Book of Lemmas (Liber Assumptorum) — Analytical Summary

Scope: Heiberg Vol. II, pp. 426–445 · Heath (1897), pp. 301–318 · Propositions 1–15

Planar Gemstones of Circle Tangency, Curvilinear Quadrature & \(\mathbb{Q}(\phi)\)

Unlike the exhaustion treatises, the Book of Lemmas achieves instantaneous exact quadratures of curvilinear figures (Arbelos in Prop. 4, Salinon in Prop. 14) by algebraic cancellation of semicircular areas (Eucl. XII. 2), alongside projective/homothetic circle-contact invariants (Props. 1, 5, 6), kinematic neusis angle trisection (Prop. 8), orthogonal-chord quadrance conservation (Prop. 11), and golden-ratio decagon geometry (Prop. 15).

Complete Register of all 15 Propositions

Prop.	Theme	Exact Geometric Statement
1	Homothetic Tangent Circles	Parallel diameters CDparallelEF in tangent circles have collinear extremities ADF through contact A
2	Semicircle Tangents	Secant AT to intersection of tangents at BD bisects altitude DE ()
3	Equal-Arc Chord Lemma	Reflection of altitude foot gives for equal arcs
4	The Arbelos (ἄρβηλος)	Curvilinear area on equals the circle on tangent altitude PN
5	Archimedes’ Twin Circles	Circles tangent to CD, outer semicircle, and either inner semicircle are equal:
6	Inscribed Arbelos Circle	Circle tangent to all 3 semicircles satisfies ( for )
7	Square Circles	Circumcircle of a square is double its incircle ()
8	Neusis Angle Trisection	Inserting between circle and extended diameter DOE gives
9	Orthogonal Secants	Arc sum identity for perpendicular chords
10	Tangent-Secant Bisection	Harmonic/angular properties of tangent & secant chords (cf. Apollonius Contacts)
11	Orthogonal Chords Quadrance	If at O, then
12	Semicircle Polar Orthocenter	Tangents at PQ meeting at T and chords AQcapBPR satisfy
13	Diameter Projections	Perpendiculars from diameter ends onto a chord cut off equal outer segments ()
14	The Salinon (σάλινον)	Symmetric 4-semicircle salt-cellar equals the circle on axial segment CF
15	Pentagon, Decagon &	Pentagon/decagon secant CDE meets diameter at E with via Eucl. XIII. 9 ()