Editing Archimedes Palimpsest (section)

===''The Method of Mechanical Theorems''===
{{Main|The Method of Mechanical Theorems}}
The palimpsest contains the only known copy of ''The Method of Mechanical Theorems''.

In his other works, Archimedes often proves the equality of two areas or volumes with [[Eudoxus of Cnidus|Eudoxus]]' [[method of exhaustion]], an ancient Greek counterpart of the modern method of limits. Since the Greeks were aware that some numbers were irrational, their notion of a [[real number]] was a quantity Q approximated by two sequences, one providing an upper bound and the other a lower bound. If one finds two sequences U and L, and U is always bigger than Q, and L always smaller than Q, and if the two sequences eventually came closer together than any prespecified amount, then Q is found, or ''exhausted'', by U and L.

Archimedes used exhaustion to prove his theorems. This involved approximating the figure whose area he wanted to compute into sections of known area, which provide upper and lower bounds for the area of the figure. He then proved that the two bounds become equal when the subdivision becomes arbitrarily fine. These proofs, still considered to be rigorous and correct, used [[geometry]] with rare brilliance. Later writers often criticized Archimedes for not explaining how he arrived at his results in the first place. This explanation is contained in ''The Method''.{{fact|date=August 2023}}

The method that Archimedes describes was based upon his investigations of [[physics]], on the [[center of mass]] and the [[Lever#Law of the lever|law of the lever]]. He compared the area or volume of a figure of which he knew the total mass and center of mass with the area or volume of another figure he did not know anything about. He viewed plane figures as made out of infinitely many lines as in the later [[method of indivisibles]], and balanced each line, or slice, of one figure against a corresponding slice of the second figure on a lever. The essential point is that the two figures are oriented differently so that the corresponding slices are at different distances from the fulcrum, and the condition that the slices balance is not the same as the condition that the figures are equal.

Once he shows that each slice of one figure balances each slice of the other figure, he concludes that the two figures balance each other. But the center of mass of one figure is known, and the total mass can be placed at this center and it still balances. The second figure has an unknown mass, but the position of its center of mass might be restricted to lie at a certain distance from the fulcrum by a geometrical argument, by symmetry. The condition that the two figures balance now allows him to calculate the total mass of the other figure. He considered this method as a useful [[heuristic]] but always made sure to prove the results he found using exhaustion, since the method did not provide upper and lower bounds.

Using this method, Archimedes was able to solve several problems now treated by [[integral calculus]], which was given its modern form in the seventeenth century by [[Isaac Newton]] and [[Gottfried Leibniz]]. Among those problems were that of calculating the [[center of gravity]] of a solid [[Sphere|hemisphere]], the center of gravity of a [[frustum]] of a circular [[paraboloid]], and the area of a region bounded by a [[parabola]] and one of its [[secant line]]s. (For explicit details, see [[Archimedes' use of infinitesimals]].)

When rigorously proving theorems involving [[volume]], Archimedes used a form of [[Cavalieri's principle]], that two volume with equal-area cross-sections are equal; the same principle forms the basis of [[Riemann sum]]s. In ''[[On the Sphere and Cylinder]]'', he gives upper and lower bounds for the surface area of a sphere by cutting the sphere into sections of equal width. He then bounds the area of each section by the area of an inscribed and circumscribed cone, which he proves have a larger and smaller area correspondingly.

But there are two essential differences between Archimedes' method and 19th-century methods:
# Archimedes did not know about differentiation, so he could not calculate any integrals other than those that came from center-of-mass considerations, by symmetry. While he had a notion of linearity, to find the volume of a sphere he had to balance two figures at the same time; he never determined how to change variables or integrate by parts.
# When calculating approximating sums, he imposed the further constraint that the sums provide rigorous upper and lower bounds. This was required because the Greeks lacked algebraic methods that could establish that error terms in an approximation are small.

A problem solved exclusively in the ''Method'' is the calculation of the volume of a cylindrical wedge, a result that reappears as theorem XVII (schema XIX) of [[Johannes Kepler|Kepler]]'s ''Stereometria''.

Some pages of the ''Method'' remained unused by the author of the palimpsest and thus they are still lost. Between them, an announced result concerned the volume of the intersection of two cylinders, a figure that [[Tom M. Apostol|Apostol]] and [[Mamikon Mnatsakanian|Mnatsakanian]] have renamed ''n&nbsp;=&nbsp;4 Archimedean globe'' (and the half of it, ''n''&nbsp;=&nbsp;4 Archimedean dome), whose volume relates to the ''n''-polygonal pyramid.