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of Ancient Mathematics
On the Equilibria of Planes, Table of Contents
I Props. 6-7
I Props. 9-10
I Prop. 8
I Prop. 8 corollary and note (this corollary may be needed but is not a part of the Equilibria of Planes depending on how we interpret I Props. 6-7)
This theorem is the fundamental theorem of the Equilibria of Planes. I 9 and I 10 are based on I 4 and 5 and their corollaries, while I 11 and 12 are based on Assumption 5, while II 1 uses I prop. 9. However, every other subsequent major proposition (i.e., every proposition except the lemma II 9) either uses it or depends on it. As messay as the statement of the theorem is, the proof is straight forward. Once it is proved, Archimedes has little need refer in this work to the general principle of the balance (I 6-7), once at I prop. 15, or to the concept of inclination again.
(diagram 1 = general diagram)
8. If from some magnitude some [other] magnitude is taken away that does not have the same center as the whole, when the straight-line that joins the centers of weight of the whole magnitude and the one taken way is extended on the same side as the center of weight of the whole magnitude and with something [else] taken out from the extended [line] that joins the mentioned centers, so that it has the same ratio to the line between the centers which the weight of the magnitude taken out has to the weight of the remaining, the center of weight of the remaining magnitude is the end-point of the [line] taken out.
(diagram 1) Let there be the center of weight, G, of some magnitude, AB, and let AD, whose center of weight is E, be taken away from AB, and with EG joined and extended, let GZ be taken out, having the same ratio to GE that magnitude AD has to DH. One must prove that the center of weight of magnitude DH is point Z.
(diagram 2 = general diagram) For otherwise, still, if it is possible, let it be point Q. And so, since the center of weight of magnitude AD is E, while that of DH is point Q, the center of weight from both magnitudes AD, DH will be on segment EQ, so that the segments of it alternate in the same ratio as the magnitudes. Thus point G will not be in the proportional cut as that mentioned. Therefore G is not the center of the magnitude composed from AD, DH, i.e., AB. But it is, for it was supposed. Therefore Q is not the center of weight of magnitude DH.
Note: The theorem does not explicitly prove that E G and Q lie on a straight line. This would follow from I 4, 5, if those theorems had proved it. On Floating Bodies II 2 refers to the required the first part of this theorem as demonstrated: “For this has been proved in the Elements of Mechanics, that if a magnitude is taken away from a magnitude that does not have the same center of weight as the whole magnitude, the center of the weight of the remaining will be on the straight-line joining the center of the whole magnitude and the magnitude taken away when it is extended on the same side where the center of weight of the whole magnitude is.”
Corollary: A very trivial corollary may required in several theorems, e.g. I prop. 13, 14, II prop. 3. Although this corollary does not appear in the Equilibria of Planes, I shall refer to it as required, although a good argument can be made that this is just what I props. 6-7 mean.
If two magnitudes are put together, the center of weight will be on the line joining them, such that the distance of the centers of the magitudes to the center of the weight of the whole are in inverse proportion to the weights.
We can imagine proving this directly from I props. 6 and 7, but the proof would probably look something like the proof of I prop. 8, anyway. So here is a very trivial proof.
Let A and B be magnitudes with centers of their respective weights a and b. Suppose that the center of weight of AB is c and that c is not on line ab. By I prop. 8, the center of B is on the extension of ac on the opposite side of c from a, such that ac : cb = B : A. If b is on the extension of ac opposite a, then c is on ab between a and b and in the required ratio.
The basic issue may be put this way. How are we to regard I Props. 6-7? Are they about establishing the principle of the balance, as most readers assume, or are they really about establishing whee the center of weight is of a body given the center of weight of its parts. It is true that the parts may be moved, tossed around the world separately, etc., but this contributes to the generality of the theorem if taken as a theorem of the centers of weight of any system of bodies. Indeed, it may be crucial to the theorem that when one conceives of the center of weight of two parts of a body that one thereby chops them in half so that one can put the center of weight in the appropriate position. Furthermore, the theorem may state that it concerns the inclination of two bodies, but the actual conclusion is that a certain point is the center of weight so that they incline equally at the point. Surely, Archimedes, so this argument goes, does not need to prove again the penultimate claim of the proof.
For the absence of this lemma as casting doubt on Archimedes' authorship of at least important parts of Book 1, cf. L. Berggren, “Spurious Theorems in Archimedes' Equilibrium of Planes: Book I,” Archive for History of Exact Sciences, 16 (1976): 87-103.