Using these six hypotheses Aristarchus tried to measure the distance to the Moon. The shadow of the Earth resembled a great cone, and Aristarchus knew that the angle of this cone was half a degree, exactly the same as the angular diameter of the Sun. When the Moon passed through the shadow cone he calculated that the distance it traveled in passing through was equal to two lunar diameters. This implied that the distance from Earth to Moon was one-third of the length of the shadow cone. The shadow cone was about 230 Earth radii, and he was able to calculate from this figure that the distance to the Moon was about 72 Earth radii. This was a good approximation, but he could not complete his calculation because he did not have an accurate figure for the radius of the Earth.
Aristarchus went on to estimate the distance to the Sun. Once again his method was very ingenious. He knew that the Moon was illuminated by the light of the Sun and that therefore when the phase of the Moon, seen from the Earth, was exactly half, then the angle of the Sun–Moon–Earth triangle was exactly 90 degrees. The angle between the Sun and the Moon could easily be measured from the Earth. If the distance from Earth to Moon was known, then the triangle could be solved and the distance from Earth to Sun could be calculated.
But the experiment was a failure. We can see from hypothesis 4 that he calculated the angle between the Sun and the Moon to be 87 degrees, when in fact it was less than one-sixth of a degree. The heavily cratered lunar surface made it impossible for him to decide when the Moon was exactly at half phase, and because of this he was only able to obtain the crudest figure of 20 lunar distances for the distance to the Sun. The lunar distance was again expressed in terms of the Earth’s radius but this, too, was an unknown quantity.
It would appear from this that all the efforts of Aristarchus came to nothing. He did not leave behind a measure of the Earth’s radius, nor of the lunar distance or the distance to the Sun. But in spite of these failures Aristarchus is remembered as one of the greatest astronomers of all time. He was an excellent practical astronomer and he was also a great theorist. His lasting claim to fame is that he was the first to propose a world model with the Sun at the center of the solar system and the planets orbiting around it. His distances may have been wrong, but he knew that the Earth was a revolving globe following an orbit around the Sun. Aristarchus was a thousand years ahead of his time, and there is a strong case for calling the Copernican System the Aristarchian System.
Eratosthenes and Measuring the Earth
Later in the same century a man called Eratosthenes (c.276–194 BC) arrived in Alexandria to take up his post as the new librarian. He was born in the town of Cyrene in the upper reaches of the River Nile, about 500 miles (800 km) to the south of Alexandria. Eratosthenes remembered when, as a child, he and his playmates peered down into the darkness of a deep well. It was possible, for a short time at noon on just one day of the year, to see a brilliant light at the bottom of the dark well. The light was the reflection of the Sun on the surface of the water far below. In fact, the light could be seen only at noon on midsummer’s day when the Sun was directly overhead. Eratosthenes knew that on the same day of the year the Sun in Alexandria did not reach the zenith. He could not repeat his childhood observations in Alexandria, but he could measure the height of the Sun at noon and show that it was an angle of 7.5 degrees away from the vertical. He knew that the Earth was a sphere and that this angle was the difference in latitude on the surface of the Earth between Alexandria and the town of Syene. The ratio of 7.5 degrees to the full circle was the same as the ratio of the distance between the two places to the circumference of the whole Earth. If, therefore, Eratosthenes could measure the distance between Alexandria and Syene, he could easily calculate the circumference of the Earth. He estimated the distance in units called stadia—each unit being the length of a games stadium, although we cannot be sure of the exact value. Using modern units, the stadium is thought to have been about 80 meters (263 ft), and the distance from Alexandria to Syene was 10,000 stadia. This gives a value for the circumference of the Earth of 23,846 miles (38,400 km)—a very accurate determination, although we have to ask if the length of the stadium has perhaps been calculated retrospectively from the known circumference of the Earth!
Eratosthenes’ work was only a generation later than that of Aristarchus. His result would have enabled Aristarchus to calculate the distance to the Moon with tolerable accuracy. By this time Aristotle’s assertion that the Earth did not move had been rejected by a few enlightened people, but it was many centuries before the negative influence of Aristotle was completely overthrown.
Earlier Attempts to Measure the Earth
Eratosthenes was the first person to measure the size of the Earth to any degree of accuracy, but it is worth mentioning that there were prior claims to this measurement. The Chaldeans of ancient Babylon had a tradition that a person could walk 30 stadia in an hour; this equates to a distance of about 1.5 miles (2.4 km) per hour. In a day, therefore, a well-organized relay team could easily cover a distance of about 37 miles (60 km). The Chaldeans claimed that if a person could walk steadily at this speed for a year then they would encompass the whole Earth. The calculation gives a figure for the circumference of the Earth of about 13,000 miles (21,000 km). In fact, this figure is only just over half the true distance, but it is of little matter for the method is very unsound scientifically and the measurement could not possibly have been carried out by marching around the Earth.
There is another much later claim. In the first century BC, the astronomer Poseidonius (c.135–51 BC) measured the circumference of the Earth. His method was very similar to that of Eratosthenes, but he made use of the stars rather than the Sun. He made observations of the star Canopus as seen from Rhodes and from Alexandria. From the elevation of the star he was able to measure the difference in latitude between the two places, and he arrived at a figure usually quoted as 240,000 stadia or 11,923 miles (19,200 km). Again, this figure was far too low, and indeed it was disputed by the Greek geographer Strabo (63 BC–AD 23). It appears from one account that when Poseidonius made his measurement on Canopus the star grazed the horizon at Rhodes. Poseidonius probably knew nothing of atmospheric refraction, so this factor would create a substantial error in his measurements. Despite the erroneous calculations often made by ancient scientists, we must nevertheless admire the ingenuity of some of the methods they used.
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THE ALMAGEST
During the second century BC, advances in astronomy included measuring the brightness of stars as well as further attempts to discover the distances between bodies in the solar system. Another highlight was the creation of the Julian calendar which survived, despite its faults, for 16 centuries. In the second century after the birth of Christ the work of the Greek mathematician, geographer and astronomer Ptolemy brought a new understanding about the Earth, the stars and the planets.
At the entrance to the harbor at Alexandria on the island of Pharos stood the most famous lighthouse in antiquity, designated as one of the Seven Wonders of the Ancient World. The Lighthouse of Alexandria is said to have been more than 110 meters (361 ft) high. But Alexandria was not the only harbor to boast one of the wonders of the world. In the ancient Greek city of Rhodes, beside Mandrákion harbor, there stood a colossal statue of the Sun god Helios. The statue, the Colossus of Rhodes, was said to be 70 cubits high (about 32 meters/105 ft).
The Brightness of the Stars
In the second century BC, when the astronomer Hipparchus (190–120 BC) worked in Rhodes, the Colossus was still remembered by the elderly residents of the city. At this time Alexandria remained unchallenged in the ancient world as the greatest center of learning. However, in Hipparchus, Rhodes had an astronomer second to none at the time. Hipparchus made valuable contributions to trigonometry. He had a sound understanding of the geometry of the sphere, and he was the first to suggest that a mesh of imaginary circles that we now call latitude and longitude were the ideal coordinates for mapping the surface of the Earth. Using
the same system he worked out a spherical coordinate system for the heavens very similar to that used for the Earth.
Hipparchus’ greatest achievement, however, was that by studying the heavens he built up a catalog of the stars and accurately recorded the positions of 850 of them in the night sky. It is a measure of his observing skill that he also assigned a degree of brightness to every star on a scale of one to six. Under his system magnitude one denoted the brightest stars and magnitude six indicated the faintest stars. It was the first attempt at a classification of magnitude, and the system was still in use centuries after his time.
Measuring the Solar System
Hipparchus was not unduly influenced by earlier thinking about the stars and planets. He believed that the Earth was a spinning globe in space and that it orbited the Sun once every year. Like Aristarchus (c.310–230 BC) and others before him Hipparchus wanted to measure the scale of the solar system—the distances from the Earth to the Moon and from the Earth to the Sun. If he could find these distances then he could easily calculate the size of the Sun and the Moon from their angular diameter of half a degree or 30 minutes of arc.
He was able to calculate the distance to the Moon without even leaving the island of Rhodes. He knew from the astronomical records that a generation before his time a total eclipse of the Sun had been observed and recorded at the Hellespont, near the city of Byzantium. He discovered that this same eclipse had also been observed at Alexandria, but because the latter was 500 miles (805 km) to the south, the eclipse was not total in Egypt. Hipparchus wanted to know how much of the Sun’s disc was covered by the Moon in the partial eclipse at Alexandria, and the Alexandrian astronomers supplied him with the one item of information he needed. At the time of maximum eclipse one-fifth of the Sun’s disc was still visible. The Alexandrians understood lengths much better than areas and, put more precisely, what they meant was that at maximum coverage, one-fifth of the Sun’s diameter was left uncovered by the Moon. This meant that the angular difference between the two observations was one-fifth of the Sun’s diameter—one-fifth of 30 minutes, or six minutes of arc. Put in another way, the triangle consisting of Byzantium, Alexandria and the Moon (BAM) had an angle of six minutes at the Moon. The distance AB between Alexandria and Byzantium could be measured by calculating the difference between their latitudes. The triangle could easily be solved to give the distance to the Moon (AM or BM). Eratosthenes’ (c.276–189 BC) estimate of the circumference of the Earth was available to help Hipparchus make this calculation, and he knew how to make the small correction from the arc connecting the two places to the length of the chord or straight line between them. Hipparchus arrived at the figure of 67 to 78 Earth radii for the lunar distance; this was equivalent to approximately 285,660 miles (460,000 km)—a little high but still an excellent result.
Elusive Sun Measurements
Hipparchus went on to try to determine the distance from the Earth to the Sun. He used an ingenious idea based on the lunar eclipse, involving an observation very similar to that used in the previous century by Aristarchus in relation to the Moon. His reasoning was based on the shadow cone of the Earth. He knew that the diameter of the Sun as seen from the Earth was 30 minutes of arc. He reasoned that the shadow cast by the Earth was a cone with almost, but not quite, the same angle at the apex. Hipparchus knew that the small difference between these angles was the key to determining the Sun’s distance. An observer at the apex of the shadow cone would be further away from the Sun and would therefore see a smaller angular diameter than an observer on the Earth. Hipparchus expected the difference between the two angles to be a few minutes of arc. His method was sound, but he was wrong about the angular difference; it was not minutes of arc he had to measure, it was a few seconds of arc and, using ancient-world technology, it was impossible to measure angles this small.
But how could Hipparchus send an observer to the apex of the Earth’s shadow to view the marvelous eclipse with the disc of the Earth just covering the Sun? How could he measure the angle he wanted? He thought up a very ingenious solution. When the Moon passed through the shadow of the Earth he calculated that it should be possible to measure the width of the shadow cone by observing the time as the Moon entered and then left the shadow. He knew the distance and the speed of the Moon’s motion, so he could calculate the length of the path through the shadow and hence the distance to the apex of the shadow cone. Finally he could calculate the angle of the cone.
As with Aristarchus’ work there was nothing wrong with Hipparchus’ logic. It was a brilliant idea, but it suffered from the same problem as the method used by Aristarchus. If the Sun had been a mere 20 times further away than the Moon then the method could have yielded results, but the difference between the angles he sought was so small that it was lost in the inevitable errors of the measurement.
Thus the problem of the true distance of the Sun was never solved in the ancient world. In fact the first reasonable estimates of this astronomical unit do not appear until the 17th century. Hipparchus did much important work, however. One of his many successful achievements was the discovery of the precession of the equinoxes. He found that the axis of the Earth’s rotation did not align with a fixed point in the sky. Over a long period of time the poles traced out a small “circle” in the sky. The Earth behaved rather like a spinning top and its axis of rotation precessed around the circle. Hipparchus was able to calculate that it took 26,000 years for the Earth’s axis to complete its cycle.
The Julian Calendar
Fifty years before the birth of Christ, Julius Caesar (100–44 BC) became emperor of Rome. He recognized that the Roman calendar needed radical reform, and he employed an astronomer called Sosigenes (fl. 46 BC) from Alexandria to create a new system, to be known as the Julian calendar. Thanks to men like Hipparchus in the century before him, Sosigenes had at his disposal all the knowledge needed to create the perfect calendar. But Sosigenes is not remembered among the great astronomers of the time, and the calendar he produced looks like a classic example of design by committee. He knew that the length of the year was a little over 365 days, so he arranged for the Julian calendar to have a leap year day every fourth year. This in itself was not a sound decision, since it was known that a leap year every four years would create an error of about three days after four centuries. Although Sosigenes knew about this discrepancy and could have adjusted for it, the Julian calendar was not reformed until 16 centuries later, by which time it was ten days in error. The Gregorian calendar, instigated by Pope Gregory XIII in 1582, made allowances for the error by decreeing that the first year of each century should only be a leap year when it is divisible exactly by 400.
Flaws in the System
The Julian calendar contained 12 months. It was not possible to make the months the same length and still add up to 365 days, so it was decided that some months should have 30 days and others 31 days. However, instead of alternate 30- and 31-day months, Sosigenes allocated the 31-day months randomly. By the time the days had been allocated for the first 11 months, February—the last month in the Julian calendar year—was left with 28 days. So Sosigenes left this short month at the end of the year with an extra day on leap years only. It is said that Caesar became jealous when he discovered his predecessor Augustus had a month with one more day in it than Caesar himself had. Caesar therefore stole a day from one of the other months, leaving the calendar in chaos and posing even more questions about the state of mind of the Roman emperors! To add to the confusion, the equinoxes and the solstices all fell on the 21st day of the month. It would have been better to have shifted the calendar so that these events took place on the first day of the month, leaving three months for each of the four seasons. Logic did not prevail, however, and the result was a catastrophic calendar design, leaving posterity forever puzzling about how many days there would be in each new month, and chanting rhymes about “thirty days hath September, April, June and November.”
Ptolemy’s Great Work
Ther
e is one other astronomer of note in the ancient world. His name was Claudius Ptolemaeus (c.85–165), better known simply as Ptolemy, and he worked at Alexandria in the second century AD. His interests extended beyond astronomy into mathematics and geography. Ptolemy’s astronomical work was enshrined in his great book the Mathematike Syntaxis (The Mathematical Collection). It eventually became known as Ho Megas Astronomos (The Great Astronomer). In the ninth century Arab astronomers used the Greek superlative Megiste to refer to the book. When the definite article al was prefixed to the term, it became known as the Almagest, the name still used today. The Almagest was a great work covering many aspects of astronomy and one that was destined to be the most important reference work for astronomers for more than a thousand years.
It is sometimes difficult to appreciate the timescales involved in the ancient world. Ptolemy’s work is separated from that of Hipparchus by 300 years. His work is over 400 years after Aristarchus and 700 years after Thales and Pythagoras. The Almagest is a work of 13 books, each of which deals with certain astronomical concepts pertaining to stars and to the solar system. In essence, it is a synthesis of all the results obtained by Greek astronomy. We know that Ptolemy drew very heavily on the earlier findings of Hipparchus for his star catalog and other aspects of his work. He has been accused of plagiarism because the bulk of his star catalog was undoubtedly the work of Hipparchus, but in his defense Ptolemy added some new stars to increase the number of entries from 850 to 1022.
The Story of Astronomy Page 3