On the Equilibrium of Planes

Plutarch represented Archimedes as declaring that Any given weight can be moved by a given force.

On the Equilibrium of Planes (Greek: Περὶ ἐπιπέδων ἱσορροπιῶν) is a treatise by Archimedes in two volumes. The first book establishes the Law of the Lever, and locates the centre of gravity of the triangle and the trapezoid.^[1][2] According to Pappus of Alexandria, Archimedes' work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth." (Greek: δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω).^[3] The second book, which contains ten propositions, examines the centres of gravity of parabolic segments.^[1]

Structure of the text

Book one contains fifteen propositions with seven postulates. In proposition six Archimedes establishes the Law of the Lever, concluding that "Magnitudes are in equilibrium at distances reciprocally proportional to their weights." In propositions ten and fourteen, respectively, Archimedes locates the centre of gravity of the parallelogram and the triangle. Additionally, in proposition 15, he establishes the centre of gravity of the trapezium. The second book, which contains ten propositions, studies parabolic segments exclusively. It examines these segments by substituting them with rectangles of equal area; an exchange made possible by results obtained in the Quadrature of the Parabola.^[1][2]

Main Theorem

Archimedes' proof of the Law of the Lever is executed within proposition six. It is for commensurable magnitudes only, and relies upon propositions four, and five, and on postulate one.^[2]

Introduction

Equal weights at equal distances.

In postulate one Archimedes states that "Equal weights at equal distances are in equilibrium" (meaning one equal weight on either side of the lever arm). At propositions four, and five, he expands this observation to include the concept of the centre of gravity; wherein it is proven that the centre of gravity of any system consisting of an even number of equal weights, equally distributed, will be located at the midpoint between the two centre weights (hence introducing multiple weights on either side of the lever arm).

Statement

Weights and levers at a ratio of four to three.

Given two unequal, but commensurable, weights and a lever arm divided into two unequal, yet commensurable, portions (see sketch opposite) proposition six states simply that if the magnitudes A and B are applied at points E and D, respectively, the system will be in equilibrium if the weights are inversely proportional to the lengths:

${\displaystyle A:B=CD:EC\,}$

Proof

Therefore, assume that lines and weights are constructed to obey the rule using a common measure (or unit) N, and at a ratio of four to three (as per the sketch). Now, double the length of ED by duplicating the longer arm on the left, and the shorter arm on the right.

For demonstration's sake, reorder the lines so that CD is adjacent to LE (the two red lines together), and compare to the original below:

Weights and levers at a ratio of eight to six.

It is clear then, that the new line is double the original, that LH has its centre at E (see adjacent red lines), and HK its centre at D. Note, additionally, that EH (which is equal to CD) carries the common divisor (or unit) N, an exact number of times, as does EC, and therefore, by inference, CH too. It remains then to prove that A applied at E, and B applied at D, will have their centre of gravity at C.

Therefore, as the ratio of LH to HK is not four to three, but eight to six, similarly divide the magnitudes A and B (a transformation that conserves their original ratio of four to three), and align them as per the diagram opposite. A centred on E, and B centred on D.

Now, since an even number of equal weights, equally spaced, have their centre of gravity between the two middle weights, A is in fact applied at E, and B at D, as the proposition requires. Further, the total system consists of an even number of equal weights equally distributed, and, therefore, following the same law, C must be the centre of gravity of the full system. Thus A applied at E, and B applied at D, have their centre of gravity at C.^[1]

Authenticity

Whilst the authenticity of book two is not doubted, a number of researches have highlighted inconsistencies within book one's presentation.^[2][4][5] Berggren, in particular, questions the validity of book one as a whole; highlighting, inter alia, the redundancy of propositions one to three, eleven, and twelve.^[2] However, Berggren follows Dijksterhuis, in rejecting Mach's criticism of proposition six. Adding that its true significance lies in the fact that it demonstrates that "if a system of weights suspended on a balance beam is in equilibrium when supported at a particular point, then any redistribution of these weights, that preserves their common centre of gravity, also preserves the equilibrium."^[2][4] Further, it should be noted that proposition seven is incomplete in its current form, so that book one demonstrates the Law of the Lever for commensurable magnitudes only.^[1][2][4]

References

1. Heath, T.L. "The Works of Archimedes (1897). The unabridged work in PDF form (19 MB)". Archive.org. Archived from the original on 6 October 2007. Retrieved 2013-01-06.
2. John Lennart Berggren (1976). "Spurious Theorems in Archimedes' Equilibrium of Planes Book I". Archive for History of Exact Sciences 16(2), 87-103. ISSN 1432-0657.
3. Quoted by Pappus of Alexandria in Synagoge, Book VIII
4. Dijksterhuis, E.J. (1987). Archimedes. Princeton University Press, Princeton. ISBN 0-691-08421-1. Republished translation of the 1938 study of Archimedes and his works by an historian of science.
5. Mach, E. (1907). The science of Mechanics a Critical and Historical Account of its Development. Open Court, Chicago. Republished translation of the 1883 original by Thomas J. McCormack. Ed. 3, rev.