**Giovanni Benedetti** was an Italian mathematician who anticipated some of Galileo's work on free fall.

- Benedetti studied the first four books of Euclid's Elements under Niccolo Fontana, better known as Tartaglia, probably about 1546-1548, although later they seem to have fallen out with each other.
- Certainly during the period that Benedetti was Tartaglia's pupil, Tartaglia was involved the public challenge with Ferrari in which he lost his reputation.
- This must have had a significant effect on his pupil and, as a consequence, it is not surprising that in Benedetti's later work he is highly critical of Tartaglia's writings.
- When he was only twenty-two years of age Benedetti wrote Resolutio omnium Euclidis problematum aliorumque ad hoc necessario inventorum una tantummodo circini data apertura which was published in Venice in 1553.
- It must be significant that the question of proving some of Euclid's theorems with only a compass of fixed opening had been posed by Tartaglia as challenges to Cardan and Ferrari so it is likely that this provided the motivation for Benedetti's study.
- Both Ferrari and Tartaglia produced results on using only a compass of fixed opening but neither produced anything as systematic or of the quality that Benedetti did.
- Tartaglia's last work was Trattato generale di numeri e misure (1560) on precisely that topic - it is not of the quality of Benedetti's work and, interestingly, does not even mention Benedetti or the Resolutio.
- This is a dedication in the form of a letter addressed to Gabriel de Guzman, a Spanish Dominican priest with whom he had held discussions in Venice in 1552 on Benedetti's theory of the free fall of bodies.
- Guzman had suggested that Benedetti publish a mathematical treatment of the speeds of free fall and this letter was his answer to Guzman.
- The answer is that Benedetti feared that his ideas would be stolen if he did not get them into print quickly, so he chose this route.
- What were these ideas of Benedetti on free fall?
- Of course at this time Aristotle's views were accepted without question and what Benedetti was putting forward was in complete contradiction to Aristotle.
- As one might expect those who read Benedetti's work did not accept his ideas.
- Benedetti himself remarks that he had critics in Rome who said that Aristotle was always right, so his work must be wrong.
- Returning to the 1550s, we record Benedetti's appointment as court mathematician to Duke Ottavio Farnese at Parma from 1558 until 1566.
- Benedetti was quite rich so when the Duke failed to pay him for long periods, he was not at all worried financially.
- In 1587 Benedetti became ducal mathematician and philosopher, employed by Emanuele Filberto the Duke of Savoy, a post he held until his death.
- Being conscious that a man knows a science so long as he continues to examine and study it, every day he used to listen to a lesson about Euclid or another writer of those sciences from a Venetian Mr Giovanni Battista Benedetti, a man who, not only in my opinion, but also according to the judgement of many expert people, is at present the best in this profession and is greatly appreciated by the Duke because, besides possessing this most excellent science, he is also such a good teacher that each of his pupils becomes very easily competent and expert in it.
- Benedetti published De gnomonum umbrarumque solarium liber (1574), a book on sundials.
- Battista Benedetti Filosofo del Sereniss.
- Carolum Emunuelem Allobrogum et Subalpinorum ducem invictissimum includes several different pieces written by Benedetti over a number of years.
- Dealing with regular polygons, Benedetti proved that Dürer's construction of the regular pentagon using compasses with a fixed opening is not exact.
- The Diversarum Speculationum contains a section on mechanics in which Benedetti again attacks Aristotle's physical concepts and also attacks Tartaglia's mechanics.
- Benedetti should certainly get the credit for being the first to publish such a claim.
- We must not give the impression that Benedetti's contributions are better than Galileo's for although he has deeper understanding on some points, he does not attempt to give the mathematical formulation of acceleration which is Galileo's great achievement.
- In Diversarum Speculationum Benedetti also considers hydrostatic pressure and the idea of hydraulic lift.
- Benedetti must have held discussions with him during this time, then written the two letters in question in 1563 shortly after da Rore left Parma for Venice.
- Proceeding thus, and asserting that the frequency of vibration of two strings under equal tension vary inversely with the strings lengths, Benedetti proposed an index of agreement obtained by multiplication of the terms of the ratios of a given consonance; by this means he could express the degree of concordance in a mathematical scale.
- Benedetti argued that since sound consists of air waves or vibrations, in the more consonant intervals the shorter more frequent waves concurred with the longer less frequent waves at regular intervals.
- Benedetti's theory was espoused in the next century by Isaac Beeckman and Marin Mersenne, who sought René Descartes' opinion of it.
- Let us now turn to Benedetti's contributions to perspective.
- the most important aspect of this originality is that unlike all his predecessors Benedetti treats the problem of perspective construction as three-dimensional.
- Benedetti has returned from the two-dimensional constructions used by artists to the actual physical configurations considered, albeit in a completely geometrical manner, in Euclid's work on Optics.
- In his very first work Benedetti included his ideas on free fall so that he could claim priority and ensure the ideas were not stolen.
- Benedetti was not so lucky.
- This is totally a piece of plagiarism, for Taisnier claims as his own the Epistola de magnete of Pierre de Maricourt and the Demonstratio, the treatise on the fall of bodies by Benedetti.
- Now, ironically, Taisnier's work became better known than that of Benedetti, particularly after it was translated into English by Richard Eden in 1578.
- However, Taisnier had stolen the first edition of Benedetti's work and Stevin criticised it, making the correction that Benedetti had made himself in his second edition 32 years earlier.
- Benedetti was not aware that Taisnier had stolen his work until after 1570 and in the Preface of De gnomonum, written in 1573, he voiced his anger.

Born 14 August 1530, Venice, Venetian States (now Italy). Died 20 January 1590, Turin, Duchy of Savoy (now Italy).

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Origin Italy

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive