Venice: Franciscum Franciscium Senensem, 1575.
First and only edition (see Rara Aritmetica) of the first clear statement of the principle 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’. Only a few of his many works were printed, the present volume containing the majority of his published scientific writings in eight treatises, each published here for the first time: De sphaera liber unus (in which he criticized Copernicus); Computus Ecclesiasticus in summam collectus; Tractatus instrumentorum astronomicorum; Tractatus de lineis horariis; Euclidis propositiones elementorum; Musicae traditione; De lineis horariis; and Arithmeticorum.
“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 is 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). In Tractatus instrumentorum astronomicorum, Maurolico described the principal astronomical instruments and discussed their theory, use, and history—a subject similar to that treated in one of his first publications, the rare and little-known tract Quadrati fabrica et eius usus (Venice, 1546).
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. Pascal, who used induction extensively in his Traité du Triangle Arithmetique, learned of the principle from Maurolico, for in the proof of one proposition he writes ‘Cela est aisé par Maurolic’. It had in fact been used earlier by Euclid, Levi ben Gerson and others, but without clearly stating the principle involved. The term ‘induction’ was first used in this context by John Wallis in 1655.
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) 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). See also Smith’s Rara Arithmetica, p.349-50.
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. Maurolico himself referred to a vast literary production, but only a few of his works were printed, although these are enough to show him to have been an outstanding scholar. In addition to writing his own books, Maurolico translated, commented upon, reconstructed, and edited works by a number of ancient authors, including Apollonius, Archimedes, Autolycus, Menelaus and Theodosius.
Honeyman 2182; Smith, Rara Arithmetica, p.349-50; Adams M919; Macclesfield 1344; Riccardi II, 141. 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 Mathematial Society, 16 (1909–1910) 70–73.
Two volumes bound as one (as issued). 4to: 214 x 160 mm. Pp. , 285; [1: blank], , 175, , . Large printer’s device on each title, woodcut initials and type ornaments, numerous woodcut diagrams in text and margins. Contemporary limp vellum, spine heavely worn but entirely unrestored. Title page with numerous contemporary inscriptions and ancient stamp. In all a fine unsophisticated copy.