On the Equilibrium of Planes — Philological & Mathematical Notes

1. Preservation of Syracusan Doric Dialect

Unlike Measurement of a Circle and On the Sphere and Cylinder, On the Equilibrium of Planes escaped the sixth-century Atticizing recension of Isidore of Miletus and preserves Archimedes' authentic Syracusan Doric Greek (heiberg:1881, Vol. II, pp. 142–241). For instance, the enunciation of Book I, Prop. 9 (Heiberg II, 162, verified from 300 DPI scan) reads:

Παντὸς παραλληλογράμμου τὸ κέντρον τοῦ βάρεός ἐστιν ἐπὶ τᾶς εὐθείας τᾶς ἐπιζευγνυούσας τὰς διχοτομίας τᾶν κατ᾽ ἐναντίον τοῦ παραλληλογράμμου πλευρᾶν.

Notice the Doric genitive singular τᾶς εὐθείας (Attic τῆς εὐθείας) and genitive plural τᾶν … πλευρᾶν (Attic τῶν … πλευρῶν), matching the Doric forms in On Spirals, The Sand-Reckoner, and Quadrature of the Parabola.

2. Mach’s Critique of Book I, Prop. 6 & Archimedes’ Implicit Postulate

Ernst Mach (Die Mechanik in ihrer Entwicklung, 1883) famously questioned whether replacing two weights A, B at E, D by spread-out arrays of unit weights centered at E and D tacitly assumes the very lever law being proved. As Heath (heath:1897, pp. 192–193) and Giovanni Vailati demonstrated, Archimedes’ derivation is strictly non-circular once one recognizes Postulate 6 (magnitudes equal to balancing magnitudes balance at the same distances) together with the definition of centre of gravity established in Archimedes’ lost preliminary tract On Balances (Περὶ ζυγῶν, cited by Pappus in pappus:synagoge:viii): replacing a subsystem of equal weights by an equal composite weight located at its midpoint centre of gravity relies solely on equal-arm symmetry (Prop. 4–5), never on unequal-arm moments!

3. GEOMETOR Cross-Corpus Concordance

• Encyclopedia of Triangle Centers (`geometor/etc`) (geometor:etc): Book I, Propositions 13–14 constitute the earliest rigorous proof in mathematical history that the three medians of a triangle are concurrent—defining Kimberling’s Centroid `X(2)` with homogeneous barycentric coordinates and trilinear coordinates . Indeed, Möbius’s entire 1827 barycentrischer Calcul is the direct algebraic descendant of Archimedes’ Book I.

• Euclid (`geometor/euclid`) (geometor:euclid): Heath cites archimedes:equilibrium-of-planes:i-7 in his commentary on Euclid Book V as the paradigm of applying Eudoxean proportion theory to continuous physical magnitudes.

• The Method (`Unit 6`): Every mechanical discovery in Quadrature of the Parabola and The Method rests directly upon Book I, Props. 6–8 & 14 and Book II, Prop. 8.