**Paul Guldin**was named Habakkuk Guldin by his parents. Although of Jewish descent, his parents were Protestants and they brought Guldin up in that faith. He became a goldsmith, after serving an apprenticeship, and worked at that trade during his teens moving between different German cities. In the second half of the 1590s, he was working in Freising and there he read a number of books which led him to have doubts about the Protestant religion he was practicing. He went to the Benedictine abbey of Weihenstephan, in Freising, and explained his doubts to the prior in the abbey. It was a hard decision, but he took the advice of the prior and renounced the Protestant religion in which he had been brought up. 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. After a few years he became a Jesuit Scholastic and, still later, he was ordained a Jesuit priest.

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. Then, in 1617, he moved to the Jesuit College in Graz but after a few years a severe health problem forced him to give up lecturing. In fact his first work was published shortly after he arrived in Graz. In *Refutatio elenchi calendarii Gregoriani a Setho Calvisio conscripti* Ⓣ (1618), Guldin defended his teacher Clavius's proposals for calendar reform. We know Guldin's views on mathematics at this time since he later published a lecture which he had delivered in 1622 [2]:-

[Also in 1622 he published a work on the centre of gravity of the Earth. He accepted the view that the centre of gravity of every large body tries to move so that will coincide with the centre of gravity of the universe. An interesting consequence was that Guldin argued that the Earth would constantly be moving.Guldin]defines mathematics according to the Aristotelian classification, as that part of philosophy lying between physics and metaphysics. Mathematics is the science that considers quantity abstracted from sensible matter. Concerning pure mathematics, arithmetic is described as the science of discrete quantity, and geometry as the science of continuous quantity. 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.

He was sent to Vienna in 1623 where he was appointed professor of mathematics at the University. In 1629 he was sent by the Jesuit Order to teach at the Jesuit Gymnasium in the Silesian principality of Sagan which had been established by Albrecht Wallenstein after he was made Prince of Sagan in 1627. 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 wrote eleven letters to Guldin between 1618 and 1628 and these are discussed in [15]. 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. [15]:-

[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. Kepler replied to Guldin showing that he was extremely grateful for the gift and sent Guldin his book detailing the discoveries he had made with it:-An]example of Kepler's trust in Guldin's help and the appreciation of his advice is shown in Kepler's letter from August the301624^{th}. He had sent Guldin a petition to be forwarded to emperor Ferdinand II(1578-1637)to promote the publication of the Rudolphinian Tables. Kepler asked him now what to bring with him to Vienna to help the progress of his affair. 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.

Guldin's most important work isTo the very reverend Father Paul Guldin, priest of the Society of Jesus, venerable and learned man, beloved patron. There is hardly anyone at this time with whom I would rather discuss matters of astronomy than with you ... Even more of a pleasure to me, therefore, was the greeting from your reverence which was delivered to me by members of your Order who are here ... I think you should receive from me the first literary fruit of the joy that I have gained from using your gift(the telescope).

*Centrobaryca seu de centro gravitatis trium specierum quantitatis continuae*Ⓣ published in 4 volumes between 1635 and 1641. Volume 1 begins with the lecture describing the mathematical sciences which he gave in 1622 (discussed above). Since the whole work is on centres of gravity it is worth giving Guldin's definition from this volume:-

In particular, in Volume 1, he discusses the centre of gravity of the Earth. Volume 2 was published in 1640 and contains his, now famous, rule on centres of gravity. The main aim of this book was to study figures obtained by rotating other figures, for example the sphere obtained by rotating a semicircle about its diameter. Here is an example of Guldin's definition:-The centre of gravity of any finite quantity is that point placed either inside that quantity, or on its boundary, or outside it, around which on all sides are parts of equal moment. For either the centre itself, or the straight line, or a plane however drawn through the centre, will always cut the proposed figure in parts of equal weight.

His famous rule can be stated in the following form:-A rotation is a simple and perfectly circular motion, around a fixed centre, or an unmoved axis, which is called the 'axis of rotation', turning around either a point, or a line, or a plane surface, which, almost as leaving a trace behind it, describes or generates a circular quantity, either a line, or a surface, or a body.

Now Guldin has been particularly unfortunate in that he has been accused of plagiarism over this result. The historian David Smith states this in no uncertain terms [14]:-If a plane figure is rotated about an axis in its plane then the volume of the solid body formed is equal to the product of the area with the distance travelled by the centre of gravity.

Historians have spent some time arguing this point, often unfortunately using fallacious information. The fairest assessment is certainly that of Ivor Bulmer-Thomas [5]:-The essence of Guldin's Theorem appears in the works of Pappus. These were published in1588,1589and1602, a generation or so before Guldin published(1641)his 'Centrobaryca' ... They were well known to scholars of that time as constituting the most important geometric work of the late Greek period. That a man like Guldin should have failed to know this important statement in such a well-known work is quite inconceivable.

Volume 3 contains work on the surface and volumes of cones, cylinders and solids of revolution. Guldin uses Volume 4 to attack other mathematicians for the methods they are using. One mathematician to be attacked in this way is Cavalieri [2]:-It is not as though Guldin was a spiteful man. The evidence of 'De centro gravitatis' is that he was zealous to give credit where credit was due. ... there are friendly references not only to Commandinus himself but to Clavius, Regiomontanus, Orontius, Freigius, Peter Ramus, and Cardan. If Guldin had been conscious of any debt to Pappus, he would surely have acknowledged it. This does not rule out a debt of which he was unconscious. It may very well be that Pappus's enunciation, read many years earlier, had sunk into his unconscious mind, and that when he came to write in1640, he genuinely believed that he was producing something original. ... 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.

The argument really centres around the fact that Guldin is a classical geometer following the methods of the ancient Greek mathematicians. His first point, however, is to accuse Cavalieri of plagiarising Kepler'sThe debate between Cavalieri and Guldin is usually mentioned in connection with the objections made by Guldin to Cavalieri's use of indivisibles. 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 ...

*Stereometria Doliorum*Ⓣ (1615) and Sover's

*Curvi ac Recti Proportio*Ⓣ (1630). There is something in his argument relating to Kepler since in that work Kepler does regard a circle as an infinite polygon composed of infinitesimals. However, Cavalieri's indivisibles are different from Kepler's infinitesimals. As to the reference to Sover, Cavalieri, in his defence, points out that he wrote his book before Sover's book was published. 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. He writes (see [2] or [13]):-

As Mancosu writes [2]:-In my opinion no geometer will grant Cavalieri that the surface is, and could, in geometrical language be called "all the lines of such a figure"; never in fact can several lines, or all the lines, be called surfaces; for, the multitude of lines, however great that might be, cannot compose even the smallest surface.

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.Guldin was a "classicist" geometer, steeped in the idea of explicit construction, sceptical of considerations of infinity in the domain of geometry, and wary of the risk of ending up with an atomistic theory of the continuum.

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. Archimedes' proofs had been strongly attacked by the humanist Joseph Scaliger in his treatise *Cyclometrica* Ⓣ (1594). David Rivaltus replied to these criticisms in 1615 with an equally strong defence of Archimedes. Guldin, in turn, attacks Rivaltus writing [13]:-

In fact in Volume 4, published in 1641, Guldin attempts a reconstruction of most of what was then considered mathematics. His aim is to prove the results without using the method of contradiction. For example, he writes (see [2] or [13]):-Indeed ostensive demonstrations have always had the applause and the victory over negatives and those reducing to absurdity or impossibility, whatever Rivaltus ... says.

One point here is worth noting. Despite the argument between Guldin and Cavalieri, both agreed that proofs by contradiction are undesirable. Perhaps surprisingly, Cavalieri justified his approach by arguing that his use of infinitesimals allowed him to prove Archimedes' results without the use of 'proof by contradiction'.We will therefore prove ostensively, in this fourth book, through our principles from rotation and originating from the centre of gravity, the main propositions proved by Archimedes on the 'Sphere and the Cylinder' and similarly on 'Conoids and Spheroids', which he himself had established by contradiction.

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. Since three books are marked as belonging to the library, yet were purchased after his death, one can deduce that he had a librarian to look after his library. What subjects interested Guldin? There are of course the works of Galileo, books on arithmetic such as that by Oronce Finé, the Alphonsine Tables, and works by Frederico Commandino. 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. Architecture also clearly interested him since he also had several works on that topic.

**Article by:** *J J O'Connor* and *E F Robertson*