4to (230 x 159 mm), pp [xvi] 128; 123 [1]; 111 [1]; 99 [1]; 80; 71 [1]; 80, with woodcut printer's device on title and numerous woodcut diagrams in text; a clean, crisp copy in early eighteenth-century English calf, spine and joints a bit worn. £55,000
The exceptionally rare first edition of Cavalieri's text containing the discovery of integration methods, one of the most important forerunners of the integral calculus. It was used by Galileo, Cavalieri's pupil Torricelli, Wallis, Pascal, and others.
The work includes the statement of 'Cavalieri's principle' for the determination of areas and volumes, which considers an area as made up on an indefinite number of equidistant parallel line segments, and a solid as made up of an indefinite number of parallel plane areas. In modern terms, the principle states that two integrals are equal if the integrands are equal. Cavalieri's principle provided a simple and speedy alternative to the ancient method of exhaustion.
'The concept of indivisibles does sometimes show up fleetingly in the history of human thought: for example, in a passage by the eleventh-century Hebrew philosopher and mathematician Abraham bar Hiyya (Savasorda); in occasional speculations - more philosophical than mathematical - by the medieval Scholastics; in a passage by Leonardo da Vinci; in Kepler's Nova stereometria doliorum... [and in] Galileo...
'In Cavalieri we come to a rational systematization of the method of indivisibles, a method that not only is deemed useful in the search for new results but also, contrary to what Archimedes assumed, is regarded as valid, when appropriately modified, for purposes of demonstrating theorems.
'At this point a primary question arises: What significance did Cavalieri attribute to his indivisibles? This mathematician, while perfectly familiar with the subtle philosophical questions connected with the problem of the possibility of constituting continuous magnitudes by indivisibles, seeks to establish a method independent of the subject's hypotheses, which would be valid whatever the concept formed in this regard... It must be further pointed out, according to L. Lombardo Radice, that the Cavalieri view of the indivisibles has given us a deeper conception of the sets: it is not necessary that the elements of the set be assigned or assignable; rather it suffices that a precise criterion exists for determining whether or not an element belongs to the set' (Ettore Carruccio in DSB, q.v. for a detailed examination of the Geometria).
Cavalieri (1598-1647) was a Jesuit and professor of mathematics at Bologna. He considered himself a disciple of Galileo, whom he had met through his teacher Benedetto Castelli. It was Galileo who urged Cavalieri to look into problems of the calculus, and who praised him by stating that 'few, if any, since Archimedes, have delved as far and as deep into the science of geometry'. Galileo included Cavalieri's theory in a discussion of the theory of matter in the First Day and in the discussion of accelerated motion in the Third Day of his Discorsi e dimostrazioni matematiche (1638).
This work amongst the rarest of major mathematical texts. NUC cites only three copies in North America, at the University of Wisconsin, the University of Michigan (imperfect) and Cornell University (NUC also incorrectly cites Bryn Mawr College, but they in fact only have the second edition of 1653). OCLC adds Brown, Burndy Library, Boston College, and Linda Hall Library. Riccardi is in error in calling for two tables; he based his collation on a copy at the University of Modena, which contains two tables inserted from another, unrelated, work. All other copies conform in collation to the above.
Provenance: the Earls of Macclesfield, Shirburn Castle, with engraved bookplate, shelfmark on front pastedown, and blindstamp Macclesfield crest on blank margins of first three leaves
Kline Mathematical Thought pp 349-350; Rouse Ball A Short Account of the History of Mathematics pp 279-282; Struik A Source Book in Mathematics pp 209-219; Parkinson p 77
GBP 55000.00
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