Archimedes

The sum of the squares on the lines drawn from any point on a sphere to the ends of a diameter is constant.

The area of an ellipse is to the area of the auxiliary circle as the minor axis to the major axis.

The volume of a segment of a hyperboloid of revolution is found by methods similar to those for the paraboloid.

The cattle problem: Compute, O friend, the number of the cattle of the sun...

How many cattle are there? They are so many... if you can tell me these things, you are not unskilled in numbers, but still you cannot be counted among the wise.

The sand is not infinite in number, but it exceeds any number that has a name.

Some people believe that the grains of sand are infinite in multitude... but I will try to show you numbers greater than the number of grains of sand.

The diameter of the universe is less than 10^14 stadia, and therefore the number of grains of sand needed to fill it is less than 10^63.

I have sent you the proofs of the theorems about the spiral, which you asked me to communicate.

Seeing moreover in you, as I say, an earnest student, a man of considerable eminence in philosophy, and an admirer of mathematical inquiry, I thought fit to write out for you and explain in detail...

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

I am persuaded that it will be of no little service, for indeed I see that many other questions can be solved by the same method.

The construction of these figures is clear from the mechanical method, but the geometrical proof must be given afterwards.

The center of gravity of any cylinder is the point of bisection of its axis.