Xu Yue

Born: about 160 in Donglai, Shandong province, China
Died: about 227 in China
We know a little of Xu Yue but the main text which bears his name, the Shushu jiyi (Notes on Traditions of Arithmetic Methods), is probably the work of a later author trying to claim a certain respectability for his writings.

Xu Yue was a pupil of Liu Hong and he studied mathematics under the famous calendar expert. Liu Hong worked at the Imperial Observatory and it was there that Xu Yue held discussions with him and also with the head of the Astronomical Bureau. Mathematics was used by Liu Hong and others at the Observatory in their studies of astronomy and the related work on the calendar which, of course, was based on the apparent motion of the sun and the moon. It was natural, therefore that Xu Yue would gain expertise from these men in astronomy and calendar science.

It is reported that Xu Yue wrote a commentary on the Nine Chapters on the Mathematical Art. This preceded the major commentary written by Liu Hui in the second half of the third century, and it would appear that Liu Hui commented on a version of the text which did not include Xu Yue's comments. As a result Xu Yue's commentary has not come down to us.

Whether Xu Yue wrote the Shushu jiyi (Notes on Traditions of Arithmetic Methods) is uncertain. It is a rather strange work filled with ideas from Buddhism and from Taoism and describes arithmetical systems intermingled with religious nuances. The version which has come down to us today has a commentary by Zhen Luan written around 566, but that does little to help understand the difficulties in the text.

Fourteen old methods of calculation are mentioned in the text. One of these uses a device resembling the abacus called ball-arithmetic. Three others also uses balls, one involving balls in columns, one involving two balls of different colours which move at right angles to each other suggesting almost the idea of Cartesian coordinates. Needham writes in [4] that Xu Yue:-

... shows an interesting appreciation of coordinate relationships.

Unfortunately there is no great clarity in the descriptions.

There is no doubt that one of the main aims of the text is to introduce a notation which will allow the representation of large numbers. Most historians believe that the aim of the author was to suggest that it was possible to represent any number, no matter how large. Three systems of powers of 10 are given. The lower system is based on the sequence of powers of 10

10, 102, 103, 104, 105, ...

the middle system on powers of 104

104, 108, 1012, 1016, ...

the upper system being based on powers of 10, each being the square of its predecessor

104, 108, 1016, 1032, ...

Finally, Xu Yue talks about calculations of the nine balls. Again what is intended here is unclear, but the commentator Zhen Luan is in no doubt that this refers to a 3 by 3 magic square.

Despite being an obscure text, after editing by Li Chunfeng, the Shushu jiyi (Notes on Traditions of Arithmetic Methods) was selected as a text for the Imperial examinations in 656 and became one of The Ten Classics in 1084.

Liu Hong

Born: 129 in Lo-yang, China
Died: 210 in China
Liu Hong was of noble birth, descended from the Imperial family of the Eastern Han Dynasty. This dynasty was established in 25 AD after the brief 15 year reign of Wang Mang's Hsin dynasty. The capital was Lo-yang where a large ornate palace was built. In Encyclopaedia Britannica the aims and achievements of the Han rulers are described:-

... the Han came to require cultural accomplishment from their public servants, making mastery of classical texts a condition of employment. The title list of the enormous imperial library is China's first bibliography. Its text included works on practical matters such as mathematics and medicine, as well as treatises on philosophy and religion and the arts. Advancement in science and technology was also sought by the rulers, and the Han invented paper, used water clocks and sundials, and developed a seismograph. Calendars were published frequently during the period.

Liu Hong became interested in astronomy as a young boy. He was appointed to the Imperial Observatory in 160 and was involved in making astronomical observations. Two works which he wrote, namely the Qi Yao Shu (The Art of Seven Planets) and a new version entitled the Ba Yuan Shu (The Art of Eight Elements) have been lost so we know little of their contents. We do know that Liu presented these works to the Emperor in 174 or 175, and we also know that they were based on Buddhist beliefs.

Perhaps the greatest of Liu's achievements was his work which led to a new calendar. This calendar was published in 187 and described the motion of the moon far more accurately than any previous Chinese calendar. His measurements of the length of the shadow of a pole at the summer and at the winter solstices give results which are accurate to within 1% of their true value.


Born: about 120 in Western India
Died: about 180 in India
Indian astrology was originally known as Jyotisha, which means "science of the stars". Until around the first century AD no real distinction was made between astrology and astronomy and in fact most astronomical theories were propounded to support the theory that the positions of the heavenly bodies directly influenced human events.

The Indian methods of computing horoscopes all date back to the translation of a Greek astrology text into Sanskrit prose by Yavanesvara in 149 AD. Yavanesvara (or Yavanaraja) literally means "Lord of the Greeks" and it was a name given to many officials in western India during the period 130 AD - 390 AD. During this period the Ksatrapas ruled Gujarat (or Madhya Pradesh) and these "Lord of the Greeks" officials acted for the Greek merchants living in the area.

The particular "Lord of the Greeks" official Yavanesvara who we are interested in here worked under Rudradaman. Rudradaman became ruler of the Ksatrapas in around 130 AD and it was during the period of his rule that Yavanesvara worked as an official and made his translation. We know of Rudradaman because information is recorded in a lengthy Sanskrit inscription at Junagadh written around 150 AD.

The Greek astrology text in question was written in Alexandria some time round about 120 BC. Yavanesvara did far more than just translate the Greek text for such a translation would have had little relevance to the Indians. He therefore not only translated the language but he translated the context too. Instead of the Greek gods who appear in the original, Yavanesvara used Hindu images. Again he worked the Indian caste system into the work and made the work one which would fit well with the Indian thought.

The work was written with the aim of letting Indians became astrologers so it had to present astronomy in a form in which it could be used for astrology. In order to do this Yavanesvara put into his work an explanation of the Greek version of the Babylonian theory of the motions of the planets. All this he wrote in Sanskrit prose but sadly the original has not survived. We do have, however, a version written in Sanskrit verse 120 years after Yavanesvara's work appeared.

Yavanesvara had an important influence on the whole of astrology in India for centuries after he made his popular translation. Although the influence was more than on astrology, as the science of astronomy split from astrology, the influence of Yavanesvara's work reached into astronomy too.

Claudius Ptolemy

Born: About AD 85 in Egypt
Died: About AD 165 in Alexandria, Egypt
One of the most influential Greek astronomers and geographers of his time, Ptolemy propounded the geocentric theory in a form that prevailed for 1400 years. However, of all the ancient Greek mathematicians, it is fair to say that his work has generated more discussion and argument than any other. We shall discuss the arguments below for, depending on which are correct, they portray Ptolemy in very different lights. The arguments of some historians show that Ptolemy was a mathematician of the very top rank, arguments of others show that he was no more than a superb expositor, but far worse, some even claim that he committed a crime against his fellow scientists by betraying the ethics and integrity of his profession.

We know very little of Ptolemy's life. He made astronomical observations from Alexandria in Egypt during the years AD 127-41. In fact the first observation which we can date exactly was made by Ptolemy on 26 March 127 while the last was made on 2 February 141. It was claimed by Theodore Meliteniotes in around 1360 that Ptolemy was born in Hermiou (which is in Upper Egypt rather than Lower Egypt where Alexandria is situated) but since this claim first appears more than one thousand years after Ptolemy lived, it must be treated as relatively unlikely to be true. In fact there is no evidence that Ptolemy was ever anywhere other than Alexandria.

His name, Claudius Ptolemy, is of course a mixture of the Greek Egyptian 'Ptolemy' and the Roman 'Claudius'. This would indicate that he was descended from a Greek family living in Egypt and that he was a citizen of Rome, which would be as a result of a Roman emperor giving that 'reward' to one of Ptolemy's ancestors.

We do know that Ptolemy used observations made by 'Theon the mathematician', and this was almost certainly Theon of Smyrna who almost certainly was his teacher. Certainly this would make sense since Theon was both an observer and a mathematician who had written on astronomical topics such as conjunctions, eclipses, occultations and transits. Most of Ptolemy's early works are dedicated to Syrus who may have also been one of his teachers in Alexandria, but nothing is known of Syrus.

If these facts about Ptolemy's teachers are correct then certainly in Theon he did not have a great scholar, for Theon seems not to have understood in any depth the astronomical work he describes. On the other hand Alexandria had a tradition for scholarship which would mean that even if Ptolemy did not have access to the best teachers, he would have access to the libraries where he would have found the valuable reference material of which he made good use.

Ptolemy's major works have survived and we shall discuss them in this article. The most important, however, is the Almagest which is a treatise in thirteen books. We should say straight away that, although the work is now almost always known as the Almagest that was not its original name. Its original Greek title translates as The Mathematical Compilation but this title was soon replaced by another Greek title which means The Greatest Compilation. This was translated into Arabic as "al-majisti" and from this the title Almagest was given to the work when it was translated from Arabic to Latin.

The Almagest is the earliest of Ptolemy's works and gives in detail the mathematical theory of the motions of the Sun, Moon, and planets. Ptolemy made his most original contribution by presenting details for the motions of each of the planets. The Almagest was not superseded until a century after Copernicus presented his heliocentric theory in the De revolutionibus of 1543. Grasshoff writes in :-

Ptolemy's "Almagest" shares with Euclid's "Elements" the glory of being the scientific text longest in use. From its conception in the second century up to the late Renaissance, this work determined astronomy as a science. During this time the "Almagest" was not only a work on astronomy; the subject was defined as what is described in the "Almagest".

Ptolemy describes himself very clearly what he is attempting to do in writing the work (see for example ):-

We shall try to note down everything which we think we have discovered up to the present time; we shall do this as concisely as possible and in a manner which can be followed by those who have already made some progress in the field. For the sake of completeness in our treatment we shall set out everything useful for the theory of the heavens in the proper order, but to avoid undue length we shall merely recount what has been adequately established by the ancients. However, those topics which have not been dealt with by our predecessors at all, or not as usefully as they might have been, will be discussed at length to the best of our ability.

Ptolemy first of all justifies his description of the universe based on the earth-centred system described by Aristotle. It is a view of the world based on a fixed earth around which the sphere of the fixed stars rotates every day, this carrying with it the spheres of the sun, moon, and planets. Ptolemy used geometric models to predict the positions of the sun, moon, and planets, using combinations of circular motion known as epicycles. Having set up this model, Ptolemy then goes on to describe the mathematics which he needs in the rest of the work. In particular he introduces trigonometrical methods based on the chord function Crd (which is related to the sine function by sin a = (Crd 2a)/120).

Ptolemy devised new geometrical proofs and theorems. He obtained, using chords of a circle and an inscribed 360-gon, the approximation

π = 3 17/120 = 3.14166

and, using √3 = chord 60,

√3 = 1.73205.

He used formulae for the Crd function which are analogous to our formulae for sin(a + b), sin(a - b) and sin a/2 to create a table of the Crd function at intervals of 1/2 a degree.

This occupies the first two of the 13 books of the Almagest and then, quoting again from the introduction, we give Ptolemy's own description of how he intended to develop the rest of the mathematical astronomy in the work (see for example ):-

[After introducing the mathematical concepts] we have to go through the motions of the sun and of the moon, and the phenomena accompanying these motions; for it would be impossible to examine the theory of the stars thoroughly without first having a grasp of these matters. Our final task in this way of approach is the theory of the stars. Here too it would be appropriate to deal first with the sphere of the so-called 'fixed stars', and follow that by treating the five 'planets', as they are called.

In examining the theory of the sun, Ptolemy compares his own observations of equinoxes with those of Hipparchus and the earlier observations Meton in 432 BC. He confirmed the length of the tropical year as 1/300 of a day less than 365 1/4 days, the precise value obtained by Hipparchus. Since, as Ptolemy himself knew, the accuracy of the rest of his data depended heavily on this value, the fact that the true value is 1/128 of a day less than 365 1/4days did produce errors in the rest of the work. We shall discuss below in more detail the accusations which have been made against Ptolemy, but this illustrates clearly the grounds for these accusations since Ptolemy had to have an error of 28 hours in his observation of the equinox to produce this error, and even given the accuracy that could be expected with ancient instruments and methods, it is essentially unbelievable that he could have made an error of this magnitude. A good discussion of this strange error is contained in the excellent article .

Based on his observations of solstices and equinoxes, Ptolemy found the lengths of the seasons and, based on these, he proposed a simple model for the sun which was a circular motion of uniform angular velocity, but the earth was not at the centre of the circle but at a distance called the eccentricity from this centre. This theory of the sun forms the subject of Book 3 of the Almagest.

In Books 4 and 5 Ptolemy gives his theory of the moon. Here he follows Hipparchus who had studied three different periods which one could associate with the motion of the moon. There is the time taken for the moon to return to the same longitude, the time taken for it to return to the same velocity (the anomaly) and the time taken for it to return to the same latitude. Ptolemy also discusses, as Hipparchus had done, the synodic month, that is the time between successive oppositions of the sun and moon. In Book 4 Ptolemy gives Hipparchus's epicycle model for the motion of the moon but he notes, as in fact Hipparchus had done himself, that there are small discrepancies between the model and the observed parameters. Although noting the discrepancies, Hipparchus seems not to have worked out a better model, but Ptolemy does this in Book 5 where the model he gives improves markedly on the one proposed by Hipparchus. An interesting discussion of Ptolemy's theory of the moon is given in .

Having given a theory for the motion of the sun and of the moon, Ptolemy was in a position to apply these to obtain a theory of eclipses which he does in Book 6. The next two books deal with the fixed stars and in Book 7 Ptolemy uses his own observations together with those of Hipparchus to justify his belief that the fixed stars always maintain the same positions relative to each other. He wrote (see for example ):-

If one were to match the above alignments against the diagrams forming the constellations on Hipparchus's celestial globe, he would find that the positions of the relevant stars on the globe resulting from the observations made at the time of Hipparchus, according to what he recorded, are very nearly the same as at present.

In these two book Ptolemy also discusses precession, the discovery of which he attributes to Hipparchus, but his figure is somewhat in error mainly because of the error in the length of the tropical year which he used. Much of Books 7 and 8 are taken up with Ptolemy's star catalogue containing over one thousand stars.

The final five books of the Almagest discuss planetary theory. This must be Ptolemy's greatest achievement in terms of an original contribution, since there does not appear to have been any satisfactory theoretical model to explain the rather complicated motions of the five planets before the Almagest. Ptolemy combined the epicycle and eccentric methods to give his model for the motions of the planets. The path of a planet P therefore consisted of circular motion on an epicycle, the centre C of the epicycle moving round a circle whose centre was offset from the earth. Ptolemy's really clever innovation here was to make the motion of C uniform not about the centre of the circle around which it moves, but around a point called the equant which is symmetrically placed on the opposite side of the centre from the earth.

The planetary theory which Ptolemy developed here is a masterpiece. He created a sophisticated mathematical model to fit observational data which before Ptolemy's time was scarce, and the model he produced, although complicated, represents the motions of the planets fairly well.

Toomer sums up the Almagest in  as follows:-

As a didactic work the "Almagest" is a masterpiece of clarity and method, superior to any ancient scientific textbook and with few peers from any period. But it is much more than that. Far from being a mere 'systemisation' of earlier Greek astronomy, as it is sometimes described, it is in many respects an original work.

We will return to discuss some of the accusations made against Ptolemy after commenting briefly on his other works. He published the tables which are scattered throughout the Almagest separately under the title Handy Tables. These were not merely lifted from the Almagest however but Ptolemy made numerous improvements in their presentation, ease of use and he even made improvements in the basic parameters to give greater accuracy. We only know details of the Handy Tables through the commentary by Theon of Alexandria but in the author shows that care is required since Theon was not fully aware of Ptolemy's procedures.

Ptolemy also did what many writers of deep scientific works have done, and still do, in writing a popular account of his results under the title Planetary Hypothesis. This work, in two books, again follows the familiar route of reducing the mathematical skills needed by a reader. Ptolemy does this rather cleverly by replacing the abstract geometrical theories by mechanical ones. Ptolemy also wrote a work on astrology. It may seem strange to the modern reader that someone who wrote such excellent scientific books should write on astrology. However, Ptolemy sees it rather differently for he claims that the Almagest allows one to find the positions of the heavenly bodies, while his astrology book he sees as a companion work describing the effects of the heavenly bodies on people's lives.

In a book entitled Analemma he discussed methods of finding the angles need to construct a sundial which involves the projection of points on the celestial sphere. In Planisphaerium he is concerned with stereographic projection of the celestial sphere onto a plane. This is discussed in  where it is stated:-

In the stereographic projection treated by Ptolemy in the "Planisphaerium" the celestial sphere is mapped onto the plane of the equator by projection from the south pole. Ptolemy does not prove the important property that circles on the sphere become circles on the plane.

Ptolemy's major work Geography, in eight books, attempts to map the known world giving coordinates of the major places in terms of latitude and longitude. It is not surprising that the maps given by Ptolemy were quite inaccurate in many places for he could not be expected to do more than use the available data and this was of very poor quality for anything outside the Roman Empire, and even parts of the Roman Empire are severely distorted. In  Ptolemy is described as:-

... a man working [on map-construction] without the support of a developed theory but within a mathematical tradition and guided by his sense of what is appropriate to the problem.

Another work on Optics is in five books and in it Ptolemy studies colour, reflection, refraction, and mirrors of various shapes. Toomer comments in :-

The establishment of theory by experiment, frequently by constructing special apparatus, is the most striking feature of Ptolemy's "Optics". Whether the subject matter is largely derived or original, "The Optics" is an impressive example of the development of a mathematical science with due regard to physical data, and is worthy of the author of the "Almagest".

An English translation, attempting to remove the inaccuracies introduced in the poor Arabic translation which is our only source of the Optics is given in .

The first to make accusations against Ptolemy was Tycho Brahe. He discovered that there was a systematic error of one degree in the longitudes of the stars in the star catalogue, and he claimed that, despite Ptolemy saying that it represented his own observations, it was merely a conversion of a catalogue due to Hipparchus corrected for precession to Ptolemy's date. There is of course definite problems comparing two star catalogues, one of which we have a copy of while the other is lost.

After comments by Laplace and Lalande, the next to attack Ptolemy vigorously was Delambre. He suggested that perhaps the errors came from Hipparchus and that Ptolemy might have done nothing more serious than to have failed to correct Hipparchus's data for the time between the equinoxes and solstices. However Delambre then goes on to say :-

One could explain everything in a less favourable but all the simpler manner by denying Ptolemy the observation of the stars and equinoxes, and by claiming that he assimilated everything from Hipparchus, using the minimal value of the latter for the precession motion.

However, Ptolemy was not without his supporters by any means and further analysis led to a belief that the accusations made against Ptolemy by Delambre were false. Boll writing in 1894 says :-

To all appearances, one will have to credit Ptolemy with giving an essentially richer picture of the Greek firmament after his eminent predecessors.

Vogt showed clearly in his important paper  that by considering Hipparchus's Commentary on Aratus and Eudoxus and making the reasonable assumption that the data given there agreed with Hipparchus's star catalogue, then Ptolemy's star catalogue cannot have been produced from the positions of the stars as given by Hipparchus, except for a small number of stars where Ptolemy does appear to have taken the data from Hipparchus. Vogt writes:-

This allows us to consider the fixed star catalogue as of his own making, just as Ptolemy himself vigorously states.

The most recent accusations of forgery made against Ptolemy came from Newton in. He begins this book by stating clearly his views:-

This is the story of a scientific crime. ... I mean a crime committed by a scientist against fellow scientists and scholars, a betrayal of the ethics and integrity of his profession that has forever deprived mankind of fundamental information about an important area of astronomy and history.

Towards the end Newton, having claimed to prove every observation claimed by Ptolemy in the Almagest was fabricated, writes:-

[Ptolemy] developed certain astronomical theories and discovered that they were not consistent with observation. Instead of abandoning the theories, he deliberately fabricated observations from the theories so that he could claim that the observations prove the validity of his theories. In every scientific or scholarly setting known, this practice is called fraud, and it is a crime against science and scholarship.

Although the evidence produced by Brahe, Delambre, Newton and others certainly do show that Ptolemy's errors are not random, this last quote from is, I [EFR] believe, a crime against Ptolemy (to use Newton's own words). The book is written to study validity of these accusations and it is a work which I strongly believe gives the correct interpretation. Grasshoff writes:-

... one has to assume that a substantial proportion of the Ptolemaic star catalogue is grounded on those Hipparchan observations which Hipparchus already used for the compilation of the second part of his "Commentary on Aratus". Although it cannot be ruled out that coordinates resulting from genuine Ptolemaic observations are included in the catalogue, they could not amount to more than half the catalogue.

... the assimilation of Hipparchan observations can no longer be discussed under the aspect of plagiarism. Ptolemy, whose intention was to develop a comprehensive theory of celestial phenomena, had no access to the methods of data evaluation using arithmetical means with which modern astronomers can derive from a set of varying measurement results, the one representative value needed to test a hypothesis. For methodological reason, then, Ptolemy was forced to choose from a set of measurements the one value corresponding best to what he had to consider as the most reliable data. When an intuitive selection among the data was no longer possible ... Ptolemy had to consider those values as 'observed' which could be confirmed by theoretical predictions.

As a final comment we quote the epigram which is accepted by many scholars to have been written by Ptolemy himself, and it appears in Book 1 of the Almagest, following the list of contents :-

Well do I know that I am mortal, a creature of one day.
But if my mind follows the winding paths of the stars
Then my feet no longer rest on earth, but standing by
Zeus himself I take my fill of ambrosia, the divine dish.

For quizzes related to ptolemy

Zhang Heng

Born: 78 in Nan-yang, China
Died: 139 in China
Zhang Heng was born at the time of the Eastern Han (sometimes called Later Han) dynasty, the second half of the longest lasting Chinese dynasty. The Eastern Han was established in 25 AD after the brief 15 year reign of Wang Mang's Hsin dynasty had replaced the Western Han dynasty. At the time of Zhang Heng's birth the Emperor was Chang-ti, the third of the Eastern Han emperors. The capital of the country had moved to Lo-yang where a large ornate palace had been built. In Encyclopaedia Britannica the aims and achievements of the Han rulers are described as follows:-

... the Han came to require cultural accomplishment from their public servants, making mastery of classical texts a condition of employment. The title list of the enormous imperial library is China's first bibliography. Its text included works on practical matters such as mathematics and medicine, as well as treatises on philosophy and religion and the arts. Advancement in science and technology was also sought by the rulers, and the Han invented paper, used water clocks and sundials, and developed a seismograph. Calendars were published frequently during the period.

Over the years that Zhang Heng grew up, Chinese influence and prestige were growing rapidly and reached their peak in around 90 when he was about 12 years old. This was the period during which Ho-ti was the emperor and the court began to be influenced by family members seeking to extend their own power. Zhang, who had been born into an important family, was educated in the moral and political philosophy of Confucianism. For ten years he studied literature and trained as a writer. He published a number of literary works which gained him considerable fame. We shall give more information below on these aspects of Zhang's achievements as well as examples of his poetry. Zhang was thirty years old before his interests turned from literature to scientific matters, and at that time he became particularly interested in astronomy.

In around 116 he was appointed an official at the Emperor's court in Lo-yang. The court, however, was beginning to provide a less efficient government due to the weakness of successive emperors who were manipulated by those around them seeking advantage for themselves. This was hardly surprising since many emperors came to the throne as children. If China began to suffer due to ambitious people seeking to further their own influence, this was certainly not the case for Zhang. His biography in The History of the Eastern Han Dynasty (see [5]) suggests that he was not as successful an official as he might have been precisely because of a lack of ambition. This seemed to stem, at least in part, from his strong moral beliefs based in Confucianism. He refused advancement in his career on several occasions when he turned down posts that were offered to him, and he also spent periods away from the capital when he lived in isolation and thought about the nature of the universe and about a wide variety of scientific topics. His highest position at court was when he became chief astrologer and minister under the emperor An-ti. He held the position of chief astrologer on a number of occasions.

We will describe below some of Zhang's outstanding scientific achievements. However, as we indicated above, he first achieved fame as a poet and writer of over twenty works, and in this capacity he had a lasting influence on Chinese culture. His works Si Chou Shi (Four Chapters of Distressed Poems) and Gui Tian Fu (To Live in Seclusion) are considered literary masterpieces. Zhang's poem, in his highly influential style of prose poetry, which we now quote comes from [1]. It is a telling criticism of the last rulers of the Western Han dynasty:-

Those who won this territory were strong;
Those who depended on it endured.
When a stream is long, its water is not easily exhausted.
When roots are deep, they do not rot easily.
Therefore, as extravagance and ostentation were given free reign,
The odour became pungent and increasingly fulsome.

Zhang wrote the Four Stanzas of Sorrow which is the first seven-syllabic poem which we know of in China. We quote (in translation of course) only the first of its four stanzas:-

In Taishan stays my dear sweetheart,
But Liangfu keeps us long apart;
Looking east, I find tears start.
She gives me a sword to my delight;
A jade I give her as requite.
I'm at a loss as she is out of sight;
Why should I trouble myself all night?

In ancient China there was a belief that an emperor received his right to rule from heaven. Changing the calendar was seen as one of the duties of the office, establishing the emperor's heavenly link on earth. After a change of ruler, and even more significantly after a change of dynasty, the new Chinese emperor would seek a new official calendar thus establishing a new rule with new celestial influences. It was natural therefore that Zhang having become an expert in astronomy should become involved in calendar reform by the year 123. In that year he corrected the calendar to bring it into line with his accurate astronomical observations.

In 132 Zhang invented the first seismograph for measuring earthquakes. One has to understand how significant earthquakes were in China at this time, not only for the destructive power which they unleashed but also because they were seen as punishment from the gods for poor governance of the country. In his role as chief astrologer he was responsible for detecting signs of bad government which were indicated by earthquakes. Zhang's device, which he called Hou Feng Di Dong Yi, was made of copper. It was in the shape of an egg with eight dragon heads around the top, each with a copper ball in its mouth, and a pendulum in the centre. Around the bottom were eight frogs, each directly under a dragon head. When an earthquake occurred, a ball fell out of a the dragon's mouth into a frog's mouth, making a noise. In fact the seismograph detected an earthquake in February of 138 and Zhang reported this fact to the Emperor despite no other evidence of the earthquake being felt in the capital Lo-yang. He was even able to indicate that the earthquake was to the west of the capital. He achieved fame when reports of an earthquake more than a thousand kilometres north west reached the capital several days later.

Zhang appears to have been the first person in China to construct an equatorial armillary sphere. It consisted of a system of rings corresponding to the great circles of the celestial sphere with a central tube which was used to line up stars and planets. With this instrument Zhang was able to make more accurate star maps than earlier Chinese astronomers. He wrote about his instrument in the work Hun-i chu where he described his version of the universe as follows:-

The sky is like a hen's egg, and is as round as a crossbow pellet, the Earth is like the yolk of the egg, lying alone at the centre. The sky is large and the Earth small.

In another work, Ling Xian (Mystical Laws), he describes the stars:-

North and south of the equator there are 124 groups which are always brightly shining. 320 stars can be named. There are in all 2500, not including those which the sailors observe. Of the very small stars there are 11520.

Only the first part of this text by Zhang has survived.

In mathematics Zhang studied 3 by 3 magic squares. He also proposed, in a treatise on inscribed and circumscribed circles of a square, that π = √10 or approximately 3.162. Although this is not particularly accurate the significance of his work is pointed out by S K Mo in [8]. As Mo notes, the significance here is that all earlier attempts to calculate were based on practical measurement, whereas the work by Zhang was based on a theoretical calculation.

Zhang also gave formulae for the volume of a sphere in terms of the volume of the circumscribing cube. These results are not very accurate and Liu Hui in his commentary on the Nine Chapters on the Mathematical Art writes:-

In an attempt to make his statement consistent and harmonise his philosophy of yin and yang, and the doctrine of odd and even, [Zhang] neglected the precision of the data.

Li Chunfeng wrote in his commentary on the Nine Chapters on the Mathematical Art:-

Zhang Heng followed the ancients blindly, making himself a laughing stock for later generations.

One interesting point to note in some of Zhang's mathematical work is that he leaves square roots as unevaluated. Some historians believe that Zhang understood the difference between rational and irrational numbers but this seems to be stretching things a bit too far.

Menelaus of Alexandria

Born: about 70 in (possibly) Alexandria, Egypt
Died: about 130
Although we know little of Menelaus of Alexandria's life Ptolemy records astronomical observations made by Menelaus in Rome on the 14th January in the year 98. These observation included that of the occultation of the star Beta Scorpii by the moon.

He also makes an appearance in a work by Plutarch who describes a conversation between Menelaus and Lucius in which Lucius apologises to Menelaus for doubting the fact that light, when reflected, obeys the law that the angle of incidence equals the angle of reflection. Lucius says (see for example [1]):-

In your presence, my dear Menelaus, I am ashamed to confute a mathematical proposition, the foundation, as it were, on which rests the subject of catoptrics. Yet it must be said that the proposition, "All reflection occurs at equal angles" is neither self evident nor an admitted fact.

This conversation is supposed to have taken place in Rome probably quite a long time after 75 AD, and indeed if our guess that Menelaus was born in 70 AD is close to being correct then it must have been many years after 75 AD.

Very little else is known of Menelaus's life, except that he is called Menelaus of Alexandria by both Pappus and Proclus. All we can deduce from this is that he spent some time in both Rome and Alexandria but the most likely scenario is that he lived in Alexandria as a young man, possibly being born there, and later moved to Rome.

An Arab register of mathematicians composed in the 10th century records Menelaus as follows (see [1]):-

He lived before Ptolemy, since the latter makes mention of him. He composed: "The Book of Spherical Propositions", "On the Knowledge of the Weights and Distribution of Different Bodies" ... Three books on the "Elements of Geometry", edited by Thabit ibn Qurra, and "The Book on the Triangle". Some of these have been translated into Arabic.

Of Menelaus's many books only Sphaerica has survived. It deals with spherical triangles and their application to astronomy. He was the first to write down the definition of a spherical triangle giving the definition at the beginning of Book I:-

A spherical triangle is the space included by arcs of great circles on the surface of a sphere ... these arcs are always less than a semicircle.

In Book I of Sphaerica he set up the basis for treating spherical triangles as Euclid treated plane triangles. He used arcs of great circles instead of arcs of parallel circles on the sphere. This marks a turning point in the development of spherical trigonometry. However, Menelaus seems unhappy with the method of proof by reductio ad absurdum which Euclid frequently uses. Menelaus avoids this way of proving theorems and, as a consequence, he gives proofs of some of the theorems where Euclid's proof could be easily adapted to the case of spherical triangles by quite different methods.

It is also worth commenting that [3]:-

In some respects his treatment is more complete than Euclid's treatment of the analogous plane case.

Book 2 applies spherical geometry to astronomy. It largely follows the propositions given by Theodosius in his Sphaerica but Menelaus give considerably better proofs.

Book 3 deals with spherical trigonometry and includes Menelaus's theorem. For plane triangles the theorem was known before Menelaus:-

... if a straight line crosses the three sides of a triangle (one of the sides is extended beyond the vertices of the triangle), then the product of three of the nonadjacent line segments thus formed is equal to the product of the three remaining line segments of the triangle.

Menelaus produced a spherical triangle version of this theorem which is today also called Menelaus's Theorem, and it appears as the first proposition in Book III. The statement is given in terms of intersecting great circles on a sphere.

Many translations and commentaries of Menelaus Sphaerica were made by the Arabs. Some of these survive but differ considerably and make an accurate reconstruction of the original quite difficult. On the other hand we do know that some of the works are commentaries on earlier commentaries so it is easy to see how the original becomes obscured. There are detailed discussions of these Arabic translations in [6], [9], and [10].

There are other works by Menelaus which are mentioned by Arab authors but which have been lost both in the Greek and in their Arabic translations. We gave a quotation above from the 10th century Arab register which records a book called Elements of Geometry which was in three volumes and was translated into Arabic by Thabit ibn Qurra. It also records another work by Menelaus was entitled Book on Triangles and although this has not survived fragments of an Arabic translation have been found.

Proclus referred to a geometrical result of Menelaus which does not appear in the work which has survived and it is thought that it must come from one of the texts just mentioned. This was a direct proof of a theorem in Euclid's Elements and given Menelaus's dislike for reductio ad absurdum in his surviving works this seems a natural line for him to follow. The new proof which Proclus attributes to Menelaus is of the theorem (in Heath's translation of Euclid):-

If two triangles have the two sides equal to two sides respectively, but have the base of one greater than the base of the other, it will also have the angle contained by the equal straight lines of the first greater than that of the other.

Another Arab reference to Menelaus suggests that his Elements of Geometry contained Archytas's solution of the problem of duplicating the cube. Paul Tannery in [8] argues that this make it likely that a curve which it is claimed by Pappus that Menelaus discussed at length was the Viviani's curve of double curvature. Bulmer-Thomas in [1] comments that:-

It is an attractive conjecture but incapable of proof on present evidence.

Menelaus is believed by a number of Arab writers to have written a text on mechanics. It is claimed that the text studied balances studied by Archimedes and those devised by Menelaus himself. In particular Menelaus was interested in specific gravities and analysing alloys.

Theon of Smyrna

Born: about 70 in Smyrna (now Izmir), Turkey
Died: about 135
Little is known of Theon of Smyrna's life. He was called 'the old Theon' by Theon of Alexandria and 'Theon the mathematician' by Ptolemy. The date of his birth is little better than a guess, but we do have some firm data about dates in his life. We know that he was making astronomical observations of Mercury and Venus between 127 and 132 since Ptolemy lists four observations which Theon made in 127, 129, 130 and 132. From these observations Theon made estimates of the greatest angular distance that Mercury and Venus can reach from the Sun. The style of his bust, dedicated by his son 'Theon the priest', gives us the date of his death to within 10 years and it is placed within the period 130-140 (hence our midpoint guess of 135).

Theon's most important work is Expositio rerum mathematicarum ad legendum Platonem utilium. This work is a handbook for philosophy students to show how prime numbers, geometrical numbers such as squares, progressions, music and astronomy are interrelated. Its rather curious title means that it was intended as an introduction to a study of the works of Plato, but this is rather fanciful. As Huxley writes in [1]:-

... the book has little to offer the specialist student of Plato's mathematics. It is, rather, a handbook for philosophy students, written to illustrate how arithmetic, geometry, stereometry, music, and astronomy are interrelated.

The most important feature of the work is the wide range of citations of earlier sources. Its worst feature is its lack of originality. Heath writes [2]:-

Theon's work is a curious medley, valuable, not intrinsically, but for the numerous historical notices which it contains.

In the introduction Theon gives his reason for writing the work:-

Everyone would agree that he could not understand the mathematical arguments used by Plato unless he were practised in this science... One who had become skilled in all geometry and all music and astronomy would be reckoned most happy on making acquaintance with the writings of Plato, but this cannot be come by easily or readily, for it calls for a very great deal of application from youth upwards. In order that those who have failed to become practised in these studies, but aim at a knowledge of his writings, should not wholly fail in their desires, I shall make a summary and concise sketch of the mathematical theorems which are specially necessary for readers of Plato....

The work begins with a collection of theorems which Theon says will be useful for the study of arithmetic, music, geometry, and astronomy in Plato. However his coverage of geometry is none too good and later in the book he makes an excuse for this saying that anyone who reads his book, or the works of Plato, will have already studied elementary geometry.

In the section on numbers Theon adopts a Pythagorean approach, writing about odd numbers, even numbers, prime numbers, composite numbers, square numbers, oblong numbers, triangular numbers, polygonal numbers, circular numbers, spherical numbers, solid numbers with three factors, pyramidal numbers, perfect numbers, deficient numbers and abundant numbers.

The best section of Expositio rerum mathematicarum is the astronomy section which teaches that the Earth is spherical, that mountains are negligible in height compared with the Earth etc. It includes knowledge of conjunctions, eclipses, occultations and transits. However, Neugebauer writes in [3]:-

It is clear that Theon's treatise does not pretend to make original contributions to astronomy. Unfortunately it is also clear that Theon has not fully digested the material he is presenting to his readers.

Theon also wrote commentaries on the main authorities of mathematics and astronomy. In particular he wrote an important work on Ptolemy and another on Plato's Republic which he refers to himself in work which survives. Whether his work on the ancestry of Plato is a separate work or a section of one of his commentaries on Plato's work, it is impossible to say.

Nicomachus of Gerasa

Born: about 60 in Gerasa, Roman Syria (now Jarash, Jordan)
Died: about 120
Nicomachus of Gerasa is mentioned in a small number of sources and we can date him fairly accurately from the information given. Nicomachus himself refers to Thrasyllus who died in 36 AD so this gives lower limits on his dates. On the other hand Apuleius, the Platonic philosopher, rhetorician and author whose dates are 124 AD to about 175 AD, translated Nicomachus's Introduction to Arithmetic into Latin so this gives an upper limit on his dates. One of the most interesting references is by Lucian, the rhetorician, pamphleteer and satirist who was born about 120 AD, who makes one of his characters say:-

You calculate like Nicomachus.

Clearly Nicomachus had achieved fame for his arithmetical work!

In the paper [7] Dillon argues that Nicomachus died in 196 AD. His argument is based on the fact that Marinus claimed that Proclus believed that he was the reincarnation of Nicomachus. Since Proclus was born in 412 AD and there was a belief among Pythagoreans that reincarnations occurred with an interval of 216 years, the date fits. Although 196 AD is not ruled out by his translator dying in 175 AD (although it comes close) the most serious objection to Dillon's theory seems to be the lack of evidence that Proclus himself believed in the 216 year interval.

Let us move from conjectures to more certain ground, and record that Nicomachus was a Pythagorean. This is obvious from his writings on numbers and music, but we are also told this by Porphyry who says that he was one of the leading members of the Pythagorean School.

Nicomachus wrote Arithmetike eisagoge (Introduction to Arithmetic) which was the first work to treat arithmetic as a separate topic from geometry. Unlike Euclid, Nicomachus gave no abstract proofs of his theorems, merely stating theorems and illustrating them with numerical examples.

However Introduction to Arithmetic does contain quite elementary errors which show that Nicomachus chose not to give proofs of his results because he did not in general have such proofs. Many of the results were known by Nicomachus to be true since they appeared with proofs in Euclid, although in a geometrical formulation. Sometimes Nicomachus stated a result which is simply false and then illustrated it with an example which happens to have the properties described in the result. We must deduce from this that some of the results are merely guesses based on the evidence of the numerical examples (and in some cases perhaps even based on one example!).

An example of this we look more closely at the results which Nicomachus quotes on perfect numbers. He states that the nth perfect number has n digits, and that all perfect numbers end in 6 and 8 alternately. These statements must be merely false deductions from the fact that there were four perfect numbers known to Nicomachus, namely 6, 28, 496 and 8128.

The work contains the first multiplication table in a Greek text. It is also remarkable in that it contains Arabic numerals, not Greek ones. However, in many respects the book is old fashioned in its style since it appears more in tune with the number theoretic ideas of Pythagoras with his mystical approach, rather than a true mathematical approach. To illustrate Nicomachus's rather strange approach to numbers, giving the moral properties, we look at his description of abundant numbers and deficient numbers. An abundant number has the sum of its proper divisors greater than the number, while a deficient number has the sum of its proper divisors less than the number. Nicomachus writes of these numbers in Introduction to Arithmetic (see [6], or [3] for a different translation):-

In the case of the too much, is produced excess, superfluity, exaggerations and abuse; in the case of too little, is produced wanting, defaults, privations and insufficiencies. And in the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort - of which the most exemplary form is that type of number which is called perfect.

He then continues his description of abundant numbers as resembling an animal:-

... with ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms, or having too many fingers on one of its hands....

while a deficient number is like an animal:-

... with a single eye, ... one armed or one of his hands has less than five fingers, or if he does not have a tongue...

For over 1000 years Introduction to Arithmetic was the standard arithmetic text. In view of the comments we have made regarding the work, this may seem a surprising fact. Mathematicians disliked the work, in particular Pappus is said to have despised it. However, several people including Boethius translated Introduction to Arithmetic into Latin and it was used as a school book. How then could a poor book become so popular. Heath tries to explain the apparent contradiction in [4], suggesting that:-

... it was at first read by philosophers rather than mathematicians, and afterwards became generally popular at a time when there were no mathematicians left, but only philosophers who incidentally took an interest in mathematics.

Arab translations of Nicomachus's Introduction to Arithmetic were important and in [5] Brentjes studies the influence of these Arabic translations. She concludes that most Arabic texts on number theory written by mathematicians were influenced by both Euclid and Nicomachus, but were mainly influenced by Euclid. However, texts by non-mathematicians were most strongly influenced by Nicomachus. This research in [5] tends to support the views of Heath on this subject.

Nicomachus also wrote two volumes Theologoumena arithmetikes (The Theology of Numbers) which was completely concerned with mystic properties of numbers. However Heath writes [4]:-

The curious farrago which has come down to us under that title and which was edited by Ast [published in Leipzig in 1817] is, however, certainly not by Nicomachus; for among the authors from whom it gives extracts is Anatolius, Bishop of Laodicaea (270 AD); but it contains quotations from Nicomachus which appear to come from the genuine work.

Another work by Nicomachus which has survived is Manual of Harmonics which is a work on music. Again Nicomachus shows the influence of Pythagoras but also Aristotle's theories of music. The work looks at musical notes and the octave. The principles of tuning a stretched string are studied as is an extension of the octave to the two-octave range.

The influences of Pythagoras's theory of music are seen from Nicomachus' (see [1]):-

... assignment of number and numerical ratios to notes and intervals, his recognition of the indivisibility of the octave and the whole tone... But, unlike Euclid, who attempts to prove musical propositions through mathematical theorems, Nicomachus seeks to show their validity by measurement of the lengths of strings.

Both Porphyry and Iamblichus wrote biographies of Pythagoras which quote from Nicomachus. From this evidence some historians have conjectured that Nicomachus also wrote a biography of Pythagoras and, although there is no direct evidence, it is indeed quite possible.


Born: 1st century AD in possibly Lysimachia, Hellespont, Greece
Died: 1st century AD
Cleomedes is known only through his book On the Circular Motions of the Celestial Bodies which is an uninspiring astronomy textbook. There are a number of points of interest in this book, however, as we shall discuss below.

We should first discuss the perplexing question of the period in which Cleomedes lived. The only certainty here is that On the Circular Motions of the Celestial Bodies discusses the work of Posidonius at length and so is clearly written after the middle of the first century BC. In fact On the Circular Motions of the Celestial Bodies ends with the words (see for example [3]):-

The preceding teachings are not the author's own opinion but collected from older or more recent summaries; much of it is taken from Posidonius.

It is hard to estimate from these words how long after Posidonius the author, Cleomedes, is writing.

Heath [2] favours a date in the middle of the first century BC. He also points out:-

As [Cleomedes] seems to know nothing of the works of Ptolemy, he can hardly... have lived later than the beginning of the second century AD.

Neugebauer, however, disagrees with these conclusions of Heath and proposes that Cleomedes wrote his text around 370 AD. His argument is based on a comment by Cleomedes in the text where he remarks than there are two bright stars (Aldebaran and Antares) such that the rising of one and the setting of the other take place at the same time. These stars Cleomedes claims lie at 15 of their sign. Using Ptolemy's positions for the stars at the time the Almagest was written and Ptolemy's value of 1 per 100 years for precession, Neugebauer gets his date of 371 AD for Cleomedes writings, to which Neugebauer estimates a maximum error of 50 years on either side.

Astronomically Neugebauer's calculations are of course perfectly correct. However they are suspect for a number of other reasons. Firstly the data in On the Circular Motions of the Celestial Bodies does not seem to be due to Cleomedes but to a variety of sources. Of course accepting this argument would make Cleomedes dates later still. Secondly the data in Cleomedes is of widely differing degrees of accuracy. Some is very good, while other data has errors of 20%. Thirdly the actual astronomical event of Aldebaran setting and Antares rising at the same instant can never be observed.

Heath's comment that Cleomedes knows nothing of the works of Ptolemy is also less certain than it might at first appear. Cleomedes is writing an elementary textbook and it is certainly not always the case that one mentions recent research in a low level textbook. For example many elementary textbooks on applied mathematics still use Newton's gravitational methods (and for good reason) 100 years after Einstein gave an improved theory of gravitation.

In [1] Dicks suggests that the most likely date for Cleomedes is the first century AD and we have taken that as the best available compromise from what is known. That is not to say that Neugebauer's date is impossible. In fact there are other features of the text which would tend to support the fourth century AD as a date, despite the lack of references to Ptolemy. Not least of these is the fact that this was a period when many second rate textbooks of this nature were written and the style is not unlike that of other fourth century AD texts, some of which give the same astronomical data as Cleomedes.

On the Circular Motions of the Celestial Bodies is a work in two volumes. We commented above that it was important for a number of reasons. The most important certainly is that it gives us the best indication that we have of the contents and the style of a text by Posidonius. As Heath comments [2]:-

... the very long first chapter of Book II (nearly half of the Book) ... seems for the most part to be copied bodily from Posidonius.

But this is not Cleomedes' main aim in writing the text. It is written to attack the Epicureans who believed among other odd beliefs, that the sun was as large as it looked, namely one foot across. Cleomedes spends much time in his text showing that this is false, but it does seem as if he is going to extremes when he compares Epicureans unfavourably with rats, reptiles and worms. Cleomedes' own philosophical views show that he is a Stoic.

Since On the Circular Motions of the Celestial Bodies is compiled from a number of sources, there is a fair variation in the quality, and in many places the book fails to be consistent where the various sources disagree. Whenever a piece of text is thought to be due to Cleomedes himself, there is much evidence that his understanding of the topic was very limited and, but for the quality of his sources, one feels that Cleomedes would not fare any better than the Epicureans for naiveté.

Lunar eclipses are described well in the text, and the conical shape of the earth's shadow shows an interesting depth of understanding (at least of Cleomedes' source). He also correctly explains the reports of lunar eclipses seen when both the sun and moon are above the horizon as being due to refraction.

Neugebauer gives samples of the sense and nonsense which are mixed in Cleomedes [3]:-

Cleomedes states that no fixed star has an apparent diameter less than one finger (a rather absurd exaggeration) while the apparent diameter of Venus should be two fingers, i.e. 1/6 of the lunar or solar diameter. Of some interest is the remark that the absolute size of fixed stars may reach, or even surpass, the sun ... it is [also] said that the earth, seen from the sun, would appear at best a very small star.

One further interest in Cleomedes' work is that it is in On the Circular Motions of the Celestial Bodies that we learn of Eratosthenes method of measuring the circumference of the earth. This is one of the best known of the achievements of early mathematical astronomy and we are indebted to Cleomedes for relating the method. Of course not everyone believes that the story of Eratosthenes' measurement is authentic but, despite this, it is widely accepted.

Finally we should remark that Neugebauer suggests from a study of certain astronomical data given in On the Circular Motions of the Celestial Bodies that Cleomedes lived in the Hellespont on the Black Sea, suggesting the city of Lysimachia. Neugebauer admits that the city of Lysimachia was destroyed in 144 BC which seems at odds with his own date of 370 AD for Cleomedes but he is able to show that despite the disaster of 144 BC records of the city certainly extend up to the fourth century AD. The weakness of Neugebauer's argument must surely be that almost all of Cleomedes' text and data is taken from the works of others so Neugebauer's arguments seem only to give strong evidence for one of Cleomedes' sources having written in Lysimachia.

Heron of Alexandria

Born: about 10 in (possibly) Alexandria, Egypt
Died: about 75
Sometimes called Hero, Heron of Alexandria was an important geometer and worker in mechanics. Perhaps the first comment worth making is how common the name Heron was around this time and it is a difficult problem in the history of mathematics to identify which references to Heron are to the mathematician described in this article and which are to others of the same name. There are additional problems of identification which we discuss below.

A major difficulty regarding Heron was to establish the date at which he lived. There were two main schools of thought on this, one believing that he lived around 150 BC and the second believing that he lived around 250 AD. The first of these was based mainly on the fact that Heron does not quote from any work later than Archimedes. The second was based on an argument which purported to show that he lived later that Ptolemy, and, since Pappus refers to Heron, before Pappus.

Both of these arguments have been shown to be wrong. There was a third date proposed which was based on the belief that Heron was a contemporary of Columella. Columella was a Roman soldier and farmer who wrote extensively on agriculture and similar subjects, hoping to foster in people a love for farming and a liking for the simple life. Columella, in a text written in about 62 AD [5]:-

... gave measurements of plane figures which agree with the formulas used by Heron, notably those for the equilateral triangle, the regular hexagon (in this case not only the formula but the actual figures agree with Heron's) and the segment of a circle which is less than a semicircle ...

However, most historians believed that both Columella and Heron were using an earlier source and claimed that the similarity did not prove any dependence. We now know that those who believed that Heron lived around the time of Columella were in fact correct, for Neugebauer in 1938 discovered that Heron referred to a recent eclipse in one of his works which, from the information given by Heron, he was able to identify with one which took place in Alexandria at 23.00 hours on 13 March 62.

From Heron's writings it is reasonable to deduce that he taught at the Museum in Alexandria. His works look like lecture notes from courses he must have given there on mathematics, physics, pneumatics, and mechanics. Some are clearly textbooks while others are perhaps drafts of lecture notes not yet worked into final form for a student textbook.

Pappus describes the contribution of Heron in Book VIII of his Mathematical Collection. Pappus writes (see for example [8]):-

The mechanicians of Heron's school say that mechanics can be divided into a theoretical and a manual part; the theoretical part is composed of geometry, arithmetic, astronomy and physics, the manual of work in metals, architecture, carpentering and painting and anything involving skill with the hands.

... the ancients also describe as mechanicians the wonder-workers, of whom some work by means of pneumatics, as Heron in his Pneumatica, some by using strings and ropes, thinking to imitate the movements of living things, as Heron in his Automata and Balancings, ... or by using water to tell the time, as Heron in his Hydria, which appears to have affinities with the science of sundials.

A large number of works by Heron have survived, although the authorship of some is disputed. We will discuss some of the disagreements in our list of Heron's works below. The works fall into several categories, technical works, mechanical works and mathematical works. The surviving works are:

On the dioptra dealing with theodolites and surveying. It contains a chapter on astronomy giving a method to find the distance between Alexandria and Rome using the difference between local times at which an eclipse of the moon is observed at each cities. The fact that Ptolemy does not appear to have known of this method led historians to mistakenly believe Heron lived after Ptolemy;

The pneumatica in two books studying mechanical devices worked by air, steam or water pressure. It is described in more detail below;

The automaton theatre describing a puppet theatre worked by strings, drums and weights;

Belopoeica describing how to construct engines of war. It has some similarities with work by Philon and also work by Vitruvius who was a Roman architect and engineer who lived in the 1st century BC;

The cheirobalistra about catapults is thought to be part of a dictionary of catapults but was almost certainly not written by Heron;

Mechanica in three books written for architects and described in more detail below;

Metrica which gives methods of measurement. We give more details below;
Definitiones contains 133 definitions of geometrical terms beginning with points, lines etc. In [15] Knorr argues convincingly that this work is in fact due to Diophantus;

Geometria seems to be a different version of the first chapter of the Metrica based entirely on examples. Although based on Heron's work it is not thought to be written by him;

Stereometrica measures three-dimensional objects and is at least in part based on the second chapter of the Metrica again based on examples. Again it is though to be based on Heron's work but greatly changed by many later editors;

Mensurae measures a whole variety of different objects and is connected with parts of Stereometrica and Metrica although it must be mainly the work of a later author;

Catoprica deals with mirrors and is attributed by some historians to Ptolemy although most now seem to believe that this is a genuine work of Heron. In this work, Heron states that vision results from light rays emitted by the eyes. He believes that these rays travel with infinite velocity.

Let us examine some of Heron's work in a little more depth. Book I of his treatise Metrica deals with areas of triangles, quadrilaterals, regular polygons of between 3 and 12 sides, surfaces of cones, cylinders, prisms, pyramids, spheres etc. A method, known to the Babylonians 2000 years before, is also given for approximating the square root of a number. Heron gives this in the following form (see for example [5]):-

Since 720 has not its side rational, we can obtain its side within a very small difference as follows. Since the next succeeding square number is 729, which has 27 for its side, divide 720 by 27. This gives 26 2/3. Add 27 to this, making 53 2/3, and take half this or 26 5/6. The side of 720 will therefore be very nearly 26 5/6. In fact, if we multiply 26 5/6 by itself, the product is 720 1/36, so the difference in the square is 1/36. If we desire to make the difference smaller still than 1/36, we shall take 720 1/36 instead of 729 (or rather we should take 26 5/6 instead of 27), and by proceeding in the same way we shall find the resulting difference much less than 1/36.

Heron also proves his famous formula in Book I of the Metrica :

if A is the area of a triangle with sides a, b and c and s = (a + b + c)/2 then
A2 = s (s - a)(s - b)(s - c).

In Book II of Metrica, Heron considers the measurement of volumes of various three dimensional figures such as spheres, cylinders, cones, prisms, pyramids etc. His preface is interesting, partly because knowledge of the work of Archimedes does not seem to be as widely known as one might expect (see for example [5]):-

After the measurement of surfaces, rectilinear or not, it is proper to proceed to solid bodies, the surfaces of which we have already measured in the preceding book, surfaces plane and spherical, conical and cylindrical, and irregular surfaces as well. The methods of dealing with these solids are, in view of their surprising character, referred to Archimedes by certain writers who give the traditional account of their origin. But whether they belong to Archimedes or another, it is necessary to give a sketch of these results as well.

Book III of Metrica deals with dividing areas and volumes according to a given ratio. This was a problem which Euclid investigated in his work On divisions of figures and Heron's Book III has a lot in common with the work of Euclid. Also in Book III, Heron gives a method to find the cube root of a number. In particular Heron finds the cube root of 100 and the authors of [9] give a general formula for the cube root of N which Heron seems to have used in his calculation:

a + b d/(b d + aD)(b - a),
where a3 < N < b3, d = N - a3, D = b3 - N.

In [9] it is remarked that this is a very accurate formula, but, unless a Byzantine copyist is to be blamed for an error, they conclude that Heron might have borrowed this accurate formula without understanding how to use it in general.

The Pneumatica is a strange work which is written in two book, the first with 43 chapters and the second with 37 chapters. Heron begins with a theoretical consideration of pressure in fluids. Some of this theory is right but, not surprisingly, some is quite wrong. Then there follows a description of a whole collection of what might best be described as mechanical toys for children [1]:-

Trick jars that give out wine or water separately or in constant proportions, singing birds and sounding trumpets, puppets that move when a fire is lit on an altar, animals that drink when they are offered water ...

Although all this seems very trivial for a scientist to be involved with, it would appear that Heron is using these toys as a vehicle for teaching physics to his students. It seems to be an attempt to make scientific theories relevant to everyday items that students of the time would be familiar with.

There is, rather remarkably, descriptions of over 100 machines such as a fire engine, a wind organ, a coin-operated machine, and a steam-powered engine called an aeolipile. Heron's aeolipile, which has much in common with a jet engine, is described in [2] as follows:-

The aeolipile was a hollow sphere mounted so that it could turn on a pair of hollow tubes that provided steam to the sphere from a cauldron. The steam escaped from the sphere from one or more bent tubes projecting from its equator, causing the sphere to revolve. The aeolipile is the first known device to transform steam into rotary motion.

Heron wrote a number of important treatises on mechanics. They give methods of lifting heavy weights and describe simple mechanical machines. In particular the Mechanica is based quite closely on ideas due to Archimedes. Book I examines how to construct three dimensional shapes in a given proportion to a given shape. It also examines the theory of motion, certain statics problems, and the theory of the balance.

In Book II Heron discusses lifting heavy objects with a lever, a pulley, a wedge, or a screw. There is a discussion on centres of gravity of plane figures. Book III examines methods of transporting objects by such means as sledges, the use of cranes, and looks at wine presses.

Other works have been attributed to Heron, and for some of these we have fragments, for others there are only references. The works for which fragments survive include one on Water clocks in four books, and Commentary on Euclid's Elements which must have covered at least the first eight books of the Elements. Works by Heron which are referred to, but no trace survives, include Camarica or On vaultings which is mentioned by Eutocius and Zygia or On balancing mentioned by Pappus. Also in the Fihrist, a tenth century survey of Islamic culture, a work by Heron on how to use an astrolabe is mentioned.

Finally it is interesting to look at the opinions that various writers have expressed as to the quality and importance of Heron. Neugebauer writes [7]:-

The decipherment of the mathematical cuneiform texts made it clear that much of the "Heronic" type of Greek mathematics is simply the last phase of the Babylonian mathematical tradition which extends over 1800 years.

Some have considered Heron to be an ignorant artisan who copied the contents of his books without understanding what he wrote. This in particular has been levelled against the Pneumatica but Drachmann, writing in [1], says:-

... to me the free flowing, rather discursive style suggests a man well versed in his subject who is giving a quick summary to an audience that knows, or who might be expected to know, a good deal about it.

Some scholars have approved of Heron's practical skills as a surveyor but claimed that his knowledge of science was negligible. However, Mahony writes in [1]:-

In the light of recent scholarship, he now appears as a well-educated and often ingenious applied mathematician, as well as a vital link in a continuous tradition of practical mathematics from the Babylonians, through the Arabs, to Renaissance Europe.

Finally Heath writes in [5]:-

The practical utility of Heron's manuals being so great, it was natural that they should have great vogue, and equally natural that the most popular of them at any rate should be re-edited, altered and added to by later writers; this was inevitable with books which, like the "Elements" of Euclid, were in regular use in Greek, Byzantine, Roman, and Arabian education for centuries.

Bryson of Heraclea

Born: about 450 BC in Heraclea (now Taranto, Italy)
Died: ?

Plato and Aristotle both mention a mathematician called Bryson, but as is often the case, there is not complete agreement among scholars as to whether these refer to the same person or to two different people.

Aristotle mentions Bryson of Heraclea, who was the son of Herodorus of Heraclea. Bryson was a Sophist and Aristotle criticises him both for his assertion that there is no such thing as indecent language, and also for his method of squaring the circle. We do know some details of this methods of squaring the circle and, despite the criticisms of Aristotle, it was an important step forward in the development of mathematics. Aristotle's criticism appears to have been based on the fact that Bryson's proof used general principles rather than on geometric ones, but it is somewhat unclear exactly what Aristotle meant by this.

Diogenes Laertius gives some other biographical details of Bryson, but these cannot all be correct since Bryson's interaction with a number of philosophers is stated, yet certain of these are impossible due to the dates during which these men lived. Perhaps the most likely of the details preserved by Diogenes Laertius is that Bryson was either a pupil of Socrates or of Euclid of Megara.

It is a little difficult to reconstruct exactly what Bryson's method of squaring the circle was. According to Alexander Aphrodisiensis, writing in about 210 AD, Bryson inscribed a square in the circle and circumscribed a second square. Bryson then constructed a third square between the inscribed and circumscribed square (but Alexander does not tell us how this third square was constructed).

Alexander then claims that Bryson's argument was that the circle was intermediate between the inscribed and circumscribed squares, the third square is also intermediate between the inscribed and circumscribed squares and therefore the third square equals the circle. Alexander then rightly points out that this argument of nonsense since, to use Alexander's example, 8 and 9 are both greater than 7 and less than 10 but 8 does not equal 9.

If indeed Alexander is right in what he attributes to Bryson then his contribution would not merit inclusion in this archive. However, other commentators attribute a much more significant argument to Bryson. Themistius, another ancient commentator, writes that Bryson claimed that the circle was greater than all inscribed polygons and less than all circumscribed polygons. It is unclear how quite how Bryson continued the argument but it seems likely that he was saying that by taking polygons with larger and larger numbers of sides then the difference the inscribed and circumscribed polygons could be made as small as we like so that a polygon intermediate between them will equal the circle to whatever degree of accuracy we chose.

This would be an improvement on Antiphon's argument and Bryson is getting close to the method of exhaustion as rigorously applied by Archimedes.

We know little else of Bryson. He wrote Diatribes which some accused Plato of stealing and indeed Bryson is claimed to have associated with Polyxenus who put forward philosophical arguments which appear in Plato's Pramenides and could well be the arguments which were claimed stolen from Bryson's Diatribes.

Democritus of Abdera

Born: about 460 BC in Abdera, Thrace, Greece
Died: about 370 BC
Democritus of Abdera is best known for his atomic theory but he was also an excellent geometer. Very little is known of his life but we know that Leucippus was his teacher.

Democritus certainly visited Athens when he was a young man, principally to visit Anaxagoras, but Democritus complained how little he was known there. He said, according to Diogenes Laertius writing in the second century AD [5]:-

I came to Athens and no one knew me.

Democritus was disappointed by his trip to Athens because Anaxagoras, then an old man, had refused to see him.

As Brumbaugh points out in [3]:-

How different he would find the trip today, where the main approach to the city from the northeast runs past the impressive "Democritus Nuclear Research Laboratory".

Certainly Democritus made many journeys other than the one to Athens. Russell in [9] writes:-

He travelled widely in southern and eastern lands in search of knowledge, he perhaps spent a considerable time in Egypt, and he certainly visited Persia. He then returned to Abdera, where he remained.

Democritus himself wrote (but some historians dispute that the quote is authentic) (see [5]):-

Of all my contemporaries I have covered the most ground in my travels, making the most exhaustive inquiries the while; I have seen the most climates and countries and listened to the greatest number of learned men.

His travels certainly took him to Egypt and Persia, as Russell suggests, but he almost certainly also travelled to Babylon, and some claim he travelled to India and Ethiopia. Certainly he was a man of great learning. As Heath writes in [7]:-

... there was no subject to which he did not notably contribute, from mathematics and physics on the one hand to ethics and poetics on the other; he even went by the name of 'wisdom'.

Although little is known of his life, quite a lot is known of his physics and philosophy. There are two main sources for our knowledge of his of physical and philosophical theories. Firstly Aristotle discusses Democritus's ideas thoroughly because he strongly disagreed with his ideas of atomism. The second source is in the work of Epicurus but, in contrast to Aristotle, Epicurus is a strong believer in Democritus's atomic theory. This work of Epicurus is preserved by Diogenes Laertius in his second century AD book [5].

Certainly Democritus was not the first to propose an atomic theory. His teacher Leucippus had proposed an atomic system, as had Anaxagoras of Clazomenae. In fact traces of an atomic theory go back further than this, perhaps to the Pythagorean notion of the regular solids playing a fundamental role in the makeup of the universe. However Democritus produced a much more elaborate and systematic view of the physical world than had any of his predecessors. His view is summarised in [2]:-

Democritus asserted that space, or the Void, had an equal right with reality, or Being, to be considered existent. He conceived of the Void as a vacuum, an infinite space in which moved an infinite number of atoms that made up Being (i.e. the physical world). These atoms are eternal and invisible; absolutely small, so small that their size cannot be diminished (hence the name atomon, or "indivisible"); absolutely full and incompressible, as they are without pores and entirely fill the space they occupy; and homogeneous, differing only in shape, arrangement, position, and magnitude.

With this as a basis to the physical world, Democritus could explain all changes in the world as changes in motion of the atoms, or changes in the way that they were packed together. This was a remarkable theory which attempted to explain the whole of physics based on a small number of ideas and also brought mathematics into a fundamental physical role since the whole of the structure proposed by Democritus was quantitative and subject to mathematical laws. Another fundamental idea in Democritus's theory is that nature behaves like a machine, it is nothing more than a highly complex mechanism.

There are then questions for Democritus to answer. Where do qualities such as warmth, colour, and taste fit into the atomic theory? To Democritus atoms differ only in quantity, and all qualitative differences are only apparent and result from impressions of an observer caused by differing configurations of atoms. The properties of warmth, colour, taste are only by convention - the only things that actually exist are atoms and the Void.

Democritus's philosophy contains an early form of the conservation of energy. In his theory atoms are eternal and so is motion. Democritus explained the origin of the universe through atoms moving randomly and colliding to form larger bodies and worlds. There was no place in his theory for divine intervention. Instead he postulated a world which had always existed, and would always exist, and was filled with atoms moving randomly. Vortex motions occurred due to collisions of the atoms and in resulting vortex motion created differentiation of the atoms into different levels due only to their differing mass. This was not a world which came about through the design or purpose of some supernatural being, but rather it was a world which came about through necessity, that is from the nature of the atoms themselves.

Democritus built an ethical theory on top of his atomist philosophy. His system was purely deterministic so he could not admit freedom of choice to individuals. To Democritus freedom of choice was an illusion since we are unaware of all the causes for a decision. Democritus believed that [3]:-

... the soul will either be disturbed, so that its motion affects the body in a violent way, or it will be at rest in which case it regulates thoughts and actions harmoniously. Freedom from disturbance is the condition that causes human happiness, and this is the ethical goal.

Democritus describes the ultimate good, which he identifies with cheerfulness, as:-

... a state in which the soul lives peacefully and tranquilly, undisturbed by fear or superstition or any other feeling.

He wanted to remove the belief in gods which were, he believed, only introduced to explain phenomena for which no scientific explanation was then available.

Very little is known for certainty about Democritus's contributions to mathematics. As stated in the Oxford Classical Dictionary :-

Little is known (although much is written) about the mathematics of Democritus.

We do know that Democritus wrote many mathematical works. Diogenes Laertius (see [5]) lists his works and gives Thrasyllus as the source of this information. He wrote On numbers, On geometry, On tangencies, On mappings, On irrationals but none of these works survive. However we do know a little from other references. Heath [7] writes:-

In the Method of Archimedes, happily discovered in 1906, we are told that Democritus was the first to state the important propositions that the volume of a cone is one third of that of a cylinder having the same base and equal height, and that the volume of a pyramid is one third of that of a prism having the same base and equal height; that is to say, Democritus enunciated these propositions some fifty years or more before they were first scientifically proved by Eudoxus.

There is another intriguing piece of information about Democritus which is given by Plutarch in his Common notions against the Stoics where he reports on a dilemma proposed by Democritus as reported by the Stoic Chrysippus (see [7], [10] or [11]).

If a cone were cut by a plane parallel to the base [by which he means a plane indefinitely close to the base], what must we think of the surfaces forming the sections? Are they equal or unequal? For, if they are unequal, they will make the cone irregular as having many indentations, like steps, and unevennesses; but, if they are equal, the sections will be equal, and the cone will appear to have the property of the cylinder and to be made up of equal, not unequal, circles, which is very absurd.

There are important ideas in this dilemma. Firstly notice, as Heath points out in [7], that Democritus has the idea of a solid being the sum of infinitely many parallel planes and he may have used this idea to find the volumes of the cone and pyramid as reported by Archimedes. This idea of Democritus may have led Archimedes later to apply the same idea to great effect. This idea would eventually lead to theories of integration.

There is much discussion in [7], [8], [10] and [11] as to whether Democritus distinguished between the geometrical continuum and the physical discrete of his atomic system. Heath points out that if Democritus carried over his atomic theory to geometrical lines then there is no dilemma for him since his cone is indeed stepped with atom sized steps. Heath certainly believed that to Democritus lines were infinitely divisible. Others, see for example [10], have come to the opposite conclusion, believing that Democritus made contributions to problems of applied mathematics but, because of his atomic theory, he could not deal with the infinitesimal questions arising.

Hippias of Elis

Born: about 460 BC in Elis, Peloponnese, Greece
Died: about 400 BC
Hippias of Elis was a statesman and philosopher who travelled from place to place taking money for his services. He lectured on poetry, grammar, history, politics, archaeology, mathematics and astronomy. Plato describes him as a vain man being both arrogant and boastful, having a wide but superficial knowledge. Heath tells us something of this character when he writes in [3]:-

He claimed ... to have gone once to the Olympian festival with everything that he wore made by himself, ring and sandal (engraved), oil-bottle, scraper, shoes, clothes, and a Persian girdle of expensive type; he also took poems, epics, tragedies, dithyrambs, and all sorts of prose works.

As to Hippias's academic achievements, Heath writes:-

He was a master of the science of calculation, geometry, astronomy, 'rhythms and harmonies and correct writing'. He also had a wonderful system of mnemonics enabling him, if he once heard a string of fifty names to remember them all.

A rather nice story, which says more of the Spartans than it does of Hippias, is that it was reported that he received no payment for the lectures he gave in Sparta since [3]:-

... the Spartans could not endure lectures on astronomy or geometry or calculation; it was only a small minority of them who could even count; what they liked was history and archaeology.

Since Hippias was reported to give lectures on archaeology, he seems to have chosen the wrong topics when he lectured in Sparta!

Hippias's only contribution to mathematics seems to be the quadratrix which may have been used by him for trisecting an angle and squaring the circle. The curve may be used for dividing an angle into any number of equal parts. Perhaps the highest compliment that we can pay to Hippias is to report on the arguments of certain historians of mathematics who have claimed that the Hippias who discovered the quadratrix cannot be Hippias of Elis since geometry was not far enough advanced at this time to have allowed him to make these discoveries. However, their arguments are not generally accepted and there is ample evidence to attribute the discovery of the quadratrix to Hippias of Elis.

Heath [3] writes:-

It was probably about 420 BC that Hippias of Elis invented the curve known as the quadratrix for the purpose of trisecting any angle.

However this is far from certain and there is some evidence to suggest that Geminus, writing in the first century BC, had in his possession a treatise by Hippias of Elis on the quadratrix which indicated how it could be used to square the circle. If this is indeed the case then the treatise by Hippias must have been lost between this time and that of Sporus in the third century AD.

Pappus wrote his major work on geometry Synagoge in 340. It is a collection of mathematical writings in eight books. Book IV contains a description of the quadratrix of Hippias.

Look at the diagram of the quadratrix.

ABCD is a square and BED is part of a circle, centre A radius AB. As the radius AB rotates about A to move to the position AD then the line BC moves at the same rate parallel to itself to end at AD. Then the locus of the point of intersection F of the rotating radius AB and the moving line BC is the quadratrix. Hence

angle EAD/angle BAD = arc ED/arc BED = FH/AB,

so, taking AB = 1,

angle EAD = arc ED = FH π/2.

To divide the angle FAD in a given ratio, say p : q, then draw a point P on the line FH dividing it in the ratio p : q.
Draw a line through P parallel to AD to meet the quadratrix at Q. Then AQ divides angle FAD in the ratio p : q.

Pappus also gives the rather more complicated version of the construction necessary to square the circle. However, Pappus reports that Sporus had two criticisms of Hippias's method with which he agrees. The second is specifically related to the construction necessary for squaring the circle which we have not described. The first however relates to the construction of the quadratrix itself. Pappus reports that Sporus writes (see [3]):-

The very thing for which the construction is thought to serve is actually assumed in the hypothesis. For how is it possible, with two points starting from B, to make one of them move along a straight line to A and the other along a circumference to D in an equal time, unless you first know the ratio of the straight line AB to the circumference BED? In fact this ratio must also be that of the speeds of motion. For, if you employ speeds not definitely adjusted to this ratio, how can you make the motions end at the same moment, unless this should sometime happen by pure chance? Is not the thing thus shown to be absurd?

The point here seems to be a question of what exactly Hippias is trying to show with his quadratrix. Certainly he knew perfectly well that he was not providing a ruler and compass construction for squaring the circle. Exactly what he has proved concerning squaring the circle is, as Pappus and Sporus suggest, far from clear.

Theodorus of Cyrene

Born: 465 BC in Cyrene (now Shahhat, Libya)
Died: 398 BC in Cyrene (now Shahhat, Libya)
Theodorus of Cyrene was a pupil of Protagoras and himself the tutor of Plato, teaching him mathematics, and also the tutor of Theaetetus. Plato travelled to and from Egypt and on such occasions he spent time with Theodorus in Cyrene. Theodorus, however, did not spend his whole life in Cyrene for he was certainly in Athens at a time when Socrates was alive.

Theodorus, in addition to his work in mathematics, was [5]:-

... distinguished ... in astronomy, arithmetic, music and all educational subjects.

A member of the society of Pythagoras, Theodorus was one of the main philosophers in the Cyrenaic school of moral philosophy. He believed that pleasures and pains are neither good nor bad. Cheerfulness and wisdom, he believed, were sufficient for happiness.

Our knowledge of Theodorus comes through Plato who wrote about him in his work Theaetetus. Theodorus is remembered by mathematicians for his contribution to the development of irrational numbers and it is this aspect of his work which Plato describes (see for example [5]):-

[Theodorus] was proving to us a certain thing about square roots, I mean the side (i.e. root) of a square of three square units and of five square units, that these roots are not commensurable in length with the unit length, and he went on in this way, taking all the separate cases up to the root of seventeen square units, at which point, for some reason, he stopped.

Our whole knowledge of Theodorus's mathematical achievements are given by this passage from Plato. Yet there are points of interest which immediately arise. The first point is that Plato does not credit Theodorus with a proof that the square root of two was irrational. This must be because √2 was proved irrational before Theodorus worked on the problem, some claim this was proved by Pythagoras himself.

There is no doubt that Theodorus would have constructed lines of length √3, √5 etc. using Pythagoras's theorem. It is also clear that Theodorus had no general result here, for Plato goes on to describe how Theodorus's results inspired Theaetetus and Socrates to look at generalisations:-

The idea occurred to the two of us (Theaetetus and Socrates), seeing that these square roots appeared to be unlimited in multitude, to try to arrive at one collective term by which we could designate all these roots....

So the question which naturally comes next is how did Theodorus prove that √3, √5, ..., √17 were irrational without giving a proof which would clearly prove that any non-square number was irrational. The usual proof that √2 is irrational, namely the one which supposes that √2 = p/q where p/q is a rational in its lowest terms and derives a contradiction by showing that p and q are both even, would have been known to Theodorus. This proof generalises easily (for a modern mathematicians thinking in terms of numbers rather than lengths) to show √n is irrational for any non-square n. It is almost impossible to conceive that Theodorus would have used this proof on each of √3, √5, ..., √17 without obtaining a general theorem long before he got to 17.

An interesting proposal was made by Zeuthen in 1915. He suggested that Theodorus may have used the result which would later appear in Euclid's Elements namely:-

If, when the lesser of two unequal magnitudes is continually subtracted in turn from the greater, that which is left never measures the one before it, the magnitudes will be incommensurable.

Heath [5] illustrates the use of this result to show that √5 is irrational. Start with 1 and √5.

√5/1 = 2 +(√5-2)
1/(√5-2) = 4 + (√5-2)2
(√5-2)/(√5-2)2= 1/(√5-2) = 4 + (√5-2)2

The process now clearly fails to terminate since the ratio 1 : (√5-2) is the same as (√5-2) : (√5-2)2. Heath [5] gives a geometric version of this, starting with a right-angled triangle with sides 1, 2 and √5 which may be close to the method that Theodorus used. However there is little chance to do more than guess at Theodorus's method.
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