Dialling, sometimes called Gaemonics, is a branch of the mixed mathematics, which treats of the construction of sun-dials. Its foundation is the astronomical theory of the sun's apparent motions; and from these its rules and operations have been deduced by the aid of geometry and trigonometry.

It may be supposed that in the early ages men would be satisfied with the divisions of the day marked by the rising and setting of the sun, and his greatest elevation above the horizon. When the gnomon, an upright pillar, the first of all astronomical instruments, had been applied to astronomy, the angular motion of its shadow might suggest that it was applicable to the division of the day.

The earliest mention that is made of a sun-dial is in the Bible. We read in the thirty-second chapter of Second Chronicles, that when Hezekiah was sick, he prayed to the Lord, and "He gave him a sign;" what that sign was is particularly told in Isaiah, chap. xxxviii. verse 8, "Behold I will bring again the shadow of the degrees which is gone down in the sun-dial of Ahaz ten degrees backward. So the sun returned ten degrees, by which degrees it was gone down." This was about 700 years before the Christian era.

The Chaldeans, among the earliest astronomers, as well as the other nations of Asia, divided the day into sixty parts. They had also a division of the day into twelve hours. The earliest of all sun-dials of which we have any certain knowledge was the Hemicycle or Hemisphere of their astronomer Berosus, who probably lived about 540 years B.C. This was the most simple and natural of all sun-dials, and therefore must have preceded the others. It has been the most generally used, but it could never be of any considerable dimensions, and was not susceptible of much accuracy. It, however, required no mathematical theory for its construction; a distinct notion of the spherical motion of the heavens was sufficient. To understand this dial, let us suppose a concave hemisphere, placed horizontally in an open place, with the concavity turned towards the zenith, and let a globule be suspended or fixed in any way at its centre; when the sun's centre rises above the horizon the shadow of the globule will enter the hemisphere, and throughout the day the shadow will trace on its inside the sun's diurnal parallel. Now, let the lines described by the shadow on the solstitial and equinoctial days be traced on the inside of the sphere, and also on as many intermediate days as may be, but in fact the tropical paths of the shadow will be sufficient; let each of these be divided into twelve equal parts, and let curve lines be drawn through corresponding points of division. These will be sensibly great circles on the inside of the hemisphere, and will two and two converge towards points in the meridian more or less distant. Here then is a sun-dial which will divide the period between sunrise and sunset into twelve portions, called temporary hours. The hours indicated by this dial were from its nature unequal, and varied from day to day. This defect, however, was not of much consequence when there were no machines for dividing time: a knowledge of geometry would have served to construct the dial so as to divide time equally, but at that remote period geometry was not known as a science.

The difference between the equinoctial hours, which were equal, and the unequal temporary hours, might not be at first observed, or might have been disregarded by a people who inhabited a climate where the elevation of the pole was small, and who, besides, attended only to the rising and setting of the sun, and, it may be, his passing the meridian, which at all times divided the day into parts nearly equal. They probably first divided each half of the day, as well as they could by estimation, into three equal parts, and these again into two. The dial of Berosus, although imperfect, was a great step towards improvement in the division of time, and was not at variance with the conjectural divisions, because in both ways the twelve hours had the same bounding limits, the rising and setting of the sun.

It would have been easy to have passed from the temporary to equal hours, which were marked by equal arcs on the equator; it was only necessary to describe semi-circles through these points of division which should cut each other in the poles of the world: these would have divided the parallels into arcs of fifteen degrees each: by carrying the divisions into fifteen degrees upon the summer tropic from the six o'clock hour circle to sunrise and sunset, there would have been got the excess of the longest above the equinoctial day. Thus a more exact notion of the length of the day and night at all seasons would have been obtained. But this would have appeared a great innovation, and not likely to have been adopted: every one knows how obstinate the common people are in adhering to old habits. Accordingly the construction of Berosus descended beyond the time of Hipparchus and Ptolemy. We find it in the year 900 among the Arabsians, who followed it in the construction of their dials, as appears from the work of Albategnius; and it is only since the invention of mechanical instruments for dividing time that it has altogether disappeared.

It is doubtful whether the Chaldeans had any mathematical theory for their dial, although it was of great simplicity. The facility, however, of its construction has probably made it the best known. Four have, in modern times, been recovered in Italy. One was discovered in the year 1746 at Tivoli. It has been supposed that this belonged to Cicero, who in one of his letters says that he had sent a dial of this kind to his villa near Tusculum. P. Zuzzeri, a Jesuit, has made this dial the subject of a memoir published at Venice. The second and third were found in 1751; one at Castel-Nuovo, and the other at Rignano; and a fourth was found in 1782 at Pompeii. It differs from the others in respect that the tropics are not expressly marked, and the equator alone is seen. G. H. Martini, the author of a dissertation, in the German language, on the Dials of the Ancients, supposes this last to be the oldest of the four, and that it is probably the primitive dial, such as was known to Berosus. Delambre, however, in his Histoire de l'Astronomie Ancienne, expresses himself of a contrary opinion, because it was more difficult to construct such a dial without than with the tropics. Martini says that the dial was made for the latitude of Memphis; it may therefore be the work of Egyptians, if it was not constructed in the school of Alexandria. It may be ascribed to the Egyptians, however, without attributing to them much knowledge in geometry; a terrestrial globe, on which were traced the equator and tropics, divided into degrees, and with its pole elevated to the latitude of the place, was all that was necessary. This being cut into two hemispheres horizontally, and great circles traced through the corresponding points of the two tropics, would furnish two convex models of the dial, from which any number of concave ones might easily be formed. The convex spherical model might even have had the parts cut away which were not to be shown on the concavity, and thus the construction of the dial might be a purely manual operation performed by an artist.

When the first dial was constructed, it was easy from that to make others. Thus, by the side of a Chaldean dial let a plane be fixed in any position, with a perpendicular gnomon; mark on the plane the position of the top of the shadow from hour to hour on the solstitial and equinoctial days, and join the corresponding horary points by lines, and it will appear that the three analogous points are always in a straight line; and thus there might be formed, without any theory, temporary dials of all kinds.

It may be supposed that every nation that cultivated astronomy had found means to divide time. It appears that the Egyptians had found in the heavens the means of attaining this object, but no sun-dial has been found among the antiquities of Egypt, and their sculptures give no indication of any having existed; they may, however, be buried in the sands, or overwhelmed in the midst of the vast ruins of their cities. It has been supposed that the numerous obelisks found everywhere in Egypt were erected in honour of the sun, and employed as gnomons. The famous circle of Osymandias might have determined the azimuths of the heavenly bodies, and thus have given the hours of the day or night.

Herodotus has recorded that the Greeks derived from the Babylonians the use of the pole and of the gnomon, and the division of the day into twelve parts; the pole was an instrument that showed the hour of the day. The Greeks by their geometry were in full possession of the means of constructing dials; and the Syntaxis of Ptolemy treats of their construction by means of his Analemma, an instrument by which the various problems of astronomy might be resolved.

The dials of the ancients marked the hour of the day by the shadow of a gnomon, but they had neither centres nor axes, and in some respects this was an advantage. The nature of their hours, which varied with the season, made it necessary to give such directions to the horary lines as prevented them from meeting in a point; they therefore were satisfied with finding the position of three points in each, although two would have been sufficient, for the lines were sensibly straight in all plane dials; the diurnal paths of the extremity of the shadow were hyperbolas, which intersected the horizon in the points of rising and setting, and of these there might be any number, but in general they were limited to the arcs of Cancer and Capricorn; the line for Aries and Libra was always straight, and furnished a middle point in the hour line. The constructions given by Ptolemy were sufficient for regular dials, the only ones he treats of. It is certain, however, that the ancients constructed vertical declining dials, for eight are yet in existence on the Tower of the Winds at Athens. Probably a part of the book of the Analemma is now lost, otherwise it would be remarkable that after Ptolemy had announced, in commencing, that he proposed to facilitate the construction of dials, he should not say a single word on the applications which may be made of his obscure methods and graphical operations, the object of which can hardly be seen, or the principles understood.

The ancient hours were called hectemoria by the Greeks, because they were sixth parts of the semidurnal arc. It does not appear that the nature of the hectemorial hour lines was ever considered by the ancients; indeed, practically, their precise nature was of little importance to them, and of no use in tracing their dials. On the sphere, when the pole is not considerably elevated, the lines which divide the semidurnal arcs into equal parts differ not much in appearance from arcs of great circles, as has already been remarked; and that they are not exactly so was first suggested by Clavius. Delambre has treated of them in his History of Astronomy, and in the Commissione des Temps for 1820. Mr Cadell has treated the subject at considerable length in the eighth volume of the Transactions of the Royal Society of Edinburgh, and, again, Mr Davis in the twelfth volume of the same work.

The most interesting monument of ancient gnomonics is the Tower of the Winds, which is yet in existence, and is figured and described by Stuart in his Antiquities of Athens. This is a regular octagon, on the faces of which the eight principal winds are represented, and over them are eight different dials; four regular, viz. east, west, north, and south; the other four have the intermediate directions. Vitruvius has described the Tower in the sixth chapter of his first book; but he has not said anything of the dials. This is remarkable, because he has described all the dials known in his time, and these are in every way more important than those of which he has mentioned the inventors. His silence gives reason to believe that the dials on the Tower have been an after thought, and are of a date later than his time, and much posterior to that of Andronicus Cyrresthes, author of the monument. It is impossible to say anything with precision as to the time when these eight dials were erected. From the impaired purity of style in the architecture of the Tower, it has been judged to be later than the time of Pericles and Anaxagoras. In their day the science of gnomonics was too little advanced to have served for the construction of the dials; however, as far as mathematical science was required, they might have been made in the time of Hipparchus, or later. It is certain they must have required a science of gnomonics, and therefore a trigonometry, unless they were made empirically by the help of the concave hemisphere of Berosus.

These dials, as described by Stuart, have the forms of those in the commentary given by Commandine on the Analemma of Ptolemy; everywhere the style is wanting, but the hole in which it had been inserted is visible in the marble. Its vertex is rarely in the axis of the hole, even in the regular dials, which are four in number. However, the height and the foot of the style are not indispensable data; they may be discovered from the principal dimensions of the dial. By a careful examination, Delambre ascertained that the south dial, the most important, was remarkably accurate, and that the height of the style was ten and a half English inches. The hours are not numbered, but they are temporary, and reckoned from sunrise to noon, and thence to sunset. The north dial is but a supplement of the south, on the same scale, and with the same style. There appear only two evening and two morning lines; and these, instead of being horary, are truly azimuthal; they only indicate the direction of the shadow. Two of these four lines are even a little too long, because that, instead of the hyperbolic arc, which should limit them, a straight line has been drawn. These slight defects are of no real importance.

The east dial is as exact as the south. It is very narrow, although it has a style double the length, almost nineteen and a fifth inches. The south-east presents the same agreement in all its parts. The height of the style is about twenty-five and a half inches. The inclination of the equinoctial to the horizontal line is $43^\circ 40'$, such as calculation makes it.

The north-east dial does not appear to have been traced with so much care, or at least success; the style is only about six and a half inches. The horary lines, three in number, are very oblique. The artist, however, may be excused, because the least error in the graphical operations might alter sensibly their length. Besides, this dial is the least important of them all; there is nothing which cannot be determined with more certainty on some one of the neighbouring dials. The three remaining dials, viz. the southwest, the west, and the north-west, are merely the counterparts of their opposite, and have not been figured in Stuart's book. The whole give the same general view of the ancient gnomonics as had been previously acquired by an examination of a dial found at Delos, and described by Delambre in the class of mathematical sciences of the Institute for 1814. They are however larger and better executed than that dial, and in their original position. Altogether they form the most complete monument of the practical gnomonics of the ancients.

There is another remarkable combination of dials in Athens, known by the name of the dials of Phaedrus. They are four in number, and are traced on the same block of pentelicque marble. It bears the following inscription: $\gamma \alpha \delta \epsilon \rho \sigma \eta \sigma \iota \sigma \tau \alpha \nu \alpha \mu \epsilon \tau \omega \nu$. From the form of the letters, M. Visconti, who communicated the designs of these dials to Delambre, has inferred that the monument may be of the second or third century of our era. Delambre described and carefully verified all the parts of these dials (Hist. de l'Astron. An. vol. ii.), and everywhere he found, if not all the conformity which he desired, at least as much as he could expect. The dials by themselves confirm what is manifest from those on the Tower of the Winds, that the Greeks had geometrical methods for vertical, and also for declining dials. To the ancient dials which have been here noticed we may add a singular one found at Portici in 1755, and described by the academicians of Naples. It has the figure of a bacon ham, and, like all the others, shows temporary hours. The theory of this dial is simple, but, considering the imperfect trigonometry of the Greeks, its construction by calculation might be laborious; probably it was made by the simple process, already described, from the Chaldean dial. The epoch of this dial is not known, nor is any dial of this kind noticed by Vitruvius, to whom we owe all the knowledge we have of other ancient dials. Vitruvius has attributed to Berosus the hemicycle hollowed in a square cut according to the inclination of the place (a description which Delambre says is incorrect). He has added, that Aristarchus of Samos invented the horologium called acrope (boat), or the hemisphere. This may have been the hemisphere of Berosus which Aristarchus taught the Greeks. He attributes also to him the disk in a plane. This may have been an equinoctial or a horizontal dial; these the Greeks could execute graphically without calculation. Vitruvius gives precepts for the construction of this dial, which are not, however, very intelligible now. He goes on to say, that The Arabian is due to Eudoxus, or, according to others, to Apollonius. This appears to have been a horizontal dial, on which circles of altitude were traced. Scopas of Syracuse invented the plinth or lambris, and he assigns to Parmenion the invention of an universal dial; and he goes on to speak of many others, the nature of which can only now be guessed. He indicates dials of suspension intended for travellers; these may have been like our ring-dials, but he does not explain them particularly. In concluding, he says that, to comprehend the theory of dials, it was necessary to know that of the Analemma.

The first sun-dial erected at Rome was in the year 290 B.C. Papirius Cursor had taken it from the Samnites. In 261 B.C. Valerius Messala placed in the forum a dial which he had taken at Catania, the latitude of which is five degrees less than that of Rome. In 164, Q. Marcius Philippus caused the first dial to be constructed; it was probably the work of a foreign artist, for no Roman has written anything on gnomonics. We have seen that the dial found at Pompeia was made for the latitude of Memphis, consequently it was less adapted to its position than that of Catania was to Rome. These facts prove that mathematical knowledge was not cultivated at Rome or in Italy.

It appears that sun-dials had been common in the days of Plautus. In a fragment of one of his comedies (The Bacchian) preserved by Aulus Gellius in his Attic Nights, he makes a parasite declaim against sun-dials in these terms:

Ut illum di perdant, primus qui horas repperit, Quoque adeo primus statuit hic solarium. Qui mihi comminuit misero articulatum diem, Nam me puerco uterum erat solarium Multum omnium istorum et verissimum, Ubi iste moneset esse, nisi quum nihil erat. Nunc etiam quod est, non est, nisi Sol luet. Itaque adeo jam oppletum est oppidum solaris. Major pars populi aridi rapit famae.

The gods confound the man who first found out How to distinguish hours! Confound him, too, Who in this place set up a sun-dial, To cut and hack my days so wretchedly Into small portions. When I was a boy, My belly was my sun-dial; one more sure, Truer, and more exact than any of them. This dial told me when 'twas proper time To go to dinner, when I had ought to eat. But now—days, why, even when I have, Don't fall-to unless the sun give leave. The town's all full of these confounded dials, The greatest part of its inhabitants, Shrunk up with hunger, creep along the streets.

Translation of Plautus by Thornton and Warner.

The Arabians, as they drew their knowledge of astronomy from the Greeks, so they also adopted their system of gnomonics without any material alteration. They studied the Analemma of Ptolemy, and found means of simplifying and diversifying his solutions. No nation attached more importance to gnomonics. Indeed, when there was no other way of knowing the hours, dials were in great request, particularly in southern climes; for in northern countries the shortness of the day for a great part of the year, and the uncertainty of sunshine, diminished much their utility. Abul-Hassan, who lived about the beginning of the thirteenth century, and one of their most modern writers, introduced among them equal hours, and taught how to trace them on dials; but he made but little use of them, and it does not appear that his invention was supported. He was also the author of a new way of describing the arcs of parallels. He invented a dial called khaphir, a word which means the hoof of a horse; the Greeks had no such dial. He made various others, to which particular names were given. He taught how to trace the progress of the shadow on plane, cylindrical, Dialling-conic, and spherical surfaces. The Greeks, however, had preceded him in the construction of dials on conic surfaces.

The writings of Abul-Hassan present a theory not to be found in the gnomonics of the Greeks, or any other Arabian writer, or in that of the moderns. He employed the properties of the conic sections to describe the arc of signs. It is true, Commandine and Clavius have also traced their arcs of signs by means deduced from the theory of these curves; but their mode of proceeding, very obscure, is not that of the Arabian. We do not find among them the universal dial of Regiomontanus, nor the analemmatic dials, which give the hour by the sun's altitude; nor had they any idea of the angles at the centre of different dials. We find these angles and different other novelties in the first European authors who have written on gnomonics; but these geometers do not say that they were the inventors of these happy innovations. There is therefore in the history of gnomonics a blank which cannot now be filled up. We see a marked progress, without knowing precisely to whom we owe the obligation. These discoveries probably preceded the invention of printing. The original works are lost, but tradition has handed down to us what was most useful in them. The oldest of these writers, Munster and Schoner, have affected to imitate the Arabs, in suppressing all demonstrations, as had been done by Albategnius and Ibn-Jounis. They have given constructions resting on principles nowhere demonstrated, and hence has arisen an obscurity not easily to be dissipated.

Among the first of the moderns who have treated of gnomonics may be reckoned John Stabius, Andrew Striborius, and John Werner, astronomers of the fifteenth century. Their works however have never been printed. To these may be added John Schoner, an astronomer of the sixteenth century, who gave in 1515 a work entitled Horarii cylindri Canonice, in which he treats of the construction of cylindrical dials. His other works on dialling were afterwards published by Andrew Schoner, his son. Some notions on modern gnomonics were given in a treatise on the Roman calendar, printed in 1518. This writer treats of a general horary square, after Regiomontanus, which supposes equal hours; so that these appear to have been established in the middle of the fifteenth century, and probably earlier. It appears from Stöffler, that in his time a new system of gnomonics had been formed, the author of which however is unknown.

Sebastian Munster, a cordelier, who had embraced the opinions of Luther, published at Basle, in 1531, a work with this title, Compositio Horologiorum in plano muro, trunco, anulo concavo cylindro, et variis quadrantibus, &c.; and again, in 1533, Horologographia post priorem editionem. The author was born at Ingelheim in 1489, and died of the plague at Basle in 1532. He employs equinoctial hours, and supposes great circles to pass through the pole of the world perpendicular to the equator, dividing it into twenty-four equal parts. A plane which cuts all these circles shows the hours by its intersection with the different planes. Here then is a complete revolution in the theory of dialling; and doubtless the change had subsisted for some time; for he speaks with contempt of these vulgar constructors who, without any theory, blindly followed the rules and tables which had been given to them. It is very remarkable that so total a change should have taken place without the author being known; and not less so that the first author who printed a book on gnomonics should have given all his precepts without demonstration. Equinoctial hours were now substituted for temporary hours, a centre was given to the dial, an axis was substituted for the perpendicular style; centres and dividing radii were now introduced; all these changes could only have been made by a skilful geometer. Among the inventions of Munster was a sun-dial. It is not certain that he was the author of all the inventions which he describes, but at any rate he was the first to publish them.

The next writer to be noticed was Andrew Schoner. His book has the date 1562, with this title: *Gnomonice Andreae Schoneri Nordbergensis, hoc est de descriptionibus horologiorum scieriorum omnium generis, &c. omnia recens nata et edita.* By the concluding words, the author seems to assert that all contained in his book was his own invention. This, however, could not be true of anything contained also in the first or second edition of Munster's book, which preceded his by twenty-nine years. Schoner speaks of many learned men that had gone before him in the same science, and regrets that their labours were unknown. He cites Regiomontanus and others, but he does not name Munster. He was an enthusiast on his particular subject, and affirmed that dials could no more be dispensed with than meat and drink. He was an obscure writer, and communicated nothing new, unless in treating of inclining-declining dials, of which no mention is made by Munster, nor among the ancients. In the same century there came out other treatises on dialling, as by Benedict in 1574, Elie Vinet in 1564, John of Padua in 1582, and Valentino Pini in 1598. These do not seem to have added much to what was previously known.

In the following century we have a work on dialling by an astronomer, La Hire. The first edition came out in 1682, another in 1698. Montucla pronounces this book obscure, and Delambre agrees in his opinion; for La Hire merely indicates his demonstrations, and his constructions are so complicated that they are with difficulty understood. He never employs calculation, unless indeed in an appendix, which might be left out, without altering the character of the work. All his operations may be performed without any idea of even plane trigonometry. He employs only compasses, the rule, and the plumb-line. A dial may be traced by his directions, without knowing whether it be horizontal, vertical, east, west, declining, or inclining; and without the latitude of the place or the sun's declination being known. This plan is not the best for practice; but it was new, and therefore remarkable and deserving of notice. His constructions are often ingenious; but they have the essential fault of not admitting of much accuracy.

There is a treatise on gnomonics by Ozanam, of date 1693. It also forms a part of his *Cours de Mathematiques,* printed in 1697. This, in the opinion of Delambre, is a perspicuous work, and much superior for practice to that of La Hire.

The subject of dialling was greatly agitated in the course of the seventeenth century by all writers on astronomy. Thus a quarto volume of 800 pages, entirely on gnomonics, forms a part of the works of Clavius, printed in 1612. This may be supposed to contain all that was known before his time, as well as his own inventions. In this work he has demonstrated both the theory and practice, after the manner of the ancient mathematicians.

The eighteenth century produced some writers on dialling; but clocks and watches had by this time superseded sun-dials; their theory was well known and explained in all works on astronomy. The art of constructing them was now considered as a mathematical recreation. At this point we shall conclude their history.

The principal writers on dials and dialling are the following:—Ptolemy, restoration of his work on the *Analemma,* by Commandine; Vitruvius, in his *Architecture*; Sebastian Munster, in his *Horologiographia*; Orentius Fincus, *De Horologis Solaribus,* &c.; Mutio Oddi da Urbino, *Gli Horologi Solari nelle Superficie Plane*; Dreyander, *De Horologiorum varia Compositione*; Conrad Gesner's *Pan-

**General Principles of Dialling.**

1. The theory of dialling, to be fully understood, requires an acquaintance with some of the more simple doctrines of astronomy; also of the elements of geometry and plane and spherical trigonometry. However, a less extensive knowledge of pure mathematics will be sufficient for the construction of the more simple and common dials. A correct notion of the nature of an angle, and a knowledge of the elementary problems of practical geometry, viz. the drawing of parallels, perpendiculars, &c., and how to make an angle of any proposed number of degrees, also to measure an angle, will suffice for the mere geometrical construction of dials. The instruments required are compasses, a scale of chords, or a protractor for measuring angles, and a straight-edged rule. A globe is useful in giving distinct notions of the doctrine of the sphere, but may be dispensed with in the practice of dialling. A dialling scale facilitates the practice; and therefore instructions will be given by which it may be constructed.

2. The preliminary astronomical knowledge has been delivered in the article *Astronomy.* To avoid repetition, we must direct the attention of the reader to that article. He will find it in the first and second chapters of the Second Part; but it is only the definition of the circles of the sphere, and the description of the apparent motions, to which we shall have occasion to refer.

3. The apparent diurnal motion of the starry heavens is perfectly uniform. The sun's apparent diurnal motion about the earth's axis, however, deviates a little from perfect equality, by reason of his unequal angular motion in the ecliptic, and its obliquity to the equator; but these need not be attended to in the construction of a sun-dial. Their joint effect produces the *equation of time,* a correction which must always be applied to the time it indicates. We have given a table of its quantity for every day in the year at the end. Atmospheric refraction likewise might be taken into account, but in practice it is neglected. The construction of a dial is a graphical operation, subject to Dialling.

The imperfection of instruments and their application. The object in view is a practical method of finding a measure of time for the ordinary affairs of life; and for this, extreme accuracy is not required.

4. The mean distance of the earth from the sun is 23,984 times its semidiameter. Hence we may infer that all the phenomena of the solar motions, as seen from any part of the earth's surface, will be almost exactly the same as if they were seen from its centre, the difference being inappreciable by the nicest instrument; just as the apparent position of a spire or other object four miles distant, in respect of neighbouring objects, will not be in the least altered by a change of a foot in the position of the eye to either side of the point from which the object was first viewed. Hence it follows, that if we place in sunshine a globe on which the circles of the sphere are delineated, or, instead of a globe, a skeleton sphere, such as is represented in fig. 1 and 2 of Plate CLXXXV., formed of twenty-four equal wire circles, which all pass through the extremities of a common diameter Pp, formed of some solid matter; and if these circles, in radiating from their intersections P, p, make equal angles round these points, just as the meridians do on the common terrestrial globe; then, if the axis Pp be placed parallel to the earth's axis; the shadow projected by the axis will fall on the wires one after another at intervals of an hour, because the apparent angular motion of the sun about the axis Pp will be uniform, just as it is about the imaginary axis of the earth.

This very simple instrument, if correctly made, and placed in a fixed position, with its axis directed to the pole of the world, which is near the pole star, and, moreover, with one of its circles in the plane of the meridian (that is, in a direction due north and south), would serve to divide the day into twenty-four equal portions. This is the most natural and elementary of all dials.

5. We have supposed our dial to be composed entirely of wires or material circles; and although the hour of the day will be known if the shadow of the axis be observed to fall on one of the circles, there will be an uncertainty as to the exact time when the shadow is passing between two circles. To remedy this, let us suppose that a plane ABCD, fig. 1, or AFCG, fig. 2, of some solid matter, is placed within the sphere, and passes through E its centre. The shadow of the axis PE will now be projected on this plane surface, and will be seen at all times in its progress from wire to wire. If now straight lines be drawn from the centre E to the points in which the plane DAB meets the wire circles, and the hours be marked on them as in the figures, it will be noon when the shadow falls along the line EXII, and eleven in the forenoon when it falls on EXI, and one in the afternoon when it falls on EI, and so on throughout the day. As the shadow on the plane will always be visible, any time intermediate between two hours may be guessed at nearly by the position of the shadow in respect to the lines on which it falls at the preceding and following hours.

6. It is evident that the plane BAD may have any position whatever within the sphere; the dial will still indicate the hours, supposing always that the axis Pp is directed towards the north and south poles of the heavens. We may assume that the plane DAB is horizontal; then the lines drawn on the plane from the centre to the points in which it cuts the wires will constitute a horizontal dial. Such a dial, constructed as has been explained, for London, in latitude 51°35', is shown by fig. 1. On the longest day the sun's centre rises at London forty-four minutes before four, and sets seventeen minutes after eight; therefore the extreme hours marked on the dial are IV. and VIII.

7. If again we suppose the plane AFG to be vertical, as in fig. 2, and that it faces the south, then we have a south

dial. In this case the plane of the dial cannot be illuminated before six in the morning, nor after six in the evening; therefore these are the extreme hours marked on it.

8. If the plane, which we have supposed to be in the inside of the sphere, were perpendicular to the axis, it would then pass through the equator in the heavens, and the result would be an equinoctial dial. In this case, the circle which forms the circumference of the dial would evidently be divided into equal parts by the wire circles; therefore the shadow would move with an uniform angular motion about the centre, just as the hour-hand of a watch does; and, moreover, it would be illuminated on the north side only in the summer, and on the south side in the winter. In the same way, by giving different positions to the plane, the student of this subject may get an exact conception of every other kind of dial.

9. It is sufficiently obvious, that when the points in which the circles cut the plane BAD (fig. 1) or FCG (fig. 2) have been once determined, and lines drawn from them to the centre, the circles are no longer necessary to the dial, which is then simply a plane; and although we have employed the hypothesis of material circles to represent the circles of the sphere, as being a convenient way of treating the subject, yet the intersection of the circles and the plane of the dial may be determined either by a geometrical construction or a numerical calculation.

10. It will now be proper to define certain terms which will frequently occur in treating this subject.

The plane ABCD (fig. 1), on which the lines that indicate the hours are drawn, is called the plane of the dial.

The material line PE, which rises out of the plane of the dial, and projects a shadow on it, and thereby indicates the hours, is called the axis of the dial. Instead of being a line or rod, it is sometimes the edge of a thin flat plate (as in fig. 3, 4, &c.) fixed on the plane of the dial; it is sometimes called the gnomon, also the style, of the dial.

The circles PpP, PpP, PpP, &c. also the circles of the celestial sphere which they represent, are hour circles (Astronomy, Part II. chap. 1). Of these, that which passes through the sun at noon is the meridian; the others are named from the hours, as the six o'clock hour circle, &c.

The lines EXI, EXII, EI, &c. which are the intersections of the hour circles and the plane of the dial, are called hour lines; that in the plane of the meridian, which indicates the hour of noon, is the meridian line.

The common intersection of the hour lines is called the centre of the dial. The angles which the hour lines make with the meridian lines are the hour angles at the centre of the dial; and the spherical angles which the hour circles make with the meridian are the hour angles at the pole. These are the same as the angles which the planes of the circles make with the meridian.

A horizontal dial is that which is delineated on a horizontal plane.

A vertical dial is that on a vertical plane. These may be north, south, east, or west, according to the quarter which they face.

Vertical declining dials are such vertical dials as do not face any of the cardinal points.

Inclining or oblique dials are those traced on planes which make oblique angles with the horizon. They are reclining when they lean backwards from an observer, and proclining when they project forward.

An equinoctial dial is that whose plane is parallel to the equator, or perpendicular to the earth's axis.

A polar dial is that traced on a plane perpendicular to the meridian, and passing through the poles.

Construction of Horizontal Dials by a Globe.

11. The manner in which the general principles of dial- Dialling have been explained shows directly how a dial may be constructed by means of a terrestrial globe. Let PEP be the axis of a globe (fig. 1), XII. E XII. the equator, and BAD its horizon. Suppose now the globe to be rectified for the latitude of any place, London for instance, which is in latitude $51^\circ$; this is done by placing it in such a position that the arc of the brazen meridian between the pole and horizon is equal to the latitude. Then it is manifest, that if at London the brazen meridian of the globe be placed in the plane of the celestial meridian, its axis will point to the poles of the world; and if any one of the meridians on the globe be brought under the brazen meridian, all the meridians, supposing there are twenty-four, will correspond to hour circles in the heavens. In short, the meridians on the globe will correspond to the sphere formed of wire circles, its wooden horizon to the plane inserted within the sphere formed by the circles, and its axis to the axis of that sphere. Assuming now the line drawn from E, the centre of the globe, to the north point on the horizon as the twelve o'clock hour line, the angle XII. E I., which the one o'clock line makes with it at the centre of the dial, will have for its measure the arc of the horizon between the arc of the meridian P XII. and the arc P L of the next hour circle. This will be about $11\frac{3}{4}$. In the very same way the hour angle XII. E II. at the centre will be measured by the arc of the horizon between the brazen meridian and the arc P II. of the two o'clock hour circle, and so on. The angles made by the forenoon hour lines and the meridian line are equal to those made by the corresponding afternoon hour lines and the meridian line; that is, the angle XI. E XII. to the angle I. E XII. and X. E XII. to II. E XII. &c. The morning and evening hour lines for VI. will both be perpendicular to the meridian; and it is easy to see that the morning hour line for V. will be the continuation of the afternoon hour line for V.; and that the hour lines for IV. and III. in the morning will be the prolongations of the lines for the same hours in the afternoon. In the same way the evening hours after VI. are determined from the morning hours.

12. If the globe have more than twenty-four circles (some have thirty-six), place the first meridian, which is that of London, under the brazen meridian, and set the moveable hour index to XII. at noon; then turn the globe westward until the index points successively to I. II. III. IV. V. and VI. in the afternoon, or until $15^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $75^\circ$, and $90^\circ$ of the equator pass under the brazen meridian; and it will appear that the first meridian of the globe cuts the horizon in the following number of degrees from the north toward the east, viz. $11\frac{3}{4}$, $24\frac{1}{4}$, $38\frac{1}{4}$, $53\frac{1}{4}$, $71\frac{1}{4}$, and $90^\circ$; these are the respective distances of the above hours from XII. on the plane of the horizon.

13. To transfer these and the rest of the hours to the plane of a horizontal dial (Plate CLXXXV., fig. 3), draw parallel straight lines ae, bd, distant from each other by the thickness of the style or gnomon; these, or the space between them, will mark the meridian or twelve o'clock line on the dial. Cross the double meridian line by the perpendicular gahb, and this will be the hour lines for VI. in the morning and evening.

About a and b as centres, with any convenient radius, describe quadrants of circles eg, fh, and divide each into ninety 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 convenient to have the centres of these quadrants at some distance from the centre of the surface on which the dial is delineated, to admit of more space for the hour lines about noon.

Lay a rule over b, and draw the hour line of I. through $11\frac{3}{4}$ in the quadrant; the hour line of II. through $24\frac{1}{4}$; of III. through $38\frac{1}{4}$; of IV. through $53\frac{1}{4}$; and of V. through $71\frac{1}{4}$. Again, lay the ruler to the centre a of the quadrant eg, and through the like divisions or degrees of that quadrant, viz. $11\frac{3}{4}$, $24\frac{1}{4}$, $38\frac{1}{4}$, $53\frac{1}{4}$, and $71\frac{1}{4}$, draw the forenoon hours of XI. IX. VIII. VII. and VI. Extend the hour lines of IV. and V. in the afternoon through b across the dial, and the prolongations will be the hour lines of the same morning hours. In like manner, the prolongation of the hour lines of VIII. VII. of the morning hours through a will give the hour lines of the same evening hours.

To form the style, draw a line from a through that degree of the quadrant eg which is the latitude of the place, in the present case $51^\circ$. This line will determine the elevation of the style, which is represented in the figure by the shaded triangle, supposed to be of solid matter, and lying on the surface of the dial. The thickness of the style must be equal to ab, the breadth between the meridian lines. Let it now be placed truly upright on the dial, so as to stand on the space between them, and the dial will be finished. The style should be of such a height that its shadow at midsummer shall reach to the space on which the hours are marked.

Note. The trouble of dividing the quadrants will be avoided if you have a scale of chords, or protractor, and know how to lay down by them an angle of any given number of degrees. This is one of the simplest problems of practical geometry.

To construct a direct South Dial by a Globe.

14. Let PFDpDF (fig. 2, Plate CLXXXV.) be a globe on which twenty-four meridians or hour circles are marked, P and p being the north and south poles, and XII. E XII. the equator. Suppose now Pp, the axis of the globe, to coincide with the axis of the world; and some one meridian on it, as PFDp, with the meridian of the place for which the dial is to serve; then if the globe be cut through the centre E, by a vertical plane AFGC, in an east and west direction, it is manifest that straight lines drawn from the centre to the points on which this plane meets the hour circles on the globe will be the hour lines of a south or north dial. The figure shows a south dial on the lower half of the circle, which is the common section of the circle and cutting plane, the hour lines being E VI. E VII. &c.; and Ep, the southern half of the axis of the globe, is the axis of the dial. A north dial would just be its counterpart on the upper and opposite side of the plane.

15. By a comparison of fig. 1 and 2, it will appear that, as in fig. 1, the angle which the axis PE makes with DAB, the plane of the dial, is the latitude of the place; so in fig. 2, the angle which the axis pE makes with the plane FAGC is the complement of the latitude, or what it wants of $90^\circ$. Hence it follows that a south dial for any given place will serve as a horizontal dial at a place whose latitude is the complement of that of the given place, and vice versa. Indeed it is easy to see that, if a dial be correctly constructed on any given plane for a given place, there will be some other place, which may be found, where that dial will serve as a horizontal dial. And the reason is this; whatever be the position of a plane, there must be some place on the earth whose horizon is parallel to that plane. Now all dials whose planes are parallel, wherever they be situated on the earth's surface, have their axes parallel; therefore, at the same instant the angular motion of the shadow will be the same on them all. If the planes of dials which are parallel be perpendicular to the same meridian, they will indicate the same hour at the same instant. If they are not, the difference between the hours they indicate at any instant will be different, because of their difference of longitude.

16. From what has been explained, it is evident how the Dialling.

Draw two vertical lines \(a\), \(b\), on the plane of the dial, for the twelve o'clock hour lines, so that the distance between them may be exactly the thickness of the style; cross them perpendicularly by the line VI. \(a\) VI.; this is the six o'clock hour line. On \(a\) and \(b\) as centres with any convenient distance describe quadrants, and divide each into ninety degrees. Rectify now the globe for the complement of the latitude of the place; bring a meridian or hour circle on the globe to the north point of the horizon, and (supposing there are twenty-four hour circles) note the degrees reckoned from the north on the horizon, in which it is intersected by the hour circles; these will be the angles which the hour lines make with the double meridian line at \(a\) and \(b\) the centre of the dial. Proceed in laying them down exactly as in making a horizontal dial; and it will only be necessary to lay down the hours from noon to six in the evening and the morning, because a vertical plane facing the south can never be illuminated earlier nor later.

17. We have explained the application of a globe to the construction of dials, less with a view to recommend it in practice, than to elucidate the theory. Other methods are better, because, with the best mounted globe, the hour angles at the centre of the dial cannot be determined with much accuracy. It will be better to compute the angles and lay them down from a scale of chords; or we may use a geometrical construction; or, lastly, we may lay down the hour lines by means of a dialling scale, the easiest method of any.

Geometrical Theory of Dialling.

18. It has been shown (art. 3 and 4) that if at any place a straight rod or wire be parallel to the earth's axis, in which position it may be considered as coinciding with the axis of the heavens, the angular motion of the sun about that rod will appear to be perfectly uniform throughout the day, and therefore the shadowy space within which the sun's light is wholly or in part intercepted by the rod will also turn uniformly about it. Hence, to construct a sun-dial, it is only necessary to place the rod so that its shadow may fall on a surface of any kind, and to trace on that surface, at any equal intervals of time (hours for instance), the line shown on the surface by the shadow at the instants which separate these intervals. These lines, numbered according to the hours, will serve to show time by the shadow at all seasons. This is the most simple way of constructing a dial, but it supposes that we have the means of dividing time into equal intervals, a thing indeed easy since the invention of clocks and watches.

To reduce the subject to a geometrical theory, let us suppose that, in the adjoining figure, OCF is the rod (considered as a material straight line) which projects the shadow, and that it is perpendicular to the plane of a circle APB at C its centre, and let CA be the position of the shadow on the circle at noon. Then, if the circumference be divided into twenty-four equal parts, beginning from A, as at 1, 2, &c., and lines be drawn from the centre through the points of division, it is manifest that the shadow will fall on the lines at the hours marked on them, and the circle will serve as a dial. And since its plane coincides with that of the equator in the heavens, it is an Equinoctial Dial.

Equinoctial Dial.

19. To construct this dial (Plate CLXXXVII. fig. 6), on C, a point in the middle of its face, as a centre, describe a circle ABDE; divide the circumference into twenty-four equal parts, and from the points of division draw straight lines to the centre; these will be the hour lines; mark the hours on them as in the figure; fix a thin and straight wire in the centre, perpendicular to the face of the dial, for its style, and place it with the style directed to the pole, and the twelve o'clock hour line in the plane of the meridian; and when the dial is illuminated by the sun, the hours will be indicated by the shadow of the style.

20. In our climate the superior side of an equinoctial dial is illuminated when the sun is on the north side of the equator, and the inferior or opposite when he is on the south side. On the equinoctial days neither side will be illuminated, because the sun is in the plane of the dial. However, if it have a ledge rising a little above the opposite sides, and the hour lines be continued on the ledge, the shadow will fall on its inside, and indicate the hour, although the direction of the sun's rays be almost parallel to its face.

21. To set up an equinoctial dial, direct the straight edge of a vertical plane towards the polar star, which is about one degree thirty-six minutes from the pole of the world; it will then nearly coincide with the axis of the sphere; but for greater accuracy the edge may be directed to the star when highest or lowest, and a line drawn on the plane, making with its edge the above angle. This will be in the true direction of the axis of the dial, the plane of which must be placed perpendicular to the line so determined, and the six o'clock hours in a horizontal line; the dial will then be properly placed.

22. An equinoctial dial may be set up about the time of either solstice without knowing either the latitude of the place or the direction of the meridian, from this property, that when truly placed, the extremity of the shadow of the axis will then describe a circle on the plane of the dial, the centre being the common intersection of the hour lines. If, therefore, the dial be placed nearly in a true position, with the six o'clock hour line exactly horizontal, by observing the line which is the extremity of the path of the shadow, it will be seen in which way the deviation from the true position lies, and by repeated adjustment it may be truly placed. About the equinoxes the daily path will deviate somewhat from a circle, by reason of the quick change in the sun's declination.

Horizontal Dial.

23. Let GEDH (see the adjoining figure) be a horizontal plane on which a dial is to be delineated, and let LOK be a meridian line on this plane; let OCF be a material line or rod in the plane of the meridian, which meets the horizontal plane in O, and makes with OK an angle equal to the latitude of the dial; this rod will be directed to the pole, and will be the axis of the dial. Let BAP be an equinoctial dial, having OCF for its axis, and C for its centre; and let CA, the meridian line on this dial, meet the meridian line on the horizontal plane in K. As has been explained, the plane of the shadow will turn uniformly about the axis OCF, meeting the equinoctial plane in some line CPQ, and the horizontal plane in a correspond- Let C1, C2, &c. be the hour lines after noon on the equatorial dial, and O I, O II, &c., the corresponding hour lines on the horizontal dial; the former will make with the meridian line OAK angles proportional to the time from noon, and will be known when the hour is given, 15 degrees being reckoned an hour. The plane of the equinoctial dial being supposed extended to meet the horizontal plane in the line QK, which will be at right angles to the meridian lines CK, OK, the problem to be now resolved is, to find the hour angle KOQ at the centre of the horizontal dial corresponding to any given angle KCQ at the centre of the equinoctial dial, which measures the time from noon.

The triangles CKQ on the equinoctial plane, and OKQ on the horizontal plane, have a common side KQ, and each a right angle at K; therefore, by trigonometry,

\[ \frac{CK}{KQ} = \tan \text{rad. tan. KCQ}, \]

and \( KQ : OK = \tan. KOQ : \tan. rad. \)

therefore \( \tan. CK : OK = \tan. KOQ : \tan. KCQ. \)

But in the triangle COK, right angled at C,

\[ \frac{CK}{OK} = \tan. \sin. OCK, \]

therefore \( \tan. rad. sin. OCK = \tan. KOQ : \tan. KCQ. \)

Hence we have this general theorem or rule for computing the hour angles at the centre of a horizontal dial: As radius to the sine of the latitude, so is the tangent of the hour from noon (reckoning 15° to an hour) to the tangent of the hour angle at the centre of a horizontal dial; or, putting \( x \) for the hour from noon in degrees, \( y \) for the hour angle at the centre of the dial, \( L \) for the latitude of the place,

\[ \tan. y = \frac{\tan. x \sin. L}{\tan. rad.}. \]

We have supposed GIIDE to be a horizontal plane, but the formula evidently applies to any plane whatever perpendicular to the meridian: all that is required for its application is the angle which the axis OF makes with the meridian line OK on the plane.

As an example, let it be required to find the hour angle at the centre of a horizontal dial for XI. or I o'clock for the latitude of London 51° 30'. In this case the hour angle from noon at the pole is 15°.

| Latitude | Mor. H. XI. | Mor. H. X. | Mor. H. IX. | M. H. VIII. | M. H. VII. | M. H. VI. | |----------|-------------|------------|-------------|-------------|------------|-----------| | 50° 0' | 11° 36' | 23° 51' | 37° 27' | 53° 0' | 70° 43' | 90° 0' | | 50° 30' | 11° 41' | 24° 1' | 37° 39' | 53° 12' | 70° 51' | 90° 0' | | 51° 0' | 11° 46' | 24° 10' | 37° 51' | 53° 23' | 70° 59' | 90° 0' | | 51° 30' | 11° 51' | 24° 19' | 38° 3' | 53° 35' | 71° 6' | 90° 0' | | 52° 0' | 11° 55' | 24° 28' | 38° 14' | 53° 46' | 71° 13' | 90° 0' | | 52° 30' | 12° 0' | 24° 37' | 38° 25' | 53° 57' | 71° 20' | 90° 0' | | 53° 0' | 12° 5' | 24° 45' | 38° 37' | 54° 8' | 71° 27' | 90° 0' | | 53° 30' | 12° 9' | 24° 54' | 38° 48' | 54° 19' | 71° 34' | 90° 0' | | 54° 0' | 12° 14' | 25° 2' | 38° 58' | 54° 29' | 71° 40' | 90° 0' | | 54° 30' | 12° 18' | 25° 10' | 39° 9' | 54° 39' | 71° 47' | 90° 0' | | 55° 0' | 12° 23' | 25° 19' | 39° 19' | 54° 49' | 71° 53' | 90° 0' | | 55° 30' | 12° 27' | 25° 27' | 39° 30' | 54° 59' | 71° 59' | 90° 0' | | 56° 0' | 12° 31' | 25° 35' | 39° 40' | 55° 9' | 72° 5' | 90° 0' | | 56° 30' | 12° 36' | 25° 43' | 39° 50' | 55° 18' | 72° 11' | 90° 0' | | 57° 0' | 12° 40' | 25° 50' | 39° 59' | 55° 27' | 72° 17' | 90° 0' | | 57° 30' | 12° 44' | 25° 58' | 40° 9' | 55° 36' | 72° 22' | 90° 0' | | 58° 0' | 12° 48' | 26° 5' | 40° 18' | 55° 45' | 72° 28' | 90° 0' | | 58° 30' | 12° 52' | 26° 13' | 40° 27' | 55° 54' | 72° 33' | 90° 0' | | 59° 0' | 12° 56' | 26° 20' | 40° 36' | 56° 2' | 72° 39' | 90° 0' | | 59° 30' | 13° 0' | 26° 27' | 45° 45' | 56° 11' | 72° 44' | 90° 0' |

In this table the angles which the hour lines of V. IV. III. make with the meridian are not put down, because they are the same as those of the like hours in the afternoon, the former being the continuation of the latter: a similar remark applies to the hour lines of VII. VIII. IX. in the evening, which are the continuation of the like morning hours.

The use of the table is obvious. For example, if the angles of the hour lines at the centre of a dial for 56° of latitude be required, the table shows that the hour lines of XI. and I. make angles of 12° 31' with the meridian, and the hour lines of X. and II. angles of 25° 35', and so on. If the latitude is not exactly contained in the table, proportional parts may be taken, without any sensible error. Thus, if the hour line angles of a horizontal dial for 55° 42' be required, the table gives 12° 37' for the angle of the XI. or I.o'clock line in lat. 55° 30', and 12° 31' in lat. 56°: the difference of the angles corresponding to 30' difference of latitude is 4': Now the proposed latitude exceeds the least of these two by 12', therefore we state this proposition 30 : 12 = \( \frac{4 \times 12}{30} = 146 \). Hence we find that 12' of addition to the latitude gives 1° 6', or nearly 2' of addition to the hour angle, which will therefore be 12° 29'. Geometrical Construction of a Horizontal Dial.

25. The formula of art. 23, namely, that the tangent of the angle which any hour line makes with the meridian line is a fourth proportional to radius, the sine of the latitude, and the tangent of the hour angle at the pole (that is, the hour from noon in degrees), reduces the construction of a dial to this geometrical problem.

Having given any angle, to find another whose tangent shall have to that of the former a given ratio. This problem may be resolved graphically in various ways, and in as many ways may the hour lines on a dial be determined.

First Construction.

26. Draw two parallel straight lines CM, CM' (Plate CLXXXV., fig. 5) at a distance equal to the thickness of the style for the double meridian line, and cross them at right angles by the six o'clock hour line VI. C'C VI. Make a right angled triangle HCK (fig. 6), having K a right angle, and C equal to the angle which the axis of the dial is to make with its plane, that is, to the latitude of the place, which for London is 51°. About C and C' (in fig. 5) as centres with a radius equal to CH, the hypothenuse of the triangle CHK (fig. 6), describe the quadrants M6, M'6; and about the same centres, with a radius equal to HK, the side of the triangle opposite to C, describe concentric quadrants, as in the figure; divide these each into six equal parts at the points 11, 10, 9, &c. on one side of the meridian line, and at 1, 2, &c. on the other side. From the points of division in the inner quadrant draw lines parallel to the meridian line, and from those in the outer quadrant draw lines parallel to the six o'clock line, so that the lines drawn from the corresponding divisions in the concentric quadrants may meet, viz. those from 11 in a, from 10 in b, &c. on one side, and those from 2 in a, from 3 in b, &c. on the other side; draw straight lines C'XI. C I. through the points a and a on each side of the meridian line, and these will be the hour lines of XI. and I. In like manner, draw straight lines C'X. C II. through b and b, and these will be the hour lines of X. and II. and so on. Extend the morning hour lines of VII. VIII. and IX. across the dial, and the prolongations will be the hour lines of the same hours in the evening; and in like manner form the hour lines of V. IV. III. in the morning from the same afternoon hours.

The style CLM (fig. 6) must have the angle at C equal to the latitude, and may be formed from the triangle CHK (see art. 23).

27. To demonstrate the truth of this construction, let CB (fig. 5) be any one of the hour lines determined by the intersection of DB a parallel, and EBA a perpendicular to the meridian line, drawn from corresponding points D, E in the concentric quadrants. The points C, D, E will be in a straight line; and from similar triangles,

\[ AE : AB = CE : CD = CH : HK \] (fig. 6).

Now AE is to AB as the tangent of ACE, the hour from noon (in degrees) to the tangent of ACB, the angle made by the hour line and the meridian, and CH is to HK as radius to the sine of the latitude: Therefore (art. 23) the hour line is rightly determined.

Second Construction.

28. In the following figure let C be the centre of the dial, CM the XII. and CA the VI. o'clock hour lines. Make the angle MCB equal to the latitude of the place. Take CB of any length, and draw BA perpendicular to CA; make CM equal to CB, and join MA; bisect MA in D; draw DO perpendicular to MA, and equal to DM or DA; join OM, OA; and about O as a centre, with OM or OA (which are equal) as a radius, describe the quadrant M3A, Dialling, and divide it into six equal parts at the points 1, 2, 3, 4, 5. These are to correspond to hours; but each may be subdivided into quarters, or other smaller parts of an hour.

Now, to determine any hour line, for example, that of II. o'clock; draw a straight line OH from O to the division 2 of the quadrant, meeting MA in E; next draw a straight line CK from C through E, and this will be the hour line for II. Exactly in the same way all the other hour lines for the afternoon, and by a like construction on the other side of the meridian those for the forenoon, may be found.

29. To demonstrate this construction, draw MK perpendicular to CM, meeting CE in K, and MH perpendicular to OM, meeting OE in H. The triangles OAE, HME, are manifestly equiangular; so also are the triangles CAE, KME;

\[ \text{therefore } MH : OA = ME : AE = MK : AC. \]

Now, by trigonometry, making MO radius,

\[ MH : OA = \tan. MOH : \tan. OMA. \]

And making MC radius; MK : AC = tan. MCK : tan. AMC.

Therefore tan. MOH : tan. OMA = tan. MCK : tan. AMC.

And by alter tan. MOH : tan. MCK = tan. OMA : tan. AMC.

That is, because tan. OMA = radius,

\[ \tan. MOE : \tan. MCE = \text{rad.} : \tan. AMC. \]

But rad. : tan. AMC = CM or CB : CA = rad. : sin. CMA;

therefore, radius is to the sine of CMA, the latitude, as the tangent of MOA, the hour angle from noon, to the tangent of MCE; hence (art. 23) CE is the hour line required.

30. If OD be made the radius of a circle, DE will evidently be the tangent of the hour angle, reckoned from three o'clock; and in general, the segment between the middle of AM and the point in which any hour line cuts it, will be the tangent of the interval in time between that hour and three o'clock. If therefore these tangents be laid down on a scale, which may be called the scale of hours, it will serve for the construction of all dials whatever.

It has been shown that CM is to CA as radius to the sine of the latitude; but CM is to CA as radius to the tangent of CMA, therefore the tangent of CMA is equal to the sine of the latitude; so that the latitude being known, the angle CMA may be found by inspection in the trigonometrical tables. Let this angle be \( e \), then AC

\[ = AM \times \frac{\sin e}{\text{rad.}} \]

is known. From this formula, a second scale, which shall show the length of AC for every degree of latitude, may be formed; this may be called the scale Dialling of Latitudes, and, with the scale of hours, will serve for the construction of all dials.

Construction of Dialling Scales.

31. Scales for the construction of dials may be either made entirely by a geometrical construction, or their divisions may be computed by trigonometry, and laid down from a scale of equal parts.

If a geometrical construction is used (Plate CLXXXV. fig. 8), divide AD a quadrant of a circle into six equal parts, at 1, 2, 3, &c. each of these may be again divided into four (these, however, are not shown in the figure). Draw a straight line from E, the centre, to 3, the middle point of division, and draw rs perpendicular to ES; also straight lines from the centre through A, D, the extremities of the quadrant, meeting the perpendicular in r and s; and against these marks put XII. and VI. Draw straight lines from E through the intermediate division 1, 2, &c. and where they meet rs mark I. II. III. IV. V. The line rs thus divided is the scale of hours.

32. Divide CD, another quadrant of the circle, into 90 equal parts (only every tenth division is shown), and from the points of division draw perpendiculars to the radius EC; these cut off distances E10, E20, &c. which are equal to the sines of 10°, 20°, &c. and EC is a line of sines.

33. Draw straight lines from D through 10, 20, &c., the bottoms of the perpendiculars on EC, and produce them to meet the circle in a, b, &c.; transfer the chords of the arcs Ba, Bb, &c. to the line BC, and mark the divisions 10, 20, &c.; there will thus be formed the scale of latitudes BC.

34. Transfer the chords of the arcs D10, D20, &c. or A10, A20, &c. arcs equal to them, to the line AB, thus forming a scale of chords AB. The scales of hours, latitudes, and chords, are shown together in fig. 9.

35. The dialling scales may also be constructed by laying down the divisions from a good scale of equal parts, and the numbers in the two following tables, of which the first gives the distances of the divisions on the scale of latitudes, in thousandth parts of the whole length of the scale; and the second gives, in the same equal parts, the distances of the divisions on the scale of hours, from the beginning, XII. to every fifth minute of each hour.