The Basic Features of Hellenistic Science as Seen from Ptolemaic Astronomy

16,670 characters2006.10.18

Basic Content and Historical Sources of Ptolemaic Astronomy

Ptolemy (Claudius Ptolemy, c. 150 CE) lived and worked in Alexandria under Roman rule in the late Hellenistic period. Ancient astronomy reached its summit in Ptolemy’s hands,

“Just as Euclid summed up the mathematics of the Greek classical age and wrote the famous Elements, Ptolemy systematically synthesized the outstanding achievements of Greek astronomy and produced the work that would be handed down through the ages, the Almagest. This 13-volume treatise was hailed by the Arabs as ‘greatest of the great,’ and as a result its title became the Almagest (Almagest).”[1]

In the Almagest, Ptolemy set out the full set of geometric models for the geocentric system, discussed the necessary mathematical tools such as spherical trigonometry, and also provided chord tables, star tables, and the like.

Three mathematical devices were used in Ptolemy’s geometric models: the “eccentric circle,” the “epicycle-deferent” (epicycle-on-deferent), and the “equant point” (equant point). Among these, the “eccentric circle” and the “epicycle-deferent” came from the scheme of Apollonius of Perga, around 200 BCE.

In the “eccentric circle” model, the planets move around the earth in uniform circular motion, but the center of the circle is not the earth; it is offset to one side. Apollonius used this model to explain the fact that the four seasons are not equal in length.

Apollonius also noticed that the above eccentric-circle model could be replaced by another geometric model[2], namely the “epicycle-deferent” model. That is, one imagines the planet moving on a small circle (the epicycle), while the center of that circle moves on a large circle (the deferent) centered on the earth.

For Apollonius, the eccentric-circle model was a special case of the epicycle-deferent model (when the angular velocities of the epicycle and the deferent are equal, it can exhibit the effect of the eccentric-circle model). And people soon discovered that by combining these two devices, it became possible to describe more complex planetary motions.

Hipparchus (c. 190 BCE to c. 120 BCE) was a great astronomer before Ptolemy, and his work is known to later generations largely because Ptolemy quoted him extensively. In addition to leaving behind a large amount of observational data and independently discovering the precession of the equinoxes, Hipparchus’s major contribution was the creation of spherical trigonometry. Greek trigonometry, after being perfected and developed by Menelaus (c. 98 CE), ultimately reached its peak in Ptolemy’s hands and became an important tool in his astronomical calculations. In addition, Hipparchus introduced astronomical records and parameters from Babylon. In particular, he introduced from the Babylonians the degree-minute-second system of angular measurement still used today, as well as the “ecliptic coordinate system,” which can precisely mark the positions of stars. These quantitative mathematical tools laid the foundation for Ptolemy’s work.

Ptolemy comprehensively inherited and developed the geometric models, observational records, and calculation techniques of his predecessors,

In trigonometry, Ptolemy first constructed a chord table more complete than Hipparchus’s. In the process, Ptolemy proved and employed the so-called “Ptolemy’s theorem”[3], obtained the sine formula for difference angles and the cosine formula for arbitrary angles, and finally, by means of approximation, produced a chord table with a step size of 1/2 degree and a precision roughly equivalent to the fifth decimal place in modern decimal notation.

In terms of models, in addition to using the eccentric circle and the epicycle at the same time, he added another design, namely the “equant.” He assumed that the epicycle did not move uniformly around the center of the deferent, but rather moved with uniform angular velocity around some “equant point.” In Ptolemy’s most typical planetary model (applicable to Venus, Mars, Jupiter, and Saturn)[4], the “equant point” is symmetric with the earth with respect to the center of the deferent.

Ptolemy’s planetary models, especially the design of the “equant,” had already seriously offended the principle, held since Plato, that celestial motion must be uniform circular motion, so much so that Copernicus in the sixteenth century denounced it as “something self-satisfied”[5]. But the key point is that Ptolemy’s system was not only highly accurate and convenient, but also “possessed strong capacity for expansion, and was able to accommodate quite well the ever-emerging new astronomical observations before the invention of the telescope. Thus it was always regarded as the best astronomical system and ruled the Western astronomical world for more than a thousand years.”[6]

Ptolemaic Astronomy and the Characteristics of Hellenistic Science

Ptolemaic astronomy represents the summit of Hellenistic science and concentrates the main characteristics of Hellenistic science.

First, “Alexander’s campaigns brought speculative Greek geometric astronomy into contact with Babylonian arithmetic astronomy, which was oriented toward state affairs and based on observation.”[7] On the one hand, Hellenistic scientists inherited the strong desire of the ancient Greek natural philosophers to “save the phenomena” through perfect uniform circular motion; on the other hand, the practical, application-oriented elements of Babylonian science profoundly altered the way Hellenistic scientists thought. In fact, one reason—perhaps the more important one—why Hellenistic scientists insisted so strongly on uniform circular motion, apart from aesthetic considerations, was precisely this: at the time, only uniform circular motion could be calculated precisely and conveniently.

The fact that Ptolemy’s “equant” could be readily accepted shows that quantitative precision had become a more important concern for Hellenistic scientists than philosophical considerations. By contrast, “the concept of precise, quantitative prediction could by no means enter Greek astronomy or any other scientific field at that time; people were satisfied with a rough, qualitative agreement between theory and observation.”[8]

For example, “because classical Greek mathematicians were unwilling to admit irrational numbers as numbers, they produced a purely qualitative geometry. The Alexandrian mathematicians followed the Babylonian practice, unhesitatingly using irrational numbers, and in fact freely applied numbers to length, area, and volume. The high point of this work was the development of trigonometry.”[9] As the historian of mathematics M. Kline said: “We may say in general that the mathematicians of Alexandria broke with philosophy and became allied with engineering.”[10]

Thus Hellenistic astronomers also “fully realized that these theories were not really design plans, but merely descriptions that could fit the observational data.”[11] For instance, Ptolemy said: “In astronomy, one should strive to make the mathematical model as simple as possible,”[12]……“In short, speaking generally, the final causes of first principles are either irrelevant or difficult to explain in essence.”[13]

However, Ptolemy was by no means entirely indifferent to connecting his mathematical system with the real universe. In the Almagest, Ptolemy even tried, on the basis of various seemingly plausible reasons, to assign a “sequence” to the major planets. In the book Hypotheses ton planomenon (Hypotheses of the Planets), Ptolemy further integrated his planetary models into a physically real system and set the size of the universe.[14] The considerations behind these settings were more aesthetic or faith-based; he wrote: “The motions of the planets are balanced rotations, like dancers holding hands and dancing in a circle, like people in a contest helping one another, cooperating with one another without colliding with opponents, not hindering one another.”[15] This pantheistic mood inherited from ancient Greece took another form in Ptolemy’s great astrological work, the Tetrabiblos (Tetrabiblos), and the like. Although no obvious astrological trace entered Ptolemy’s astronomical system, Ptolemy did indeed take “studying the motions of the gods can bring the astronomer closer to the heavenly gods” as a motive for learning astronomy[16]. We can see that from Ptolemy to Newton, religious sentiment has always been one of the main driving forces that led many scientists to devote themselves to research.

Finally, like the Elements, the Almagest is a magnificent comprehensive enterprise, far beyond the power of a single person to complete. Take Ptolemy’s chord table as an example: although Ptolemy employed various trigonometric techniques, the amount of calculation required to compile such an accurate and complete chord table was still astonishing: “… this calculation is quite complex, and since Ptolemy for many such calculations merely presented the results, it is believed that Ptolemy must have relied on a large number of ‘computers’ to carry out this lengthy, tedious, yet indispensable work.”[17] It was precisely the comprehensively supported research institution of the Hellenistic period—the Mouseion (Museum)—and the vast holdings of its library that created the conditions for this large-scale, systematic research work.

October 18, 2006

Bifengtang

References

[UK] edited by Michelle Hoskin: Cambridge Illustrated History of Astronomy, translated by Jiang Xiaoyuan, Guan Zengjian, and Niu Weixing, Shandong Pictorial Publishing House, 2003

Wu Guosheng: The Course of Science (Second Edition), Peking University Press, 2002

[US] David Lindberg: The Beginnings of Western Science, translated by Wang Jun, Liu Xiaofeng, Zhou Wenfeng, and Wang Xirong, China Translation & Publishing Corporation, 2001

[US] James E. McClellan III and Harold Dorn: Science and Technology in World History, translated by Wang Mingyang, Shanghai Science and Technology Education Press, 2003

[US] Morris Kline: Mathematical Thought from Ancient to Modern Times (Volume 1), translated by Zhang Lijing, Zhang Jinyan, and Jiang Zehan, proofread by Xu Mingwei and Li Wenlin, Shanghai Scientific & Technical Publishers, 2002

Victor J. Katz: A History of Mathematics: An Introduction, translated by Li Wenlin, Zou Jiancheng, Xu Mingwei, and others, proofread by Xu Mingwei and Li Wenlin, Higher Education Press, 2nd edition, 2004

[Austria] Reiterb: Zhang Heng, Science and Religion, Social Sciences Academic Press, 2000

Notes

[1] Wu Guosheng: The Course of Science (Second Edition), Peking University Press, 2002, p. 95

[2] Victor J. Katz: A History of Mathematics: An Introduction, translated by Li Wenlin et al., proofread by Xu Mingwei and Li Wenlin, Higher Education Press, 2nd edition, 2004, p. 111

[3] Its plane-geometry form is: “Given any cyclic quadrilateral, the product of its diagonals equals the sum of the products of the two pairs of opposite sides.” For Ptolemy’s trigonometry, see Victor J. Katz: A History of Mathematics: An Introduction, translated by Li Wenlin, Zou Jiancheng, Xu Mingwei, et al., proofread by Xu Mingwei and Li Wenlin, Higher Education Press, 2nd edition, 2004, pp. 116ff.

[4] See [US] David Lindberg: The Beginnings of Western Science, translated by Wang Jun et al., China Translation & Publishing Corporation, 2001, p. 108 (p104)

[5] [UK] edited by Michelle Hoskin: Cambridge Illustrated History of Astronomy, translated by Jiang Xiaoyuan, Guan Zengjian, and Niu Weixing, Shandong Pictorial Publishing House, 2003. p. 39

[6] Wu Guosheng: The Course of Science (Second Edition), Peking University Press, 2002, p. 96

[7] [UK] edited by Michelle Hoskin: Cambridge Illustrated History of Astronomy, translated by Jiang Xiaoyuan, Guan Zengjian, and Niu Weixing, Shandong Pictorial Publishing House, 2003, p. 32

[8] [US] David Lindberg: The Beginnings of Western Science, translated by Wang Jun, Liu Xiaofeng, Zhou Wenfeng, and Wang Xirong, China Translation & Publishing Corporation, 2001, p. 99 (p95)

[9] [US] Morris Kline: Mathematical Thought from Ancient to Modern Times (Volume 1), translated by Zhang Lijing, Zhang Jinyan, and Jiang Zehan, Shanghai Scientific & Technical Publishers, 2002, p. 118

[10] [US] Morris Kline: Mathematical Thought from Ancient to Modern Times (Volume 1), translated by Zhang Lijing, Zhang Jinyan, and Jiang Zehan, Shanghai Scientific & Technical Publishers, 2002, pp. 118–119

[11] [US] Morris Kline: Mathematical Thought from Ancient to Modern Times (Volume 1), translated by Zhang Lijing, Zhang Jinyan, and Jiang Zehan, Shanghai Scientific & Technical Publishers, 2002, p. 182

[12] The end of Chapter 2 of Book VIII of the Almagest, cited in [US] Morris Kline: Mathematical Thought from Ancient to Modern Times (Volume 1), translated by Zhang Lijing, Zhang Jinyan, and Jiang Zehan, Shanghai Scientific & Technical Publishers, 2002, p. 182

[13] Book IX of the Almagest, ibid.

[14] See [UK] edited by Michelle Hoskin: Cambridge Illustrated History of Astronomy, translated by Jiang Xiaoyuan, Guan Zengjian, and Niu Weixing, Shandong Pictorial Publishing House, 2003, pp. 40–41

[15] Hypotheses of the Planets 2.12, cited in [Austria] Reiterb: Zhang Heng, Science and Religion, Social Sciences Academic Press, 2000, p. 158

[16] [Austria] Reiterb: Zhang Heng, Science and Religion, Social Sciences Academic Press, 2000, p. 159

[17] Victor J. Katz: A History of Mathematics: An Introduction, translated by Li Wenlin, Zou Jiancheng, Xu Mingwei, et al., proofread by Xu Mingwei and Li Wenlin, Higher Education Press, 2nd edition, 2004. p. 117

Latest Comments

Gu
2006-10-19 21:28:27

By the way, I recommend it: Cambridge Illustrated History of Astronomy is indeed very good, worthy of being a book that Teacher Jiang Xiaoyuan “resolutely” decided to translate
Gu
2006-10-28 12:23:49

sigh…… This paper only got 88 points, and that already counts as Wu laoshi being generous with the grading. I realized that one of Lao Yang’s suggestions was partly right: he said that I write everything strictly along my own line of thought, without paying attention to inferring the teacher’s line of thought behind the question. The first half of that was correct, but in fact I am not incapable of inferring the teacher’s thinking. In fact, before I even wrote my own paper, I had already mentioned on the course BBS in replies to other students: first, Ptolemy’s astronomy is not simply Greek mathematical astronomy. Ptolemy belongs to the Hellenistic period. This assignment’s topic was “The Basic Features of Hellenistic Science Seen from Ptolemaic Astronomy,” so the first thing to clarify is what Ptolemaic astronomy actually is. Since the task is to look at the features of Hellenistic science from it, rather than the features of Greek science, one needs to pay attention to some characteristics of the Hellenistic period that differ from the Greek period; second, from another angle, the main features of Hellenistic science were mostly inherited from the Greek spirit, so basically it is not wrong to write about the characteristics reflected in Greek science. I knew this paper should first seize the main features of Hellenistic science, namely the Greek tradition, and then highlight the differences between the two. But when I wrote it myself, my line of thought seemed more like writing one chapter in a general history of science: it felt as though the characteristics of Greek science had already been covered “earlier,” and that when it came to Ptolemy, I only needed to emphasize the additional features he introduced. As a result, I neglected the main Greek characteristics and did not mention Ptolemy’s connections with Pythagoras, Plato, and Eudoxus, which in Wu laoshi’s view was the most important part.

Of course, every one of my papers must contain at least a little of my own original line of thought or viewpoint. Here, I considered an element that Wu laoshi did not mention: namely, the influence of mathematical techniques (trigonometry). Without trigonometry, with only Apollonius’s geometric models, it would have been impossible to establish such a precise system. In addition, the sexagesimal numeral system from Babylon and ecliptic coordinates were also important basic tools. From the perspective of the history of mathematics, that is what is original in my paper.
古雴
2007-10-19 01:19:29

Wow~~~ the readership of this article………………
Wu laoshi still assigned this topic this year—no wonder all the students would search their way here…………
This article was not written well… the opinions in my own second comment are actually worth looking at. If they really did provide a reference for your assignment, then in keeping with academic integrity, please note them in the references~
12
2007-12-26 22:14:35 Anonymous 124.17.17.22

Could you provide some of the previous exam questions? Thanks
古
2007-12-26 23:58:39 Anonymous 125.34.52.84

My memory is bad; once the exam is over, I forget it. In any case, the exam wasn’t hard—the terminology explanations were all things you could find on the PPT. I only remember that the first big question was to briefly describe the historical shifts in the center of scientific activity (which is nothing more than moving from ancient Babylon and ancient Greece to Britain, France, Germany, and the United States). Those who consistently attend class usually don’t need any special preparation for the exam; those who always skip class might as well go to Wu laoshi’s personal homepage and read a few of his essays to get a feel for Wu laoshi’s basic style, and grasp the two words “freedom”……
古雴
2009-03-12 21:56:06

This topic is back again this year……
According to Wu laoshi’s preference, “equant point” should be translated as “off-center uniform-motion point.” Nothing else much. This article is the worst one I wrote among all Wu laoshi’s papers; fortunately, it seems that it was the TA who graded it at the time…… which kept my paper scores under Wu laoshi’s hand at 95.
A reminder to younger classmates: my papers and even the existence of this blog are all known to Wu laoshi, so plagiarism will be a suicidal act。。
古雴
2009-03-18 20:48:54

Let me give some more hints. If I had to write this assignment again now, I would definitely do a better job:
Actually, it is very easy to organize a line of thought for this topic:
First, ignore everything else and give an overall account of the basic content of Ptolemaic astronomy.
Then look at each clause, each item, each feature; basically they only have two possibilities: one, they were inherited and developed from ancient Greece; two, they were added in the Hellenistic period (and then they were probably mostly from the East). Then point them out one by one, and that will do.
You can either follow a reorganized order, or interweave the explanations. In short, for every sentence you use to describe the content of Ptolemaic astronomy, you can add a second-order interpretation, identifying what kind of intellectual feature it reflects and where it came from.
Written this way, the structure is clear and the key points won’t be missed.
Of course, one still needs to consult and read a large amount of literature to support it.
fog
2009-03-18 23:46:40

Is another history-of-science assignment being given out?
One function of a senior fellow is to write the assignments for you*^_*^