Famous Celebrities: December 2011

Thursday, 1 December 2011

Hipparchus of Nicaea

Hipparchus of Nicaea Biography
Little is known of Hipparchus's life, but he is known to have been born in Nicaea in Bithynia. The town of Nicaea is now called Iznik and is situated in north-western Turkey. Founded in the 4th Century BC, Nicaea lies on the eastern shore of Lake Iznik. Reasonably enough Hipparchus is often referred to as Hipparchus of Nicaea or Hipparchus of Bithynia and he is listed among the famous men of Bithynia by Strabo, the Greek geographer and historian who lived from about 64 BC to about 24 AD. There are coins from Nicaea which depict Hipparchus sitting looking at a globe and his image appears on coins minted under five different Roman emperors between 138 AD and 253 AD.

This seems to firmly place Hipparchus in Nicaea and indeed Ptolemy does describe Hipparchus as observing in Bithynia, and one would naturally assume that in fact he was observing in Nicaea. However, of the observations which are said to have been made by Hipparchus, some were made in the north of the island of Rhodes and several (although only one is definitely due to Hipparchus himself) were made in Alexandria. If these are indeed as they appear we can say with certainty that Hipparchus was in Alexandria in 146 BC and in Rhodes near the end of his career in 127 BC and 126 BC.

It is not too unusual to have few details of the life of a Greek mathematician, but with Hipparchus the position is a little unusual for, despite Hipparchus being a mathematician and astronomer of major importance, we have disappointingly few definite details of his work. Only one work by Hipparchus has survived, namely Commentary on Aratus and Eudoxus and this is certainly not one of his major works. It is however important in that it gives us the only source of Hipparchus's own writings.

Most of the information which we have about the work of Hipparchus comes from Ptolemy's Almagest but, as Toomer writes in [1]:-

... although Ptolemy obviously had studied Hipparchus's writings thoroughly and had a deep respect for his work, his main concern was not to transmit it to posterity but to use it and, where possible, improve upon it in constructing his own astronomical system.

Where one might hope for more information about Hipparchus would be in the commentaries on Ptolemy's Almagest. There are two in particular by the excellent commentators Theon of Alexandria and by Pappus, but unfortunately these follow Ptolemy's text fairly closely and fail to add the expected information about Hipparchus. Since when Ptolemy refers to results of Hipparchus he does so often in an obscure way, at least he seems to assume that the reader will have access to the original writings by Hipparchus, and it is certainly surprising that neither Theon or Pappus fills in the details. One can only assume that neither of them had access to the information about Hipparchus on which we would have liked them to report.

Let us first summarise the main contribution of Hipparchus and then examine them in more detail. He made an early contribution to trigonometry producing a table of chords, an early example of a trigonometric table; indeed some historians go so far as to say that trigonometry was invented by him. The purpose of this table of chords was to give a method for solving triangles which avoided solving each triangle from first principles. He also introduced the division of a circle into 360 degrees into Greece.

Hipparchus calculated the length of the year to within 6.5 minutes and discovered the precession of the equinoxes. Hipparchus's value of 46" for the annual precession is good compared with the modern value of 50.26" and much better than the figure of 36" that Ptolemy was to obtain nearly 300 years later. We believe that Hipparchus's star catalogue contained about 850 stars, probably not listed in a systematic coordinate system but using various different ways to designate the position of a star. His star catalogue, probably completed in 129 BC, has been claimed to have been used by Ptolemy as the basis of his own star catalogue. However, Vogt shows clearly in his important paper [26] that by considering the Commentary on Aratus and Eudoxus and making the reasonable assumption that the data given there agreed with his star catalogue, then Ptolemy's star catalogue cannot have been produced from the positions of the stars as given by Hipparchus.

This last point shows that in any detailed discussion of the achievements of Hipparchus we have to delve more deeply than just assuming that everything in the Ptolemy's Almagest which he does not claim as his own must be due to Hipparchus. This view was taken for many years but since Vogt's 1925 paper [26] there has been much research done trying to ascertain exactly what Hipparchus achieved. So major shifts have taken place in our understanding of Hipparchus, first it was assumed that his discoveries were all set out by Ptolemy, then once it was realised that this was not so there was a feeling that it would be impossible to ever have detailed knowledge of his achievements, but now we are in a third stage where it is realised that it is possible to gain a good knowledge of his work but only with much effort and research.

Let us begin our detailed description of Hipparchus's achievements by looking at the only work which has survived. Hipparchus's Commentary on Aratus and Eudoxus was written in three books as a commentary on three different writings. Firstly there was a treatise by Eudoxus (unfortunately now lost) in which he named and described the constellations. Aratus wrote a poem called Phaenomena which was based on the treatise by Eudoxus and proved to be a work of great popularity. This poem has survived and we have its text. Thirdly there was commentary on Aratus by Attalus of Rhodes, written shortly before the time of Hipparchus.

It is certainly unfortunate that of all of the writings of Hipparchus this was the one to survive since the three books on which Hipparchus was writing a commentary contained no mathematical astronomy. As a result of this Hipparchus chose to write at the same qualitative level in the first book and also for much of the second of his three book. However towards the end of the second book, continuing through the whole of the third book, Hipparchus gives his own account of the rising and setting of the constellations. Towards the end of Book 3 Hipparchus gives a list of bright stars always visible for the purpose of enabling the time at night to be accurately determined. As we noted above Hipparchus does not use a single consistent coordinate system to denote stellar positions, rather using a mixture of different coordinates. He uses some equatorial coordinates, although often in a rather strange way as for example saying that a star (see [1]):-

... occupies three degrees of Leo along its parallel circle...

He has therefore divided each small circle parallel to the equator into 12 portions of 30° each and this means that the right ascension of the star referred to in the quotation is 123°. The data in the Commentary on Aratus and Eudoxus has been analysed by many authors. In particular the authors of [15] argue that Hipparchus used a mobile celestial sphere with the stars pictured on the sphere. They claim that the data was taken from on a star catalogue constructed around 140 BC based on observations accurate to a third of a degree or even better. In the earlier work [16] by the same authors, they suggest that the observations were made at a latitude of 36° 15' which corresponds to that of northern Rhodes. This would tend to confirm that this work by Hipparchus was done near the end of his career. As Toomer writes in [1]:-

Far from being a "work of his youth", as it is frequently described, the commentary on Aratus reveals Hipparchus as one who had already compiled a large number of observations, invented methods for solving problems in spherical astronomy, and developed the highly significant idea of mathematically fixing the positions of the stars...

There is of course no agreement on many of the points discussed here. For example Maeyama in [13] sees major differences between the accuracy of the data in Commentary on Aratus and Eudoxus (claimed to be written around 140 BC) and Hipparchus's star catalogue (claimed to be produced around 130 BC). Maeyama writes [13]:-

... Hipparchus's "Commentary" contains his own observations of the stellar positions, great in number but inaccurate in operation, despite all his ability for accurate observations. ... the observational accuracy [of] his two different epochs have nothing in common, as if they dealt with two different observers. Within an interval of 10 years everything can happen, particularly in the case of a man like Hipparchus. Those views which consider Hipparchus's astronomical activities at his two different epochs as similar are completely unfounded.

Perhaps the discovery for which Hipparchus is most famous is the discovery of precession which is due to the slow change in direction of the axis of rotation of the earth. This work came from Hipparchus's attempts to calculate the length of the year with a high degree of accuracy. There are two different definitions of a 'year' for one might take the time that the sun takes to return to the same place amongst the fixed stars or one could take the length of time before the seasons repeated which is a length of time defined by considering the equinoxes. The first of these is called the sidereal year while the second is called the tropical year.

Of course the data needed by Hipparchus to calculate the length of these two different years was not something that he could find over a few years of observations. Swerdlow [20] suggests that Hipparchus calculated the length of the tropical year using Babylonian data to arrive at the value of 1/300 of a day less than 3651/4 days. He then checked this against observations of equinoxes and solstices including his own data and those of Aristarchus in 280 BC and Meton in 432 BC. Hipparchus also calculated the length of the sidereal year, again using older Babylonian data, and arrived at the highly accurate figure of 1/144 days longer than 3651/4 days. This gives his rate of precession of 1° per century.

Hipparchus also made a careful study of the motion of the moon. There are difficult problems in such a study for there are three different periods which one could determine. There is the time taken for the moon to return to the same longitude, the time taken for it to return to the same velocity (the anomaly) and the time taken for it to return to the same latitude. In addition there is the synodic month, that is the time between successive oppositions of the sun and moon. Toomer [22] writes:-

For his lunar theory [Hipparchus] needed to establish the mean motions of the Moon in longitude, anomaly and latitude. The best data available to him were the Babylonian parameters. But he was not content merely to accept them: he wanted to test them empirically, and so he constructed (purely arithmetically) the eclipse period of 126007 days 1 hour, then looked in the observational material available to him for pairs of eclipses which would confirm that this was indeed an eclipse period. The observations thus played a real role, but that role was not discovery, but confirmation.

In calculating the distance of the moon, Hipparchus not only made excellent use of both mathematical techniques and observational techniques but he also gave a range of values within which be calculated that the true distance must lie. Although Hipparchus's treatise On sizes and distances has not survived details given by Ptolemy, Pappus, and others allow us to reconstruct his methods and results.

The reconstruction of Hipparchus's techniques is beautifully presented in [24] where the author shows that Hipparchus based his calculations on an eclipse which occurred on 14 March 190 BC. Hipparchus's calculations led him to a value for the distance to the moon of between 59 and 67 earth radii which is quite remarkable (the correct distance is 60 earth radii). The main reason for his range of values was that he was unable to determine the parallax of the sun, only managing to give an upper value. Hipparchus appears to know that 67 earth radii for the distance of the moon comes from this upper limit of solar parallax, while the lower value of 59 earth radii corresponds to the sun being at infinity.

Hipparchus not only gave observational data for the moon which enabled him to compute accurately the various periods, but he developed a theoretical model of the motion of the moon based on epicycles. He showed that his model did not agree totally with observations but it seems to be Ptolemy who was the first to correct the model to take these discrepancies into account. Hipparchus was also able to give an epicycle model for the motion of the sun (which is easier), but he did not attempt to give an epicycle model for the motion of the planets.

Finally let us examine the contributions which Hipparchus made to trigonometry. Heath writes in [6]:-

Even if he did not invent it, Hipparchus is the first person whose systematic use of trigonometry we have documentary evidence.

The documentary evidence comes from Ptolemy and Theon of Alexandria who explicitly says that Hipparchus wrote a work on chords in 12 books. However, Neugebauer [7] points out that:-

... this number is obvious nonsense since 13 books sufficed for the whole of the "Almagest" or of Euclid's "Elements"...

Toomer ([1] or [23]) reconstructs Hipparchus's table of chords, and the mathematical means by which Hipparchus calculated it. The table was based on a circle divided into 360 degrees with each degree divided into 60 minutes. The radius of the circle is then 360.60/2π = 3438 minutes and the chord function Crd of Hipparchus is related to the sine function by

(Crd 2a)/2 = 3438 sin a.

Toomer claims that Hipparchus defined his Crd function at 7.5° intervals (1/48 of the circle) and used linear interpolation to find the value at intermediate points. He then goes on to show that the table can be computed from some basic formulae which would be known to Hipparchus, one of which is the supplementary angle theorem, essentially Pythagoras's theorem, and the half-angle theorem. The only trace of Hipparchus's tables that survives is in Indian tables which are thought to have been based on that of that of Hipparchus.

Toomer summarises the contributions of Hipparchus in this area when he writes in [1]:-

... it seems highly probable that Hipparchus was the first to construct a table of chords and thus provide a general solution for trigonometrical problems. A corollary of this is that, before Hipparchus, astronomical tables based on Greek geometrical methods did not exist. If this is so, Hipparchus was not only the founder of trigonometry but also the man who transformed Greek astronomy from a purely theoretical into a practical predictive science.

Hipparchus of Nicaea

science project yay.wmv

Eratosthenes of Cyrene

SCIENTIST ERATOSTHENES:

Eratosthenes was born in Cyrene which is now in Libya in North Africa. His teachers included the scholar Lysanias of Cyrene and the philosopher Ariston of Chios who had studied under Zeno, the founder of the Stoic school of philosophy. Eratosthenes also studied under the poet and scholar Callimachus who had also been born in Cyrene. Eratosthenes then spent some years studying in Athens.

The library at Alexandria was planned by Ptolemy I Soter and the project came to fruition under his son Ptolemy II Philadelphus. The library was based on copies of the works in the library of Aristotle. Ptolemy II Philadelphus appointed one of Eratosthenes' teachers Callimachus as the second librarian. When Ptolemy III Euergetes succeeded his father in 245 BC and he persuaded Eratosthenes to go to Alexandria as the tutor of his son Philopator. On the death of Callimachus in about 240 BC, Eratosthenes became the third librarian at Alexandria, in the library in a temple of the Muses called the Mouseion. The library is said to have contained hundreds of thousands of papyrus and vellum scrolls.

Despite being a leading all-round scholar, Eratosthenes was considered to fall short of the highest rank. Heath writes [4]:-

[Eratosthenes] was, indeed, recognised by his contemporaries as a man of great distinction in all branches of knowledge, though in each subject he just fell short of the highest place. On the latter ground he was called Beta, and another nickname applied to him, Pentathlos, has the same implication, representing as it does an all-round athlete who was not the first runner or wrestler but took the second prize in these contests as well as others.

Certainly this is a harsh nickname to give to a man whose accomplishments in many different areas are remembered today not only as historically important but, remarkably in many cases, still providing a basis for modern scientific methods.

One of the important works of Eratosthenes was Platonicus which dealt with the mathematics which underlie Plato's philosophy. This work was heavily used by Theon of Smyrna when he wrote Expositio rerum mathematicarum and, although Platonicus is now lost, Theon of Smyrna tells us that Eratosthenes' work studied the basic definitions of geometry and arithmetic, as well as covering such topics as music.

One rather surprising source of information concerning Eratosthenes is from a forged letter. In his commentary on Proposition 1 of Archimedes' Sphere and cylinder Book II, Eutocius reproduces a letter reputed to have been written by Eratosthenes to Ptolemy III Euergetes. The letter describes the history of the problem of the duplication of the cube and, in particular, it describes a mechanical device invented by Eratosthenes to find line segments x and y so that, for given segments a and b,

a : x = x : y = y : b.

By the famous result of Hippocrates it was known that solving the problem of finding two mean proportionals between a number and its double was equivalent to solving the problem of duplicating the cube. Although the letter is a forgery, parts of it are taken from Eratosthenes' own writing. The letter, which occupies an important place in the history of mathematics, is discussed in detail in [14]. An original Arabic text of this letter was once kept in the library of the St Joseph University in Beirut. However it has now vanished and the details given in [14] come from photographs taken of the letter before its disappearance.

Other details of what Eratosthenes wrote in Platonicus are given by Theon of Smyrna. In particular he described there the history of the problem of duplicating the cube (see Heath [4]):-

... when the god proclaimed to the Delians through the oracle that, in order to get rid of a plague, they should construct an alter double that of the existing one, their craftsmen fell into great perplexity in their efforts to discover how a solid could be made the double of a similar solid; they therefore went to ask Plato about it, and he replied that the oracle meant, not that the god wanted an alter of double the size, but that he wished, in setting them the task, to shame the Greeks for their neglect of mathematics and their contempt of geometry.

Eratosthenes erected a column at Alexandria with an epigram inscribed on it relating to his own mechanical solution to the problem of doubling the cube [4]:-

If, good friend, thou mindest to obtain from any small cube a cube the double of it, and duly to change any solid figure into another, this is in thy power; thou canst find the measure of a fold, a pit, or the broad basin of a hollow well, by this method, that is, if thou thus catch between two rulers two means with their extreme ends converging. Do not thou seek to do the difficult business of Archytas's cylinders, or to cut the cone in the triads of Menaechmus, or to compass such a curved form of lines as is described by the god-fearing Eudoxus. Nay thou couldst, on these tablets, easily find a myriad of means, beginning from a small base. Happy art thou, Ptolemy, in that, as a father the equal of his son in youthful vigour, thou hast thyself given him all that is dear to muses and Kings, and may be in the future, O Zeus, god of heaven, also receive the sceptre at thy hands. Thus may it be, and let any one who sees this offering say "This is the gift of Eratosthenes of Cyrene".

Eratosthenes also worked on prime numbers. He is remembered for his prime number sieve, the 'Sieve of Eratosthenes' which, in modified form, is still an important tool in number theory research. The sieve appears in the Introduction to arithmetic by Nicomedes.

Another book written by Eratosthenes was On means and, although it is now lost, it is mentioned by Pappus as one of the great books of geometry. In the field of geodesy, however, Eratosthenes will always be remembered for his measurements of the Earth.

Eratosthenes made a surprisingly accurate measurement of the circumference of the Earth. Details were given in his treatise On the measurement of the Earth which is now lost. However, some details of these calculations appear in works by other authors such as Cleomedes, Theon of Smyrna and Strabo. Eratosthenes compared the noon shadow at midsummer between Syene (now Aswan on the Nile in Egypt) and Alexandria. He assumed that the sun was so far away that its rays were essentially parallel, and then with a knowledge of the distance between Syene and Alexandria, he gave the length of the circumference of the Earth as 250,000 stadia.

Of course how accurate this value is depends on the length of the stadium and scholars have argued over this for a long time. The article [11] discusses the various values scholars have given for the stadium. It is certainly true that Eratosthenes obtained a good result, even a remarkable result if one takes 157.2 metres for the stadium as some have deduced from values given by Pliny. It is less good if 166.7 metres was the value used by Eratosthenes as Gulbekian suggests in [11].

Several of the papers referenced, for example [10], [15] and [16], discuss the accuracy of Eratosthenes' result. The paper [15] is particularly interesting. In it Rawlins argues convincingly that the only measurement which Eratosthenes made himself in his calculations was the zenith distance on the summer solstice at Alexandria, and that he obtained the value of 7°12'. Rawlins argues that this is in error by 16' while other data which Eratosthenes used, from unknown sources, was considerably more accurate.

Eratosthenes also measured the distance to the sun as 804,000,000 stadia and the distance to the Moon as 780,000 stadia. He computed these distances using data obtained during lunar eclipses. Ptolemy tells us that Eratosthenes measured the tilt of the Earth's axis with great accuracy obtaining the value of 11/83 of 180°, namely 23° 51' 15".

The value 11/83 has fascinated historians of mathematics, for example the papers [9] and [17] are written just to examine the source of this value. Perhaps the most commonly held view is that the value 11/83 is due to Ptolemy and not to Eratosthenes. Heath [4] argues that Eratosthenes used 24° and that 11/83 of 180° was a refinement due to Ptolemy. Taisbak [17] agrees with attributing 11/83 to Ptolemy although he believes that Eratosthenes used the value 2/15 of 180°. However Rawlins [15] believes that a continued fraction method was used to calculate the value 11/83 while Fowler [9] proposes that the anthyphairesis (or Euclidean algorithm) method was used (see also [3]).

Eratosthenes made many other major contributions to the progress of science. He worked out a calendar that included leap years, and he laid the foundations of a systematic chronography of the world when he tried to give the dates of literary and political events from the time of the siege of Troy. He is also said to have compiled a star catalogue containing 675 stars.

Eratosthenes made major contributions to geography. He sketched, quite accurately, the route of the Nile to Khartoum, showing the two Ethiopian tributaries. He also suggested that lakes were the source of the river. A study of the Nile had been made by many scholars before Eratosthenes and they had attempted to explain the rather strange behaviour of the river, but most like Thales were quite wrong in their explanations. Eratosthenes was the first to give what is essentially the correct answer when he suggested that heavy rains sometimes fell in regions near the source of the river and that these would explain the flooding lower down the river. Another contribution that Eratosthenes made to geography was his description of the region "Eudaimon Arabia", now the Yemen, as inhabited by four different races. The situation was somewhat more complicated than that proposed by Eratosthenes, but today the names for the races proposed by Eratosthenes, namely Minaeans, Sabaeans, Qatabanians, and Hadramites, are still used.

Eratosthenes writings include the poem Hermes, inspired by astronomy, as well as literary works on the theatre and on ethics which was a favourite topic of the Greeks. Eratosthenes is said to have became blind in old age and it has been claimed that he committed suicide by starvation.

ERATOSTHENES

ERATOSTHENES: The Librarian Who Measured the Earth

Archimedes of Syracuse

SCIENTIST ARCHIMEDES:

Archimedes' father was Phidias, an astronomer. We know nothing else about Phidias other than this one fact and we only know this since Archimedes gives us this information in one of his works, The Sandreckoner. A friend of Archimedes called Heracleides wrote a biography of him but sadly this work is lost. How our knowledge of Archimedes would be transformed if this lost work were ever found, or even extracts found in the writing of others.

Archimedes was a native of Syracuse, Sicily. It is reported by some authors that he visited Egypt and there invented a device now known as Archimedes' screw. This is a pump, still used in many parts of the world. It is highly likely that, when he was a young man, Archimedes studied with the successors of Euclid in Alexandria. Certainly he was completely familiar with the mathematics developed there, but what makes this conjecture much more certain, he knew personally the mathematicians working there and he sent his results to Alexandria with personal messages. He regarded Conon of Samos, one of the mathematicians at Alexandria, both very highly for his abilities as a mathematician and he also regarded him as a close friend.

In the preface to On spirals Archimedes relates an amusing story regarding his friends in Alexandria. He tells us that he was in the habit of sending them statements of his latest theorems, but without giving proofs. Apparently some of the mathematicians there had claimed the results as their own so Archimedes says that on the last occasion when he sent them theorems he included two which were false [3]:-

... so that those who claim to discover everything, but produce no proofs of the same, may be confuted as having pretended to discover the impossible.

Other than in the prefaces to his works, information about Archimedes comes to us from a number of sources such as in stories from Plutarch, Livy, and others. Plutarch tells us that Archimedes was related to King Hieron II of Syracuse (see for example [3]):-

Archimedes ... in writing to King Hiero, whose friend and near relation he was....

Again evidence of at least his friendship with the family of King Hieron II comes from the fact that The Sandreckoner was dedicated to Gelon, the son of King Hieron.

There are, in fact, quite a number of references to Archimedes in the writings of the time for he had gained a reputation in his own time which few other mathematicians of this period achieved. The reason for this was not a widespread interest in new mathematical ideas but rather that Archimedes had invented many machines which were used as engines of war. These were particularly effective in the defence of Syracuse when it was attacked by the Romans under the command of Marcellus.

Plutarch writes in his work on Marcellus, the Roman commander, about how Archimedes' engines of war were used against the Romans in the siege of 212 BC:-

... when Archimedes began to ply his engines, he at once shot against the land forces all sorts of missile weapons, and immense masses of stone that came down with incredible noise and violence; against which no man could stand; for they knocked down those upon whom they fell in heaps, breaking all their ranks and files. In the meantime huge poles thrust out from the walls over the ships and sunk some by great weights which they let down from on high upon them; others they lifted up into the air by an iron hand or beak like a crane's beak and, when they had drawn them up by the prow, and set them on end upon the poop, they plunged them to the bottom of the sea; or else the ships, drawn by engines within, and whirled about, were dashed against steep rocks that stood jutting out under the walls, with great destruction of the soldiers that were aboard them. A ship was frequently lifted up to a great height in the air (a dreadful thing to behold), and was rolled to and fro, and kept swinging, until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall.

Archimedes had been persuaded by his friend and relation King Hieron to build such machines:-

These machines [Archimedes] had designed and contrived, not as matters of any importance, but as mere amusements in geometry; in compliance with King Hiero's desire and request, some little time before, that he should reduce to practice some part of his admirable speculation in science, and by accommodating the theoretic truth to sensation and ordinary use, bring it more within the appreciation of the people in general.

Perhaps it is sad that engines of war were appreciated by the people of this time in a way that theoretical mathematics was not, but one would have to remark that the world is not a very different place at the end of the second millenium AD. Other inventions of Archimedes such as the compound pulley also brought him great fame among his contemporaries. Again we quote Plutarch:-

[Archimedes] had stated [in a letter to King Hieron] that given the force, any given weight might be moved, and even boasted, we are told, relying on the strength of demonstration, that if there were another earth, by going into it he could remove this. Hiero being struck with amazement at this, and entreating him to make good this problem by actual experiment, and show some great weight moved by a small engine, he fixed accordingly upon a ship of burden out of the king's arsenal, which could not be drawn out of the dock without great labour and many men; and, loading her with many passengers and a full freight, sitting himself the while far off, with no great endeavour, but only holding the head of the pulley in his hand and drawing the cords by degrees, he drew the ship in a straight line, as smoothly and evenly as if she had been in the sea.

Yet Archimedes, although he achieved fame by his mechanical inventions, believed that pure mathematics was the only worthy pursuit. Again Plutarch describes beautifully Archimedes attitude, yet we shall see later that Archimedes did in fact use some very practical methods to discover results from pure geometry:-

Archimedes possessed so high a spirit, so profound a soul, and such treasures of scientific knowledge, that though these inventions had now obtained him the renown of more than human sagacity, he yet would not deign to leave behind him any commentary or writing on such subjects; but, repudiating as sordid and ignoble the whole trade of engineering, and every sort of art that lends itself to mere use and profit, he placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life; studies, the superiority of which to all others is unquestioned, and in which the only doubt can be whether the beauty and grandeur of the subjects examined, of the precision and cogency of the methods and means of proof, most deserve our admiration.

His fascination with geometry is beautifully described by Plutarch:-

Oftimes Archimedes' servants got him against his will to the baths, to wash and anoint him, and yet being there, he would ever be drawing out of the geometrical figures, even in the very embers of the chimney. And while they were anointing of him with oils and sweet savours, with his fingers he drew lines upon his naked body, so far was he taken from himself, and brought into ecstasy or trance, with the delight he had in the study of geometry.

The achievements of Archimedes are quite outstanding. He is considered by most historians of mathematics as one of the greatest mathematicians of all time. He perfected a method of integration which allowed him to find areas, volumes and surface areas of many bodies. Chasles said that Archimedes' work on integration (see [7]):-

... gave birth to the calculus of the infinite conceived and brought to perfection by Kepler, Cavalieri, Fermat, Leibniz and Newton.

Archimedes was able to apply the method of exhaustion, which is the early form of integration, to obtain a whole range of important results and we mention some of these in the descriptions of his works below. Archimedes also gave an accurate approximation to π and showed that he could approximate square roots accurately. He invented a system for expressing large numbers. In mechanics Archimedes discovered fundamental theorems concerning the centre of gravity of plane figures and solids. His most famous theorem gives the weight of a body immersed in a liquid, called Archimedes' principle.

The works of Archimedes which have survived are as follows. On plane equilibriums (two books), Quadrature of the parabola, On the sphere and cylinder (two books), On spirals, On conoids and spheroids, On floating bodies (two books), Measurement of a circle, and The Sandreckoner. In the summer of 1906, J L Heiberg, professor of classical philology at the University of Copenhagen, discovered a 10th century manuscript which included Archimedes' work The method. This provides a remarkable insight into how Archimedes discovered many of his results and we will discuss this below once we have given further details of what is in the surviving books.

The order in which Archimedes wrote his works is not known for certain. We have used the chronological order suggested by Heath in [7] in listing these works above, except for The Method which Heath has placed immediately before On the sphere and cylinder. The paper [47] looks at arguments for a different chronological order of Archimedes' works.

The treatise On plane equilibriums sets out the fundamental principles of mechanics, using the methods of geometry. Archimedes discovered fundamental theorems concerning the centre of gravity of plane figures and these are given in this work. In particular he finds, in book 1, the centre of gravity of a parallelogram, a triangle, and a trapezium. Book two is devoted entirely to finding the centre of gravity of a segment of a parabola. In the Quadrature of the parabola Archimedes finds the area of a segment of a parabola cut off by any chord.

In the first book of On the sphere and cylinder Archimedes shows that the surface of a sphere is four times that of a great circle, he finds the area of any segment of a sphere, he shows that the volume of a sphere is two-thirds the volume of a circumscribed cylinder, and that the surface of a sphere is two-thirds the surface of a circumscribed cylinder including its bases. A good discussion of how Archimedes may have been led to some of these results using infinitesimals is given in [14]. In the second book of this work Archimedes' most important result is to show how to cut a given sphere by a plane so that the ratio of the volumes of the two segments has a prescribed ratio.

In On spirals Archimedes defines a spiral, he gives fundamental properties connecting the length of the radius vector with the angles through which it has revolved. He gives results on tangents to the spiral as well as finding the area of portions of the spiral. In the work On conoids and spheroids Archimedes examines paraboloids of revolution, hyperboloids of revolution, and spheroids obtained by rotating an ellipse either about its major axis or about its minor axis. The main purpose of the work is to investigate the volume of segments of these three-dimensional figures. Some claim there is a lack of rigour in certain of the results of this work but the interesting discussion in [43] attributes this to a modern day reconstruction.

On floating bodies is a work in which Archimedes lays down the basic principles of hydrostatics. His most famous theorem which gives the weight of a body immersed in a liquid, called Archimedes' principle, is contained in this work. He also studied the stability of various floating bodies of different shapes and different specific gravities. In Measurement of the Circle Archimedes shows that the exact value of π lies between the values 310/71 and 31/7. This he obtained by circumscribing and inscribing a circle with regular polygons having 96 sides.

The Sandreckoner is a remarkable work in which Archimedes proposes a number system capable of expressing numbers up to 8 × 1063 in modern notation. He argues in this work that this number is large enough to count the number of grains of sand which could be fitted into the universe. There are also important historical remarks in this work, for Archimedes has to give the dimensions of the universe to be able to count the number of grains of sand which it could contain. He states that Aristarchus has proposed a system with the sun at the centre and the planets, including the Earth, revolving round it. In quoting results on the dimensions he states results due to Eudoxus, Phidias (his father), and to Aristarchus. There are other sources which mention Archimedes' work on distances to the heavenly bodies. For example in [59] Osborne reconstructs and discusses:-

...a theory of the distances of the heavenly bodies ascribed to Archimedes, but the corrupt state of the numerals in the sole surviving manuscript [due to Hippolytus of Rome, about 220 AD] means that the material is difficult to handle.

In the Method, Archimedes described the way in which he discovered many of his geometrical results (see [7]):-

... certain things first became clear to me by a mechanical method, although they had to be proved by geometry afterwards because their investigation by the said method did not furnish an actual proof. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.

Perhaps the brilliance of Archimedes' geometrical results is best summed up by Plutarch, who writes:-

It is not possible to find in all geometry more difficult and intricate questions, or more simple and lucid explanations. Some ascribe this to his natural genius; while others think that incredible effort and toil produced these, to all appearances, easy and unlaboured results. No amount of investigation of yours would succeed in attaining the proof, and yet, once seen, you immediately believe you would have discovered it; by so smooth and so rapid a path he leads you to the conclusion required.

Heath adds his opinion of the quality of Archimedes' work [7]:-

The treatises are, without exception, monuments of mathematical exposition; the gradual revelation of the plan of attack, the masterly ordering of the propositions, the stern elimination of everything not immediately relevant to the purpose, the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.

There are references to other works of Archimedes which are now lost. Pappus refers to a work by Archimedes on semi-regular polyhedra, Archimedes himself refers to a work on the number system which he proposed in the Sandreckoner, Pappus mentions a treatise On balances and levers, and Theon mentions a treatise by Archimedes about mirrors. Evidence for further lost works are discussed in [67] but the evidence is not totally convincing.

Archimedes was killed in 212 BC during the capture of Syracuse by the Romans in the Second Punic War after all his efforts to keep the Romans at bay with his machines of war had failed. Plutarch recounts three versions of the story of his killing which had come down to him. The first version:-

Archimedes ... was ..., as fate would have it, intent upon working out some problem by a diagram, and having fixed his mind alike and his eyes upon the subject of his speculation, he never noticed the incursion of the Romans, nor that the city was taken. In this transport of study and contemplation, a soldier, unexpectedly coming up to him, commanded him to follow to Marcellus; which he declining to do before he had worked out his problem to a demonstration, the soldier, enraged, drew his sword and ran him through.

The second version:-

... a Roman soldier, running upon him with a drawn sword, offered to kill him; and that Archimedes, looking back, earnestly besought him to hold his hand a little while, that he might not leave what he was then at work upon inconclusive and imperfect; but the soldier, nothing moved by his entreaty, instantly killed him.

Finally, the third version that Plutarch had heard:-

... as Archimedes was carrying to Marcellus mathematical instruments, dials, spheres, and angles, by which the magnitude of the sun might be measured to the sight, some soldiers seeing him, and thinking that he carried gold in a vessel, slew him.

Archimedes considered his most significant accomplishments were those concerning a cylinder circumscribing a sphere, and he asked for a representation of this together with his result on the ratio of the two, to be inscribed on his tomb. Cicero was in Sicily in 75 BC and he writes how he searched for Archimedes tomb (see for example [1]):-

... and found it enclosed all around and covered with brambles and thickets; for I remembered certain doggerel lines inscribed, as I had heard, upon his tomb, which stated that a sphere along with a cylinder had been put on top of his grave. Accordingly, after taking a good look all around ..., I noticed a small column arising a little above the bushes, on which there was a figure of a sphere and a cylinder... . Slaves were sent in with sickles ... and when a passage to the place was opened we approached the pedestal in front of us; the epigram was traceable with about half of the lines legible, as the latter portion was worn away.

It is perhaps surprising that the mathematical works of Archimedes were relatively little known immediately after his death. As Clagett writes in [1]:-

Unlike the Elements of Euclid, the works of Archimedes were not widely known in antiquity. ... It is true that ... individual works of Archimedes were obviously studied at Alexandria, since Archimedes was often quoted by three eminent mathematicians of Alexandria: Heron, Pappus and Theon.

Only after Eutocius brought out editions of some of Archimedes works, with commentaries, in the sixth century AD were the remarkable treatises to become more widely known. Finally, it is worth remarking that the test used today to determine how close to the original text the various versions of his treatises of Archimedes are, is to determine whether they have retained Archimedes' Dorian dialect.

ARCHIMEDES

Archimedes Of Syracuse

Archimedes' secrets 4/5

Euclid

SCIENTIST EUCLID:

Euclid of Alexandria is the most prominent mathematician of antiquity best known for his treatise on mathematics The Elements. The long lasting nature of The Elements must make Euclid the leading mathematics teacher of all time. However little is known of Euclid's life except that he taught at Alexandria in Egypt. Proclus, the last major Greek philosopher, who lived around 450 AD wrote (see [1] or [9] or many other sources):-

Not much younger than these [pupils of Plato] is Euclid, who put together the "Elements", arranging in order many of Eudoxus's theorems, perfecting many of Theaetetus's, and also bringing to irrefutable demonstration the things which had been only loosely proved by his predecessors. This man lived in the time of the first Ptolemy; for Archimedes, who followed closely upon the first Ptolemy makes mention of Euclid, and further they say that Ptolemy once asked him if there were a shorted way to study geometry than the Elements, to which he replied that there was no royal road to geometry. He is therefore younger than Plato's circle, but older than Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes somewhere says. In his aim he was a Platonist, being in sympathy with this philosophy, whence he made the end of the whole "Elements" the construction of the so-called Platonic figures.

There is other information about Euclid given by certain authors but it is not thought to be reliable. Two different types of this extra information exists. The first type of extra information is that given by Arabian authors who state that Euclid was the son of Naucrates and that he was born in Tyre. It is believed by historians of mathematics that this is entirely fictitious and was merely invented by the authors.

The second type of information is that Euclid was born at Megara. This is due to an error on the part of the authors who first gave this information. In fact there was a Euclid of Megara, who was a philosopher who lived about 100 years before the mathematician Euclid of Alexandria. It is not quite the coincidence that it might seem that there were two learned men called Euclid. In fact Euclid was a very common name around this period and this is one further complication that makes it difficult to discover information concerning Euclid of Alexandria since there are references to numerous men called Euclid in the literature of this period.

Returning to the quotation from Proclus given above, the first point to make is that there is nothing inconsistent in the dating given. However, although we do not know for certain exactly what reference to Euclid in Archimedes' work Proclus is referring to, in what has come down to us there is only one reference to Euclid and this occurs in On the sphere and the cylinder. The obvious conclusion, therefore, is that all is well with the argument of Proclus and this was assumed until challenged by Hjelmslev in [48]. He argued that the reference to Euclid was added to Archimedes' book at a later stage, and indeed it is a rather surprising reference. It was not the tradition of the time to give such references, moreover there are many other places in Archimedes where it would be appropriate to refer to Euclid and there is no such reference. Despite Hjelmslev's claims that the passage has been added later, Bulmer-Thomas writes in [1]:-

Although it is no longer possible to rely on this reference, a general consideration of Euclid's works ... still shows that he must have written after such pupils of Plato as Eudoxus and before Archimedes.

For further discussion on dating Euclid, see for example [8]. This is far from an end to the arguments about Euclid the mathematician. The situation is best summed up by Itard [11] who gives three possible hypotheses.

(i) Euclid was an historical character who wrote the Elements and the other works attributed to him.

(ii) Euclid was the leader of a team of mathematicians working at Alexandria. They all contributed to writing the 'complete works of Euclid', even continuing to write books under Euclid's name after his death.

(iii) Euclid was not an historical character. The 'complete works of Euclid' were written by a team of mathematicians at Alexandria who took the name Euclid from the historical character Euclid of Megara who had lived about 100 years earlier.

It is worth remarking that Itard, who accepts Hjelmslev's claims that the passage about Euclid was added to Archimedes, favours the second of the three possibilities that we listed above. We should, however, make some comments on the three possibilities which, it is fair to say, sum up pretty well all possible current theories.

There is some strong evidence to accept (i). It was accepted without question by everyone for over 2000 years and there is little evidence which is inconsistent with this hypothesis. It is true that there are differences in style between some of the books of the Elements yet many authors vary their style. Again the fact that Euclid undoubtedly based the Elements on previous works means that it would be rather remarkable if no trace of the style of the original author remained.

Even if we accept (i) then there is little doubt that Euclid built up a vigorous school of mathematics at Alexandria. He therefore would have had some able pupils who may have helped out in writing the books. However hypothesis (ii) goes much further than this and would suggest that different books were written by different mathematicians. Other than the differences in style referred to above, there is little direct evidence of this.

Although on the face of it (iii) might seem the most fanciful of the three suggestions, nevertheless the 20th century example of Bourbaki shows that it is far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley and Alexander Grothendieck wrote collectively under the name of Bourbaki and Bourbaki's Eléments de mathématiques contains more than 30 volumes. Of course if (iii) were the correct hypothesis then Apollonius, who studied with the pupils of Euclid in Alexandria, must have known there was no person 'Euclid' but the fact that he wrote:-

.... Euclid did not work out the syntheses of the locus with respect to three and four lines, but only a chance portion of it ...

certainly does not prove that Euclid was an historical character since there are many similar references to Bourbaki by mathematicians who knew perfectly well that Bourbaki was fictitious. Nevertheless the mathematicians who made up the Bourbaki team are all well known in their own right and this may be the greatest argument against hypothesis (iii) in that the 'Euclid team' would have to have consisted of outstanding mathematicians. So who were they?

We shall assume in this article that hypothesis (i) is true but, having no knowledge of Euclid, we must concentrate on his works after making a few comments on possible historical events. Euclid must have studied in Plato's Academy in Athens to have learnt of the geometry of Eudoxus and Theaetetus of which he was so familiar.

None of Euclid's works have a preface, at least none has come down to us so it is highly unlikely that any ever existed, so we cannot see any of his character, as we can of some other Greek mathematicians, from the nature of their prefaces. Pappus writes (see for example [1]) that Euclid was:-

... most fair and well disposed towards all who were able in any measure to advance mathematics, careful in no way to give offence, and although an exact scholar not vaunting himself.

Some claim these words have been added to Pappus, and certainly the point of the passage (in a continuation which we have not quoted) is to speak harshly (and almost certainly unfairly) of Apollonius. The picture of Euclid drawn by Pappus is, however, certainly in line with the evidence from his mathematical texts. Another story told by Stobaeus [9] is the following:-

... someone who had begun to learn geometry with Euclid, when he had learnt the first theorem, asked Euclid "What shall I get by learning these things?" Euclid called his slave and said "Give him threepence since he must make gain out of what he learns".

Euclid's most famous work is his treatise on mathematics The Elements. The book was a compilation of knowledge that became the centre of mathematical teaching for 2000 years. Probably no results in The Elements were first proved by Euclid but the organisation of the material and its exposition are certainly due to him. In fact there is ample evidence that Euclid is using earlier textbooks as he writes the Elements since he introduces quite a number of definitions which are never used such as that of an oblong, a rhombus, and a rhomboid.

The Elements begins with definitions and five postulates. The first three postulates are postulates of construction, for example the first postulate states that it is possible to draw a straight line between any two points. These postulates also implicitly assume the existence of points, lines and circles and then the existence of other geometric objects are deduced from the fact that these exist. There are other assumptions in the postulates which are not explicit. For example it is assumed that there is a unique line joining any two points. Similarly postulates two and three, on producing straight lines and drawing circles, respectively, assume the uniqueness of the objects the possibility of whose construction is being postulated.

The fourth and fifth postulates are of a different nature. Postulate four states that all right angles are equal. This may seem "obvious" but it actually assumes that space in homogeneous - by this we mean that a figure will be independent of the position in space in which it is placed. The famous fifth, or parallel, postulate states that one and only one line can be drawn through a point parallel to a given line. Euclid's decision to make this a postulate led to Euclidean geometry. It was not until the 19th century that this postulate was dropped and non-euclidean geometries were studied.

There are also axioms which Euclid calls 'common notions'. These are not specific geometrical properties but rather general assumptions which allow mathematics to proceed as a deductive science. For example:-

Things which are equal to the same thing are equal to each other.

Zeno of Sidon, about 250 years after Euclid wrote the Elements, seems to have been the first to show that Euclid's propositions were not deduced from the postulates and axioms alone, and Euclid does make other subtle assumptions.

The Elements is divided into 13 books. Books one to six deal with plane geometry. In particular books one and two set out basic properties of triangles, parallels, parallelograms, rectangles and squares. Book three studies properties of the circle while book four deals with problems about circles and is thought largely to set out work of the followers of Pythagoras. Book five lays out the work of Eudoxus on proportion applied to commensurable and incommensurable magnitudes. Heath says [9]:-

Greek mathematics can boast no finer discovery than this theory, which put on a sound footing so much of geometry as depended on the use of proportion.

Book six looks at applications of the results of book five to plane geometry.

Books seven to nine deal with number theory. In particular book seven is a self-contained introduction to number theory and contains the Euclidean algorithm for finding the greatest common divisor of two numbers. Book eight looks at numbers in geometrical progression but van der Waerden writes in [2] that it contains:-

... cumbersome enunciations, needless repetitions, and even logical fallacies. Apparently Euclid's exposition excelled only in those parts in which he had excellent sources at his disposal.

Book ten deals with the theory of irrational numbers and is mainly the work of Theaetetus. Euclid changed the proofs of several theorems in this book so that they fitted the new definition of proportion given by Eudoxus.

Books eleven to thirteen deal with three-dimensional geometry. In book eleven the basic definitions needed for the three books together are given. The theorems then follow a fairly similar pattern to the two-dimensional analogues previously given in books one and four. The main results of book twelve are that circles are to one another as the squares of their diameters and that spheres are to each other as the cubes of their diameters. These results are certainly due to Eudoxus. Euclid proves these theorems using the "method of exhaustion" as invented by Eudoxus. The Elements ends with book thirteen which discusses the properties of the five regular polyhedra and gives a proof that there are precisely five. This book appears to be based largely on an earlier treatise by Theaetetus.

Euclid's Elements is remarkable for the clarity with which the theorems are stated and proved. The standard of rigour was to become a goal for the inventors of the calculus centuries later. As Heath writes in [9]:-

This wonderful book, with all its imperfections, which are indeed slight enough when account is taken of the date it appeared, is and will doubtless remain the greatest mathematical textbook of all time. ... Even in Greek times the most accomplished mathematicians occupied themselves with it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, altering the language here and there, mostly with a view to greater clearness and consistency...

It is a fascinating story how the Elements has survived from Euclid's time and this is told well by Fowler in [7]. He describes the earliest material relating to the Elements which has survived:-

Our earliest glimpse of Euclidean material will be the most remarkable for a thousand years, six fragmentary ostraca containing text and a figure ... found on Elephantine Island in 1906/07 and 1907/08... These texts are early, though still more than 100 years after the death of Plato (they are dated on palaeographic grounds to the third quarter of the third century BC); advanced (they deal with the results found in the "Elements" [book thirteen] ... on the pentagon, hexagon, decagon, and icosahedron); and they do not follow the text of the Elements. ... So they give evidence of someone in the third century BC, located more than 500 miles south of Alexandria, working through this difficult material... this may be an attempt to understand the mathematics, and not a slavish copying ...

The next fragment that we have dates from 75 - 125 AD and again appears to be notes by someone trying to understand the material of the Elements.

More than one thousand editions of The Elements have been published since it was first printed in 1482. Heath [9] discusses many of the editions and describes the likely changes to the text over the years.

B L van der Waerden assesses the importance of the Elements in [2]:-

Almost from the time of its writing and lasting almost to the present, the Elements has exerted a continuous and major influence on human affairs. It was the primary source of geometric reasoning, theorems, and methods at least until the advent of non-Euclidean geometry in the 19th century. It is sometimes said that, next to the Bible, the "Elements" may be the most translated, published, and studied of all the books produced in the Western world.

Euclid also wrote the following books which have survived: Data (with 94 propositions), which looks at what properties of figures can be deduced when other properties are given; On Divisions which looks at constructions to divide a figure into two parts with areas of given ratio; Optics which is the first Greek work on perspective; and Phaenomena which is an elementary introduction to mathematical astronomy and gives results on the times stars in certain positions will rise and set. Euclid's following books have all been lost: Surface Loci (two books), Porisms (a three book work with, according to Pappus, 171 theorems and 38 lemmas), Conics (four books), Book of Fallacies and Elements of Music. The Book of Fallacies is described by Proclus [1]:-

Since many things seem to conform with the truth and to follow from scientific principles, but lead astray from the principles and deceive the more superficial, [Euclid] has handed down methods for the clear-sighted understanding of these matters also ... The treatise in which he gave this machinery to us is entitled Fallacies, enumerating in order the various kinds, exercising our intelligence in each case by theorems of all sorts, setting the true side by side with the false, and combining the refutation of the error with practical illustration.

Elements of Music is a work which is attributed to Euclid by Proclus. We have two treatises on music which have survived, and have by some authors attributed to Euclid, but it is now thought that they are not the work on music referred to by Proclus.

Euclid may not have been a first class mathematician but the long lasting nature of The Elements must make him the leading mathematics teacher of antiquity or perhaps of all time. As a final personal note let me add that my [EFR] own introduction to mathematics at school in the 1950s was from an edition of part of Euclid's Elements and the work provided a logical basis for mathematics and the concept of proof which seem to be lacking in school mathematics today.

EUCLID

Euclid

MathFoundations19: Euclid's Elements