Archimedes Quotes
130 Archimedes Quotes
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The diameter of the sun is greater than the side of the chiliagon inscribed in the greatest circle in the (sphere of the) universe. I make this assumption because Aristarchus discovered that the sun appeared to be about – nth part of the circle of the zodiac, and I myself tried, by a method which I will now describe, to find experimentally the angle subtended by the sun and having its vertex at the eye.
Archimedes
I conceive that these things, King Gelon, will appear incredible to the great majority of people who have not studied mathematics, but that to those who are conversant therewith and have given thought to the question of the distances and sizes of the earth, the sun and moon and the whole universe, the proof will carry conviction.
Archimedes
I thought the subject would not be inappropriate for your consideration.
Archimedes
Of course it is easier to establish a proof if one has in this way previously obtained a conception of the questions, than for him to seek it without such a preliminary notion.
Archimedes
It is true that this is not proved by what we have said here; but it indicates that the result is correct.
Archimedes
[On The Cattle Problem] If thou art diligent and wise, O stranger, compute the number of cattle of the Sun, who once upon a time grazed on the fields of the Thrinacian isle of Sicily, divided into four herds of different colours, one milk white, another a glossy black, a third yellow and the last dappled. In each herd were bulls, mighty in number according to these proportions: Understand, stranger, that the white bulls were equal to a half and a third of the black together with the whole of the yellow, while the black were equal to the fourth part of the dappled and a fifth, together with, once more, the whole of the yellow. Observe further that the remaining bulls, the dappled, were equal to a sixth part of the white and a seventh, together with all of the yellow.
Archimedes
If thou canst accurately tell, O stranger, the number of cattle of the Sun, giving
separately the number of well-fed bulls and again the number of females according to each colour, thou wouldst not be called unskilled or ignorant of numbers, but not yet shalt thou be numbered among the wise.
Archimedes
But come, understand also all these conditions regarding the cattle of the Sun. When the white bulls mingled their number with the black, they stood firm, equal in depth and breadth, and the plains of Thrinacia, stretching far in all ways, were filled with their multitude. Again, when the yellow and the dappled bulls were gathered into one herd they stood in such a manner that their number, beginning from one, grew slowly greater till it completed a triangular figure, there being no bulls of other colours in their midst nor none of them lacking. If thou art able, O stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in this species of wisdom.
Archimedes
As we have just seen that it has not been proved but rather conjectured that the result is correct we have devised a geometrical demonstration which we made known sometime ago and will again bring forward farther on.
Archimedes
In the same way it may be perceived that any segment of an ellipsoid cut off by a perpendicular plane, bears the same ratio to a cone having the same base and the same axis, as half of the axis of the axis of the ellipsoid + the axis of the opposite segment bears to the axis of the opposite segment.
Archimedes
Now the fact here stated is not actually demonstrated by the argument used; but that argument has given a sort of indication that the conclusion is true. Seeing then that the theorm is not demonstrated, but at the same time suspecting the conclusion is true, we shall have recourse to the geometrical demonstration which I myself discovered and have already published.
Archimedes
Any sphere is (in respect of solid content) four times the cone with base equal to a great circle of the sphere and height equal to its radius.
Archimedes
The cylinder with base equal to a great circle of the sphere and height equal to the diameter is 1 ½ times the sphere.
Archimedes
I considered the notion that the surface of any sphere is four times as great as a great circle in it…
Archimedes
A cylinder with base equal to the greatest circle in a spheroid and height equal to the axis of the spheroid is 1 ½ time sthe spheroid.
Archimedes
If any spheroid be cut by a plane through the centre and at nright angles to the axis, the half of the spheroid is double of the cone which has the same base and the same axis as the segment. (IE. The half of the spheroid.)
Archimedes
Any segment of a right-angled conoid (ie a paraboloid of revolution) cut off by a plane at right angles to the axis is 1 ½ times the cone which has the same base and the same axis as the segment.
Archimedes
The centre of gravity of a segment of a right-angled conoid (ie a paraboloid of revolution) cut off by a plane at right angles to the axis is on the straight line in wuch a way that the portion of it adjacent to the vertex is double of the remaining portion.
Archimedes
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.
Archimedes
[Any segment of a sphere has] to the cone [with the same base and height the ratio which the sum of the radius of the sphere and the height of the complementary segment has to the height of the complementary segment.]
Archimedes
The centre of gravity of any segment of a sphere is on the straight line which is the axis of the segment, and divides this straight line in such a way that the part of it adjacent to the vertex of the segment has to the remaining part the ratio which the sum of the axis of the segment and four times the axis of the complementary segment has to the sum of the segment and double the axis of the complementary segment.
Archimedes
[A segment of an obtuse-angled conoid (ie. A hyperboloid of revolution) has to the cone which has] the same base [as the segment and equal height the same ratio as the sum of the axis of the segment and the three times] the ‘annex to the axis’ (ie. Half the transverse axis of the hyperbolic section through the axis of the hyperboloid or in other words, the distance between the vertex of the segment and the vertex of the enveloping cone) has to the sum of the axis of the segment and double of the ‘annex’.
Archimedes
If in a right prism with square bases a cylinder be inscribed having its bases in opposite square-faces and touching with its surface the remaining four parallelogrammic faces, and if through the centre of the circle which is the based of the cylinder and one side of the opposite square face a plane be drawn, the figure cut off by the plane so drawn is one sixth part of the whole prism.
Archimedes
Seeing in you, as I say, an earnest student, a man of considerable eminence in philosophy and an admirer of mathematical inquiry when it comes your way, I have thought fit to write out for you and explain in detail in the same book the peculiarity of a certain method, which, when you see it, will put you in possession of a means whereby you can investigate some of the problems of mathematics by mechanics.
Archimedes
For certain things which first became clear to me by a mechanical method had afterwards to be demonstrated by geometry, because their investigation by the said method did not furnish an actual demonstration. 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 the proof without any previous knowledge. This is a reason why, in the case of the theorems the proof of which Eudoxus was the first to discover, namely, that the cone is a third part of the cylinder, and the pyramid a third part of the prism, having the same base and equal height, we should give no small share of the credit to Democritus, who was the first to assert this truth with regard to the said figures, though he did not prove it.
Archimedes
I am myself in the position of having made the discovery of the theorem now to be published in the same way as I made my earlier discoveries; and I thought it desirable now to write out and publish the method, partly because I have already spoken of it and I do not want to be thought to have uttered vain words, but partly also because I am persuaded that it 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 occurred to me.
Archimedes
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
Archimedes
Given a cylinder inscribed in a rectangular parallelepiped on a square base in such a way that the two bases of the cylinder are circles inscribed in the opposite square faces, suppose a plane drawn through one side of the square containing one base of the cylinder and through the parallel diameter of the opposite base of the cylinder. The plane cuts off a solid with a surface resembling that of a horse’s hoof.
Archimedes
A cylinder is inscribed in a cube in such a way that the bases of the cylinder are circles inscribed in two opposite square faces. Another cylinder is inscribed which is similarly related to another pair of opposite faces. The two cylinders include between them a solid with all its angles rounded off; and Archimedes proves that the volume of this solid is two-thirds of that of the cube.
Archimedes
That the surface of a sphere is equal to four times its greatest circle [i.e. what we call a “great circle” of the sphere];
Archimedes
That the surface of any segment of a sphere is equal to a circle with radius equal to the straight line drawn from the vertex of the segment to a point on the circle which is the base of the segment
Archimedes
That, if we have a cylinder circumscribed to a sphere and with height equal to the diameter, then (a) the volume of the cylinder is 1½ times that of the sphere and (b) the surface of the cylinder, including its bases, is 1½ times the surface of the sphere.
Archimedes
Of all lines which have the same extremities the straight line is the least
Archimedes
Of unequal lines, unequal surfaces and unequal solids the greater exceeds the less by such a magnitude as, when (continually) added to itself, can be made to exceed any assigned magnitude among those which are comparable[with it and] with one another [i.e. are of the same kind]. This is the Postulate of Archimedes.
Archimedes
Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium, but the system in that case“inclines towards the weight which is at the greater distance. [in other words, the action of the weight which is at the greater distance produces motion in the direction in which it acts]
Archimedes
If when weights are in equilibrium something is added to or subtracted from one of the weights, the system will “incline” towards the weight which is added to or the weight from which nothing is taken respectively
Archimedes
If equal and similar figures be applied to one another so as to coincide throughout, their centres of gravity also coincide; if figures be unequal but similar, their centres of gravity are similarly situated with regard to the figures.
Archimedes
Let us assume that a fluid has the property that, if its parts lie evenly and are continuous, the part which is less compressed is expelled by that which is more compressed, and each of its parts is compressed by the fluid above it perpendicularly, unless the fluid is shut up in something and compressed by something else.
Archimedes
Let us assume that any body which is borne upwards in water is carried along the perpendicular [to the surface] which passes through the centre of gravity of the body.
Archimedes
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 [ie 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.
Archimedes
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