Book III — Architecture, Nuremberg Sundials, Optical Scaling & Typography

[Folio N1525 p. 077–107]

Columns, Entasis, Enveloped-Sphere Helices & Monumental Towers (Figs. 1–19)

Having treated lines in Book I and plane figures in Book II, Dürer turns in Book III to corporeal architectural and monumental forms generated by carrying planar ground plans (nidergedruckte Gründ) into the third dimension:

  1. Prismatic Shafts, Pyramids & Rib-Vault Piers (pp. 077–081, Figs. 1–4): Lifting a circle, triangle, square, pentagon, hexagon, or compound rotated polygon vertically generates cylindrical, prismatic, or helically twisted columns (Seülen, Fig. 1); drawing those same ground plans up to an apex generates upright, right-angled, or oblique cones and pyramids (Piramides, Figs. 2–3). Setting pyramids in due proportion atop columns forms towers, while Late Gothic pier cross-sections (Pfeyler, Fig. 4) interpenetrate convex shafts (Stäbe), concave scotias (Holkelen), and chamfers (Fasen) to spring complex rib vaults.

  2. Praise of Vitruvius & the Six Elementary Molding Profiles (pp. 082–085, Figs. 5–6): Paying homage to “the ancient Roman :person:`Vitruvius <vitruvius>`, who wrote so artfully and masterfully in his books concerning the durability, utility, and ornament of buildings (:cite:`vitruvius:architectura`), wherefore he is to be followed before others,” Dürer also acknowledges the Northern builder’s delight in fresh invention (“der deütschen gemüt / dann gewonlich alle die etwas newes bauwen wöllen / wolten auch geren ein newe fatzon dar zuᵒ haben”). He reduces all lathe-turned and planed moldings to six geometric primitives (p. 085, Fig. 6):

    • flat fascia (blatte Fasen),

    • concave scotia / cavetto (Holkelen),

    • convex torus / ovolo (auspogne Wellen),

    • salient corner (Eck),

    • recessed angle (Winckel), and

    • inflected serpentine cyma (Schlangenlini).

  3. Entasis & the 60-Station Twisted Column of Enveloped Spheres (pp. 086–107, Figs. 7–19): After two geometric methods for column entasis (Bauch, swelling at one-third height via a three-point arc or stepped ordinates, Fig. 7), Dürer constructs a sinuous serpentine column (pp. 090–093, Figs. 9–10) by winding its axis through 60 stations on three nested circles in plan coupled to a projectively stretched 60-part vertical scale—and determines its true 3D contour by conceiving every station on the curved spine as the center of an inscribed sphere (ein yeklicher punct in der gewunden axt der seulen ein centrum einer kugel)! He completes the architectural section with three famous triumphal memorials—for a victory, for vanquished rebellious peasants (1525), and for a drunkard—alongside a monumental market tower with a quadruple helical ramp.

[Folio N1525 p. 108–114]

Nuremberg Sundials ($49^\circ$ Pole) & Optical Tower Scaling (Figs. 20–27)

  1. The Nuremberg Analemma & Truncated-Cube Polyhedral Dial (pp. 108–113, Figs. 20–27): Dividing quadrant b.a.c. into $90^\circ$ ($90 = 3 \times 3 \times 2 \times 5$, Fig. 21), Dürer sets the altitude of the celestial pole at Nuremberg to $49^\circ$ (“also zuᵒ Nüremberg ist sein höch neun vnd vierzig grad”), establishing the right-angled wooden or brass gnomonic triangle a.e.d. containing the polar Axis mundi a.d. ($49^\circ$), the vertical Mauerlini d.e., and the perpendicular Equinoctialis e.f. ($41^\circ$). Projecting a 24-sector equinoctial wheel (Fig. 22) onto tangent lines p.q. and r.s. and transferring radii d.e. and a.e. constructs both the vertical South wall dial (an die aufrecht mauer) and the horizontal dial (auf den orizon, Fig. 23), culminating in an octahedrally chamfered polyhedral cube sundial (Figs. 24–25) whose facets face East, West, South, North, the $49^\circ$ Arctic/Antarctic poles, and the $41^\circ$ equinoctial plane.

  2. Equal-Visual-Angle Scaling of High Inscriptions (p. 114): When carving lettering or statues up a high tower wall a.b. so that the uppermost line reads at the same apparent visual angle from eye point c as the lowest line, Dürer divides a circular arc b.e. centered at c into equal angular increments and projects straight sight-rays from c onto wall a.b.—noting that any parallel rod f.g. slicing the ray pencil is divided in identical projective proportion.

[Folio N1525 p. 115–141]

Roman Antiqua Capitals (A–Z) & Modular Gothic Textura

Because builders and painters inscribe letters on high walls and monuments, Dürer devotes pages 115–141 to the first printed northern Renaissance geometric construction of both Roman capitals (*ein Lateinisch abc*) and Gothic blackletter (*Textur*).

Every Roman capital is inscribed in a rechte Fierung a.b.c.d. of side $10$ units (with a secondary $9$-unit canon on p. 130), governed by broad-edged pen contrast: thick strokes equal $\frac{1}{10}$ of the square ($w = 1$), and thin strokes equal one-third of the thick stroke ($t = \frac{1}{3}$):

  • Capital A (p. 115): Bisect square a–b–d–c by vertical midline e–f and horizontal midline g–h, and step foot points i and k inward by one-tenth ($1$ unit) from bottom corners c and d. Draw the thin left leg ($t = \frac{1}{3}$) up from i and the thick right leg ($w = 1$) down to k so top midpoint e falls midway inside the apex; place the crossbar ($t = \frac{1}{3}$) immediately below midline g–h, enclosing triangular counter apex V. Flare the outer feet to corners c and d with tangent serif arcs centered at SL and SR of radius $\frac{10}{7}$, fillet the inner feet with arcs of diameter $1$ (radius $\frac{1}{2}$), and trim the overhanging apex with a concave Schlangenlini.

a: (0, 10) b: (10, 10) c: (0, 0) d: (10, 0) e: (5, 10) f: (5, 0) g: (0, 5) h: (10, 5) i: (1, 0) k: (9, 0) V: (\frac{14}{3}, \frac{200}{21}) S_L: (\frac{3}{2} - \frac{\sqrt{449}}{14}, \frac{10}{7}) S_R: (\frac{\sqrt{449}}{14} + \frac{17}{2}, \frac{10}{7}) a b c d e f g h i k V SL SR Book III, p. 115 — Roman Antiqua A (10x10 Square, 1 : 1/3 Contrast)
Dürer's compass-and-straightedge construction of Roman *Antiqua* capital **A** inside the $10 \times 10$ square (:g:`a`, :g:`b`, :g:`d`, :g:`c`, Book III, p. 115), showing $1 : \frac{1}{3}$ stroke contrast, counter apex :g:`V` $= (\frac{14}{3}, \frac{200}{21})$, and tangent serif fillet circles centered at :g:`S_L` and :g:`S_R`.

  • Shifted-Diagonal Capitals O & Q (pp. 124–125): Unlike Luca Pacioli's four-circle construction (pacioli:1509), Dürer constructs both the outer contour and inner counter of O with just two congruent circles! Draw the $45^\circ$ diagonal c–b (Ortstrich) through square center i, mark centers f and g symmetrically astride i along c–b separated by the thick stroke width $|fg| = 1$, and draw from each center a circle tangent to two adjacent sides of the square ($r = 5 - frac{sqrt{2}}{4}$); only the short thin hairlines near lens intersections P1 and P2 are rounded smoothly by hand. Appending a sweeping tail through corner d to tip h transforms O immediately into Q.

a: (0, 10) b: (10, 10) c: (0, 0) d: (10, 0) i: (5, 5) f: (5 - \frac{\sqrt{2}}{4}, 5 - \frac{\sqrt{2}}{4}) g: (\frac{\sqrt{2}}{4} + 5, \frac{\sqrt{2}}{4} + 5) P_1: (5 - \frac{\sqrt{199 - 20 \sqrt{2}}}{4}, \frac{\sqrt{199 - 20 \sqrt{2}}}{4} + 5) P_2: (\frac{\sqrt{199 - 20 \sqrt{2}}}{4} + 5, 5 - \frac{\sqrt{199 - 20 \sqrt{2}}}{4}) h: (18, - \frac{10}{3}) a b c d i f g P1 P2 h Book III, pp. 124–125 — Two-Circle Shifted Diagonal O & Q (|fg| = 1, r = 5 - sqrt(2)/4)
Dürer's two-circle shifted-diagonal Roman **O** and **Q** (Book III, pp. 124–125): congruent circles around :g:`f` and :g:`g` on $45^\circ$ diagonal :g:`c`–:g:`b` ($|fg| = 1$, radius $5 - \frac{\sqrt{2}}{4}$), meeting at lens points :g:`P_1`, :g:`P_2`, with **Q**'s tail sweeping through :g:`d` to :g:`h`.

  • Modular Combinatorial Gothic Textura (pp. 136–141): For traditional alte Textur, Dürer deliberately begins not with a but with minuscule i (“darumb das schier alle pustaben darauß gemacht mögen werden”): stack three unit squares vertically, trisect top and bottom edges, and cap the stem with two $45^\circ$-rotated unit diamond squares (Ort-Fierungen) set at the front upper and rear lower trisection points. Joining two such i-stems corner-to-corner yields n, and three yields m—automatically producing an exact irrational stem pitch of $\sqrt{2}$ and an interior counter width of $\sqrt{2} - 1 \approx 0.414 < 1$, explaining algebraically why Dürer observes that “the white space between the strokes becomes narrower than the breadth of a stroke”!

i (3 squares + 2 ort-fierungen) n (pitch = sqrt(2)) m (3 modular stems)
Dürer's purely modular *alte Textur* minuscules **i**, **n**, and **m** (Book III, p. 136): stacked unit squares capped by $45^\circ$ *Ort-Fierungen* touching corner-to-corner with exact stem pitch $\sqrt{2}$ and interior counter $\sqrt{2} - 1$.