Bohemian Rapsody

A sketchy story of mathematics at Charles-Ferdinand University

The university of Prague was created in 1348 after an imperial decree of Charles IV. It included four nationes: the Bavarians, the Saxonian, the Polish and the Bohemian, which turned out to be one with more political weight from the 15th century onwards. A second college, the Academia Ferdinandea, was created in 1556 and was run by the Jesuits. It is better known by the name Klementinum. In 1654, Charles University and the Jesuit Clementinum were unified in a single institution, then called “Charles-Ferdinand university”. From that time until well into 19th century the university of Prague was organized in a traditional way, around four disciplines: philosophy, theology, law and medicine. The teaching of the first two, philosophy and theology, was monopoly of the Jesuits at the Clementinum, while the superior faculties of law and medicine were lay faculties, controlled by the state.

Mathematics was taught at the faculty of philosophy, and since attending philosophy courses was compulsory for all students, the dissemination of mathematical sciences in Prague and Bohemia was monopolized by the Jesuit order for about two centuries, from the first half of the XVIIth to the disbanding of the order in 1773.

The place of mathematics in the Jesuit curriculum had been a hotly debated topic among Jesuit educators and reformers soon after the creation of the order. Eventually, Christophorus Clavius’ strenuous campaign to obtain a revered place for mathematics in the curriculum won: “owing almost entirely to his dogged and tireless leadership – A. Alexander recounts – only a few decades later, Jesuits were setting the standard for the study of mathematics in Europe.” And, within Europe, in Prague too.

But what did students learn when they studied maths under the jesuits at the Clementinum? One thing that should be remarked is that the organization of the teaching kept pretty much stable until the dissolution of the order, and included, for example, disciplines we would not treat as part of mathematics today: civil and military architecture, optics (catoptrics and dioptrics), hydrostatics.


The Klementinum  is a complex of buildings located at the heart of Prague’s old town, laying across  Charles Bridge and the astronomical clock. It was for century the nerve centre of the Jesuit province of Bohemia. Nowadays it hosts the collections of the National Library of the Czech Republic (Autor: VitVit – Vlastní dílo, CC BY-SA 4.0,


Before the Jesuits order was suppressed in the Habsburg empire in 1773-74, the system of university education had already undergone a gradual process of modernization and specialization through several reforms, starting from 1741. If we focus on the teaching of mathematical sciences, a significant change brought about by the reorganization of the curricula from the half of 18th century was an emphasis on mathematics and physics. One of the main consequence of these reformes was that new chairs were introduced, and the teaching of disciplines which did not belong to the traditional Jesuit cursus was promoted. For instance, alongside with the traditional chair of elementary mathematics, a chair of advanced mathematics (mathesis sublimior) was created in 1762. Concomitantly, the traditional Aristotelian framework which dominated the teaching of natural sciences was abandoned and substituted by an approach oriented towards experimental sciences.

Such reforms in the scientific education began, not without difficulties, under the leadership of the Jesuit Joseph Stepling (1716-1777), director of the philosophical faculty at Prague university from 1763 to 1776, and continued with his successor and most brilliant student, Jan Tessánek (1728-1788). Another important change was the introduction of the German as the official language for teaching in 1784, which opened the door to the circulation of German textbooks. Meanwhile, the modernization of technical higher education was prompted by the creation of a special chair of practical mathematics (1784), whose first professor was Franz A. L. Herget (1741-1800), followed by Joseph Havle (1763-1840) and A. Bittnar (1777-1844). Finally, the creation of a polytechnical school upon the model of the French Ecole polytechnique, in 1803, crowned the ongoing process of modernization in the 18th century Czech lands.


Ignac Platzer, monument to Stepling in the courtyard of Klementinum (Autor: marv1N – Vlastní dílo, CC BY-SA 3.0,

The table below summarizes the historical evolution in the teaching of mathematics between 1760 and the end of the 18th century. We note that the disbanding of the society of Jesus, although in some cases caused the removal of Jesuit professors from their chair, did not seem to have changed the structure of mathematical education, which continued to be part of the faculty of philosophy, except for an independent programme in advanced mathematics.


Chair: elementary mathematics.

Elementary mathematics course (3 years).

Included a teaching in applied mathematics (mathesis mixta) from the second year (introduced in 1775).

Chair: Advanced mathematics (from 1762).

Teaching of higher mathematics for those who excelled in elementary mathematics.

Chair: Practical mathematics (from 1784)

Before 1784, a course in practical mathematics was lectured during the 3rd year of the elementary mathematics course.

General content

Algebra, arithmetic, trigonometry, geometry. “Applied mathematics”: possibly with elements of differential and integral calculus.

Analytic geometry, differential and integral calculus, mechanics, hydrodynamics and astronomy.

Land surveying, and trigonometry, both theoretical and practical. Topics related to construction and engineering.


Joseph Bergmann (1723-1786) 1761-1767;

Franciscus Zeno (1734-1781) 1767-1772 (or 1774?);

Stanislav Vydra (1741-1804) till 1772/1774-1804;1

Ladislav Jandera (1776-1857) from 1804.

Joseph Bergmann 1762- 1766;

Franciscus Zeno 1766- 1772 (or 1774?);

Jan Tesanek from 1774 to 1786 (although sources indicate that Tessanek was professor of mathematics since 1763);

Gerstner from 1787 (appointed regular professor in 1789) till 1823.

Before 1784, presumably the teachers of elementary or advanced mathematics. From 1784:

Herget 1784-1799

Havle 1800-1802

Bittnar 1802-1804.

Texbooks used for lectures

Franciscus Zeno, Elementa algebrae, geometriae ac trigonometriae. Prague, typis Josephi Emmanueli Diesbach. Veneunt apud Antonium Elsenwanger, 1769.

Joseph Anton Nagel, Mathesis Wolfiana in usum juventutis scholasticae, Wien, Johann. Thomae de Trattner, 1776.

S. Vydra, Elementa calculi differentialis et integrali, Pragae et Viennae, Schönfeld, 1783 (possibly used for applied mathematics).

A. G. Kästner, Anfangsgrunde der Arithmetik, Geometrie, ebenen und sphaerischen Trigonometrie und Perspectiv, 1758 (Used in Prague after 1784).

J. Bergmann, Lectiones mathematicae in usum auditorum, Prague, typis Joannae Pruscha, 1765.

L. Euler, Introductio in analysin infinitorum, 1748 (used in German translations, [Michelsen, 1788]). Used in Prague after 1789.

W. J. Karsten, Lehrbegriff der gesamte Mathematik. Used after 1789.

Not known.

(Bolzano’s notes of Herget lectures are extant).

1 We do not have consistent information about the years 1772-1774, probably due to diverse ways in which the ban of the Jesuits from university teaching was implemented.

When Bernard Bolzano, a key figure in the history of mathematics, logic and philosophy, stepped into Prague university as a teenager, at the end of 18th century, the education system of the jesuits was still in place in its general lines, even though the order was then gone for more than 20 years.

The disestablishment of an organization that had ruled for centuries was a process that affected the practical and intellectual aspects of the lives  of the people involved. A paradigmatic change of sort, which has too often been neglected when studying the emergence of modern mathematics, such as in the case of Bolzano.


Reading hearts between the lines

My fascination with secret messages goes back to these lines of Ovid’s Ars Amatoria and his tricks and subterfuges to exchange secret vows:

Characters written in fresh milk are a well-known means of secret communication. Touch them with a little powdered charcoal and you will read them.

But I haven’t inked my pen in milk or discovered any romantic words among the lines of the manuscripts I examined this week. Although this is not the whole truth, because I have at least discovered a heart. And within a manuscript.

The manuscript   in question, LH 35, XIII, 2c (162-163) was written by Leibniz and contains some notes on differential calculus and transcendental curves.


As we can see, there are figures to be seen and calculations to be studied, and also a lot to be read between the lines, not only metaphorically.

Yet, is there a heart in this stream of mathematical thoughts? Invisible to the human eye, there is one. It lies hidden as a motif printed on the paper, a watermark. Watermarks are invisible in a normal scanned copy, like the one above, but a manuscript brought onto a light table (i.e. illuminated from below) can reveal these hidden treasures. For instance, in the paper reproduced above, it showed that at the centre of the left sheet (which is indeed a quarter of the original sheet) stands an heart-shaped pattern which contains a horse, with a crown above it. Likewise, at the centre of the right quarter we can find two letters, “A” and “B”, and a sort of “reversed” heart between them. Below I have enlarged a picture in which the motifs appear visible (and reversed as in a mirror image).


A heart containing a horse.


Letters A and B (reversed).

Here is a reproduction that can be checked against the originals:


Watermarks do not merely embellish a piece of paper, but contribute to tell its story. For instance, who made it and when? They also give precious information on when and where the story which a piece of paper carries with it was written.

Let us take Leibniz’s manuscript again. The same heart-shaped symbol can be found in two other pieces of Leibniz, now published in Band VI, 4, 99 and VI4, 270 (all available here). These are philosophical notes which, as the watermark tells us, happen to have been drafted probably on the same stack of papers Leibniz used for our mathematical notes. The editors of the philosophical works have established that the paper bringing such particular watermark was produced from 1679 to 1680 in lower Saxony, which gives a relatively precise dating for Leibniz’s manuscripts, including the mathematical one (LH 35, XIII, 2c, 162).

Of course, other elements will contribute to a better dating such as the content, the kind of symbols used and possibly the variations in the handwriting. But this is a promising start. And our heart has still carried its own secret message, even if without milk or powdered charcoal.



Activity report: a break from laziness

I am such a lazy writer that I have even forgotten the url of this site. But since someone has expressed, as a new year wish, to read me more constantly,  I thought I should also write more constantly. Looking forwaed to writing more lengthy posts, here is a brief list of the reviews I have written during the year 2017 for Zentralblatt fur Mathematik. It is a great honour for me to be one of their reviewers, firstly because it looks like I am doing something useful for the mathematical community, and secondly because it looks like that even mathematicians can trust me.

Anyway, these papers are all quite interesting (more than my reviews), and since I didn’t know them before they were sent to me, maybe you have missed them too for some reason. So here’s a good occasion to catch up:

Leibniz’s ontological proof of the existence of God and the problem of “impossible objects“.

The myth of Leibniz’s proof of the fundamental theorem of calculus.

From Euler to Navier-Stokes: a spatial analysis of conceptual changes in nineteenth-century fluid dynamics

And perhaps with time I could add something about them here. But not now. Oh, and this is another review, but for Historia Mathematica:

Book review of: G. W. Leibniz, Sämtliche Schriften und Briefe. Reihe 7. Mathematische Schriften. Band 6. 1673–1676. Arithmetische Kreisquadratur.

The Philosophy of Mathematical Practice, whither and whence

I want now to criticize, as briefly as I can, the philosophy of mathematical practice. I don’t want to criticize the numerous contributions which have been filed under this name recently, and are often interesting and serious studies, but rather the idea that this approach to the philosophy of mathematics should become a veritable “discipline” and the idea that the philosophy of mathematical practice is something new and young, as it is often claimed.

Philosophy of mathematical practice, whither now?

Around ten years ago a book appeared, edited by Paolo Mancosu, entitled “The philosophy of mathematical practice”. Soon after an Association was created, to stimulate:

research in philosophy of mathematics from the perspective of mathematical practice … [by bringing] together researchers that work on a variety of topics ranging from the way mathematics is done and evaluated to the study of its epistemology, its history and the educational strategies associated to it.

The passage just quoted is vaguely circular, but its message is clear: to understand what philosophy of mathematical practice is about one should look at epistemological or cognitive problems stemming from actual or past mathematics. This loose characterization is not obvious, even if it may seem so, especially if considered against what has been referred to as “mainstream” philosophy of mathematics. Paolo Mancosu contrasts the perspective of the mainstream philosophy of mathematics of analytical inspiration with the perspective brought aboyt by the philosophy of mathematical practice:

Contemporary philosophy of mathematics offers us an embarrassment of riches. Anyone even partially familiar  with it is certainly aware of the recent work on neo-logicism, nominalism, indispensability arguments, structuralism, and so on. Much of this work can be  seen as an attempt to address a set of epistemological and ontological problems  that were raised with great lucidity in two classic articles by Paul Benacerraf. Benacerraf’s articles have been rightly quite influential, but their influence has also had the unwelcome consequence of crowding other important topics off the table. In particular, the agenda set by Benacerraf’s writings for philosophy of mathematics was that of explaining how, if there are abstract objects, we could have access to them. And this, by and large, has been the problem that philosophers of mathematics have been pursuing for the last fifty years.

As Mancosu put it, although “Contemporary philosophy of mathematics”  (an overview of which can be found here) may have gained a considerable number of riches, it surely did so at the price of keeping references to history of mathematics and to mathematics itself to a minimum. This attitude resonates with a usual feature of analytic philosophy, which also disparages history in favour of fully-fledged rational arguments.

However, if such an  operation might have the benefits of curtailing the subject matter of philosophy of mathematics in a way that allows one to clearly formulate an agenda of issues and questions, easily identify a “field” of research in order to get published in international journals and being recognised as a legitimate member of a group, community or party, this comes with the price of impoverishing the spectrum of interests and of cutting the connections with the discipline in flesh and bone (another unofficial “manifesto” of the movement is emphatically called: “Towards a philosophy of real mathematics”). Freed from historical encrustations but also from many of its connections with matters mathematical from the past and the present, mainstream philosophy of mathematics, even if addresses questions of general interest about the nature of mathematical knowledge and its objects, tends to fossilize into a vast amount of technical discussions whose interest for the working mathematician, the educator or the student is often next to nil. Too often the impression of philosophy of mathematics that one gathers from the outside is that of a self-contained discipline, focussed on discussions which end up having a life of their own, loosely connected or wholly separated from the actual body of mathematics, its history and its context. Rueben Hersch pinned down the difficult relationship between mathematics and its philosophy in terms which, although written more than 30 years ago, seem to be valid still today:

It has to be said that if a mathematician, uncomfortable with his philosophical confusion, looks for help in the books and journals in his library, he will be badly disappointed. Some philosophers who write about mathematics seem unacquainted with any mathematics more advanced than arithmetic and elementary geometry. Others are specialists in logic or axiomatic set theory; their work seems as narrowly technical as that in any other mathematical specialty.

I think that the difficulty in finding any mathematical insight in contemporary, mainstream philosophy of mathematics, i.e. to find any enlightment that philosophical debates may give to the mathematician or the historian of mathematics, should be regarded as a legitimate and serious concern, instead of being looked down as a naive question, when such a question is actually raised (has philosophy ever clarified mathematics?).

We can only welcome the philosophy of mathematical practice if, as its promoters often proclaim, will bring grist to the philosopher’s mill by reconciling existing philosophical debates about mathematics with the content of actual mathematics or its history, or by finding new themes worth of philosophical reflection out of the body of mathematics itself. One could imagine that philosophy fruitfully interacts with mathematics in at least three ways:

i)   Mathematical practice may offer arguments, or raise problems in order to rediscuss classical issues in philosophy of mathematics (what are mathematical objects, how do we know them?).

ii) mathematical practice may provide new philosophical issues for the philosopher to reflect upon.

iii) philosophy of mathematics could be use in a normative sense, i.e. when it dictates its own agenda upon mathematical practice (what is good mathematics? whatnot?) or, more humbly, could be used as an instrument for cautious self-criticism, for instance by establishing which kind of philosophical stance can damage less the development of a certain area of mathematics.

I believe that all, or most of these points are already included in the agenda of the philosophy of mathematical practice. However, philosophers of mathematics (or mathematical practice?) are not happy with simple but clear descriptions, and show a tendency to discuss details to the extreme and to cut the hair into four, often only to achieve ridicolously sophisticated results. Thus Philip Kitcher has even formulated a very complicated attempt to define “mathematical practice”, and has come up with a mathematical structure – a quintuple, composed by Language, Metamathematical views, accepted Statements,  Reasoning methods and Questions. All this is very sophisticated, but at the same time I wonder how it can illuminate mathematics itself and the way it developed, or the way its objects are apprehended and known.

Discussions like those around Kitcher’s “quintuple model” replicate the vice of analytic philosophy of mathematics: they pose esoteric problems for the sake of clarifying matter for themselves and a restricted number of people in the business of philosophy of mathematical practice. In short, these discussions show the dangerous inclination to turning a broad reflection on mathematics and its practice into a self-referring discipline with its rank of experts and selected problems of little interest beyond the discipline itself.

Philosophy of mathematical practice, whence now?

Together with the danger of reproducing the problems of analytic philosophy of mathematics, i see another danger in the philmath approach. Many have pointed out at the novelty brought about by the agenda of the Philosophy of Mathematical Practice, but few have paused to examine it critically. I find this attitude a bit puzzling, especially if it comes from people who profess an attention for history. Without a proper justification, the repeated claims that philosophy of mathematical practice is “new” or that it brings “fresh air to the philosophy of mathematics” (as stated here) , or that it is a “young discipline” , as stated by J. P. Van Bendegem (See van Bendegem, J.P. (2014). The Impact of the Philosophy of Mathematical Practice to the Philosophy of Mathematics. In Soler, L., Zwart, S., Lynch, M., & Israel-Jost, V. (Eds.), Science after the Practice Turn in the Philosophy, History, and Social Studies of Science. New York- London: Routledge, 215-226), sound almost like propaganda to me.

So let us go back to our steps and search where this fresh air comes from. Very simply the contrast between the contemporary analytic philosophy of mathematics and the philosophy of mathematical practice as dressed up by Mancosu considerations, closely reminds of an essay by Lakatos, Proofs and Refutations, and particularly of the preface of the book. Lakatos sternly criticised there a “formalist” or “dogmatic” component of mathematical philosophy:

An abstraction of mathematics in which mathematical theories are replaced by formal systems, proofs by certain sequences of well-formed formulas, definitions by ‘abbreviators devices’, which are ‘theoretically dispensable’ but ‘typographically convenient’.

Formalist philosophy of mathematics “disconnects philosophy from history”, and leaves problems related to aspects like growth of mathematics or the “situational logic of problem-solving” aside. Much of what the critics of the analytic philosophy of maths objet to the latter. So it looks as if the underlying and polemical motive of philosophy of mathematical practice – namely that philosophy of mathematics should include themes and questions from mathematical research and neighbouring disciplines, mainly history of mathematics and logic – is not such a novelty and it was already perfectly realised in the past. To become convinced of that, let us take a look at the index of Lakatos’ problems in philosophy of mathematics, published in 1967. The book contains indeed a collection of papers from a conference held in Summer 1965. Surprisingly, perhaps, the problems of philosophy of mathematics discussed in the book are the following (I am merely copying and pasting the index): Euclidean dialectics and Greek Axiomatics, the metaphysics of the calculus; problems in set theory; Godel and philosophy of maths; informal rigor; a discussion on logic and mathematical education and a section on Frege. The collection also contains a series of contributions with the encompassing title: “Foundations of mathematics, whither now?”

Ironically, I have titled this section “the philosophy of mathematical practice, whence now?”, as it seems to me that enthousiasts and supporters of this approach to the philosophy of maths have forgotten their recent past too quickly.

What then the “philosophy of mathematics” was supposed to be in 1965-67? Certainly it was also a gamut of questions from history, logic and foundations of mathematics. The impression that I got is that it was possible, also back at the time, in bringing together discussions in logic, Frege, Greek philosophy of mathematics or Leibniz under the encompassing name of Philosophy of Mathematics. Perhaps this was not a dominant view, but certainly an existing one.

On the other hand, and this is the main point that I want to make here, I believe that many of the authors to Lakatos’ volume would have had no qualms in praising the agenda of the philosophy of mathematical practice, and recognizing it as their own agenda. We might perhaps extend the scope of our subject matter, and consider scientists-philosophers such as Poincaré, Einstein, Mach or Hermann Weil. Would have they found that the only reasonable way to do philosophy of mathematics was by referring philosophical problems to their actual practice as scientists? Probably this was their only conceivable way of doing philosophy of mathematics.

Hence, should we call all of  them philosophers of mathematical practice? Perhaps it is the case, and the philosophy of mathematical practice is not such a young discipline, after all. But then, why avoiding something simpler like “philosophy of science, or of mathematics” to denote philosophical reflection upon science or upon mathematics? Given that there seems to be no real qualitative difference between what some have been doing in the past, when doing philosophy of mathematics, and what they are doing today, when doing Philosophy of mathematical practice, why conjure up a new expression to denote an approach that is not that new, or even to claim that a new discipline is there, when it fact it has existed for a long time, even if no need was felt to turn it into a special discipline, properly speaking? After all what we have been calling “the philosophy of mathematical practice” in the recent years was simply “philosophy of mathematics” in 1967 or earlier, and why not continuing to do so?

I do not know how and when things precisely started to change, so that philosophy of mathematics had narrowed down its focus to the limited and esoteric array of problems that we know of today. Certainly the philosophy of mathematics needs not be confined within such esoteric problems, and was not so even in our recent past. To bring fresh air to the philosophy of mathematics, it may be perhaps more useful to admit that the philosophy of mathematical practice is, after all, nothing more and nothing less than philosophy of mathematics.




On infinitesimals and the alleged decline of Italian mathematics

In his book Infinitesimal, how a dangerous mathematical theory shaped the modern world, Amir Alexander argues that the Jesuits fought a veritable war to ban infinitely small quantities from mathematics and thus preserve their ideas of truth and rational universal order. Such order was exemplified at its best by the edifice of Euclid’s Elements which, as we know, do not admit of infinitesimal magnitudes.

Unfortunately, this thesis has several weaknesses (for detailed criticism, see Blåsjö and Grabiner). I will choose here only one line of criticism. According to Alexander,  by refusing the infinitely small the Jesuits should be also held accountable for the “extinction of the Italian mathematical tradition” at the end of the 1600s. This is the tradition of Galileo, Cavalieri, Torricelli and their pupils, with whom the mathematics of indivisibles and infinitesimals thrived and paved the way to calculus. As a result of a social and political struggle by which the Jesuits managed to remove the pupils of Galileo from all vital academic institutions, “… the leadership in mathematical innovation now shifted decisively, moving beyond the Alps, to Germany, France, England and Switzerland. It was in the Northern lands that Cavalieri and Torricelli’s ‘method of indivisibles’ would be developed first into ‘infinitesimal calculus’ and then into the broad mathematical field known as ‘analysis’ “. Eventually, Alexander notes: ” creativity and innovation … came to an end [in Italy] around the close of the seventeenth century”.

With the exception of those quoted above, reviewers have praised Alexander’s book (for instance, FixSwetzGerovitch), sometimes endorsing the story about the mathematical decline in Italy (see Sherry).

Against this opinion, I think that Alexander is largely exaggerating the negative influence of the Jesuits’ refusal of the infinitely small. I do not deny that the Jesuits exerted a tenacious and long-standing hegemony upon scientific and mathematical education in Italy, provoking a decline in mathematics by the end of the 17th century. However,  there is a vast evidence proving that such a control, as rigid as it might have been, did not cause a continual stagnation in the Italian peninsula from the end of the 17th century onwards, but stirred important reactions which are ignored by Alexander and that I shall summarise below.

Thus it is worth recalling that during the first half of the 18th century a systematic programme to spread scientific knowledge was implemented, especially in Northern Italy (having its centre in the Republic of Venice), in order to establish a “Republic of letters” after the ideal promoted by Ludovico Muratori. This project was carried out by several intellectuals such as Antonio Vallisnieri, Scipione Maffei, Apostolo Zeno, Bernardino Ramazzini and Michalengelo Fardella, and recruited mathematicians like Ramiro Rampinelli, Guido Grandi, Maria Gaetana Agnesi, Giulio Fagnano, the Riccati family, Giovanni Poleni and several others. If these names may sound unfamiliar to many, the illuminating (Roero 2013) and (Roero 2015) can give an idea of the extent of the debates in which these intellectuals were engaged. In particular two journals, published in Venice, stood in the forefront in this work of knowledge dissemination: the Giornale dei letterati d’Italia, active between 1710 and1740, and the Raccolta d’opuscoli scientifici e filologici, between 1728 and 1757. As regards to mathematics, the Giornale dei letterati was one of the main vectors of the diffusion of the Leibnizian calculus in Italy, the other being the presence and teaching of Jakob Hermann (1678-1733) and Nicolaus I Bernoulli (1687-1759).

As a result of such a cultural fervour, remarkable mathematical contributions appeared in the pens of Italian mathematicians. D. Tournès has shown that Jacopo and Vincenzo Riccati (himself a Jesuit, by the way!) developed an ingenious geometrical approach to the solution of differential equations, using tractional motion, while studies in the construction of transcendental equations were carried out by Giovanni Poleni and Giambattista Suardi. Correspondence networks which connected Poleni, Jakob and Daniel Bernoulli and Euler stand as a further evidence of the international dimension of the mathematics cultivated in Italy.

Certainly much more work remains to be done, especially in the light of the vast amount of manuscripts preserved in several libraries in Padua, Venice, Udine, Bologna, etc., in order to uncover networks of correspondents, controversies and new mathematical pieces. Amir Alexander, who perhaps ignores these manuscripts, certainly should not ignore the books and articles mentioned above, all available to the willing reader.

The Italian Republic of Letters was ephemeral, thus it is perfectly legitimate to ask how and why it eventually died. Could it be that, after all, the polemics about the nature and the legitimacy of infinitesimals had any impact in the decline of the Republic of Letters? Unlikely. As we read, for example, in Undías, by the half of the 18th Century differential and integral calculus were fully integrated within the Jesuit system of teaching over most Europe. Discussions on how to define and present infinitesimal quantities certainly are extant in Jesuit manuals, but the option of banning infinitesimal objects in order to save the order of the Euclidean cosmos was never contemplated: the differential and integral calculus had proved their power so much that any mathematician would have considered as a foolish thing to reject them on the basis of mere ideology. In the light of these considerations, Alexander’s statements that a) the end of the 17th Century marked the beginning of an irreversible decline in the Italian scientific environment, and b) infinitesimals were one of the major causes of this decline appear quite implausible.


A gentleman’s mathematical library

As my week of patient manuscript-digging at the Marciana Library is coming to an end, I would like to share a discovery that was certainly lucky and delightful, even if not of the utmost scientific important. It is a letter written by the Paduan professor of mathematics Giovanni Poleni (1683-1761) to a certain Lamberti, Librarian in Rio Terà (Venezia), about the “most certain method and most renown writers” through which one could learn mathematics. Poleni had little doubt on the path to follow: the safest way to begin the journey through mathematics is via Euclid’s Elements.

Poleni also gave other, more contemporary suggestions, like Wolff (as a vademecum for more specialized studies) and Newton (concerning physics). And, even if it is not so surprising that Euclid’s Elements constituted the cornerstone of mathematical education, it is worth stressing that books VI, VIII,  IX and X were not included among the necessary readings of a beginner. In other words, all Euclid’s arithmetic, including the difficult theory of irrational lines of Book X, was separated from the corpus of the Elements and either considered outdated or too  advanced.

I shall reproduce below the whole letter, with my rough English translation.

Lettera al signor Lamberti libraio di Ravenna, Venezia. (Biblioteca Marciana, IT. X 284 (6576), fol 76)

26 Settembre  1750, a Padova

Dalla gentil di lei lettera rilevo, essermi da un Cavaliere comandato d’esprimere in poche righe il più sicuro metodo, ed i più accredenziati Scrittori, da cui possa a fondo tutte apprendere le Matematiche. A mio debol parere molto saggiamente pensano quelli, i quali affermano, non potersi lo studio delle Matematiche principiar meglio, che dall’imparare a perfezione gli Elementi della Geometria d’Euclide: ma essere a sufficienza li sei primi, e l’undecimo, e il duodecimo. Egli è vero, che l’Analisi ha un gran merito; ma giova principiare dalla Sintesi; e ben conoscere quale sia il sodo e puro raziocinio Geometrico. Aperta essa regia porta, sicuramente s’entrerà per tutte le strade conducenti alle Scienze matematiche, che ormai si noverano molte.

E come di tutte giova averne gli Elementi, così io credo un’utile partito l’andare seguitando gli Elementi della Matematica, che ci ha dati Cristiano Wolfio. O almeno (dopo il primo Tomo, che tutto e necessario) andar sciegliendo quelli, nè quali più inclinazione si abbia. Avvertirò, che nella Geometria il Wolfio non ha seguitato l’ordine di Euclide : ma chi di questo sommo antico Autore averà ben comprese le Proporzioni, troverà indi facilissima la Wolffiana geometria. Se si vorrà poi studiare di più in alcuna delle parti della matematica, dopo lo studio fatto per quella Parte negli Elementi del Wolffio, riuscirà piano il progredire ulteriormente nella Scienza. Come sento, che il Cavaliere pensa di volersi anche applicare alla filosofia del Newton, di essa io credo la miglior edizione sia quella, che illustrata fù co’ perpetui Comentarj dei Dottissimi e celebratissimi Padri Le Sueur, et Jacquier. Eccole quanto ho creduto poter alla di Lei domanda servir di Risposta, senza entrar di più nelle cose dette tra quelli, che del metodo di ben studiare le Scienze matematiche hanno ragionato: e tanto più dovetti scrivere strettamente, quanto in somma ristrettezza di tempo mi ritrovo. Quel cavaliere si degnerà di gradire il poco, che potrei, Sono

Di Lei mio

Attmo Affmo

Giovanni Poleni


Letter to Lamberti, librarian.

Padua, 26 September 1750.

From your kind letter I note that I was requested by a Gentleman  to express, in few lines, what are the most certain methods and the most renown  writers,  from whom one can fully learn the whole of mathematics. In my humble opinion, they do very wisely think those who claim that there is no better way to begin the  study of mathematics than by perfectly learning the Elements of Euclid’s Geometry. But I think that the first six books are sufficient, together with the Eleventh and the Twelfth. It is also true that Analysis has great merit; but it is more useful to begin with the Synthesis, and know well what the correct and pure geometrical reasoning is. Once this royal door has been opened, one will certaily enter all the roads leading to Mathematical Sciences, of which nowadays we count many.

And since it is a good thing to possess the elements of all of them, I think it is a useful strategy to follow the Elements of Mathematics which Christiaan Wolff has offered to us. Or, at least, after the first Tome (all necessary) it is useful to continue by choosing those for which one feels more inclined to. I shall warn you that, regarding geometry, Wolff has not followed the same order as Euclid’s. However, those who have well understood, through this celebrated ancient author, the theory of proportions will thus find the Wolffian geometry very easy.

Then, if one wants to study more in some part of geometry, after the study of that part made on the textbook of Wolff, any other advance in that science will proceed smoothly. As I hear that our Gentleman wants to apply himself to the philosophy of Newton, I think that the best edition is the one which was embellished with the everlasting commentaries of the very Learned and Celebrated Fathers Le Sueur and Jacquier.

This is what I deemed useful  as an answer to your question, without entering any further into the things said by those, who have reasoned about the method of well-studying Mathematical Sciences. And the more I had to write concisely, the scarcest my time is at the moment. I hope that our Gentleman will be pleased with the little I could. I am,

of  You, my Lord, devoted and affectionate,

Giovanni  Poleni


Beauty in the margin: a discovery

Beauty sometimes lurks in the margin. And a marginal note I found some years ago keeps coming to my mind whenever I think of mathematical beauty, or perhaps of beauty in general.

The handwritten note lies hidden in a copy of Pappus’ Mathematical Collection, preserved in the Museum of history of Science of Florence (you can find the copy here).

More precisely, it is a copy of the famous translation prepared by Commandinus, which appeared posthumously in 1588 with the title: Pappi Alexandrini Mathematicae collectiones / à Federico Commandino … in Latinum conuersae et commentariis illustratae. The original Greek text dates back to the 4th century BC and stands out as a treasure of mathematical results, as well as our most precious source of information concerning mathematics in antiquity and late antiquity. The publication of the first Latin translation – a language which, unlike Greek, most mathematicians from the 16th and 17th century perfectly mastered – marked the impressive fortune of Pappus’ text for the whole subsequent century. Suffice it to say that it was studied (some of its eight books more than others, in particular books 3, 4 and 8) by Clavius, Descartes, Fermat, Pascal, Leibniz and Newton… and it deeply influenced the development of mathematics and mechanics during the early modern period.

The annotation was drafted by an unknown hand. At first sight it has little or nothing to do with mathematics.

Screen Shot 2017-03-11 at 20.05.45Marginal note in F. Commandinus, Pappi Alexandrini Mathematicae collectiones, 1588, carta di guardia 3r. The copy is preserved in the Library of Museo Galileo, Firenze.

It is written in Italian except for the concluding Latin motto. It says :

Venerdì 13 di Aprile 1647 alle hore 4 et 3 minuti in Roma la Luna apparente dimidiata si congiugneva con Giove coprendo col punto luminoso septentrionale l’istessa stella.

Dulcia non meruit qui non gustavit amara

A rough translation might go like this (suggestions for improvement are welcome!):

On Friday 13th April 1647, at 4 hours and 3 minutes in Rome, the Moon, which appeared as a half, was in conjunction with Jupiter, covering with its northern brilliant point that very star.

He does not deserve the sweet, who will not taste the sour.

The annotation records an astronomical phenomenon, probably an occultation, i.e. the passing of the Moon, in its first quarter (the “half moon”), in front of Jupiter. The spatio-temporal coordinates of this astronomical event are also precisely recorded: according to our observer, it happened on 13th April 1647, at 4.03, in Rome. I don’t think the hour should be read as “4.03am”, like we would do today, but rather as 4h and 3m after the sunset, according to custom of the period. Since the sunset occurred at about 18:37 back on 12th April 1647, then the recorded conjunction must have happened –  in universal time – on 12th April 1647, at about 22,41.

It is easy to check the correctness of my interpretation. We can rewind the sky above Rome back until the night of 12th April 1647 (for instance, using this software) and verify that the Moon and Jupiter (in yellow in the chart below) were actually very close in the sky. But there’s something more to it. If my reconstruction is correct, the chart also shows that Mars – the red circle close to Jupiter in the map – “covers” the Moon, indicating another possible conjunction, or occultation during the same night. Why our author did not mention it?

Screen Shot 2017-03-11 at 19.37.37

The sky on the night of 12th April 1647, around 22.41. We can see an unusual configuration Jupiter-Mars-Moon.

Anyway, the accurateness of our reader’s astronomical observations does not disturb the beauty and mystery of the annotation. In particular, I have been wondering about the sense of the concluding motto: Dulcia non meruit qui non gustavit amara. How does it relate to Pappus and to the extraordinary configuration of planets happening on that night so long ago?

I can think of one possible explanation. I imagine our unknown reader spending night after night sleepless, perhaps during the Winter and the Spring of 1647, pondering over Pappus’ beautiful and difficult geometrical Collection. Then one night, tired of brooding over the generation of the quadratrix or the spiral (for example!), he (or she?) must have raised the eyes to the serene night sky … where the Moon, Jupiter and Mars formed a splendid, unusual configuration. Who knows, perhaps such a marvellous view would have remained unnoticed if Pappus had not kept our studious reader awake. Who, aware of the extraordinary event, might have taken it as a metaphor or remainder of a promised beauty, a beauty one can only contemplate – secretly and unexpectedly – after hours and hours of difficult mathematical labour and sleepless nights:

He does not deserve the sweet, who will not taste the sour.