Lost books and forgotten theorems from ancient Greek

Lost books and forgotten theorems from ancient Greek Geometry

At least 300 known ancient Greek geometers Thales (~580 B. C. ), Pythagoras (~550 B. C. ), Anaxagoras (~480 B. C. ), Hippias (~430 B. C. ), Hippocrates, Theodorus, Archytas, Thaetetus, (Plato), Eudoxus, Menaechmus, Euclid (~300 B. C. ), Archimedes, Eratosthenes, Apollonius, Heron, Ptolemy, Pappus, Proclus, Eutocius (550 A. D. )

Ancient Histories of Mathematics Xenocrates: On Geometry (5 books) Theophrastus: History of Geometry Eudemus: History of Arithmetic History of Geometry History of Astronomy Geminus: History of Mathematics Proclus: Commentary on Euclid lost fragments lost survives

Surviving books on Geometry mostly Byzantine copies or Latin/Arabic translations Euclid: Elements, Data, Catoptrics etc Archimedes: On sphere and cylinder, Method, Measurement of circle, On spirals, On floating bodies, On equilibria, Book of Lemmas, Squaring of parabola, etc Apollonius: Conic sections Diocles: On burning mirrors (only in Arabic) (cont. )

Theodosius: Heron: Metrics, Dioptra etc. Menelaos: Ptolemy: Sphaerics (only in Arabic and Hebrew) Almagest (mostly astronomical) Serenus: Conic sections, Cylinder sections Hypsicles: Elements XIV and XV Pappus: Collectio Proclus: Commentary on book I of Elements Anthemius: On paradoxical devices Eutocius: Commentary on Archimedes

Commentaries: Pappus, Proclus, Eutocius, Simplicius various spread out An-Nayrizi (Anaritius, 865 -922) FIHRIST of al-Nadim (10ος αι. )

Lost books of Geometry - Everything before the Elements. In particular all the books of the Pythagoreans. - Anaximander: Geometry Thymaridas: History of Pythagoreans Protagoras: On Mathematics Hippocrates: Elements Democritus: On Geometry, Development Aristaeus: Conics etc. Speusippus: On Mathematics, On triangles, On pyramids Philipp Mendaes: On circles Theudius of Magna: Elements Hermotimus: On locii Leon: Elements Theophrastus: History of Geometry, On Spirals (cont. )

and, contemporary or later that Euclid -Ermarchus: On Mathematics - Euclid: Conics, Plane locii, Porisms, On Divisions, Pseudaria -Archimedes: Isoperimetrics, Semiregular polyhedra, Blocks and cylinders, On parallels etc - Demetrius Lacon: On Geometry - Apollonius: Cutting of a ratio, Cutting of an area, Given section, On vergings, On cochlias, On tangencies, The general treatise, Plane loci - Zenodorus: On isoperimetrc figures - Perseus: On spirals - Hipparchus: On chords - Charmandos: Plane locii Heron, Pappus, Porhyry, Simplicius Commmentaries on the Elements

Papyrus

The path of manuscripts from antiquity to printing ancient original (lost) Byzantine copy (9 th, 10 th c. ) Arabic translation (10 th c. and later) Latin translation (12 th , 13 th c. ) Latin translation (15 th c. ) Printed editions

A forgotten theorem

Euclid’s proof in the Elements uses the following figure

Heron Commentary on Euclid (lost) Few pages survive in the Commentary of An-Nayrizi from where ο Albertus Magnus (1193 -1280) draws, in a unique Latin manuscript with a commentary, today in Vienna

gnomon

K applications A G B L I E M H F G K D C H A E F 1 1 2 x B Heron’s theorem C bisectors K K M L A G H B L H A E F M 1 1 x C G E F B N medians x 1 C altitudes

A G B I E F K D H C Heron’s theorem generalisation

Regular (Platonic) and semiregular solids Pythagoras discovered the five solid figures, which are called mathematical, … Aetius, Collection on Philosophers And the figures on the sphere are five… Stobeus, Florigentum

Archimedes by Fetti

Ruler and compass constructions In Pappus’ Collectio Book 8

Elements Ι, 2 Transfer a segment Byrne, Euclid’s Elements

Constructions with a rusty compass Abul Wefa (940 -998) Ludovico Ferrari (1522 -1565)

… in the so called mechanical constructions, outside the geometrical rules, such as those that can be done with a single segment Pappus, Collection 8 the Greek original is lost from there onwards but an Arabic translation of Banu Musa in paragraphs 44, 45 has constructions with a «small opening compas» and with «fixed opening compass» .

perpendicular at Α Α Β Α Β midpoint of ΑΒ Μ Α Β

Equal and parallel magnitudes at unequal distances from the eye are not seen propotrionate Α Γ Ζ Ε Β θ φ ΗΔ Ο triangle (ΟΕΓ) > sector (ΟΕΖ) triangle (ΟΕΔ) < sector (ΟΗΕ) triangle (ΟΓΔ) triangle (ΟΕΔ) > sector(ΟΖΗ) sector (ΟΗΕ)

Modern trigonometric table How much is sin 1;

value of sin 1

Ptolemy’s table of chords Α sinφ Β sinθ φ θ Γ θ φ Δ

Table of chords from Almagest Γ Β Α Δ

Ptolemy’s table of chords θ φ Γ Β θ Α φ Δ

The table