Unit V: Book V — Color-Coded Symbolic Algebra of Eudoxian Proportions¶
Abstract Magnitude Tokens & Equimultiple Calculus (Pickering pp. 153–212, Leaves 187–246)
Byrne’s Radical Leap in Book V: Abstract Colored Magnitude Tokens (p. 154)¶
In Book V, Eudoxus and Euclid generalize ratio and proportion from spatial lines to arbitrary continuous magnitudes of the same kind—lengths, planar areas, polyhedral volumes, angles, or time intervals. Recognizing that line diagrams obscure this universality, Oliver Byrne abandons spatial figures altogether across pages 153–212 and introduces four typed colored magnitude symbols:
First magnitude: (
:byrne-mag:`red circle`)Second magnitude: (
:byrne-mag:`yellow square`)Third magnitude: (
:byrne-mag:`blue diamond`)Fourth magnitude: (
:byrne-mag:`black shield`)
Definition V (:just:`V.def.5` · Eudoxus’s Criterion of Same Ratio, p. 154)
Four magnitudes are in the same ratio,
: ∷ : ,
when for every pair of positive integer equimultiples $m, n \in \mathbb{N}^+$:
if $m \cdot$ ⊏ $n \cdot$ , then $m \cdot$ ⊏ $n \cdot$ ;
if $m \cdot$ = $n \cdot$ , then $m \cdot$ = $n \cdot$ ; and
if $m \cdot$ ⊐ $n \cdot$ , then $m \cdot$ ⊐ $n \cdot$ .
Core Laws of Colored Ratio Algebra (V.11, V.16, V.18, V.22)¶
Proposition XI (Transitivity of Ratios · ``G.271``). If : ∷ : and : ∷ : , ∴ : ∷ : . Q.E.D.
Proposition XVI (:term:`Alternando <alternando>`, p. 187 · ``G.276``). If four magnitudes of the same kind be proportionals, they shall also be proportionals when taken alternately.
Given. : ∷ : .
Demonstration (:just:`V.15`, :just:`V.11`, :just:`V.14`, :just:`V.def.5`). For arbitrary equimultiples $m, n$:
$m \cdot$ : $m \cdot$ ∷ : ∷ $n \cdot$ : $n \cdot$ .
By V.14, according as $m \cdot$ is ⊏, =, or ⊐ $n \cdot$ , so is $m \cdot$ ⊏, =, or ⊐ $n \cdot$ .
∴ : ∷ : . Q.E.D.
Proposition XVIII (*Componendo* · ``G.278``). If : ∷ : , ∴ ( + ) : ∷ ( + ) : . Q.E.D.
Proposition XXII (*Ex Aequali* · ``G.282``). If (, , ) and (, , ) have pairwise equal ratios, ∴ : ∷ : . Q.E.D.