Science in the Renaissance

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Template:Renaissance

Science in the Renaissance was predominantly an extension of medieval traditions of natural philosophy but also included a radical revolution in the European understanding of the Earth's geography and produced new ideas and methods in anatomy, mathematics, and astronomy. The collection of ancient scientific texts began in earnest at the start of the 15th century and continued up to the Fall of Constantinople in 1453, and the invention of printing allowed a faster propagation of new ideas. Some scholars have argued that the era was scientifically backward because Renaissance humanists were excessively deferential to ancient authorities or because they favored human-centered subjects like politics and history over study of natural philosophy or applied mathematics. More recently, however, scholars have acknowledged the positive impact of the rediscovery of lost or obscure texts and the increased focus on the study of language and the correct reading of texts,[1][2] while also emphasizing how the invention and rapid spread of the moveable type printing press and the encounter with the Americas served as necessary preconditions for the Scientific Revolution of the 17th century.

## Contents

* 1 Renaissance humanism
* 2 Important developments
* 2.1 Printing
* 2.2 Geography and the New World
* 2.3 Alchemy and chemistry
* 2.4 Astronomy
* 2.5 Mathematics
* 2.6 Physics
* 2.7 Medicine
* 3 See also
* 4 Notes
* 5 References

## Renaissance humanism

Renaissance humanism was a "movement to recover, interpret, and assimilate the language, literature, learning and values of ancient Greece and Rome",[3] which asserted "the genius of man ... the unique and extraordinary ability of the human mind".[4] The scholar, poet and Catholic priest Petrarch's rediscovery of Cicero's letters in the 14th century is often credited with initiating the movement.[5] In Italy, the humanist educational program (whose subjects came to be known as the "humanities") won rapid acceptance, and by the mid-15th century many of the upper classes had received humanist educations. Some of the highest officials of the Catholic Church were humanists (with the resources to amass important libraries), and there were several 15th-century and early-16th-century humanist popes.[6]

The science historian A. Rupert Hall writes that "there are important reservations to be made concerning Renaissance humanism as a force making for innovation in science":

> ... [I]n their revision of classical authors for the press the scholar scientists were if anything less prone to comment adversely upon them than their medieval predecessors. Among the humanists intense admiration for the work of antiquity led to the belief than human talent and achievement had consistently deteriorated after the golden age of Hellenistic civilization; that the upward ascent demanded imitation of this remote past, rather than an adventure along strange paths ... [H]umanism sometimes made it more difficult to enunciate a new idea, or to criticize the splendid inheritance from antiquity.[7]

Science historian Herbert Butterfield makes a similar point about the deference Renaissance humanists showed to ancient authorities:

> A primary aspect of the Renaissance ... is the fact that it completes and brings to its climax the long process by which the thought of antiquity was being recovered in the middle ages. It even carries to what at times is a ludicrous extreme the spirit of an exaggerated subservience to antiquity, the spirit that helped to turn Latin into a dead language. Ideas may have appeared in new combinations, but we cannot say that essentially new ingredients were introduced into our civilization in the Renaissance.[8]

## Important developments

### Printing

The spread of printing between 1440 and 1500.

From a single print shop in Mainz, Germany, around 1440, the movable type printing-press had spread to around 270 cities in central, western and eastern Europe and had already produced more than 20 million volumes by the end of the 15th century.[9] Printing made scholarly books much more widely accessible, but most of the initial demand for the new printed books during the Renaissance was for the works of long-dead authors rather than living scholars.[10]

By the late 16th century, however, the manuscript culture of the Middle Ages, where facts were few and far between, had begun to give way to a printing culture where living scholars could compare their own observations with those of other (living) scholars.[11] That shift decisively transformed natural philosophy into a collective and cumulative enterprise where reliable and documented facts rapidly proliferated, creating the intellectual scaffolding for the Scientific Revolution and a secure foundation for new scientific knowledge.[12]

### Geography and the New World

In the history of geography, the key classical text was the Geographia of Claudius Ptolemy (2nd century). It was translated into Latin in the 15th century by Jacopo d'Angelo.[13] It was widely read in manuscript and went through many print editions after it was first printed in 1475. Regiomontanus worked on preparing an edition for print prior to his death; his manuscripts were consulted by later mathematicians in Nuremberg. Ptolemy's Geographia became the basis for most maps made in Europe throughout the 15th century.[13] Even as new knowledge began to replace the content of old maps, the rediscovery of Ptolemy's mapping system, including the use of coordinates and projection, helped to redefine the overall field of cartography as a scientific pursuit rather than an artistic one.[13]

European voyages of the Age of Discovery

The information provided by Ptolemy, as well as Pliny the Elder and other classical sources, was soon seen to be in contradiction to the lands explored in the Age of Discovery.[13] The new discoveries revealed shortcomings in classical knowledge and opened the European imagination to new possibilities.[14][15] In particular, Christopher Columbus' voyage to the New World in 1492 helped set the tone for what would soon after become a wave of European expansion.[16] Thomas More's Utopia was inspired partly by the discovery of the New World. Most maps developed prior to this period grossly underestimated the extent of the lands separating Europe from India on a westward route through the New World; however, through contributions of explorers such as Ferdinand Magellan, efforts were made to create more accurate maps during this period.[17]

### Alchemy and chemistry

Diagram of Alchemical Elements

While differing in some respects, alchemy and chemistry often had similar goals during the Renaissance period, and together they are sometimes referred to as chymistry.[18] Alchemy is the study of the transmutation of materials through obscure processes. Although it is often viewed as a pseudoscientific endeavor, many of its practitioners utilized widely accepted scientific theories of their times to formulate hypotheses about the constituents of matter and the ways matter could be changed.[19] One of the main aims of alchemists was to find a method of creating gold and other precious metals from the transmutation of base materials.[19] A common belief of alchemists was that there is an essential substance from which all other substances formed, and that if you could reduce a substance to this original material, you could then construct it into another substance, like lead to gold.[18] Medieval alchemists worked with two main elements or "principles", sulphur and mercury.[18]

Paracelsus was a chymist and physician of the Renaissance period who believed that, in addition to sulphur and mercury, salt served as one of the primary alchemical principles from which everything else was made.[20] Paracelsus was also instrumental in helping to put chemical practices to practical medicinal use through a recognition that the body operates through processes which may be seen as chemical in nature.[20] These lines of thinking directly conflicted with many long-held traditional beliefs, such as those popularized by Aristotle; however, Paracelsus was insistent that questioning principles of nature was essential to continue the general growth of knowledge.[20] Hall describes him as a "picturesque ranter" whose "mystical conception of nature [was] entirely alien to that of natural science."[21]

Despite its frequent basis in what may be considered scientific practices by modern standards, numerous factors caused chymistry as a discipline to remain separate from general academia until near the end of the Renaissance, when it finally began appearing as a portion of some university education.[18][22]: 104–115 The commercial nature of chymistry at the time, along with the lack of classical basis for the practice, were some of the contributing factors which led to the general view of the discipline as a craft rather than a respectable academic discipline.[18]

### Astronomy

Pages from 1550 Annotazione on Sacrobosco's De sphaera mundi, showing the Ptolemaic system

The astronomy of the late Middle Ages was based on the geocentric model described by Claudius Ptolemy in antiquity. Probably very few practicing astronomers or astrologers actually read Ptolemy's Almagest, which had been translated into Latin by Gerard of Cremona in the 12th century. Instead they relied on introductions to the Ptolemaic system such as the De sphaera mundi of Johannes de Sacrobosco and the genre of textbooks known as Theorica planetarum. For the task of predicting planetary motions they turned to the Alfonsine tables, a set of astronomical tables based on the Almagest models but incorporating some later modifications, mainly the trepidation model attributed to Thabit ibn Qurra. Contrary to popular belief, astronomers of the Middle Ages and Renaissance did not resort to "epicycles on epicycles" in order to correct the original Ptolemaic models—until one comes to Copernicus himself.

Sometime around 1450, mathematician Georg Purbach (1423–1461) began a series of lectures on astronomy at the University of Vienna. Regiomontanus (1436–1476), who was then one of his students, collected his notes on the lecture and later published them as Theoricae novae planetarum in the 1470s. This "New Theorica" replaced the older theorica as the textbook of advanced astronomy. Purbach also began to prepare a summary and commentary on the Almagest. He died after completing only six books, however, and Regiomontanus continued the task, consulting a Greek manuscript brought from Constantinople by Cardinal Bessarion. When it was published in 1496, the Epitome of the Almagest made the highest levels of Ptolemaic astronomy widely accessible to many European astronomers for the first time.

Nicolaus Copernicus

The last major event in Renaissance astronomy was the work of Nicolaus Copernicus (1473–1543). He was among the first generation of astronomers to be trained with the Theoricae novae and the Epitome. Shortly before 1514 he began to revive Aristarchus's idea that the Earth revolves around the Sun. He spent the rest of his life attempting a mathematical proof of heliocentrism. In De revolutionibus orbium coelestium, published in 1543, Copernicus attempted to align his work as closely as possible with Ptolemaic tradition. A comparison of his work with the Almagest shows that he followed Ptolemy's methods[23] and even his order of presentation.[24] Yet, in order to purge astronomy of the equant—which violated the theological and philosophical ideal that all celestial motion must be "perfect" (i.e. circular) and uniform[25]—Copernicus challenged Ptolemy's geocentrism, an orthodoxy that had prevailed for over a millennium. Copernicus' heliostatic model (with a stationary Sun located near, though not precisely at, the mathematical center of the heavens[26][27][28]) retained several false Ptolemaic assumptions such as the planets' circular orbits, epicycles, and uniform speeds,[29] but also included such accurate ideas as:

* The Earth is one of several planets revolving around the Sun in a determined order.
* The Earth rotates daily on its axis and revolves annually about the Sun.Template:Sfnq
* Retrograde motion of the planets is explained by the Earth's motion.
* The distance from the Earth to the Sun is small compared to the distance from the Sun to the stars.

### Mathematics

Archimedes' proofs of the area of a parabolic segment in Quadrature of the Parabola inspired work on quadratures and cubatures in the 15th and 16th centuries.[30]

The accomplishments of Greek mathematicians survived throughout Late Antiquity and the Middle Ages through a long and indirect history. Much of the work of Euclid, Archimedes, and Apollonius, along with later authors such as Hero and Pappus, were copied and studied in both Byzantine culture and in Islamic centers of learning. Translations of these works began already in the 12th century, with the work of translators in Spain and Sicily, working mostly from Arabic and Greek sources into Latin. Two of the most prolific were Gerard of Cremona and William of Moerbeke.

The greatest of all translation efforts, however, took place in the 15th and 16th centuries in Italy, as attested by the numerous manuscripts dating from this period currently found in European libraries. Virtually all leading mathematicians of the era were obsessed with the need for restoring the mathematical works of the ancients. Not only did humanists assist mathematicians with the retrieval of Greek manuscripts, they also took an active role in translating these work into Latin, often commissioned by religious leaders such as Nicholas V and Cardinal Bessarion.[31][32]

Some of the leading figures in this effort include Regiomontanus, who made a copy of the Latin Archimedes and had a program for printing mathematical works; Commandino (1509–1575), who likewise produced an edition of Archimedes, as well as editions of works by Euclid, Hero, and Pappus; and Maurolyco (1494–1575), who not only translated the work of ancient mathematicians but added much of his own work to these. Their translations ensured that the next generation of mathematicians would be in possession of techniques far in advance of what it was generally available during the Middle Ages.[33]

It must be borne in mind that the mathematical output of the 15th and 16th centuries was not exclusively limited to the works of the ancient Greeks. Some mathematicians, such as Tartaglia and Luca Paccioli, welcomed and expanded on the medieval traditions of both Islamic scholars and people like Jordanus and Fibonnacci.[34][35] Giordano Bruno was also one to critique the works of people like Aristotle, whom he believed to have a flawed logic and developed a mathematical doctrine for the computation of partial physics, with Bruno attempting to transform theories of nature.[36]

### Physics

The progress being made in mathematics was complemented by advancements in physics, with people attempting to bridge the gap between the two fields and question Aristotelian ideas.[37] The revived investigation of physics opened up many opportunities in subfields like mechanics, optics, navigation, and cartography.[22]: 79–89

Mechanical theories had originated with the Greeks, especially Aristotle and Archimedes.[22]: 79–82 Mechanics and philosophy had been related disciplines in ancient Greece, and only in the Renaissance did the two subjects begin to split.[22]: 79–82 A lot of the work of developing new mechanical ideas and theories was carried out by Italians such as Rafael Bombelli, though the Fleming Simon Stevin also provided many ideas.[22]: 79–82

Navigation was an important topic of the time, and many innovations were made that, with the introduction of better ships and applications of the compass, would later lead to geographical discoveries.[22]: 89–91 The calculations involved in navigation proved to be difficult, with the technology of the time unable to accuately predict weather or determine one's geographic position. Determining one's longitude proved especially challenging, since one's local time need to be calculated on the basis of an astronomical observation.[22]: 89–91 One theory that was tested was to record the time of an eclipse and use Regiomontanus' Ephemerides to compare it with Nuremberg time or Zacuto's Almanach perpetuum to compare it with Salamanca time, though the margin of error in such calculations was unacceptably great (around 25.5 degrees).[22]: 89–91 Until longitude could be accurately determined, navigators had to rely on dead reckoning, with its many uncertainties.[22]: 89–91

### Medicine

Main article: History:Medical Renaissance

With the Renaissance came an increase in experimental investigation, principally in the field of dissection and body examination, thus advancing our knowledge of human anatomy.[38] The development of modern neurology began in the 16th century with Andreas Vesalius, who described the anatomy of the brain and other organs; he had little knowledge of the brain's function, thinking that it resided mainly in the ventricles. Understanding of medical sciences and diagnosis improved, but with little direct benefit to health care. Few effective drugs existed, beyond opium and quinine. William Harvey provided a refined and complete description of the circulatory system. The most useful tomes in medicine, used both by students and expert physicians, were materiae medicae and pharmacopoeiae.

## See also

* Continuity thesis
* The Copernican Question
* Renaissance magic
* Renaissance technology

## Notes

1. ↑ Anglin, W. S.; Lambek, J. (1995), "Mathematics in the Renaissance", in Anglin, W. S.; Lambek, J. (in en), The Heritage of Thales, Undergraduate Texts in Mathematics, New York, NY: Springer, pp. 125–131, doi:10.1007/978-1-4612-0803-7_25, ISBN 978-1-4612-0803-7, https://doi.org/10.1007/978-1-4612-0803-7_25, retrieved 2021-04-09
2. ↑ Jayawardene, S. A. (June 1978). "The Italian Renaissance of Mathematics: Studies on Humanists and Mathematicians from Petrarch to Galileo. Paul Lawrence Rose". Isis 69 (2): 298–300. doi:10.1086/352043. ISSN 0021-1753. https://www.journals.uchicago.edu/doi/pdfplus/10.1086/352043.
3. ↑ Burke, P., "The spread of Italian humanism", in The Impact of Humanism on Western Europe, ed. A. Goodman and A. MacKay, London, 1990, p. 2.
4. ↑ As asserted by Gianozzo Manetti in On the Dignity and Excellence of Man, cited in Clare, J., Italian Renaissance.
5. ↑ Wootton, David (1996). Modern Political Thought: Readings from Machiavelli to Nietzsche. Hackett Publishing. p. 1. ISBN 978-0-87220-341-9. https://archive.org/details/modernpoliticalt00woot. Retrieved 27 August 2013.
6. ↑ They include Innocent VII, Nicholas V, Pius II, Sixtus IV, Alexander VI, Julius II and Leo X. Innocent VII, patron of Leonardo Bruni, is considered the first humanist pope. See James Hankins, Plato in the Italian Renaissance (New York: Columbia Studies in the Classical Tradition, 1990), p. 49; for the others, see their respective entries in Sir John Hale's Concise Encyclopaedia of the Italian Renaissance (Oxford University Press, 1981).
7. ↑ Hall, A. Rupert (1970). The Scientific Revolution, 1500–1800: The Formation of the Modern Scientific Attitude (2nd ed.). Boston: Beacon Press. pp. 8–9. ISBN 0-8070-5093-8.
8. ↑ Butterfield, Herbert (1957). The Origins of Modern Science. Free Press. p. 40. ISBN 978-0-6848-3637-9.
9. ↑ Febvre, Lucien; Martin, Henri-Jean (1976). The Coming of the Book: The Impact of Printing 1450–1800. London: New Left Books. Quoted in: Anderson, Benedict. Comunidades Imaginadas. Reflexiones sobre el origen y la difusión del nacionalismo. Fondo de cultura económica, Mexico, 1993. ISBN 978-968-16-3867-2. pp. 58f.
10. ↑ Hall, A. Rupert (1970). The Scientific Revolution, 1500–1800: The Formation of the Modern Scientific Attitude (2nd ed.). Boston: Beacon Press. pp. 71–72. ISBN 0-8070-5093-8. "A hasty reference to the output of the printing press may act as a guide to the nature of contemporary [Renaissance] taste and estimations; the most sought after and respected books were still those composed long before the invention of printing. Neither the library, nor the academic training, of the medieval world were suddenly outmoded by a vast efflux of novel aspirations and methods. The mythical "renaissance man" of the early sixteenth century, though his tastes might be more hedonistic than those of his forefathers, though he might be more enamored of the powers and potentialities of this world and less regardful of the next, was still largely limited in his science to the achievements of the medieval renaissance: his very classicism only attached him more deeply to the roots of western learning."
11. ↑ "Printing made it possible for Brahe to survey a wide range of publications (there were over a hundred on the comet of 1577, though many were merely astrological prognostications) and demonstrate that the four best observers had produced results compatible with his own. It also ensured that Brahe's new system was quickly known throughout Europe, so that his arguments could be tested against the nova of 1604 and the comets of 1618. Printing created a community of astronomers working on common problems with common methods and reaching agreed solutions." Wootton, David. The Invention of Science: A New History of the Scientific Revolution (Penguin, 2015). pp.197-198. ISBN 0-06-175952-X
12. ↑ Wootton, David (2015). The Invention of Science: A New History of the Scientific Revolution. Penguin. p. 282. ISBN 978-0-06-175952-9. "... a manuscript culture, in which experience is unspecific, indirect, and amorphous ... a print culture, in which experience is specific, direct, documented and retrievable. ... In comparison to the world of print, manuscript culture is one of rumour and gossip. The printing press represents an information revolution, and secure facts are its consequence."
13. ↑ 13.0 13.1 13.2 13.3 Hunt, Arthur (2000). "2000 Years of Map Making". Geography 85 (1): 3–14. doi:10.1080/20436564.2000.12219726. ISSN 0016-7487. https://www.jstor.org/stable/40573370.
14. ↑ Daston, Lorraine; Park, Katherine (1998). Wonders and the Order of Nature. USA: New York: Zone Books. p. 147. "...the revelation that the world included whole new continents undreamed of by the ancients opened up a fissure in time as well as space. Compared to earlier writing on travel, the work from the decades after 1492 demonstrate a heightened sense of novelty and possibility – of just how new and different things were able to be."
15. ↑ "...before Columbus discovered America in 1492 there was no clear-cut and well-established idea of discovery; the idea of discovery is, as will become apparent, a precondition for the invention of science....the discovery of America disproved a central claim about the world which had been generally accepted before 1492...This was a crucial precondition for the astronomical revolution which followed." Wootton, David. The Invention of Science: A New History of the Scientific Revolution (Penguin, 2015) . xiv + 769 pp. ISBN 0-06-175952-X
16. ↑ Cortada, James W. (1974). "Who Was Christopher Columbus?". Renaissance and Reformation / Renaissance et Réforme 10 (2): 99–102. doi:10.33137/rr.v10i2.13735. ISSN 0034-429X.
17. ↑ Heawood, Edward (1921). "The World Map Before and After Magellan's Voyage". The Geographical Journal 57 (6): 431–442. doi:10.2307/1780791. ISSN 0016-7398. Bibcode: 1921GeogJ..57..431H. https://archive.org/details/heawood-1921-sim-geographical-journal-1921-06-57-6.
18. ↑ 18.0 18.1 18.2 18.3 18.4 Principe, Lawrence (2011). The Scientific Revolution: A Very Short Introduction. Very Short Introductions (1st ed.). Oxford: Oxford University Press. ISBN 978-0-19-956741-6.
19. ↑ 19.0 19.1 Lindberg, David C. (2007). The Beginnings of Western Science: The European Scientific Tradition in Philosophical, Religious, and Institutional Context, Prehistory to A.D. 1450 (2nd ed.). Chicago: University of Chicago Press. pp. 290–294. ISBN 978-0-226-48205-7.
20. ↑ 20.0 20.1 20.2 Moran, Bruce T. (2019). Paracelsus: An Alchemical Life. Renaissance lives. London, UK: Reaktion Books. ISBN 978-1-78914-144-3.
21. ↑ Hall, A. Rupert (1970). The Scientific Revolution, 1500–1800: The Formation of the Modern Scientific Attitude (2nd ed.). Boston: Beacon Press. pp. 73. ISBN 0-8070-5093-8. "Paracelsus, whatever his other merits, was a picturesque ranter and it is futile to portray him as a herald of the scientific revolution. The texture of his thought - in which it is difficult to see any precise pattern - was woven upon a mystical conception of nature entirely alien to that of natural science."
22. ↑ 22.0 22.1 22.2 22.3 22.4 22.5 22.6 22.7 22.8 Sarton, George (1967). Six Wings: Men of Science in the Renaissance. Bloomington: Indiana University Press.
23. ↑ "... there was not a single proof or mathematical procedure in the De Revolutionibus which did not have its exact replica in the Almagest. ... Copernicus again and again demonstrated by numerical computation that his model agreed with Ptolemy's." Neugebauer, Otto (1968). "On the Planetary Theory of Copernicus". Vistas in Astronomy 10: 103.
24. ↑ "Copernicus admired Ptolemy enormously; in organizing his book, ordering the different chapters and choosing the sequence in which various topics were introduced, he followed Ptolemy's Almagest. Cohen, Bernard. The Birth of New Physics (W. W. Norton & Company, 1985). p.35. ISBN 9780393019940
25. ↑ "All movement within it (outside of the immediate vicinity of the Earth) is determined by the fundamental principle that heavenly movement is circular and therefore unchanging. Ptolemy, Copernicus thought, had betrayed this principle not ... by adding epicycles to deferents in order to explain why the planets sometimes appear to move backwards in the sky, but by introducing the equant in order to speed them up and slow them down." Wootton, David. The Invention of Science: A New History of the Scientific Revolution (Penguin, 2015). p.152. ISBN 0-06-175952-X
26. ↑ "The man who had deposed the earth from its proud position as the centre of the universe and had recognized it to be merely one of the planets, had yet felt compelled to give it quite an exceptional position in his new system. Though he had said 'in the midst of all stand the sun,' he had in his planetary theories assumed the centre of all movements to be the centre of the earth's orbit, where the sun was not." Dreyer, J.L.E. A History of Astronomy from Thales to Kepler, (Dover Publications, 1953). p.343. ISBN 9780486600796
27. ↑ Butterfield, Herbert (1957). The Origins of Modern Science. Free Press. p. 40. ISBN 978-0-6848-3637-9. "It was a disadvantage of [Copernicus'] system that it was not quite heliocentric after all - the Earth did not describe an exact circle with the sun at its centre, and, in fact, all the movements of the skies were reckoned not from the sun itself, but from the centre of the earth's orbit, which came somewhat to the side."
28. ↑ "Copernicus's universe is different from Ptolemy's in that the sun, not the Earth, lies at (or rather, to be exact, very close to) its centre." Wootton, David. The Invention of Science: A New History of the Scientific Revolution (Penguin, 2015). p.152. ISBN 0-06-175952-X
29. ↑ Kuhn, Thomas S. (1957). The Copernican Revolution: Planetary Astronomy in the Development of Western Thought. Cambridge, Massachusetts: Harvard University Press. ISBN 0-674-17103-9.
30. ↑ Høyrup, Jens (2019), Sgarbi, Marco, ed. (in en), Archimedes: Reception in the Renaissance, Cham: Springer International Publishing, pp. 1–7, doi:10.1007/978-3-319-02848-4_892-1, ISBN 978-3-319-02848-4, https://doi.org/10.1007/978-3-319-02848-4_892-1, retrieved 2021-04-23
31. ↑ "Mathematics - Rome Reborn: The Vatican Library & Renaissance Culture | Exhibitions - Library of Congress". 1993-01-08. https://www.loc.gov/exhibits/vatican/math.html.
32. ↑ Gouwens, Kenneth (1996-09-22). "Rome Reborn: The Vatican Library and Renaissance Culture" (in English). Renaissance Quarterly 49 (3): 618–620. doi:10.2307/2863370. https://go.gale.com/ps/i.do?p=AONE&sw=w&issn=00344338&v=2.1&it=r&id=GALE|A18881400&sid=googleScholar&linkaccess=abs.
33. ↑ Rose, Paul Lawrence (1973). "Humanist Culture and Renaissance Mathematics: The Italian Libraries of the Quattrocento". Studies in the Renaissance 20: 46–105. doi:10.2307/2857013. ISSN 0081-8658. https://www.jstor.org/stable/2857013.
34. ↑ Malet, Antoni (2006-02-01). "Renaissance notions of number and magnitude" (in en). Historia Mathematica 33 (1): 63–81. doi:10.1016/j.hm.2004.11.011. ISSN 0315-0860.
35. ↑ Høyrup, Jens (2003). Practitioners – school teachers – "mathematicians": The divisions of pre-Modern mathematics and its actors.
36. ↑ Gatti, Hilary (1999). Giordano Bruno and Renaissance Science. Cornell University. pp. 144. ISBN 0-8014-3529-3.
37. ↑ Singleton, Charles Southward, ed (1970). Art, Science, and History in the Renaissance. The Johns Hopkins Humanities Seminars (2. printing ed.). Baltimore, Md.: Johns Hopkins Press. pp. 310–318. ISBN 978-0-8018-0602-5.
38. ↑ Siraisi, N. G. (2012). "Medicine, 1450–1620, and the History of Science". Isis 103 (3): 491–514. doi:10.1086/667970. PMID 23286188.

## References

* Dear, Peter. Revolutionizing the Sciences: European Knowledge and Its Ambitions, 1500–1700. Princeton: Princeton University Press, 2001.
* Debus, Allen G. Man and Nature in the Renaissance. Cambridge: Cambridge University Press, 1978.
* Grafton, Anthony, et al. New Worlds, Ancient Texts: The Power of Tradition and the Shock of Discovery. Cambridge: Belknap Press of Harvard University Press, 1992.
* Grant, Edward (1971) (in en). Physical Science in the Middle Ages. Cambridge University Press. ISBN 9780521292948. https://books.google.com/books?id=bD2himiEabQC.
* Hall, Marie Boas. The Scientific Renaissance, 1450–1630. New York: Dover Publications, 1962, 1994.

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