7From 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. 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.
2Hence in whatever place a dial is to be erected, we must first obtain the equinoctial shadow. If, as 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.
3Then 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.
4A 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 H the points K and L, and opposite to C, F, and A, will be the point N.
5Diameters 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, where 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.
6In 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.
7Having proceeded with the diagram and its formation, the hour 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 the 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.