1. From the doctrines of the philosophers above mentioned, are extracted the principles of dialling, and the explanation of the increase and decrease of the days in the different months. The sun at the times of the equinoxes, that is when he is in Aries or Libra, casts a shadow in the latitude of Rome equal to eight ninths of the length of the gnomon.k At Athens the length of the shadow is three fourths of that of the gnomon. At Rhodes five sevenths; at Tarentum nine elevenths; at Alexandria three fifths; and thus at all other places the shadow of the gnomon at the equinoxes naturally differs.
2. Hence in whatever place a dial is to be erected, we must first obtain the equinoctial shadow. If, at Rome, the shadow be eight ninths of the gnomon, let a line be drawn on a plane surface, in the center whereof is raised a perpendicular thereto; this is called the gnomon, and from the line on the plane in the direction of the gnomon, let nine equal parts be measured. Let the end of the ninth part A, be considered as a centre, and extending the compasses from that centre to the extremity B of the said line, let a circle be described. This is called the meridian.
3. Then of those nine parts between the plane and the point of the gnomon, let eight be allotted to the line on the plane, whose extremity is marked C. This will be the equinoctial shadow of the gnomon. From the point C through the centre A let a line be drawn, and it will be a ray of the sun at the equinoxes. Then extend the compasses from the centre to the line on the plane, and mark on the left an equidistant point E, and on the right another, lettered I, and join them by a line through the centre, which will divide the circle into two semicircles. This line by mathematicians is called the horizon.
4. A fifteenth part of the whole circumference is to be then taken, and placing the point of the compasses in that point of the circumference F, where the equinoctial ray is cut, mark with it to the right and left the points G and H. From these, through the centre, draw lines to the plane where the letters T and R are placed, thus one ray of the sun is obtained for the winter, and the other for the summer. Opposite the point E, will be found the point I, in which a line drawn through the centre, cuts the circumference; and opposite to G and the points K and L, and opposite to C, F, and A, will be the point N.
5. Diameters are then to be drawn from G to L, and from H to K. The lower one will determine the summer, and the upper the winter portion. These diameters are to be equally divided in the middle at the points M and O, and the points being thus marked, through them and the centre A a line must be drawn to the circumference, were the letters P and Q are placed. This line will be perpendicular to the equinoctial ray, and is called in mathematical disquisitions, the Axon. From the last obtained points as centres (M and O) extending the compasses to the extremity of the diameter, two semicircles are to be described, one of which will be for summer, the other for winter.
6. In respect of those points where the two parallels cut that line which is called the horizon; on the right hand is placed the letter S, and on the left the letter V, and at the extremity of the semicircle, lettered G, a line parallel to the Axon is drawn to the extremity on the left, lettered H. This parallel line is called Lacotomus. Finally, let the point of the compasses be placed in that point where this line is cut by the equinoctial ray, and letter the point X, and let the other point be extended to that where the summer ray cuts the circumference, and be lettered H. Then with a distance equal to that from the summer interval on the equinoctial point, as a centre, describe the circle of the months, which is called Manacus. Thus will the analemma be completed.
7. Having proceeded with the diagram and its formation, the our lines may be projected on the analemma according to the place, either by winter lines, or summer lines, or equinoctial lines, or lines of the months, and as many varieties and species of dials as can be desired, may be constructed by this ingenious method. In all figures and diagrams the effect will be the same, that is to say, the equinoctial as well as the solstitial days, will always be divided into twelve equal parts. These matters, however, I pass over, not from indolence, but to avoid prolixity. I will merely add, by whom the different species and figures of dials were invented; for I have not been able to invent a new sort, neither will I pass off the inventions of others as my own. I shall therefore mention those of which I have any information, and by whom they were invented.
1. • Berosus the Chaldean, was the inventor of the semicircle, hollowed in a square, and inclined according to the climate. Aristarchus the Samian, of the Scaphe or Hemisphere, as also of the discus on a plane. The Arachne was the invention of Eudoxus the astrologer, although some attribute it to Apollonius. The Plinthium or Lacunar, an example of which is to be seen in the Circus Flaminius, was invented by Scopas the Syracusan. The sort called Πρὸς τὰ ἱστορούμενα, by Parmenio. That called Πρὸς πᾶν κλίμα, by Theodosius and Andrias. The Pelicinon by Patrocles. The Cone by Dionysodorus. The Quiver by Apollonius. The persons above mentioned not only invented other sorts; but the inventions of others have come down to us, such as the Gonarche, the Engonatos, and the Antiboreus.• Many also have left instructions for constructing the portable pendulous dials.
2. Ctesibius Alexandrinus was the first who found out the properties of the wind, and of pneumatic power, the origin of which inventions is worthy of being known. Ctesibius, whose father was a barber, was born at Alexandria. Endowed with extraordinary talent and industry, he acquired great reputation by his taste for his mechanical contrivances. Wishing to suspend a mirror in his father's shop, in such a way that it might easily be raised and lowered by means of a concealed cord, he used the following expedient.
3. Fixing a wooden tube under the beam, he attached pulleys to it upon which the cord passed and made an angle in descending into the wood which he had hollowed out: there he placed small tubes, within which a leaden ball attached to the cord was made to descend. It happened that the weight, in passing through the narrow parts of the tube, pressed on the inclosed air, and violently driving out at its mouth the quantity of air compressed in the tubes, produced by obstruction and contact a distinct sound.
4. Ctesibius having thus observed that by the compression and concussion of the air, sounds might be produced, he made use of the discovery in his application of it to hydraulic machines, to those automata which act by the power of inclosed water, to lever and turning engines, and to many other entertaining devices, but principally to water dials. First he made a perforation in a piece of gold or a smooth gem, because these materials are not liable to be worn by the action of the water, nor to collect filth, by which the passage of the water might be obstructed:
5. the water flowing through the hole equably, raises an inverted bowl, called by the workmen phellos, or the tympanum, with which are connected a rule and revolving drum wheels with perfectly equal teeth, which teeth, acting on one another, produce revolutions and measured motion. There are other rules and other wheels, toothed in a similar manner, which acted upon by the same force in their revolutions, produce different species of motion, by which figures are made to move, cones are turned round, stones or oviform bodies are ejected, trumpets sounded, and similar conceits effected.
6. On these also, either on columns or pillars, the hours are marked, to which a figure, holding a wand and rising from the lower part, points throughout the day, the increase and decrease whereof is daily and monthly adjusted, by adding or taking away certain wedges. To regulate the flow of the water, stoppers are thus formed. Two cones are prepared, one convex, the other concave, and rounded so as to fit exactly into each other. A rod, by elongating these, or bringing them together, increases or diminishes the flow of water into the vessel. In this manner, and according to the principles of this machine, water-dials for winter are constructed.
7. If the addition or removal of the wedges should not be attended by a correspondent increase or decrease in the days, for the wedges are frequently imperfect, it is thus to be remedied. Let the hours from the analemma be placed on the column transversely, and let the lines of the months be also marked thereon. The column is to turn round, so that, in its continual revolution, the wand of the figure, as it rises, points to the hours, and, according to the respective months, makes the hours long or short.
8. Other kinds of winter-dials are made, which are called Anaporica. They are constructed as follows. With the aid of the analemma the hours are marked by brazen rods on their face, beginning from the centre, whereon circles are drawn, shewing the limits of the months. Behind these rods a wheel is placed, on which are measured and painted the heavens and the zodiac with the figures of the twelve celestial signs, by drawing lines from the centre, which mark the greater and smaller spaces of each sign. On the back part of the middle of the wheel is fixed a revolving axis, round which a pliable brass chain is coiled, at one of whose ends a phellos or tympanum hangs, which is raised by the water, and at the other end a counterpoise of sand equal to the weight of the phellos.
9. Thus as the phellos ascends by the action of the water, the counterpoise of sand descends and turns the axis, as does that the wheel, whose rotation causes at times the greater part of the circle of the zodiac to be in motion, and at other times the smaller; thus adjusting the hours to the seasons. Moreover in the sign of each month are as many holes as there are days in it, and the index which in dials is generally a representation of the sun, shews the spaces of the hours; and whilst passing from one hole to another, it completes the period of the month.
10. Wherefore, as the sun passing through the signs, lengthens and shortens the days and hours, so the index of the dial, entering by the points opposite the centre round which the wheel turns, by its daily motions, sometimes in greater, at other times in less periods, will pass through the limits of the months and days. The management of the water, and its equable flow, is thus regulated.
11. Inside, behind the face of the dial, a cistern is placed, into which the water is conveyed by a pipe. In its bottom is a hole, at whose side is fixed a brazen tympanum, with a hole in it, through which the water in the cistern may pass into it. Within this is inclosed a lesser tympanum attached to the greater, with male and female joints rounded, so that the lesser tympanum turning within the greater, similar to a stopple, fits closely, though it moves easily. Moreover, on the lip of the greater tympanum are three hundred and sixty-five points, at equal distances. On the circumference of the smaller tympanum a tongue is fixed, whose tip points to the marks. In this smaller tympanum a proportionable hole is made, through which the water passes into the tympanum, and serves the work.
12. On the lip of the large tympanum, which is fixed, are the figures of the celestial signs; above, is the figure of Cancer, and opposite to it, below, that of Capricornus. On the right of the spectator is Libra, on his left Aries.All the other signs are arranged in the spaces between these, as they are seen in the heavens.
13. Thus, when the sun is in the portion of the circle occupied by Capricornus, the tongue stands in that part of the larger tympanum where Capricornus is placed, touching a different point every day: and as it then vertically bears the great weight of the running water, this passes with great velocity through the hole into the vase, which, receiving it, and being soon filled, diminishes and contracts the lengths of the days and hours. When, by the diurnal revolution of the lesser tympanum, the tongue enters Aquarius, all the holes fall perpendicular, and the flow of water being thus lessened, it runs off more slowly; whence the vase receiving the water with less velocity, the length of the hours is increased.
14. Thus, going gradually through the points of Aquarius and Pisces, as soon as the hole of the small tympanum touches the eighth part of Aries, the water flows more gently, and produces the equinoctial hours. From Aries, through the spaces of Taurus and Gemini, advancing to the upper points where the Crab is placed, the hole or tympanum touching it at its eighth division, and arriving at the summit, the power is lessened; and hence running more slowly, its stay is lengthened, and the solstitial hours are thereby formed. When it descends from Cancer, and passes through Leo and Virgo, returning to the point of the eighth part of Libra, its stay is shortened by degrees, and the hours diminished, till, arriving at the same point of Libra, it again indicates the equinoctial hours.
15. The hole being lowered through the space of Scorpio and Sagittarius, in its revolution it returns to the eighth division of Capricornus, and, by the velocity of the water, the winter hours are produced. To the best of my ability I have explained the construction and proportions of dials, so that they may be easily set up. It now remains for me to speak of machines, and the principles which govern them. These will be found in the following book, and will complete this Treatise on Architecture.