and the other through the subfile.

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

If the whole earth APEP were transparent, and plate hollow, like a sphere of glass, and had its equator LXXXIX divided into 24 equal parts by so many meridian fig. 1, semicircles, a, b, c, d, e, f, g, &c. one of which is the geographical meridian of any given place, as London (which is supposed to be at the point a;) and if which dist-

the hours 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 opake axis, as PEP, 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 show the time at London, and at all other places on the meridian of London.

If this sphere was cut through the middle by a solid horizontal plane ABCD, in the rational horizon of London, one dial 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 Vertical 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 EP 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 fill 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 to decline by any given number of degrees from the meridian toward the east or west; provided the declination be less than 90 degrees, or the reclinatior be less than the co-latitude of the place; and the axis of the sphere will be a gnomon, or tile, for the dial. But it cannot be a gnomon, when the declination is quite 90 degrees, nor when the reclinatior 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 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 show 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 dialing, by joining 24 semi-circles 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 XII, 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 brass 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 brass 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 brass 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° degrees north, elevate the north pole of the globe 51° 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 brass meridian, you will find that the first meridian of the globe cuts the horizon in the following numbers of degrees from the north towards the east, viz. 11°, 24°, 38°, 53°, 71°, 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 Plate horizontal plane, draw the parallel right lines a c and LXXXIX. d b, upon that plane, as far from each other as 8g. 3° is equal to the intended thickness of the gnomon or tile 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 six o'clock line g b, 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, setting one foot in the intersection b, as a centre, with the same radius describe the quadrant f b, 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 G, 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; IIII, through \( 53\frac{1}{2} \); and V, through \( 71\frac{1}{2} \); and because the sun rises about four in the morning, on the longest days at London, continue the hour-lines of IV 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 \( e g \); and through the like divisions or degrees of that quadrant, viz. \( 11\frac{1}{2}, 24\frac{1}{2}, 38\frac{1}{2}, 53\frac{1}{2}, 71\frac{1}{2} \), 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 days, 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, through \( 51\frac{1}{2} \) degrees of either quadrant, and from its centre, draw the right line \( a g \) for the hypotenuse or axis of the gnomon \( a g i \); and from \( g \), let fall the perpendicular \( g i \), upon the meridian line \( a i \), and there will be a triangle made, whose sides are \( a g, g i, \) and \( i a \). If a plate similar to this triangle be made as thick as the distance between the lines \( a c \) and \( b d \), and set upright between them, touching at \( a \) and \( b \), its hypotenuse \( a g \) 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, such as that on the top of Plate XC.; 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 line, 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 hypotenuse \( a g \) of the gnomon \( a g i \), 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 six in the morning until six 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 brass 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 brass 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 brass 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 brass 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 toward 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 point toward 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, 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. 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 subtilar-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, file and subtilar, 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 toward 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 affit him, and may probably not understand the method of doing it by logarithmic calculation; we shall show how to perform it by the plain dialing lines, or scale of latitudes and hours; such as those on the top of Plate XC, 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 \(a\), \(b\), \(c\), \(d\), on the plane intended for a horizontal dial, and crossed it at right angles by the six o'clock line \(f\), \(e\) (as in fig. 3.) take the latitude of your place with the compasses, in the scale of latitudes, and set that extent from \(c\) to \(e\), and from \(a\) to \(f\), on the six o'clock line: then, taking the whole six hours between the points of the compasses in the scale of hours, with that extent set one foot in the point \(c\), and let the other foot fall where it will upon the meridian line \(c\), \(d\), as at \(a\). Do the same from \(f\) to \(b\), and draw the right lines \(e\), \(d\) and \(f\), 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 on the scale, lay off these extents from \(d\) to \(e\) for the afternoon hours, and from \(b\) to \(f\) for those of the forenoon: 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 IIII 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 forenoon 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 south-dial, take the co-latitude of your place from the scale of latitudes, and then proceed in all respects for the hour-lines, 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 dialist 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 90 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 \(Z\), \(L\), describe the two semicircles \(I\), \(F\), \(K\) and \(L\), \(Q\), \(K\), upon the centres \(Z\) and \(z\), 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 need be divided;) then connect the divisions which are equidistant from \(L\), by the parallel lines \(KM\), \(IN\), \(HO\), \(GP\), and \(FQ\). Draw \(VZ\) for the hypotenuse of the file, making the angle \(VZE\) equal to the latitude of your place; and continue the line \(VZ\) to \(R\). Draw the line \(Rr\) parallel to the fix o'clock line, and set off the distance \(a\) \(K\) from \(Z\) to \(T\), the distance \(b\) \(I\) from \(Z\) to \(X\), \(c\) \(H\) from \(Z\) to \(W\), \(d\) \(G\) from \(Z\) to \(T\), and \(e\) \(F\) from \(Z\) to \(S\). Then draw the lines \(St\), \(Ti\), \(Wv\), \(Xn\), and \(Yj\), each parallel to \(Rr\). Set off the distance \(y\) \(T\) from \(a\) to \(t\), and from \(f\) to \(i\); the distance \(x\) \(F\) from \(b\) to \(o\), and from \(g\) to \(z\); \(w\) \(W\) from \(c\) to \(g\), and from \(h\) to \(3\); \(iT\) 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 \(t\), \(i\), \(o\), \(g\), \(8\), \(7\); and laying it to the centre \(z\), 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 IIII and V through the centre \(z\), 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 south 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 dialing lines, and some others; which is as follows.

With any opening of the compasses, as \(E\), \(A\), according to the intended length of the scale, describe lines how the circle \(ADC\), and cross it at right angles by constructed the diameters \(CEA\) and \(DEB\). Divide the quadrant \(AB\) 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... these divisions to the line $AFB$ by the area $10$, $20$, $30$, &c., and this will be a line of chords, divided into 90 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 $E$ 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 $Ba$, $Bb$, &c., shown by the right lines $10a$, $20b$, $30c$, &c., drawn along the edge of the ruler. If the right line $BC$ be drawn, subtending this quadrant, and the nearest distances $Ba$, $Bb$, $Bc$, &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 $PGA$, subtending the quadrant $DA$; and parallel to it, draw the right line $r$, touching the quadrant $DA$ at the numeral figure 3. Divide this quadrant into six equal parts, as $1$, $2$, $3$, &c., and through these points of division draw right lines from the centre $E$ to the line $r$, which will divide it at the points where the six hours are to be placed, as in the figure. If every sixth 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 $r$, will divide each hour upon it into quarters.

In Fig. 2, we have the representation of a portable dial, which may be easily drawn on a card, and carried in a pocket-book. The lines $a$, $b$, $c$, &c., of the gnomon, must be cut quite through the card; and as the end $ab$ of the gnomon is raised occasionally above the plane of the dial, it turns upon the uncut line $cd$ as on a hinge. The dotted line $AB$ 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 shewing the hour of the day.

To rectify this dial, let 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 shew 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. 3, as he reads the following explanation of that figure.

Draw the occult line $AB$ parallel to the top of Fig. 3, the card, and cross it at right angles with the six o'clock line $ECD$; then upon $C$, as a centre, with the radius $CA$, describe the semicircle $AEI$, and divide it into 12 equal parts (beginning at $A$), as $Ar$, $As$, &c., and from these points of division draw the hour-lines $r$, $s$, $t$, $u$, $v$, $E$, &c., and $x$, all parallel to the six 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 $ASD$, 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 $23\frac{1}{2}$ degrees, for the sun's greatest declination; and divide them into $43\frac{1}{2}$ equal parts, as in fig. 2. Through the intersection $D$ of the lines $ECD$ and $AD$, draw the right line $FDG$ at right angles to $AD$. Lay a ruler to the points $A$ and $R$, and draw the line $ARF$ through $23\frac{1}{2}$ degrees of south declination in the arc $SR$; and then laying the ruler to the points $A$ and $T$, draw the line $ATG$ through $23\frac{1}{2}$ degrees of north declination in the arc $ST$; so that the lines $ARF$ and $ATG$ cut the line $FDG$ in the proper length for the scale of months. Upon the centre $D$, with the radius $DF$, describe the semicircle $FDG$; which divide into six equal parts, $Em$, $m$, $n$, &c., and from these points of division draw the right lines $mb$, $nb$, $pb$, and $qb$, each parallel to $SD$. 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 $23\frac{1}{2}$; with the same extent, setting one foot in $G$, describe the arc $AEO$ for the tropic of $23\frac{1}{2}$. Next setting one foot in the point $h$, and extending the other to $A$, describe the arc $ACI$ for the beginnings of the signs $\text{\textsc{xx}}$ and $\text{\textsc{p}}$; and with the same extent, setting one foot in the point... Dial. I, describe the arc AN for the beginnings of the signs II and Ω. Set one foot in the point i, and having extended the other to A, describe the arc AK for the beginnings of the signs X and m; and with the same extent, set one foot in k, and describe the arc AM for the beginnings of the signs ψ and ρ. Then setting one foot in the point D, and extending the other to A, describe the curve AL for the beginnings of ϑ and ϕ; 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 FDG, make 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, which are called universal, because they serve for all latitudes. Of these, the best is Mr Pardie's, which consists of three principal parts; the first whereof is called the horizontal plane (A), 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 twenty-third of September to the twentieth of March, the upper surface of the circle must touch both the centre C of the semicircle, and the line of ϑ and ϕ; and from the twentieth of March to the twenty-third 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 sun-shine, and adjusted it by the levelling screws k and l, until the plumb-line hangs over the black line upon the edge of the quadrant, and parallel to the said edge; move the semicircle in the quadrant, until the line of ϑ and ϕ (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 falls precisely on the day of the month in the semicircle; and then the meridional plane will be due north and south, the axis plate XC. 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 26th of March to the 23rd of September; and from the 23rd of September to the 26th 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 Fig. 5. upon the semicircle is as follows. Draw the right-line ACB, equal to the diameter of the semicircle 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 Ch and Ci, each equal to 23½ degrees, and divide them accordingly into that number, for the sun's declination. Then laying the edge of a ruler over the centre B, 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 abcd ef g; which divide into six equal parts, and through the points of division draw right lines, parallel to CD, for the beginning of the signs (of which one half are on one side of the semicircle, and the other half on the other), and set the characters of the signs to their proper lines, as in the figure.

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

The construction of sun-dials on all planes whatever, may be included in one general rule: intelligible, if that of a horizontal dial for any given latitude be well understood. For there is no plane, however obliquely situated with respect to any given place, but what is parallel to the horizon of some other place; and therefore, if we can find that other place by a problem on the terrestrial globe, or by a trigonometrical calculation, and construct a horizontal dial for it; that dial applied to the plane where it is to serve, will be a true dial for that place.—Thus, an erect direct south dial in 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 brats 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 abovementioned 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 brats meridian: then, the degree of the brats meridian that stands directly over the place, is its latitude; and the number of degrees in the equator, which are intercepted between the meridian of London and the brats meridian, is the place's difference of longitude.

Thus, as the latitude of London is 51° 5' degrees north, and the declination of the place is 36 degrees west; elevate the north pole 51° 5' 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 brats meridian, it will be found to be under 30° 3' degrees in south latitude: keeping it there, count in the equator the number of degrees between the meridian of London and the brats meridian (which now becomes the meridian of the required place) and you will find it to be 42°.

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° 3' degrees south of the equator, and longitude 42° 3' 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° 3' degrees; save only, that the reckoning of the hours is to anticipate the reckoning on the horizontal dial, by 2 hours 51 minutes: for so much sooner will the sun come to the meridian of London, than to the meridian of any place whose longitude is 42° 3' degrees west from London.

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

Let NESW be the horizon of London, whose zenith is Z, and P the north pole of the sphere; and let Zb be the position of a vertical plane at

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

Let WQR be the equinoctial, whose elevation above the horizon of Z (London) is 38° 5' degrees; and PRD be the meridian of the place D, cutting the equinoctial in R. Then it is evident, that the arc RD is the latitude of the place D (where the plane Zb would be horizontal) and the arc RQ is the difference of longitude of the planes Zb and 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 RWD measures the elevation of the equinoctial above the horizon of Z, namely, 38° 5' degrees. Say therefore, As radius . . . . . . . . . 10.00000 To co-sine 36° 5' = RQ . . . . . . . . 9.90796 So co-sine 51° 30' = QZ . . . . . . . . 9.79415

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

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

To the logarithmic fine of WR= 54° 5' 9.90796. Add the logarithmic fine of RD= + 38° 30' 9.79415.

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

To find the difference of longitude of the places D and Z, say, As radius is to the co-sine of 38° 5' degrees, the height of the equinoctial at Z, so is the co-tangent of 36 degrees, the plane's declination, to the co-tangent of the difference of longitudes. Thus,

To the logarithmic fine of 51° 30' = 9.89354 Add the logarithmic tang. of + 54° 5' = 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 two 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 30° 14' south; but anticipating the time at D by two hours 51 minutes (neglecting the 1/2 min. in practice)

because

The co-sine of 36° 5' or of RQ. + The co-sine of 51° 30' or of QZ. The co-tangent of 36° 5' or of DR. because D is so far westward in longitude from the meridian of London; and this will be a true vertical dial at London, declining westward 36 degrees.

Assume any right line CSL, for the subfile of the dial, and make the angle KCP equal to the latitude of the place (viz. 30° 14') to whose horizon the plane of the dial is parallel; then CRP will be the axis of the 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 subfile 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° 52'), set off the same way, will give a point between K and N, through which the hour-line of II is to be drawn. The tangent of 2° 8' (the difference between 45° and 42° 52') placed on the other side of CL, will determine the point through which the hour-line of III is to be drawn: to which 2° 8', if the tangent of 15° be added, it will make 17° 8'; and this set off from K towards Q on the line EQ, will give the point for the hour-line of IV: and so of the rest.—The forenoon hour-lines are drawn the same way, by the continual addition of the tangents 15°, 30°, 45°, &c. to 42° 52' (=the tangent of KM) for the hours of XI, X, IX, &c. as far as necessary; that is, until there be five hours on each side of the subfile. The sixth hour, accounted 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 ho, 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 let perpendicular to the horizon, the dial will have its true position.

5. If the plane of the dial had declined by an equal angle toward the east, its description would have differed only in this, that the hour-line of XII would have fallen on the other side of the 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 be equal to the arc SD on the horizon; Fig. 1, and its reclination be equal to the arc Dd of the vertical circle DZ: then it is plain, that if the quadrant of altitude ZdD on the globe cuts the point D in the horizon, and the reclination is counted upon the quadrant from D to d; the intersection of the hour circle PRd, with the equinoctial WQE, will determine Rd, the latitude of the place d, whose horizon is parallel to the given plane Zb at Z; and RQ will be the difference in longitude of the places at d and Z.

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

Thus, if the latitude of the place Z be 52° 10' north; the declination SD of the plane Zb (which would be horizontal at d) be 36°, and the reclination be 15°, or equal to the arc Dd; the fourth latitude of the place Z, that is, the arc Rd, will be 15° 9': and RQ, the difference of the longitude, 36° 2'. From these data, therefore, let the dial (fig. 2.) be described, as in the former example.

8. There are several other things requisite in the practice of dialing; the chief of which shall be given in the form of arithmetical rules, simple and easy to those who have learned the elements of trigonometry. For in practical arts of this kind, arithmetic should be used as far as it can go; and scales never trusted to, except in the final 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 is to KM in the ratio compounded of the plate XC ratio of KC to KG (=KR) and of KG to KM; which fig. 7, making CK the radius 10,000,000, or 10,000, or 10, or 1, are the ratio of 10,000,000, or of 10,000, 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 the afternoon, from XII:

---

That is, of 15°, 30°, 45°, 60°, 75°, for the hours of I, II, III, IIII, V, in the afternoon; and XI, X, IX, VIII, VII, in the afternoon. In all horizontal dials, and erect north or south dials, the subfile and meridian are the same: but in all declining dials, the subfile line makes an angle with the meridian. To the logarithmic sine of $51^\circ 30'$ 9.89354† Add the logarithmic tang. of $15^\circ 0'$ 9.42805

The sun—radius is 9.32159—the logarithmic tangent of $11^\circ 50'$, 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 sine of the latitude, and the tangents of $30^\circ$, $45^\circ$, $60^\circ$, and $75^\circ$, for the hours of II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, and XIII; and VII in the afternoon; or of X, IX, VIII, and VII in the forenoon; you will find their angular distances from XII to be $24^\circ 18'$, $38^\circ 3'$, $53^\circ 35'$, and $71^\circ 6'$; 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 $11^\circ 50'$), and setting them from XII to XI and to I, on the line b, 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 $51^\circ 30'$) for the angle of the stile's height, which is equal to the latitude of the place.

**Rule II.** The latitude of the place, the sun's declination, and his hour-distance from the meridian, being given, to find (1.) his altitude; (2.) his azimuth.

1. Let d be the sun's place, DR his declination; and, in the triangle PZd, PD the sun, or the difference, of dR, and the quadrant PR, being given by the supposition, as also the complement of the latitude PZ, and the angle dPZ, which measures the horary distance of d from the meridian; we shall (by Case 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 sine Zd : sine PD : : sine dPZ : dZP, or of its supplement DZS, the azimuthal distance from the south.

Or the practical rule may be as follows.

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

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

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

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

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

Which theorems will be found useful, and expedi-

† In which case, the radius GK is supposed to be divided into 100000 equal parts.

‡ The co-declination of the sun.

EXAMPLE I.

Suppose the latitude of the place to be $51^\circ$ degrees north; the time five hours distant from XII, that is, an hour after VI in the morning, or before VI in the evening; and the sun's declination $20^\circ$ north. Required the sun's altitude?

Then $\log_L = \log_{\text{fin.}} 51^\circ 30' = 1.89354^*$

add $\log_D = \log_{\text{fin.}} 20^\circ 0' = 1.53405$

Their sum 1.42759 gives

$LD =$ logarithm of $0.267664$, in the natural fines.

And, to $\log_H = \log_{\text{fin.}} 15^\circ 0' = 1.41300$

add $\begin{cases} \log_I = \log_{\text{fin.}} 38^\circ 0' = 1.79414 \\ \log_d = \log_{\text{fin.}} 70^\circ 0' = 1.97300 \end{cases}$

Their sum 1.18015 gives

$Hld =$ 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 $24^\circ 47'$, the sun's altitude sought.

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

But when the declination is $20^\circ$ south (or towards the depressed pole) the difference $Hld - LD$ becomes negative; and thereby shews, that, an hour before VI in the morning, or past VI in the evening, the sun's centre is $6^\circ 40'$ below the horizon.

EXAMPLE II.

From the same data, to find the sun's azimuth.

If H, L, and D are given, then (by par. 2. of Rule II.) from H having found the altitude and its complement Zd; and the arc PD (the distance from the pole) being given; say, As the co-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 $51^\circ 30'$ north, the declination $15^\circ 9'$ south, and the time II h. 24 m. in the afternoon, when the sun begins to illuminate a vertical wall, and it is required to find the position of the wall.

Then, by the foregoing theorems, the complement of the altitude will be $81^\circ 32' \frac{1}{2}$, and PD the distance from the pole being $109^\circ 5'$, and the horary distance from the meridian, or the angle dPZ, $36^\circ$.

To $\log_{\text{fin.}} 74^\circ 51' = 1.98464$

Add $\log_{\text{fin.}} 36^\circ 0' = 1.76922$

And from the sum 1.75386

Take the log. fin. $81^\circ 32' \frac{1}{2} = 1.99525$

Remains 1.75861 = log. fin.

$35^\circ$, the azimuth distance sought.

When 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.

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

To the gnomonic projection, there is sometimes added a stereographic projection of the hour-circles, and the parallels of the sun's declination, on the same horizontal plane; the upright side of the gnomon being flopped 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 falling thro' the sun, in the stereographic projection.

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

Let \(d\) be the sun's place in the stereographic projection, \(x\), \(y\), \(z\) the parallel of the sun's declination, \(Z\), \(d\) a vertical circle through the sun'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 sun's present declination.

The Babylonian and Italian dials reckon the hours, not from the meridian, as with us, but from the sun's rising and setting. Thus, in Italy, an hour before sun-set is reckoned the 23rd hour; two hours before sun-set the 22nd hour; and so of the rest. And the shadow that marks them on the hour-lines, is that of the point of a file. This occasions a perpetual variation between their dials and clocks, which they must correct from time to time, before it arises to any sensible quantity, by setting their clocks so much faster or slower. And in Italy, they begin their day, and regulate their clocks, not from sun-set, but from about mid-twilight, when the Ave-Maria is said; which corrects the difference that would otherwise be between the clock and the dial.

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

Let fig. 3. represent an erect direct south wall, on which a Babylonian dial is to be drawn, shewing 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 stile of the dial; the shadow of whose point R is to mark the hours, when SR is set upright on the plane of the dial.

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

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

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

Of the right placing of dials, and having a true meridian line for the regulating of clocks and watches.

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

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

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

Thus, for example, suppose you had taken the sun's altitude when it was 20 minutes past VIII in the morning 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 h. 20 m. in the morning, and III. h. 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 co-latitude of your place, as nearly as you can guess: for then the plate will face the sun directly at noon on the equinoctial days. Let the sun shine freely thro' the hole into the room; and hang a plumb line to the ceiling of the room, at least five or six feet from the window, in such a place as that the sun's rays, transmitted through the hole, may fall upon the line when it is noon by the clock; and having marked the said place on the ceiling, take away the line.

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

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

N. B. The rays transmitted through the hole, will cast but a faint image of the sun, even on a white screen, unless the room be so darkened that no sunshine may be allowed to enter but what comes 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.

An Universal Dial, shewing 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 file of each hollow coming out from the bottom thereof, as far as the ends of the hollows project. The corners are cut out into angles, in the insides of which the hours are also marked; and the edge of the end of each side of the angle serves as a file 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 file 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 file due south, by means of the needle; which will be, when the needle points as far from the north fleur-de-lis toward the west, as it declines westward, at your place. Then bend the pillar in the joint, till the black line on the pillar comes to the latitude of your place in the quadrant.

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

The same hour will then be 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 23rd of September; and, if the meridian of your place on the globe be set even with the meridian file, all the parts of the globe that the sun shines upon, will answer to those places of the real earth which are then enlightened by the sun. The places where the shade is just coming upon the globe, answer to all those places of the earth to which the sun is then setting; as the places where it is going off, and the light coming on, answer to all the places of the earth where the sun is then rising. And lastly, if the hour of VI be marked on the equator in the meridian of your place (as it is marked on the meridian of London in the figure) the division of the light and shade on the globe will shew the time of the day. The northern side of the dial (opposite to the southern or meridian one) is hid from sight in the figure, by the axis of the globe. The hours in the hollow to which that side belongs, are also supposed to be hid by the oblique view of the figure; but they are the same as the hours in the front-hollow. Those also in the right and left hand semicircular hollows are mostly hid from sight; and so also are all those on the sides next the eye of the four acute angles.

The construction of this dial is as follows:

On a thick square piece of wood, or metal, draw the lines \(a c\) and \(b d\), as far from each other as you intend for the thickness of the side \(a b c d\); and in the same manner, draw the like thickness of the other three sides, \(e f g h\), \(i k l m\), 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 leave proper strength of stuff when \(K I\) is equal to \(a A\)), set one foot in \(a\), as a centre, and with the other foot describe the quadrant arc \(A c\). Then, without altering the compasses, set one foot in \(b\) 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 \(c f\), \(i k\), 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 \(K p\), \(L q\) 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 sides 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 arcs must be subdivided into four equal parts, for the half hours and quarters.—Do the like for the other three angles, and draw the dotted lines, and set the hours in the sides 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 sides, (as in Plate XCII.), and set in their gnomons, which must be as broad as the dial is thick; and this breadth and thickness must be large enough to keep the shadows of the gnomons from ever falling quite out at the sides of the hollows, even when the sun's declination is at the greatest.

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

As the sun goes round, the broad end of the shadow of the side \(a b c d\) 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 side \(e f g h\) in the quadrant \(D g\) (in the long days) will show the hours from sunrise till VI in the morning; and the shadow of the end \(N\) will show the morning-hours, on the side \(O r\), from III to VII.

Just as the shadow of the northern side \(a b c d\) goes off the quadrant \(A c\), the shadow of the southern side \(i k l m\) begins to fall within the quadrant \(F h\), 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 of the end \(P\) shows the time from IX in the morning till I o'clock in the afternoon, on the side \(Q s\).

At noon, when the shadow of the eastern side \(e f g h\) 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 side \(n o p q\) 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 side \(a b c d\) shows the time from VI in the evening till the sun sets.

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