Archimedes’ “On the Equilibrium of Planes”, Prop 3

 

by Corry Shores
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Archimedes

On the Equilibrium of Planes

or

The Centres of Gravity of Planes,

Book I

Proposition 3

[The following is quotation]

 

Proposition 3

P3. Unequal weights will balance at unequal distances, the greater weight being at the lesser distance.

Let A, B he two unequal weights (of which A is the greater) balancing about G at distances AC, BC respectively.

Then shall AC be less than BC. For, if not, take away from A the weight (A – B). The remainders will then incline | towards B [Post. 3]. But this is impossible, for (1) if AC = CB, the equal remainders will balance, or (2) if AC > CB, they will incline towards A at the greater distance [Post. 1].

Hence AC < CB.

Conversely, if the weights balance, and AC < CB, then A > B. (Heath 190-191)

 

From:

Archimedes. “On the Equilibrium of Planes or The Centres of Gravity of Planes, Book I”. In The Works of Archimedes. Ed. T.L. Heath. Cambridge UP, 1897. Obtained at

https://archive.org/details/worksofarchimede00arch