Abstract
This chapter concerns the Renaissance, from the XV to XVI century. This is the time when the science of motion began to take its modern form with an injection of a strong dose of mathematics in the philosophy of nature of time. Niccolò Tartaglia ‘invented’ ballistics, the science of the motion of projectiles, based on some empirical observations and Euclidean geometry. After a fairly detailed presentation of Tartaglia’s Nova scientia of 1537, the chapter summarizes the conceptions of the motion of projectiles of the Italian scientists, who were the reference point of the whole of Europe. Among them, Leonardo da Vinci and Girolamo Cardano, both supporters of the theory of impetus. Another supporter of the theory of impetus was Giovanni Battista Benedetti, dealt with in the last part of the chapter, who for the first time in history suggested, with the use of mathematical arguments, that in a vacuum all bodies fall at the same speed.
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Notes
- 1.
For the history of science of this period refer to [369, 375, 416, 433, 596, 651].
- 2.
p. 9.
- 3.
Printed for Aldo Manuzio’s types, De rebus expetendis et fugiendis consisted of 49 books, 30 of which were devoted to sciences. The first book presents a classification of philosophy, within which the mathematical sciences plays a dominant role as given on the basis of the commentary to Euclid’s Elements made by ProclusProclus. Valla’s book contains references to Archimedes’ works.
- 4.
The last book of some importance toward the end of the XIV century was Questiones super tractatum de ponderibus of Biagio Pelacani da ParmaBiagio Pelacani da Parma (1365–1416).
- 5.
The Summa is composed of 308 cards in folio. The cards numbered from 1 to 150 are involved in speculative and practical arithmetic, operations with radicals and algebra. The last 158 cards instead contain a treatise on commercial mathematics, accounting and a treatise on geometry. The work appears as a monumental compendium of materials belonging to four distinct fields of mathematics. To complete the picture of mathematics known at that time, besides Ptolemaic astronomy, trigonometry only is missing.
- 6.
The De divina proportione is well known also for the famous Leonardo da Vinci’s engravings it contains.
- 7.
Byzantine humanist. Archbishop of Nicaea he was in Italy at the Council of Ferrara-Florence. He was named cardinal of the catholic church in 1439. He gave several codes in Latin and Greek to the republic of Venice.
- 8.
p. 104.
- 9.
p. 110.
- 10.
For the role of European universities in the XV century refer to [455], vol. X, [428, 499]. For the Italian universities see the Annals of the history of Italian universities. CLUEB, Bologna and [502].
- 11.
From the website of Padua University.
- 12.
In particular, di Giorgio Martini introduced elements of theory of machines and construction contained in Book X.
- 13.
This is the case for example of Padua, where the introduction of mathematics into the undergraduate curriculum preceded that of astronomy-astrology related to medicine.
- 14.
Considering the small number of chairs of mathematics in the university of Padua and Bologna compared to those of medicine until the time of Galileo, it can be seen as the academic discipline was marginal [400, p. 261].
- 15.
pp. 266–271.
- 16.
p. 24.
- 17.
p. 212.
- 18.
Dedicatory epistle. My translation. The reference to the old testament is Winsdom of Salomon, 11.20: “but thou hast ordered all things in measure, and number, and weight.”
- 19.
p. 130.
- 20.
pp. 5r–5v. My translation.
- 21.
pp. 46–47.
- 22.
p. 331.
- 23.
p. 61.
- 24.
p. 78r. Translation in [441].
- 25.
p. 78r. Translation in [328].
- 26.
pp. 78v–79r. Translation in [441].
- 27.
p. 82v. Translation in [441].
- 28.
p. 82v. Translation in [441].
- 29.
pp. 81–82.
- 30.
p. 400. Since the 1550 edition there were additions to the original text.
- 31.
p. 392.
- 32.
An indirect evidence of this access can be found in Book 7 of Quesiti et inventioni diverse. Here the figures Tartaglia reproduces for the balance are very close to that drawn by Leonico.
- 33.
pp. LV, LVI.
- 34.
p. 189.
- 35.
Dedicatory epistle.
- 36.
pp. 630–632.
- 37.
Tartaglia published the writings of Archimedes on mathematics and mechanics only in 1543, however, before of the 1550 edition of Nova scientia.
- 38.
p. 197.
- 39.
p. 194.
- 40.
pp. 238–245.
- 41.
pp. 349–380.
- 42.
Tartaglia went into possession, probably in 1539, of a manuscript of the Liber de ratione ponderis by Jordanus de Nemore [441], p. 16. He left it to Curtio Troiano to be published after his death.
- 43.
pp. 75–76.
- 44.
p. 377.
- 45.
pp. 1r–2v. In the first definition Drabkin and Drake translated ‘egualmente grave’ with ‘uniformly heavy’, I have preferred ‘constant heaviness’.
- 46.
pp. 1r.
- 47.
I, 1, 76b.
- 48.
p. 3r.
- 49.
p. 3v.
- 50.
pp. 3v–7v.
- 51.
pp. 58–59.
- 52.
p. 199.
- 53.
p. 4r. Translation adapted from [441].
- 54.
pp. 75–76.
- 55.
The subject had been previously dealt with by Albertus de Saxonia [2], Liber II, quaestio XIV, p. 66 and Nicole Oresme [275, p 144].
- 56.
p. 6r. Translation from [441].
- 57.
pp. 7r, 7v.
- 58.
pp. 10v–11r.
- 59.
In the literature it is often found the wrong assertion that Tartaglia in Nova scientia considered perfectly straight trajectories, to recognize only in the Quesiti et inventioni diverse that they were everywhere curvilinear.
- 60.
p.11r. Translation in [441].
- 61.
p. 90.
- 62.
pp. 13v–18r.
- 63.
pp. 197–200.
- 64.
f. 589v, new numeration. My translation.
- 65.
The knowledge of Albertus de Saxonia by Leonardo is documented as he named him in various occasions [109].
- 66.
f. 8r. My translation.
- 67.
f. 29v. My translation.
- 68.
That a projectile goes increasing its speed at the beginning of the motion to reach a maximum at a certain distance from the launch point was one thing debated in antiquity; in the Middle Ages it was shared by Oresme too (see Sect. 2.5.2). Tartaglia correctly believed that this was not true [325, 328], Book I. Benedetti was silent, merely to criticize the explanation proposed by Tartaglia.
- 69.
vol. 1, Man. A, 43v. My translation.
- 70.
vol. 1, Man. A, 4r. My translation.
- 71.
vol. 1, Man. A, 4r.
- 72.
III, 42.
- 73.
pp. 31–32. My translation.
- 74.
pp. 253, 254. My translation.
- 75.
p. 11r. Translation in [441].
- 76.
p. 11v. Translation in [441].
- 77.
p. 9r.
- 78.
p. 9v. Translation in [441].
- 79.
23r.
- 80.
p. 115r.
- 81.
Ferrari, Primo cartello, p. 2.
- 82.
p. 68. My translation.
- 83.
p. 69. My translation.
- 84.
p. 70. My translation.
- 85.
pp. 286–287. Translation in [441].
- 86.
p. 184. Translation in [441].
- 87.
p.180.
- 88.
pp. 258–259.
- 89.
p. 259.
- 90.
p. 161.
- 91.
p. 161.
- 92.
p. 116.
- 93.
Chapter 2. See also [441, p. 199].
- 94.
From the discussion of this statement it is clear that Benedetti is concerned not with absolute weight but specific weight.
- 95.
p. 17. Not numbered in the text. Translation in [441].
- 96.
p. 19. Not numbered in the text. Translation in [441].
- 97.
Here the most recent version (the second one) of Demonstratio proportionum motuum localium is referred to [563], pp. XXVI–XXVII.
- 98.
p. 71. Translation in [441].
- 99.
Benedetti disproved Aristotle on vacuum in [35] (Disputationes de quibusdam placitis Aristotelis), p. 168.
- 100.
p. 71. Translation in [441].
- 101.
De motu [151], p. 265; Postille alle Esercitazioni filosofiche di Antonio Rocco [161], vol. VII, pp. 731–732; Discorsi e dimostrazioni matematiche [157], pp. 107–108.
- 102.
pp. 174–175. Translation in [441].
- 103.
For a fairly extensive discussion on the problems of equilibrium see [381].
- 104.
III, 62–64.
- 105.
For example Euclid’s Elements were known in the Latin translation by Giovanni Campano of 1255.
- 106.
Thomas Kuhn says that De revolutionibus orbium coelestium was a text able to promote a revolution more than a revolutionary text [545], Chapt. 5.
- 107.
For example [433, 541, 545]. For a detailed bibliography see [427], pp. 409–442.
- 108.
Book 1, cap1, p. 6v. My translation.
- 109.
Book 1, cap1, p. 7r. My translation.
- 110.
This is also the opinion of Anna De Pace [427], p. 195.
- 111.
Author’s introduction. Translation in [203].
- 112.
To be precise Tartaglia died in 1543, while the Diversarum speculationum mathematicarum was of 1585.
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Capecchi, D. (2014). Humanism and Renaissance. In: The Problem of the Motion of Bodies. History of Mechanism and Machine Science, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-319-04840-6_3
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