Critical & Symbolic Notes — Unit I: The Four Classical Centers (§§ 1–9)¶
Philological Apparatus, 1767 Folio Errata & Symbolic Verification
1. Concurrency Foundations: Euclid, Ceva (1678) & Euler (1765)¶
In § 1 (p. 103), Euler takes for granted the concurrency of all four cevian and perpendicular families. Historically:
The concurrency of the internal angle bisectors at \(G = X(1)\) and of the perpendicular bisectors at \(H = X(3)\) appears explicitly in Euclid’s Elements [IV. 4] and [IV. 5].
The concurrency of the medians at the center of gravity \(F = X(2)\) goes back to Archimedes (On the Equilibrium of Planes, I.13–14).
Unified ratio concurrency for medians, altitudes, and internal/external angle bisectors was established synthetically via static moments and ratio products by Giovanni Ceva in De lineis rectis se invicem secantibus statica constructio (Milan, 1678), paired with transversal collinearity from Menelaus of Alexandria’s Sphaerica (Book III, Lemma 1). Notably, Euclid’s Elements contains no proposition naming the orthocenter \(E = X(4)\), making Euler’s uniform treatment of \((E, F, G, H)\) in E325 the foundational watershed of modern triangle-center geometry.
2. Three Typographical Slips Verified on the 1767 St. Petersburg Folio¶
Direct visual inspection of the 1767 Novi Commentarii page scans reveals three minor compositorial slips in §§ 4–8, all preserved with [sic; lege ...] annotations in the Latin track:
p. 104, § 4, line 5: Printed
rectae AM, BN et CN in latera opposita sunt normales. Because the altitude from \(C\) onto \(AB\) has foot \(P\) (as Euler immediately writes in § 5EPand § 7CP),CNis a slip forCP.p. 106, § 7, lines 7–10: In computing the abscissa of the centroid \(F\), Euler denotes the midpoint of \(AB = c\) by lowercase \(c\), writes \(cP = \frac{bb - aa}{2c}\), and then dividing by \(3\) prints
bcinstead of6cthree times in succession:\[cQ = \frac{bb - aa}{bc} \quad [\text{lege } 6c], \qquad AQ = \frac{3cc + bb - aa}{bc} \quad [\text{lege } 6c].\]Handwritten
6andbare nearly indistinguishable in 18th-century cursive; Euler uses the true denominator6cwhen expanding \(EF^2\) on p. 109 (§ 12) and \(AQ - AS = \frac{bb - aa}{6c}\) on p. 112 (§ 16).p. 106, § 8, last line: Printed
si a latere BC ex A aequale interuallum c - x rescindatur. Since \(A\) does not lie on \(BC\) and Euler has just noted \(BR = c - x\),ex Ais a slip forex B.
3. Exact Rational Verification on the Canonical (15, 13, 14) Triangle¶
In geometor.feuerbach.euler.construct_euler_1765_centers(), placing \(A = (0, 0)\), \(B = (14, 0)\), \(C = (5, 12)\) gives integer side lengths \(a = 15\), \(b = 13\), \(c = 14\) and integer area \(\mathcal{A} = 84\). Evaluating Euler’s four coordinate pairs from §§ 6–9 yields exact rational points with zero radicals: