On Archimedes-The method
• On the treatise “The measurement of the circle” and “On the sphere” have
generally been known to Archimedes commentators in the sixth century.
• Almost all modern od Archimdes’ work stem from a single Greek manuscript
copied from an earlier orginal in Constantinopel in the ninth or tenth century.
• “The method” was discovered in Constantinople 1906 as a palimpsest (over-
writting of liturgy). It is evidence that Archimedes did not conceal its ideas
(reproach by later mathematicians).
• Discovery of results by heuristic method. Quote from the preface of the “The
method”:
For certain things became clear to me by a mechanical method although
they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish ac 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
without any previous knowledge .... I am presuaded that it will be of
no little service to mathematics; for I apprehend that some, either of my
contemporaries or my successors, will, by means of the method whence
once established, be able to discover other theorems in addition, which
have not yet occurred to me.
1
2
The simple heuristic of the lever is easy: If certain weights wi are placed on one side
of the lever with distance di from the midpoint, and vj with distances kj . Then
!
!
wi di =
v j kj
i
j
leads to a balanced lever L with origin 0.
Before Archimedes applies the method to geometric objects he first analyzes what
we call today the center gravity.
• The center of gravity for a triangle is the intersection of the medians, connecting vertices with the midpoint of the opposite side.
• Note: By triangularization one can find midpoint many planar figures.
The “method” starts from two planar figures P and Q, both over an interval [a, b]
on our line L.
For every x we denote by Px the length of the line perpendicular to L.
3
For every x we require that Px be moved to the left of the lever at fixed distance
d(A, O) = k and such
k|Px | = d(x, O)|Qx |
Theorem: Assume that the center of gravity xQ lies on L. Then
karea(P ) = d(0, xQ )area(Q) .
Proof. Assume that O is the origin. Recall that the center of gravity with respect
to O is
xQ areaQ =
=
!
!
zdz =
Q
b
a
!
b
(
a
!
Qx /2
dt)xdx
−Qx /2
d(x, 0)|Qx |dx = k
= k area(P ) .
!
b
a
|Px |dx
!