[Caption title:] De proportione proportionum disputatio.

[Colophon:] Rome: Giacomo Mazzochi, 27 September 1516.

First edition, exceptionally rare, of this influential and controversial but under-studied work, an important step towards the arithmetization of the mediaeval theory of proportion, which was a crucial first step in the increasing mathematization of the description of nature. In the Eudoxian theory of proportion, treated in Book V of Euclid’s Elements, proportion is a relation between magnitudes of the same kind, and is not identified with a numerical value. Thus, the ratio of the diagonal of a square to a side is a valid concept in the Euclidean theory of proportion, although it is not identified with a number as this number would (in modern terms) be irrational. The evolution from proportion to numerical ratio began with Umar al-Khayyam in the 12th century and was continued by Thomas Bradwardine and Nicole Oresme in the 14th. The most decisive progress in the modern era was made by Clavius in his great commentary on Euclid (first published in 1574). Clavius there criticized and completed the Euclidean theory of proportion. According to Rommevaux (p. 72), Clavius’s critique was ‘certainly inspired’ by the present work of Rodulphus, which treats proportions as quantities, so that proportions are just the same as proportions of quantities. Rodulphus’s work was attacked by Jean Fernel in his De proportionibus libri duo (1528), who argued that treating proportions as quantities was inconsistent with Euclid, Book V, and that it fails to see when these proportions of proportions are irrational (although understanding this had been one of the major achievements of Oresme). History found in Rodulphus’s favour, however, when the arithmetization of the theory of proportion was completed in the next century, starting with Clavius. OCLC lists six copies worldwide (Brown, Columbia, and Madison Wisconsin in US, BNF, BNCR, Tübingen). ABPC/RBH lists only this copy since Honeyman (Sotheby’s, April 11, 2002, lot 727, £4,112 = $5,934).

Provenance: Arnaud de Vitry d’Avaucourt (1926-2012), French engineering executive (sale of his library, Sotheby’s, April 11, 2002).

An important role in the mediaeval theory of ratios and proportionalities was played by the Tractatus de proportionibus seu de proportionibus velocitatum in motibus of Thomas Bradwardine, a fellow of Merton College at the beginning of the fourteenth century. In this treatise, written in 1328, published in 1481 in Paris and in 1505 in Venice, Bradwardine explained how the velocity of a moving body (motum) moved by a mover (motor) can be expressed in terms of the power of the mover and the resistance of the body. Precisely, he said that if one compares two motions, the velocities and the ratios between the powers and the resistances are geometrically proportional. Bradwardineʼs theory was taken up by Albert of Saxony in his De proportionibus, published several times at the end of the fifteenth century.

Bradwardine’s rule of motion is expressed by ratios and so the first chapter of his book is devoted to the theory of proportion – as he said, the theory is essential for an understanding of this rule. This first chapter is a summary of this theory, based on Euclidʼs Elements, Boethiusʼ Arithmetica and Jordanusʼ Arithmetica. Bradwardineʼs treatise soon became a textbook, used in universities in Paris, in the north of Italy, in Vienna and so on, until the beginning of the seventeenth century. Often the theory of proportion was presented as a part of the study of motion, but sometimes, the theory was presented on its own, with Bradwardineʼs text as an explicit or implicit source.

Bradwardine defined the ‘composition’ of two ratios as follows: if A bears a ratio to quantity B, and B another ratio to C, where A, B, C are quantities of the same kind (e.g., lengths, velocities, etc.), the composite of these two rations is that of A to C. Composition was, somewhat confusingly from a modern point of view, often referred to as ‘addition’ of the two ratios, for example in Jean Fernel’s De proportionibus libri duo. The ratio between ratios was defined in a similar way.

In a disputation in Rome in 1516, published as the present work, Rodulphus rejected Bradwardineʼs theory of proportion. In particular, he condemned the use of the vocabulary of addition to express composition. He also took an important step toward the arithmetization of the theory of proportion. The ‘denomination’ of a ratio between quantities A and B of the same kind was a number r (assumed to be rational) such that A = rB. Rodulphus defined the composite of two ratios as the product of their denominations, and the ratio between ratios as the ratio between their denominations, as we would do today.

Fernel defended Bradwardine’s theory of proportion, although he ascribes not to Bradwardine in particular, but rather to Euclid, Campanus, Jordanus, and other distinguished mathematicians. To double a proportion, he says, is not to multiply it by two, but rather to multiply it by itself. Similarly to triple a proportion one multiplies the proportion by itself three times, so that triple a proportion of 3 to 1 is the proportion of 27 to 1. He also relates this understanding of the multiplication of proportions to his previous discussion of the addition of proportions, according to which if the proportion of 3 to 1 is added to the proportion of 3 to 1 the result is 9 to 1, which is double the proportion of 3 to 1.

“After explaining the correct understanding of the proportions of proportions, Fernel turns to a refutation of the false opinion concerning proportionalities that had been proposed by Volumnius Rodulphus [who] treat[ed] proportions as quantities, so that proportions of proportions are just the same as proportions of quantities – which they could not have done had they paid attention to Euclid’s Elements, Book V … one of his main concerns is that the false view of the proportions of proportions put forth by Bassanus Politus and Volumnius Rodulphus leads to mistaken results with regard to the proportions of proportions in the sense that it fails to see when these proportions of proportions are irrational (the understanding of which had been one of Nicole Oresme’s major achievements in his De proportionibus proportionum) … by defending what he considered to be the view of compounding proportions held by Euclid, Jordanus, and Campanus, Fernel was implicitly defending the foundations of the Bradwardinian rule for the relations of velocities and proportions of force to resistance. He said nothing about what happens to velocities when forces or resistances increase or decrease, but when Fernel’s camp lost the battle on how proportions of proportions were to be understood, as was to happen within the next century, the mathematical foundations of the Bradwardinian approach to the proportions of velocities in motions would be undermined as well” (Sylla).

Nothing seems to be known of the author, except that he is sometimes referred to as ‘Spoletano’, presumably meaning that he was from the town of Spoleto in Umbria.

Censimento CNC 30331, listing copies at Rome and Urbino; Honeyman 3301; Smith, Rara Arithmetica, addenda p. 11; Riccardi I, ii, 387.Rommevaux, Clavius: Une Clé pour Euclide au XVI Siècle, 2006. Sylla, The Origin and Fate of Thomas Bradwardine’s Deproportionibus velocitatum in motibus in Relation to the History of Mathematics, in: Mechanics and Natural Philosophy before the Scientific Revolution, Laird & Roux (eds.), 2008. Smith, Rara Arithmetica, addenda, p. 11; Riccardi I 387; STC Italian, Vol. 3, p. 35.



4to, ff. [28], 39 lines, several woodcut diagrams printed in the margins. Nineteenth-century boards (spine a little worn), housed in a brown morocco-backed folding box.

Item #3078

Price: $17,500.00