Unit II Notes: Windmill Geometry, Stroke Disambiguation & The 1847 Title-Page Anomaly

Archival & Symbolic Commentary on Book I

Archival Note: The 1847 Title-Page Transposition of I.47

Direct multimodal inspection of the 1847 Pickering volume reveals a delightful bibliographic detail between Leaf 7 (Title Page) and Leaves 82–83 (pp. 48–49):

• On pp. 48–49 (the actual proof of I.47), the narrow left hypotenuse rectangle under the short leg is printed in ochre yellow (, equal to the black leg square ) and the wide right hypotenuse rectangle under the long leg is printed in Prussian blue (, equal to the red leg square ).

• On the 1847 Title Page emblem (Leaf 7), Whittingham's pressman transposed the yellow and blue woodblocks on the bottom square (coloring the narrow left strip blue and the wide right rectangle yellow)! Our symbolic i47.svg plate faithfully follows the mathematically canonical proof plate on pp. 48–49.

Boundary Stroke Modulation in I.47

In I.47, the solid red square and solid blue rectangle must also have their individual outer sides cited as 1D segments in the SAS congruence step ([I.4]). Because a red solid line would vanish into a red solid square, Byrne prints the outer edge of the red square as and the right edge of the blue rectangle as —a textbook resolution of chromatic register pressure.

Rational Trigonometry Invariants in geometor.byrne

In geometor.byrne.constructions.build_i47(), I.47 is modeled over pure rational coordinates $\mathbb{Q}$ ($B = (0, 0)$, $C = (5, 0)$, $A = (\frac{9}{5}, \frac{12}{5})$), giving exact integer quadrances $Q(A, B) = 9$, $Q(A, C) = 16$, $Q(B, C) = 25$, perpendicular spread $s(AB, AC) = 1$, and exact windmill-chord quadrance $Q(A, D) = Q(F, C) = \frac{290}{5} = 58$.