Definitiones, Catoptrica & Dioptra: Genetic Axiomatics, Universal Gnomon, Shortest-Path Optics & Mountain Tunneling

Hero’s Genetic Definitions & Universal Gnomon (§58), Proclus on Hero’s Direct Proofs, the Variational Shortest-Path Proof of Reflection (Catoptrica Prop. 4), and Two-Portal Mountain Tunneling (Dioptra Ch. 15) — Hultsch (1864), pp. 7–23 · Schmidt (1900), pp. 324–372 · Schöne (1903), pp. 238–240

Note

Edition Track (English): Original GEOMETOR scholarly translations directly from Friedrich Hultsch (1864, collated with J. L. Heiberg 1912), Wilhelm Schmidt (Teubner 1900), and Hermann Schöne (Teubner 1903), verified against 300 DPI archival page scans. Hover or click any highlighted geometric symbol across the three interactive diagrams (AEFD, CZK, BND) to inspect Hero’s geometry in motion. Switch tracks above for Greek (Teubner/Hultsch), Summary, or Notes.

Definitiones: Genetic Motion, Axial Symmetry & Postulate Economy

[Folio Hultsch 7 | p. 7]

Addressed to Dionysius as a systematic philosophical primer to Euclid's Elements, Hero's Definitions of the Terms of Geometry (Ὅροι τῶν γεωμετρίας ὀνομάτων) enriches Euclid’s static Platonic boundary definitions with kinematic generation, Archimedean minimal-length characterization, and rotational invariance:

  • Definition 1 — Point (*Σημεῖον*, Heiberg §1 / Hultsch p. 7 §1–2): “A point is that of which there is no part (:just:`I.def.1`), or a dimensionless limit or boundary of a line, incorporeal and partless and magnitudeless, possessing position alone, apprehensible by intellectual intuition (*νοητῇ μόνῃ ἐπιβολῇ) alone.”*

  • Definition 2 — Flowing Line (*Γραμμή*, Heiberg §2 / Hultsch p. 7 §3): “A line is breadthless length (:just:`I.def.2`), or the first thing taking subsistence in magnitude, or that which is extended and divisible in one dimension; and it arises from a point flowing (*σημείου ῥυέντος) from above downward under the conception of continuity, and is bounded by points, being itself the boundary of a surface.”*

  • Definition 4 — Straight Line via Tension, Shortest Path & Axial Rotation (*Εὐθεῖα γραμμή*, Heiberg §4 / Hultsch p. 8 §5): “A straight line is one which lies evenly with the points on itself (:just:`I.def.4`), being upright and as it were stretched tight toward its extremities; or the **shortest line among those having the same extremities* (Archimedes, Sphere and Cylinder I, Assump. 1); or that which, when its extremities remain fixed, as it is rotated (στρεφομένη) in the same space about its extremities as about poles, always occupies the very same place—for all parts of a straight line superpose upon equal parts.”*

[Folio Hultsch 15 | p. 15]

  • Definition 27 — Genetic Compass Definition of Circle (*Κύκλος*, Heiberg §27 / Hultsch p. 15 §29): “A circle is a plane figure contained by one line, called the circumference, unto which all straight lines falling upon it from one point among those lying within the figure are equal to one another (:just:`I.def.15`). And a circle comes into being when, a straight line remaining in the same plane, **one extremity remains motionless while the other moves and is carried round* until it is restored again to the same place whence it began to be carried; the motionless extremity is called the center, and the line traced by the other extremity the circumference.”*

  • Definition 67 — Diagonal / Diameter (*Διάμετρος / Διαγώνιος*, Heiberg §67 / Hultsch p. 22 §68): “A diameter of a rectilinear and four-sided and polygonal figure is the straight line drawn from an angle to the opposite angle; and the very same line is also called a diagonal.”

  • Definition 76 — Sphere (*Σφαῖρα*, Heiberg §76 / Hultsch p. 23 §77): Unites Euclid’s stereometric revolution of a semicircle ([XI.def.14]) with the locus of equidistant radii in 3D space—a synthesis praised by Proclus and echoed by Bertrand Russell and David Hilbert.

Proclus’s Testimonies to Hero’s Commentary on Euclid: In his Commentary on the First Book of Euclid's Elements (Friedlein pp. 196, 305–307, 340–344), Proclus records two foundational axiomatic reforms from Hero’s lost Elements commentary (partially preserved in Arabic by an-Nayrīzī):

  1. Three-Postulate Economy: Hero restricted the postulates exclusively to constructive operations (Postulates 1, 2, 3: straightedge joining, straightedge extension, compass circle), proving Postulate 4 (equality of all right angles) deductively by superposition and adopting Posidonius's equidistant definition of parallels.

  2. Direct Ostensive Proofs over *Reductio ad Absurdum*: In propositions such as [I.19] and [I.25], Hero replaced Euclid’s indirect proofs by contradiction with direct constructive proofs, and proved all ten geometric-algebra identities of Book II ([II.2]–[II.10]) deductively from [II.1] without needing separate multi-square diagrams!

Definitiones §58: The Universal Gnomon & Golden Self-Similarity

[Folio Hultsch 21 | p. 21]

Definition 58 — Universal Gnomon (*Καθόλου Γνώμων*, Heiberg §58 / Hultsch p. 21 §59):

“Primarily, in a parallelogrammic area, a :term:`gnomon <gnomon>` is any one of the parallelograms about the diameter together with the two complements (:just:`II.def.2`). **Universally, however, a gnomon is anything whatever which, when added to anything—whether number or figure—makes the whole similar to that to which it has been added* (καθόλου δὲ γνώμων ἐστὶ πᾶν ὃ προσλαβὸν ὁτιοῦν, ἀριθμὸν ἢ σχῆμα, ποιεῖ τὸ ὅλον ὅμοιον ᾧ προσείληφεν).”*

This single sentence generalizes Euclid’s parallelogram gnomon into the universal law of scale-invariant morphogenesis across arithmetic (polygonal numbers), geometry, and botanical phyllotaxis. Its supreme planar realization is the Golden Rectangle over \(\mathbb{Q}(\phi)\) (\(\phi = \frac{1 + \sqrt{5}}{2}\)):

Hero Definitiones §58 — Universal Gnomon Realized on the Golden Rectangle in Q(phi) ( M C ): y**2 + (x - 1/2)**2 - 5/4 = 0 ( E B ): y**2 + (x - sqrt(5)/2 - 1/2)**2 - (-1/2 + sqrt(5)/2)**2 = 0 [ A B ]: y = 0 [ D E ]: x = 0 [ G H ]: x - 1 = 0 [ D C ]: y - 1 = 0 [ P Q ]: x - sqrt(5)/2 - 1/2 = 0 [ U V ]: y - sqrt(5)/2 + 1/2 = 0 / A E /: |seg| = 1/2 + sqrt(5)/2 / B E /: |seg| = -1/2 + sqrt(5)/2 A = (0, 0) B = (1, 0) D = (0, 1) C = (1, 1) M = (1/2, 0) E = (1/2 + sqrt(5)/2, 0) F = (1/2 + sqrt(5)/2, 1) G = (1/2 + sqrt(5)/2, -1/2 + sqrt(5)/2) H = (1, -1/2 + sqrt(5)/2) E1 = (1/2 - sqrt(5)/2, 0) A B D C M E F G H E1

  1. Construct the unit square gnomon ABCD (\(AB = BC = 1\)), bisect AB at M, and swing compass circle (MC) of radius \(MC = \frac{\sqrt{5}}{2}\) onto AB extended to E ([II.11]), giving AE \(= \frac{1}{2} + \frac{\sqrt{5}}{2} = \phi\) and BE \(= \phi - 1 = \frac{1}{\phi}\).

  2. Erect the perpendicular at E to complete the whole rectangle AEFD. Adjoining the unit square gnomon ABCD to the seed rectangle BEFC (sides \(1\) and \(\phi - 1\)) produces a whole rectangle AEFD (sides \(\phi\) and \(1\)) strictly similar to the seed:

    \[\frac{AE}{AD} = \frac{\phi}{1} = \frac{1}{\phi - 1} = \frac{BC}{BE}, \qquad \frac{Q(A, E)}{Q(A, D)} = \frac{Q(B, C)}{Q(B, E)} = \phi^2 = \frac{3 + \sqrt{5}}{2}.\]

  3. Swinging circle (EB) inside BEFC cuts off a second square gnomon \(BEGH\) (G, H), leaving residual rectangle \(HGFC\) with the exact same aspect quadrance \(\phi^2\) ad infinitum, while circle (MC) also meets AB at the conjugate point E1 (\(x = -\phi^{-1}\)), forming a golden projective range \((E_1, A; B, E)\)!

Catoptrica Prop. 4: Variational Shortest-Path Proof of the Law of Reflection

[Folio Schmidt 368 | p. 368]

In his Catoptrica—preserved in medieval Latin translation by William of Moerbeke (Schmidt 1900, pp. 324–329) and verbatim in the original Greek by Olympiodorus (Commentary on Aristotle’s Meteorology III.2, Schmidt 1900, pp. 368–372, Fig. 92)—Hero delivers the first variational extremum principle in the history of physics, sixteen centuries before Fermat’s Principle of Least Time (1662):

“Straight visual rays proceeding from the eye to mirrors and the objects seen are bent (:term:`reflected <anaklasis>`) at equal angles to themselves. Why then are they not also bent at unequal angles? To this we shall reply that **Nature does nothing in vain nor labors pointlessly* (ἡ φύσις οὐδὲν μάτην ἐργάζεται οὐδὲ ματαιοπονεῖ); and if she refuses to labor pointlessly, it is manifest that she sees by the shortest paths (διʼ ἐλαχίστων) and not by long detours. And since straight lines bent at equal angles are shorter than straight lines bent at unequal angles, for this reason reflection takes place at equal angles, and not at unequal angles.”*

(Diagram Mapping: AB = plane mirror; eye C (Hero’s \(\Gamma\) ); seen object D (Hero’s \(\Delta\) ); perpendicular foot H; virtual reflected point K; equal-angle point E; arbitrary unequal-angle point Z ).

Hero Catoptrica Prop. 4 — Variational Shortest-Path Proof of the Law of Reflection [ A B ]: y = 0 [ D F ]: x - 10 = 0 [ C K ]: 9*x + 9*y - 36 = 0 [ E D ]: -6*x + 6*y + 24 = 0 / C E /: |seg| = 3*sqrt(2) / E D /: |seg| = 6*sqrt(2) / E K /: |seg| = 6*sqrt(2) / C Z /: |seg| = 3*sqrt(5) / Z D /: |seg| = 3*sqrt(5) / Z K /: |seg| = 3*sqrt(5) / D H /: |seg| = 6 / H K /: |seg| = 6 A = (0, 0) B = (12, 0) C = (1, 3) D = (10, 6) Z = (7, 0) H = (10, 0) K = (10, -6) E = (4, 0) A B C D Z H K E

Hero’s Geometric Proof (Schmidt pp. 370–372):

  1. Let the plane mirror be AB, the eye C, and the object seen D. Let ray CE strike the mirror at E and reflect as ED making equal grazing angles with the mirror (\(\angle \Gamma EA = \angle \Delta EB\)); and let another ray CZ strike any other point Z and reflect as ZD at unequal angles. I say that \(\Gamma E + E\Delta < \Gamma Z + Z\Delta\).

  2. Drop the perpendicular DH from D onto mirror AB ([I.12]), produce it straight through to K so that HK = DH ([I.3]), and join EK and ZK.

  3. In right triangles \(\Delta HE\) and \(KHE\), DH = HK, \(HE\) is common, and the angles at H are right; therefore by [I.4], ED = EK and \(\angle \Delta EH = \angle KEH\).

  4. Unfolding Collinearity: By hypothesis \(\angle \Delta EH = \angle \Gamma EA\); hence \(\angle \Gamma EA = \angle KEH\)—they are vertical angles, so :g:`CEK <g2:[ C K ]>` is a single unbroken straight line ([I.14]–[I.15])!

  5. Again, in right triangles \(\Delta HZ\) and \(KHZ\), DH = HK and \(ZH\) is common, so ZD = ZK ([I.4]).

  6. Triangle Inequality Climax: In triangle CZK, any two sides are strictly greater than the remaining side ([I.20]):

    \[\Gamma Z + ZK > \Gamma K.\]

    Substituting \(ZK = Z\Delta\) and \(\Gamma K = \Gamma E + EK = \Gamma E + E\Delta\) yields immediately:

    \[\Gamma Z + Z\Delta \;>\; \Gamma E + E\Delta. \qquad \text{Q.E.D.}\]

(In our exact coordinates \(C=(1,3), D=(10,6), E=(4,0), Z=(7,0)`*, the unfolded path has squared length* :math:`(|CE|+|ED|)^2 = (9\sqrt{2})^2 = 162\) with equal spreads \(s = \frac{1}{2}`*, whereas the perturbed path has squared length* :math:`(|CZ|+|ZD|)^2 = (6\sqrt{5})^2 = 180 > 162\) !)

Dioptra Ch. 15: Tunneling a Mountain from Both Ends simultaneously

[Folio Schöne 238 | p. 238]

In his treatise On the Dioptra (Περὶ διόπτρας, Schöne 1903, pp. 188–314), Hero designs a precision geared sighting and leveling theodolite (dioptra) and demonstrates in Chapter 15 (pp. 238–240, Fig. 95) how engineers can bore a straight underground tunnel or aqueduct (ὑπόνομος) through an opaque mountain simultaneously from both portals—providing the mathematical paradigm behind masterpieces like Eupalinus of Megara's Tunnel of Samos:

Problem 15: “To bore straight through a mountain when the mouths (:g:`B <g3:B>`, :g:`D <g3:D>`) of the tunnel upon the mountain are given.”

(Diagram Mapping: Portals B \(= (0,0)\) and D \(= (12,5)\) have true tunnel length \(13\); orthogonal dioptra traverse stations E, Z, H, G (Hero’s \(\Theta\) ), K, M skirt the mountain foot; virtual vertex N \(= (12,0)\) completes the hidden right triangle BND; and exterior similar triangles BOX and DPQ align both digging crews along XQ ).

Hero Dioptra Ch. 15 — Two-Portal Mountain Tunneling by Orthogonal Traverse [ E B ]: y = 0 [ E Z ]: x + 2 = 0 [ Z H ]: y + 3 = 0 [ H G ]: x - 16 = 0 [ G K ]: y - 8 = 0 [ C F ]: x - 14 = 0 [ D J ]: y - 5 = 0 [ D N ]: x - 12 = 0 [ X Q ]: -10*x + 24*y = 0 / B D /: |seg| = 13 / B E /: |seg| = 2 / E Z /: |seg| = 3 / Z H /: |seg| = 18 / H G /: |seg| = 11 / G K /: |seg| = 2 / K M /: |seg| = 3 / M D /: |seg| = 2 B = (0, 0) D = (12, 5) E = (-2, 0) Z = (-2, -3) H = (16, -3) G = (16, 8) K = (14, 8) M = (14, 5) N = (12, 0) O = (-6, 0) X = (-6, -5/2) P = (18, 5) Q = (18, 15/2) B D E Z H G K M N O X P Q

Hero’s Survey & Alignment Algorithm:

  1. Orthogonal Traverse Around the Mountain: Starting from portal B, sight an arbitrary baseline BE with the dioptra, turn successively through exact right angles around the base of the mountain along segments EZ, ZH, HG, GK, and slide the dioptra along the perpendicular line through K until the second portal D appears directly through the orthogonal sights at station M (giving MD parallel to BE and ZH).

  2. Signed Coordinate Summation: Summing the signed East–West and North–South legs determines the invisible right-angled triangle BND inside the rock:

    \[BN = \sum \Delta x = (-2) + 18 + (-2) + (-2) = 12, \qquad ND = \sum \Delta y = (-3) + 11 + (-3) = 5,\]

    whence by [I.47] the exact underground tunnel distance BD is \(\sqrt{12^2 + 5^2} = 13\)!

  3. Exterior Similar-Triangle Sighting: Outside portal B, produce \(NB\) to O and erect perpendicular \(OX\) in the exact ratio \(BO : OX = BN : ND = 12 : 5\), constructing right triangle BOX. Likewise outside portal D, produce \(MD\) to P and erect \(PQ\) with \(DP : PQ = 12 : 5\), constructing DPQ.

  4. Guaranteed Underground Rendezvous: Because BOX and DPQ are similar to BND ([VI.4]), their hypotenuses \(XB\) and \(DQ\) lie on the exact same straight line XQ as the subterranean chord BD ([I.14]). Sighting backward along straightedge rules aligned with \(XB\) and \(QD\) (κανόνας παραβάλλοντες), both excavation crews meet face-to-face in the heart of the mountain—and if vertical ventilation shafts (φρέατα) are needed from the ridge above, the same ratio locates every shaft directly over the tunnel axis!