The first part of this article discusses Sundials. The second part of the article discusses Water Clocks.
For this second part see Water Clocks
In the time of the ancient Greeks and Romans, the earth was considered the centre of the universe, which was itself a sphere containing all the stars. This celestial sphere rotated from east to west, carrying not only the stars but also the sun and the planets. Therefore, the sun revolved around the earth. This is what caused day and night. The earth did not rotate. For the purpose of understanding sundials, it is perfectly acceptable and convenient to adopt this geocentric view. The sun did not travel around the earth in a circle at right angles to the earth's axis (which was also the axis of the celestial sphere) as the stars did. Rather, the sun traced a circle along the celestial sphere, centred on the earth, known as the ecliptic.
The ecliptic plane meets the equatorial plane at approximately 23.5°. This is known as the obliquity of the ecliptic. The circle of the ecliptic more or less intersects the twelve constellations of the zodiac, and the time of year (corresponding to modern months) was reckoned by what sign of the zodiac the sun was traversing. (Regardless of the exact location of the zodiac constellations, the ecliptic was divided into 12 equal arcs of 30° each, leaving most of the constellations off-centred and often not entirely in their designated 30° region.) The sun's motion along the ecliptic circle takes a (solar) year. The dual motion of the sun (on the celestial sphere and along the ecliptic) means that the sun follows a different path in the sky each day. From the perspective of the northern hemisphere, during the summer, the sun is higher in the sky and remains visible for a longer period of time. Since the ancients always divided the daylight into twelve equal hours, these summertime hours were longer. In the winter months, the sun is lower in the sky and visible for a shorter period of time. Consequently, the winter hours were also shorter.
Time in the ancient world was first measured by naturally occurring events, such as sunrise, sunset, and meal times :-
In the early ages of Rome and even down to the middle of the fifth century after the foundation of the city no other divisions of the day were known than sunrise, sunset and midday, which were marked by the arrival of the Sun between the Rostra and a place called Graecostasis.The single greatest literary source that exists for the sundials of Greece and Rome is Vitruvius's Ten Books on Architecture written about 25 B.C. In Book 9, Vitruvius gives a list of a variety of dials and their inventors :-
Berosus the Chaldaean is said to have invented the semicircular one carved out of a squared block and undercut to follow the earth's tilt. The hemisphere, or scaphê, is attributed to Aristarchus of Samos, and he also invented the disk on a plane. The Spider was invented by Eudoxus the astronomer; some say by Apollonius. The Plinth or Coffer, of which an example is set in the [region of the City known as the] Circus of Flaminius, was invented by Scopinas of Syracuse; Parmenion invented the "Sundial for Examination"; Theodosius and Andrias the sundial "For Every Climate," Patrocles the Axe, Dionysodorus the Cone, Apollonius the Quiver. The men named here invented other kinds, and many others have left us still other kinds, like the Spider-Cone, the Hollowed Plinth, and the Antiboreus ("Opposite the North"). Many, moreover, have left behind written directions for making portable and hanging versions of these kinds. Anyone who wants to may find additional information in their books, so long as they know how to set up an analemma.Vitruvius's analemma is the system of lines and curves that denote the changing hours and months on the face of a sundial. His previous chapter is devoted to determining the analemma based upon the observance of the shadow of a gnomon at noon on the equinox. (The gnomon was the upright stick that cast its shadow on the dial face. Depending on the design of the dial, either the side of the shadow's length or the position of the tip of the shadow was used to determine the time.) Unfortunately, Vitruvius ends his discussion of sundials with the list given above and writes of water clocks for the rest of Book 9.
Before the Greeks developed the sundial into the forms Vitruvius lists, the more ancient civilizations of Egypt and Mesopotamia had shadow measuring devices as early as 1500 B.C. Though this is the date of the earliest surviving sundials :-
... it is possible that sundials were invented as early as the third millennium when Egyptian priests began to divide the night and day each into twelve equal parts.A funerary text from 1290 B.C., referring to astronomical events in the 19th century B.C., gives instructions on how to construct a "shadow stick."
This shadow clock consisted of a base with an upright stick at one end. Because of the angular shift in the shadow over the course of the day, it has been speculated that the upright had a crossbar added to it to widen the shadow so that it would always fall on the clock. Neither the funerary text nor surviving examples have the crossbar, though one specimen has holes on either side of its upright which may suggest such an addition.
In practice, the shadow clock needed to be rotated once a day at noon in order to be able to mark the time in both the morning and afternoon :-
With the head to the east 4 hours are marked off by decreasing shadow lengths after which the instrument is reversed with head to the west to mark 4 afternoon hours.Two hours are said to have occurred before the sun struck the clock in the morning, and another two hours passed after the sun left the clock but before night began. The assumption is that the morning twilight before sunrise was counted as one hour, and that another hour passed between sunrise and when the upright cast an observable shadow on the clock. (The shadow at sunrise would be infinite in length, and so useless for marking the hour.) Two hours similarly passed in the evening. The markings on the clock indicating the four hours were very inaccurate, and were possibly not based on observation but rather some fallacy of celestial geometry.
Sundials resembling the kind of which Vitruvius speaks were in use in Egypt from at least 1200 B.C. These were vertical hanging sundials, semicircular in shape with a horizontal gnomon at the top and centre. "The shadow would sweep around such a dial more rapidly in the early morning and late afternoon than around midday, but the Egyptians simply divided the dial into 12 15° sectors or 'hours'. This is perhaps the crudest order of gnomon use and provides little of either theoretical or empirical interest for the Greeks." Further Egyptian development in timekeeping seems to have waned until the Assyrian invasion in the 7th century B.C.
A near complete sundial was found at Kantara, Egypt dating back to approximately 320 B.C., well over a thousand years after the shadow clocks were in operation :-
The gnomon was a perpendicular block rising at the foot of the sloping face, its height and width being the same as those of the latter. On one side was an arrangement whereby a plummet could be hung so as to swing free of the base. The instrument was put down on a flat surface, and whenever it was to be used, was turned so that it faced the sun directly. The shadow of the gnomon then fell upon the face. The spaces marked off by the parallel lines running from top to bottom of the face showed where the shadow was to be read during the different months of the year, starting with the summer solstice at one edge and turning back again with the winter solstice at the other.Along the face was a set of obliquely drawn lines sloping from the winter solstice edge to the summer solstice edge.
At six in the morning the shadow would strike the top of the dial; as the sun rose higher the shadow would decrease in length until at noon it touched the lowest line; it reached the top of the dial again at six in the evening.This sundial and others of similar design that survive today are not terribly accurate :-
Certain modifications would have been necessary if they were to tell the correct time. Part of this inexactness may have been due to their being representations of larger or more accurate instruments, although dials of this type must either have been small enough to handle or else have had some sort of arrangement whereby they could be turned about easily.In the Greek world, the earliest sundials "consisted of a gnomon in the form of a vertical post or peg set in a flat surface, upon which the shadow of the gnomon served to indicate the time." This is as opposed to modern designs that have their gnomon slanted parallel to the earth's axis. In this modern system, lines on the dial face denoting the hours issue from a central point and remain straight. It is the shadow of the edge of the gnomon lying on these lines that gives the time. Seasonal variations are practically immaterial :-
In the ancient dials with vertical gnomon, the direction of the shadow at any given time of day varied with the seasons. Thus it was the position of the tip of the shadow that was essential to the determination of the hour. The shadow's tip traced a curve on the dial plane as the sun moved, a curve which changed from summer to winter.The curves traced out on the dial of such a sundial may have led to the discovery of the conic sections, as attributed to Menaechmus in the fourth century B.C.
The sun traces a circular path in the sky in its daily motion. The tip of the gnomon is the vertex of a cone with the sun's rays as elements, and since the dial plane cuts the cone, the shadow path is a conic section. If Menaechmus or someone else marked this path with a series of dots on a given day, he would 'discover' a hyperbola.It should be noted that the solstitial curves are only hyperbolae between the Arctic and Antarctic circles. The equinoctial curve is a line at every latitude except the poles. The solstitial lines at the Arctic and Antarctic circles would be parabolae and within the circles, they would be ellipses. The ellipse is easy to see as during the Arctic day, the sun makes its full circuit above the horizon, and thus a gnomon's shadow would map out the closed conic section.
The paths of the tip of the gnomon's shadow as traced out on these horizontal sundials formed a pattern resembling an axe called a pelekinon (derived from the Greek word for axe).
The pattern consisted of a hyperbola tracing the shadow's path at the winter solstice, a second for the summer solstice, and a straight east-west line in between marking the equinoctial shadows. A line from the base of the gnomon at the south of the dial running due north denoted noontime. (Since the shadow of the tip of the gnomon was the time telling device, the gnomon may have been inclined. The angle of the gnomon is irrelevant. In such a dial, the noon line would run from the base of a perpendicular line between the gnomon's tip and the dial surface.) The hyperbolae were centred on this noon line. The winter hyperbola opened north, the summer hyperbola south (assuming the dial is in the northern hemisphere). In addition to the centre noon line, additional oblique lines were added on either side to denote the hours of daylight before and after noon :-
It is obvious from preserved examples of horizontal dials that straight lines which connect hour points on the summer solstice, equinox, and winter solstice served to approximate these hour lines in Graeco-Roman antiquity.Furthermore:-
... the solstitial day curves on almost all preserved horizontal dials have been approximated by broken lines which connect the hour points. This seems to indicate that the dialer located these hour points on the dial face before engraving the day curves.It is still a matter of debate whether some, if not all, sundials of this type were drawn by observation or calculation. Evidence does exist which suggests methods of projection were used to determine the hour points :-
Both Vitruvius and Ptolemy describe analemmas which for given solar positions serve to determine length and direction of the shadow cast by a gnomon on the face of a planar sundial.
Specifically, in his book, 'On the Analemma', Ptolemy gives methods for deriving, both by trigonometric and also by graphic means, three pairs of spherical coordinates for the sun relative to a given place on earth, given solar declination, terrestrial latitude, and hour of the day. Though he does not say so explicitly, each pair of spherical coordinates is singularly suited for finding the length and direction of a gnomon's shadow for a type of plane sundial.To complicate matters, the exact specifications of a sundial's network of curves varied with the sundial's latitude. If mathematical means were used to create the pattern on a sundial, it should be expected that the intended latitude would be taken into account. However, sundials have been found in latitudes that vary as much as 7 degrees latitude (a distance of over 700 kilometres). The most significant instance of such a discrepancy was the sundial that was the first official timepiece of Rome. The Romans captured a sundial during a war on Sicily in 264 B.C. Notwithstanding the difference of about 4 degrees latitude, the sundial served Rome for almost one hundred years before a new dial calibrated for the city was set up. This was despite the time being in observable error :-
Although the shadow of a stick in the ground appears to be the simplest form of timekeeper, the horizontal dial is more complex to mark off into the hour spaces for the temporary hour system than are the dials of spherical or conical section ... since a basic understanding of the origins of the hyperbolic shadow paths on the plane surface is necessary in order to adapt the geometrical figure needed to make it.The geometrical figure is the analemma which Vitruvius spoke of above. The analemma is the:-
... projection of the celestial sphere into one plane, from which in turn the positions of the hours on the dial's surface were deduced. Vitruvius describes the basic figure ... though his text at this point is somewhat obscure and he may well have not clearly understood what he was describing in any case.After describing how the equinoctial line can be found, as well as the point of noon on the solstices, Vitruvius closes his thoughts on the analemma as follows :-
Once this construction has been drawn and executed as specified, for the winter lines and for the summer, for the equinoctial lines and the monthly lines, then, in addition, the system of hours should be inscribed along the form of the analemma. To these can be added many varieties and kinds of sundial, and they are all marked off by these inventive methods. However, the result of all these figures and their delineation is identical: namely, that the day at the equinox and at the winter solstice, and again at the summer solstice, is equally divided into twelve parts. Therefore, I have not chosen to omit these matters as if I were deterred by laziness, but so as not to cause annoyance by writing too much ... . Therefore I shall simply tell about the kinds that have been handed down to us, and by whom they were invented.Vitruvius's dismissiveness supports the claim that he did not fully understand the adaptation of the analemma to the sundial. And though he states that any sundial can be constructed from the analemma, it is only later authors who give the details of such constructions.
Whilst the initial construction required a greater effort, the ease with which day and hour lines could be drawn made spherical sundials in antiquity more popular than their flat counterparts. The basic principle of the spherical sundial was that it mirrored the celestial sphere in which the sun travels. The basic construction involved hollowing out a hemisphere (or smaller wedge of a sphere) with its top parallel to the horizon. A gnomon was set up so that its point was at the centre of the hemisphere flush with the plane of the horizon. On any given day, the shadow cast by the tip of the gnomon would trace out the arc of a circle on the surface of the dial.
The arc of the summer solstice was farthest towards the bottom of the hemisphere. As the seasons shifted toward winter, these arcs were closer and closer to the upper edge of the hemisphere. These daily arcs were all parallel, and the arc of the equinox was half of a circle with the same centre as the hemisphere (a great circle). The hour lines were not circular curves, with the exception of those at the horizons (marking sunrise and sunset) and the noon line. These were great circles which ran perpendicular to the equinoctial circle :-
In spite of their noncircular nature, for latitudes below 45° [which includes the whole of the Mediterranean Sea] the seasonal hour lines between meridian and horizon are very closely approximated by the great circles which do pass through corresponding seasonal hour points on solstitial and equinoctial curves. The engraved hour lines on preserved spherical sundials appear to be such great circle approximations. The deviation of the hour lines from great circles cannot even be detected on the few dials where more than three day curves have been divided.Thus the marking of the hour lines needed neither careful observations nor complicated mathematics. All that was needed was to divide the area of the hemisphere that received the gnomon's shadow into twelve equal parts using great circles, in much the same way as a modern globe is divided into lines of longitude. To simplify the spherical dial even more, the day curves did not need to correspond to the equinoxes or solstices if the dial's only purpose was to act as a clock. Two or three parallel circular arcs were all that were needed for ease of reading (being the corresponding lines of "latitude"). Several examples of such dials were found in such sites as Pompeii, Herculaneum, Ostia, and Rome. It was only when the dial was to serve as a calendar that these lines needed to correspond to the equinoxes and solstices.
The second part of the article discusses Water Clocks.
For this second part see Water Clocks
Article by: Daniel Mintz (University of St Andrews)
MacTutor History of Mathematics