# Archimedes

Greek mathematician, physicist, engineer, inventor, and astronomer
Give me a place to stand, and I shall move the world.

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

## QuotesEdit

I have found it!
• εὕρηκα [heureka]
• 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. [Dos moi pa sto, kai tan gan kinaso.]
• 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.

### On Spirals (225 B.C.)Edit

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 PlanesEdit

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 TheoremsEdit

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
• Proprosition 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.

sorted chronologically

Circuli dimensio, 1544
• 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.
• 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.
• 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."
• 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.
• 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 ${\displaystyle \int y~dx}$  as the sum of infinitely many terms ${\displaystyle y~dx}$ . 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.
• 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)
• 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.
• 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).