Archimedes

I will try to show... that the numbers named by me and given in the work which I sent to Zeuxippus can fill not only the world but also the universe.

The diameter of the Earth is greater than the diameter of the Moon and less than the diameter of the Sun.

The ratio of the circumference of any circle to its diameter is less than 3 1/7 but greater than 3 10/71.

The area of a circle is to the square on its diameter as 11 to 14.

Spiral lines are described by a point moving uniformly along a straight line while the line itself rotates uniformly about a fixed point.

I conceive that these properties will be of no small service to mathematicians, for I perceive that many other theorems, and more important ones, may be derived from them.

The area bounded by the first turn of the spiral and the initial line is one-third of the area of the 'first circle'.

The cylinder circumscribed about a sphere has a volume and surface area each one and a half times that of the sphere.

I thought fit to write out for you and explain in detail the peculiarity of a certain method, by which it will be possible for you to get a start to enable you to investigate some of the problems in mathematics by means of mechanics.

Certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards.

It is easier to supply the proof when we have previously acquired, by the method, some knowledge of the questions than it is to find it without any previous knowledge.

The weight of any floating body is equal to the weight of the liquid displaced.

A body heavier than a fluid will, if placed in it, descend to the bottom, and the body will be lighter in the fluid by the weight of the fluid displaced.

I have found by experiment that the crown is not of pure gold, but contains silver alloy.

The properties of the balance depend on the principle of the lever.

Equal weights at equal distances are in equilibrium.

Unequal weights will balance at unequal distances, the greater weight being at the lesser distance.

The center of gravity of a triangle is at the intersection of lines drawn from any two angles to the midpoints of the opposite sides.

The quadrature of the parabola is four-thirds of the triangle with the same base and equal height.

Any segment of a right-angled conoid (paraboloid) cut off by a plane at right angles to the axis is one-and-a-half times the cone which has the same base and axis.