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Thales of Miletus

A compendium of articles

In the beginning was a question. Thales asked ‘What is the world made of?’ With that question, and by Thales insistence that it NOT be answered by a story about the gods, Thales planted the seed that would grow into Western Science. Thales based his speculations on observation and reason rather than revelation – Thales ideas were criticized and the dialogue prompted resulted in an open clash of competing ideas – with the ultimate appeal not to the whims of the gods but to nature and reason – this marks the birth of science.

There is considerable agreement that Thales was born in Miletus in Greek Ionia in the mid 620s BCE and died in about 546 BCE, but even those dates are indefinite. Aristotle, the major source for Thales's philosophy and science, identified Thales as the first person to investigate the basic principles, the question of the originating substances of matter and, therefore, as the founder of the school of natural philosophy. Thales was interested in almost everything, investigating almost all areas of knowledge, philosophy, history, science, mathematics, engineering, geography, and politics. He proposed theories to explain many of the events of nature, the primary substance, the support of the earth, and the cause of change. Thales was much involved in the problems of astronomy and provided a number of explanations of cosmological events which traditionally involved supernatural entities. His questioning approach to the understanding of heavenly phenomena was the beginning of Greek astronomy. Thales's hypotheses were new and bold, and in freeing phenomena from godly intervention, he paved the way towards scientific endeavour. He founded the Milesian school of natural philosophy, developed the scientific method, and initiated the first western enlightenment. A number of anecdotes is closely connected to Thales's investigations of the cosmos. When considered in association with his hypotheses they take on added meaning and are most enlightening. Thales was highly esteemed in ancient times, and a letter cited by Diogenes Laertius, and purporting to be from Anaximenes to Pythagoras, advised that all our discourse should begin with a reference to Thales.

The articles are:

A.Thales: By J J O'Connor and E F Robertson

Articles B to S are taken from the Internet Encyclopaedia of Philosophy.

B.The Writings of Thales
C.Possible Sources for Aristotle
D.Thales says Water is the Primary Principle
E.Thales and Mythology
F. Thales's Primary Principle

G. New Ideas about the earth
        a. The Earth Floats on Water
        b. Thales's Spherical Earth
        c.  Earthquake Theory

H. All Things are Full of God

I. Thales's Astronomy
        a. The Eclipse of Thales
        b. Setting the Solstices
        c. Thales's Discovery of the Seasons
        d. Thales's Determination of the Diameters of the Sun and the Moon
        e. Ursa Minor
        f. Falling into a Well

J. Mathematics
        a. The Theorems Attributed to Thales

K. Crossing the Halys
L. The Possible Travels of Thales
M.The Milesian School
N. The Seven Sages of Ancient Greece
O. Corner in Oil
P.The Heritage of Thales
Q.Bibliography to B to P
R. Abbreviations in B to P

S.  Greek astronomy: By J J O'Connor and E F Robertson
T. The trigonometric functions:   By J J O'Connor and E F Robertson
U. The history of cartography:     By J J O'Connor and E F Robertson

A. Thales: By J J O'Connor and E F Robertson

Thales of Miletus was the son of Examyes and Cleobuline. His parents are said by some to be from Miletus but others report that they were Phoenicians. J Longrigg writes in [1]:-
But the majority opinion considered him a true Milesian by descent, and of a distinguished family.
Thales seems to be the first known Greek philosopher, scientist and mathematician although his occupation was that of an engineer. He is believed to have been the teacher of Anaximander (611 BC - 545 BC) and he was the first natural philosopher in the Milesian School. However, none of his writing survives so it is difficult to determine his views or to be certain about his mathematical discoveries. Indeed it is unclear whether he wrote any works at all and if he did they were certainly lost by the time of Aristotle who did not have access to any writings of Thales. On the other hand there are claims that he wrote a book on navigation but these are based on little evidence. In the book on navigation it is suggested that he used the constellation Ursa Minor, which he defined, as an important feature in his navigation techniques. Even if the book is fictitious, it is quite probable that Thales did indeed define the constellation Ursa Minor.

Proclus, the last major Greek philosopher, who lived around 450 AD, wrote:-
[Thales] first went to Egypt and thence introduced this study [geometry] into Greece. He discovered many propositions himself, and instructed his successors in the principles underlying many others, his method of attacking problems had greater generality in some cases and was more in the nature of simple inspection and observation in other cases.

There is a difficulty in writing about Thales and others from a similar period. Although there are numerous references to Thales which would enable us to reconstruct quite a number of details, the sources must be treated with care since it was the habit of the time to credit famous men with discoveries they did not make. Partly this was as a result of the legendary status that men like Thales achieved, and partly it was the result of scientists with relatively little history behind their subjects trying to increase the status of their topic with giving it an historical background.

Certainly Thales was a figure of enormous prestige, being the only philosopher before Socrates to be among the Seven Sages. Plutarch, writing of these Seven Sages, says that (see [8]):-
[Thales] was apparently the only one of these whose wisdom stepped, in speculation, beyond the limits of practical utility, the rest acquired the reputation of wisdom in politics.

This comment by Plutarch should not be seen as saying that Thales did not function as a politician. Indeed he did. He persuaded the separate states of Ionia to form a federation with a capital at Teos. He dissuaded his compatriots from accepting an alliance with Croesus and, as a result, saved the city.

It is reported that Thales predicted an eclipse of the Sun in 585 BC. The cycle of about 19 years for eclipses of the Moon was well known at this time but the cycle for eclipses of the Sun was harder to spot since eclipses were visible at different places on Earth. Thales's prediction of the 585 BC eclipse was probably a guess based on the knowledge that an eclipse around that time was possible. The claims that Thales used the Babylonian saros, a cycle of length 18 years 10 days 8 hours, to predict the eclipse has been shown by Neugebauer to be highly unlikely since Neugebauer shows in [11] that the saros was an invention of Halley. Neugebauer wrote [11]:-
... there exists no cycle for solar eclipses visible at a given place: all modern cycles concern the earth as a whole. No Babylonian theory for predicting a solar eclipse existed at 600 BC, as one can see from the very unsatisfactory situation 400 years later, nor did the Babylonians ever develop any theory which took the influence of geographical latitude into account.

After the eclipse on 28 May, 585 BC Herodotus wrote:-
... day was all of a sudden changed into night. This event had been foretold by Thales, the Milesian, who forewarned the Ionians of it, fixing for it the very year in which it took place. The Medes and Lydians, when they observed the change, ceased fighting, and were alike anxious to have terms of peace agreed on.

Longrigg in [1] even doubts that Thales predicted the eclipse by guessing, writing:-
... a more likely explanation seems to be simply that Thales happened to be the savant around at the time when this striking astronomical phenomenon occurred and the assumption was made that as a savant he must have been able to predict it.

There are several accounts of how Thales measured the height of pyramids. Diogenes Laertius writing in the second century AD quotes Hieronymus, a pupil of Aristotle [6] (or see [8]):-
Hieronymus says that [Thales] even succeeded in measuring the pyramids by observation of the length of their shadow at the moment when our shadows are equal to our own height.

This appears to contain no subtle geometrical knowledge, merely an empirical observation that at the instant when the length of the shadow of one object coincides with its height, then the same will be true for all other objects. A similar statement is made by Pliny (see [8]):-
Thales discovered how to obtain the height of pyramids and all other similar objects, namely, by measuring the shadow of the object at the time when a body and its shadow are equal in length.

Plutarch however recounts the story in a form which, if accurate, would mean that Thales was getting close to the idea of similar triangles:-
... without trouble or the assistance of any instrument [he] merely set up a stick at the extremity of the shadow cast by the pyramid and, having thus made two triangles by the impact of the sun's rays, ... showed that the pyramid has to the stick the same ratio which the shadow [of the pyramid] has to the shadow [of the stick]

Of course Thales could have used these geometrical methods for solving practical problems, having merely observed the properties and having no appreciation of what it means to prove a geometrical theorem. This is in line with the views of Russell who writes of Thales contributions to mathematics in [12]:-
Thales is said to have travelled in Egypt, and to have thence brought to the Greeks the science of geometry. What Egyptians knew of geometry was mainly rules of thumb, and there is no reason to believe that Thales arrived at deductive proofs, such as later Greeks discovered.

On the other hand B L van der Waerden [16] claims that Thales put geometry on a logical footing and was well aware of the notion of proving a geometrical theorem. However, although there is much evidence to suggest that Thales made some fundamental contributions to geometry, it is easy to interpret his contributions in the light of our own knowledge, thereby believing that Thales had a fuller appreciation of geometry than he could possibly have achieved. In many textbooks on the history of mathematics Thales is credited with five theorems of elementary geometry:-

i. A circle is bisected by any diameter.
ii.The base angles of an isosceles triangle are equal.
iii.The angles between two intersecting straight lines are equal.
iv.Two triangles are congruent if they have two angles and one side equal.
v. An angle in a semicircle is a right angle.

What is the basis for these claims? Proclus, writing around 450 AD, is the basis for the first four of these claims, in the third and fourth cases quoting the work History of Geometry by Eudemus of Rhodes, who was a pupil of Aristotle, as his source. The History of Geometry by Eudemus is now lost but there is no reason to doubt Proclus. The fifth theorem is believed to be due to Thales because of a passage from Diogenes Laertius book Lives of eminent philosophers written in the second century AD [6]:-
Pamphile says that Thales, who learnt geometry from the Egyptians, was the first to describe on a circle a triangle which shall be right-angled, and that he sacrificed an ox (on the strength of the discovery). Others, however, including Apollodorus the calculator, say that it was Pythagoras.

A deeper examination of the sources, however, shows that, even if they are accurate, we may be crediting Thales with too much. For example Proclus uses a word meaning something closer to 'similar' rather than 'equal- in describing (ii). It is quite likely that Thales did not even have a way of measuring angles so 'equal- angles would have not been a concept he would have understood precisely. He may have claimed no more than "The base angles of an isosceles triangle look similar". The theorem (iv) was attributed to Thales by Eudemus for less than completely convincing reasons. Proclus writes (see [8]):-
[Eudemus] says that the method by which Thales showed how to find the distances of ships from the shore necessarily involves the use of this theorem.

Heath in [8] gives three different methods which Thales might have used to calculate the distance to a ship at sea. The method which he thinks it most likely that Thales used was to have an instrument consisting of two sticks nailed into a cross so that they could be rotated about the nail. An observer then went to the top of a tower, positioned one stick vertically (using say a plumb line) and then rotating the second stick about the nail until it point at the ship. Then the observer rotates the instrument, keeping it fixed and vertical, until the movable stick points at a suitable point on the land. The distance of this point from the base of the tower is equal to the distance to the ship.

Although theorem (iv) underlies this application, it would have been quite possible for Thales to devise such a method without appreciating anything of 'congruent triangles'.

As a final comment on these five theorems, there are conflicting stories regarding theorem (iv) as Diogenes Laertius himself is aware. Also even Pamphile cannot be taken as an authority since she lived in the first century AD, long after the time of Thales. Others have attributed the story about the sacrifice of an ox to Pythagoras on discovering Pythagoras's theorem. Certainly there is much confusion, and little certainty.

Our knowledge of the philosophy of Thales is due to Aristotle who wrote in his Metaphysics :-
Thales of Miletus taught that 'all things are water'.

This, as Brumbaugh writes [5]:-
...may seem an unpromising beginning for science and philosophy as we know them today; but, against the background of mythology from which it arose, it was revolutionary.

Sambursky writes in [15]:-
It was Thales who first conceived the principle of explaining the multitude of phenomena by a small number of hypotheses for all the various manifestations of matter.

Thales believed that the Earth floats on water and all things come to be from water. For him the Earth was a flat disc floating on an infinite ocean. It has also been claimed that Thales explained earthquakes from the fact that the Earth floats on water. Again the importance of Thales' idea is that he is the first recorded person who tried to explain such phenomena by rational rather than by supernatural means.

It is interesting that Thales has both stories told about his great practical skills and also about him being an unworldly dreamer. Aristotle, for example, relates a story of how Thales used his skills to deduce that the next season's olive crop would be a very large one. He therefore bought all the olive presses and then was able to make a fortune when the bumper olive crop did indeed arrive. On the other hand Plato tells a story of how one night Thales was gazing at the sky as he walked and fell into a ditch. A pretty servant girl lifted him out and said to him "How do you expect to understand what is going on up in the sky if you do not even see what is at your feet". As Brumbaugh says, perhaps this is the first absent-minded professor joke in the West!

The bust of Thales shown above is in the Capitoline Museum in Rome, but is not contemporary with Thales and is unlikely to bear any resemblance to him.

B. The Writings of Thales

Doubts have always existed about whether Thales wrote anything, but a number of ancient reports credit him with writings. Simplicius (Diels, Dox. p. 475) specifically attributed to Thales authorship of the so-called Nautical Star-guide. Diogenes Laertius raised doubts about authenticity, but wrote that 'according to others [Thales] wrote nothing but two treatises, one On the Solstice and one On the Equinox' (D.L. I.23). Lobon of Argus asserted that the writings of Thales amounted to two hundred lines (D.L. I.34), and Plutarch associated Thales with opinions and accounts expressed in verse (Plutarch, De Pyth. or. 18. 402 E). Hesychius, recorded that '[Thales] wrote on celestial matters in epic verse, on the equinox, and much else' (DK, 11A2). Callimachus credited Thales with the sage advice that navigators should navigate by Ursa Minor (D.L. I.23), advice which may have been in writing.

Diogenes mentions a poet, Choerilus, who declared that '[Thales] was the first to maintain the immortality of the soul' (D.L. I.24), and in De Anima, Aristotle's words 'from what is recorded about [Thales]', indicate that Aristotle was working from a written source. Diogenes recorded that '[Thales] seems by some accounts to have been the first to study astronomy, the first to predict eclipses of the sun and to fix the solstices; so Eudemus in his History of Astronomy. It was this which gained for him the admiration of Xenophanes and Herodotus and the notice of Heraclitus and Democritus' (D.L. I.23). Eudemus who wrote a History of Astronomy, and also on geometry and theology, must be considered as a possible source for the hypotheses of Thales. The information provided by Diogenes is the sort of material which he would have included in his History of Astronomy, and it is possible that the titles On the Solstice, and On the Equinox were available to Eudemus.

Xenophanes, Herodotus, Heraclitus and Democritus were familiar with the work of Thales, and may have had a work by Thales available to them.

Proclus recorded that Thales was followed by a great wealth of geometers, most of whom remain as honoured names. They commence with Mamercus, who was a pupil of Thales, and include Hippias of Elis, Pythagoras, Anaxagoras, Eudoxus of Cnidus, Philippus of Mende, Euclid, and Eudemus, a friend of Aristotle, who wrote histories of arithmetic, of astronomy, and of geometry, and many lesser known names. It is possible that writings of Thales were available to some of these men.

Any records which Thales may have kept would have been an advantage in his own work. This is especially true of mathematics, of the dates and times determined when fixing the solstices, the positions of stars, and in financial transactions. It is difficult to believe that Thales would not have written down the information he had gathered in his travels, particularly the geometry he investigated in Egypt and his measuring of the height of the pyramid, his hypotheses about nature, and the cause of change.

Proclus acknowledged Thales as the discoverer of a number of specific theorems (A Commentary on the First Book of Euclid's Elements 65. 8-9; 250. 16-17). This suggests that Eudemus, Proclus's source had before him the written records of Thales's discoveries. How did Thales 'prove' his theorems if not in written words and sketches? The works On the Solstice, On the Equinox, which were attributed to Thales (D.L. I.23), and the 'Nautical Star-guide, to which Simplicius referred, may have been sources for the History of Astronomy of Eudemus (D.L. I.23).

C. Possible Sources for Aristotle

There is no direct evidence that any written material of Thales was available to Plato and Aristotle, but there is a surprisingly long list of early writers who could have known Thales, or had access to his works, and these must be considered as possible sources for Plato, Aristotle, and the philosophers and commentators who followed them. Aristotle's wording, 'Thales says', is assertive wording which suggests a reliable source, perhaps writings of Thales himself. Anaximander and Anaximenes were associates of Thales, and would have been familiar with his ideas. Both produced written work. Anaximander wrote in a poetical style (Theophr. ap. Simpl. Phys. fr. 2), and the writing of Anaximenes was simple and unaffected (D.L. II.3). Other philosophers who were credited with written works, who worked on topics similar to those of Thales, and who may have provided material for later writers, are Heraclitus of Ephesus, Anaxagoras of Clazomenae, Alcmaeon, Hippo of Samos, and Hippias of Elis.

D. Thales says Water is the Primary Principle

Aristotle defined wisdom as knowledge of certain principles and causes (Metaph. 982 a2-3). He commenced his investigation of the wisdom of the philosophers who preceded him, with Thales, the first philosopher, and described Thales as the founder of natural philosophy (Metaph. 983 b21-22). He recorded: 'Thales says that it is water'. 'it' is the nature, the archê, the originating principle. For Thales, this nature was a single material substance, water. Despite the more advanced terminology which Aristotle and Plato had created, Aristotle recorded the doctrines of Thales in terms which were available to Thales in the sixth century BCE Aristotle made a definite statement, and presented it with confidence. It was only when Aristotle attempted to provide the reasons for the opinions that Thales held, and for the theories that he proposed, that he sometimes displayed caution.

E. Thales and Mythology

Those who believe that Thales inherited his views from Greek or Near-Eastern sources are wrong. Thales was esteemed in his times as an original thinker, and one who broke with tradition and not as one who conveyed existing mythologies. Aristotle unequivocally recorded Thales's hypothesis on the nature of matter, and proffered a number of conjectures based on observation in favour of Thales's declaration (Metaph. 983 b20-28). His report provided the testimony that Thales supplanted myth in his explanations of the behaviour of natural phenomena. Thales did not derive his thesis from either Greek or non-Greek mythological traditions.

Thales would have been familiar with Homer's acknowledgements of divine progenitors but he never attributed organization or control of the cosmos to the gods. Aristotle recognized the similarity between Thales's doctrine about water and the ancient legend which associates water with Oceanus and Tethys, but he reported that Thales declared water to be the nature of all things.

Aristotle pointed to a similarity to traditional beliefs, not a dependency upon them. Aristotle did not call Thales a theologian in the sense in which he designated 'the old poets' (Metaph. 1091 b4) and others, such as Pherecydes, as 'mixed theologians' who did not use 'mythical language throughout' (Metaph. 1091 b9). To Aristotle, the theories of Thales were so obviously different from all that had gone before that they stood out from earlier explanations. Thales's views were not ancient and primitive. They were new and exciting, and the genesis of scientific conjecture about natural phenomena. It was the view for which Aristotle acknowledged Thales as the founder of natural philosophy.

F. Thales's Primary Principle

The problem of the nature of matter, and its transformation into the myriad things of which the universe is made, engaged the natural philosophers, commencing with Thales. For his hypothesis to be credible, it was essential that he could explain how all things could come into being from water, and return ultimately to the originating material. It is inherent in Thales's hypotheses that water had the potentiality to change to the myriad things of which the universe is made, the botanical, physiological, meteorological and geological states. In Timaeus, 49B-C, Plato had Timaeus relate a cyclic process. The passage commences with 'that which we now call "water" ', and describes a theory which was possibly that of Thales. Thales would have recognized evaporation, and have been familiar with traditional views, such as the nutritive capacity of mist and ancient theories about spontaneous generation, phenomena which he may have 'observed', just as Aristotle believed he, himself had (Hist. An. 569 b1; Gen. An. 762 a9-763 a34), and about which Diodorus Siculus (I.7.3-5; 1.10.6), Epicurus (ap. Censorinus, D.N. IV.9), Lucretius (De Rerum Natura , V.783-808) and Ovid (Met. I.416-437) wrote.

When Aristotle reported Thales's pronouncement that the primary principle is water, he made a precise statement: 'Thales says that it [the nature of things] is water' (Metaph. 983 b20), but he became tentative when he proposed reasons which might have justified Thales's decision: '[Thales's] supposition may have arisen from observation . . . ' (Metaph. 983 b22). It was Aristotle's opinion that Thales may have observed, 'that the nurture of all creatures is moist, and that warmth itself is generated from moisture and lives by it; and that from which all things come to be is their first principle' (Metaph. 983 b23-25). Then, in the lines 983 b26-27, Aristotle's tone changed towards greater confidence. He declared: 'Besides this, another reason for the supposition would be that the semina of all things have a moist nature . . . ' (Metaph. 983 b26-27). In continuing the criticism of Thales, Aristotle wrote: 'That from which all things come to be is their first principle' (Metaph. 983 b25).

Simple metallurgy had been practised long before Thales presented his hypotheses, so Thales knew that heat could return metals to a liquid state. Water exhibits sensible changes more obviously than any of the other so-called elements, and can readily be observed in the three states of liquid, vapour and ice. The understanding that water could generate into earth is basic to Thales's watery thesis. At Miletus it could readily be observed that water had the capacity to thicken into earth. Miletus stood on the Gulf of Lade through which the Maeander river emptied its waters. Within living memory, older Milesians had witnessed the island of Lade increasing in size within the Gulf, and the river banks encroaching into the river to such an extent that at Priene, across the gulf from Miletus the warehouses had to be rebuilt closer to the water's edge. The ruins of the once prosperous city-port of Miletus are now ten kilometres distant from the coast and the Island of Lade now forms part of a rich agricultural plain. There would have been opportunity to observe other areas where earth generated from water, for example, the deltas of the Halys, the Ister, about which Hesiod wrote (Theogony, 341), now called the Danube, the Tigris-Euphrates, and almost certainly the Nile.

This coming-into-being of land would have provided substantiation of Thales's doctrine. To Thales water held the potentialities for the nourishment and generation of the entire cosmos. Aëtius attributed to Thales the concept that 'even the very fire of the sun and the stars, and indeed the cosmos itself is nourished by evaporation of the waters' (Aëtius, Placita, I.3).

It is not known how Thales explained his watery thesis, but Aristotle believed that the reasons he proposed were probably the persuasive factors in Thales's considerations. Thales gave no role to the Olympian gods. Belief in generation of earth from water was not proven to be wrong until A.D. 1769 following experiments of Antoine Lavoisier, and spontaneous generation was not disproved until the nineteenth century as a result of the work of Louis Pasteur.

G. New Ideas about the Earth

Thales proposed answers to a number of questions about the earth: the question of its support; its shape; its size; and the cause of earthquakes; the dates of the solstices; the size of the sun and moon.

a. The Earth Floats on Water

In De Caelo Aristotle wrote: 'This [opinion that the earth rests on water] is the most ancient explanation which has come down to us, and is attributed to Thales of Miletus (Cael. 294 a28-30).

He explained his theory by adding the analogy that the earth is at rest because it is of the nature of wood and similar substances which have the capacity to float on water, although not on air (Cael. 294 a30-b1). In Metaphysics (983 b21) Aristotle stated, quite unequivocally: 'Thales . . . declared that the earth rests on water'. This concept does appear to be at odds with natural expectations, and Aristotle expressed his difficulty with Thales's theory (Cael. 294 a33-294 b6).

Perhaps Thales anticipated problems with acceptance because he explained that it floated because of a particular quality, a quality of buoyancy similar to that of wood. At the busy city-port of Miletus, Thales had unlimited opportunities to observe the arrival and departure of ships with their heavier-than-water cargoes, and recognized an analogy to floating logs. Thales may have envisaged some quality, common to ships and earth, a quality of 'floatiness', or buoyancy. It seems that Thales's hypothesis was substantiated by sound observation and reasoned considerations.

Indeed, Seneca reported that Thales had land supported by water and carried along like a boat (Sen. QNat. III.14). Aristotle's lines in Metaphysics indicate his understanding that Thales believed that, because water was the permanent entity, the earth floats on water.

Thales may have reasoned that as a modification of water, earth must be the lighter substance, and floating islands do exist. Herodotus (The Histories, II.156) was impressed when he saw Chemmis, a floating island, about thirty-eight kilometres north-east of Naucratis, the Egyptian trading concession which Thales probably visited. Seneca described floating islands in Lydia: 'There are many light, pumice-like stones of which islands are composed, namely those which float in Lydia' (Sen. QNat., III.25. 7-10). Pliny described several floating islands, the most relevant being the Reed Islands, in Lydia (HN, II.XCVII), and Pliny (the Younger) (Ep. VIII.XX) described a circular floating island, its buoyancy, and the way it moved. Thales could have visited the near-by Reed Islands. He might have considered such readily visible examples to be models of his theory, and he could well have claimed that the observation that certain islands had the capacity to float substantiated his hypothesis that water has the capacity to support earth.

Again it is understood that Thales did not mention any of the gods who were traditionally associated with the simple bodies; we do not hear of Oceanus or Gaia: we read of water and earth. The idea that Thales would have resurrected the gods is quite contrary to the bold, new, non-mythical theories which Thales proposed.

b. Thales's Spherical Earth

Modern commentators assume that Thales regarded the earth as flat, thin, and circular, but there is no ancient testimony to support that opinion. On the contrary, Aristotle may have attributed knowledge of the sphericity of the earth to Thales, an opinion which was later reported by Aëtius (Aët. III. 9-10) and followed by Ps.-Plutarch (Epit. III.10). Aristotle wrote that some think it spherical, others flat and shaped like a drum (Arist. Cael. 293 b33-294 a1), and then attributed belief in a flat earth to Anaximenes, Anaxagoras, and Democritus (Arist. Cael. 294 b14-15). If following chronological order, Aristotle's words, 'some think it spherical', referred to the theory of Thales. Aristotle then followed with the theory of Thales's immediate Milesian successor, Anaximander, and then reported the flat earth view of Anaximenes, the third of the Milesian natural philosophers.

There are several good reasons to accept that Thales envisaged the earth as spherical. Aristotle used these arguments to support his own view (Arist. Cael. 297 b25-298 a8). First is the fact that during a solar eclipse, the shadow caused by the interposition of the earth between the sun and the moon is always convex; therefore the earth must be spherical. In other words, if the earth were a flat disk, the shadow cast during an eclipse would be elliptical. Second, Thales, who is acknowledged as an observer of the heavens, would have observed that stars which are visible in a certain locality may not be visible further to the north or south, a phenomena which could be explained within the understanding of a spherical earth. Third, from mere observation the earth has the appearance of being curved. From observation, it appears that the earth is covered by a dome. When observed from an elevated site, the sky seems to surround the earth, like a dome, to meet the apparently curved horizon. If observed over the seasons, the dome would appear to revolve, with many of the heavenly bodies changing their position in varying degrees, but returning annually to a similar place in the heavens.

Through his work in astronomy Thales would almost certainly have become familiar with the night sky and the motion of the heavenly bodies. There is evidence that he gave advice to navigate by Ursa Minor, and was so involved in observation of the stars that he fell into a well. As a result of observations made over a long period of time, Thales could have realized that the motions of the fixed stars could not be explained within the idea of the observable hemispherical dome. During the determination of the size of the rising sun, and again while watching its risings and settings during his work on fixing the solstices, Thales may have realized that much natural phenomena could be explained only within the understanding of the earth as a sphere.

From the shore, a ship can be seen to be descending, gradually, below the horizon, with the hull disappearing from view first, to be followed by masts and sails. If one had a companion observing from a higher point, the companion would see the ship for a long period before it disappeared from view.

Aëtius recorded the different opinions of the shape of the earth that were held by Thales, Anaximander and Anaximenes (III.9-10; III.10; and III.10). Cicero attributed to Thales the earliest construction of a solid celestial globe (Rep. I.XIII.22). Thales's immediate successors proposed theories about the shape of the earth which were quite different from each other, but that is no reason to reject the view that Thales hypothesized a spherical earth. It is not the only occasion on which Anaximander and Anaximenes failed to follow the theories of Thales. That they did not do so is the main argument in favour of accepting that the scientific method commenced in the Milesian School. There is testimony that Thales knew the earth to be spherical, but no evidence to suggest that he proposed any other shape.

c. Earthquake Theory

Thales's theory about the cause of earthquakes is consistent with his hypothesis that earth floats upon water. It seems that he applied his floating on water simile to the natural phenomena of earthquakes. Aëtius recorded that Thales and Democritus found in water the cause of earthquakes (Aët. III.15), and Seneca attributed to Thales a theory that on the occasions when the earth is said to quake it is fluctuating because of the roughness of oceans (QNat. III.14; 6.6). Although the theory is wrong, Thales's hypothesis is rational because it provides an explanation which does not invoke hidden entities. It is an advance upon the traditional Homeric view that they resulted from an angry supernatural god, Poseidon, shaking the earth through his rapid striding.

H. All Things are Full of God

The question of whether Thales endowed the gods with a role in his theories is fundamental to his hypotheses. The relevant text from Aristotle reads: 'Thales, too, to judge from what is recorded of his views, seems to suppose that the soul is in a sense the cause of movement, since he says that a stone [magnet, or lodestone] has a soul because it causes movement to iron' (De An. 405 a20-22); 'Some think that the soul pervades the whole universe, whence perhaps came Thales's view that everything is full of gods' (De An. 411 a7-8). In reference to the clause in the first passage 'to judge from what is recorded of his views', Snell convincingly argued that Aristotle had before him the actual sentence recording Thales's views about the lodestone (Snell, 1944, 170). In the second passage the 'some' to whom Aristotle refers are Leucippus, Democritus, Diogenes of Apollonia, Heraclitus, and Alcmaeon, philosophers who were later than Thales. They adopted and adapted the earlier view of Thales that soul was the cause of motion, permeating and enlivening the entire cosmos. The order in which Aristotle discussed Thales's hypothesis obscures the issue.

The source for Aristotle's report that Thales held all things to be full of gods is unknown, but some presume that it was Plato. Thales is not mentioned in the relevant lines in Plato, but there is a popular misconception that they refer to the belief of Thales. This is wrong. Thales had rejected the old gods. In a passage in Apology(26 C) Socrates identified the heavenly bodies as gods, and pointed out that that was the general understanding. In Cratylus(399 D-E) Plato had Socrates explain a relationship between soul as a life-giving force, the capacity to breathe, and the reviving force. In Timaeus 34B) Plato had Timaeus relate a theory which described soul as pervading the whole universe. Then, in Laws Plato has the Athenian Stranger say: 'Everyone . . . who has not reached the utmost verge of folly is bound to regard the soul as a god. Concerning all the stars and the moon, and concerning the years and months and all seasons, what other account shall we give than this very same, - namely, that, inasmuch as it has been shown that they are all caused by one or more souls . . . we shall declare these souls to be gods . . .? Is there any man that agrees with this view who will stand hearing it denied that 'all things are full of gods'?

The response is: 'No man is so wrong-headed as that' (Laws, 899 A-B). Plato had the Athenian Stranger extend his ideas into a theological theory. He used a sleight of hand method to express his own ideas about divine spiritual beings. With the exception of gods in the scheme of things, these passages reflect the beliefs which formed the Thalean hypothesis, but Plato did not have the Athenian Stranger attribute the crucial clause 'all things are full of gods' to Thales. Thales is not mentioned.

Aristotle's text not the earliest extant testimony. Diogenes preserved a report from Hippias: 'Aristotle and Hippias affirm that, arguing from the magnet and from amber, [Thales] attributed a soul or life even to inanimate objects' (D.L. I.24). This early report does not mention godly entities. The later commentators, Cicero (Nat. D. I.X.25), and Stobaeus (Ecl. I.1.11) included gods in Thales's theory. However, their views post-date Stoicism and are distorted by theistic doctrines.

Plato converted the idea of soul into a theory that 'all things are full of gods', and this may have been Aristotle's source, but the idea of gods is contrary to Thales's materialism. When Thales defined reality, he chose an element, not a god. The motive force was not a supernatural being. It was a force within the universe itself. Thales never invoked a power that was not present in nature itself, because he believed that he had recognized a force which underpinned the events of nature.

I. Thales's Astronomy

a. The Eclipse of Thales

Thales is acclaimed for having predicted an eclipse of the sun which occurred on 28 May 585 BCE The earliest extant account of the eclipse is from Herodotus: 'On one occasion [the Medes and the Lydians] had an unexpected battle in the dark, an event which occurred after five years of indecisive warfare: the two armies had already engaged and the fight was in progress, when day was suddenly turned into night. This change from daylight to darkness had been foretold to the Ionians by Thales of Miletus, who fixed the date for it within the limits of the year in which it did, in fact, take place' (Hdt. I.74). The vital points are: Thales foretold a solar eclipse; it did occur within the period he specified. How Thales foretold the eclipse is not known but there is strong opinion that he was able to perform this remarkable feat through knowledge of a cycle known as the Saros, with some attributing his success to use of the Exeligmos cycle. It is not known how Thales was able to predict the Eclipse, if indeed he did, but he could not have predicted the Eclipse by using the Saros or the Exeligmos cycles.

In addition to Herodotus, the successful prediction of the eclipse was accepted by Eudemus in his History of Astronomy and acknowledged by a number of other writers of ancient times (Cicero, Pliny, Dercyllides, Clement, Eusebius). This is how Diogenes Laertius recorded the event: '[Thales] seems by some accounts to have been the first to study astronomy, the first to predict eclipses of the sun, and to fix the solstices; so Eudemus in his History of Astronomy. It was this which gained for him the admiration of Xenophanes and Herodotus and the notice of Heraclitus and Democritus' (D.L. I.23). Diogenes asserted that Herodotus knew of Thales's work, and in naming Xenophanes, Heraclitus, and Democritus, he nominated three of the great pre-Socratics, eminent philosophers who were familiar with the work of Thales.

Modern astronomy confirms that the eclipse did occur, and was total. According to Herodotus's report, the umbra of the eclipse of Thales must have passed over the battle field. The "un-naturalness" of a solar eclipse is eerie and chilling. All becomes hushed and there is a strong uncanny sensation of impending disaster, of being within the control of some awful power. In ancient times, the awesome phenomenon must have aroused great fear, anxiety and wonder. The combatants saw the eclipse as disapproval of their warfare, and as a warning. They ceased fighting and a peace agreement was reached between the two kings.

It is not known why Thales turned away from the traditional beliefs which attributed all natural events and man's fortunes and misfortunes to the great family of Olympian gods, but Miletus was the most prosperous of the Ionian cities, and it cannot be doubted that the flourishing merchants believed that their prosperity resulted from their own initiative and endeavours. Thales's great philosophical pronouncement that water is the basic principle shows that Thales gave no acknowledgement to the gods as instigators and controllers of phenomena. Thales's hypotheses indicate that he envisaged phenomena as natural events with natural causes and possible of explanation. From his new perspective of observation and reasoning, Thales studied the heavens and sought explanations of heavenly phenomena.

It is widely accepted that Thales acquired information from Near-Eastern sources and gained access to the extensive records which dated from the time of Nabonassar (747 BCE) and which were later used by Ptolemy (Alm. III.7. H 254). Some commentators have suggested that Thales predicted the solar eclipse of 585 BCE through knowledge of the Saros period, a cycle of 223 lunar months (18 years, 10-11 days plus 0.321124 of a day) after which eclipses both of the sun and moon repeat themselves with very little change, or through knowledge of the Exeligmos cycle which is exactly three times the length of the Saros (Ptolemy, Alm. IV.2. H270). The ancients could not have predicted solar eclipses on the basis of those periodic cycles because eclipses of the sun do not repeat themselves with very little change. The extra 0.321124 of a day means that each recurring solar eclipse will be visible to the west, just under one-third of the circumference of the earth, being a period of time of almost 7.7 hours. This regression to the west could not have been known to the ancient astrologers, a fact which seems not to have been taken into account by the philosophers who attribute Thales's success to application of one of those two cycles.

The following important fact should be noted. Some commentators and philosophers believe that Thales may have witnessed the solar eclipse of 18th May 603 BCE or have had heard of it. They accepted that he had predicted the solar eclipse of 28 May 585 BCE and reasoned from the astronomical fact of the Saros cycles and the fact that the two solar eclipses had been separated by the period of 18 years, 10 days, and 7.7 hours, and concluded that Thales had been able to predict a solar eclipse based upon the knowledge of that cycle. Two facts discount rebut those claims. First, recent research shows that the solar eclipse of 18th May 603 BCE would not have been visible in Egypt, nor in the Babylonian observation cities where the astronomers watched the heavens for expected and unusual heavenly events. The eclipse of 603 passed over the Persian Gulf, too far to the south for observation (Stephenson, personal communication, March 1999; and Stephenson, "Long-term Fluctuations", 165-202). Even if the eclipse of 603 had been visible to the Near-Eastern astronomers, it is not possible to recognize a pattern from witnessing one event, or indeed, from witnessing two events. One may suggest a pattern after witnessing three events that are separated by equal periods of time, but the eclipse which preceded that of 603, and which occurred on 6th May 621, was not visible in Near-Eastern regions. Consequently, it could not have been recorded by the astrologer/priests who watched for unusual heavenly phenomena, and could not have been seen as forming a pattern.

It is quite wrong to say that eclipses repeat themselves with very little change, because each solar eclipse in a particular Saros occurs about 7.7 hours later than in the previous eclipse in the same Saros, and that is about 1/3 of the circumference of the earth's circumference. Adding to the difficulty of recognizing a particular cycle is the fact that about forty-two periodic cycles are in progress continuously, and overlapping at any time. Every series in a periodic cycle lasts about 1,300 years and comprises 73 eclipses. Eclipses which occur in one periodic cycle are unrelated to eclipses in other periodic cycles.

The ancient letters prove that the Babylonians and Assyrians knew that lunar eclipses can occur only at full moon, and solar eclipses only at new moon, and also that eclipses occur at intervals of five or six months. However, while lunar eclipses are visible over about half the globe, solar eclipses are visible from only small areas of the earth's surface. Recent opinion is that, as early as 650 BCE the Assyrian astronomers seem to have recognized the six months-five months period by which they could isolate eclipse possibilities (Steele, "Eclipse Prediction", 429).

In other recent research Britton has analysed a text known as Text S, which provides considerable detail and fine analysis of lunar phenomena dating from Nabonassar in 747 BCE The text points to knowledge of the six-month five month periods. Britton believes that the Saros cycle was known before 525 BCE (Britton, "Scientific Astronomy", 62) but, although the text identifies a particular Saros cycle, and graphically depicts the number of eclipse possibilities, the ancient commentary of Text S does not attest to an actual observation (Britton, "An Early Function", 32).

There is no evidence that the Saros could have been used for the prediction of solar eclipses in the sixth century BCE, but it remains possible that forthcoming research, and the transliteration of more of the vast stock of ancient tablets will prove that the Babylonians and Assyrians had a greater knowledge of eclipse phenomena than is now known.

The Babylonian and Assyrian astronomers knew of the Saros period in relation to lunar eclipses, and had some success in predicting lunar eclipses but, in the sixth century BCE when Thales lived and worked, neither the Saros nor the Exeligmos cycles could be used to predict solar eclipses.
It is testified that Thales knew that the sun is eclipsed when the moon passes in front of it, the day of eclipse - called the thirtieth by some, new moon by others (The Oxyrhynchus Papyri, 3710). Aëtius (II.28) recorded: [Thales] says that eclipses of the sun take place when the moon passes across it in a direct line, since the moon is earthy in character; and it seems to the eye to be laid on the disc of the sun'.

There is a possibility that, through analysis of ancient eclipse records, Thales identified another cycle, the lunar eclipse-solar eclipse cycle of 23 1/2 months, the fact that a solar eclipse is a possibility 23 1/2 months after a lunar eclipse. However, lunar eclipses are not always followed by solar eclipses. Although the possibility is about 57% it is important to note that the total solar eclipse of 28th May, 585, occurred 23 1/2months after the total lunar eclipse of 4th July, 587. The wording of the report of the eclipse by Herodotus: 'Thales . . . fixed the date for the eclipse within the limits of the year' is precise, and suggests that Thales's prediction was based upon a definite eclipse theory.

b. Setting the Solstices

A report from Theon of Smyrna ap. Dercyllides states that: 'Eudemus relates in the Astronomy that Thales was the first to discover the eclipse of the sun and that its period with respect to the solstices is not always constant' (DK, 11 A 17). Diogenes Laertius (I.24) recorded that [Thales] was the first to determine the sun's course from solstice to solstice, and also acknowledged the Astronomy of Eudemus as his source.

Solstices are natural phenomena which occur on June 21 or 22, and December 21 or 22, but the determination of the precise date on which they occur is difficult. This is because the sun seems to 'stand still' for several days because there is no discernible difference in its position in the sky. It is the reason why the precise determination of the solstices was so difficult. It was a problem which engaged the early astronomers, and more than seven centuries later, Ptolemy acknowledged the difficulty (Alm. III.1. H203).

It is not known how Thales proceeded with his determination, but the testimony of Flavius Philostratus is that: '[Thales] observed the heavenly bodies . . . from [Mount] Mycale which was close by his home' (Philostratus, Life of Apollonius , II.V). This suggests that Thales observed the rising and setting of the sun for many days at mid-summer and mid-winter (and, necessarily, over many years). Mount Mycale, being the highest point in the locality of Miletus, would provide the perfect vantage point from which to make observations. Another method which Thales could have employed was to measure the length of the noon-day sun around mid-summer and mid-winter. Again this would require observations to be made, and records kept over many days near the solstice period, and over many years.

c. Thales's Discovery of the Seasons

From Diogenes Laertius we have the report: '[Thales] is said to have discovered the seasons of the year and divided it into 365 days' (D.L. I.27). Because Thales had determined the solstices, he would have known of the number of days between say, summer solstices, and therefore have known the length of a solar year. It is consistent with his determination of the solstices that he should be credited with discovering that 365 days comprise a year. It is also a fact that had long been known to the Egyptians who set their year by the more reliable indicator of the annual rising of the star Sirius in July. Thales may have first gained the knowledge of the length of the year from the Egyptians, and perhaps have attempted to clarify the matter by using a different procedure. Thales certainly did not 'discover' the seasons, but he may have identified the relationship between the solstices, the changing position during the year of the sun in the sky, and associated this with seasonal climatic changes.

d. Thales's Determination of the Diameters of the Sun and the Moon

Apuleius wrote that 'Thales in his declining years devised a marvellous calculation about the sun, showing how often the sun measures by its own size the circle which it describes'. (Apul. Florida, 18). Following soon after Apuleius, Cleomedes explained that the calculation could be made by running a water-clock, from which the result was obtained: the diameter of the sun is found to be one seven-hundred-and-fiftieth of its own orbit (Cleomedes, De Motu circulari corporum caelestium, II.75). The third report is from Diogenes: 'According to some [Thales was] the first to declare the size of the sun to be one seven hundred and twentieth part of the solar circle, and the size of the moon to be the same fraction of the lunar circle' (D.L. I.24). Little credence can be given to the water-clock method for reaching this determination, because there is an inbuilt likelihood of repeated errors over the 24 hour period. Even Ptolemy, who flourished in the second century A.D., rejected all measurements which were made by means of water-clocks, because of the impossibility of attaining accuracy by such means (Alm. V.14. H416).

In his work in geometry, Thales was engaged in circles and angles, and their characteristics, and he could have arrived at his solution to the problem by applying the geometrical knowledge he had acquired. There is no evidence to support a suggestion that Thales was familiar with measurements by degrees but he could have learnt, from the Babylonians, that a circle is divided into 3600. The figure of 720, which was given by Diogenes for Thales, is double 360, and this is related to the Babylonian sexagesimal system.

To establish the dates of the solstices, Thales probably made repeated observations of the risings and settings of the sun. From such experiments he could have observed that the angle which was subtended by the elevation of the rising sun is 1/20 and with 3600 in a circle, the ratio of 1:720 is determined.

Of the report from Diogenes Laertius (D.L. I.24) that Thales also determined the orbit of the moon in relation to the size of its diameter, Thales would repeat the method to calculate the orbit of the moon.

e. Ursa Minor

Callimachus (D.L. I.22) reported that Thales 'discovered' Ursa Minor. This means only that he recognized the advantages of navigating by Ursa Minor, rather than by Ursa Major, as was the preferred method of the Greeks. Ursa Minor, a constellation of six stars, has a smaller orbit than does the Great Bear, which means that, as it circles the North Pole, Ursa Minor changes its position in the sky to a lesser degree than does the Great Bear. Thales offered this sage advice to the mariners of Miletus, to whom it should have been of special value because Miletus had developed a maritime trade of economic importance.

f. Falling into a Well

In Theaetetus (174 A) Plato had Socrates relate a story that Thales was so intent upon watching the stars that he failed to watch where he was walking, and fell into a well. The story is also related by Hippolytus (Diels, Dox. 555), and by Diogenes Laertius (D.L. II.4-5). Irony and jest abound in Plato's writing and he loved to make fun of the pre-Socratics, but he is not likely to have invented the episode, especially as he had Socrates relate the event. Aristotle wrote that viewing the heavens through a tube 'enables one to see further' (Gen. An. 780 b19-21), and Pliny (HN, II.XI) wrote that:
 'The sun's radiance makes the fixed stars invisible in daytime, although they are shining as much as in the night, which becomes manifest at a solar eclipse and also when the star is reflected in a very deep well'.

Thales was renowned and admired for his astronomical studies, and he was credited with the 'discovery' of Ursa Minor (D.L. I.23). If Thales had heard that stars could be viewed to greater advantage from wells, either during day or night, he would surely have made an opportunity to test the theory, and to take advantage of a method that could assist him in his observations. The possibility that the story was based on fact should not be overlooked. Plato had information which associated Thales with stars, a well, and an accident. Whether Thales fell into a well, or tripped when he was getting in or out of a well, the story grew up around a mishap.

J. Mathematics

The practical skill of land measurement was invented in Egypt because of the necessity frequently to re-measure plots of land after destructive inundations. The phenomena is well described by Herodotus (II.93-109). Egypt was believed to be the source of much wisdom and reports tell us that many Greeks, including Thales, Pythagoras, Solon, Herodotus, Plato, Democritus, and Euclid, visited that ancient land to see the wonders for themselves.

The Egyptians had little to offer in the way of abstract thought. The surveyors were able to measure and to calculate and they had outstanding practical skills. In Egypt Thales would have observed the land surveyors, those who used a knotted cord to make their measurements, and were known as rope-stretchers. Egyptian mathematics had already reached its heights when The Rhind Mathematical Papyrus was written in about 1800 BCE More than a thousand years later, Thales would have watched the surveyors as they went about their work in the same manner, measuring the land with the aid of a knotted rope which they stretched to measure lengths and to form angles.

The development of geometry is preserved in a work of Proclus, A Commentary on the First Book of Euclid's Elements (64.12-65.13). Proclus provided a remarkable amount of intriguing information, the vital points of which are the following: Geometry originated in Egypt where it developed out of necessity; it was adopted by Thales who had visited Egypt, and was introduced into Greece by him.

The Commentary of Proclus indicates that he had access to the work of Euclid and also to The History of Geometry which was written by Eudemus of Rhodes, a pupil of Aristotle, but which is no longer extant. His wording makes it clear that he was familiar with the views of those writers who had earlier written about the origin of geometry. He affirmed the earlier views that the rudiments of geometry developed in Egypt because of the need to re-define the boundaries, just as Herodotus stated.

a. The Theorems Attributed to Thales

Five Euclidean theorems have been explicitly attributed to Thales, and the testimony is that Thales successfully applied two theorems to the solution of practical problems.

Thales did not formulate proofs in the formal sense. What Thales did was to put forward certain propositions which, it seems, he could have 'proven' by induction: he observed the similar results of his calculations: he showed by repeated experiment that his propositions and theorems were correct, and if none of his calculations resulted in contrary outcomes, he probably felt justified in accepting his results as proof. Thalean 'proof' was often really inductive demonstration. The process Thales used was the method of exhaustion. This seems to be the evidence from Proclus who declared that Thales 'attacked some problems in a general way and others more empirically'
DEFINITION I.17: A diameter of the circle is a straight line drawn through the centre and terminated in both directions by the circumference of the circle; and such a straight line also bisects the circle (Proclus, 124).

PROPOSITION I.5: In isosceles triangles the angles at the base are equal; and if the equal straight lines are produced further, the angles under the base will be equal (Proclus, 244). It seems that Thales discovered only the first part of this theorem for Proclus reported: We are indebted to old Thales for the discovery of this and many other theorems. For he, it is said, was the first to notice and assert that in every isosceles the angles at the base are equal, though in somewhat archaic fashion he called the equal angles similar (Proclus, 250.18-251.2).

PROPOSITION I.15: 'If two straight lines cut one another, they make the vertical angles equal to one another' (Proclus, 298.12-13). This theorem is positively attributed to Thales. Proof of the theorem dates from the Elements of Euclid (Proclus, 299.2-5).

PROPOSITION I.26: 'If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining sides equal to the remaining sides and the remaining angle equal to the remaining angle' (Proclus, 347.13-16). 'Eudemus in his history of geometry attributes the theorem itself to Thales, saying that the method by which he is reported to have determined the distance of ships at sea shows that he must have used it' (Proclus, 352.12-15). Thales applied this theorem to determine the height of a pyramid. The great pyramid was already over two thousand years old when Thales visited Gizeh, but its height was not known. Diogenes recorded that 'Hieronymus informs us that [Thales] measured the height of the pyramids by the shadow they cast, taking the observation at the hour when our shadow is of the same length as ourselves' (D.L. I.27). Pliny (HN, XXXVI.XVII.82) and Plutarch (Conv. sept. sap. 147) also recorded versions of the event. Thales was alerted by the similarity of the two triangles, the 'quality of proportionality'. He introduced the concept of ratio, and recognized its application as a general principle. Thales's accomplishment of measuring the height of the pyramid is a beautiful piece of mathematics. It is considered that the general principle in Euclid I.26 was applied to the ship at sea problem, would have general application to other distant objects or land features which posed difficulties in the calculation of their distances.

PROPOSITION III.31: 'The angle in a semicircle is a right angle'. Diogenes Laertius (I.27) recorded: 'Pamphila states that, having learnt geometry from the Egyptians, [Thales] was the first to inscribe a right-angled triangle in a circle, whereupon he sacrificed an ox'. Aristotle was intrigued by the fact that the angle in a semi-circle is always right. In two works, he asked the question: 'Why is the angle in a semicircle always a right angle?' (An. Post. 94 a27-33; Metaph. 1051 a28). Aristotle described the conditions which are necessary if the conclusion is to hold, but did not add anything that assists with this problem.

It is testified that it was from Egypt that Thales acquired the rudiments of geometry. However, the evidence is that the Egyptian skills were in orientation, measurement, and calculation. Thales's unique ability was with the characteristics of lines, angles and circles. He recognized, noticed and apprehended certain principles which he probably 'proved' through repeated demonstration.

K. Crossing the Halys

Herodotus recorded 'the general belief of the Greeks' that Thales assisted Croesus in transporting his troops across the Halys river (Hdt. I.75) on his advance into Capadoccia to engage the great Persian conqueror, Cyrus who threatened from the east. Herodotus provided a detailed description of the reported crossing which many of the Greeks supposed had been accomplished through Thales's engineering skills and ingenuity (Hdt. I.75). Herodotus had been told that Thales advised Croesus to divide the river into two parts. The story is that Thales directed the digging so that the river was diverted into two smaller streams, each of which could then be forded. The story from Herodotus describes a formation similar to an oxbow lake. The work could have been undertaken by the men of Croesus's army, and directed by Thales. With both channels then being fordable, Croesus could lead his army across the Halys. This description complies with 'the general belief of the Greeks' which Herodotus related.

However, Herodotus did not accept that story, because he believed that bridges crossed the river at that time (I.74). Herodotus's misgivings were well founded. There is considerable support for the argument that Croesus and his army crossed the Halys by the bridge which already existed and travelled by the Royal Road which provided the main access to the East. Herodotus explained that at the Halys there were gates which had to be passed before one crossed the river, which formed the border, with the post being strongly guarded (Hdt. V.52).

The town of Cesnir Kopru, or Tcheshnir Keupreu, is a feasible site for a crossing. Before the industrialization of the area, a mediaeval bridge was observed, underneath which, when the river was low, could be seen not only the remains of its Roman predecessor but the roughly hewn blocks of a much earlier bridge (Garstang, 1959, 2). Any clues that may have helped to provide an answer to the question of whether there were bridges in the time of Croesus are now submerged by the hydroelectric plants which have been built in the area. Herodotus recorded the details that he had obtained, but used his own different understanding of the situation to discount the report.

L. The Possible Travels of Thales

Establishing whether or not Thales travelled and what countries he visited is important because we may be able to establish what information he could have acquired from other sources. In Epinomis 987 E) Plato made the point that the Greeks took from foreigners what was of value and developed their notions into better ideas.

Eudemus, who was one of Aristotle's students, believed that Thales had travelled to Egypt (Eudemus ap. Proclus, 65.7). A number of ancient sources support that opinion, including Pamphila who held that he spent time with the Egyptian priests (D.L. I.24), Hieronymus from whose report we learn that Thales measured the height of the pyramids by the shadow they cast (D.L. I.27), and Plutarch (De Is. et Os. 131). Thales gave an explanation for the inundation (D.L. I.37). He may have devised this explanation after witnessing the phenomena, which Herodotus later described (Hdt. II.97).

By 620 BCE, and perhaps earlier, Miletus held a trading concession at Naucratis (Hdt. II.178, Strab. 17.1.18) on the Canopic mouth of the Nile, and it is possible that Thales visited Egypt on a trading mission. Travel to Egypt would not have been difficult. Homer had Ulysses sailing from Crete to the Nile in five days, and Ernle Bradford recently made a similar journey, proving the trip to be feasible (Bradford, Ulysses Found, 26, and passim). The wealth of Miletus was the result of its success as a trading centre, and there would have been no difficulty in arranging passage on one of the many vessels which traded through of Miletus.

Josephus (Contra Apionem I.2) wrote that Thales was a disciple of the Egyptians and the Chaldeans which suggests that he visited the Near-East. It is thought that Thales visited the Babylonians and Chaldeans and had access to the astrological records which enabled him to predict the solar eclipse of 585 BCE

Miletus had founded many colonies around the Mediterranean and especially along the coasts of the Black Sea. Pliny (HN, V.31.112) gives the number as ninety. The Milesians traded their goods for raw materials, especially iron and timber, and tunny fish. Strabo made mention of 'a sheep-industry', and the yield of 'soft wool' (Strabo, 12.3.13), and Aristophanes mentioned the fine and luxurious Milesian wool (Lysistrata, 729; Frogs, 543). The Milesian traders had access to the hinterland. The land around the mouth of the Halys was fertile, 'productive of everything . . . and planted with olive trees' (Strabo, 12.3.12-13). Thales was associated with a commercial venture in the production of olive oil in Miletus and Chios, but his interests may have extended beyond those two places. Olive oil was a basic item in the Mediterranean diet, and was probably a trading commodity of some importance to Milesian commerce.

It is likely that Thales was one of the 'great teachers' who, according to Herodotus, visited Croesus in the Lydian capital, Sardis (Hdt. I.30). From Sardis, he could have joined a caravan to make the three-month journey along the well used Royal Road (Hdt. V.53), to visit the observatories in Babylonia, and seek the astronomical knowledge which they had accumulated over centuries of observation of heavenly phenomena. In about 547 BCE late in his life, Thales travelled into Cappadocia with Croesus, and, according to some belief, devised a scheme by which the army of Croesus was able to cross the River Halys. Milesian merchantmen continually plied the Black Sea, and gaining a passage could have been easily arranged. From any number of ports Thales could have sought information, and from Sinope he may have ventured on the long journey to Babylonia, perhaps travelling along the valley of the Tigris, as Xenophon did in 401-399 BCE

In a letter purported to be from Thales to Pherecydes, Thales stated that he and Solon had both visited Crete, and Egypt to confer with the priests and astronomers, and all over Hellas and Asia (D.L. I.43-44). All that should be gleaned from such reports, is that travel was not exceptional, with many reports affirming the visits of mainly notable people to foreign lands. Alcaeus visited Egypt' (Strabo, 1.2.30), and his brother, Antimenidas, served in Judaea in the army of the Babylonian monarch, King Nebuchadrezzar. Sappho went into exile in Sicily, her brother,Charaxus, spent some time in Egypt, and a number of friends of Sappho visited Sardis where they lived in Lydian society. There must have been any number of people who visited foreign lands, about whom we know nothing.

Very little about the travels of Thales may be stated with certainty, but it seems probable that he would have sought information from any sources of knowledge and wisdom, particularly the centres of learning in the Near-East. It is accepted that there was ample opportunity for travel.

M. Milesian School

Thales was the founder of a new school of philosophy (Arist. Metaph. 983 b20). His two fellow Milesians who also engaged in the new questioning approach to the understanding of the universe, were Anaximander, his disciple (D.L. I.13), and Anaximenes, who was the disciple of Anaximenes (D.L. II.2). Anaximander was about ten years younger than Thales, but survived him by only a year, dying in about 545. Anaximenes was born in 585 and died in about 528. Their lives all overlapped. Through their association they comprised the Milesian School: They all worked on similar problems, the nature of matter and the nature of change, but they each proposed a different material as the primary principle, which indicates that there was no necessity to follow the master's teachings or attribute their discoveries to him. Each proposed a different support for the earth. Thales was held in high regard for his wisdom, being acclaimed as the most eminent of the Wise Men of Ancient Greece, but he was not regarded as a god, as Pythagoras was. Anaximander and Anaximenes were free to pursue their own ideas and to express them in writing. This surely suggests that they engaged in critical discussion of the theories of each other. The Greeks are a sociable people, and their willingness to converse brought rewards in knowledge gained, as Plato remarked (Epinomis, 987E). Critical discussion implies more than familiarity with other views, and more than mere disagreement with other theories. It is the adoption, or in this case, the development, of a new style of discussion. It is a procedure which encourages questioning, debate, explanation, justification and criticism. There was a unique relationship between the three Milesians and it is highly probable that the critical method developed in the Milesian School under the leadership of Thales.

N. The Seven Sages of Ancient Greece

The earliest reference to the Seven Sages of Ancient Greece is in Plato's Protagoras in which he listed seven names: 'A man's ability to utter such remarks [notable, short and compressed] is to be ascribed to his perfect education. Such men were Thales of Miletus, Pittacus of Mitylene, Bias of Priene, Solon of our city [Athens], Cleobulus of Lindus, Myson of Chen, and, last of the traditional seven, Chilon of Sparta. . . . and you can recognize that character in their wisdom by the short memorable sayings that fell from each of them' (Protagoras, 342 E-343 A).

Diogenes recorded that 'Thales was the first to receive the name of Sage in the archonship of Damasias at Athens, when the term was applied to all the Seven Sages, as Demetrius of Phalerum [born. ca. 350 B.C] mentions in his List of Archons (D.L. I.22). Demetrius cannot have been the source for Plato, who died when Demetrius was only three years old. Perhaps there was a source common to both Plato and Demetrius, but it is unknown.

Damasias was archon in 582/1. It may be significant that at this time the Pythian Games were re-organized. More events were added and, for the first time, they were to be held at intervals of four years, in the third year of the Olympiad, instead of the previous eight-yearly intervals. Whether there is an association between the re-organization of the Pythian Games and the inauguration of the Seven Sages in not known but, as Pausanias indicates, the Seven were selected from all around Greece: 'These [the sages] were: from Ionia, Thales of Miletus and Bias of Priene; of the Aeolians in Lesbos, Pittacus of Mitylene; of the Dorians in Asia, Cleobulus of Lindus; Solon of Athens and Chilon of Sparta; the seventh sage, according to the list of Plato, the son of Ariston is not Periander, the son of Cypselus, but Myson of Chenae, a village on Mount Oeta' (Paus. 14.1). The purpose of Damasias may have been aimed at establishing unity between the city-states.

It is difficult to believe that the Seven all assembled at Delphi, although the dates just allow it. Plato wrote that their notable maxims were featured at Delphi: 'They [the Sages], assembled together and dedicated these [short memorable sayings] as the first-fruits of their lore to Apollo in his Delphic temple, inscribing there those maxims which are on every tongue - "Know thyself' and "Nothing overmuch" ' (Pl. Prt. 343 A-B).

Plato regarded wise maxims as the most essential of the criteria for a sage, and associated them with wisdom and with good education, but he has Socrates say: 'Think again of all the ingenious devices in arts or other achievements, such as you might expect in one of practical ability; you might remember Thales of Miletus and Anacharsis the Scythian' (Respublica , 600 A). Practical ability was clearly important.

Several other lists were compiled: Hippobotus (D.L. I.42); Pittacus (D.L. I.42); and Diogenes (D.L. I.13. They omitted some names and adding others. In his work On the Sages, Hermippus reckons seventeen, which included most of the names listed by other compilers.

Many commentators state that Thales was named as Sage because of the practical advice he gave to Miletus in particular, and to Ionia in general. The earlier advice was to his fellow Milesians. In 560, the thirty-five year old Croesus (Hdt. I.25) succeeded his father Alyattes and continued the efforts begun by his father to subdue the Milesians, but without success. Diogenes tells us that 'when Croesus sent to Miletus offering terms of alliance, [Thales] frustrated the plan' (D.L. I.25). The second occasion was at an even later date, when the power of Cyrus loomed as a threat from the east. Thales's advice to the Ionian states was to unite in a political alliance, so that their unified strength could be a defence against the might of Cyrus. This can hardly have been prior to 550 BCE which is thirty years later than the promulgation of the Seven Sages. Thales was not named as a Sage because of any political advice which is extant.

One of the few dates in Thales's life which can be known with certainty is the date of the Eclipse of 585 BCE It brought to a halt the battle being fought between Alyattes and the Mede, Cyaxares and, in addition, brought peace to the region after 'five years of indecisive warfare' (Hdt. I.74). The Greeks believed that Thales had predicted the Eclipse, and perhaps even regarded him as being influential in causing the phenomenon to occur. This was reason enough to declare Thales to be a man of great wisdom and to designate him as the first of the Seven Sages of Ancient Greece.

O. Corner in Oil

Thales's reputation for wisdom is further enhanced in a story which was related by Aristotle. (Politics, 1259 a 6-23). Somehow, through observation of the heavenly bodies, Thales concluded that there would be a bumper crop of olives. He raised the money to put a deposit on the olive presses of Miletus and Chios, so that when the harvest was ready, he was able to let them out at a rate which brought him considerable profit. In this way, Thales answered those who reproached him for his poverty. As Aristotle points out, the scheme has universal application, being nothing more than a monopoly. There need not have been a bumper harvest for the scheme to have been successful. It is quite likely that Thales was involved in commercial ventures, possibly the export of olive oil, and Plutarch reported that Thales was said to have engaged in trade (Plut. Vit. Sol. II.4).

P. The Heritage of Thales

Thales is the first person about whom we know to propose explanations of natural phenomena which were materialistic rather than mythological or theological. His theories were new, bold, exciting, comprehensible, and possible of explanation. He did not speak in riddles as did Heraclitus, and had no need to invent an undefined non-substance, as Anaximander did. Because he gave no role to mythical beings, Thales's theories could be refuted. Arguments could be put forward in attempts to discredit them. Thales's hypotheses were rational and scientific. Aristotle acknowledged Thales as the first philosopher, and criticized his hypotheses in a scientific manner.
The most outstanding aspects of Thales's heritage are: The search for knowledge for its own sake; the development of the scientific method; the adoption of practical methods and their development into general principles; his curiosity and conjectural approach to the questions of natural phenomena - In the sixth century BCE Thales asked the question, 'What is the basic material of the cosmos?' The answer is yet to be discovered.


Q. Bibliography in B to P

Ernle Bradford. Ulysses Found. London: Hodder and Stoughton, 1964.

Britton, John P. "An Early Function for Eclipse Magnitudes in Babylonian Astronomy." Centaurus, 32 (1989): 32.

Britton, John P. "Scientific Astronomy in Pre-Seleucid Babylon." Chapter in H.D. Galter, Die Rolle der Astronomy in den Kulteren Mesopotamiens. Graz: 1993.

Garstang, John and O.R. Gurney. The Geography of the Hittite Empire. Occasional Publications of

The British Institute of Archaeology in Ankara, no. 5. London: The British Institute of Archaeology at Ankara, 1959.

Proclus. A Commentary on the First Book of Euclid's Elements. Translated with an Introduction and

Notes by Glenn R Morrow. Princeton: Princeton University Press, 1970.

Ptolemy. Ptolemy'snAlmagest. Translated and Annotated by G.J. Toomer. London: Duckworth, 1984.

Snell, Bruno. "Die Nachrichten über die Lehren des Thales und die Anfänge der griechischen Philosophie - und Literaturgeschichte." [The News about the Teachings of Thales and the Beginnings of the Greek History of Philosophy and Literature], Philologus 96 (1944): 170-182.

Steele, John M."Eclipse Prediction in Mesopotamia." Archive for History of Exact Science 54 (5) (2000):421-454.

Stephenson, F. Richard, and L.V. Morrison. "Long-term fluctuations in the Earth's rotation: 700 BC to AD 1990." Philosophical Transactions of the Royal Society of London351 (1995): 165-202.

R. Abbreviations in B to P

Aristotle, An. Post., Analytica Posteriora; Cael., De Caelo; De An., De Anima; Gen An., De Generatione Animalium; Hist. An., Historia Animalium; Metaph., Metaphysics; Pol., Politics; Hist. An.; Historia Animalium
Cicero, Rep., De Republica; Nat. D., De Natura Deorum
D.L., Diogenes Laertius, Lives of Eminent Philosophers
Diels,Dox., H. Diels, Doxographi Graeci
DK, Diels, Hermann and Walther Kranz.Die Fragmente der Vorsokratiker. Zurich: Weidmann, 1985.
Epicurus, ap.Censorinus, D.N.; Censorinus, De die natali
Ovid,Met., Metamorphoses
Plutarch,Plut. De Is. et Os., De Iside et Osiride; De Pyth. or., De Pythiae oraculis; Conv. sept. sap., Convivium septem sapientium, [The Dinner of the Seven Wise Men];; Vit. Sol., Vitae Parallelae, Solon
Pliny (the Elder), HN: Naturalis Historia
Pliny (the Younger), Ep: Epistulae
Ps.-Plutarch, Epit;Pseudo-Plutarch, Epitome
Seneca, QNat., Quaestiones Naturales
Stobaeus, Ecl., jEklogaiv ['Selections']
Theophr. ap. Simpl. Phys., Theophrastus, ap. Simplicius, in Physics


S. Greek astronomy: By J J O'Connor and E F Robertson

Today the study of astronomy requires a deep understanding of mathematics and physics. It is important to realise that Greek astronomy (we are interested in the topic during the 1000 years between 700 BC and 300 AD) did not involve physics. Indeed, as Pannekoek points out in [7], a Greek astronomer aimed only to describe the heavens while a Greek physicist sought out physical truth. Mathematics provided the means of description, so astronomy during the 1000 years that interest us in this article was one of the branches of mathematics.

The Greeks began to think of philosophy from the time of Thales in about 600 BC. Thales himself, although famed for his prediction of an eclipse, probably had little knowledge of astronomy, yet he brought back from Egypt knowledge of mathematics into the Greek world and possibly also some knowledge of Babylonian astronomy. It is reasonable to begin by looking at what 'astronomy' was in Greece around this time. However we begin by looking further back than this to around 700 BC.

Basically at this time astronomy was all to do with time keeping. It is natural that astronomical events such as the day would make a natural period of time and likewise the periodic phases of the moon make the next natural time span. Indeed these provided the basic methods of time keeping around the period of 700 BC yet, of course, another important period of time, the year, was not easy to determine in terms of months. Yet a knowledge of the approximate length of the year was vital for food production and so schemes had to be devised. Farmers at this time would base their planting strategies on the rising and setting of the constellations, that is the times when certain constellations would first become visible before sunrise or were last visible after sunset.

Hesiad, one of the earliest Greek poets, often called the "father of Greek didactic poetry" wrote around 700 BC. Two of his complete epics have survived, the one relevant to us here is Works and Days describing peasant life. In this work Hesiad writes that (see [5], also [1] and [7]):-

... when the Pleiades rise it is time to use the sickle, but the plough when they are setting; 40 days they stay away from heaven; when Arcturus ascends from the sea and, rising in the evening, remain visible for the entire night, the grapes must be pruned; but when Orion and Sirius come in the middle of heaven and the rosy fingered Eos sees Arcturus, the grapes must be picked; when the Pleiades, the Hyades, and Orion are setting, then mind the plough; when the Pleiades, fleeing Orion, plunge into the dark sea, storms may be expected; 50 days after the sun's turning is the right time for man to navigate; when Orion appears, Demeter's gift has to be brought to the well-smoothed threshing floor.

For many hundreds of years astronomers would write works on such rising and setting of constellations indicating that the type of advice given by Hesiad continued to be used.

An early time scale based on 12 months of 30 days did not work well since the moon rapidly gets out of phase with a 30 day month. So by 600 BC this had been replaced by a year of 6 'full' months of 30 days and 6 'empty' months of 29 days. This improvement in keeping the moon in phase with the month had the unfortunate effect of taking the year even further out of phase with the period of the recurring seasons. About the same time as Thales was making the first steps in philosophy, Solon, a statesman in Athens who became known as one of the Seven Wise Men of Greece, introduced an improved calendar.

Solon's calendar was based on a two yearly cycle. There were 13 months of 30 days and 12 months of 29 days in each period of two years so this gave a year of about 369 days and a month of 291/2 days. However, the Greeks relied mainly on the moon as their time-keeper and frequent adjustments to the calendar were necessary to keep it in phase with the moon and the seasons. Astronomy was clearly a subject of major practical importance in sorting out the mess of these calendars and so observations began to be made to enable better schemes to be devised.

Pythagoras, around 500 BC, made a number of important advances in astronomy. He recognised that the earth was a sphere, probably more because he believed that a sphere was the most perfect shape than for genuine scientific reasons. He also recognised that the orbit of the Moon was inclined to the equator of the Earth and he was one of the first to realise that Venus as an evening star was the same planet as Venus as a morning star. There is a pleasing appeal to observational evidence in these discoveries, but Pythagoras had a philosophy based on mathematical 'perfection' which tended to work against a proper scientific approach. On the other side there is an important idea in the Pythagorean philosophy which had a lasting impact, namely the idea that all complex phenomena must reduce to simple ones. One should not underestimate the importance of this idea which has proved so powerful throughout the development of science, being a fundamental driving force to the great scientists such as Newton and particularly Einstein.

Around 450 BC Oenopides is said to have discovered the ecliptic made an angle of 24 with the equator, which was accepted in Greece until refined by Eratosthenes in around 250 BC. Some scholars accept that he discovered that the ecliptic was at an angle but doubt that he measured the angle. Whether he learnt of the 12 signs of the zodiac from scholars in Mesopotamia or whether his discoveries were independent Greek discoveries is unknown. Oenopides is also credited with suggesting a calendar involving a 59 year cycle with 730 months. Other schemes proposed were 8 year cycles, with extra months in three of the eight years and there is evidence that this scheme was adopted.

About the same time as Oenopides proposed his 59 year cycle, Philolaus who was a Pythagorean, also proposed a 59 year cycle based on 729 months. This seemed to owe more to the numerology of the Pythagoreans than to astronomy since 729 is 272, 27 being the Pythagorean number for the moon, while it is also 93, 9 being the Pythagorean number associated with the earth. Philolaus is also famed as the first person who we know to propose that the earth moves. He did not have it orbiting the sun, however, but rather all the heavenly bodies went in circles round a central fire which one could never see since there was a counter earth between the earth and the fire. This model, certainly not suggested by any observational evidence, is more likely to have been proposed so that there were 10 heavenly bodies, for 10 was the most perfect of all numbers to the Pythagoreans.
Meton, in 432 BC, introduced a calendar based on a 19 year cycle but again this is similar to one devised in Mesopotamia some years earlier. Meton worked in Athens with another astronomer Euctemon, and they made a series of observations of the solstices (the points at which the sun is at greatest distance from the equator) in order to determine the length of the tropical year. Again we do not know if the 19 year cycle was an independent discovery or whether Greek advances were still based on earlier advances in Mesopotamia. Meton's calendar never seems to have been adopted in practice but his observations proved extremely useful to later Greek astronomers such as Hipparchus and Ptolemy.

That Meton was famous and widely known is seen from the play Birds written by Aristopenes in about 414 BC. Two characters are speaking, one is Meton [see D Barrett (trs.), Aristophanes, Birds (London, 1978)]:-

Meton: I propose to survey the air for you: it will have to be marked out in acres.
Peisthetaerus: Good lord, who do you think you are?
Meton: Who am I? Why Meton. THE Meton. Famous throughout the Hellenic world - you must have heard of my hydraulic clock at Colonus?

Meton and Euctemon are associated with another important astronomical invention of the time, namely a parapegma. A parapegma was a stone tablet with movable pegs and an inscription to indicate the approximate correspondence between, for example, the rising of a particular star and the civil date. Because the calendar had to be changed regularly to keep the civil calendar in phase with the astronomical one, the parapegma had movable pegs which could be adjusted as necessary. A parapegma soon also contained meteorological forecasts associated with the risings and settings of the stars and not only were stone parapegma constructed but also ones on papyri. Meton and Euctemon are usually acknowledged as the inventors of parapegmata and certainly many later astronomers compiled the data nessary for their construction.

There is evidence for other observational work being undertaken around this time, for Vitruvius claims that Democritus of Abdera, famed for his atomic theory, devised a star catalogue. We have no knowledge of the form this catalogue took but Democritus may well have described the major constellations in some way.

The beginning of the 4th century BC was the time that Plato began his teachings and his writing was to have a major influence of Greek thought. As far as astronomy is concerned Plato had a negative effect, for although he mentions the topic many times, no dialogue is devoted to astronomy. Worse still, Plato did not believe in astronomy as a practical subject, and condemned as lowering the spirit the actual observation of the heavenly bodies. Plato only believed in astronomy to the extent that it encouraged the study of mathematics and suggested beautiful geometrical theories.

Perhaps we should digress for a moment to think about how the ideas of philosophy which were being developed by Plato and others affected the development of astronomy. Neugebauer [6] feels that philosophy had a detrimental affect:-
I see no need for considering Greek philosophy as an early stage in the development of science ...

One need only read the gibberish of Proclus's introduction to his huge commentary on Book I of Euclid's Elements to get a vivid picture of what would have become of science in the hands of philosophers. The real "Greek miracle" is the fact that a scientific methodology was developed, and survived, in spite of a widely admired dogmatic philosophy.

Although there is some truth in what Neugebauer writes here, I [EFR] feel that he has overstated his case. It is true that philosophers came up with ideas about the universe which were not based on what we would call today the scientific method. However, the very fact that theories were proposed which could be shown to be false by making observations, must have provided a climate where the scientific approach could show its strength. Also the fact the philosophy taught that one should question all things, even "obvious" truths, was highly beneficial. Another important philosophical idea which had important consequences from the time of Pythagoras, and was emphasised by Plato, was that complex phenomena must be consequences of basic simple phenomena. As Theon of Smyrna expressed it, writing in the first century AD:-
The changing aspects of the revolution of the planets is because, being fixed in their own circles or in their own shperes whose movements they follow, they are carried across the zodiac, just as Pythagoras had first understood it, by a regulated simple and equal revolution but which results by combination in a movement that appears variable and unequal.

This led Theon of Smyrna to write:-
It is natural and necessary that all the heavenly bodies have a uniform and regular movement.
Perhaps the most telling argument against the above claim by Neugebauer is that our present idea of space-time, as developed from Einstein's theory of relativity, was suggested more by the basic philosophy of simplicity than by experimental evidence.

The advances made not long after the time of Plato by Eudoxus, incorporating the idea of basic simplicity as expressed in Pythagorean and Platonic philosophy, were made by an outstanding mathematician and astronomer. In fact Eudoxus marks the beginning of a new phase in Greek astronomy and must figure as one of a small number of remarkable innovators in astronomical thought. Eudoxus was the first to propose a model whereby the apparently complex motions of the heavenly bodies did indeed result from simple circular motion. He built an observatory on Cnidus and from there he observed the star Canopus. The star Canopus played an important role in early astronomy, for it is seen to set and rise in Cnidus yet one does no have to go much further north from there before it can never be seen. The observations made at Eudoxus's observatory in Cnidus, as well as those made at an observatory near Heliopolis, formed the basis of a book concerning the rising and setting of the constellations. Eudoxus, another who followed Pythagorean doctrines, proposed a beautiful mathematical theory of concentric spheres to describe the motion of the heavenly bodies. It is clear that Eudoxus thought of this as a mathematical theory, and did not believe in the spheres as physical objects.

Although a beautiful mathematical theory, Eudoxus's model would not have stood the test of the simplest of observational data. Callippus, who was a pupil of Polemarchus himself a pupil of Eudoxus, refined this system as presented by Eudoxus. The reason that we have so much information about the spheres of Eudoxus and Callippus is that Aristotle accepted the theory, not not as a mathematical model as originally proposed, but rather as spheres which have physical reality. He discussed the interactions of one sphere on another, but there is no way that he could have had enough understanding of physics to get anywhere near describing the effects of such an interaction. Although in many areas Aristotle advocated a modern scientific approach and he collected data in a scientific way, this was unfortunately not the case in astronomy. As Berry writes [2]:-
There are also in Aristotle's writings a number of astronomical speculations, founded on no solid evidence and of little value ... his original contributions are not comparable with his contributions to the mental and moral sciences, but are inferior in value to his work in other natural sciences ...

As Berry goes on to say, this was very unfortunate for astronomy since the influence of the writings of Aristotle had an authority for many centuries which meant that astronomers had a harder battle than they might otherwise have had in getting the truth accepted.

The next development which was absolutely necessary for progress in astronomy took place in geometry. Spherical geometry was developed by a number of mathematicians with an important text being written by Autolycus in Athens around 330 BC. Some claim that Autolycus based his work on spherical geometry On the Moving Sphere on an earlier work by Eudoxus. Whether or not this is the case there is no doubt that Autolycus was strongly influenced by the views of Eudoxus on astronomy. Like so many astronomers, Autolycus wrote a work On Risings and Settings which is a book on observational astronomy.

After Autolycus the main place for major developments in astronomy seemed to move to Alexandria. There Euclid worked and wrote on geometry in general but also making an important contribution to spherical geometry. Euclid also wrote Phaenomena which is an elementary introduction to mathematical astronomy and gives results on the times stars in certain positions will rise and set.

Aristarchus, Timocharis and Aristyllus were three astronomers who all worked at Alexandria and their lives certainly overlapped. Aristyllus was a pupil of Timocharis and in Maeyama [23] analyses 18 of their observations and shows that Timocharis observed around 290 BC while Aristyllus observed a generation later around 260 BC. He also reports an astounding accuracy of 5' for Aristyllus' observations. Maeyama writes [23]:-
The order of accuracy is an essential measure for the development of natural sciences. accuracy is in fact more than the mere operation of measuring. Accuracy increases only by virtue of active measuring. There cannot exist a high order of observational accuracy which is not connected with a high order of observations. Hence my assumption is that there must have been abundant accurate observations of the fixed stars made at least at the epochs 300 BC - 250 BC in Alexandria. They must have disappeared in the fires which frequently raged there.

Maeyama also points out that this is the period when the coordinate systems for giving stellar positions originated. Both the equatorial and the ecliptic systems appear at this time. But why were these observations being made? This is a difficult question to answer for on the face of it there seems little point in the astronomers of Alexandria striving for observational accuracy at this time. In [34] van der Waerden makes an interesting suggestion related to the other important astronomer who worked in Alexandria around this time, namely Aristarchus.

We know that Aristarchus measured the ratio of the distances to the moon and to the sun and, although his methods could never yield accurate results, they did show that the sun was much further from the earth than was the moon. His results also showed that the sun was much larger than the earth, although again his measurements were very inaccurate. Some historians believe that this knowledge that the sun was the largest of the three bodies, earth, moon and sun, led him to propose his heliocentric theory. Certainly it is for this theory, as reported by Archimedes, that Aristarchus has achieved fame. His sun-centred universe found little favour with the Greeks, however, who continued to develop more and more sophisticated models based on an earth centred universe.

Now Goldstein and Bowen in [16] attempt to answer the question of why Timocharis and Aristyllus made their accurate observations. These authors do not find a clear purpose for the observations, such as the marking of a globe. However van der Waerden in [34] suggests that the observations were made to determine the constants in the heliocentric theory of Aristarchus. Although this theory has strong attractions, and makes one want to believe in it, all the evidence suggests that Timocharis certainly began his observations some time before Aristarchus proposed his heliocentric universe.

Goldstein and Bowen in [16] make other interesting suggestions. They believe that the observations of Timocharis and Aristyllus recorded the distance from the pole, and the distances between stars. They argued that the observations were made by means of an instrument similar to Heron's dioptra. These are interesting observations since the work of Timocharis and Aristyllus strongly influenced the most important of all of the Greek astronomers, namely Hipparchus, who made his major contribution about 100 years later. During these 100 years, however, there were a number of advances. Archimedes measured the apparent diameter of the sun and also is said to have designed a planetarium.

Eratosthenes made important measurements of the size of the earth, accurately measured the angle of the ecliptic and improved the calendar. Apollonius used his geometric skills to mathematically develop the epicycle theory which would reach its full importance in the work of Ptolemy.

The contributions of Hipparchus are the most important of all the ancient astronomers and it is fair to say that he made the most important contribution before that of Copernicus in the early sixteenth century. As Berry writes in [2]:-
An immense advance in astronomy was made by Hipparchus, whom all competent critics have agreed to rank far above any other astronomers of the ancient world, and who must stand side by side with the greatest astronomers of all time.

It is Hipparchus's approach to science that ranks him far above other ancient astronomers. His approach, based on data from accurate observations, is essentially modern in that he collected his data and then formed his theories to fit the observed facts. Most telling regarding his understanding of the scientific method is the fact that he proposed a theory of the motion of the sun and the moon yet he was not prepared to propose such a theory for the planets. He realised that his data was not sufficiently good or sufficiently plentiful to allow him to base a theory on it. However, he made observations to help his successors to develop such a theory. Delambre, in his famous work on the history of astronomy, writes:-
When we consider all that Hipparchus invented or perfected, and reflect upon the number of his works and the mass of calculations which they imply, we must regard him as one of the most astonishing men of antiquity, and as the greatest of all in the sciences which are not purely speculative, and which require a combination of geometrical knowledge with a knowledge of phenomena, to be observed only by diligent attention and refined instruments.

Although a great innovator, Hipparchus gained important understanding from the Babylonians. As Jones writes in [21]:-

For Hipparchus, the availability of the Babylonian predictive methods was a boon.  We will not describe the contributions of Hipparchus and Ptolemy in detail in this article since these are given fully in their biographies in our archive. Suffice to end this article with a quotation from [6]:-
Alexandria in the second century AD saw the publication of Ptolemy's remarkable works, the Almagest and the Handy Tables, the Geography, the Tetrabiblos, the Optics, the Harmonics, treatises on logic, on sundials, on stereographic projection, all masterfully written, products of one of the greatest scientific minds of all times. The eminence of these works, in particular the Almagest, had been evident already to Ptolemy's contemporaries. this caused an almost total obliteration of the prehistory of the Ptolemaic astronomy.

Ptolemy had no successor. What is extant from the later Roman times is rather sad.....

T. The trigonometric functions: By J J O'Connor and E F Robertson

The use of trigonometric functions arises from the early connection between mathematics and astronomy. Early work with spherical triangles was as important as plane triangles.

The first work on trigonometric functions related to chords of a circle. Given a circle of fixed radius, 60 units were often used in early calculations, then the problem was to find the length of the chord subtended by a given angle. For a circle of unit radius the length of the chord subtended by the angle x was 2sin (x/2). The first known table of chords was produced by the Greek mathematician Hipparchus in about 140 BC. Although these tables have not survived, it is claimed that twelve books of tables of chords were written by Hipparchus. This makes Hipparchus the founder of trigonometry.

The next Greek mathematician to produce a table of chords was Menelaus in about 100 AD. Menelaus worked in Rome producing six books of tables of chords which have been lost but his work on spherics has survived and is the earliest known work on spherical trigonometry. Menelaus proved a property of plane triangles and the corresponding spherical triangle property known the regula sex quantitatum.

Menelaus' Theorom

Ptolemy was the next author of a book of chords, showing the same Babylonian influence as Hipparchus, dividing the circle into 360 and the diameter into 120 parts. The suggestion here is that he was following earlier practice when the approximation 3 for ? was used. Ptolemy, together with the earlier writers, used a form of the relation sin2 x + cos2 x = 1, although of course they did not actually use sines and cosines but chords.

Similarly, in terms of chords rather than sin and cos, Ptolemy knew the formulas

sin(x + y) = sinx cos y + cosx sin y
a/sin A = b/sin B = c/sin C.

Ptolemy calculated chords by first inscribing regular polygons of 3, 4, 5, 6 and 10 sides in a circle. This allowed him to calculate the chord subtended by angles of 36, 72, 60, 90 and 120. He then found a method of finding the cord subtended by half the arc of a known chord and this, together with interpolation allowed him to calculate chords with a good degree of accuracy. Using these methods Ptolemy found that sin 30' (30' = half of 1) which is the chord of 1 was, as a number to base 60, 0 31' 25". Converted to decimals this is 0.0087268 which is correct to 6 decimal places, the answer to 7 decimal places being 0.0087265.

The first actual appearance of the sine of an angle appears in the work of the Hindus. Aryabhata, in about 500, gave tables of half chords which now really are sine tables and used jya for our sin. This same table was reproduced in the work of Brahmagupta (in 628) and detailed method for constructing a table of sines for any angle were give by Bhaskara in 1150.

The Arabs worked with sines and cosines and by 980 Abu'l-Wafa knew that

sin 2x = 2 sin x cos x

although it could have easily have been deduced from Ptolemy's formula

sin(x + y) = sin x cos y + cos x sin y with x = y.

The Hindu word jya for the sine was adopted by the Arabs who called the sine jiba, a meaningless word with the same sound as jya. Now jiba became jaib in later Arab writings and this word does have a meaning, namely a 'fold'. When European authors translated the Arabic mathematical works into Latin they translated jaib into the word sinus meaning fold in Latin. In particular Fibonacci's use of the term sinus rectus arcus soon encouraged the universal use of sine.

Chapters of Copernicus's book giving all the trigonometry relevant to astronomy was published in 1542 by Rheticus. Rheticus also produced substantial tables of sines and cosines which were published after his death. In 1533 Regiomontanus's work De triangulis omnimodis was published. This contained work on planar and spherical trigonometry originally done much earlier in about 1464. The book is particularly strong on the sine and its inverse.

The term sine certainly was not accepted straight away as the standard notation by all authors. In times when mathematical notation was in itself a new idea many used their own notation. Edmund Gunter was the first to use the abbreviation sin in 1624 in a drawing. The first use of sin in a book was in 1634 by the French mathematician Hérigone while Cavalieri used Si and Oughtred S.

It is perhaps surprising that the second most important trigonometrical function during the period we have discussed was the versed sine, a function now hardly used at all. The versine is related to the sine by the formula

versin x = 1 - cos x.

It is just the sine turned (versed) through 90.

The cosine follows a similar course of development in notation as the sine. Viète used the term sinus residuae for the cosine, Gunter (1620) suggested co-sinus. The notation Si.2 was used by Cavalieri, s co arc by Oughtred and S by Wallis.

Viète knew formulas for sin nx in terms of sin x and cos x. He gave explicitly the formulas (due to Pitiscus)

sin 3x = 3 cos 2x sin x - sin 3 x
cos 3x = cos 3x - 3 sin 2x cos x.

The tangent and cotangent came via a different route from the chord approach of the sine. These developed together and were not at first associated with angles. They became important for calculating heights from the length of the shadow that the object cast. The length of shadows was also of importance in the sundial. Thales used the lengths of shadows to calculate the heights of pyramids.

The first known tables of shadows were produced by the Arabs around 860 and used two measures translated into Latin as umbra recta and umbra versa. Viète used the terms amsinus and prosinus. The name tangent was first used by Thomas Fincke in 1583. The term cotangens was first used by Edmund Gunter in 1620.

Abbreviations for the tan and cot followed a similar development to those of the sin and cos. Cavalieri used Ta and Ta.2, Oughtred used t arc and t co arc while Wallis used T and t. The common abbreviation used today is tan by we write tan whereas the first occurrence of this abbreviation was used by Albert Girard in 1626, but tan was written over the angle


cot was first used by Jonas Moore in 1674.

The secant and cosecant were not used by the early astronomers or surveyors. These came into their own when navigators around the 15th Century started to prepare tables. Copernicus knew of the secant which he called the hypotenusa. Viète knew the results

cosec x/sec x = cot x = 1/tan x
1/cosec x = cos x/cot x = sin x.

The abbreviations used by various authors were similar to the trigonometric functions already discussed. Cavalieri used Se and Se.2, Oughtred used se arc and sec co arc while Wallis used s and . Albert Girard used sec, written above the angle as he did for the tan.
The term 'trigonometry' first appears as the title of a book Trigonometria by B Pitiscus, published in 1595. Pitiscus also discovered the formulas for sin 2x, sin 3x, cos 2x, cos 3x.

The 18th Century saw trigonometric functions of a complex variable being studied. Johann Bernoulli found the relation between sin-1z and log z in 1702 while Cotes, in a work published in 1722 after his death, showed that

ix = log(cos x + i sin x ).

De Moivre published his famous theorem

(cos x + i sin x )n = cos nx + i sin nx

in 1722 while Euler, in 1748, gave the formula (equivalent to that of Cotes)

exp(ix) = cos x + i sin x .

The hyperbolic trigonometric functions were introduced by Lambert.

U. The history of cartography:

This article describes how map making has played an important role in the development of mathematics. It is hardly surprising that cartography should be considered as a mathematical discipline in early times since cartography measures positions of places (mathematics was the science of measurement) and represents a the surface of a sphere on a two dimensional map.
Of course what constitutes a map is hard to say, especially when one goes back to the very earliest times. In around 6200 BC in Catal Hyük in Anatolia a wall painting was made depicting the positions of the streets and houses of the town together with surrounding features such as the volcano close to the town. The wall painting was discovered in 1963 near the modern Ankara in Turkey. Whether it is a map or a stylised painting is a matter of debate.

Town Plan from Catal Hyük (6200 B.C.)
Catal Hyük Town Plan  - earliest known map

The Earliest Known Map  DATE: 6,200 B.C. The human activity of graphically translating one's perception of his world is now generally recognized as a universally acquired skill and one that pre-dates virtually all other forms of written communication.

Early attempts at maps were severely limited by lack of knowledge of anything other than very local features. In Egypt geometry was used from very early times to help measure land. The annual flooding by the Nile meant that without such measurements it was impossible to reconstruct the boundaries that had existed before the flood. Such measurements, however, seem only to have been of local use and there is no evidence that the Egyptians integrated the measurements into maps of large areas. The oldest extant example of a local Egyptian map is the Turin papyrus which dates from around 1300 BC.

Early world maps reflect the religious beliefs of the form of the world. For example maps have been discovered on Babylonian clay tablets dating from around 600 BC. One such map shows Babylon and the surrounding area in a stylised form with Babylon represented by a rectangle and the Euphrates river by vertical lines. The area shown is depicted as circular surrounded by water which fits the religious image of the world in which the Babylonians believed.

  Babylonian map of the world.
Babylonian map

The earliest ancient Greek who is said to have constructed a map of the world is Anaximander, who was born in 610 BC in Miletus (now in Turkey), and died in 546 BC. He is said to have studied under Thales but sadly no details of his map have survived. Of course, although only a very limited portion of the Earth was known to these ancient Greeks, the shape of the Earth was always going to be of fundamental importance in world maps. Pythagoras, in the 6th century BC, is believed to be the first to put forward a belief in a spherical Earth while Parmenides certainly argued in favour of this in the following century. Around 350 BC Aristotle put forward six arguments to prove that the Earth was spherical and from that time on scholars generally accepted that indeed it was a sphere.

Eratosthenes, around 250 BC, made major contributions to cartography. He measured the circumference of the Earth with great accuracy. He sketched, quite precisely, the route of the Nile to Khartoum, showing the two Ethiopian tributaries. He made another important contribution in using a grid to locate positions of places on the Earth. He was not the first to use such a grid for Dicaearchus, a follower of Aristotle, had devised one about 50 years earlier. Today we use latitude and longitude to determine such coordinates and Eratosthenes' grid was of a similar nature. Note, of course, that the use of such positional grids are an early form of Cartesian geometry. Following Dicaearchus, Eratosthenes chose a line through Rhodes and the Pillars of Hercules (present day Gibraltar) to form one of the principal lines of his grid. This line is, to a quite high degree of accuracy, 36 north and Eratosthenes chose it since it divided the world as he knew it into two fairly equal parts and defined the longest east-west extent known. He chose a defining line for the north-south lines of his grid through Rhodes and drew seven parallel lines to each of his defining lines to form a rectangular grid.

Eratosthenes' map of the world.
Eratosthene's map of the world

Hipparchus was critical of the grid defined by Eratosthenes, saying reasonably enough that it was chosen arbitrarily. He suggested that a grid should be chosen with astronomical significance so that, for example, points on the same line would all have the same length of longest day. Although Hipparchus never constructed a map as far as we know, he did make astronomical observations to describe eleven parallels given by his astronomical definition. Although no copies of the work by Eratosthenes and Hipparchus survives, we know of it through the Geographical Sketches of Strabo which was completed in about 23 A.D. Although Strabo gives a critical account of earlier contributions to cartography, he devotes only a small discussion to the problem of projecting a sphere onto a plane. He states clearly that his work is not aimed at mathematicians, rather at statesmen who need to know about the customs of the people and the natural resources of the land.

The final ancient Greek contribution we consider was the most important and, unlike that of Strabo, was written by a noted mathematician. In about 140 A.D. Ptolemy wrote his major work Guide to Geography, in eight books, which attempted to map the known world giving coordinates of the major places in terms of what are essentially latitude and longitude. The first volume gives the basic principles of cartography and considers the problem of map projection, that is mapping the sphere onto the plane. He gave two examples of projections, and also described the construction of globes. Right at the beginning Ptolemy identifies two distinct types of cartography, the first being [1]:-
... an imitation through drawing of the entire known part of the world together with the things which are, broadly speaking, connected with it.

The second type is [1]:-
... an independent discipline [which] sets out the individual localities.

Now the main part of Geography consisted of maps but Ptolemy knew that although a scribe could copy a text fairly accurately, there was little chance that maps could be successfully copied. He therefore ensured that the work contained the data and the information necessary for someone to redraw the maps. He followed previous cartographers in dividing the circle of the equator into 360 and took the equator as the basis for the north-south coordinate system. Thus the line of latitude through Rhodes and the Pillars of Hercules (present day Gibraltar) was 36 and this line divided the world as Ptolemy knew it fairly equally into two. The problem of defining lines of longitude is more difficult. It required the choice of an arbitrary zero but it also required a knowledge of the circumference of the Earth in order to have degrees correspond correctly to distance. Ptolemy chose the Fortune Islands (which we believe are the Canary Islands) as longitude zero since it was the most western point known to him. He then marked off where the lines of longitude crossed the parallel of Rhodes, taking 400 stadia per degree.

Ptolemy's map of the world.
Ptolemy's map of the world

Had Ptolemy taken the value of the circumference of the Earth worked out by Eratosthenes then his coordinates would have been very accurate. However he used the later value computed by Posidonius around 100 BC which, although computer using the correct mathematical theory, is highly inaccurate. Therefore instead of the Mediterranean covering 42 as it should, it covers 62 in Ptolemy's coordinates. Books 2 to 7 of Geography contain the coordinates of about 8000 places, but although he knew the correct mathematical theory to compute such coordinates accurately from astronomical observations, there were only a handful of places for which such information existed. 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, for anything outside the Roman Empire, this was of very poor quality with even some parts of the Roman Empire severely distorted.

Ptolemy used data from most of his predecessors, particularly Marinus of Tyre. He used information from reports of travellers who gave such information as "after ten days travel north we reached ...". In order to estimate distances from such data, Ptolemy had to estimate the difficulty of the terrain, how straight the route the travellers taken had been, and many other unknowns. Given the way that he gathered the data it is certainly not surprising that the maps were inaccurate but they represented a considerable advance on all previous maps and it would be many centuries before more accurate world maps would be drawn.

Little progress was made in cartography over the next centuries. That the Romans made few contributions is slightly strange given their skills at road building which required accurate surveying measurements. Also the very precise military strategies which their commanders used would seem to give them the motivation to create maps to help their military campaigns. Perhaps it was the mathematical nature of a map which prevented the non-mathematical Romans from advancing the subject. In China, however, there is evidence that mathematics had been used an a major way in surveying and cartography. In [12] Liu Hui's 3rd century work the Sea Island mathematical manual is studied. The book gives a good insight into the history of surveying in China and its links with cartography. The main driving force in China to survey and draw maps was often for military reasons but also for problems such as water conservancy.

Once Christianity spread across Europe those of learning were Churchmen and the truth about the world, they argued, was contained in the Bible and not to be found by scientific investigation. Where Bible quotations appeared to contradict pre-Christian scientific discoveries, then good science was dismissed as pagan foolishness. Biblical quotations convinced some that the Earth was a circle, certainly not a sphere, while for others quotations such as "the four corners of the Earth" in Isaiah proved that the Earth was rectangular.

In the Arabic/Persian/Muslim world, progress was made in cartography, however, and in fact far more progress than was realised for a long time, for it is only in recent years that the full significance of these contributions has been realised. Ptolemy's Geography was translated into Arabic in the 9th century and soon improvements were being made using data obtained from the explorations being carried out. Al-Khwarizmi wrote a major work on cartography which gave the latitudes and longitudes for 2402 localities as a basis for his world map. The book, which was based on Ptolemy's Geography, lists with latitudes and longitudes, cities, mountains, seas, islands, geographical regions, and rivers. The manuscript does include maps which are more accurate than those of Ptolemy, in particular it is clear that where more local knowledge was available to al-Khwarizmi such as in the regions of Islam, Africa and the Far East then his work is considerably more accurate than that of Ptolemy, but for Europe al-Khwarizmi seems on the whole to have used Ptolemy's data.

The major work by Sezgin, see [8], [9], and [10], has done much to demonstrate that the medieval Islamic geographers had an important influence on the development of geography in Europe up to 1800. In [8] he presents a reconstruction of al-Khwarizmi's map of the world which he believes used a stereographic projection of the terrestrial hemisphere, with pole on the terrestrial equator. Sezgin also argues that Ptolemy's Geography may not have included a world map, and that some later world maps are based, at least in part, on Islamic sources.

The next important Islamic scholar we should mention is al-Biruni who wrote his Cartography in around 995. In it he discussed map projections due to other scientists, then gives his own interesting mapping of a hemisphere onto a plane. A detailed description of this projection is given in [17]. Al-Biruni wrote a textbook on the general solution of spherical triangles around 1000 then, some time after 1010, he applied these methods on spherical triangles to geographical problems. He introduced techniques to measure the Earth and distances on it using triangulation. He computed very accurate values for the differences in longitude and latitude between Ghazni in Afghanistan and Mecca. He found the radius of the earth to be 6339.6 km, a value not obtained in the West until the 16th century. His Masudic canon contains a table giving the coordinates of six hundred places, some of which were measured by al-Biruni himself, some being taken al-Khwarizmi's work referred to above.

At a time when Christian Europe was producing religious representations of the world rather than scientific maps, another type of map, or perhaps more accurately chart, for the use of sailors began to appear. These were called portolan maps (from the Italian word for a sailing manual) and were produced by sailors using a magnetic compass. The earliest examples we know about date from the beginning of the 14th century, and were Italian or Catalan portolan maps. The earliest portolan maps covered the Mediterranean and Black Sea and showed wind directions and such information useful to sailors. The coast lines shown on these maps are by far the most accurate to have been produced up to that time. The Catalan World Map produced in 1375 was the work of Abraham Cresques from Palma in Majorca. He was a skilled cartographer and instrument maker and the map was commissioned by Charles V of France. The western part of his map was partly based on portolan maps while the eastern part was based on Ptolemy's data.

A 'portolan' map of the North Atlantic.
Portolan map of the North Atlantic
The Catalan World Map.

Catalan World Map

The 15th century saw cartography revolutionised in Europe. The first steps involved the translation of Ptolemy's Geography into Latin which was begun by Emmanuel Chrysoloras and completed in 1410 by Jacobus Angelus. The main motivation to improve cartography came with the discoveries of new lands made by the Portuguese explorers of the 15th century. Brother Mauro, a monk from Murano near Venice, had an excellent reputation in cartography by the middle of the 15th century. In 1457 he was commissioned by the King of Portugal to produce a new world map containing details of the new lands discovered by the Portuguese explorers, and charts drawn by these explorers were sent to him. Producing a map which did not follow Ptolemy clearly worried Mauro who wrote (see for example [5]):-
I do not think it derogatory to Ptolemy if I do not follow his 'Cosmografia', because, to have observed his meridians or parallels or degrees, it would be necessary in respect to the setting out of the known parts of this circumference, to leave out many provinces not mentioned by Ptolemy. But principally in latitude, that is from south to north, he has much 'terra incognita', because in his time it was unknown.

Brother Mauro added the new discoveries to his maps but he made no improvements in the science of cartography. Despite 1300 years passing since Ptolemy's time, Mauro is still not able to give a good approximation for the circumference of the Earth writing:-
I have found various opinions regarding this circumference, but it not possible to verify them ... they are not of much authenticity, since they have not been tested.

The means to make maps widely available also happened in the 15th century with the invention of the printing press around the middle to the century. The first printed version of Ptolemy's Geography appeared in 1475 being the Latin translation referred to above. This edition only contained the text and not maps. The date of the first edition to contain maps is still disputed but may be the one printed in Rome in 1478 which contained 27 maps. Many printed editions with maps followed in quick succession, and newly discovered lands were soon included. New maps were added to various editions to include more accurate and detailed information about Europe, the first being in the Florence edition of 1480 which contained new maps of France, Italy, Spain and Palestine based on recent knowledge. The first to show the New World was a new edition of the 1475 Rome edition, which appeared in 1508 with 34 maps. The edition which many consider to be the first modern atlas (although the term 'atlas' was not used until Gerardus Mercator coined it around 1578) was published in Strasburg in 1513 with 27 maps of the ancient world and 20 new maps based on recent knowledge produced by Martin Waldseemüller. He made a clear distinction between the two parts (see [7] where the following quotation is given):-
We have confined the Geography of Ptolemy to the first part of the work, in order that its antiquity may remain intact and separate.

Waldseemüller's map of the world was the first to cover 360 of longitude and to show the complete coast of Africa. Another first for Waldseemüller occurred in an earlier work in 1507 in which he proposed the naming of America (see [16] where the following quotation is given):-
Since another fourth part [of the world] has been discovered by Americus Vesputius, I do not see why anyone should object to its being called after Americus the discoverer, a man of natural wisdom, Land of Americus or America, since both Europe and Asia have derived their names from women.

Waldseemüller also made important contributions to the science of cartography. He wrote on surveying and perspective and produced a booklet on how to use globes.

Arabic science continued to flourish, now along side European science, and mathematical geography saw important developments with Sulayman al-Mahri's Tuhfat al-fuhul fi tamhid al-usul and the commentary on it Kitab sarh written in the early sixteenth-century. Al-Mahri used astronomical observations of the height of stars to determine the difference in latitude between two places. Trigonometric methods allowed differences in longitude to be calculated. He also developed a good understanding of how to compensate for the errors caused by short-cuts in his mathematical calculations and also for errors caused by inaccurate data.

It was the 16th century which saw the first major mathematical improvements in cartography in Europe although Regiomontanus had led the way towards the end of the 15th century. He set up a new press in Nuremburg in 1472 and announced his intention to publish maps and books including Ptolemy's Geography. With an interest in trigonometry, mathematical instruments, astronomy, and geography, Regiomontanus was in a good position to give a lead. He set up a workshop in Nuremburg to make mathematical instruments, and published works giving details of the use of the instruments. He realised that accurate coordinates of places were required to draw accurate maps and that the greatest problem was in computing the longitude. He proposed the method of lunar distances to determine longitude which was an important proposal. Johann Werner was a follower of Regiomontanus from Nuremburg. Werner's most famous work on geography is In Hoc Opere Haec Cotinentur Moua Translatio Primi Libri Geographicae Cl'Ptolomaei written in 1514. This book contains a description of an instrument with an angular scale on a staff from which degrees could be read off. It also gives a method to determine longitude based on eclipses of the Moon and makes a study of map projections. This work by Werner strongly influenced Gerardus Mercator.
Albrecht Dürer visited Regiomontanus' workshops in Nuremburg when he was young lad. He was fascinated with the ideas of projecting a sphere and also of what the Earth would look like if viewed from the heavens. He employed his ideas of perspective on maps, and in particular he collaborated with Johann Stabius in the construction of globes in 1515. Apianus, a noted mathematician, began his publishing career with producing a world map Typus orbis universalis which he based on an earlier 1520 world map by Martin Waldseemüller. His 1524 publication Cosmographia seu descriptio totis orbis was a work based largely on Ptolemy which provided an introduction to astronomy, geography, cartography, surveying, navigation, weather and climate, the shape of the earth, map projections, and mathematical instruments.

Gemma Frisius was another mathematician who made significant contributions to cartography. He produced his own version of Apianus's Cosmographia a few years after the original edition. In 1530 he published On the Principles of Astronomy and Cosmography, with Instruction for the Use of Globes, and Information on the World and on Islands and Other Places Recently Discovered which made major contributions to cartography. In particular he described how longitude could be calculated using a clock to determine the difference in local and absolute times, being the first to make such a proposal. In 1533 Gemma Frisius published Libellus de locurum which described the theory of trigonometric surveying and in particular contains the first proposal to use triangulation as a method of accurately locating places. This provided an accurate means of surveying using relatively few observations. Positions of places were fixed as the point of intersection of two lines and, as Frisius pointed out, only one accurate measurement of actual distance was required to fix the scale. Not only did Frisius propose an efficient theoretical method for surveying which was needed to produce accurate maps, but he also produced the instruments with which to undertake the surveys and he published accurate maps using the data gathered from such surveys.

An example of Gemma Frisius's triangulation.
Example of Gemma Frisius' triangulation
Following Gemma Frisius, major contributions were made by Gerardus Mercator who studied under Frisius. Mercator made many new maps and globes, but his greatest contribution to cartography must be the Mercator projection. He realised that sailors incorrectly assumed that following a particular compass course would have them travel in a straight line. A ship sailing towards the same point of the compass would follow a curve called a loxodrome (also called a rhumb line or spherical helix), a curve which Pedro Nunes, a mathematician greatly admired by Mercator, had studied shortly before. A new globe which Gerardus Mercator produced in 1541 was the first to have rhumb lines shown on it. This work was an important stage in his developing the idea of the Mercator projection which he first used in 1569 for a wall map of the world on 18 separate sheets. The 'Mercator projection' has the property that lines of longitude, lines of latitude and rhomb lines all appear as straight lines on the map. In this projection the meridians are vertical and parallels having increased spacing in proportion to the secant of the latitude. Edward Wright published mathematical tables to be used in calculating Mercator's projection in 1599, see [20] for details.

Mercator's world map.Mercator's World Map

Mercator's map of Europe.
Mercator's map of Europe
Mercator's map of the Americas.
Mercator's map of Americas

Mercator's map of Asia.
Mercator's map of Asia

Of course the Mercator projection has the property that distances near the poles are greatly distorted so it was not easy to use the map to measure distances. Gerardus Mercator gave instructions on the map so that for two places if one knew any two of the following four pieces of data:

• difference on longitudes,
• difference in latitudes,
• direction between them,
• distance between them,

then his formula allowed one to find the other two. It is interesting to realise that on a map of the world drawn with the Mercator projection, Greenland (whose area is about 2 million km2) appears to be larger than Africa (whose area is about 30 million km2). As a world map the Mercator projection then has considerable disadvantages (as necessarily do all projections) but for sea charts it is undoubtedly the best projection and was eventually adopted by all sailors.

Abraham Ortel, known by his Latinised name of Ortelius, was born in Antwerp on 4 April 1527. He studied Greek, Latin and mathematics and, strongly influenced by Gerardus Mercator, went on to open a map making business. He published the Theatrum orbis terrarum in 1570 which consisted of 70 maps presented in a uniform style using the most up-to-date data. It was the most popular atlas of its time, and it is important in the history of cartography partly because Ortelius quotes 87 authorities for the data on which his maps are based. It appeared a few years before the atlas of Mercator began publication and many argue that Mercator delayed in order to let his younger friend have priority. This, however, seems unlikely and it is much more probable that Mercator's work was delayed, for by the 1570s he was an old man with health problems.

By the 17th and 18th centuries scientific advances had paved the way for further improvements in cartography. Not only were new methods being developed, but there were also arguments to produce a different type of map. For example Thomas Burnet in The theory of the earth (London, 1684) wrote:-
I do not doubt but that it would be of very good use to have natural maps of the Earth . . . as well as civil. . . . Our common maps I call civil, which note the distinction of countries and of cities, and represent the artificial Earth as inhabited and cultivated: But natural maps leave out all that, and represent the Earth as it would be if there were not an Inhabitant upon it, nor ever had been - the skeleton of the Earth, as I may so say, with the site of all its parts. Methinks also a Prince should have such a draught of his country and dominions, to see how the ground lies in the several parts of them, which highest, which lowest; what respect they have to one another, and to the sea; how the rivers flow, and why; how the mountains lie, how heaths, and how the marches. Such a map or survey would be useful both in time of war and peace, and many good observations might be made by it, not only as to natural history and philosophy, but also in order to the perfect improvement of a country.

Progress in cartography now became dependent on having the means of accurately determining the position of places in the world. Calculating latitude was easy, and had long been achieved with a sextant, but the problem of accurately calculating the longitude proved a great challenge. The story of attempts at solving this problem are given in our two essays Longitude and the Académie Royale and English attack on the Longitude Problem and it is to these essays that we refer the reader for information on many later developments in cartography. The Low Countries had dominated developments in cartography through the 16th and early 17th centuries. However after this the centre of activity moved to France where a national survey based on a mathematical approach to trigonometric surveying led the way.

There is another problem with longitude, other than methods to calculate it, namely that a zero needs to be set arbitrarily. At first, as is to be expected, several different places were chosen as the zero such as Paris, Cadiz, Naples, Pulkova, Stockholm and London. International agreement was needed to set cartographic standards and the International Meridian Conference held in Washington D.C. USA in 1884 had delegates from 26 countries. They standardised the Greenwich Meridian as the zero for longitude and, after some delay, all countries adopted this and the equator as the basic reference lines.

There is, of course, another decision to be taken in order to standardise maps, namely how the map is oriented. It is fairly logical to have either north or south at the top, but which is chosen is a completely arbitrary decision. Early Christian maps had north at the top while early Arabic/Muslim maps had south at the top. Without any international agreement, it has become standard practice to have north at the top of a map. Other collaborative international projects have been less successful. In 1891 there was an International Geographical Congress in Bern which established the International Map of the World. Standards were set and a symbol convention was chosen. The scale was to be 1:1000000 and several nations agreed to cooperate to produce a world map to this standard. Some, but not all, of the proposed maps have been produced but the project has never been completed.

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