**Paul Guldin** was a Swiss mathematician who wrote on volumes and centres of gravity.

- Although of Jewish descent, his parents were Protestants and they brought Guldin up in that faith.
- At this point he changed his name from Habakkuk (a Jewish name coming from one of the twelve minor Prophets) to Paul since he saw Paul as the Jew who took Christianity to the Gentiles.
- Guldin became a convert to Catholicism at the age of 20 and joined the Jesuit Order in Munich as a Coadjutor Brother.
- Up to this point, Guldin would certainly not have studied mathematics, in fact it is doubtful if he had received much education beyond being able to read and write.
- The Jesuits, however, were an Order committed to rigorous education and Guldin went through the lengthy educational process leading to a doctorate in divinity.
- Since Guldin showed considerable mathematical abilities so, in 1609, he was sent to the Jesuit Collegio Romano in Rome to study under Clavius who was the professor of mathematics there.
- Although not known for mathematical discoveries, nevertheless Clavius was an exceptionally good teacher and Guldin gained deep mathematical understanding from his lectures.
- Clavius was, however, a classical mathematician teaching only Euclid's geometric methods and Guldin would also take this classical approach and oppose the newer ideas of the calculus which were beginning to appear around this time.
- After being instructed by Clavius, Guldin taught mathematics at the Jesuit College in Rome.
- In Refutatio elenchi calendarii Gregoriani a Setho Calvisio conscripti Ⓣ(Refutation of the Gregorian calendar lists written by Sethus Calvisius) (1618), Guldin defended his teacher Clavius's proposals for calendar reform.
- It is interesting to note the addition of algebra to the traditional disciplines of pure mathematics; of course, Guldin's conception of algebra depends in large part on the work of Viète.
- An interesting consequence was that Guldin argued that the Earth would constantly be moving.
- After teaching there for some time, Guldin returned to his professorship in Vienna where he remained until 1637 when he returned to Graz.
- One interesting correspondence which Guldin entered into was with Johannes Kepler.
- Unfortunately only Kepler's letters to Guldin have been preserved but, nevertheless, they give us interesting information.
- Kepler sought Guldin's advice both on scientific matters and on religious matters, and he also asks Guldin to use his influence in the court.
- This fact also illustrates that Guldin was a very influential person at the imperial court in Vienna.
- Kepler's last two letters to Guldin express his uneasiness concerning Guldin's expectation of Kepler's possible conversion to the Catholic Church.
- Kepler's financial position was poor throughout the period of their correspondence and Guldin was concerned that Kepler could not afford a telescope to carry out scientific work.
- One of Guldin's Jesuit friends, Nicolas Zucchi, was a telescope maker and Guldin asked him to give Kepler one of his telescopes.
- Guldin's most important work is Centrobaryca seu de centro gravitatis trium specierum quantitatis continuae Ⓣ(On the centres of gravity of the three shapes) published in 4 volumes between 1635 and 1641.
- Now Guldin has been particularly unfortunate in that he has been accused of plagiarism over this result.
- These were published in 1588, 1589 and 1602, a generation or so before Guldin published (1641) his 'Centrobaryca' ...
- That a man like Guldin should have failed to know this important statement in such a well-known work is quite inconceivable.
- If Guldin had been conscious of any debt to Pappus, he would surely have acknowledged it.
- In the light of all the evidence, it seems best to give Guldin the benefit of the doubt - to acquit him of conscious plagiarism, but to accept that he may have brought forth from his subconscious mind the fruit of his distant reading.
- Guldin uses Volume 4 to attack other mathematicians for the methods they are using.
- Although that is probably the main issue between Cavalieri and Guldin, a more careful reading of the debate will allow us to indicate the existence of other interesting issues ...
- The argument really centres around the fact that Guldin is a classical geometer following the methods of the ancient Greek mathematicians.
- Guldin attacks Cavalieri's indivisibles by arguing that when a surface is generated by rotating a line about the axis, the surface is not just a set of lines.
- If one asks whether Guldin or Cavalieri is right, then the answer must be Cavalieri.
- However, this does not make Guldin's work useless for in his support of the classical approach, he forced innovators like Cavalieri to think more deeply and to justify their methods more rigorously.
- In order to understand why Guldin attacks David Rivaltus in Volume 4 we must realise that, following the line taken by his teacher Clavius, Guldin does not believe that Archimedes' results are proved in a satisfactory manner.
- In Volume 4, published in 1641, Guldin attempts a reconstruction of most of what was then considered mathematics.
- Despite the argument between Guldin and Cavalieri, both agreed that proofs by contradiction are undesirable.
- Finally, let us look at some details of Guldin's library which still exists.
- The first fact to note is that Guldin was allowed to keep his own library, something which was very unusual for those within the Jesuit Order.
- Around 300 volumes have been identified as belonging to Guldin's library although many other books which have been subsequently rebound may have originally belonged to the library but have lost the evidence in the rebinding.
- Most of the books appear to have had no previous owners and so Guldin was buying new books.
- What subjects interested Guldin?
- Guldin was clearly very interested in mechanics because he had a whole series of books on this subject including ones on military equipment, fortifications, artillery and pyrotechnics.

Born 12 June 1577, St Gall (now Sankt Gallen), Switzerland. Died 3 November 1643, Graz, Austria.

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Astronomy, Origin Switzerland

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