Winterberg’s Critical Introduction (1889)

Preface and Historical-Mathematical Introduction by Constantin Winterberg (Vienna, 1889)

Series Title Page (Quellenschriften für Kunstgeschichte, Vienna, 1889)

Quellenschriften für Kunstgeschichte und Kunsttechnik des Mittelalters und der Neuzeit, published with the support of the Imperial-Royal Austrian Ministry for Worship and Education, founded by Rudolf Eitelberger von Edelberg, continued by Albert Ilg. New Series, Volume II (Neue Folge, II. Band): Fra Luca Pacioli, Divina Proportione: Die Lehre vom Goldenen Schnitt (The Doctrine of the Golden Section), newly edited, translated, and elucidated from the Venice Edition of the year 1509 by Constantin Winterberg (Vienna: Published by Carl Graeser, printed by Carl Fromme, 1889).

Preface (Vorwort, Karlsruhe, March 1889)

The efforts of modern scholarship have long been directed toward recovering the lost technical foundations that continually recur in the pictorial and architectural composition of the masterpieces of the Renaissance, so that they might once again exert a stimulating influence upon contemporary art. To have advanced the solution of this task through the careful publication of the written traditions of the foremost masters of that era is the great merit of the Quellenschriften für Kunstgeschichte und Kunsttechnik des Mittelalters und der Neuzeit.

Indeed, it is precisely the technical treatises that have suffered the undeserved fate of either mouldering unused and half-forgotten in the libraries of Italy, or—when published in earlier centuries—standing so far removed from modern habits of expression that only a few scholars undertake their study. This applies above all to Fra Luca Pacioli's treatises, De Divina Proportione and the Libellus de Quinque Corporibus Regularibus. Modest as purely abstract scientific works, their chief value lies rather in the fact that, having arisen in the closest connection with the needs of artistic practice, they cast a flood of light upon the manner in which mathematical disciplines were applied in the visual arts during the first great flowering of the Early Renaissance. The cumbersome, archaic mode of expression and the dense, nearly indecipherable scribal and typographical abbreviations that characterize the surviving early sixteenth-century edition (Venice, 1509) explain why these treatises have hitherto remained virtually inaccessible to practicing artists, quite apart from the extreme rarity of the printed volumes themselves.

To remedy this deficiency at least in part through the present new edition, to offer a fresh contribution to the understanding of the artistic creation of a great epoch, and to dedicate a tribute of gratitude to the memory of the late Councillor Rudolf von Eitelberger—who alongside his friend Heinrich Ludwig in Rome gave the first impetus to this undertaking—affords the editor special satisfaction.

Karlsruhe i. B., March 1889 — Constantin Winterberg

1. Pacioli, Piero della Francesca, and Leonardo da Vinci

Among the few writings of the Early Renaissance that treat technical and mathematical disciplines in their direct relation to artistic practice, none deserves the attention of both the historian and the specialist more than the celebrated treatises of Fra Luca Pacioli. Even the greatest and most famous artistic compilation of that age, Leonardo da Vinci’s Trattato della Pittura (Book of Painting / Malerbuch), consists essentially of loosely connected notes extending across the auxiliary sciences of painting without a unified systematic architecture. It is otherwise with the writings of Pacioli, which—despite all stylistic defects—form a complete, systematically worked-out whole. Their interest is heightened by the fact that they arose in immediate response to the practical needs of Renaissance workshops; indeed, the companion treatise on the five regular bodies (Libellus de Quinque Corporibus Regularibus) derives its substantive authorship from Piero della Francesca, a practicing master of the highest rank. Only in reading these treatises does one gain a full view of the formidable difficulties that confronted Renaissance artists even in elementary problems of solid geometry, and one cannot sufficiently admire the energy and insight with which the foremost masters strove to solve the problems required for their art using the mathematical tools inherited from antiquity.

Fra Luca Pacioli, a friar of the Franciscan Order, was born (according to Guillaume Libri, Histoire des sciences mathématiques en Italie, Vol. III) at Borgo Sansepolcro in the Tuscan Apennines—the very same town where Piero della Francesca first saw the light of day. According to Jean-Étienne Montucla (Histoire des mathématiques), Pacioli undertook travels to the Orient at the command of his Franciscan superiors, where he may have become more closely acquainted with Arabic mathematical science. The fruits of Pacioli's mathematical studies were first gathered in his monumental Summa de Arithmetica, Geometria, Proportioni et Proportionalita (Venice, 1494), dedicated to Guidobaldo da Montefeltro, Duke of Urbino. Beside its algebraic treatise, the Summa de Arithmetica includes an extensive geometric section in which continuous proportions are accorded special practical utility, and in which the golden section already appears as a privileged case.

Throughout his career, as Giorgio Vasari reports in his Vite, Pacioli stood in close scientific intercourse with the foremost artists of his age:

From this living exchange between theory and workshop practice, between science and art, emerged De Divina Proportione—completed in manuscript at Milan in 1498 for Duke Ludovico Maria Sforza (il Moro) and first printed at Venice by Paganino de' Paganini in June 1509.

2. The Golden Section, Vitruvian Harmonic Ratios, and Aesthetic Rhythm

In his Histoire des sciences mathématiques en Italie (Vol. III), Guillaume Libri asserted that Pacioli sought in De Divina Proportione to establish the golden section as a universal morphological and aesthetic law governing the human figure, architecture, and even the construction of the Roman alphabet. Closer examination of the text shows that this claim requires careful qualification:

  • What Pacioli sets forth in the Tractato de l'Architectura concerning the proportions of the human body (Chapters I–III) and the classical column orders (Chapters IV–XX) is substantially a faithful restatement of Vitruvius (De Architectura, Books III–V). In Vitruvius, as in Renaissance architects such as Leon Battista Alberti (De Re Aedificatoria) and Sebastiano Serlio (Sette Libri dell'Architettura), it is not the irrational golden section, but rather rational harmonic ratios ($1:2$, $2:3$, $3:4$, $3:4:5$) derived from musical consonances and aliquot body modules ($\frac{1}{6}$, $\frac{1}{8}$, $\frac{1}{10}$, $\frac{1}{12}$) that govern architectural commensuration.

  • Likewise, the twenty-three capital letters of the Roman Alphabet (Appendix I) are constructed from rational subdivisions ($\frac{1}{9}$, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$) of the square and inscribed circle.

  • Nevertheless, within the domain of solid geometry, Pacioli shows that not only the five Platonic solids (above all the dodecahedron and icosahedron), but also their truncated and stellated derivatives, are governed directly or indirectly by the proportion having a mean and two extremes. In Chapter LIV (and Chapter XIX of the Tractato de l'Architectura), Pacioli even conceives the progressive truncation and subdivision of polyhedra (in agibilibus, misprinted “in ogibilibus” in the 1509 edition) as a geometric method of exhaustion—an anticipation of the differential calculus—approaching curved vaults and the sphere itself, and urges architects and sculptors to follow the example of Phidias in applying the regular solids.

As Euclid already knew in the Elements, all the mathematical theorems Pacioli adduces on the Divina Proportione follow as necessary consequences from its definition (Euclid VI.def.3, Euclid II.11, Euclid VI.30, Euclid XIII.1–Euclid XIII.18). When in the mid-nineteenth century Adolf Zeising (Neue Lehre von den Proportionen des menschlichen Körpers, 1854) revived the golden section as a universal morphological and aesthetic canon (Kanon) of the human figure, empirical measurements and canonical systems—whether constructed geometrically after the golden section by Adolf Zeising and Brocheneck (Die männliche und weibliche Normalgestalt), derived from the module of the vertebral column (Wirbelsäule) by Carl Gustav Carus (Die Proportionslehre der menschlichen Gestalt), based on an arbitrary geometric key by Carl Schmidt (Proportionsschlüssel), or determined purely empirically by probability and mean values by Adolphe Quetelet (Anthropométrie) and after him J. J. Trost (Die Proportionslehre Dürer's)—showed that while human proportions approximate the golden ratio within modest tolerances, any continuous geometric division adjusted to empirical means yields comparable external approximations, as Emil Harless (Lehrbuch der plastischen Anatomie) rightly observed.

Where the golden section possesses a genuine and unique aesthetic preeminence, however, is in visual perception and rhythmic articulation:

  1. For the eye to perceive lawful articulation (gesetzmässige Gliederung) across the parts of a whole without artificial measuring aids, the governing law must announce itself immediately as a continuous rhythm.

  2. In an arithmetic series generated additively from two initial terms ($c = a + b$, $d = b + c$, $e = c + d$, $\dots$), every term is the sum of its two predecessors—an organic law of growth (as shown in Constantin Winterberg’s study "Das Malerbuch Lionardo da Vinci's" in Jahrbücher der königlich preussischen Kunstsammlungen, 1887, and exemplified both in the branching of trees and in the parallelogram composition of resultant forces $c, c'$ meeting at an angle).

  3. In a discontinuous geometric proportion $\frac{a}{b} = \frac{c}{d}$ among four segments, two terms remain independent, creating a perceptual gap as the eye passes from $b$ to $c$. Only when the proportion becomes continuous—so that each neighbor-ratio equals the preceding ratio, $\frac{b}{a} = \frac{c}{b}$—can the eye follow an unbroken rhythmic progression.

  4. When we unite both requirements in a single three-term relation—demanding simultaneously the continuous geometric proportion $a : b = b : c$ and the additive arithmetic relation $a + b = c$—we obtain uniquely the Golden Section (der goldene Schnitt):

    \[\frac{b}{a} = \frac{a + b}{b} = \varphi = \frac{1 + \sqrt{5}}{2}.\]

Because every triad of consecutive terms satisfies both the multiplicative ratio and the additive sum of parts into the whole, the eye apprehends the unity of the whole and its parts immediately and without auxiliary perspective grids.

3. Mathematical Structure of the Treatise & Critique of the Mutual Inscriptions

According to its mathematical content, Part I (Compendio de Divina Proportione) divides into four principal groups:

  1. The Thirteen Effects of the Divine Proportion (Chapters I–XXIII): After the dedication to Ludovico Sforza, the defense of mathematics, and the five divine attributes of the proportion (Chapters I–VI), Pacioli presents thirteen theorems on lines cut in extreme and mean ratio and their application to the regular pentagon, decagon, and hexagon (Chapters VII–XXIII, following Euclid, Elements Books II, IV, VI, and Euclid XIII.1–Euclid XIII.12). Because Pacioli states several of these theorems primarily through numerical examples (with $a+b = 10$, $a = \sqrt{125}-5$, $b = 15-\sqrt{125}$), concise algebraic proofs are supplied in Anmerkungen 1–9.

  2. Construction, Sphere Inscription, and Uniqueness of the Five Regular Solids (Chapters XXIV–XXXIV): Using the theorem of Pythagoras (Euclid I.47) and the chord and tangent theorems of the circle in Euclid, Pacioli demonstrates why at most five regular polyhedra can exist in nature (Euclid XI.21), constructs each solid in the sphere (Euclid XIII.13–Euclid XIII.18), and determines the ratios of their edges, altitudes, and sphere diameters (Anmerkungen 10–14).

  3. The Twelve Mutual Inscriptions and the Truncated/Elevated Dependent Bodies (Chapters XXXV–LV): In Chapters XXXV–XLVI, Pacioli treats twelve problems of inscribing one regular solid inside another, following Euclid Book XV (Euclid XV.1–Euclid XV.12) and his medieval predecessor Campanus of Novara (Elementa Euclidis cum Commentariis Campani). Here Pacioli repeats an error already criticized by Niccolò Tartaglia in his General Trattato di Numeri et Misure (General-Trattato di Matematiche): not only does he omit several valid inscriptions (such as the icosahedron in the cube, which Tartaglia and Piero della Francesca’s Libellus de Quinque Corporibus Regularibus both treat), but in Chapter XLVI he explicitly attempts to prove—on Plato’s cosmological grounds (Timaeus)—that the octahedron cannot be inscribed in the icosahedron! As Niccolò Tartaglia proved by far clearer and shorter methods than Pacioli's, and as shown in Anmerkung 15, this claim is false: joining the midpoints of the three pairs of opposite parallel edges of the icosahedron yields three mutually perpendicular axes of equal length, whose six endpoints form a regular octahedron inscribed in the icosahedron. And as demonstrated in full in Anmerkung 17, the modern projective theory of poles and polar planes with respect to a concentric sphere ($R = \sqrt{r\rho}$) immediately resolves the complete system of all 21 possible mutual inscriptions of the five regular solids by pairing each direct inscription (vertices on vertices, faces, or edges) with its polar-reciprocal dual (faces on faces, vertices, or edges). Chapters XLVII–LV then describe the truncated (abscisi) and elevated/stellated (elevati) dependent bodies (Anmerkung 16) and the 26-base and 72-base solids.

  4. The Sphere, Oblong Prisms/Columns, Pyramids/Cones, and Practical Mensuration (Chapters LVI–LXXI): After demonstrating how all five regular solids are carved within the sphere (Chapter LVII), Pacioli treats prismatic columns and pyramids, proves that the pyramid is one-third of the prism of equal base and height (Euclid XII.7), and concludes with the panegyric to Ludovico Sforza, the ordering of Leonardo da Vinci’s plates, and a glossary of mathematical terms, serving as a theoretical preparation for the problems in the Libellus de Quinque Corporibus Regularibus.

Already among his sixteenth-century contemporaries—most famously Annibale Caro (Apologia)—Pacioli's diffuse, half-Latinized vernacular prose style, mingling mathematical theorems with scholastic and theological digressions, was disparagingly dubbed "Ceneraccio" (“ash-lye” / Aschenlauge, Schmutzbrühe). Yet fairness requires recognizing that the Franciscan friar, addressing courtly audiences of princes, nobles, and artists in his public lectures at Florence, Milan, Rome, and Venice, deliberately wove historical and practical episodes into his discourse to sustain the interest of his listeners.

4. Editorial Principles and Sources (1509 Venice Print & Vatican Manuscript)

The primary purpose of this 1889 edition was first to render readable again the archaic text of the 1509 Venice edition—which had become almost unreadable through scribal and typographical abbreviations of every kind—while correcting obvious printing corruptions in the mathematical letters and numbers, and second to provide a critical German translation and mathematical commentary using the methods of modern geometry, so that Leonardo da Vinci’s polyhedral constructions may once again inspire contemporary artists, architects, and engineers.

The copy of the 1509 Venice edition (Paganino de' Paganini, Paganinus de Paganinis de Brixia) used as the basis of this work belongs to the library of the k. k. Österreichisches Museum für Kunst und Industrie in Vienna, kindly placed at the editor’s disposal by the late Director Hofrat Rudolf von Eitelberger; in doubtful passages, the illuminated manuscript in the Vatican Library (Cod. Vat. Lat. 3129) was also consulted.

Winterberg’s Footnotes to the Introduction (Einleitung, pp. 1–15)

  1. Footnote 1 (p. 2): Guillaume Libri, Histoire des sciences mathématiques en Italie, Vol. III.

  2. Footnote 2 (p. 2): Jean-Étienne Montucla, Histoire des mathématiques.

  3. Footnote 3 (p. 2): Giorgio Vasari, Vite de' più eccellenti pittori, scultori, e architettori.

  4. Footnote 4 (p. 4): Adolf Zeising, Neue Lehre von den Proportionen des menschlichen Körpers (Leipzig, 1854).

  5. Footnote 5 (p. 5): Unless a misprint in the 1509 Venice edition (“in ogibilibus”) is to be assumed and read as “:term:`in agibilibus <in-agibilibus>”` (compare Compendio Chapter LIV and Architectura Chapter XIX).

  6. Footnote 6 (p. 9): Constantin Winterberg, "Das Malerbuch Lionardo da Vinci's", in Jahrbücher der königlich preussischen Kunstsammlungen (1887).

  7. Footnote 7 (p. 9): If by $a, b$ and $a', b'$ we understand not numerical magnitudes, but pairs of linear forces meeting at an angle, and by $c, c'$ their resultants in magnitude and direction, then the composition of the latter two yields a new resultant force $c + c'$, and so forth.

  8. Footnote 8 (p. 11): Niccolò Tartaglia, General Trattato di Numeri et Misure (General-Trattato di Matematiche, Venice, 1556–1560).