Archimedes

Archimedes

EDUARD JAN DIJKSTERHUIS
translated by C. Dikshoorn
With a new bibliographic essay by Wilbur R. Knorr
Copyright Date: 1987
Pages: 460
https://www.jstor.org/stable/j.ctt7ztpbp
  • Cite this Item
  • Book Info
    Archimedes
    Book Description:

    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.

    ThePrinceton Legacy Libraryuses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

    eISBN: 978-1-4008-5861-3
    Subjects: Mathematics, Philosophy

Table of Contents

  1. Front Matter
    (pp. 1-4)
  2. Table of Contents
    (pp. 5-6)
  3. PREFACE
    (pp. 7-8)
  4. CHAPTER I. THE LIFE OF ARCHIMEDES
    (pp. 9-32)

    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...

  5. CHAPTER II. THE WORKS OF ARCHIMEDES. MANUSCRIPTS AND EDITIONS
    (pp. 33-49)

    As we already saw, Archimedes was accustomed to send his mathematical writings to his colleagues at Alexandria: Conon, Dositheus, Eratosthenes, who apparently saw to their further distribution. The documents were accompanied by introductory letters, which contained a summary of the propositons to be proved and not infrequently referred back to statements made on earlier occasions. In this way we know some particulars about the order in which he made his discoveries and about the way in which he published them.

    It appears that he would usually first send the propositions only, with the request to discuss them with other mathematicians...

  6. CHAPTER III THE ELEMENTS OF THE WORK OF ARCHIMEDES
    (pp. 49-141)

    Owing to the completion of theElementsof Euclid, the Greek mathematicians had gained possession of a systematically arranged collection of the fundamental mathematical propositions on which they could base their further investigations. They were thus absolved from the obligation to revert in their works to matters of an elementary,i.e.fundamental nature: when a proposition was contained in theElements, it was sufficient, in view of the apparently universal diffusion of the work, merely to mention it.

    If the reader was to become acquainted with the work of a writer of Archimedes’ level, however, an understanding of theElements...

  7. CHAPTER IV. ON THE SPHERE AND CYLINDER. BOOK I.
    (pp. 141-187)

    The first of the two books into which the treatiseOn the Sphere and Cylinderis divided opens with a letter to Dositheus, in which Archimedes reminds him that on a former occasion he already sent him the proof of the proposition that any segment of an orthotome is four-thirds of the triangle with the same base and equal height¹). He is now going to demonstrate new propositions:

    a) The surface of any sphere is four times its greatest circle²).

    b) The surface of any segment of a sphere is equal to that of a circle whose radius is equal...

  8. CHAPTER V. ON THE SPHERE AND CYLINDER. Book II
    (pp. 188-221)

    The second book of the treatiseOn the Sphere and Cylinderopens again with a letter by Archimedes to Dositheus, in which he sums up once more the principal propositions of Book I, announces the solution of some related problems for Book II, and holds out a prospect of sending him soon the promised propositions on spirals and conoids.

    The first of the problems dealt with in Book II concerns the construction of a circle whose surface is equal to that of a given sphere; the solution of this problem follows at once from I, 33. This is followed by...

  9. CHAPTER VI. MEASUREMENT OF THE CIRCLE
    (pp. 222-240)

    1. The work on the measurement of the circle, in which Archimedes derives the ratio between the circumference and the diameter of a circle, which was to become one of the most popular results of his mathematical investigations, is a very short treatise comprising only three propositions. As appears both from its language, from which all traces of the Siculo-Dorian dialect have vanished, and from the argumentation, which is scrappy and rather careless, it has not come down to us in its original form. It is quite possible that the fragment we possess formed part of a longer work, which is...

  10. CHAPTER VII. ON CONOIDS AND SPHEROIDS
    (pp. 240-263)

    This treatise deals with theorems on the volumes of segments of conoids and spheroids,i.e.of the solids comprehended by a plane and the surface of either a right-angled conoid (i.e.a paraboloid of revolution) or an obtuse-angled conoid (i.e.a sheet of a hyperboloid of revolution of two sheets), or an oblong or flat spheroid (i.e.an ellipsoid of revolution). Each time Archimedes first discusses the case that the segment is right,i.e.that the cutting plane is at right angles to the axis of revolution, and subsequently devotes a new proposition to the oblique segment, with the cutting...

  11. CHAPTER VIII. ON SPIRALS
    (pp. 264-285)

    1. The discussion of the so-called Archimedean spiral, to which the whole of the treatiseOn Spiralsis devoted, after eleven introductory propositions, which have already been dealt with in Chapter III (10.1–2; 9; 7.30; 7.50), opens with a number of definitions, the first of which is rendered herein extenso.

    If in a plane a straight line be drawn and, while one of its extremities remains fixed, after performing any number of revolutions at a uniform rate return again to the position from which it started, while at the same time a point moves at a uniform rate along...

  12. CHAPTER IX. ON THE EQUILIBRIUM OF PLANES OR CENTRES OF GRAVITY OF PLANES. Book I.
    (pp. 286-313)

    1. The treatise on the equilibrium of planes occupies a place apart in the work of Archimedes. In fact, whereas in all his mathematical treatises he builds on foundations long ago established, in this work he concerns himself with an investigation into the very foundations; moreover he leaves the domain of pure mathematics for that of natural science considered from the mathematical point of view: he sets forth certain postulates on which he bases a chapter from the theory of equilibrium, and he is thus the first to establish the close interrelation between mathematics and mechanics, which was to become of...

  13. CHAPTER X THE METHOD OF MECHANICAL THEOREMS
    (pp. 313-336)

    1. We shall now, deviating from the order in which Archimedes’ works appear in Heiberg’s edition of the text, first discuss the treatiseThe Method of Mechanical Theorems, for Eratosthenes,to be briefly designated asThe Method. In fact, when we know this work, it is easier to understand theQuadrature of the Parabola, the contents of which in turn are assumed as known in Book II ofOn the Equilibrium of Planes.

    The discovery and decipherment of the manuscript of theMethodhas already been discussed in Chapter II. The object of the work becomes clear from the introductory letter...

  14. CHAPTER XI QUADRATURE OF THE PARABOLA
    (pp. 336-345)

    1. The theorem on the volume of the cylinder hoof already enabled us to see how Archimedes arrived, from a surmise gained by mechanical means with the aid of the method of indivisibles, at a mathematical proof which satisfied all his requirements of exactness.

    An even more beautiful example of such a logical confirmation of an intuitively gained insight is furnished by the theorem on the area of any segment of an orthotome, which formed the subject of the first proposition of theMethod. In fact, it was to the mathematical proof of this theorem that Archimedes devoted a separate treatise,...

  15. CHAPTER XII ON THE EQUILIBRIUM OF PLANES. Book II
    (pp. 346-360)

    The main object of this book is the determination of the centre of gravity of any segment of an orthotome, which object is attained in Prop. 8.

    This determination is a combined application of the principles of the barycentric theory and the theory of the quadrature of the orthotome, dealt with in Book I; it is further based on various properties of this curve, which we have already discussed in Chapter III.

    At first sight it seems rather strange to find the argument starting with a proposition which is nothing but a particular case of the lever principle proved in...

  16. CHAPTER XIII THE SAND-RECKONER
    (pp. 360-373)

    1. The workThe Sand-Reckoner, though meant by the author as a contribution to Greek arithmetic, owes its historical interest not only to what it contains as such; it is no less valuable as a document of Archimedes’ astronomical activity. It was of course to be expected that he engaged in astronomy, though he has not left any work exclusively devoted to it: astronomy and mathematics in his day were scarcely distinguished as two different branches of science; his father Pheidias had already cultivated this science, and his friend Conon had even gained a great reputation in it. He himself also...

  17. CHAPTER XIV FLOATING BODIES
    (pp. 373-398)

    1. As we saw in Chapter II, the Greek text of this work has only been known since 1899; however, it still exhibits considerable lacunae. Before that time scholars always had to make shift with a Latin translation by William of Moerbeke; the latter is still used to supply the undecipherable or missing parts of the Greek text.

    2. Book I of the work opens with a postulate the precise meaning of which may be subject to doubt and which on that account we quote in the original.

    ὑποϰείσϑω τὸ ὑγϱὸν φύσιν ἔχον τοιαύταν, ὥστε τῶν μεϱέων αὐτοῦ τῶν ἐξ ἴσου ϰειμένων...

  18. CHAPTER XV MISCELLANEOUS
    (pp. 398-416)

    In this last chapter we will discuss some minor treatises by Archimedes, which have only been preserved either in fragments or by references in other writers.

    This problem²), which was already famous in antiquity, is contained in an epigram which, as appears from the heading, was sent by Archimedes to Eratosthenes with instructions to submit it to the Alexandrian mathematicians. It is considered improbable by philologists that it was written by Archimedes himself in the form in which it has come down to us¹). This, however, does not exlude the possibility that the problem may quite well originate from him....

  19. BIBLIOGRAPHY
    (pp. 417-418)
  20. ARCHIMEDES AFTER DIJKSTERHUIS: A GUIDE TO RECENT STUDIES
    (pp. 419-451)
    WILBUR R. KNORR

    E. J. Dijksterhuis (1892–1965) gained wide respect for his historical studies in the exact sciences. These included major contributions on the work of the sixteenth-century Flemish physicist-engineer Simon Stevin (see Dijksterhuis [1955; 1970]), and most notably, his expansive survey of the history of physical science from antiquity to Newton,The Mechanization of the World Picture(1950; 1961). (For further particulars on Dijksterhuis’ life and work, I refer the reader to the admirable portraits by Struik [1986] and Hooykaas [1966].)

    In these works Dijksterhuis criticizes the ancients for building mechanical science on the foundation of formal geometry, thus elevating considerations...

  21. INDEX OF NAMES
    (pp. 452-455)
  22. ERRATA
    (pp. 456-458)
  23. Back Matter
    (pp. 459-459)