- εὕρηκα [heureka]
- I have found it! or I have got it!
- 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 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.
- Variant translations:
- Give me a place to stand, and I shall move the world.
- Give me a fulcrum, and I shall move the world.
- Give me a stick long enough and a pivot and I shall move the world.
- Give me a firm spot on which to stand, and I shall move the earth.
- Give me a lever and a place to stand and I will move the earth.
- 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.
- 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.
- 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.
- 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.
- The centre of gravity of a parallelogram is the point of intersection of its diagonals.
- In any triangle the centre of gravity lies on the straight line joining any angle to the middle point of the opposite side.
- 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.
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.
- 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
- 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.
Quotes about ArchimedesEdit
- 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.
- 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.