In *De Architectura*, Book IX, Chapter VII, Vitruvius explains how to construct a diagram called the *analemma*, from which a sundial for any season of the year can be laid out. My text is that of the Loeb Classical Library, edited and translated by Frank Granger, pages 248-255, illustrated by Plate L. The translation proceeds well through the construction of the first part of the diagram, that shown in Plate L, but then falls apart, as the words receive no adequate interpretation. In this article, I want to attempt to explain clearly what Vitruvius was about, and show how to complete the *analemma*. I shall lay out the drawing in a modern style for ease of comprehension. The exact form of Vitruvius's *analemma* cannot be determined from the text., but this construction was well-known at the time, and Vitruvius is merely explaining it for the student. Diodorus, a contemporary of Vitruvius and a maker of sundials, wrote on the *analemma*, as later did Ptolemy. Ptolemy's work is known only in fragments, but involved the orthogonal projection of the celestial sphere on three mutually perpendicular planes [Ivor Thomas, *Greek Mathematical Works,* I, p. 300].

The modern *analemma* is the figure-eight diagram that shows how much the apparent sun is ahead or behind the mean sun, so that mean time with equal or equinoctial hours can be read from the sundial. The *analemmatic sundial*, in which the gnomon or style was movable to account for the equation of time, was popular in France. The *analemma* was also a kind of armillary sphere used for spherical trigonometry by Ptolemy. An *analemma*, in general, appears to be any kind of assistance in used for astronomical calculations. Vitruvius's *analemma* was a figure from which the direction of the sun's rays could be determined at certain times throughout the year, for each of the variable hours of the day. The time from sunrise to sunset was divided into twelve equal hours. These hours were longer at the summer solstice, and shorter at the winter solstice. Once the direction of the sun's rays was known, any kind of sundial could be designed. To construct the *analemma*, the most basic form of sundial was assumed, the vertical gnomon, but the results were applicable to any of the various popular forms of dial.

The construction of modern sundials is presented by R. R. J. Rohr in *Sundials: History, Theory, and Practice* (Dover, 1996) and by A. E. Waugh in *Sundials, Their Theory and Construction* (Dover, 1973). The history given in these books is traditional and sketchy. Surprisingly little is known, and the mentions by Herodotus and Vitruvius are the major sources. Vitruvius's *analemma* is quite similar to the equatorial polar sundial, and serves the same purpose in designing dials. Vitruvius describes 13 types of sundials, but not in enough detail to reconstruct them accurately. They were well-known to his readers, and a detailed description was not, unfortunately, necessary. The most popular type of sundial in classical times was the hemispherical basin with a vertical gnomon.

This explanation will be much clearer if the reader has a grasp of the apparent motion of the sun through the day and year, and understands the gnomon, all of which is clearly explained by Rohr. I have elsewhere described how to make a gnomon and use it for observing the altitude of the sun throughout the year, an exercise that will be found very instructive. One finds that measurements can be made to an accuracy of about half a degree without difficulty, and that the observer's latitude and the obliquity of the ecliptic can be determined easily. Together with knowing the cardinal directions (north, east, south, and west), this is all that is required to construct the *analemma*, using precisely the kind of instruments and data available to Vitruvius.

The latitude of Rome is given as the angle whose tangent is 8/9, or 41.6 degrees . The precise latitude of the Rome Observatory is 41.9 degrees , so this is within the accuracy expected from the gnomon. A slightly better tangent would have been 9/10, but the difference is not significant. Vitruvius takes the obliquity of the ecliptic as 24 degrees , or 1/15 of a circle, which is close enough to the precise value of 23.5 degrees. The drawing will use these values, which would be accurate enough for designing sundials.

Roman engineers used a method of graphical computation and design that was very much in the style of Monge's descriptive geometry of the 19th century, until recently taught to every engineering student. This procedure makes use of two or more views, each showing two dimensions, and the combination showing all three dimensions necessary to describe a body in space. Vitruvius would probably have made his views overlap, in spite of the confusion of lines that results, to save space. We shall separate the two views in the modern manner so that each can be appreciated alone. The graphical construction called the *épure* in France for the design of sundials is a similar procedure.

Vitruvius first describes the elevation view. This is the view shown in Figure L, and clearly described in Granger's translation. It shows the gnomon, the flat base on which it is mounted, and directions in the meridian plane, defined by points on a circle whose center is at the point of the gnomon, A. The horizontal line, called, in fact, the *horizon*, and the line in the direction of the north celestial pole, called the *axis*, are very important. Lines perpendicular to the axis are the projections of the paths of the sun through day and night. Three are drawn, those corresponding to the equinoxes, the summer solstice, and the winter solstice. The path of the sun at any season could be represented if desired, since the gnomon gives the declination of the sun for any date by a simple observation at noon. Sunrise and sunset occur when these lines cross the horizon line, as will soon be made clear.

Now we come to the important part. Vitruvius constructs what would now be called an *auxiliary view* looking directly down the axis. Circles are drawn with the projection of point A as center whose radii are given by the lines in the elevation view that are the projections of these circles for the equinoxes and the solstices. These circles are the paths of the sun about the axis, each described in one solar day from noon to noon. The points of crossing of the horizon are projected back into the auxiliary view to show the positions of the sun at sunrise and sunset. Then, the portions of each circle that correspond to the sun's being above the horizon are now divided into twelve equal parts for the hours. This is easily done with dividers, dividing in half twice, and then each interval by three. These points are, in turn, projected back into the elevation view onto the lines representing their corresponding circles. Now we have the locations of the sun at each hour of the day for the equinoxes and solstices represented on two views at right angles to each other. The lines joining each of these points and point A in the two views give the direction of the sun's rays for the corresponding hours. As difficult as this may be to grasp in verbal description, anyone versed in orthographic projection will see immediately what is going on.

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Composed by J. B. Calvert

Last revised 3 July 1999