## Opuscula mathematica; nunc primum in lucem aedita, cum rerum omnium notatu dignarum. Indice locupletissimo. Pagell huic proxime contigua, eorum catalogus est. [Volume 2:] Arithmeticorum libri duo, nunc primum in lucem editi. Cum rerum omnium notabilium. Indice copiossisimo.

Venice: Franciscum Franciscium Senensem, 1575.

First and only edition, rare, containing seven ‘opuscula,’ and, with separate title and pagination, the *Arithmeticorum libri duo*; all of these works were published here for the first time. One of the ‘opuscula’ contains the most important original work on conic sections published in the sixteenth century; the *Arithmeticorum *includes the first published use of mathematical induction. “Francesco Maurolico is generally recognized to have been one of the foremost mathematicians of the sixteenth century” (Rosen). Cajori has called him “the greatest geometer of the sixteenth century”, although many of his works remain unpublished even today. “The greatest number of Maurolico’s mathematical writings are gathered in the *Opuscula mathematica**;* indeed, the second volume of that work, *Arithmeticorum libri duo*, is wholly devoted to that subject and contains, among other things, some notable research on the theory of numbers. This includes, in particular, a treatment of polygonal numbers that is more complete than that of Diophantus, to which Maurolico added a number of simple and ingenious proofs. L. E. Dickson has remarked upon Maurolico’s argument that every perfect number in hexagonal, and therefore traingular, while Baldassarre Boncompagni noted his proof of a peculiarity of the succession of odd numbers … Among the topics related to mathematics in the *Opuscula* are chronology (the treatise ‘Computus ecclesiasticus’) and gnomonics (in two treatises, both entitled ‘De lineis horariis,’ one of which also discusses conics). The work also contains writing on Euclid’s *Elements* … Of particular interest, too is a passage on a correlation between regular polyhedrons, which was commented upon by J.H.T. Müller, and later by Moritz Cantor” (DSB IX, 191). Referring to the second of the horological works, Rose (p. 176) remarks that “Apart from Werner’s *Libellus* of 1522, the ‘Libri tres’ is the first original European treatise on conic sections. It is of greater significance than Werner’s book and had a wide influence in the sixteenth century” (Rose, p. 176). Several of the results on polygonal and square numbers in the *Arithmeticorum* are proved by making use of the principle of mathematical induction (e.g. Book I, Proposition 15), the first time this principle had been clearly stated in print. The manuscript of the book was sent to the bookseller Giovanni Comisino in Venice in 1569, but the book was not in press until November 1574. Even then there were further delays, and the book was not finally published until 26 July 1575, four days after Maurolico’s death. Some bibliographies mention editions of 1574, 1580 and 1585, but in ‘The Editions of Maurolico’s mathematical works’ (*Scripta Mathematica* 24 (1959), pp. 59-76), Edward Rosen has shown that there was only one edition of the *Opuscula* and its ‘companion piece’ the *Arithmeticorum* (though one or two of the works were reprinted separately later). Only two copes (one in a modern binding) have sold at auction since the Macclesfield copy, which realised £9,600 in 2005.

Maurolico’s family came from Greece, from which they had fled to Sicily to escape the Turks. Maurolico learned Greek, as well as astronomy, from his father. In 1521 he was ordained priest, and in 1550 the governor of Messina conferred upon Maurolico the abbey of Santa Maia del Parto. Maurolico also held a number of civil commissions in Messina, and like his father became master of the Messina mint. Most importantly, he gave public lectures on mathematics at the University of Messina, where he was appointed professor in 1569.

“In 1569 the outlook for the publication of Maurolico’s works improved with his appointment as lecturer in mathematics at the Jesuit University of Messina. This new educational affiliation inspired Maurolico to attempt again to publish new textbooks on astronomy which would replace Scarobosco in the universities. With the encouragement of his friends, Maurolico wrote to the Jesuit General Francesco Borgia on 16 April 1569, pointing out the need for such *compendia* and asking for help with their publication. The General’s reply was favourable: if Maurolico would send his books to Rome, the Order would immediately forward them to a printer in Venice. To help Maurolico with his preparation of the *compendia*, Juan Marques, a young Jesuit, was sent to Sicily in 1570, followed by the Jesuit mathematician Christopher Clavius in April 1574. Maurolico and Clavius had apparently been long in contact with one another, and the Jesuit seems to have derived some benefit from Maurolico’s studies. Thus, some of Maurolico’s proofs are used in the Clavius Euclid (Rome, 1574) and Clavius in his *Gnomonices libri octo* (Rome, 1581) thanks his friend for sending him the autograph manuscript of *De lineis horariis* [which was published in the *Opuscula*] … After lecturing on the fifth and sixth books of Euclid at Messina, Clavius returned to Rome in September, 1574, taking with him the autograph manuscripts of the *Photismi* and *Diaphana* [eventually published in 1611 and 1613] … In the meantime, Mauurolico had given a collection of other works to the bookseller Giovanni Comenzino to take to Venice, where indeed they were published in August 1575, a few weeks after the author’s death (21 or 22 July 1575).

“The publication (which was dedicated by the printer to Commandino’s patron, the Duke of Urbino), consists of two separate volumes. The first of these is the *Opuscula Mathematica* which comprises several works of Maurolico. It includes a *De Sphaera Liber Unus* [pp. 1-26] wherein Maurolico makes his notorious remark that Copernicus is so incorrigible in his errors that he is more deserving of a whipping than a reprimand [p. 26]. The remark is certainly unfortunate, particularly as its context puts Copernicus in the same group of ‘ignoramuses’ as Gerard of Cremona. It should be remembered, however, that Maurolico’s remark was uttered out of a genuine horror at the introduction of what seemed to be new errors into astronomy. Maurolico was a first rate mathematician and his anti-Copernicanism cannot be regarded as motivated by the same sort of ignorance that inspired other opponents of the geokinetic system” (Rose, pp. 175-6).

*De Sphaera Liber Unus* deals principally with the planets, but also contains sections on geography and some topics in natural history. Part of the passage relating to Copernicus, which appears at the end of the work, is quoted by Rosen (pp. 183-4): “I wrote the foregoing work, gentle reader, not so that you would peruse only my treatment and ignore all the others, but so that you would understand the others better because of my discussion, and from it learn what was omitted by the others. I have no doubt that on the basis of my elementary exposition you will read more circumspectly, and judge more perspicaciously, what you see in Sacrobosco, Robert [Grosseteste] or Campanus. Grosseteste did not put an end to the reading of Sacrobosco, nor did Campanus put an end to the reading of Grosseteste, as perhaps he thought he did. In like manner, Peurbach’s *Theoricae*, although extremely accurate and worked out in accordance with the Ptolemaic system, could not completely eliminate the teachings of Al-Bitruji (Alpetragius) and the ravings of [Gerard of] Cremona. Georg [Peurbach] and Regiomontanus contented themselves with warning their readers to learn to the best of their ability what to reject and what to accept. But not even Atlas, who supports the heavens, despite all his vigor would have the strength to correct every mistake that has been made and to lead everyone’s mind to [the path of] truth. There is toleration even for Nicholas Copernicus, who maintained that the sun is still and the earth has a circular motion; and [yet] he deserves a whip or a scourge rather than a refutation. Let us therefore go on to the remaining topics, lest we waste our time for nothing.” Thus, Copernicus deserves a whipping rather than a refutation “because nobody has the energy to rectify all the errors that are committed” (*ibid*., p. 184)

“The second of the ‘opuscula,’ “‘Computus Ecclesiasticus’ [pp. 26-47], datable to 1568, deals with the ecclesiastical calendar and its defects. While Maurolico was fully aware of the calendar’s errors with respect to the date of the equinox and the full moon, he found it inappropriate to try and change the traditions ‘which were once handed down to us by our ancestors.’ Accordingly, he had no sympathy for those chronologists who had in his eyes wasted dozens or hundreds of pages on fanciful proposals to correct the calculation of Easter. He expressed the same contempt for attempts to make the date of Christ’s Passion match up with the astronomical full moon. Once again, Maurolico preferred to stick to ecclesiastical tradition, which in his mind prescribed a crucifixion on 26 March, AD 34. Knowing that this had been the date of a full moon, he nonchalantly noted that

‘it is no wonder if the full moon preceded that date by two or three days, for due to the inequality of lunar motion, the 14^{th} day of the lunation does not always correspond exactly to the feast [of Passover], as the curiosity of many would make them believe. And thus, according to the true and simple calculation, Christ suffered in the 34^{th} year of salvation, which has the Sunday letter C and the golden number 16, on the 26^{th} of March, so that he rose from the dead on the 28^{th} of the same month, which was a Sunday. Mistaken are hence Paul of Middelburg, Johannes Lucidus, and all others who think differently and who busily twist and turn the years according to their views’” (Nothaft, pp. 256-7).

One wonders what Maurolico would have thought, had he lived to see the Julian calendar replaced by the Gregorian just seven years after the publication of the *Opuscula*.

The next chapter of the *Opuscula*, entitled ‘Tractatus instrumentorum astronomicorum’ (pp. 48-79), is devoted to a description of the geometrical models of the celestial system, and of astronomical instruments used for measuring and modelling the celestial system, including the quadrant, astrolabe and armillary sphere. It gives an explanation of the necessary mathematical and geometrical principles, and how to apply the instruments, to solve problems in positional astronomy, as well as the correct application of these instruments to avoid observational errors. Corresponding to their specific usage, these instruments were equipped with pointers, graduation plates, pinholes, and mirrors.

There are two works devoted to horology, ‘Tractatus de Lineis horariis’ (pp. 80-102) and the much more extensive ‘De lineis horariis libri tres’ (pp. 161-285), dated 1553. The second work contains Maurolico’s important treatise on conic sections, the second original treatise on the subject published in Europe, following that of Johann Werner contained in his *Libellus* of 1522. In 1547 Maurolico had completed a translation (of Books I-IV) and ‘restitution’ (of Books V and VI) of Apollonius’ *Conics*, but it was not published until 1654 and so had little impact on the historical development of the subject.

Werner’s treatise, in 22 propositions, contained only those properties of the parabola and hyperbola he needed to give his solution of the problem of the duplication of the cube. Maurolico’s treatise, “also originating from a practical need, that of applying conics to gnomonics, although also restricted to a very small number of pages [pp. 263-285], is much richer and more complete than that of Werner and had no small influence on further publications. Between the publication of these two treatises appeared the work of Dürer, in which the possibility was shown of obtaining a construction of the three conics from the intersection of a straight circular cone with a plane using the method of descriptive geometry. But this method was either not understood or was forgotten and had litle effect on the progress of the theory of conics. Maurolico’s treatise, on the other hand, marks a new direction for the rapid direct construction of conics, independent of the cone, and the theory devised by him was immediately followed and improved, and provided the impetus for all the new developments in the theory of conics that soon took place in Italy and France. One of those who most highlighted the merit of Maurolico was Giovanni Alfonso Borelli who, in the introduction to his *Elementa conica* (1679), says that Maurolico devises excellent demonstrations on the tangent lines and on the asymptotes of conics, and at the end of prop. 32 adds that: ‘The properties of the tangents of the conic sections that Apollonius demonstrates with abstruse and prolix methods, are shown with admirable ease by Maurolico’, and that he tried to imitate the genius of this man in his own treatment of the subject” (freely translated from Amodeo, pp. 123-4). The new method pioneered by Maurolico, and a century later taken up by Desargues and Pascal, was to derive the properties of the conic sections from those of the circle using perspective.

In the preface to his work on conics (pp. 263-265), Maurolico deplores the fact that earlier writers on gnomonics were ignorant of the doctrine of conic sections. He notes that, having already had occasion in the two preceding books of ‘De lineis horariis’ regarding the description of hour lines to say something about the species of the conics, their axes and their descriptions, he now proposed to expose what is necessary to clarify those things, such as the properties of diameters and asymptotes and the construction of the different conics – some results will be quoted from Apollonius, while others he will demonstrate in the simplest way known to him.

The other purely geometrical work in the *Opuscula *is ‘Euclidis propositiones elementorum, libri tredecimi solidorum tertij, regularium corporum primi’ (pp. 103-144), a version of Euclid books XIII-XV. These books deal mainly with the properties of the five regular Platonic solids. Book XIII discusses the lengths of the sides of such a solid compared to the radius of a sphere in which the solid is inscribed. Book XIV continues Euclid’s comparison of regular solids inscribed in spheres, the chief result being that the ratio of the surface areas of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes. Book XV covers topics such as counting the number of edges and solid angles in the regular solids, and finding the measure of dihedral angles of faces that meet at an edge. It is now known that books XIV and XV are not by Euclid: book XIV was probably written by Hypsicles (ca. 200 BC) on the basis of a treatise by Apollonius, and book XV is by a pupil of Isidore of Miletus, probably from the sixth century AD.

“In 1532 Maurolico had given lessons on books I-XII of Euclid to his patron Girolamo Vartesi at Messina. He then seems to have started a condensed edition or epitome of Euclid which occupied him for the next thirty years. Although only books XIII-XV were actually published (posthumously in 1575), nearly all of the *Elementorum Euclidis epitome* exists in various drafts in autograph manuscripts … In the 1532 preface to books XIII-XV, Maurolico states his dissatisfaction with both the Campanus [1482] and Zamberti [1505] editions. In language recalling Regiomontanus’ *Programme*, he indicates that some of the *placita *of Campanus are to be rejected, others admitted. But he is openly scornful of Zamberti’s lack of mathematics which had made him a slave to the Greek text. This point was entirely missed by Bernardino Baldi who, saying that Maurolico had castigated Zamberti’s ignorance of mathematics, cited Commandino’s view that errors of the Venetian translator had been due rather to the poor texts with which he worked. Maurolico’s point was that a better knowledge of mathematics would have allowed Zamberti to correct the Greek texts, no matter how bad they were” (Rose, p. 165).

The final work of the *Opuscula* is the brief ‘Musicae traditiones’ (pp. 145-160), on which Grove’s Dictionary comments as follows. “He was co-opted by the Jesuits in 1569 to lecture on mathematics at the University of Messina, and the theory of music was among the topics on which he was expected to teach. Among his printed *Opuscula mathematica* (Venice, 1575) is the treatise *Musicae traditiones carptim collectae*, which contains an epitome of Boethius’s *De institutione musica* together with a number of brief essays on the nature of sounds, musical notes and intervals, the seven-string lyre, the rudiments of counterpoint, the inventors of musical instruments and, most originally, the calculation of the proportions of intervals. The last topic bulks large in Maurolico’s manuscript notebooks.”

The *Opuscula* opens with Maurolico’s classification of philosophy, ‘De Scientiarum divisione’ (pp. 2-4). “Here Maurolico offers three alternative systems of classification. The second of these, based on the objectives of the sciences, has a special interest in that it classifies *mathematica* and physics as the two sciences of nature … in the same scheme geometry and arithmetic and their derivatives astronomy, music, *metrica *and geography are grouped with logic as belonging to art rather than nature … while theology is beyond human understanding, and physics is a science of uncertainty, mathematics and astronomy possess the greatest certainty and nobility. Maurolico’s ‘De Scientiarum divisione’ was the nearest he came to giving an encyclopaedic account of Renaissance culture” (Rose, p. 176).

“The other work besides the *Opuscula* to be published in the last year of Maurolio’s life was his *Arithmeticorum libri duo*, which had been written in 1557 … This important work is essentially an extension of the *Arithmetica* of Diophantus. The most striking aspect of the *Arithmeticorum* is Maurolico’s use of letters in place of numbers, although he does not, it is true, use these letters in combination with coefficients or exponents. The general effect of this procedure is to try to raise numerical operations to the same generality and abstraction as the graphic operations of geometry. As such, it might be taken once again as evidence of Maurolico’s perpetual search for mathematical rigour and truth. From the point of view of algebra, on the other hand, the *Arithmeticorum* is notable for its introduction of the rules of algorithms into algebra.

“One feature of the *Arithmeticorum *which demonstrates its author’s perpetual passion for more cogent and shorter methods of proof is the use of mathematical induction. In the ‘Prolegomena’ to the book Maurolico says that he will not simply repeat what previous authors have said, but will try to show these things and others besides by a *faciliori via* [simple method]. This method is best seen in book I, propositions 13 and 15, and it was later adapted by Pascal who is usually credited with its discovery. However, Pascal himself knew of Maurolico’s application of the method, for he remarks in a letter that ‘Cela est aisé par Maurolic’.

At the end of the *Arithmeticorum* is a version of Maurolico’s *Index** lucubrationum*, a list of works then in manuscript which he expected to be published. He had first produced a version of this list in 1540, probably basing it on Regiomontanus’ *Programme*, and updated it in his 1558 collection on spherics. The version in the *Opuscula* is “much fuller than that in the 1558 edition and expanded by a series of religious and humanistic titles including several epitomes of classical works. Maurolico had compiled this revised list in 1568 and in 1570 he made out a suggested guide for reading the mathematical works in his programme. After beginning with Euclid’s geometry, the spherics of Theodosius and Menelaus and the arithmetical works of Jordanus and Maurolico, the student was to progress to Archimedes and Apollonius. Next came various texts on perspective and astronomy, including Peurbach, Regiomontanus, Ptolemy and various Arabic authors. Almost at the end was mechanics, represented by Pappus, Hero and Vitruvius” (Rose, pp. 176-7).

Adams M919; Honeyman 2182; Macclesfield 1344; Riccardi II, 141; Smith, *Rara Arithmetica*, pp. 349-50. Amodeo, ‘Il Trattato del Coniche di Francesco Maurolico,’ *Bibliotheca Mathematica*, 3. Folge, 9. Bd. (1908-9), pp. 123-138. Nothaft, *Dating the Passion*: *The Life of Jesus and the Emergence of Scientific Chronology* (200–1600), 2011. Rose, *The Italian Renaissance of Mathematics*, 1975. Rosen, ‘Maurolico’s attitude toward Copenicus,’ *Proceedings of the American Philosophical Society* 101 (1957), pp. 177-194. On the use of the principle of mathematical induction by Maurolico, see G. Vacca, ‘Maurolycus, the First Discover of the Principle of Mathematical Induction,’ *Bulletin of the American Mathematical Society***16** (1909–1910), pp. 70–73.

Two volumes bound in one (as issued), 4to (214 x 160 mm), pp. [xx], 285; [1, blank], [viii], 175, [1], [18], large printer’s device on each title, woodcut initials and type ornaments, numerous woodcut diagrams in text and margins (title page with numerous contemporary inscriptions and ancient stamp, slight dampstaining). Contemporary limp vellum (spine heavily worn but entirely unrestored). A genuine unsophisticated copy.

Item #2613

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Price:
$18,500.00
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