DIALLING

DIALLING.

DIALLING, sometimes called Gnomonics, 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 Berossus, 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

Dialling. 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 each into two. The dial of Berossus, 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 semicircles 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 Berossus descended beyond the time of Hipparchus and Ptolemy. We find it in the year 900 among the Arabians, 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 1762 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 Berossus. 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 Osmandias 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 semidiurnal 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 semidiurnal 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 Connoissance 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 any thing 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 Cyresthes, author of the monument. It is impossible to say any thing 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 $42^\{\backslash circ\}\; 40\text{'}$, 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 south-west, 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 Phœdru. They are four in number, and are traced on the same block of pentelic marble. It bears the following inscription: φειδρος Ἰσίδωρ πατρὶνος ἔσται. 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.² 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 scaphē (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

² These interesting dials are now in the British Museum; Elgin Saloon, No. 186.

Dialling. the aranea 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 Bacchan) 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,
Quicquid adeo primus statuit hic solarium.
Qui mihi comminuit misero articulum diem,
Nam me pueri uterus erat solarium
Multo onnium istorum et verissimum,
Ubi iste monebat esse, nisi quum nihil erat.
Nunc etiam quod est, non est, nisi Soli lubet.
Itaque adeo jam oppletum est oppidum solaris.
Major pars populi aridi raptant fame.

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 aught to eat.
But now-a-days, why, even when I have,
I can't fall-to unless the sun give leave.
The town's so full of these confounded dials,
The greatest part of its inhabitants,
Shrunk up with hunger, creep along the streets.

Translation of Plautus by Thoroton and Werner.

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, 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 Stridborius, 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 Canones, 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, truncis, 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

Dialling. radii were now introduced; all these changes could only have been made by a skilful geometer. Among the inventions of Munster was a moon-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: Gnomonica Andreae Schoneri Noribergensis, hoc est de descriptionibus horologiorum sciatricorum omnis 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 any thing 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 Mathématiques, 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 Horologio-graphia; Oronotius Fineus, De Horologiis Solaribus, &c.; Mutio Oddi da Urbino, Gli Horologi Solari nelle Superficie Piane; Dryander, De Horologiorum varia Compositione; Conrad Gesner's Pan-

doctæ; Andrew Schoner's Gnomonica; F. Commandine, Dialling. De Horologiorum Descriptione; Joan. Bapt. Benedictus, De Gnomonum Umbrarum Solarium Usu; Joannes Georgius Schomberg, Exegesis Fundamentorum Gnomonicorum; Solomon de Caus, Traité de Horologes Solaires; Joan. Bapt. Trota, Praxis Horologiorum; Desargues, Manière Universelle pour poser l'Essieu et placer les Heures et autres Choses au Cadran Solaires; Ath. Kircher, Ars magna Lucis et Umbra; Hallum, Explicitio Horologii in Horto Regio Londini; Joan. Mark, Tractatus Horologiorum; Clavius, Gnomonices de Horologiis; the treatises of Deschales, Ozanam, Schottus, Wolfus (in his Elementa), Picard, Lahire, Walper, with additions by Sturmius, &c. In the German language there are treatises by Paterson, Michael, and Müller. Of English writers there are, Foster, Horologographia (in his works); Wells, The Art of Shadows; Collins, Leadbetter, Jones, Leybourn, and others. At a later period we have treatises by Emerson, by Ferguson in his Lectures, and others.

The theory of dialling has sometimes been treated as a branch of perspective, as by Gravesand in Essai de Perspective, and Dr Horsley in his Tracts on the Projection of the Sphere; also, as a branch of analysis, by Kæstner in his Gnomonica Universalis Analytica; by MM. Sejour and Godin, in Recherches Gnomoniques, &c. Lastly, it has been referred to descriptive geometry, by Hachette, Cours de Géométrie Descriptive. Lefrançois, Journal d'École Polytechnique, 2d cahier. Delambre has treated of the subject in different parts of his Histoire de l'Astronomie, and in his Astronomie Théorique et Pratique. Biot has added a tract on it by Berroyer to his Astronomie Physique.

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 figs. 1 and 2 of Plate CC. 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 $E\; XII$, and eleven in the forenoon when it falls on $E\; XI$, and one in the afternoon when it falls on $E\; I$, 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 $51\backslash frac\{1\}\{2\}^\backslash circ$, 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.

Dialling. 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 $AFCG$ (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 $Pap$, $Pbp$, $Pcp$, &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 $E\; XI$, $E\; XII$, $E\; I$, &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 declining 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-

ling 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 $51\backslash frac\{1\}\{2\}^\backslash circ$; 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 I. of the next hour circle. This will be about $11\backslash frac\{1\}\{2\}^\backslash circ$. 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 $15^\backslash circ$, $30^\backslash circ$, $45^\backslash circ$, $60^\backslash circ$, $75^\backslash circ$, and $90^\backslash circ$ 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. $11\backslash frac\{1\}\{2\}^\backslash circ$, $24\backslash frac\{1\}\{2\}^\backslash circ$, $38\backslash frac\{1\}\{2\}^\backslash circ$, $53\backslash frac\{1\}\{2\}^\backslash circ$, $71\backslash frac\{1\}\{2\}^\backslash circ$, and $90^\backslash circ$; 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 CC. fig. 3), draw parallel straight lines $ac$, $bd$, 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 $gab$, 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 $11\backslash frac\{1\}\{2\}^\backslash circ$ in the quadrant; the hour line of II. through $24\backslash frac\{1\}\{2\}^\backslash circ$; of III. through $38\backslash frac\{1\}\{2\}^\backslash circ$; of IV. through $53\backslash frac\{1\}\{2\}^\backslash circ$; and of V.

through $71\backslash frac\{1\}\{2\}^\backslash circ$. Again, lay the ruler to the centre $a$ of the quadrant $eg$, and through the like divisions or degrees of that quadrant, viz. $11\backslash frac\{1\}\{2\}^\backslash circ$, $24\backslash frac\{1\}\{2\}^\backslash circ$, $38\backslash frac\{1\}\{2\}^\backslash circ$, $53\backslash frac\{1\}\{2\}^\backslash circ$, and $71\backslash frac\{1\}\{2\}^\backslash circ$, draw the forenoon hours of XI. X. IX. VIII. and VII. 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 $51\backslash frac\{1\}\{2\}^\backslash circ$. 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 $PBpDF$ (fig. 2. Plate CC.) 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 $90^\backslash circ$. 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. south dial (fig. 7) is to be constructed for a given place, London for instance. Draw two vertical lines $ac,\; bd$, 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

ference 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 CCII. 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-

circle $APB$ at $C$ its centre, and let $CA$ be the position of the shadow on the circle at noon. Then, if the circum-

ing line OQ. Let C1, C2, &c. be the hour lines after noon on the equinoctial 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,

therefore ex. æq. $CK\; :\; OK\; =\; \backslash tan.\; KOQ\; :\; \backslash tan.\; KCQ$.

But in the triangle COK, right angled at C,

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,

We have supposed GHDE 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 $51\backslash frac\{1\}\{2\}^\backslash circ$. In this case the hour angle from noon at the pole is $15^\backslash circ$.

Here we have found that the hour lines of XI. and I. must each make an angle of $11^\backslash circ\; 51\text{'}$ with the meridian line at the centre of the dial.

24. In this way the following table has been computed for every half degree of latitude from $50^\backslash circ$ to $59^\backslash circ\; 30\text{'}$, which are the limits of Britain.

Table of the Angles which the Hour Lines on a Horizontal Dial make with the Meridian for every half degree of latitude from $50^\backslash circ$ to $59\backslash frac\{1\}\{2\}^\backslash circ$.

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 $56^\backslash circ$ of latitude be required, the table shows that the hour lines of XI. and I. make angles of $12^\backslash circ\; 31\text{'}$ with the meridian, and the hour lines of X. and II. angles of $25^\backslash circ\; 35\text{'}$, 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 $55^\backslash circ\; 42\text{'}$ be required, the table gives $12^\backslash circ\; 27\text{'}$ for the angle of the XI. or I. o'clock line in lat. $55^\backslash circ\; 30\text{'}$, and $12^\backslash circ\; 31\text{'}$ in lat. $56^\backslash circ$: the difference of the angles corresponding to $30\text{'}$ difference of latitude is $4\text{'}$. Now the proposed latitude exceeds the least of these two by $12\text{'}$, therefore we state this

proposition $30\; :\; 12\; =\; 4\text{'}\; :\; \backslash frac\{4\; \backslash times\; 12\}\{30\}\; =\; 1.6$. Hence we find

that $12\text{'}$ of addition to the latitude gives $1.6$, or nearly 2 of addition to the hour angle, which will therefore be $12^\backslash circ\; 29\text{'}$.

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.

26. Draw two parallel straight lines $CM$, $CM\text{'}$ (Plate CC. 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 $VI$. $C\text{'}C$ $VI$. 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 $51\backslash frac\{1\}\{2\}^\backslash circ$. About $C$ and $C\text{'}$ (in fig. 5) as centres with a radius equal to $CH$, the hypotenuse of the triangle $CKH$ (fig. 6), describe the quadrants $M6$, $M\text{'}6$; 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 $C\text{'}XI$. $C\text{'}I$ 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 $C\text{'}X$. $C\text{'}II$. 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,

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.

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 $M3A$, 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 $CAE$, $KME$;

Now, by trigonometry, making $MO$ radius,

And making $MC$ radius; $MK\; :\; AC\; =\; \backslash tan.\; MCK\; :\; \backslash tan.\; AMC$. Therefore $\backslash tan.\; MOH\; :\; \backslash tan.\; OMA\; =\; \backslash tan.\; MCK\; :\; \backslash tan.\; AMC$. And by alter. $\backslash tan.\; MOH\; :\; \backslash tan.\; MCK\; =\; \backslash tan.\; OMA\; :\; \backslash tan.\; AMC$.

That is, because $\backslash tan.\; OMA\; =\; \backslash text\{radius\}$,

But $\backslash text\{rad.\}\; :\; \backslash tan.\; AMC\; =\; CM$ or $CB\; :\; CA\; =\; \backslash text\{rad.\}\; :\; \backslash sin.\; CMA$; 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 $v$, then $AC$

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 CC. 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 $r3s$ perpendicular to $E3$; 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 $10^\backslash circ$, $20^\backslash circ$, &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 of 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.

Distance of Divisions from beginning of Scale of Latitudes.

Distance of Divisions from beginning of Scale of Hours. Dialling.

Construction of a Horizontal Dial by the Scales or the Tables.

36. Draw two parallel straight lines $ae$, $bd$, at a distance equal to the thickness of the style of the dial (Plate CC. fig. 4), for the double meridian line, and across them, at right angles, draw $fabe$, the six o'clock hour line. Supposing the dial is to be for London, in latitude $51\backslash frac\{1\}\{2\}^\backslash circ$, make $be$ and $af$ each equal to the distance from 0 to $51\backslash frac\{1\}\{2\}^\backslash circ$ on the scale of latitudes (fig. 9), and place between the points $e$, $f$ and the meridian lines, straight lines $ed$, $fc$, each equal to the whole length of the scale of hours, viz. from XII. to VI.; make $e1$ and $d1$ each equal to the distance from XII. to I.; and through the points 1, 1 draw $\backslash alpha\; X1$ and $\backslash beta\; 1$ for the eleven and one o'clock hour lines. In like manner, make $e2$ and $d2$ equal to the distance XII. II., and draw the hour lines $\backslash alpha\; X2$, $\backslash beta\; 2$, and so on, for the remaining hour lines; the quarters and lesser divisions are to be laid down in the same way. The lines for the hours before VI. in the morning, and after VI. in the evening, are to be drawn as directed in art. 26.

37. If a common scale of equal parts and the tables be used, then make $be$ and $af$ each equal to 871.5 equal parts, this being the mean between the numbers for $51^\backslash circ$ and $52^\backslash circ$ in the table for the scale of latitudes; the lines $ed$ and $ae$ must in all cases be each 1414.2 from the same scale; and the distances $d1$ and $e1$ each 298.8; and $d2$, $e2$ each 517.7, these being the distances for 1 and 2 from the beginning of the scale (see Table). The distances of 3, 4, 5 from $e$ and $d$ must be 701.7, 896.5, and 1154.4 equal parts respectively, as appears by the table; and in this way the points in $de$, $cf$, through which the hour lines pass, also the lines for every fifth minute, may be readily found, and the corresponding divisions transferred to the circumference of the dial.

Vertical North and South Dials.

38. These dials are represented in figs. 1 and 2 of Plate CII. It has been explained in par. 15 that south and north dials for a given latitude would be horizontal

Dialling. dials if carried to a place whose latitude is the complement of that latitude: so that a south dial for latitude $51\backslash frac\{1\}\{2\}^\backslash circ$ would be a horizontal dial for $38\backslash frac\{1\}\{2\}^\backslash circ$. This principle, first discovered by the Arabian writers on gnomonics, enables us to apply all that has been here taught concerning horizontal dials to those directly facing the south and north.

39. If we put $x$ to denote the hour angle in the heavens, reckoned from noon either way, and $y$ for the angle which any hour line on a north or south dial makes with the meridian line, and $L$ for the latitude; the formula for determining trigonometrically the hour lines on a north and south dial will be

In north latitudes, a north dial can only be illuminated when the sun has north declination, that is, from about 21st March to 22d September; and the nearest times to noon which can be shown by it are those at which the sun crosses the prime vertical on the day of the summer solstice. A south dial can never be illuminated before six in the morning, nor after six in the evening; because when the sun rises earlier and sets later, he never crosses the prime vertical so early as six in the morning, nor so late as six in the evening. It will therefore be needless to describe on either more hours than can be wanted.

40. Supposing a dial to be traced on a thin material plane, it is easy to understand that a direct north dial is only the back of a direct south one; so that if the hour lines of one of them were cut through it, and extended, they would form on the opposite side the hour lines of the other, and the style of the one dial produced would form the style of the other dial.

Vertical East and West Dials.

41. These are described on vertical planes facing due east and west; therefore their planes coincide with the meridian, and pass through the poles of the world. Fig. 8 and 9, in Plate CCII. represent east and west dials. In fig. 8, let HR be a horizontal line on a plane facing due east, and $ab$ a line in the direction of the earth's axis crossing it at $c$. Through this point draw EQ perpendicular to $ab$, then EQ will be in the plane of the equinoctial circle in the heavens.

Suppose now that at any two points $a,\; b$ in the line directed to the pole there are wires of equal length fixed perpendicular to the plane of the dial, and that their tops are joined by a third wire, so as to form a rectangle $AabB$ (see fig. 10), one side of which $ab$ is on the face of the dial, and the opposite side AB is parallel to it and directed to the pole. The earth's radius being as nothing when compared with the sun's distance, it follows that the angular motion of the sun about the line AB will be uniform, just as it is about the earth's axis, and therefore that the shadow of the line will turn uniformly about it as an axis, and its projection on the dial will always be perpendicular to the line EQ. It will now be easy to understand, that at six in the morning the shadowy plane will be perpendicular to the plane of the dial, and will fall on the line $ab$; and that, as it turns uniformly about its axis, it will cut off from $cQ$ lines which will be the tangents of the angles generated by the shadow, that is, the tangents of the hour angles from six.

The same will be true of a west dial (see fig. 9); the hour lines of both will be perpendicular to EQ, the line in which the plane of the equator intersects the plane of the dial; and if $d$ be put for the height of the rod AB above the dial, $x$ for the hour angle from six o'clock, and $y$ for the distance of the hour line reckoned on EQ either way from $c$, then, supposing radius = 1, the formula for the dial will be

Geometrical Construction of Vertical East and West Dials. Dialling.

42. On the east and west vertical plane (fig. 8, 9) draw the horizontal line HR, and at an assumed point $c$ in that line draw a straight line $acb$, so as to make with it an angle equal to the latitude of the place, and directed, in each dial, towards the north. This line is to serve as the six o'clock hour line, over which the rod or wire AB is to be placed on its supports, $Aa,\; Bb$, as the style of the dial. In the line $acb$ take $cb$ equal to $Bb$, the height of the style. About $b$ as a centre, with a radius equal to $bc$, describe a semicircle, and divide each quadrant into six equal parts: draw $EcQ$ perpendicular to $bc$, and from $b$, through the points of division of the circle, draw straight lines, producing them until they meet the line EQ. Through the points of intersection draw perpendiculars to EQ, which will be the hour lines; against these the hours are to be written as in the figures. At the points $a$ and $b$ erect the supports of the style perpendicular to the plane of the dial, and each equal in length to the line $bc$, and over them place the rod AB, and the dial is finished.

43. The east dial will show the hours from sunrise until near noon, and the west dial from a little after noon until sunset; but neither can indicate the exact time of noon by a shadow, because then it goes off parallel to their planes.

Polar Dial.

44. This dial is shown in fig. 3, Plate CCII. It is described on a plane perpendicular to the meridian, and passing through the poles. It has a great affinity with east and west dials; for if a prism with six rectangular faces were placed with its axis directed to the pole, and two of its faces due east and west, the remaining two would form planes for polar dials, and the face directed to the pole would serve as the plane of an equinoctial dial.

45. The style of a polar dial, like that of an east or west dial, must be parallel to its face, and may have the same form (see fig. 10). To construct it, draw $ab$ in the plane of the meridian for the twelve o'clock hour line, and cross it at right angles by the horizontal line HR. The hour lines are to be found exactly as in the east and west dials, and marked with the hours as in the figure.

A polar dial may, if it be of sufficient extent, show time from a little after six in the morning to a little before six in the evening. At the hours of six the sun is in the plane of the dial, and the shadow parallel to its face.

Putting $x$ to denote the hour from noon, and $y$ for the distance of the hour line from the meridian line, and $d$ for the height of the style above the plane of the dial; the formula for a polar dial is

Vertical Declining Dials.

46. It seldom happens that an upright wall faces exactly one of the cardinal points; therefore, in general, a dial described on the plane of a wall will be a vertical declining dial.

47. The declination of a plane is an arch of the horizon between the plane and the prime vertical; or it is the arch between the meridian and a plane perpendicular to the dial plane, and is always reckoned from the south or north.

48. The meridian of a plane is the meridian perpendicular to the plane of the dial. This differs from the meridian of the place, which is the meridian that is perpendicular to the horizon.

The substyle of a dial is the common section of its plane and the plane of its meridian, or it is the line in which perpendiculars drawn from every point in the axis of the dial meet its plane. In horizontal and in vertical south and north dials, the substyle coincides with the twelve o'clock hour line; but not in declining dials.

Dialling. The difference of longitude of a dial plane is the angle which the plane of its meridian makes with the meridian of the place.

The latitude of any dial plane is the angle which the axis makes with the plane; which is also the latitude of the place where the dial would be a horizontal one.

49. In the adjoining figure let AB be a line drawn on a vertical plane, a wall, for instance. It may have any aspect, but, to fix our ideas, let us suppose it to face some point between the south and west; let CE be the style or axis fixed at C in the wall in the direction of the earth's axis; draw the vertical line CD on the plane, and from E, the extremity of the style, draw ED perpendicular to CD; the plane of the triangle CDE will manifestly coincide with the meridian, and CD will be the twelve o'clock hour line on the dial.

Let NCF be a horizontal line passing through C due east and west; conceive a plane to pass along NF and the vertical line CD; this will coincide with the prime vertical in the heavens. Let us now suppose that at some hour, for example two in the afternoon, the horary plane (that is, the plane passing through the sun and the axis CE) meets the plane of the dial in the line CH, and the plane of the prime vertical in Ch; the first of these lines will be the hour line of two on the dial to be constructed, and the second the line of the same hour on the prime vertical, that is, on a vertical south dial. Now by the theory of that dial (art. 38) the angle DC$\backslash hat\{h\}$ will be known; it is, however, the angle DCH that is required in order to construct the dial under consideration.

50. Suppose a horizontal plane to pass along ED, and meet the horary plane in the line EH, the plane of the dial in DH, and the prime vertical in D$\backslash hat\{h\}$; the plane ED$\backslash hat\{h\}$ may now be regarded as that of a horizontal dial, of which EC is the axis, E the centre, and ED the meridian line; and on this dial EH will be the hour line of two; but for any given hour, the angle DEH will be known by formula 1, art. 23. And because the horizontal lines DE, D$\backslash hat\{h\}$ lie, one in the plane of the meridian CDE, and the other in the plane of the prime vertical CD$\backslash hat\{h\}$, which is perpendicular to the former, the angle ED$\backslash hat\{h\}$ will be a right angle; now the angle HD$\backslash hat\{h\}$, or its equal BCE, is the declination of the dial plane (art. 46), and therefore is given, or may be found; therefore the angle EDH, its complement, is known; and hence

51. All the angles of the triangle DEH are known.

Let L denote the latitude of the place for which the dial is to be made; D, the angle HD$\backslash hat\{h\}$ or BCE, the declination of the plane; and E the angle made by the meridian ED of the assumed horizontal dial, and EH the hour line; then, in the triangle DEH we have the angle at E (denoted by E), the angle EDH = 90 - D, and therefore the third angle DHE = 180 - (E + 90 - D) = 90 - (E - D).

In the right-angled triangles CDE, CDH (which have

CD, one of the sides about the right angle, common to Dialling. both), by trigonometry,

But in the triangle DEH,

therefore, $\backslash cos.\; (E\; -\; D)\; :\; \backslash sin.\; E\; =\; \backslash cot.\; DEC\; :\; \backslash tan.\; DCH$. Now the first three terms of this proposition are known, because the angles E and D are given, and also the angle DEC, which is the latitude; therefore the fourth term, viz. the tangent of DCH, is known; and hence the angle DCH, between the hour line CH on the dial and the meridian line CD, is known. Thus may all the angles made by the hour lines and the meridian of the dial be determined by plane trigonometry.

52. This way of finding the hour lines requires two operations for each, viz. one to find the angle DEH at the centre of the auxiliary horizontal dial; and a second to find the angle DCH at the centre of the dial to be constructed. We shall now investigate a formula which gives each angle by a single operation, when two subsidiary quantities common to them all have been found.

For any vertical declining dial let

L = latitude of the place;

D = BCE, the declination of the dial plane reckoned from the east towards the south;

x the variable horary angle described by the sun since noon;

y the corresponding angle DCH at the centre of the dial; and, as before, put E for the angle described at the centre of a horizontal dial for the same latitude in the time x.

From what has been shown in the preceding articles, we have

In formula (1), instead of $\backslash cos.\; (E\; -\; D)$, put its equal $\backslash cos.\; E\; \backslash cos.\; D\; +\; \backslash sin.\; E\; \backslash sin.\; D$ (ALGEBRA, art. 239), then divide the numerator and denominator by $\backslash cos.\; E$, and lastly,

put $\backslash tan.\; E$ instead of $\backslash frac\{\backslash sin.\; E\}\{\backslash cos.\; E\}$; the formula will then be transformed to this,

Now put $\backslash sin.\; L\; \backslash tan.\; x$ instead of $\backslash tan.\; E$; and again $\backslash frac\{\backslash sin.\; x\}\{\backslash cos.\; x\}$ instead of $\backslash tan.\; x$; we then have

Let P denote a subsidiary angle, such, that

this value of $\backslash cos.\; D$ being substituted, instead of it, in formula 3, we get

The denominator of this last fraction is manifestly $\backslash cos.\; (P\; -\; x)$. Hence we have the following simple formula for computing the angle which the shadow of the axis of any vertical declining dial describes in any time before or after noon.

This formula gives the afternoon hours on a vertical

Dialling. south dial declining towards the west; but we have only to suppose the angles $D$ and $x$ to vary, and attend to the change of sign of $\backslash sin.\; x$ and $\backslash cos.\; (x-P)$, and the formula will be adapted to the forenoon hours.

53. If we suppose $D$, the declination of the plane, to decrease, the dial will approach to a south dial; and since $\backslash frac\{\backslash tan.\; P\}\{\backslash tan.\; D\}\; =\; \backslash frac\{\backslash sin.\; L\}\{\backslash text\{rad.\}\}$, as $D$ decreases, $P$ will decrease (since $L$ is constant), and they will vanish together, so that ultimately the fraction

becomes simply $\backslash frac\{\backslash sin.\; P\}\{\backslash sin.\; D\}\; =\; \backslash sin.\; L\; =\; \backslash frac\{\backslash cos.\; L\}\{\backslash cot.\; L\}$, and the preceding formula becomes, when $D\; =\; 0$,

agreeing with the formula for a south dial (art. 39).

54. When the dial declines to the east instead of the west, that the formula may apply, we assume that $D$ is negative; then $\backslash sin.\; D$ and $\backslash tan.\; D$ will be negative (ALGEBRA, art. 225); and since $\backslash tan.\; P\; =\; \backslash sin.\; L\; \backslash tan.\; D$, we must in the formula make the sign of $P$ negative, and it will in this case be

for the afternoon hours.

55. On the whole, we have this formula for the construction of all south vertical declining dials.

Let $L$ denote the latitude of the dial,

$D$ the declination, which may be west or east,

$x$ the hour angle from noon ($15^\backslash circ$ to an hour),

$y$ the hour line angle from the meridian line on the dial.

Find these two auxiliary quantities, viz. an angle $P$, and the tangent of an angle $Q$ (of which only the logarithm is required), such, that

These quantities, $P$ and $\backslash tan.\; P$, may be called the constants of the dial. Then, the forenoon hours on the west declining dial, and the afternoon hours on the east, will be found from the formula

and the morning hours on the east declining dial, and afternoon hours on the west, by the formula

56. We shall now give examples of the application of the formulae.

Ex. 1. Let it be required to find the angles which the hour lines make with the meridian line on a vertical south dial that declines $36^\backslash circ$ westward, the latitude being $54\backslash frac\{1\}\{2\}^\backslash circ$.

In this dial, which is represented in Plate CCII. fig. 5, $L\; =\; 54^\backslash circ\; 30\text{'}$; $D\; =\; 36^\backslash circ$ west.

Computation of $P$ and $\backslash tan.\; Q$.

Having determined the constants, the hour line angles may be found as follows:

Calculation of hour line angle of XI. A. M.

Calculation of hour line angle of I. p. m.:

The following table exhibits at one view the elements of the dial and the hour line angles for its construction:

57. If the hour lines of this dial were traced on a transparent plane and extended, and if the style were produced through the dial, the reverse would show a north dial declining eastward $36^\backslash circ$.

Ex. 2. Suppose a vertical south dial decline east $49^\backslash circ$ in the latitude $51^\backslash circ\; 30\text{'}$; to determine the hour line angles. (See fig. 4 of Plate CCII.)

In this case $L\; =\; 51^\backslash circ\; 30\text{'}$; $D\; =\; 49^\backslash circ$ east. We now apply formula $\backslash beta$ of (5). The data of the dial, the constants, and hour line angles, are exhibited in the following table:—

In the calculation for the hour line angle of III. a. m. the cosine of $x\; -\; P\; =\; 93^\backslash circ$ is negative; this makes the sign of $y$ negative, and therefore $y$ an angle between $90^\backslash circ$ and $180^\backslash circ$.

58. If this dial were traced on a plane, and the hour

Dialling. lines extended and continued through it, and if the style also were produced through the plane, the reverse would be a north vertical dial, declining to the westward 49°.

In general, to make a north declining dial, we have only to make a south declining dial whose declination is the same and lies the same way, and then turn it upside down, and it will be the dial required; but the hours must be numbered the contrary way. Therefore these two examples, duly considered, will serve for examples of all declining dials.

59. It has been already observed (art. 15), that a dial on any plane whatever, given in position at a given place, will be a horizontal dial at some other place, which may be found. This principle gives another method of constructing a declining vertical dial; for if the latitude of that place, and also the difference between its longitude and that of the place where the dial is to show the time, be found, the former will be the angle which the style must make with the plane of the dial; that is, the angle it makes with the substyle; and the latter will give the time the shadow takes to pass between the twelve o'clock line and the substyle, from which the angle they contain may be found. These three elements being known, viz. the latitude and longitude of the place where the dial would be horizontal, and the angle contained by the meridian line and substyle, the construction is reduced to that of a horizontal dial.

60. These elements may be found by spherical trigonometry as follows: let $SZNz$ be the meridian, in which $Z$ and $z$ are the zenith and nadir, and $P,\; p$ the poles: Let $SEN$ be the horizon, $S$ and $N$ being the south and north points, and $E$ the east: let $ZFz$ be a vertical plane or great circle of the sphere on which the dial is to be drawn; let this plane cut the horizon in $F$, and the plane of the meridian in the straight line $Zz$; and let it be cut perpendicularly in the line $Aa$ by a circle $PAp$ which passes through the poles. Then $Cp$ or $CP$ will be the axis of the dial, according as it faces the south or the north, $Cz$ or $CZ$ the twelve o'clock hour line, and $Ca$ or $CA$ the substyle (art. 48).

In the spherical triangle $ZPA$, right angled at $A$, $PZ$, the complement of the latitude of the place where the dial is to show time is given; so also is the angle $PZA$, for it is measured by $FZ$, the arc of the horizon of which the complement is the declination of the dial (art. 47); from these, $AP$, the measure of the angle contained by $PC$ the axis, and $AC$ the substyle, that is, the latitude of the dial, (art. 48), also $AZ$, the measure of the angle contained by the substyle $AC$ and the vertical or twelve o'clock hour line, lastly, the angle $ZPA$, the difference of longitude of the planes $ZPC$, $APC$ (art. 73), may be all found.

61. By the principles of spherics (see SPHERICAL TRIGONOMETRY),

Let $L$ be the latitude of the place where the dial is to Dialling. serve,

$D$ the declination of the dial (art. 47),

$l$ the latitude of the dial, that is, of the place where it would be horizontal (48),

$a$ the longitude of the dial,

$b$ the angle between the substyle and the vertical.

Then from the above proportions there is got

These formulae, when $L$ and $D$ are known, give the three elements for the construction of the dial, which is now reduced to that of a horizontal dial. The hour line angles must, however, be so found that one of them shall fall on the vertical or twelve o'clock hour line.

62. Let us again take Ex. 1 of art. 56, and suppose that a vertical south dial, declining 36° to the west, is to be constructed, the latitude being 54° 30' (fig. 5 of Plate CCII.) In this case $L\; =\; 54^\backslash circ\; 30\text{'}$, $D\; =\; 36^\backslash circ$.

The construction of our south declining dial for latitude 54° 30' is now reduced to that of a horizontal dial for latitude 28° 1'; for the sake of brevity, let us call the former place $A$ and the latter $B$. The earliest hour that can be shown at $A$, on the dial, is about IX. a. m. that is, 45° of an hour angle from noon. Now when it is 45° from noon at $A$, because of the difference of longitude, it will be $45^\backslash circ\; +\; a\; =\; 86^\backslash circ\; 45\text{'}$ at $B$; the hour line angle with the meridian corresponding to this will, by the formula for a horizontal dial, be found by this proportion:

Here we have found that at the place $B$, when it is nine in the morning at $A$, the hour line will make with the meridian line there an angle of 83° 6'; this is the angle which the hour line of IX. at $A$ will make with the substyle. In the same manner may the angles which the remaining hour lines on the dial make with the substyle be found. The whole are shown in this table.

Dialling. The angles which the hour lines make with the vertical or meridian line at A are found from the like angles at B, by taking the difference between them and the angle $b\; =\; 22^\{\backslash circ\}\; 45\text{'}$ for all those on one side of the substyle, and the sum of each and that angle when they lie on the other side. The agreement of the results in this table with those in article 56 will appear by inspection.

To trace a Meridian Line on any Plane.

63. In constructing a dial it is always necessary to determine the line in which the plane of the meridian meets the plane of the dial. On an assumed point in the meridian line as a centre describe several concentric circles with any distances in the compasses: at this point fix a wire truly perpendicular to the plane, and of such a length that when the sun shines its shadow shall extend beyond the circles. Watch now the instant when the extremity of the shadow exactly reaches some one of the circles in the forenoon, and mark the point in which it crosses the circle. In the afternoon, mark in like manner the point in which it again crosses the same circle: and any point that is equally distant from the two points so determined will be in a meridian line passing through the centre of the circle, which may now be drawn. By a like attention to the path of the extremity of the shadow when it crosses the other circles, points may be found in a meridian line passing through their common centre; and a mean position of all the meridian lines thus found will be sufficiently accurate for a dial.

The theory of this method is sufficiently obvious. At equal intervals of time each way from noon, the shadow is nearly of the same length, and makes nearly equal angles with the meridian line; the deviation from absolute equality arises chiefly from the continual change in the sun's declination. This, however, is but little in the course of a day near the solstices. Accordingly, greater accuracy will be obtained about midsummer or midwinter than nearer the equinoxes.

64. A good compass will give an approximation to the true position of the meridian; but the variation, which is not the same at all places and at all times, must be known. In 1823 it was about $24^\{\backslash circ\}\; 10\text{'}$ west at London, and in the same year $27^\{\backslash circ\}\; 49\text{'}$ west at Edinburgh. If the direction of the meridian be known at one place, the variation may be there determined; and thence the direction of the meridian at another place not very distant may be found.

65. If a good watch be set to true time, or its deviation from it be found, by taking equal altitudes of the sun with a sextant, or by a single altitude, or any other astronomical observation, a dial may thereby be set truly in the meridian.

66. The pole star is distant from the pole about $1^\{\backslash circ\}\; 36\text{'}$; hence, knowing the latitude, its greater azimuth may be found. At London its greatest deviation from the meridian either way is about $2^\{\backslash circ\}\; 34\text{'}$ in azimuth. If now two plumb-lines be suspended at some distance from each other, and in such a position that the pole star may appear in the same plane with them, that plane will be nearly in the meridian. If one of the plumb-lines be moveable, and it be shifted, following the change of position in the star, the two extreme positions of the plumb-line will, with the other plumb-line, determine two vertical planes, which deviate equally from the meridian in opposite directions. If lines were now traced on a horizontal plane in the directions of the vertical planes thus found, these would form with the meridian line sought equal angles on opposite sides of it, and the meridian would be found by bisecting the angle they contain. The pole star comes to the meridian about noon and midnight in the beginning of October. It may then be seen twice in the same night at

its greatest distance on opposite sides of the meridian; but one observed maximum distance, and the computed deviation in azimuth, would serve to find the meridian at any time.

The time of the meridian passage of the pole star on any day of the year may be found as taught in works on astronomy. At this time, as shown by a watch, two plumb-lines may be placed in the meridian, or one, and the corner intersection of two walls truly perpendicular instead of another. Great accuracy in the time of the observation is not required, the motion of the star in azimuth being only 7 minutes of a degree in ten minutes of time at London.

67. A meridian line may be traced by observing when two stars which have the same right ascension, or which differ in right ascension by twelve hours, come into the same vertical plane; for then they are both in the meridian.

The pole star and the star $\backslash epsilon$ (the first of the three in the tail) of the great bear have nearly this relative position. On the 1st of January 1834, their mean right ascension, as given in our table of the places of the fixed stars (ASTRONOMY), will be,

The difference from twelve hours is about forty-six minutes of time. By the revolution of the heavens the star $\backslash epsilon$ comes into the meridian under the pole; but the pole star is not then exactly in the meridian. Afterwards they come into the same vertical, and then the pole star is very near the meridian, and, for finding a meridian line, may be considered as exactly so. The azimuth of the pole star, at the instant when the stars are in the same vertical, may be found by calculation, and the approximate meridian thereby corrected.

There are other pairs of stars which come into the meridian nearly at the same time. The stars $\backslash beta$ Draconis and $\backslash alpha$ Ophiuchi are such. The stars $\backslash delta$ of the lesser bear, and $\backslash alpha$ of the lyre, are also well adapted to the end in view, in latitudes exceeding the polar distance of the most remote star, which is about $51\backslash frac\{1\}\{2\}^\{\backslash circ\}$.

68. In whatever way a meridian line has been found on a horizontal plane, two plumb-lines hung over it, or so that their lines may pass through it, will indicate the position of the plane of the meridian in space. The line in which it cuts any other plane, as a vertical wall not far from the meridian line, may be found by placing the eye in the meridional plane, with the plumb-lines between it and the wall, and noting points on the wall which their lines cover from the eye. The ingenious dialist may now, with a little dexterity in practical geometry, fix the style of his dial in the wall or other plane, with its edge in the meridian plane, and making with a vertical line the proper angle, so that it may point to the pole. A perpendicular drawn now from the end of the style on the plane, will give a point in the substyle, which may now be drawn to the centre. Thus two important elements in a dial will be determined.

To find the Declination of a Vertical Plane.

69. Place a board in a position truly horizontal, with a straight side in contact with the plane; trace a meridian line on the board; and the angle made by this line and the line in which the board meets the vertical plane will be the inclination of the plane to the meridian.

To find the Inclination of any Plane to the Horizon.

70. Extend a surface truly level until it meet the plane;

Dialling. the line of their intersection will be horizontal. Draw perpendiculars to this line on the two planes from any point in it, and the angle which these form will be the inclination of the plane to the horizon.

If the plane were made to form one side of a temporary trough into which water was poured, the surface of the water would accurately mark the horizontal line. The inclination of the plane to the surface of the water, which is horizontal, might be found by various expedients too obvious to require being pointed out.

When the position of the meridian and the axis of the dial are truly determined, the finding of the inclination and declination of the plane are problems in practical geometry which may be resolved by the application of levels, or plumb-lines and squares, and those simple principles with which most workmen are familiar.

Inclining Dials.

71. We come now to consider inclining dials, or those which stand oblique to the horizon, either projecting forwards from the perpendicular, or retiring backwards.

From what has been taught, any dial plane being given in position, a meridian line may be traced on it; also, at a point in that line an axis may be fixed pointing to the pole of the world. Suppose now that in the following figure, OB is a meridian line drawn on an inclining dial plane, and OC

the axis directed to the pole; the angle COB will be the complement of the latitude of that place where the dial, being vertical, would show true time: this, therefore, is known. If, in addition, we knew the declination of the dial, the hour line angles might be found by the formulæ of art. 55. Now it is easy to find the declination, for since the horizon must be perpendicular to the vertical OB, if CB be drawn in the plane of the meridian, and HB in the dial plane, both perpendicular to OB, these lines will be in the plane of the horizon of the place where the dial, being vertical, would show true time; and the angle CBH will be the inclination of the plane of the dial to the meridian, which is the complement of the declination (art. 47). If now we make BH = Bb of any length, and join CH, Ch, we have, by trigonometry,

Hence, by measuring the lines on a scale of equal parts, the inclination may be found.

72. Thus, having the position of the meridian, and the axis, on a dial plane making a given angle with the horizon, we may find the latitude of the place where it would be a vertical declining dial, and also its declination at that place; and with these data the angles which the hour lines make with the meridian may be found by the formulæ for vertical dials, art. 55; or else we may find the latitude and longitude of that place where the dial would be horizontal, and construct it by the rules for a horizontal dial.

73. To begin with the first of these methods, let SZNH be the meridian (see fig. in next column), Pp the axis of the sphere, Z the zenith of a place where the dial is to stand, SEN the horizon, S, E, and N, the south, east, and north points respectively; also let HFI be a circle of the sphere on the plane of which the dial is to be constructed, and

let it meet the horizon in F, and the plane of the meridian in the line Hh; the line POp will be the axis of the dial, and HOA the meridian line. The arch EF between the east point of the horizon and the plane will be its declination, and the spherical angle HFN its inclination to the horizon, or the complement of its reclination from the vertical position.

Now, let SEN be the horizon of a place having H for its zenith, which of course is the place where the dial would be a vertical dial, and let it cut the dial plane circle in f; because the two horizons are perpendicular to the meridian, their intersection will be in the east point in both.

Let L = PN, the given latitude of the place where the dial is to be made;

D = FE, the declination of the dial;

R = complement of angle fEF, its reclination;

l = the latitude of the place where the dial would be vertical, which is sought;

d = Ef, its declination there, also sought;

then l - L = Nn, the measure of the angle fEf.

By spherical trigonometry, in the triangle EFf, right angled at f,

From these proportions we find

Hence the angles d and l are determined; and these being substituted in the formula of art. 55, instead of L and D, it will become a general expression for the angle y, which the shadow makes with the meridian line on the reclining dial.

EXAMPLE.—To find the hour lines on a south dial plane FHh, that declines westward 25°, and reclines 15°, in latitude 54° 30'.

Here L = 54° 30', D = 25°, R = 15°.

Since l - L = 16° 27', and L = 54° 30', therefore l = 70° 57'. The dial declines to the west; therefore (art. 55),

and hence $P\; =\; 22^\{\backslash circ\}\; 55\text{'}$, and $\backslash log.\; \backslash tan.\; Q\; =\; 9.51759$.

The elements of the dial and the results of the calculation are exhibited in this table.

The hour lines are to be drawn on the dial, so as to make with the XII. o'clock hour line the angles in this table. The dial is represented in fig. 7 of Plate CCII.

74. To construct the dial by finding the latitude and longitude of a place where it would be horizontal, we must find the angle which the substyle makes with the meridian, and also the latitude and longitude of the plane.

Resuming the figure of art. 73, let a great circle passing through the pole, and perpendicular to the plane of the dial, meet it in the line OA; this will be the substyle, and AOH the angle which it makes with the meridian or XII. o'clock hour line; AOP the angle which the axis makes with the plane of the dial, and the spherical angle APH its difference of longitude.

In the spherical triangle AHP right-angled at A, $AHP\; =\; fhs$ is measured by the arc $fs$; but this arc is the complement of the arc $fE$, which we have denoted by $d$, therefore the angle AHP is the complement of $d$. Again, the arc HP is the complement of the arc $Pn$, which was expressed by $l$. Now formulæ have been given (in the preceding article) for the computation of $d$ and $l$, therefore the angle AHP and the side HP of the spherical triangle may be considered as known. The three remaining parts of the triangle may be found by spherical trigonometry by these proportions,

By substituting the symbols $d$ and $l$ in these proportions, they give

These formulæ, applied to the example of last article, give

The dial may now be constructed as horizontal, and for latitude $17^\{\backslash circ\}\; 20\text{'}$; and since the meridian line lies to the east of the XII. o'clock line, the hour lines of $2^\{\backslash text\{h\}\}.\; 41\backslash frac\{1\}\{3\}\backslash text\{m\}$,

$3^\{\backslash text\{h\}\}.\; 41\backslash frac\{1\}\{3\}\backslash text\{m\}$, &c., reckoned from the substyle of the dial, must be found, and made the hour lines of XI, X, &c. in the forenoon. Also, the hour line of $41\backslash frac\{1\}\{3\}\backslash text\{m\}$. must be made the hour line of I. The hour line of II. will be on the other side of the substyle, and will correspond to $18\backslash frac\{1\}\{3\}\backslash text\{m\}$, that of III. to $1^\{\backslash text\{h\}\}.\; 18\backslash frac\{1\}\{3\}\backslash text\{m\}$, and so on.

Of the Time when the Sun begins or ceases to shine on a Dial on a given Day.

75. The solution of this problem is wanted, that it may be known what hours should be inscribed on a dial. It has various cases; but as all are to be resolved on the same principles, it will be sufficient if we consider one.

Retaining the construction of the fig. of art. 73, let PI be the hour circle passing through the sun when he is in the plane of the dial on a given day. In the spherical triangle HAP, we have found AP, the measure of the angle contained by the axis and substyle; and HPA, the difference of longitude. Again, in the spherical triangle IAP, right-angled at A, besides AP, there is known IP, the sun's distance from the pole on the given day. Hence the angle IPA may be found by this proportion:

The angle API, expressed in time, is half the period the sun shines on the plane; and the hour angle HPI, in time, is the interval between noon and the sun's leaving the plane.

76. Taking the dial constructed in art. 73 as an example, it will be found that when the sun is in the northern tropic, that is, when $PI\; =\; 66^\{\backslash circ\}\; 30\text{'}$, the angle IPA = $82^\{\backslash circ\}\; 12\text{'}$. Now (art. 74), HPA = $25^\{\backslash circ\}\; 20\text{'}$; therefore, when the sun ceases to shine on the plane, the horary angle IPH from noon is $107^\{\backslash circ\}\; 32\text{'}\; =\; 7$ hours 10 minutes, the sum of the two arcs; and when he begins to shine on it, the horary angle is $56^\{\backslash circ\}\; 52\text{'}\; =\; 3$ hours 47 minutes, their difference. Hence it will be needless to trace on the dial any hour line earlier than VIII. in the morning, or later than VIII. in the evening.

Of the Line described by the extremity of a Shadow on a Plane.

77. Sometimes the line which is the boundary of the space passed over by the shadow of the axis of a dial on given days of the year is traced on its plane. The path of the extremity of the shadow when the sun enters the different signs of the ecliptic is an elegant appendage to a dial, and the geometrical problem which determines it is interesting.

78. Setting aside the considerations of the change in the sun's declination, the apparent diurnal path of the sun in the heavens is, in the theory of dialling, considered to be a circle parallel to the equator; therefore any fixed point that projects a shadow throughout a day may be considered as the vertex of a cone, the base of which is the diurnal circle described by the sun on that day, and the space bounded by the shadow will be the same cone continued beyond its vertex. Thus it appears that the shadow of a fixed point in space generates the surface of a cone whose axis passes through the pole of the world; and hence the path of the shadow on any plane will be a conic section, because it is the line in which the surface of the cone meets the plane.

Let O be the centre of a horizontal dial, OF the axis, O XII. the meridian line, and OB the shadow of the axis at any time; join FB, draw FA perpendicular to O XII., and AD perpendicular to OB, meeting OB in D; and join FD. Because FA is perpendicular to the plane BO XII. the plane of the triangle FAD is perpendicular to that plane. Now, by construction, BD is perpendicular to their common section DA, therefore it is perpendicular to the plane

of the triangle $FAD$, and consequently $FD$ is perpendicular to $OB$.

Let $a\; =\; OF$ the length of the axis;
$r\; =\; OB$ the length of its shadow;
$v\; =\; \backslash begin\{cases\}\; \backslash text\{the\; angle\; \}\; BOA\; \backslash text\{\; contained\; by\; the\; shadow\}\; \backslash \backslash \; \backslash text\{and\; the\; meridian;\}\; \backslash end\{cases\}$
$x\; =\; \backslash begin\{cases\}\; \backslash text\{the\; angle\; \}\; FOB\; \backslash text\{\; contained\; by\; the\; shadow\}\; \backslash \backslash \; \backslash text\{and\; the\; axis;\}\; \backslash end\{cases\}$
$L\; =\; \backslash text\{the\; angle\; \}\; FOA,\; \backslash text\{\; the\; latitude;\}\; \backslash \backslash \; D\; =\; \backslash text\{the\; angle\; \}\; OFB,\; \backslash text\{\; the\; sun\text{'}s\; polar\; distance;\}\; \backslash \backslash \; \backslash text\{in\; the\; triangles\; \}\; ODF,\; ODA,\; \backslash text\{\; right-angled\; at\; \}\; D,\; \backslash \backslash \; OD\; :\; OF\; =\; \backslash cos.\; FOD\; :\; \backslash text\{rad.\}\; \backslash \backslash \; OA\; :\; OD\; =\; \backslash text\{rad.\}\; :\; \backslash cos.\; AOD,\; \backslash \backslash \; \backslash text\{therefore\; \}\; OA\; :\; OF\; =\; \backslash cos.\; FOD\; :\; \backslash cos.\; AOD;\; \backslash \backslash \; \backslash text\{but\; in\; the\; triangle\; \}\; OAF,\; \backslash \backslash \; OA\; :\; OF\; =\; \backslash cos.\; FOA\; :\; \backslash text\{rad.\}\; \backslash \backslash \; \backslash text\{therefore\; \}\; \backslash cos.\; FOA\; :\; \backslash text\{rad.\}\; =\; \backslash cos.\; FOD\; :\; \backslash cos.\; AOD.\; \backslash \backslash \; \backslash text\{From\; the\; triangle\; \}\; FOB\; \backslash text\{\; we\; get\; this\; other\; proportion,\}\; \backslash \backslash \; \backslash sin.\; (F\; +\; O)\; \backslash text\{\; or\; \}\; \backslash sin.\; B\; :\; \backslash sin.\; F\; =\; OF\; :\; OB.$

The last two proportions in symbols are
$\backslash cos.\; L\; :\; \backslash text\{rad.\}\; =\; \backslash cos.\; v\; :\; \backslash cos.\; y,$
$\backslash sin.\; (D\; +\; v)\; :\; \backslash sin.\; D\; =\; a\; :\; r.$

By the first proportion, we may determine $v$ from $y$, the angle the shadow makes with the meridian; and by the second we get $r$, the length of the shadow. We may also express its length by the hour angle from noon; for if we put $x$ to denote that angle, then (art. 23),

From the first and last of these three proportions, it is easy to infer, by spherical trigonometry, that the arcs $L$ and $y$ are the sides of a right-angled spherical triangle, of which $v$ is the hypotenuse, and that $x$ is the angle opposite the side $y$; hence it follows that

To determine the length of the shadow, we have now these formulae,

By these, and the formulae

the position of the shadow, and its length, may be found at any time on any given day; the sun's declination being taken from the table which concludes this article.

79. It has been shown (art. 78) generally, that the line described by the shadow is a conic section: the formulae just found serve to determine its nature in any given case.

By the calculus of sines (ALGEBRA, art. 239), $\backslash sin.\; (D\; +\; v)\; =\; \backslash sin.\; D\; \backslash cos.\; v\; +\; \backslash cos.\; D\; \backslash sin.\; v$; now $\backslash sin.\; v\; =\; \backslash sqrt\{1\; -\; \backslash cos^2\; v\}$, therefore, from formulae (2) and (3), putting $\backslash text\{rad.\}\; =\; 1$, we have

This is the polar equation of the line which limits the shadow; it may also have this form,

Supposing now $t$ and $u$ to be rectangular co-ordinates, which have their origin at $O$; we have $t\; =\; r\; \backslash cos.\; y$, and $r^2\; =\; t^2\; +\; u^2$; therefore, by substitution, and taking the squares of both sides, we find

an expression which by the calculus of sines may be transformed to

This is the equation of the line described by the shadow referred to axes perpendicular to each other; it contains the first and second powers of the co-ordinates and constants, therefore the line is, in general, a conic section.

80. If $L\; =\; D$, then $\backslash sin.\; (L\; -\; D)\; =\; 0$. In this case the term which contains $t^2$ vanishes, and the remaining terms indicate that the curve is a parabola.

If $L$ is greater than $D$, so that the sine of $D\; -\; L$ is negative, the equation belongs to an ellipse; but if $L$ be less than $D$, the equation belongs to a hyperbola. In each case the meridian line is the transverse axis of the curve.

The path of the shadow is an ellipse at any place within the polar circle on the days when the sun does not set; it is a parabola at that place on the day that the sun just touches the horizon at midnight; it is a straight line at all places of the earth on the equinoctial days; and in every other case it is a hyperbola.

81. The points in which the curve crosses its axis may be found from its polar equation by making $u\; =\; 0$, and $\backslash theta\; =\; 180^\backslash circ$. If $r\text{'}$ and $r\text{'}\text{'}$ denote the distances from the centre, when $u\; =\; 0$, then $r\text{'}\; =\; \backslash frac\{a\; \backslash sin.\; D\}\{\backslash sin.\; (D\; +\; L)\}$; $r\text{'}\text{'}\; =\; \backslash frac\{a\; \backslash sin.\; D\}\{\backslash sin.\; (D\; -\; L)\}$.

The first of these is the length of the shadow at noon. The vertices of the curve lie on the same side of the centre of the dial when it is a hyperbola, but on opposite sides when it is an ellipse. The other elements of the curve may be found in the same way, and the curve may be described as a conic section. In practice it will however be sufficient to find the points in which the curve crosses the hour lines of the dial, and then trace the curve through them by hand. The intersection of the curve and any hour line may be found by this construction.

Let $O$ be the centre of the dial, and $O\; XII$ the meridian line. Take any two lines, $OM,\; ON$, in the proportion of the cosine of the latitude to radius; and about $O$ as a centre, with these distances, describe circles. Take $OF$ in the meridian line equal to the axis of the dial, and make the angle $OFH$ equal to the sun's distance from the

Dialling. pole. Let K be the intersection of any hour line and the lesser circle. Draw KL perpendicular to the meridian, meeting the greater circle in G. Draw OG, meeting FH in H. In the hour line OY, take OB equal to OH, and B is the point in which the hour line meets the path of the shadow.

For, by trigonometry,
$OG\; :\; OK\; =\; \backslash sin.\; OKL\; :\; \backslash sin.\; OGL\; =\; \backslash cos.\; KOL\; :\; \backslash cos.\; GOL$;
that is, because $KOL\; =\; YOL\; =\; y$,
$\backslash text\{rad.\}\; :\; \backslash cos.\; L\; =\; \backslash cos.\; y\; :\; \backslash cos.\; HOL$.
Hence the angle $HOL\; =\; v$ (formula 2 of art. 78); now
$HFO\; =\; D$, and $OF\; =\; a$, and $\backslash sin.\; H\; :\; \backslash sin.\; F\; =\; OF\; :\; OH$,
that is, $\backslash sin.\; (D\; +\; v)\; :\; \backslash sin.\; D\; =\; a\; :\; OH$;
therefore $OH\; =\; OB$ is the length of the shadow (formula 3).

Whatever has been shown regarding the shadow of a horizontal dial, will apply to any dial whatever, if L be put for the latitude of the place where the dial would be horizontal, and the substyle be taken for the meridian line.

Dials with variable Centres.

82. Dials of this kind are not common; yet they are deserving of attention, because of the elegance of their geometrical theory. Their construction depends on this principle.

It is possible to determine a system of hour points on a plane, such, that if a style be placed in the plane of the meridian at certain points, to be found, corresponding to the days of the year, and making with the horizon any given angle, its shadow shall pass through the hour points at the times they indicate, and in this way show the time of the day.

83. To establish the truth of this proposition, the following problem is to be resolved.

Having given the sun's declination, the time from noon, and the latitude of a place; to find the angle which the shadow of a style in the plane of the meridian, and inclined at a given angle, makes with the meridian line on a horizontal plane.

Let LMN be the horizon, LPN the meridian, P the pole, CQ the style, which, being produced, meets the celestial meridian in Q. Let S be the sun in the hour circle PS, and MSQA a great circle passing through S and Q, and cutting the horizon in the line MCA.

Put $L\; =\; PN$, the given latitude;
$E\; =\; QN$, the given elevation of the style;
$D\; =\; PS$, the sun's distance from the pole;
$x\; =\; QPS$, the hour angle;
$z$ = the angle $LQM$;
$y$ = angle $ACN$, or arc $LM$, which is to be found.
In the spherical triangle PSQ, by spherics,
$\backslash cot.\; PS\; \backslash cdot\; \backslash sin.\; PQ\; =\; \backslash cot.\; Q\; \backslash sin.\; P\; +\; \backslash cos.\; PQ\; \backslash cos.\; P$;
and in the spherical triangle QLM, right-angled at L,
$\backslash cot.\; LQM\; =\; \backslash cot.\; LM\; \backslash sin.\; QL$.
Now, in the first of these formulae,

$PS\; =\; 90^\backslash circ\; -\; D$, $PQ\; =\; E\; -\; L$, $Q\; =\; 180^\backslash circ\; -\; z$, $P\; =\; x$;
and in the second,

$LQM\; =\; z$, $LM\; =\; y$, $\backslash sin.\; QL\; =\; \backslash sin.\; QN\; =\; \backslash sin.\; E$;
therefore, by substituting the symbols in the formulae, we have

$\backslash cot.\; z\; =\; \backslash cot.\; y\; \backslash sin.\; E\; \backslash dots\; \backslash dots\; \backslash dots\; (2.)$
Let the value of $\backslash cot.\; z$ be substituted instead of it in the first equation, then, deducing from the result the value of $\backslash cot.\; y$, we find

This formula gives the value of $y$, the angle made by the shadow and the meridian of the dial, which was required.

84. Fig. 13 of Plate CCII. represents a dial with a moveable centre, O XII. being the meridian line, and the hour points as shown in the figure. Let C be the variable centre, which is the position of the bottom of the style on any given day, A any one of the hour points, and AC the position of the shadow at that hour on the given day; then, $x$ denoting the hour angle from noon, and $y$ the hour line angle AC XII., the preceding formula gives the value of $y$. Draw AB perpendicular to the meridian, put $OB\; =\; t$, $AB\; =\; u$, $t$ and $u$ being the co-ordinates of an hour point; and put $r$ for OC, the variable distance of the bottom of the style from the fixed point C in the plane of the

dial; then $Be\; =\; t\; -\; r$. And because $\backslash cot.\; C\; =\; \backslash frac\{Be\}\{BA\}$, that is,

therefore

and hence, deducing the value of $r$,

By the nature of the dial, the position of the style must depend entirely on the sun's declination, and be altogether independent of the hour of the day. These conditions will be satisfied if we make

for by this assumption $t$ and $u$ are independent of $D$, the sun's declination; and $r\; =\; a\; \backslash tan.\; D$ independent of $x$. By resolving the equations, we obtain

These equations express the nature of every dial of this kind.

85. To construct the dial, we must assume a line of any convenient length, as a scale on which the co-ordinates of the hour points may be measured. The values of $t$ and $u$ are now to be computed by making $x\; =\; 15^\backslash circ$ for the hours of XI. and I.; again, $x\; =\; 30^\backslash circ$ for the hours of X. and II., and so on. A graduated scale must be formed along the meridian line, proceeding both ways from O, that point being the position of the style at the equinoxes; and since it appears from the figure in art. 83, that for any given hour angle $x$, the angle $y$ at the variable centre $c$ ought to increase as the sun approaches the north pole, the scale of declination for the north side of the equator

Dialling. must lie on the north side of O; and that for the south, on the opposite side. Lastly, the months and days of the year ought to be placed on the scale opposite to the degrees of declination to which they correspond.

86. We may investigate, from the equations, the nature of the curve that is the locus of the dial points, or which has $t$ and $u$ for rectangular co-ordinates. This will be obtained by eliminating the angle $x$.

we have now

hence it appears that the locus of the hour points is an ellipse, of which the axes are $m$ and $n$.

Azimuth or Analemmatic Dial.

87. This dial, represented by fig. 16, Plate CCII., is of the kind having a variable centre. Its style is vertical, therefore, by what has been shown in par. 84, the equations of the hour points on the dial will be

The dimensions of the ellipse which is the locus of the hour points are these:

These values of $t$, $u$, $r$, serve to find the hour points by calculation. They may also be found by this construction, fig. 15.

Draw two straight lines $Aa$, $Bb$, intersecting at right angles in O; and in OA, one of these lines, take OD of any suitable length for the eccentricity of the dial; draw DB so as to make with DO an angle equal to the latitude of the place; then CB will be half the lesser axis, and B the twelve o'clock hour point.

In OD, take OA and Oa each equal to DB; and Aa will be the hour points for six o'clock in the morning and evening.

On O as a centre, with OA and OB as radii, describe circles, and divide the quadrants in the same angle, each into six equal parts. From K, any one of the divisions of the outer circle, draw KL perpendicular to Oa; and from $k$, the corresponding point in the inner circle, draw $kN$ parallel to Oa, meeting KL in N; this will be one of the hour points, and in the same way may all the others be found. At the point D make angles ODE, ODe, each $23\backslash frac\{1\}\{2\}$ degrees, the sun's greatest declination; and E, s shall be the positions of the bottom of the style at the solstices, that of the summer being on the north of O, the middle of the dial. Describe a circle with DO as a radius, and find the tangents of the series of arcs $1^\backslash circ$, $2^\backslash circ$, $3^\backslash circ$, &c. of that circle, and lay them down as a scale from O to E and e, on each side. Find, in the tables which conclude this article, the sun's declination on the first day of every month, and mark the beginning of the month on the scale E opposite to its corresponding degree of declination. As many of the intermediate days may in like manner be laid down as there is room for.

The style must now, by some contrivance, be placed over the scale, so as to admit of sliding along it, and being set to any day, and the dial is finished.

If a dial of this kind be united to a horizontal dial, as they can only show the same hour when their meridian lines are in their true position, the compound dial may be set to show time without the help of a compass or meridian line.

Lambert's Dial.

88. This is a particular case of the class of dials with a variable centre. It was given by M. Lambert in the Berlin Ephemerides for 1777.

It has been found (art. 86), that L being put to denote the latitude, and E the angle which the style is to make with the meridian line, the semiaxes of the ellipse, which is the locus of the hour points, are

Now these will be equal, if the cosine of $E\; -\; L$ be equal to the sine of E, that is, if $E\; -\; L\; +\; E\; =\; 90^\backslash circ$, or $E\; =\; \backslash frac\{1\}\{2\}(90\; +\; L)$. When E, the elevation of the style, has this value, the hour points are in the circumference of a circle. At London E would be $70^\backslash circ\; 45\text{'}$.

Lambert's dial may be constructed geometrically thus: (see fig. 14 of Plate CCII.)

Take a straight line OD of any length, and draw a perpendicular OB. At the point D, make the angle ODB equal to half the sum of $90^\backslash circ$ and the latitude of the place, and OB will be the radius of the dial.

Describe a circle about O as a centre, with the radius OB, and divide the quadrants each into six equal parts, and the points of division will be the hour points on the dial.

Draw the lines DE, De, and make a scale of tangents of the sun's declination from O to E and e, and against the divisions of the scale write the days of the month exactly as directed in the Analemmatic dial. Also place the style over the meridian, so that it may be adjusted to the time of the year, and make with the meridian line an angle equal to BDO, and the dial will be constructed.

Note.—The style must be on the north or south of the point O, according as the sun is on the north or south side of the equator.

In these dials, instead of making the style moveable, it might be fixed, and different sets of hour points found for different days of the year. This, however, would make the construction laborious.

Thus far we have given the theory of the dials, along with their construction. In what follows, we shall, for the sake of brevity, simply give their construction. In general, the theory is just an application of what has been already delivered.

Portable Dial on a Card.

89. This dial, represented by fig. 11, Plate CC., has been called the Capuchin, because of a supposed resemblance it has to the head of a capuchin friar with his cowl inverted. It may be constructed as follows:

Draw a straight line ACB parallel to the top of the card (fig. 10), and another DCE bisecting the former at right angles; on C as a centre, with any convenient radius CA, describe a semicircle, and divide it into twelve equal parts, at the points From the points draw lines perpendicular to ACB, the diameter of the circle, and these will be the hour lines, viz. the line through $r$ will be the hour line of XI. I. (see fig. 11), that through $s$ the hour line of X. II. &c., the hour of XII. noon being at A. The half hours and quarters may also be laid down by subdivision. At A, the extremity of the diameter, draw a line AD, making with AC an angle equal to the latitude of the place, and meeting the six o'clock line in D, through which draw a line FDG at right angles to AD. At the point A, draw lines AF, AG, to make with AD, on opposite sides, angles of $23\backslash frac\{1\}\{2\}$ degrees, the sun's greatest

Dialling. declination: these determine the line FDG, the scale of months. Describe a semicircle on FG as a diameter; divide it into six equal parts at $m,\; n,\; o,\; p,\; q$, and draw $mh,\; nl,\; od,\; pk,\; qi$ perpendicular to FG; these points are the centres of the arcs of the signs.

On F and G as centres, through A, describe the arcs AH, AO, for the tropics of Capricorn and Cancer, the former of which is marked $\backslash gamma$ and the latter $\backslash omega$ (fig. 11); also on $h$ and $i$, as centres through A, describe the arcs AI, AN; the one will be the arc of the signs Aquarius $\backslash pi$ and Sagittarius $\backslash zeta$; and the other the arcs of Gemini $\backslash pi$ and Leo $\backslash sigma$; and on $l$ and $k$, as centres through A, describe the arcs AK and AM, one for the signs Pisces $\backslash chi$ and Scorpio $\backslash mu$, and the other for Taurus $\backslash delta$ and Virgo $\backslash sigma$; and lastly, on D as a centre, through A, describe the arc AL for the signs Aries $\backslash gamma$ and Libra $\backslash omega$. On A as a centre, with any convenient radius, describe an arc RST, terminating in the lines AF, AG, and divide each half into 23$\backslash frac\{1\}\{2\}$ equal parts for the scale of the sun's declination. Find from the tables which conclude this article the sun's declination for every fifth day of the year, and, laying a ruler over A, and the degree of each day on the scale RT, mark the point in which the ruler meets the line FG, thereby forming the scale for the months and days of the year, as shown in fig. 11 (between A and B), observing that the days from 21st of March to 23d September must be on the right hand side of D, the middle of the scale. Cut a slit through the card along the line AB in fig. 11, corresponding to FG in fig. 10, and through it put a thread having a bead $c$ sliding along it, and a plummet D at one end, which may hang freely along the face of the dial when it is held vertically; and make a knot on the thread at the back of the dial, so that it may not be drawn through the slit. Draw the shadow line (fig. 11) parallel to the top of the card, and on $ab$, a part of it, form the rectangle $abcd$ for the gnomon. Cut the card through, along the three sides $ba,\; ad,\; dc$ of the rectangle, and partly through along $bc$, the fourth, on the back of the card, so as to admit of the rectangle turning on $bc$ as a hinge. The hours, the scale of months, and scale of declinations, being drawn on the card, as shown in fig. 11, the dial will be finished.

90. To rectify the dial, set the thread in the slit AB against the day in the month, and stretch it over the point where the circles of the signs meet at XII.; then slide the bead $c$ along the thread until it be on that point, and the dial is rectified.

To find the hour of the day, raise the gnomon by turning it about the line $bc$, so that when the card is held vertically with its edge towards the sun, the side $ab$ of the gnomon may cast a shadow along the shadow line; then, the thread being brought into a vertical line, passing along the face of the card by the weight of the plummet, the hour line on which the bead falls indicates the time of the day before or after noon.

Note. About noon the hour will be shown with uncertainty; this is an inconvenience inseparable from all dials that show the hour by the sun's altitude.

To find the time of sun rising and setting: Having rectified the dial for the given day, move the thread about its end in the slit until it be parallel to the hour lines; it will then cut the time of sun rising among the forenoon hours, and of setting among those of the afternoon.

To find the sun's declination: Stretch the thread from the day of the month over the hour point at XII., and it will indicate the declination on the graduated arch.

To find on what days the sun enters the signs: When the bead, as above rectified, moves along any one of the arcs of the signs, the sun enters that sign on the day pointed out by the thread on the scale of months.

Dial on a Cylinder.

91. This dial is represented by fig. 4, Plate CCI. It shows the hour of the day, the sun's place in the ecliptic, and his altitude at the time of observation. The dial is constructed by tracing the lines on a rectangular piece of paper (fig. 5), and pasting it on the surface of the cylinder. The lines may be traced by the following rules. Having formed on paper a rectangle ABDC, of such a size that, when wrapped round the cylinder, its opposite sides AC, BD, may just coincide; produce the side BA to $a$, any convenient distance, and on $a$ as a centre, with the distance $aA$, describe the quadrantal arc AE, and divide it into ninety equal parts for degrees; draw lines from the centre through as many divisions of the quadrant as there are degrees of the sun's altitude in the longest day of the year at noon of the place for which the dial is to serve, which at London is 62°, and continue these lines until they meet the tangent line AC. From the points of meeting draw lines across the rectangle parallel to AB, as shown in the figure, and these will be the parallels of the sun's altitude, in whole degrees, from sunrise to sunset, on all the days of the year.

Divide AB, CD, the top and bottom sides of the rectangle, into twelve equal parts for the signs of the ecliptic, and draw straight lines, joining the points of division, on which, at the bottom, place the characters of the signs, as in the figure. The twelve divisions should be subdivided by parallel lines into halves, and, if there be room, into quarters.

At the top of the rectangle make a scale of the months and days of the year, so as that the division of the scale for each day may stand over the division for the sun's place on that day in the scale of signs. The sun's place in the ecliptic each day of the year 1830, the second after leap year, is given at the end of this article, and may serve for this scale. Compute the sun's altitude for every hour in the latitude of the place, when he is in the beginning and middle of each sign, and in the upright parallel lines make marks for these computed altitudes among the parallels of altitude, reckoning them downward according to the order of the numeral figures set to them along the side BD, answering to the divisions of the quadrant on the opposite side; and through these marks draw the curve hour lines, and set the hours to them, as in the figure, reckoning the forenoon hours downwards, and the afternoon hours upwards. The sun's altitude should also be computed for the half hours; as to the quarter hour points, they may be put nearly in their proper place by estimation with the eye.

The scales and hour lines being constructed, the part of the paper on which the quadrant was drawn is to be cut off along the line AC, and that on the opposite side along BD. The superfluous parts at the top and bottom are also to be removed, and it will be fit for being pasted round the cylinder.

The cylinder, reduced in size, is represented by ABCD, fig. 4. It should be hollow, to hold the style DE when it is not used. The crooked end of the style is put into a hole in the top AD of the cylinder, which fits on it tightly, and may be turned round, like the lid of a paper snuff-box. The style must stand straight out, perpendicular to the side of the cylinder, just over the line AB, round the top from which the parallels of the sun's altitude are reckoned; and the length of the style, projecting beyond the cylinder, must be exactly equal to $Aa$, the radius of the quadrant in fig. 5.

To use the dial, place the base of the cylinder on a level table where the sun shines, and turn the top round till the style stand directly over the day of the month; then turn

Dialling. the cylinder about on the table till the shadow falls on it parallel to those upright lines which divide the signs, that is, till the shadow be parallel to the (supposed) axis of the cylinder; and then the point or lowest shadow will fall upon the time of the day among the curve hour lines, and will show the sun's altitude at that time, among the cross parallels of his altitude which go round the cylinder; at the same time it will indicate the sign of the ecliptic in which the sun is.

The dial may also, when used, be suspended by the ring F in its top. Dials of this kind, of rather rude construction, are sometimes used by the rural population in France.

92. The construction of this dial requires that the sun's altitude be found for any given hour of the day, knowing the latitude of the place and the sun's declination.

Let $D$ be the sun's declination,

$L$ the latitude,

$x$ the hour angle at the pole,

$z$ the sun's altitude.

Then $90^\backslash circ\; -\; D$, $90^\backslash circ\; -\; L$, $90^\backslash circ\; -\; z$, are the three sides of a spherical triangle, and $x$ is the angle opposite to the last of these sides; hence, by spherics,

This is equivalent to

From this formula, by giving different values to $x$, the hour from noon, there will be found corresponding values of $z$, the sun's altitude.

Ex. To find the sun's altitude at ten in the forenoon or two in the afternoon on the 21st of May at London, in latitude $51^\backslash circ\; 31\text{'}$, the sun then nearly entering the sign gemini. In this case, from the table at the end, $D\; =\; 20^\backslash circ\; 8\text{'}$, $S\text{'}$, we have also $L\; =\; 51^\backslash circ\; 31\text{'}$, and $x\; =\; 2^h\; =\; 30^\backslash circ$.

The formula supposes that the latitude and declination are of the same name, that is, both north or both south. When one is north and the other south, then the first term will be negative, and we have for this case

$\backslash sin.\; z\; =\; \backslash sin.\; L\; \backslash sin.\; D\; (\backslash cot.\; L\; \backslash cos.\; D\; \backslash cos.\; x\; -\; 1)\; \backslash dots\; (3.)$; and when the hour angle exceeds $90^\backslash circ$, it must be remembered that $\backslash cos.\; x$ is a negative quantity, so that the expression in which it occurs is subtractive.

Ring Dial.

93. This kind of dial is shown in figure 7. It is formed of a brass ring or rim, usually about two inches in diameter, and one third of an inch in breadth. In a point of this rim there is a hole, through which the sun's light passes, and forms a lucid speck on the opposite concave surface, on which the hours of the day are marked.

These divisions are made by describing a circle to represent the ring, and drawing a horizontal chord $EF$; with this as a radius describe a quadrant $FD$, and divide it into degrees, and through those which mark the sun's altitude at every hour of the day, at the time of either equinox, draw lines from $E$ to the opposite side of the circle, and there put the corresponding figures. Thus for a place whose latitude is $51^\backslash circ\; 32\text{'}$, the XII. o'clock line will pass through $38^\backslash circ\; 38\text{'}$ the colatitude, and the XI. and I. line through $36^\backslash circ\; 56\text{'}$, the X. and II. through $32^\backslash circ\; 36\text{'}$, the IX. and III. through $26^\backslash circ\; 5\text{'}$, the VIII. and IV. through $18^\backslash circ\; 8\text{'}$, and the VII. and V. through $9^\backslash circ\; 17\text{'}$. The dial, thus constructed, would only show time at the equinox; but to make it perform the whole year, the hole is made

moveable, and the signs of the ecliptic or the days of the month are marked on the convex surface of the ring by taking $ET$ and $Et$, on each side of $E$, equal to twice the sun's declination when he enters any particular sign, as Taurus and Pisces, and there marking the character of the sign or corresponding month, and so for all the others; and by these the dial is rectified for the time of the year. From the figure it appears that $E\; XII.\; \backslash delta$ is equal to $FE\; XII.$ or the altitude of the sun in the equinox; but $T\; XII.\; E$ is equal to the sun's declination in Taurus, because it is an angle at the circumference standing on an arc which is double the declination; therefore $T\; XII.\; \backslash delta$ is equal to the meridian altitude when the sun enters Taurus, and a ray passing through the hole at $T$ will mark the XII. o'clock hour of that day. But this dial will not show the other hours exactly, because $T\; III.\; d$ exceeds $E\; III.\; d$, the equinoctial hour, by the angle of declination, and the same holds of the rest. To remedy this defect, the concave surface of the ring has sometimes been made broader, and seven circles described upon it, the middle one to represent the equinoctial, and the extremes the tropics; and on these circles the hours have been marked as shown in a table of altitudes.

To use the dial, set the moveable hole to the day of the month, or the degree of the sun's place in the ecliptic; then, suspending it by the ring, turn it towards the sun till his rays point out the hour among the divisions in the inside.

Universal or Astronomical Ring Dial.

94. This is represented in figure 6 of Plate CCI. It serves to show the time of the day in any part of the earth, whereas the former is only adapted to a particular latitude. It consists of two rings or flat circles from two to six inches in diameter. The outer ring $A$ represents the meridian of any place. Two of its opposite quadrants are divided into $90^\backslash circ$, serving, the one for north and the other for south latitude. The inner ring represents the equator, and turns within the outer on two pivots at the extremities of a diameter, where the hour XII. is marked. A thin reglet or bridge goes across the circles, and in its middle there is an opening, along which there slides a cursor $C$, having a small hole in it for the sun to shine through. The middle of this bridge represents the axis of the celestial sphere, and its extremities the poles; on one side of it the signs of the zodiac are drawn, and on the other the days of the month. A piece, to which a ring $H$ is fixed, slides along the meridian; and the dial when used is suspended by the ring.

In this dial, the divisions laid down on the axis on either side of the centre are the tangents of the angles of the sun's declination, to a radius equal to that of the brass circle, which represents the equator. These may be laid down from a scale of equal parts, of which 1000 answer to the length of the semiaxes from the centre to the equinoctial ring; and then the extreme divisions on the axis will be at the distance of 434 of these equal parts (the tangent of $23^\backslash circ\; 29\text{'}$) from the division at the centre.

To use this dial, place the line marked on the sliding piece (opposite to the supporting ring) over the degree of the latitude of the place, and put the line which crosses the hole of the cursor to the degree of the sign or to the day of the month. Turn the inner ring on its pivots, and put it at right angles to the outer; and suspend the instrument by the ring $H$, so that the middle of the bridge (which is the axis of the dial) may be parallel to the axis of the world; then turn its flat side towards the sun, so that his rays passing through the little hole in the cursor may fall exactly on a line drawn round the middle of the concave surface of the inner ring; and in this posi-

Dialling. tion the bright spot shows the hour of the day on that surface.

The hour of XII. is not shown by this dial, because the outer circle, being then in the plane of the meridian, hinders the sun's rays from falling on the inner. Nor will it serve to show the hour on the equinoctial days, because his rays then pass parallel to the plane of the dial.

Universal Dial on a Cross.

95. This dial, represented by figure 1 of Plate CCI. is moveable on a joint C for elevating it to any latitude on the quadrant CO 90, as it stands on the horizontal board A. The arms of the cross stand perpendicularly to the middle part, and the top of it from a to n is of equal length with either of the arms mk, nt.

Having set the board A level, and the line ut (on the middle of a side of the cross) to the latitude of the place on the quadrant, and the point N on the compass north by the needle, allowing for the variation; the plane of the cross will then be parallel to the plane of the equator, and the dial rectified so as to show time.

The morning hours from III. till VI. will be shown by the upper edge kl of the arm io casting a shadow on a face of the arm em; from VI. to IX. the lower edge i of the arm io will cast a shadow on the hours of the side og; from IX. to XII. noon, the edge ab of the upper part an will cast a shadow on the arm nf; from XII. to III. in the afternoon, the edge cd of the top part will cast a shadow on the hours of the arm klm; and from III. to VI. the edge gh will cast a shadow on the hours on the part pu; and from VI. to IX. the shadow of the edge ef will show the time on the top part an.

The breadth of each part ab, ef, &c. must be so great as never to let the shadow fall entirely without the part or arm on which the hours are marked, when the sun is at his greatest declination from the equator. To determine the breadth of the sides of the arms which contain the hours, make an angle ABC (fig. 2) of $23\backslash frac\{1\}\{2\}^\backslash circ$, the sun's greatest declination; and in Be take Bd, equal to the intended length of each arm, from the side of the long middle part, which is also the length of the top part above the arms, and draw the perpendicular def.¹ Then, as the edges of the shadow from each arm will be parallel to BA when the sun's declination is $23\backslash frac\{1\}\{2\}^\backslash circ$, it is plain that if the length of the shadow be Bd, the least breadth it can have to keep the edge Be of the shadow Begd from going off the side of the arm before it comes to its end, must be de or Bd; but in order to keep the shadow within the quarter divisions of the hours when it comes near the end of the arm, the breadth thereof should be still greater, so as to be almost doubled, on account of the distance between the tips of the arms.

To place the hours right on the arms, lay down the shape and size of the cross abcd on paper (fig. 3), and on a as a centre, with ae as a radius, describe the quadrantal arcs ef. Divide this arc into six equal parts, and through the divisions draw lines ag, ah, &c. continuing three of them to the arm ae, which are all that fall on it, and they will meet the arms in the points through which the hour lines are to be drawn right across it.

Divide each arm in the same manner, and set the hours to their proper places, as shown in the figure; each of the hour spaces should be divided into four equal parts for the half hours and quarters in the quadrant ef; and straight lines should be drawn through the division marks of the quadrant to the arms of the cross, in order to de-

termine the places where the subdivisions of the hours Dialling. must be marked.

An universal Dial, showing the Hours by a Globe, and by several Gnomons.

96. The dial is represented in fig. 9, Plate CCI. It may be made of a thick square piece of wood or hollow metal. The sides are cut into semicircular hollows, on which the hours are placed, the style of each coming out from its bottom as far as the ends of the hollow projects. The corners are cut into angles, in the insides of which the hours are also marked; and the edge at the end of each side of the angle serves as a style for casting a shadow on the hours marked on the opposite side. In the middle of the uppermost plane there is an equinoctial dial, in the centre of which an upright wire is fixed for casting a shadow on the hours of that dial, and supporting a small terrestrial globe on its top. The whole dial stands on a pillar rising out of a circular base, in which there is a compass and magnetic needle for placing the meridian style towards the south. The pillar has a joint with a graduated quadrant on it (supposed to be laid from sight under the dial in its representation) for setting the dial to the latitude of any given place. The equator of the globe is divided into twenty-four equal parts, and the hours are laid down on it: the time of the day may be shown by these hours when the sun shines on the globe.

To construct the dial: On a square piece of wood or metal of proper thickness draw the lines ac, bd (fig. 8) at a distance equal to the intended thickness of the style abcd; and in the same manner trace out the form of the three other styles efgh, iklm, nopq, all directed towards the centre of the square. On a and b as centres, with such a radius as will leave sufficient strength of stuff, when the distance KI is equal to Aa, describe the quadrantal arcs Ae, Bd. In like manner, with the same radius describe arcs in all the quadrantal openings, leaving room, however, for the equinoctial dial in the middle. Divide each quadrant into six equal parts for as many hours, as in the figure, and subdivide these for half hours and quarters, numbering the whole eight quadrants as in the figure. To lay down the hours in the angular spaces in the corners: On K and I as centres, with a radius equal to KI, which is equal to Aa, describe the arcs Kt, It, meeting in t. Divide each arc into four equal parts, and from the centres through the points of division draw the right lines I3, I4, I5, I6, I7, and K2, K1, K12, K11; and they will meet the sides of the angle where the hours should be marked. These hour spaces should also be subdivided into quarters. Do the like for the other three angles, determining in this way the hours shown as in the figure, in which it will be observed that the dotted lines which are parallel are all directed to the same hour of the day. The angular and quadrantal spaces should now be cut out, quite through the solid material of which the dial is formed, and the gnomons inserted in their proper position, observing that these should be as broad as the dial is thick, and this breadth and thickness such as to keep the shadows of the gnomons from ever falling quite out of the hollows, even when the sun's declination is greatest. Lastly, construct the equinoctial dial in the middle, agreeably to its theory in art. 19, and the dial will be finished.

To rectify and use the dial: Place it with its base level, and the gnomons of the quadrants directed to the cardinal points by means of the compass; then bend the pillar in the joint, till the axis be inclined to the horizon at an

¹ Place dashes (omitted in the engraving) over the letters d and e in the perpendicular to Be, thus making them d', e'; and in fig. 3 observe that the letter a ought to be over the figure 3 on the adjoining line. Also in fig. 1 write δ on the top opposite to c.

Dialling. angle equal to the latitude of the place. The plane of the equinoctial dial will then be parallel to the equator, the axis of the globe directed to the pole of the heavens; and when the sun shines on the dial, the hours will be indicated by all the parallel edges which cast a shadow, as well as by the axis of the globe and the globe itself, which will show to what places of the earth the sun is rising, setting, and in the meridian.

Universal Mechanical Dial, Plate CCII. Fig. 11.

96. This is an equinoctial dial, which may be applied to the construction of a dial on any kind of plane, by a process requiring no calculation. Suppose, for example, the plane ABCD is horizontal, and that GEF is the meridian line, assumed if the plane be moveable, or found if it be fixed. Place the equinoctial dial H with its axis GI directed to the pole, or, in the present case, so as to make with the meridian line an angle equal to the latitude of the place, with the twelve o'clock hour line over the meridian line. If now EF, the lower edge of the upright triangular support KEF, lie along the meridian, and the edge EK make with EF an angle equal to the latitude, the dial H will be in its proper position when its axis GI is parallel to EK. Supposing now the whole apparatus to be in a dark place, let a lighted candle be carried round GI, the axis of the equinoctial dial, and let the position of the shadow on the horizontal plane be noted when it falls on the successive hours of the equinoctial dial, then lines from G through these positions will be the hour lines on the horizontal dial. In this way a dial may be described on any plane whatever.

Dials on three different Planes, Plate CCII. Fig. 12.

97. This combination of dials shows the hour at the same time on the equinoctial dial IK, the horizontal dial ABC, and the vertical south dial; the same axis or gnomon FHG serving all the three. The figure sufficiently explains its construction and application. The method of tracing the hour lines proposed in the preceding article may be applied to produce the horizontal and south dials from the equinoctial dial.

Babylonian, Italian, and Jewish Hours.

The dials which have been described in this article are all intended to show astronomical or apparent time, of which the hours are equal, if we set aside the equation of time. Such are the hours by which we reckon time; but besides these, there are Babylonian, Italian, and Jewish hours.

The Babylonian hours are reckoned from sun rising to sun rising, and are twenty-four equal hours, nearly of the same length as the common hours, only they are differently numbered.

The Italian hours begin at sunset, and are numbered to twenty-four at sunset next day. They also are equal, and nearly the same as the common hours.

The Jewish hours, otherwise called the old unequal planetary hours, are reckoned from sunrise; and the day, from sunrise to sunset, is divided into twelve equal parts or hours. The hours of one day, however, will not be

equal to those of another, at least in our climate, because Dialling. of the inequality of the length of the days.

The older writers on dialling teach how to construct dials which shall show time according to all these ways of dividing the day. They are, however, mere matters of curiosity, and therefore we think it sufficient, in this work, to have thus briefly noticed them. They are fully explained by Emerson in his Dialling; see also Ferguson's Lectures.

Table of the Sun's Longitude and Declination, and the Equation of Time, for every day in the year.

The construction of some of the dials which have been described requires that the sun's place in the ecliptic, that is, his longitude reckoned from the beginning of the sign Aries, and his declination for the different days of the year, should be known. And in order to convert apparent time into true time, that is, the time shown by the sun into the time shown by a good clock or watch going uniformly, the equation of time is wanted for every day of the year. None of these, however, can be exactly the same on the same day of every year at any given place. Thus, at Greenwich, for four succeeding years beginning with 1820, they were at noon on the first of March as follows:

There are various reasons why the above three elements are not the same on the same day of every year: the principal one is, because the sun requires nearly six hours more than 365 days to complete a revolution in the ecliptic; and this annual deficiency being compensated by the intercalary day in every fourth year, after this interval the elements return nearly, but not exactly, to the same value on the same day of the year. Our table corresponds to 1830, the second after leap year. This will be sufficiently accurate for the construction of dials, and finding true time by them for a considerable number of years to come.

The columns of the table containing the sun's longitude and declination require no explanation. The numbers in that for the equation of time have the signs of addition (+) and of subtraction (—) annexed to them. Thus the sign + joined to 3^rd. 50^th, the equation for the first of January, is understood to belong to all the days from the beginning of the year to the 15th of April inclusive, and to indicate that the equation is to be added to the time shown by the dial to get true time, or that shown by the clock. Again, the sign — belongs to all days from 16th April to 15th June inclusive, and shows that the equation must be subtracted from the time shown by the dial, and so on throughout the year.¹ (II. M.—2.)

¹ See also the article DIPLEIDOSCOPE.

TABLE showing the Sun's Longitude and Declination, and the Equation of Time, for every day of the second year after leap year.