Content of type 'notes'

Measurement of a Circle — Philological & Mathematical Notes

Fragmentary Byzantine Recension: As Heiberg (heiberg:1880) and Heath (heath:1897, pp. xxiv, 93) observe, our surviving Greek text of Κύκλου μέτρησις bears marks of a late editor or epitomator (connected with the sixth-century school of Isidore of Miletus): the Syracusan Doric dialect has been stripped in favor of Attic/Koine forms (κύκλος, περιφέρεια, διάμετρος), and Proposition 2 () stands illogically before Proposition 3, upon whose bound it depends, while lacking Archimedes’ characteristic inequality phrasing (“approximately” or upper/lower bounds).

On the Equilibrium of Planes — Philological & Mathematical Notes

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:

Quadrature of the Parabola — Philological & Mathematical Notes

As Heath (heath:1897, p. 233) notes, the manuscript title Τετραγωνισμὸς παραβολῆς cannot have been Archimedes' own original heading: throughout the body of the treatise (and in Conoids and Spheroids and The Method), Archimedes never writes παραβολή for the curve—a term coined shortly afterward by Apollonius of Perga from Pythagorean application of areas—but always ὀρθογωνίου κώνου τομή (section of a right-angled cone, τμῆμα ὀρθογωνίου κώνου τομῆς).

Sphere and Cylinder, On Spirals & Conoids — Philological & Mathematical Notes

In Tusculan Disputations V.23, Cicero records how, serving as quaestor in Sicily in 75 BC, he cleared away brambles outside the Agrigentine Gate of Syracuse and rediscovered the forgotten tomb of Archimedes by recognizing on a small column the sphere and circumscribed cylinder of Sphere and Cylinder I.34 Corollary, accompanied by worn Greek iambic verses declaring that the cylinder’s volume and surface stood in the sesquialter ratio to the sphere. As The Method, Prop. 2 reveals in Archimedes’ own words, he discovered the volume first by weighing slices on a virtual balance, and then conjectured by analogy with Measurement of a Circle Prop. 1 ()!

Book of Lemmas (Liber Assumptorum) — Philological & Mathematical Notes

No Greek manuscript of the Book of Lemmas survives. Its preservation is owed to ninth-century Baghdad geometer Thābit ibn Qurra (836–901 AD), whose Arabic translation (Kitāb al-Maʾkhūdhāt li-Arshimīdis) was commented upon in the tenth century by Abū al-Ḥasan ʿAlī ibn Aḥmad al-Nasawī (“Almochtasso”). In 1661, Abraham Ecchellensis and Giovanni Alfonso Borelli discovered an Arabic codex in the Laurentian Medici Library at Florence and published the Latin editio princeps (ecchellensis-borelli:1661), later re-edited by Heiberg (heiberg:1881, Vol. II, pp. 426–445; 2nd ed. pp. 510–525).

Semi-Regular Solids, Stomachion & Sand-Reckoner — Philological & Mathematical Notes

In Synagoge V.19 (pappus:synagoge:v), Pappus lists the thirteen solids in increasing order of complexity within the tetrahedral/cubic and icosahedral families (8, 14, 14, 14, 26, 26, 38, 32, 32, 32, 62, 62, 92 faces) and computes their vertex and edge numbers by face-incidence counting (). During the Renaissance, Piero della Francesca and Luca Pacioli (geometor:pacioli, De Divina Proportione, Venice 1509, with woodcuts after Leonardo da Vinci’s corpora vacua skeletal drawings) rediscovered the truncated and elevated (abscisi / elevati) members directly from the five Platonic solids of Euclid Book XIII, before Johannes Kepler (Harmonices Mundi, Linz 1619) restored all thirteen—including the enantiomorphic snub cube and snub dodecahedron—by systematic angular-defect analysis ( at every convex vertex, Eucl. XI. 21).