Notes — Book III: Algebraic Typography, Pacioli vs. Dürer & Gnomonics¶
1. Dürer vs. Luca Pacioli on the Roman Alphabet (1509 vs. 1525)¶
In Divina Proportione (pacioli:1509), Luca Pacioli framed Roman capitals inside a square with a $1 : 9$ stroke-to-height ratio and constructed round letters such as O using four distinct circles (two concentric outer/inner circles or paired ovals). In Book III of the Underweysung (pp. 115–135), Dürer makes three decisive innovations:
Dual Proportional Canons ($1 : 10$ and $1 : 9$): He establishes the primary $10 \times 10$ square ($w = 1$, thin stroke $t = \frac{1}{3}$) for bold legibility, followed explicitly on p. 130 by the slenderer $9 \times 9$ canon (and $1 : 5$ for small handwritten capitals).
Calligraphic Broad-Nib Axis via Shifted Congruent Circles: In O and Q (
construct_durer_roman_letter_O(),construct_durer_roman_letter_Q()), shifting a single compass radius along the $45^\circ$ diagonal $cb$ ($y = x$) from center $f$ to center $g$ with $|fg| = 1$:$$f = \left(5 - \frac{\sqrt{2}}{4},\; 5 - \frac{\sqrt{2}}{4}\right), \qquad g = \left(5 + \frac{\sqrt{2}}{4},\; 5 + \frac{\sqrt{2}}{4}\right), \qquad r = 5 - \frac{\sqrt{2}}{4},$$
makes $\bigcirc(f, r)$ tangent to the left and bottom edges ($x = 0, y = 0$) and $\bigcirc(g, r)$ tangent to the right and top edges ($x = 10, y = 10$), while automatically tilting the maximum bowl stress to $45^\circ$ just like a humanist broad-edged pen! Their lens intersections lie at exact nested radical coordinates in $\mathbb{Q}(\sqrt{2}, \sqrt{80\sqrt{2} - 6})$:
$$P_{1,2} = \left(5 \mp \frac{\sqrt{10\sqrt{2} - \frac{3}{4}}}{2},\; 5 \pm \frac{\sqrt{10\sqrt{2} - \frac{3}{4}}}{2}\right).$$
Exact Coordinates of Roman Capital A: In
construct_durer_roman_letter_A(), placing foot points $i = (1, 0)$ and $k = (9, 0)$ with top midpoint $e = (5, 10)$ inside the leg junction yields exact rational coordinates for the triangular counter apex and tangent serif centers:$$V_{\text{counter}} = \left(\frac{14}{3},\; \frac{200}{21}\right), \qquad r_{\text{outer}} = \frac{10}{7}, \qquad r_{\text{inner}} = \frac{1}{2},$$
with outer fillet circle centers lying over $\mathbb{Q}(\sqrt{449})$ so each arc is simultaneously tangent to the $y = 0$ baseline and the oblique leg!
2. Combinatorial Proof of Dürer’s Textura Counter Inequality¶
On p. 136, Dürer constructs minuscule i (construct_durer_textura_minuscules())
from three unit squares of width $1$ capped by $45^\circ$-tilted unit squares (Ort-Fierungen) of
diagonal $\sqrt{2}$, and immediately remarks that when two stems touch corner-to-corner to form
n, “das felt [wirt] schlemer zwischen den zügen dann ein zug des pustaben preyt ist” (“the
interior white counter becomes narrower than a stroke is wide”).
Why? Because each $45^\circ$-rotated unit square has horizontal span equal to its diagonal $d = \sqrt{2}$! When two stems meet corner-to-corner at the diamond tips, the center-to-center stem pitch is exactly $p = \sqrt{2}$, so subtracting one stem width ($w = 1$) leaves an exact interior white counter width of
$$c_{\text{Textura}} = \sqrt{2} - 1 \approx 0.41421356 < 1,$$
giving black-to-white rhythm $1 : (\sqrt{2} - 1) = (\sqrt{2} + 1) : 1$ (the silver ratio $\delta_S = 1 + \sqrt{2}$)! Five pages later (p. 138), Dürer contrasts this dense medieval picket fence with the modernized freye Textur (developed in Nuremberg with Johann Neudörfer and Hieronymus Andreae), which widens the counter to equal the full stroke width $1$.
3. Projective Collineation in the Tower Inscription & Analemma¶
Both Dürer’s Nuremberg gnomonic projection (Figs. 21–23, mapping uniform $15^\circ$ equatorial hour angles onto tangent lines via $\tan(k \cdot 15^\circ)$ scaled by $\sin 49^\circ$ on the horizontal plane and $\cos 49^\circ$ on a prime-vertical South wall) and his tower-inscription ray pencil (p. 114, mapping uniform visual angles $\theta_k$ onto heights $h_k = D\tan\theta_k$) prepare the reader directly for the central-projection perspective theory of Book IV, and inspired D'Arcy Wentworth Thompson's coordinate transformation grids (thompson:1942).