Archimedes

Archimedes of Syracuse (c. 287 BC – c. 212 BC) was a Greek mathematician, philosopher, scientist and engineer.

Give me a place to stand, and I shall move the world.

The Death of Archimedes (1815) by Thomas Degeorge

Quotes

• εὕρηκα [heúrēka]
  • I have found it! or I have got it!, commonly quoted as Eureka!
  • What he exclaimed as he ran naked from his bath, realizing that by measuring the displacement of water an object produced, compared to its weight, he could measure its density (and thus determine the proportion of gold that was used in making a king's crown); as quoted by Vitruvius Pollio in De Architectura, ix.215;

• δῶς^{[citation needed]} μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω. [Dôs moi pâ stô, kaì tàn gân kinásō.]
  • Give me the place to stand, and I shall move the earth.
    • Said to be his assertion in demonstrating the principle of the lever; as quoted by Pappus of Alexandria, Synagoge, Book VIII, c. AD 340; also found in Chiliades (12th century) by John Tzetzes, II.130. This and "Give me a place to stand, and I shall move the world" are the most commonly quoted translations.
  • Variant translations:
  • Give me a place to stand and with a lever I will move the whole world.
    • This variant derives from an earlier source than Pappus: The Library of History of Diodorus Siculus, Fragments of Book XXVI, as translated by F. R. Walton, in Loeb Classical Library (1957) Vol. XI. In Doric Greek this may have originally been Πᾷ βῶ, καὶ χαριστίωνι τὰν γᾶν κινήσω πᾶσαν [Pā bō, kai kharistiōni tan gān kinēsō [variant kinasō] pāsan].
  • Give me a lever and a place to stand and I will move the earth.
  • Give me a fulcrum, and I shall move the world.
  • Give me a firm spot on which to stand, and I shall move the earth.

• Noli turbare circulos meos. or Noli tangere circulos meos.
  • Do not disturb my circles!
  • Original form: "noli … istum disturbare" ("Do not … disturb that (sand)") — Valerius Maximus, Memorable Doings and Sayings, Book VIII.7.ext.7 (See Chris Rorres (Courant Institute of Mathematical Sciences) – "Death of Archimedesː Sources"). This quote survives only in its Latin version or translation. In modern era, it was paraphrased as Noli turbare circulos meos and then translated to Katharevousa Greek as "μὴ μου τοὺς κύκλους τάραττε".
  • Reportedly his last words, said to a Roman soldier who, despite being given orders not to, killed Archimedes during the conquest of Syracuse; as quoted in World Literature: An Anthology of Human Experience (1947) by Arthur Christy, p. 655.

On Spirals (225 B.C.)

As translated by T. L. Heath, The Works of Archimedes (1897) unless otherwise indicated.

• How many theorems in geometry which have seemed at first impracticable are in time successfully worked out!

• Those who claim to discover everything but produce no proofs of the same may be confuted as having actually pretended to discover the impossible.

On the Equilibrium of Planes

or The Centres of Gravity of Planes

as translated by T. L. Heath, The Works of Archimedes (1897) unless otherwise indicated.

• Equal weights at equal distances are in equilibrium and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance.
  • Book 1, Postulate 1.

• If two equal weights have not the same centre of gravity, the centre of gravity of both taken together is at the middle point of the line joining their centres of gravity.
  • Book 1, Proposition 4.

• Two magnitudes whether commensurable or incommensurable, balance at distances reciprocally proportional to the magnitudes.
  • Book 1, Propositions 6 & 7, The Law of the Lever.

• The centre of gravity of any parallelogram lies on the straight line joining the middle points of opposite sides.
  • Book 1, Proposition 9.

• The centre of gravity of a parallelogram is the point of intersection of its diagonals.
  • Book 1, Proposition 10.

• In any triangle the centre of gravity lies on the straight line joining any angle to the middle point of the opposite side.
  • Book 1, Proposition 13.

• It follows at once from the last proposition that the centre of gravity of any triangle is at the intersection of the lines drawn from any two angles to the middle points of the opposite sides respectively.
  • Book 1, Proposition 14.

The Method of Mechanical Theorems

As proven by Archimedes, the area of the parabolic segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure.

Figure for Proposition 4

As quoteed in The Method of Archimedes, recently discovered by Heiberg: a supplement to the Works of Archimedes (1912) Ed. T. L. Heath unless otherwise indicated.

• I thought fit to... explain in detail in the same book the peculiarity of a certain method, by which it will be possible... to investigate some of the problems in mathematics by means of mechanics. This procedure is... no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards... 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.

• I am persuaded that it [The Method of Mechanical Theorems] will be of no little service to mathematics; for I apprehend that some, either of my contemporaries or of my successors, will, by means of the method when once established, be able to discover other theorems in addition, which have not yet occurred to me.

• First then I will set out the very first theorem which became known to me by means of mechanics, namely that
    Any segment of a section of a right angled cone (i.e., a parabola) is four-thirds of the triangle which has the same base and equal height,
  and after this I will give each of the other theorems investigated by the same method. Then at the end of the book I will give the geometrical [proofs of the propositions]...

• The centre of gravity of any cylinder is the point of bisection of the axis.
  • Proposition presumed from previous work.

• The centre of gravity of any cone is [the point which divides its axis so that] the portion [adjacent to the vertex is] triple [of the portion adjacent to the base].
  • Proposition presumed from previous work.

• Any segment of a right-angled conoid (i.e., a paraboloid of revolution) cut off by a plane at right angles to the axis is 1½ times the cone which has the same base and the same axis as the segment
  • Proposition 4.

• The centre of gravity of any hemisphere [is on the straight line which] is its axis, and divides the said straight line in such a way that the portion of it adjacent to the surface of the hemisphere has to the remaining portion the ratio which 5 has to 3.
  • Proposition 6.

Quotes about Archimedes

sorted chronologically

Circuli dimensio, 1544

• Archimedes said, “Give to me a fulcrum on which to plant my lever, and I will move the world.” And I say, give to woman the ballot, the political fulcrum, on which to plant her moral lever, and she will lift the world into a nobler purer atmosphere.
  • Susan B. Anthony from here 1870s

• Shall we not make an end to this fighting against this geometrical Briareus who uses our ships like cups to ladle water from the sea, drives off our sambuca ignominiously with cudgel-blows, and by the multitude of missiles that he hurls at us all at once outdoes the hundred-handed giants of mythology?
  • Marcus Claudius Marcellus, during the Siege of Syracuse (214–212 BC) as quoted by Thomas Little Heath, A History of Greek Mathematics (1921) Vol. 2, p. 17, citing Plutarch, Marcellus.

• When... the Romans assaulted the walls in two places at once, fear and consternation stupefied the Syracusans.... But when Archimedes began to ply his engines, he at once shot against the land forces all sorts of missile weapons... that came down with incredible noise and violence... they knocked down those upon whom they fell in heaps, breaking all their ranks and files. ...huge poles thrust out from the walls, over the ships, sunk some by the great weights... from on high... others they lifted up into the air by an iron hand or beak like a crane's... and... plunged them to the bottom of the sea; or else the ships, drawn by engines within, and whirled about, were dashed against steep rocks... under the walls, with great destruction of the soldiers... aboard them. A ship was frequently lifted up to a great height in the air... and was rolled to and fro... until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall. At the engine [called Sambuca] that Marcellus brought upon the bridge of ships... while it was as yet approaching the wall, there was discharged a... rock of ten talents [600-700 lb. total] weight, then a second and a third, which, striking upon it with immense force and a noise like thunder, broke all its foundation to pieces... and completely dislodged it from the bridge. So Marcellus... drew off his ships to a safer distance, and sounded a retreat... They then took a resolution of coming up under the walls... in the night; thinking that as Archimedes used ropes stretched at length in playing his engines, the soldiers would now be under the shot, and the darts would... fly over their heads... But he... had... framed... engines accommodated to any distance, and shorter weapons; and... with engines of a shorter range, unexpected blows were inflicted on the assailants. Thus... instantly a shower of darts and other missile weapons was again cast upon them. And when stones came tumbling down... upon their heads, and... the whole wall shot out arrows at them, they retired. ...as they were going off, arrows and darts of a longer range inflicted a great slaughter among them, and their ships were driven one against another; while they themselves were not able to retaliate... For Archimedes had provided and fixed most of his engines immediately under the wall; whence the Romans, seeing that indefinite mischief overwhelmed them from no visible means, began to think they were fighting with the gods.
  • Plutarch (75 AD) describing the Siege of Syracuse (214–212 BC) in "Marcellus," Parallel Lives Tr. John Dryden.

• Abstract enquiries into the most puzzling problems did not arise in the brain of Archimedes as a spontaneous and hitherto untouched subject, but rather as a reflection of prior enquiries in the same direction and by men separated from his days by as long a period — and far longer — than the one which separates you from the great Syracusian.
  • Koot Hoomi, as quoted in The Mahatma Letters to A. P. Sinnett ~1880 and 1884, p. 2, (1923).

• Archimedes originally solved the problem of finding the solid content of a sphere before that of finding its surface, and he inferred the result of the latter problem from that of the former. ...another illustration of the fact that the order of propositions in the treatises of the Greek geometers as finally elaborated does not necessarily follow the order of discovery.
  • Thomas Little Heath, The Method of Archimedes, recently discovered by Heiberg: a supplement to the Works of Archimedes (1897) footnote, p. 21.

• Some of the later Greeks, such as Archimedes, had just views on the elementary phenomena of hydrostatics and optics. Indeed, Archimedes, who combined a genius for mathematics with physical insight, must rank with Newton, who lived nearly two thousand years later, as one of the founders of mathematical physics.
  • Alfred North Whitehead, An Introduction to Mathematics (1911) p. 37.

    Archimedes: specific gravity
Alfred North Whitehead
Introduction to Mathematics (1911)

• In these days an infinite number of chemical tests would be available. But then Archimedes had to think... afresh. The solution flashed upon him as he lay in his bath. He jumped up and ran through the streets to the palace, shouting Eureka! Eureka! (I have found it! ...) This day... ought to be celebrated as the birthday of mathematical physics; the science came of age when Newton sat in his orchard. Archimedes... had made a great discovery. He saw that a body when immersed in water is pressed upwards by the surrounding water with a resultant force equal to the weight of the water it displaces. ...Hence if $W$ lb. be the [known] weight of the crown, as weighed in air, and $w$ lb. be the [unknown] weight of the water which it displaces when completely immersed, $W-w$ [from which (knowing $W$) the weight w of the equal volume of water can be derived,] would be the extra upward force necessary to sustain the crown as it hung in the water. [Alternatively, the weight of water, equaling the volume of the crown, and overflowing a tub, could be weighed directly.]
  Now, this upward force can easily be obtained by weighing the body as it hangs in the water [Fig. 3]...But ${\frac {w}{W}}$ ...is the same for any lump of metal of the same material: it is now called the specific gravity... Archimedes had only to take a lump of indisputably pure gold and find its specific gravity by the same process. ...[N]ot only is it the first precise example of the application of mathematical ideas to physics, but also... a perfect and simple example of what must be the method and spirit of the science for all time. The discovery of the theory of specific gravity marks a genius of the first rank.
  • Alfred North Whitehead, An Introduction to Mathematics (1911) pp. 38-40.

• 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.
  • Thomas Little Heath, A History of Greek Mathematics II (1931).

• There is here, as in all great Greek mathematical masterpieces, no hint as to the kind of analysis by which the results were first arrived at; for it is clear that they were not discovered by the steps which led up to them in the finished treatise. If the geometrical treatises had stood alone, Archimedes might seem, as Wallis said, "as it were of set purpose to have covered up the traces of his investigations, as if he has grudged posterity the secret of his method of inquiry, while he wished to extort from them assent to his results."
  • Thomas Little Heath, A Manual of Greek Mathematics (1931) p. 281.

• Modern mathematics was born with Archimedes and died with him for all of two thousand years. It came to life again with Descartes and Newton.
  • Eric Temple Bell, The Development of Mathematics (1940)

• To conceive of a parabolic segment or of a triangle as the sum of infinitely many line segments, is closely akin to the idea of Leibniz, who thought of the integral $\int y~dx$ as the sum of infinitely many terms $y~dx$. But, in contrast to Leibniz, Archimedes is fully aware that this conception is... incorrect and that the heuristic derivation should be supplemented by a rigorous proof.
  • Bartel Leendert van der Waerden, Ontwakende wetenschap (1950) translated as Science Awakening (1954) by Arnold Dresden.

• The estimations, which occur in the summing of infinite series and in limiting operations, the "epsilontics", as the calculation with an arbitrarily small ε is sometimes called, were for Archimedes an open book. In this respect, his thinking is entirely modern.
  • Bartel Leendert van der Waerden, Ontwakende wetenschap (1950) translated as Science Awakening (1954) by Arnold Dresden.

• In Euclidean geometry the infinitely small was rejected and in the classical treatises of Archimedes we have the finest example of mathematical rigour in antiquity. Notwithstanding, in the discovery method we find him manipulating line and surface indivisibles skilfully, imaginatively and non-rigorously
  • Margaret E. Baron, The Origins of the Infinitesimal Calculus (1969)

• Using his masterful understanding of mechanics, equilibrium, and the principles of the lever, he weighed in his mind solids or figures whose volumes or areas he was attempting to find against ones he already knew. After determining in this way the answer...he found it much easier to prove geometrically... Consequently The Method starts with a number of statements on centers of gravity and only then proceeds to the geometrical propositions and their proofs. ...[He] essentially introduced the concept of a thought experiment into rigorous research. ...[He] freed mathematics from the somewhat artificial chains that Euclid and Plato had put on it. ...He did not hesitate to explore and exploit the connections between the abstract mathematical objects (the Platonic forms) and physical reality (actual solids and flat objects) to advance his mathematics.
  • Mario Livio, Is God a Mathematician? (2009)

• Archimedes was the earliest thinker to develop the apparatus of an infinite series with a finite limit ...starting on the conceptual path toward calculus. Of the giants on whose shoulders Isaac Newton would eventually perch, Archimedes was the first.
  • Alex Bellos, The Grapes of Math (2014)

• Archimedes was a brilliant inventor and a mathematician.  He says to the people around him, "Don't just live in the lap of the gods.  Don't be dominated by Mother Nature.  You, as a man, can take control of your own destiny."
  • A&E Television Networks, LLC., "Ancient Einsteins," Ancient Impossible (July 27, 2014).

• According to legend, nothing could get between him [Archimedes] and his work, and sometimes he would even forget to eat.  Ideas would come to him at any moment, and he would scribble them on any available surface.  Famously, he was in the bath when he discovered the laws of buoyancy, leading him to run naked through the streets shouting "Eureka!"  …  Eureka means "I have found it," and it could be argued that Archimedes found out more than anyone else before or since.
  • A&E Television Networks, LLC., "Ancient Einsteins," Ancient Impossible (July 27, 2014).

• Tragically for all of us, he [Archimedes] was cut down by a Roman soldier because he refused to stop working.  …  If Archimedes hadn't been killed before his time, what could have he achieved?  The industrial revolution could have happened two thousand years earlier.  He might have kick-started the modern age.
  • A&E Television Networks, LLC., "Ancient Einsteins," Ancient Impossible (July 27, 2014).

See also

• Archimedean point

External links

• Archimedes Home Page
• "Archimedes - The Greatest Scientist Ever?"
• "Archimedes of Syracuse"
• Archimedes' Book of Lemmas
• "Archimedes and the Rhombicuboctahedron" by Antonio Gutierrez
• Archimedes - The Golden Crown
• Archimedes' Quadrature Of The Parabola
• Archimedes' On The Measurement Of The Circle
• The Works Of Archimedes
• Works by Archimedes at Project Gutenberg
• NOVA program on Archimedes Palimpsest
• The Archimedes Palimpsest
• The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland
• Archimedes and his Burning Mirrors, Reality or Fantasy?
• Squaring the circle History Topic at MacTutor
• Biography of Archimedes
• Archimedes - The Greek mathematician and his Eureka moments" on BBC 4 (25 January 2007) (requires RealPlayer).