The clock is one of the most influential discoveries in the history of western science. The division of time into regular, predictable units is fundamental to the operation of society. Even in ancient times, humanity recognized the necessity of an orderly system of chronology. Hesiod, writing in the 8th century BC., used celestial bodies to indicate agricultural cycles: "When the Pleiads, Atlas' daughters, start to rise begin your harvest; plough when they go down" ( Hesiod 71). Later Greek scientists, such as Archimedes, developed complicated models of the heavens-celestial spheres-that illustrated the "wandering" of the sun, the moon, and the planets against the fixed position of the stars. Shortly after Archimedes, Ctesibus created the Clepsydra in the 2nd century BC. A more elaborate version of the common water clock, the Clepsydra was quite popular in ancient Greece. However, the development of stereography by Hipparchos in 150 BC. radically altered physical representations of the heavens. By integrating stereography with the Clepsydra and the celestial sphere, humanity was capable of creating more practical and accurate devices for measuring time-the anaphoric clock and the astrolabe. Although Ptolemy was familiar with both the anaphoric clock and the astrolabe, I believe that the development of the anaphoric clock preceded the development of the astrolabe.

The earliest example, in western culture, of a celestial sphere is attributed to the presocratic philosopher Thales. Unfortunately, little is known about Thales' sphere beyond Cicero's description in the De re publica:
For Gallus told us that the other kind of celestial globe, which was solid and contained no hollow space, was a very early invention, the first one of that kind having been constructed by Thales of Mileus, and later marked by Eudoxus with the constellations and stars which are fixed in the sky. (Price 56)
This description is helpful for understanding the basic form of Thales' sphere, and for pinpointing its creation at a specific point in time. However, it is clearly a simplification of events that occurred several hundred years before Cicero's lifetime. Why would Thales' create a spherical representation of the heavens and neglect to indicate the stars? Of what use is a bowling ball for locating celestial bodies? Considering Eudoxus' preoccupation with systems of concentric spheres, a more logical explanation is that Thales marked his sphere with stars, and Eudoxus later traced the ecliptic and the paths of the planets on the exterior. The celestial sphere in question probably resembled this early Persian example.

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Perhaps the most famous celestial sphere is the mechanized globe attributed to Archimedes. Cicero was especially impressed by this invention because of its ability to imitate "the motions of the sun and moon and of those five stars which are called wanderers" with a single rotational focus (Price 56). By turning a crank, one could observe the "natural" course of the sun, moon and planets around the earth. The sphere was also remarkable for a second reason. Unlike a stationary globe, like that of Thales' and Eudoxus, a mechanized sphere requires gears to accurately represent the motion of the heavens. According to Prof. Derek Price, the mean period of Saturn can be mechanically represented by a gear ratio of 30 to 1. In other words, for every revolution of the sun around the earth, Saturn will only accomplish 1/30th of its revolution around the earth. The mean period of Jupiter can be represented by a gear ratio of 12 to 1, and Mars can be represented by a gear ratio of 2.5 to 1.

An interesting problem arises when one attempts to mechanically represent the synodic month. A gear ratio of 235 to 19 is required for an accurate representation. However, this is impossible to achieve directly, presenting a serious challenge to Archimedes and other Greek scientists. Prof. Price claims that two different gear arrangements can be used to create this ratio. First, one may simply use a more intricate combination of gears, as Archimedes did in his mechanical sphere. The second solution is one of the greatest innovations in Greek engineering; the development and incorporation of a differential gear. In addition to having been the first mechanized globe, Archimedes' sphere became a model for later Greek astronomers. For example, Posidonios of Rhodes, a contemporary of Cicero, built a mechanical globe based on Archimedes' sphere. Members of the school of Posidonios created a device to compute the positions of the sun and the moon-what we now call "The Antikythera Mechanism." Challenged by the same, mechanical difficulty Archimedes faced in representing the synodic month, these scientists developed the first differential gear to solve the problem. Archaeological evidence suggests that after the Antikythera Mechanism was lost in a shipwreck, the differential gear essentially disappeared from western knowledge until 1575, when it reappeared in a globe clock designed by Jobst Brgi. The differential gear later became a critical component of the cotton gin, a late 18th century invention that marked the beginning of the industrial revolution. However, devices such a the Antikythera Mechanism were quite rare. The celestial sphere was the most common form of celestial representation, prompting a number of structural modifications.

Because of the difficulty in imagining the position of the earth within a solid representation of the heavens, the celestial globe assumed a more skeletal appearance over time. This new model of the heavens, the armillary sphere, quickly began to replace the more ambiguous celestial globe. However, the method of locating celestial bodies remained the same. Greek astronomers continued to use an ecliptical system for specifying the position of the stars and planets. To understand how this system works it is first necessary to explain a few terms, and to remember that we are assuming that the earth is in the center of the universe-we are using a geocentric model of the universe. The ecliptic measures the annual rotation of the sun around the earth, and is inclined 23deg. from the celestial equator. It is not a representation of the daily rising and setting of the sun. The Greeks divided the ecliptic into twelve sections, and each section was named after the constellation it contained-Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius, and Pices respectively. The ecliptic, divided in this fashion, is called the zodiac. The Greeks further divided each of these twelve sections into thirty units, effectively graduating the entire circle for longitudinal measurement (30 multiplied by 12 is equal to 360). The system began at the vernal equinox, the intersection of the ecliptic with the celestial equator in the constellation of Ares, and completed a 360deg. circle around the circumference of the celestial sphere. The Greeks used the ecliptical to measure a star's horizontal, angular displacement from the vernal equinox. Vertical, angular displacement was measured by constructing a graduated circle perpendicular to the ecliptical. If you are completely confused by my written description, take a look at the diagram I have created. Ecliptical coordinates were used by Hipparchos and Ptolemy in their star catalogues, and were the standard of celestial measurement until the Renaissance, when they were replaced by the equatorial coordinate system. The equatorial coordinate system is identical to the ecliptical system, except that it uses the celestial equator for horizontal measurement instead of the ecliptic. Because the celestial equator is simply a projection of the earth's equator, the equatorial coordinate system is analogous to terrestrial longitude and latitude, and provides a more accurate system of measurement. This 17th century armillary sphere is graduated for both ecliptic and equatorial coordinate systems-notice how each sign of the zodiac contains thirty degrees of the circle.

Measuring time on an armillary sphere is a simple matter. First, imagine that you live on the earth's equator. From this position, the ecliptic is almost a perfect arch over your head. As the earth rotates, the sun will rise and set in a twenty-four hour period. Please remember that this is not the ecliptic-the ecliptic will only determine where, on the horizon, the sun will rise and set each day. In antiquity, every day is a complete rotation of the sun around the earth. Time may be measured simply by dividing this rotation into twenty-four hours. If the rotation is a circle of 360deg., dividing it into 24 sections results in hours that are 15deg. long. In other words, if we know where the sun will rise on the horizon, according to the ecliptic, every fifteen degrees that the sun travels across the sky marks the end of an hour. Given a constant source of motion, it is possible to create a clock-an accurate representation of the heavens, from an armillary sphere. Although the Greeks had the means of producing the necessary motion, the shape and intricacy of an "armillary sphere clock" may have prevented rigorous experimentation until the development of stereography.

Until the development of stereography by Hipparchos in the middle of the second century BC., the Greeks measured time with various types of water clocks. The most simple water clock consisted of a large urn that had a small hole located near the base, and a graduated stick attached to a floating base. The hole would be plugged while the urn was being filled with water, and then the stick would be inserted into the urn. The stick would float perpendicular to the surface of the water, and when the hole at the base of the urn was unplugged, the passage of time was measured as the stick descended farther into the urn. These early clocks were used when equal measurements of time needed to be established. For example, if two orators were to be allotted the same amount of time to speak before an assembly, a water clock of this nature would have been constructed for the occasion. In the second century BC., a man named Ctesibus created a more elaborate water clock for measuring the time of day. The Clepsydra, as it is called, consisted of four major parts: a vessel for providing a constant supply of water (B), a reservoir and notched floatation rod (F), a display (G), and a device for adjusting the flow of water into the vessel (D).

Water was continually poured into the vessel (B), with the overflow escaping from a pipe (I). Water flowed from this vessel into the reservoir at a constant rate. As the reservoir filled with water, the floating, notched rod ascended at a constant rate. This rod was attached to the display (G), which indicated the time of day. The Greeks divided the day into twelve hours of unequal length to insure an equal division of day and night. Because the Greeks divided the day into hours of unequal length, it was necessary to include a device (D) to regulate the flow of water from the vessel (B) into the reservoir (F). By raising the flat, circular cap in the conical vessel (B), the flow of water could be increased, decreasing the length of an hour. In the summer, the day is longer than the night, and in the winter the reciprocal is true. Therefore, in the summer, the clock would be adjusted to extend the length of each day hour. A second way the Greeks standardized the length of a day was by modifying the clock display.

A cylinder with sloping hour lines was used instead of a circular face. The mechanism worked as follows: as water collected in the reservoir, a pointer would raise as the cylindrical display rotated. In this manner, the pointer would gradually trace the course of the adjusted hours on the cylindrical display. However, the former example, the circular face, is more important because of the modifications made to it after the discovery of stereography by Hipparchos.

Stereography is a technique by which three dimensional objects are projected on two dimensional surfaces. Hipparchos used stereography to create a projection of the celestial sphere from its southern celestial pole to its equatorial plane. In other words, he created a two dimensional image of a three dimensional model-a planispheric projection of the heavens. By separating the projection of the stars and the ecliptic from the projection of the horizon and the equator, Greek scientists could simultaneously represent the progression of the sun along the ecliptic and the daily rotation of the sun around the earth. In essence, by separating the two projections scientists recreated the rotational components of an armillary sphere on a two dimensional surface. By incorporating these two planispheric projections of the sky into the display of a clepsydra, the Greeks discovered a way for providing the constant source of motion necessary for an accurate representation of time. Recall that an armillary sphere can be used to tell time because it allows one to divide the daily rotation of the sun around the earth into 24 hours, with each hour equal to 15 degrees of the complete rotation. The problem with keeping time on an armillary sphere is that a constant source of motion is required for the sphere to mimic the actual motion of the sun around the earth. By using stereography, scientists were able to project the armillary sphere on two disks-the first provided the means for measuring sun's position in the sky, and the second disk illustrated the sun's actual path across the sky. There are two advantages to having the heavens projected on two disks, as opposed to a single sphere. First, it is easier to construct a two dimensional model than a complicated sphere. Second, it is easy to provide constant motion for two disks by using a clepsydra. By incorporating planispheric projections of the heavens into the clepsydra, the Greeks created the first anaphoric clocks.

The anaphoric clock consists of a rotating star map behind a fixed, wire representation of the meridian, the horizon, the equator and the two tropics. The fixed disk consists of several concentric circles, divided into twenty-four sections by a series of small arcs. Each section represents one hour of the day. Because the long arc extending from one end of the disk to the other is the horizon, the first hour of the day begins on the right side of the disk at the horizon. The twelve hours of the day are above the horizon, and the twelve hours of the night are below the horizon. A stereographic map of the ecliptic was attached behind this fixed representation. Although circular in shape, the ecliptic did not rotate around its center. To accurately represent the daily path of the sun, the ecliptic rotated around a point approximately halfway between the center and the bottom edge of the circle. The ecliptic would complete one rotation around this point every day. Furthermore, the ecliptic was fashioned with 365 holes around its circumference, one for every day of the year, in which was placed a peg to represent the sun. The year began at the vernal equinox, and after each daily rotation of the ecliptic the peg would advance to the next hole along the perimeter of the ecliptic. However, the ecliptic was reset each day so that the peg always began at the horizon. The anaphoric clock was both a clock and a calendar, illustrating the both the time of day and the progression of the sun along the ecliptic.

A second product of stereography is the astrolabe, a device for locating the position of the stars at any point in time. The astrolabe consists of three major parts: First, there is a fixed disk called a tympanum on which one can measure the position of the stars. The tympanum is an engraved plate, making it easier to use than the wire mesh of the anaphoric clock, but because the position of the horizon differs from place to place, each astrolabe typically contained a number of tympanum. Only one tympanum was used at a time, and the inclusion of several tympanum insured that the astrolabe could be used at a variety of positions on the earth. Second, a skeletal projection of the stars-called a rete-was fastened over the tympanum. The third primary component of an astrolabe is a simple device for measuring the distance of a star above the horizon-usually a rod attached to the back of the astrolabe. One could produce a map of the sky on any given night by locating a known star, measuring its angular distance above the horizon, and rotating the rete until the representation of the star was aligned with its angular distance on the tympanum. During the Renaissance, the astrolabe was also included in clock designs such as this one by Janus Reinhold.

The evolution of the anaphoric clock depended on several hundred years of Greek science. Thales' crude, spherical representation of the heavens laid a foundation for other Greek scientists to build on. After the construction of the first celestial sphere by Eudoxus, Archimedes created the first mechanical representation of the heavens using a complicated series of gears. However, armillary spheres were more commonly used to study the heavens. Shortly after the construction of Archimedes' sphere, Ctesibus built the first clepsydra. Although it is possible to observe the time on an armillary sphere, it is quite difficult to perpetually mimic the motion of the sun around the earth. The invention of stereography by Hipparchos made the construction of a dynamic representation of the heavens possible through the combination of planispheric projections with the clepsydra. The anaphoric clock and its cousin, the astrolabe, not only helped Ptolemy create the extensive catalogue in the Almagest, but also established the foundation of modern time keeping.