This classic study by the eminent Dutch historian of science E. J. Dijksterhuis (1892-1965) presents the work of the Greek mathematician and mechanical engineer to the modern reader. With meticulous scholarship, Dijksterhuis surveys the whole range of evidence on Archimedes' life and the 2000-year history of the manuscripts and editions of the text, and then undertakes a comprehensive examination of all the extant writings.
Originally published in 1987.
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By E. J. Dijksterhuis, C. Dikshoorn
PRINCETON UNIVERSITY PRESSCopyright © 1987 Princeton University Press
All rights reserved.
THE LIFE OF ARCHIMEDES
1. The Personality.
In our scanty knowledge of Greek mathematics the information available about the lives of those who were engaged in its pursuit is one of the weakest points. Reliable reports about the lives and fortunes of hardly any of them are in our possession. The period in which they worked can generally be only roughly approximated; the places where they dwelt are frequently altogether unknown.
To this general rule Archimedes is only an apparent exception. It is true that already in antiquity a great many stories about him were current, which have remained indissolubly bound up with his name down to our own time, but these reports are usually very fanciful and seldom stand the test of historical criticism. It is therefore no more possible to write a coherent and reliable biography) of Archimedes than of any other Greek mathematician. The writer cannot do much beyond arranging the traditional reports and accurately stating the sources from which they originate; only occasionally will it be possible to assess the value of these reports.
The year of his birth already is not established with absolute certainty; it is usually stated to be 287 B.C., because he is reported by the Byzantine polyhistor Tzetzes) to have been seventy-five when he was killed in 212 B.C. during the Roman conquest of Syracuse.
About his parentage, too, little can be said with certainty. All classical writers agree in calling him a Syracusan by birth, but whereas Cicero) and Silus Italicus) create the impression that he was poor and of humble birth, Plutarch) reports on his family connection and intimate relations with King Hieron II) of Syracuse, which seems to point to quite a different position in the life of the city. Nor need there be any absolute inconsistency between the two possibilities. Indeed, Hieron, who was said to be the illegitimate son of a nobleman by one of his female slaves, apparently owed his brilliant career to his personal merits rather than to hereditary privileges, and his original relations with Archimedes may very well have remained unchanged, the wise mathematician having no desire to share in the power and prosperity of the tyrant.
According to an originally incomprehensible statement by Archimedes himself (a passage in his work The Sand-reckoner, which is meaningless in the version of the manuscripts as handed down to us, but has been intelligently emended by F. Blass)) he was the son of an astronomer, Phidias, of whose work we know nothing beyond an estimation of the ratio of the diameters of sun and moon, which Archimedes refers to in the passage in question.
Besides these rather vague biographical particulars there are some others of a more positive nature, which, however, are not at all worthy of credit); the fact that an Arabian writer calls him a son of Pythagoras is merely mentioned here as an example of the unchecked fancy with which the Arabian mathematicians were wont to write about the lives of their Greek predecessors. And that, as Linceo Mirabello has it, he should have been a pupil of Plato's in his youth at Syracuse must be due to a chronological error, because Plato had been dead for sixty years when Archimedes was born. About his descendants there is a report which is equally strange: Rivault), who in 1615 edited a Latin translation of his collected works, says in his biographical introduction that he has heard from a learned Greek friend that the Sicilian martyr Santa Lucia had been a descendant of the great mathematician.
This much is certain about the life of Archimedes that he spent some time in Egypt on at least one occasion; it is not saying much that Arabian writers mention this, but Diodorus) confirms it in two passages of his Bibliotheca Historical), where he refers to the cochlias (a hydraulic machine to which we shall revert in § 4) as an invention of Archimedes, which he was said to have made in Egypt. Moreover it appears from the prefaces to his works that from Syracuse he always maintained very friendly relations with several scholars at Alexandria; it is quite natural to suppose that these ties date from the period when he stayed for his studies at the then centre of Greek science, which must always have exercised a peculiar attraction on mathematicians because at the Museum) the tradition of Euclid lived on. Of his Alexandrian colleagues he seems to have had the highest regard for the astronomer Conon) of Samos, to whom, until the latter's death, he used to send his mathematical discoveries before their publication, and about whom he always spoke with the greatest admiration). These scholars further included the many-sided Eratosthenes of Cyrene), for whom the Method was written, and Conon's pupil, Dositheus of Pelusium), to whom the works On the Sphere and Cylinder, On Conoids and Spheroids, and On Spirals are dedicated.
According to a statement in the biography with which J. Torelli introduces his great Archimedes edition) Archimedes is said after his return from Egypt to have visited other countries as well. There is in particular a story about a voyage to Spain in a note by Leonardo da Vinci), in which the latter mentions that he has read in a history of the Spaniards that the Syracusan Archimedes aided Ecliderides, King of the Cilodastri, in a maritime war against the English through the invention of a device for spouting burning pitch on to the ships of his opponents. I t is, however, altogether unknown in what work Leonardo can have read this, and authorities on Spanish history ignore both King Ecliderides and the people of the Cilodastri. In connection with the possibility of a voyage to Spain, a statement by Diodorus about the use in the Spanish silver mines of the cochlias, which he alleges had been invented by Archimedes), deserves some attention.
It is possible that Archimedes also returned once more to Egypt, and that on that occasion he constructed the great works concerned with dike and bridge building and regulation of the Nile about which there are some reports in Arabian sources). However, nothing can be said with certainty about this either, and considering the large amount of work of a mathematical and technical kind achieved by him in his native city it is rather more likely that he spent the greater part of his life at Syracuse itself, and that it was from there that his great fame spread in antiquity.
As is understandable from the nature of mathematical work, this fame is not based in the first place on the writings which he sent to his mathematical friends and which have secured him the admiration of the ages, ever since the revival of mathematical science. Even before this, his astronomical work was appreciated in wider circles. He is, however, mainly mentioned in non-mathematical literature on account of the manifestations of his technical ingenuity, and these manifestations appear to have made) a great impression in the militaristic Roman Empire chiefly because they made possible the military action — to be discussed more fully in § 5 — which he developed during the defence of Syracuse.
If Plutarch is to be believed, he himself seems to have considered the whole of his technical activity as an occupation of a lower order: as it is reported in the Life of Marcellus), "he did not deign to leave behind him any written work on such subjects; he regarded as sordid and ignoble the construction of instruments, and in general every art directed to use and profit, and he only strove after those things which, in their beauty and excellence, remain beyond all contact with the common needs of life".
In pure mathematics he found the possibility to satisfy this desire to the full, and if we rely once more on what Plutarch tells us further), there seldom lived anyone who was so much preoccupied with mathematics as he was: "continually bewitched by a Siren who always accompanied him, he forgot to nourish himself and omitted to care for his body; and when, as would often happen, he was urged by force to bathe and anoint himself, he would still be drawing geometrical figures in the ashes or with his finger would draw lines on his anointed body, being possessed by a great ecstasy and in truth a thrall to the Muses".
Elsewhere) Plutarch refers once more to the slight importance which Archimedes himself attached to his technical inventions: "most of them were the diversions of a geometry at play which he had practised formerly, when King Hieron had emphatically requested and persuaded him to direct his art a little away from the abstract and towards the concrete, and to reveal his mind to the ordinary man by occupying himself in some tangible manner with the demands of reality".
2. [TEXT NOT REPRODUCIBLE IN ASCII] ... (Give me a place to stand on ...).
An occasion on which Archimedes was able to comply with Hieron's request seems to have presented itself during the construction of the famous ship Syracusia (later Alexandras), which the king, who was known for his love of display and his propensity to have big projects carried out, had ordered to be built so that after its completion he might present it, loaded with articles of food, to King Ptolemy of Egypt. Detailed reports about the appointments of this famous ship, the size of which is estimated at 4,200 tons and which is said to have been fitted with all conceivable wonders of luxury and technology, are to be found in the Deipnosophistae of Athenaeus), who also states that its construction was directed by Archias of Corinth under the superintendence of Archimedes. He further relates that, when no one knew how to launch the ship, Archimedes succeeded in this all alone, with the aid of a few instruments. The same story is told by Proclus), who represents Hieron as operating the device himself and calling out in amazement: "From this day forth Archimedes is to be believed in everything that he may say").
In Plutarch the description of the launching of a big ship appears in a slightly different form). Archimedes is said to have declared to Hieron that it was possible to move a given weight by a given force. Hieron then invited him to demonstrate this on a ship from the royal fleet, which had been drawn on land with difficulty and had there been loaded with a large crew and the customary freight; upon which Archimedes with quiet movements of his hand put into operation a device which drew the ship to the sea as smoothly as if she were already moving through the water. The story has become famous in particular because of the familiar saying which Archimedes is reported to have uttered on this occasion, viz.: "Give me a place to stand on, and I will move the earth".
In the traditional form in which we have given them here the reports about the setting in motion of a ship are of course purely fantastic; devices capable of achieving what Archimedes must have required of them only exist in the ideal realm of rational mechanics, in which friction and resistance of the air are eliminated. It is, however, hardly to be doubted that the stories about it have a real basis in the invention or demonstration of a device by which force was economized in moving heavy weights. And thus it is after all to the point to speculate with the classical writers on the question what device Archimedes may, theoretically speaking, have used to draw the heavy ship into the sea, and how he proposed to lift the earth off its hinges if only he could have found a place to stand on while doing so.
The writers who deal with this question are anything but unanimous in their answers. Plutarch says) that the device used was a polyspaston ([TEXT NOT REPRODUCIBLE IN ASCII]), i.e. a tackle with a large number of sheaves in each of the two pulley blocks. Tzetzes) speaks in particular of a trispaston ([TEXT NOT REPRODUCIBLE IN ASCII]), i.e. a tackle with two pulley blocks, each of three sheaves. With this apparatus the ideal ratio between effort and load would be 1:6, which does not very well agree with his assertion that a weight of 50,000 medimns) of wheat had been moved with it. More casually the trispaston is also mentioned by the physician Oribasius); he states that Archimedes used the device to draw ships into the sea, and that the medical profession had begun to apply a small scale model of it for reducing fractures and dislocations.
Heron, on the contrary, for the general solution of the problem To move a given weight by a given force, discusses the so-called Barulcus, i.e. an instrument in which a windlass is rotated through a system of toothed wheels, the last of which is driven by an endless screw). Probably Athenaeus is thinking of such a device when in the Deipnosophistae he represents Archimedes as using a screw to move the ship Syracusia).
About the moving of the earth we read in Pappus), who mentions the familiar formulation quoted above ([TEXT NOT REPRODUCIBLE IN ASCII]); in Simplicius, in the commentary on the Physica of Aristotle), where he says that Archimedes, after having constructed a weighing machine, the so-called charistion ([TEXT NOT REPRODUCIBLE IN ASCII]), is reported to have exclaimed: [TEXT NOT REPRODUCIBLE IN ASCII] (a place to stand on! and I will move the earth); and in two passages in Tzetzes, who on one occassion mentions the trispaston), and on the other the charistion), and who represents Archimedes as having said in the Dorian dialect of Syracuse: [TEXT NOT REPRODUCIBLE IN ASCII] (a place to stand on! and I will move the entire earth by means of a charistion).
Now we have in the first place to ask ourselves what is to be understood by the charistion, the device which Simplicius calls a weighing machine, which Tzetzes does not define, but which is to be found described in Simon Stevin1) as comprising "assen met vijsen" (shafts with screws), so that he seems to imagine something in the style of the Barulcus. Now it can hardly be doubted that in this difference of opinion it is Simplicius who is right; indeed, it is fairly natural to suppose with P. Duhem) that the charistion is identical with the so-called charasto (also called carasto, canisto or baracto), which is discussed in the work Liber Charastonis, translated into Latin by Gherard of Cremona from the Arabic version of Thabit b. Qurra, and which is found to be the balance with unequal arms, which the Romans called statera, which is still known in French as balance romaine, and in English as steelyard. The term charistion, about whose origin there has been a good deal of controversy, may originally have been the name of the inventor or of the first describer of the device; this is at least the simplest way to account for the striking fact that a mechanical instrument bears a name derived from a word ([TEXT NOT REPRODUCIBLE IN ASCII]) which does not have the slightest connection with technical matters and which on the other hand frequently appears in proper names).
If this interpretation is correct, the hyperbolic words of Archimedes about the moving of the earth must have been inspired by the lever principle manifested in the charistion, which consists in the inverse proportion of effort and load to the distances from the fulcrum to their lines of action. And indeed, in view of the fact that in a work by ArcKimedes dealing with mechanics (On the Equilibrium, of Planes) the theory of the lever is discussed, the famous saying was in later days as a rule associated with this theory. Nevertheless it seems doubtful whether this is the right interpretation of the classical tradition; indeed, Plutarch and Heron, who speak about a polyspaston or a trispaston, and Proclus, who does not mention any particular apparatus at all, were acquainted with the properties of levers just as well as later writers, and there would not have been the slightest reason for their deviating from the tradition, if this had been unambiguous in its reports about the use of a charistion.
Excerpted from Archimedes by E. J. Dijksterhuis, C. Dikshoorn. Copyright © 1987 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents
- FrontMatter, pg. 1
- Contents, pg. 5
- Preface, pg. 7
- Chapter I. The Life of Archimedes, pg. 9
- Chapter II. The Works of Archimedes. Manuscripts and Edition, pg. 33
- Chapter III. The Elements of the Work of Archimedes, pg. 49
- Chapter IV. On the Sphere and Cylinder. Book 1, pg. 141
- Chapter V. On the Sphere and Cylinder. Book II, pg. 188
- Chapter VI. Measurement of a Circle, pg. 222
- Chapter VII. On Conoids and Spheroids, pg. 240
- Chapter VIII. On Spirals, pg. 264
- Chapter IX. On the Equilibrium of Planes or Centres of Gravity of Planes. Book I, pg. 286
- Chapter X. The Method of Mechanical Theorems, pg. 313
- Chapter XI. Quadrature of the Parabola, pg. 336
- Chapter XII. On the Equilibrium of Planes. Book II, pg. 346
- Chapter XIII. The Sand-Reckoner, pg. 360
- Chapter XIV. Floating Bodies, pg. 373
- Chapter XV. Miscellaneous, pg. 398
- Bibliography, pg. 417
- Archimedes after Dijksterhuis: A Guide to Recent Studies, pg. 419
- Index of Names, pg. 452
- Errata, pg. 456