Byrne–Simson–Heath–G-Index Concordance

Cross-Repository Alignment Across geometor/byrne, geometor/euclid, geometor/elements & geometor/codes

Because Oliver Byrne adopted Robert Simson's 1756 Glasgow numbering, several Book I definitions and axioms carry numbers distinct from the critical Heiberg/Heath text (1908, geometor/euclid) and the geometor/elements G-Index.

Simson / Byrne (1847) vs. Heiberg / Heath First Principles

Book I Definitions, Postulates & Axioms Alignment

Byrne (1847)

Simson (1756)

Heath / euclid

Mathematical Role & G-Index

Defs. I–XXVI

I.Def.1–26

I.def.1–21

Point, line, surface, circle, rectilinear triangles.

Def. XXVII

I.Def.27

I.def.22

Square (right-angled equilateral four-sided figure).

Def. XXXV

I.Def.35

I.def.23

Parallel straight lines (∥).

Postulate I

Post. 1

I.post.1

Straightedge join (G.24).

Postulate II

Post. 2

I.post.2

Straightedge extension (G.25).

Postulate III

Post. 3

I.post.3

Compass circle construction (G.26).

Axioms I–VII

Ax. 1–7

C.N.1–3 (+ interp.)

Transitivity, addition, subtraction, doubles, halves.

Axiom VIII

Ax. 8

C.N.4

Coincidence (epharmozein) implies equality.

Axiom IX

Ax. 9

C.N.5

The whole is greater (⊏) than its part.

Axiom X

Ax. 10

C.N.9 (interp.)

Two straight lines cannot enclose a space.

Axiom XI

Ax. 11

I.post.4

All right angles () are equal (G.27).

Axiom XII

Ax. 12

I.post.5

Parallel Postulate (< , G.28).

Showcase Symbolic Plates & Algebraic Number Fields

17 Calibrated Symbolic Plates + Book V Eudoxian Calculus

Byrne

Pickering Page

euclid

elements

Exact SymPy Field

Codex Unit

I.1

  1. 1 (leaf 35)

elem.1.1

G.84

$\mathbb{Q}(\sqrt{3})$

Unit II

I.4

  1. 4 (leaf 38)

elem.1.4

G.90

$\mathbb{Q}$

Unit II

I.5

  1. 5 (leaf 39)

elem.1.5

G.91

$\mathbb{Q}$

Unit II

I.15

  1. 15 (leaf 49)

elem.1.15

G.104

$\mathbb{Q}$

Unit II

I.32

  1. 33 (leaf 67)

elem.1.32

G.125

$\mathbb{Q}$

Unit II

I.46

  1. 47 (leaf 81)

elem.1.46

G.139

$\mathbb{Q}$

Unit II

I.47

pp. 48–49 (leaves 82–83)

elem.1.47

G.140

$\mathbb{Q}$

Unit II

II.4

pp. 56–57 (leaves 90–91)

elem.2.4

G.147

$\mathbb{Q}$

Unit III

II.5

  1. 58 (leaf 92)

elem.2.5

G.148

$\mathbb{Q}$

Unit III

II.11

  1. 66 (leaf 100)

elem.2.11

G.154

$\mathbb{Q}(\phi)$

Unit III

III.20

  1. 102 (leaf 136)

elem.3.20

G.185

$\mathbb{Q}$

Unit IV

III.31

  1. 115 (leaf 149)

elem.3.31

G.196

$\mathbb{Q}$

Unit IV

III.36

  1. 121 (leaf 155)

elem.3.36

G.201

$\mathbb{Q}$

Unit IV

IV.4

  1. 129 (leaf 163)

elem.4.4

G.217

$\mathbb{Q}$

Unit IV

IV.10

pp. 135–136 (leaves 169–170)

elem.4.10

G.223

$\mathbb{Q}(\phi, \sqrt{10 - 2\sqrt{5}})$

Unit IV

IV.11

  1. 137 (leaf 171)

elem.4.11

G.224

$\mathbb{Q}(\phi, \sqrt{10 - 2\sqrt{5}})$

Unit IV

V.16

  1. 187 (leaf 221)

elem.5.16

G.276

Eudoxian Semigroup (geometor/codes)

Unit V

VI.30

  1. 258 (leaf 292)

elem.6.30

G.328

$\mathbb{Q}(\phi)$

Unit VI