Feature Column


The Method of Archimedes

4. The plane geometry behind the argument


The areas of the three discs in question are proportional tothe square of their radii. With points labeled as in this diagram(adapted from T. L. Heath, The Works of Archimedes withThe Method of Archimedes, Dover, New York), the equationm d = M D from the previous pagebecomes

MS2.SA = (OS2 + QS2).AH .

Since MS = CA and SQ = SA, we have MS.SQ = CA.SA . Now OS is the altitude of the right triangle OSA and therefore OS2 = CS.SA . The Pythagorean theorem applied to OSA gives OA2 = OS2 + SA2, so

OA2 = CS.SA + SA2 = CA.SA,

yielding MS.SQ = OA2 = OS2 + SQ2.

Since AH = CA it follows that
AH/SA = CA/SA = MS/SQ (substitutingequals for equals) = MS2/MS.SQ = MS2/(OS2 + SQ2). Cross-multipling gives the desired equation.

Welcome to the
Feature Column!

These web essays are designed for those who have already discovered the joys of mathematics as well as for those who may be uncomfortable with mathematics.
Read more . . .

Search Feature Column

Feature Column at a glance

Show Archive

Browse subjects

American Mathematical Society