In around 1588 Ghetaldi completed his high school education and, coming from an affluent important family, he was able to spend the next couple of years in independent study while moving in a circle of friends which included the leading scholars from the city. The topics he studied were scientific and as well as reading the latest mathematical papers he also studied mathematical astronomy. Beginning in 1590, Ghetaldi carried out a number of tasks in administration including working in the State Office for Weapons and working in the Office for the Sale of Salt. After acting as a clerk in various government departments, he left Dubrovnik in 1595 with his friend Marin Gucetic to visit various European countries. Ghetaldi first went to Rome before travelling extensively in Europe. In Rome he attended lectures by Christopher Clavius on the parabola. He then spent two years in England around 1597. Then he studied at Antwerp with Michel Coignet, being there in 1599, following which he went to Paris in 1600 where he was greatly influenced by François Viète. Ghetaldi certainly impressed Viète who gave him many of his papers to study. Despite not having any publications at this time, Viète gave him one of his unpublished papers and asked him to revise it and edit it. Repeatedly encouraged by Viète, he began to prepare some of work for publication, particularly results he had obtained on the parabola.
Leaving Paris, Ghetaldi returned to Italy, spending some time in Padua where he came into contact with Galileo in 1600. This was an important opportunity for Ghetaldi who attended Galileo's lectures on mathematics, mechanics and astronomy. Galileo showed him his geometric and military compass, and Ghetaldi decided that after he returned to Dubrovnik he would make one for himself. He left Padua in 1601 and spent time in other centres of scientific research before arriving back in Rome in 1602. Ghetaldi's first paper Promotus Archimedes seu de variis corporum generibus gravitate et magnitudine comparatis Ⓣ appeared in Rome in 1603 and it was on the physics of Archimedes - it is, in fact, Ghetaldi's only physics paper. In this he gave an accurate table of specific weights of solids and liquids, in particular of gold, mercury, lead, silver, copper, iron, tin, honey, water, wine, wax and oil. In a second work Nonnullae propositiones de parabola Ⓣ, also published in Rome in 1603, he studied parabolas obtained as sections of a right circular cone. In the middle of 1603 Ghetaldi seems to have got into difficulties in Rome but the exact nature of the problem has never been ascertained. It was sufficiently serious to mean that he hurriedly departed, seemingly in fear of prosecution, and returned to his native Dubrovnik via Venice.
There are some puzzles in this biography which are not fully understood by historians. For example who funded Ghetaldi during these many years of travel and study? He had certainly been employed in the administration of Dubrovnik before he left, but it seems unlikely that this would have given him funds for six years of travel. One suggestion has been that he inherited considerable property from a wealthy nobleman living in London and this allowed him to finance his travels. Another puzzle relates to the role played by his friend Marin Gucetic who seems to have spent the whole six years with him. In fact Gucetic wrote about his years travelling with Ghetaldi:-
We travelled together through northern and southern Germany, we stayed in England for two years, no parts of France remained unknown to us, and we passed through the whole of Italy.Back in Dubrovnik he was elected as a judge of the Appellate Court but in 1604 he was sent by the authorities of the Dubrovnik Republic to supervise the building of the semicircular Pozvizd tower, the most important defensive tower in the fortification system of the town of Ston. This town, on the coast to the north of Dubrovnik, had an inner wall of about 900 metres and an outer wall of about 5 km designed to protect its salt production, a main source of revenue for the Dubrovnik Republic. Ghetaldi found this a very difficult assignment for the area was rife with malaria and there were frequent attacks by bands of Uskok soldiers, mainly involving acts of piracy, while he was helping with the defences. After four months he returned to Dubrovnik but, after the years he had spent with leading scientists throughout Europe, he found himself very isolated from the latest developments. He attempted to get round this isolation by carrying out a frequent correspondence with leading mathematicians and physicists such as Christopher Clavius, Christopher Grienberger, Galileo Galilei and Paul Guldin. In one letter Ghetaldi writes that he is:-
... in the corner of the world where you cannot see any mathematical Gazette.However, after a while back in Dubrovnik, he was sent on a government mission :-
In June 1606 the government of Ragusa charged Ghetaldi with a mission to the sultan of Constantinople. The task absorbed him considerably, and to it must be attributed the break in his scientific work that coincides with this period. The mission must have had its dangers, since rumours of his death began to circulate. So persistent were these rumours that even J E Montucla, in his 'Histoire des mathématiques', gave Ghetaldi's date of death as about 1609, "in the course of his mission to the [Sublime] Porte."When Ghetaldi had been in Paris he had learnt that Viète was working on constructing Apollonius's lost works. In fact Viète was often known as "Apollonius Gallus" because of this. Ghetaldi took over this work of Viète. He followed Pappus's description of the contents of certain lost books and to do this he had to solve the problems which the books were supposed to contain. He published Apollonius redivivus seu restituta Apollonii Pergaei inclinationum geometria Ⓣ and Supplementum Apollonii Galli seu exsuscitata Apollonii Pergaei tactionum geometriae pars reliqua Ⓣ both in Venice in 1607. Another reconstruction of works of Apollonius was Apollonius redivivus. Seu Restituta Apollonii Pergaei De inclinationibus geometriae liber secundus Ⓣ (1613). Also in 1607 Ghetaldi produced a pamphlet Variorum problematum collectio Ⓣ with 42 problems with solutions. These contain early application of algebra to geometry. It is worth noting that in a letter to Christopher Grienberger, written in September 1604, he says that the three papers which we mentioned above with a 1607 publication date:-
... are almost all finished, but I do not mean to send them to the printer for some time.In fact he waited for two years before having them printed. In the same letter Ghetaldi wrote:-
It seems that our century is happier than the past, because it has produced a mathematician who, without a doubt, can be compared with those of the ancients.While on the mission to Constantinople, Ghetaldi had tried to find an Arabic translation of the works of Apollonius. Despite offering to pay well for such a copy, he was unable to find one. After he returned to Dubrovnik following his mission, Ghetaldi became interested in developing scientific instruments. In particular he built instruments which he had learnt about from Galileo, particularly optical instruments such as a refracting telescope which survived and is today on view in the National Maritime Museum in London. He conducted his experiments with mirrors in Bete's cave, a location on the coast outside the city so named because Ghetaldi was known by the nickname of "Bete". We must not think of Ghetaldi's interest in mirrors and mathematics as being distinct for he certainly used the interplay between the physical construction of mirrors and the mathematical theory of the parabola as being two-way. In a letter to Christopher Clavius on 20 May 1608, he says that with his latest parabolic mirror:-
... the sun melts not only lead, but silver.Some of Ghetaldi's work is described in Pierre Hérigone's 1634 work Cursus mathematicus Ⓣ.
After his return from Constantinople, Ghetaldi held various government posts, working as an official in the Office for Wine, in the Office for Processing Wool, as a consul for civil litigation, and again as a judge of the Appellate Court. Whatever had been his crime when in Rome, he was pardoned and this allowed him to return to this city again around 1620. He lived in Rome for about a year and was elected to the Accademia dei Lincei in 1621. However, by this time he had left Rome and returned to Dubrovnik and, since the Academy was unable to contact him, he was never properly admitted to membership. It is worth noting that the Accademia dei Lincei had been founded in 1603 by Federico Cesi, the son of the Duke of Acquasparta, and Ghetaldi had dedicated his treatise on Apollonius to Paolo Emilio Cesi, a distant cousin of Federico.
Ghetaldi married Marijom Sorkocevic; they had three daughters Anicu, Franicu and Mariju but Marijom died shortly after the birth of their third daughter. When he felt that he was near the end of his life, he tried to secure the future of his daughters in a will he drew up appointing his brother as executor. Shortly before his death he was working on problems associated to the triangle and their application to measuring the diameter of the Earth. He wrote to Christopher Grienberger on 15 November 1625 saying that Grienberger was the last of his old friends to still be alive. In this letter he explained how his work on the triangle would lead to two methods of calculating the diameter of the Earth and he explained how he would conduct experiments with two Jesuits in the spring of 1626. However, he died before he could carry out these experiments.
It is reasonable to ask: what is the most impressive ideas contained in Ghetaldi's work? Without doubt, it is his application of algebraic methods to the solution of problems in geometry. We now think of Descartes as founding the application of algebra to geometry, and although Ghetaldi never quite managed to achieve this breakthrough (nowhere in his work are there algebraic equations for geometric objects) nevertheless he came very close. Perhaps it is just as well since somehow 'Ghetaldian geometry' does not quite have the same ring as 'Cartesian geometry'. He certainly used such algebraic geometry in Variorum problematum collectio Ⓣ but his main contributions in this area are contained in his book De resolutione et de compositione mathematica, libri quinque Ⓣ published in 1630, four years after his death. We shall never know how much this book influenced Descartes, but we do know that he read it.
It is interesting to look at the kind of person Ghetaldi was. He turned down a chair at Louvain when he was a young man. Descriptions of him say he had the morals of an angel and to be a Ragusan gentleman of discernment. Paolo Scarpi wrote:-
In mathematics he was like a demon, and in his heart - like an angel.
Article by: J J O'Connor and E F Robertson
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