antiquity, sacrifices performed by the flamen dialis. See Flamen.

Dialing,

The art of drawing dials on the surface of any given body or plane. The Greeks and the Latins called this art gnomonica and sciatherica, by reason it distinguishes the hours by the shadow of the gnomon. Some call it photo-sciathereca, because the hours are sometimes shown by the light of the sun. Lastly, others call it horolography.

Dialing 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 flop; 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, almanacs, 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 Miletius; and others to Thales. Vitruvius mentions one made by the ancient Chaldee historian Berosus, 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 design of Aristarchus was a horizontal dial, with its limb raised up all around, to prevent the shadows stretching too far.

But it was late ere the Romans became acquainted with dials. The first sun-dial at Rome was set up by Papirius Cursor, about the year of the city 460; before which time, says Pliny, there is no mention of any account. 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. Witness, 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., verse 8.

The first professed writer on dialing 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 give 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 Dialing, printed in 1683, a geometrical method of drawing hour-lines from certain points determined by observation. Eberhardus Welperus, in 1625, published his Dialing, 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 Münster, in his Rudimenta Mathematica, published in 1551. Sturmius, in 1672, published a new edition of Welperus's Dialing, with the addition of a whole second part, about inclining and declining dials, &c.

In 1708, 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 dialing, in the German tongue; Coetius in his Horologiographia Plana, printed in 1689; Gauppenius, in his Gnomonica Mechanica; Bion, in his Use of Mathematical Instruments; the late ingenious Mr Ferguson, in his Select Lectures; Mr Emerlom, in his Dialing; and Mr W. Jones, in his Instrumental Dialing.

Definitions. 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 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 fulfile 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 divided into 24 equal parts by so many meridian semicircles, a, b, c, d, e, f, g, &c. one of which is the geographical meridian of any given place, as London's principal (which is supposed to be at the point a); and if 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 opaque 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 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, AFGG, 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 dial 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, 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 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 reclination be less than the co-latitude of the place: and the axis of the sphere will be a gnomon, or stile, for the dial. But it cannot be a gnomon, when the declination is quite 90 degrees, nor when the reclination 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 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 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 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 semicircles 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° 30' degrees north, elevate the north pole of the globe 51° 30' 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° 30', 24° 30', 38° 30', 53° 30', 71° 30', 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 horizontal plane, draw the parallel right lines ac and db, upon that 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. Crost this meridian at right angles with the six o'clock line gh, and setting one foot of your compasses in the intersection a, as a centre, describe the quadrant ge 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 fb, 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 b) draw the hour-line of I through $11\frac{3}{4}$ 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 IIII and V in the afternoon through the centre b to the opposite fide 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. $11\frac{3}{4}$, $24\frac{1}{2}$, $38\frac{1}{2}$, $53\frac{1}{2}$, and $71\frac{1}{2}$, draw the forenoon hour-lines of XI, X, IX, VIII, and VII; and because the sun sets not before six 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 hypothenuse 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 hypothenuse 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 (as represented on the plate); for if you extend the compasses from o 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 a g of the gnomon a g f, 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 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 point 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 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 hori- zon, 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. 4. A south dial declining west.

Only, placing the proper number of hours, and the stile or gnomon respectively, upon each plate. 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, stile and subtilar, quite through the centre; the axis of the stile (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 (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 \(a, b, c, d\), on the plane intended for a horizontal dial, and crossed it at right angles by the fix o'clock line \(f\) (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 fix 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 \(e\), 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 \(c, d\) 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 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 \(e\), 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-line as in the horizontal dial; only reversing the hours, as in fig. 4, and making the angle of the stile'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 stile's elevation; which is easily done.

With any opening of the compasses, as \(ZL\), de. Fig. 6. Scribe the two semicircles \(LFk\) and \(LQk\), 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 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 hypotenuse of the stile, 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 \(aK\) from \(Z\) to \(Y\), the distance \(bI\) from \(Z\) to \(X\), \(cH\) from \(Z\) to \(W\), \(dG\) from \(Z\) to \(T\), and \(eF\) from \(Z\) to \(S\). Then draw the lines \(St, Th, Ww, Xx,\) and \(Yy\), each parallel to \(Rr\). Set off the distance \(yY\) from \(a\) to \(i\), and from \(f\) to \(i\); the distance \(xX\) from \(b\) to \(o\), and from \(g\) to \(z\); \(wW\) from \(c\) to \(q\), and from \(h\) to \(s\); \(tT\) from \(d\) to \(8\), and from \(i\) to \(4\); \(sS\) 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 \(ii, io, 9, 8, 7\); and laying it to the centre \(z\), draw the afternoon lines through the points \(i, 2, 3, 4, 5\); 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 stile 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, reverting 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 \(EA\), according to the intended length of the scale, describe lines, how the circle \(ADC\), and cross it at right angles by the 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. 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 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 fines; and although these are seldom put among the dialing 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 fines, 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 $DGA$, subtending the quadrant $DA$; and parallel to it, draw the right line $rs$, touching the quadrant $DA$ at the numeral figure 3. Divide this quadrant into five equal parts, as 1, 2, 3, &c. and through these points of division draw right lines from the centre $E$ to the line $rs$, which will divide it at the points where the fix 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 $rs$, 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 $a$, $d$, $ab$, and $bc$ 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 thro' 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. 1 of Plate CLIX., as he reads the following explanation of that figure.

Draw the occult line $AB$ 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 $Ar$, $At$, &c. and from these points of division draw the hour lines $r$, $s$, $t$, $u$, $v$, $E$, $w$, and $x$, 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 $ASDo$, 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 $23\frac{1}{2}$ equal parts, as in fig. 2. Thro' the intersection $D$ of the lines $ECD$ and $ADo$, draw the right line $FDG$ at right angles to $ADo$. 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 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 $FoG$; which divide into six equal parts, $Fm$, $mn$, $no$, &c. and from these points of division draw the right lines $mb$, $ni$, $pk$, and $ql$, each parallel to $oD$. Then setting one foot of the compasses in the point $F$, extend the other to $A$, and describe... describe the arc AZH for the tropic of γφ; with the same extent, setting one foot in G, describe the arc AEO for the tropic of δσ. Next setting one foot in the point b, and extending the other to A, describe the arc ACI for the beginnings of the signs ξξ and ψ; and with the same extent, setting one foot in the point l, describe the arc AN for the beginnings of the signs ιι and ου. Set one foot in the point i, and having extended the other to A, describe the arc AK for the beginnings of the signs χ and μ; and with the same extent, set one foot in k, and describe the arc AM for the beginnings of the signs ψ and νν. Then setting one foot in the point D, and extending the other to A, describe the curve AL for the beginnings of γφ and ω; 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 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), 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 five months of the year are laid down, according to the sun's declination for their respective days, and on the other side the last five 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 23rd 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 γφ and ω; and from the 20th of March to the 23rd 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 back line upon the edge of the quadrant, and parallel to the said edge; move the semicircle in the quadrant, until the line of γφ and ω (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 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 23rd of September; and from the 23rd 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 Fig. 3. 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 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 six equal parts, and through the points of division draw right lines parallel to CD, for the beginning of the fines (of which one half are on one side of the semicircle and the other half on the other), and set the characters of the fines to their proper lines, as in the figure.

A 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, and latitude arch C, are Wright 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