Levers and Archimdes

Archimedes (287-212 BCE) has more information known about him then any other Greek mathematician. As the son of an astronomer, Phidias, he is credited with the invention, that is used for raising water for irrigation the Archimedian screw. He spent his time solving various problems for Hiero and his successor and it is noted that Archimedes, being very dedicated to his work, would neglect his health and hygiene. For month it was Archimedes military engineering that kept the Roman army  at bay for months during the siege of Syracuse, however eventually the Roman entered the city. During all this chaos Archimedes was working on a math problem and did not obey the Roman solider,  who found him, to come follow him to Marcellus. The solider ended taking his life.

No one before Archimedes had created a mathematical model of the lever by which one could derive a mathematical proof of the law of levers
4 of the 7 Postulate that Archimedes stated in his Planes in Equilibrium

A and B)
Equal weights at equal distances are in equilibrium, and equal weights at unequal distances will have the lever incline towards the weight at a greater distance

C)
When weights are at certain distances are in equilibrium and if more weight is added to one side then the lever will incline towards the side were the weight was added

E)
When weights are at certain distances and weight is taken away from one then the lever will incline towards the side where no weight is taken away.

F)
If magnitudes at certain distances are in equilibrium, other magnitudes equal to them will also be in equilibrium at the same distances

 The Law of the Lever is stated in Proposition 6 and 7:

Two magnitudes, whether commensurable [prop 6] or incommensurable [prop 7], balance at distances inversely proportional to the magnitudes.

Archimedes gave the first proof of the law of the lever and with that law Archimedes found the center of gravity for various figures

Propositions leading to the law of lever

Prop1
Weights which balance at equal distances are equal

Prop2
Unequal weights at equal distances will incline toward the greater weight

Prop3
Suppose A and Bare unequal weights with A> B which balance at point C. Letting AC=a and BC=b then a<b. Conversely, if the weights balance and a<b then A>B

Unanswered Question

1.  What other application does the law of the lever have besides finding the center of gravity?

Evaluation:
complexity: 7
interesting: 7
quality:  8