by Corry Shores
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[Mathematics, Calculus, Geometry, Entry Directory]
[“On the Equilibrium of Planes”, Entry Directory]
Archimedes
On the Equilibrium of Planes
or
The Centres of Gravity of Planes,
Book I
Proposition 6, 7
[The following is quotation]
Propositions 6, 7.
P6, 7. Two magnitudes, whether commensurable [Prop. 6] or incommensurable [Prop. 7], balance at distances reciprocally proportional to the magnitudes.
I. Suppose the magnitudes A, B to be commensurable, and the points A, B to be their centres of gravity. Let DE be a straight line so divided at C that
A : B = DC : CE.
We have then to prove that, if A be placed at E and B at D, C is the centre of gravity of the two taken together.
Since A, B are commensurable, so are DC, CE. Let N be a common measure of DC, CE. Make DH, DK each equal to CE, and EL (on CE produced) equal to CD. Then EH= CD, since DH = CE. Therefore LH is bisected at E, as HK is bisected at D.
Thus LH, HK must each contain N an even number of times.
Take a magnitude O such that O is contained as many times in A as N is contained in LH, whence
A : O = LH : N.
But
B : A = CE : DC
= HK : LH.
Hence, ex aequali, B : O = HK : N, or O is contained in B as many times as N is contained in HK.
Thus O is a common measure of A, B.
(Heath 192)
Divide LH, HK into parts each equal to N, and A, B into parts each equal to O. The parts of A will therefore be equal in number to those of LH, and the parts of B equal in number to those of HK. Place one of the parts of A at the middle point of each of the parts N of LH, and one of the parts of B at the middle point of each of the parts N of HK.
Then the centre of gravity of the parts of A placed at equal distances on LH will be at E, the middle point of LH [Prop. 5, Cor. 2], and the centre of gravity of the parts of B placed at equal distances along HK will be at D, the middle point of HK.
Thus we may suppose A itself applied at E, and B itself applied at D.
But the system formed by the parts O of A and B together is a system of equal magnitudes even in number and placed at equal distances along LK. And, since LE = CD, and EG = DK, LC = CK, so that C is the middle point of LK. Therefore G is the centre of gravity of the system ranged along LK.
Therefore A acting at E and B acting at D balance about the point C.
II. Suppose the magnitudes to be incommensurable, and let them be (A + a) and B respectively. Let DE be a line divided at C so that
(A + a) : B = DC : CE.
Then, if (A + a) placed at E and B placed at D do not balance about C, (A + a) is either too great to balance B, or not great enough.
Suppose, if possible, that (A + a) is too great to balance B. Take from (A + a) a magnitude a smaller than the deduction which would make the remainder balance B, but such that the remainder A and the magnitude B are commensurable.
(Heath 193)
Then, since A, B are commensurable, and
A B < DC : CE,
A and B will not balance [Prop. 6], but D will be depressed. But this is impossible, since the deduction a was an insufficient deduction from (A + a) to produce equilibrium, so that E was still depressed.
Therefore (A + a) is not too great to balance B; and similarly it may be proved that B is not too great to balance (A + a).
Hence (A + a), B taken together have their centre of gravity at C.
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
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