## 10 Apr 2014

### Archimedes’ “On the Equilibria of Planes” Postulates

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Archimedes

On the Equilibrium of Planes

or

The Centres of Gravity of Planes,

Book I

Postulates

We will quote from the Heath text, but we will use Henry Mendell’s formulations. To do this, we will need to understand the meanings of his terms and symbols.

“inclination” and “balance” / “equal-inclination” / “equilibrium”

One weight is set against another on a lever. One weight might incline the lever downward on that side. Or, the lever might remain level, in which case it is balanced, or in equalibrium. Heath uses the term equalibrium, while Mendell uses “equal-inclination”.

A, B

Capital letters A and B are the counter-balancing weights.

a, b

Lower case letters a and b are the distances of A and B respectively to the fulcrum.

R

R is inclination of one side of the lever. R(A, a) is the inclination of weight A at length a. So if R(A,a) = R(B,b), that means weights A and B equally incline at distances a and b, respectively. If R(A,a) > R(B,b), then weight A inclines more than B at distances a and b, respectively.

Mendell then formulates one basic but unstated assumption in the text, which he names CW1:

CW1: Every magnitude or collection of magnitudes has exactly one center of weight. (Mendell)

Mendell also offers definitions for ‘center of weight. Archimedes’ text never defines this term, and Eutocius’ formulation is inadequate.

Center of Weight: “a point from which a freely hanging body is stable, no matter how it is positioned about the point.” (Mendell)

Center of Weight: “a point such that if a body is hung freely from any point on the body, a perpendicular from the point of suspension to the horizon will go through it.” (Mendell)

Postulates / Assumptions [A]

[All of the following is quotation.]

Postulate 1

A1a. Equal weights at equal distances from the fulcrum are in equilibrium. (Heath)

1a. A = B & a = b => R(A,a) = R(B,b)

(Mendell)

A1b. … and equal weights at unequal distances are not in equilibrium
but incline towards the weight which is at the greater distance. (Heath)

A1b. A = B & a > b => R(A,a) > R(B,b)

(Mendell)

2. If, when weights at certain distances are in equilibrium, something be added to one of the weights, they are not in equilibrium but incline towards that weight to which the addition was made. (Heath)

A2. R(A,a) = R(B,b) => R(A+C,a) > R(B,b)

(Mendell)

3. Similarly, if anything be taken away from one of the weights, they are not in equilibrium but incline towards the weight from which nothing was taken. (Heath)

A3. R(A,a) = R(B,b) => R(A-C,a) < R(B,b)

(Mendell)

Postulate 4
4. When equal and similar plane figures coincide if applied to one another, their centres of gravity similarly coincide. (Heath)

4. When equal and similar plane figures fit on one another the centers of weights also fit on one another. (Mendell)

[See Mendell’s Eutocius image:]

http://web.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Archimedes/De%20Planorum%20%20Aeqilibriis/Intro.I.Props1-5/Intro.I.Props1-5.html#Eutocius2

Postulate 5
5. In figures which are unequal but similar the centres of gravity will be similarly situated. By points similarly situated in relation to similar figures I mean points such that, if straight lines be drawn from them to the equal angles, they make equal angles with the corresponding sides. (Heath 189)

[See Mendell’s Eutocius image:]

http://web.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Archimedes/De%20Planorum%20%20Aeqilibriis/Intro.I.Props1-5/Intro.I.Props1-5.html#Eutocius2

6. If magnitudes at certain distances be in equilibrium, (other) magnitudes equal to them will also be in equilibrium at the same distances. (Heath 190)

A6. R(A,a) = R(B,b) & C = A => R(C,a) = R(B,b)

(Mendell)

Postulate 7
7. In any figure whose perimeter is concave in (one and) the same direction the centre of gravity must be within the figure."

[See Mendell’s diagram:]

http://web.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Archimedes/De%20Planorum%20%20Aeqilibriis/Intro.I.Props1-5/Intro.I.Props1-5.html#Eutocius4

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

http://web.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Archimedes/De%20Planorum%20%20Aeqilibriis/ArchimedesEP.Contents.html

see other great works by Mendell on ancient mathematics here:

Vignettes of Ancient Mathematics

http://web.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/VignettesAncientMath.html