The Mathematical Works of Leon Battista Alberti

Leon Battista Alberti (1404–72) was an archetypical Renaissance figure whose interests and works ranged over the cultural and technical landscape of his day. An early fascination with human nature produced a comedy in the antique style, a light-hearted collection of dinner pieces, a dissertation on the life of the intellectual, a discourse on the character of love and another on virtue and family relations. After these, came a careful study of the law as well as a proposal to reform the Tuscan language through the introduction of Latin words and phrases. Alberti’s fame, however, rests not on these achievements but on the results of his mathematical intelligence, technical curiosity and inventor’s skill applied to the nature of art – to painting, sculpture and architecture – as well as the art of writing in code. Alberti’s most well-known works are De re Aedificatoria (“On Architecture”), the first comprehensive treatment of the subject since Vitruvius, and De Pictoria (“Of Painting”), his systematic exposition of the method and practice of vanishing-point perspective that allowed a three-dimensional space to be rendered in a two-dimensional painting. Though both these treatises rely on the arithmetical and geometrical properties of angles, lines, and measured proportions they are less mathematically framed than the works offered in the book under review. “The Mathematical Works of Leon Battista Alberti” contains English translations and extensive scholarly commentary on three major contributions: Ex ludis rerum mathematicarum (title left untranslated from the Latin), Elementi di pittura (“Elements of Painting”) and De Componendis cifris (“On Writing in Cyphers”). In addition the editors include a very short work, De lunularam quadrature (“On Squaring the Lune”), of uncertain origin that has long been attributed to Alberti.

In spite of its Latin title, Ex ludis rerum mathematicarum was written in Italian and known as Ludi matematici or simply Ludi. The version given here is bilingual with English and Italian on opposite pages. Ludi was written for a member of the powerful Este family and seems intended for private circulation; it is the most informal of Alberti’s mathematical works and the only one of Alberti’s works that contains diagrams. It consists of 23 practical mathematical problems and solutions, or games as the editors, after the Latin and Italian, call them, and they suggest that the Latin title could be rendered as “Cool Things You Can Do with Mathematics.”(4). Some of the problems seem to have originated with Alberti, others are traditional with Vetruvius and Fibonacci cited as sources. About half are measuring tasks. These include “Measuring the height of a tower, knowing the distance to its foot as well as its height up to a certain point”; “Measuring the width of a river from one bank”; “Measuring the height of a tower, only the top of which is in view”; “Measuring the weight of heavy loads”, and so on. Others have to do with using/making a device: such as “Constructing a circular instrument for measuring angles in order to draw the map of a city and to estimate distances“; “Constructing a fountain powered by gravity”; “Constructing an equilibra for leveling surfaces and weighing objects”; “Constructing an instrument for measuring the speed of a ship using the wind”. Some of Alberti’s solutions are less clear than they might be, partly because he isn’t always concerned to fill in the steps, partly because of linguistic ambiguities and corruptions in the manuscripts by copyists in the era before printing, and on at least one occasion a logical error by Alberti. Against these hazards the editors offer a detailed, deeply researched commentary helpfully attentive to every conceivable aspect of a problem and its proposed solution; a commentary which like others in the volume, incorporates the work of contemporary French and Italian Alberti scholarship.

Alberti’s “Of Painting” consisted of three books (as he called them). The first and most mathematical lays out the basic practical steps for perspective drawing whilst the following books offer accounts of various aspects of painting such as composition and light. “Elements of Painting”, strips Book I down to essentials; its mere 2000 words has the spare, formal style of Euclid’s Elements – a work Alberti studied and greatly admired. In contrast to the style of the Ludi , “Elements of Painting” consists of lists of Euclid-like definitions such as “A point is said to be that which cannot be divided at all into any parts”, as well as more obscure ones, “A comminuted area we call that which is similar to the surface viewed, when it is placed in such a way that it will appear smaller in some of its parts.”, and then of actions, detailed procedures that the would-be painter will find necessary to perform in the course of a geometrically correct painting. For example, “”Enclosing a concentric, angular and proportionally larger area in one that is concentric and proportionally larger” or “ Drawing a rectangular area similar to a comminuted surface in a rectangular concentric area” and dozens more all conveyed in the same terse minimal manner with no elaboration and not a single diagram. Unsurprisingly, the commentary needed to make Alberti’s text comprehensible to the contemporary reader is considerable and its author, Stephen Wassel, uses many diagrams, some after Alberti drawn from his other works, some after modern scholars who have grappled with the text, as well as constructing interpretations from later, more transparent works influenced by Alberti. The result, a careful and convincing opening up of Alberti’s work, is not without unresolved difficulties. After much discussion about the meaning of the term “commensurate” comes the admission “It would be false to conclude that we understand exactly what Alberti meant by a commensurate point” (168)

“On Writing in Cyphers” is in two halves. The first is a detailed examination of the frequencies of vowels and consonants and their 2-, 3- and 4-letter combinations in the Latin language (with occasional forays into Tuscan), together with the need to scramble the structure of an encoded message “so that to an outsider the words appear like the leaves of a tree blown about by the wind that have been raked into a pile and set there.” (180) In the second half Alberti explains his original contribution to cryptography, a device he calls the “Formula”. It consists of two metallic parts: an inner movable disc and a fixed concentric outer ring. Each is divided into 24 equal spaces. The ring contains the 20 capital letters of the Roman alphabet in alphabetic order (the infrequently used H, K, Y are omitted) and the numerals 1, 2, 3, 4 filling the spaces. The inner disc contains all 23 Roman letters in lower case and the ampersand sign ‘&’ arranged randomly. Coder and Decoder possess identical copies of the Formula and agree on a designated element of the disc — the ‘index’. To code a message one chooses a capital letter and aligns it with the index by rotating the disc. The choice is signaled to the decoder at the head of the coded message, and the letters to be coded are replaced by those corresponding to them on the outer ring. After every few words Alberti recommends repeating the procedure: choosing another letter to align the index to, signaling the choice, and further coding. The result, he asserts, is that “no other cipher is more expedient, easier to use, faster to write with, none quicker and more rapid to read and none […] more difficult to decipher” (180), a claim that held true until the mid 19th century.

Asserting the “Opinion of the many who say that figures composed of lines that are curved and circular cannot be squared”, to be mistaken, the final work presented here, “On squaring the Lune”, offers “the perfect squaring of … the figure with two cusps in shape of the moon.” (204) What is presented is a neatly conceived geometrical proof that does indeed show that a certain constructed square has the same area as the lune – the quarter moon figure in question. However, the work’s closing assertion, “it is likewise possible to square the circle”, cannot of course be correct, since one knows from the fact that pi is a transcendental number that such a task is impossible.

Introducing their volume, the editors disclaim any intention to produce a critical edition, but rather through their translations and copious commentaries to make Alberti’s treatises “readable and understandable” (4) in English to a wide range of readers. In this, as well as in adding to an understanding of Alberti as a “true Renaissance man”, they admirably succeed.

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