Archimedes on mechanical and geometric methods
In the summer of 1906, J L Heiberg, professor of classical philology at the University of Copenhagen, discovered a 10th century manuscript which included Archimedes' work The method. This provides a remarkable insight into how Archimedes discovered many of his results and we will discuss this below once we have given further details of what is in the surviving books. Below we give an extract from the Introduction to The Method in which Archimedes discusses mechanical and geometric methods:-
Archimedes to Eratosthenes greeting.
I sent you on a former occasion some of the theorems discovered by me, merely writing out the enunciations and inviting you to discover the proofs, which at the moment I did not give. The enunciations of the theorems that I sent were as follows.
- If in a right prism with a parallelogrammic base a cylinder be inscribed which has its bases in the opposite parallelograms [in fact squares], and its sides [i.e., four generators] on the remaining planes (faces) of the prism, and if through the centre of the circle which is the base of the cylinder and (through) one side of the square in the plane opposite to it a plane be drawn, the plane so drawn will cut off from the cylinder a segment which is bounded by two planes, and the surface of the cylinder, one of the two planes being the plane which has been drawn and the other the plane in which the base of the cylinder is, and the surface being that which is between the said planes; and the segment cut off from the cylinder is one sixth part of the whole prism.
- If in a cube a cylinder be inscribed which has its bases in the opposite parallelograms [in fact squares] and touches with its surface the remaining four planes (faces), and if there also be inscribed in the same cube another cylinder which has its bases in other parallelograms and touches with its surface the remaining four planes (faces), then the figure bounded by the surfaces of the cylinders, which is within both cylinders, is two-thirds of the whole cube.
First then I will set out the very first theorem that 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] ...
[I premise the following propositions that I shall use in the course of the work.]
- If from [one magnitude another magnitude be subtracted which has not the same centre of gravity, the centre of gravity of the remainder is found by] producing [the straight line joining the centres of gravity of the whole magnitude and of the subtracted part in the direction of the centre of gravity of the whole] and cutting off from it a length which has to the distance between the said centres of gravity the ratio which the weight of the subtracted magnitude has to the weight of the remainder.
- If the centres of gravity of any, number of magnitudes whatever be on the same straight line, the centre of gravity of the magnitude made up of all of them will be on the same straight line.
- The centre of gravity of any straight line is the point of bisection of the straight line.
- The centre of gravity of any triangle is the point in which the straight lines drawn from the angular points of the triangle to the middle points of the (opposite) sides cut one another.
- The centre of gravity of any parallelogram is the point in which the diagonals meet.
- The centre of gravity of a circle is the point that is also the centre [of the circle].
- The centre of gravity of any cylinder is the point of bisection of the axis.
- 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].
If in two series of magnitudes those of the first series are, in order, proportional to those of the second series and further] the magnitudes [of the first series], either all or some of them, are in any ratio whatever [to those of a third series], and if the magnitudes of the second series are in the same ratio to the corresponding magnitudes [of a fourth series], then the sum of the magnitudes of the first series has to the sum of the selected magnitudes of the third series the same ratio which the sum of the magnitudes of the second series has to the sum of the (correspondingly) selected magnitudes of the fourth series.