IL COMPASSO GEOMETRICO E MILITARE
The invention's originality, scientific value and economic utilisation

Roberto Vergara Caffarelli

(Translated by Katiuscia Mariottini)

1. The invention's originality

2. Scientific Value of the Compass

3. Economic utilisation

The introduction to Galileo's brief treatises¹, including the first two printed by him, gives the opportunity for some considerations, which can be grouped from a logical point of view into three parts: an examination of the originality of the invention of the compass is the first step; only after that can the invention's scientific value be discussed. Finally, it is interesting to mark another aspect which, due to its complexity, can be summed up by referring generically to its economic utilisation.

The compass was extremely important for Galileo, so we should not be surprised by his sharp and cutting words of protest against Baldassarre Capra's plagiarism in Usus et fabrica circini, which led to the suppression of almost every copy of that book² and to the writing of the Difesa.

It is strange that, after a positive verdict concluding an official inquiry, doubts still exist on Galileo's paternity of the geometrical compass. In fact, the verdict refers to the book and only indirectly to the instruments. Moreover, as often happens when studying the history of an invention, the compass too has multiple paternities, because of prior writings and ancient compasses showing the same scale shown by the compass of Galileo as well.

The application of the pendulum to the clock, the thermoscope, and the microscope: almost every Galilean invention has been the object of controversy.

A clear example is what happened after the discoveries he made with the telescope: it was not by chance that he was compared with another debated discoverer, Christopher Colombo³.

A remark on the subject: were it proved that Galileo built the compass after having knowledge of its existence and use, his claims and defences should not be judged insincere for this, because they would be coherent with the meaning he gave to the word invention.

Do you remember what Galileo says about the invention of the telescope? He was told that a Flemish man had made an "occhiale" (eye-glass) by which far away objects could be seen as if they were very near. He was driven to look for the reasons and to find out how to reach such an instrument's invention: "quod tandem in causa fuit, ut ad rationes inquirendas, necnon media escogitanda, per quae ad consimilis Organi ionventionem devenirem me totum convertem"⁴. The telescope as well was called an invention.

After sixteen years, it is Galileo himself who tells of the circumstances of the plagiarism: "Yet I don't want to keep silent anymore and I will tell you about the second theft⁵, which was intentionally committed with too much boldness by the same man who did the first one many years ago, when he claimed the invention of the compass, even if I had already shown it to a great number of Gentlemen, and finally made it of public knowledge by its publication. And please for this time forgive me for my too sharp grudge, which is contrary to my nature, habit and intention, and for my exclaiming what I have been silent about for many years. I am talking of Simon Mario Guntzehusano, who in Padua, where I was at that time, already translated the commentary about the use of my compass into Latin, and claimed its paternity, asking one of his pupils to publish it under his name. After that, perhaps in order to flee any punishment, he immediately went to his country. However, he left his pupil in trouble, because, since Simon Mario was not there, I was compelled to act against him by writing and publishing Difesa.

Since Usus et fabrica circini's dedication to the Margravio of Ansbach-Brandeburg dates from March 7th 1607, whereas the dedication of Compasso to Cosimo dates from July 10th 1606, one knows that Baldassarre Capra had no more than eight months of time to carry out the plagiarism. However, the existence of a previous draft of the translation cannot be excluded.

Galileo had been dealing with the subject of "priority" already in the introduction to Compasso⁶, where he claims that the reason for a 60-copy private issue was his need to make himself sure of his attribution of the invention of the compass by the testimony of publication. In order to achieve this target, he published at its own expense a few copies, making a present, together with the instrument, to the young prince of Tuscany (who the treatise is dedicated to) and to other lords⁷.

In his memorial of April 9th 1607, together with a petition against Capra addressed to the "Riformatori dello Studio di Padova", which was a censure office for the press as well, Galileo confirms what he had already written in Compasso: since he felt that someone else would attribute to himself all the honour for the invention if he did not take any steps, he decided to publish a few copies in Padua, in order to get in the way of anybody who would assume all the credit of the fruits of his labour⁸.

It is not possible to know exactly who this "someone" was; it may be remembered that Eutel Zugmesser is mentioned in Difesa as a German appearing in Padua around 1603 carrying an instrument very similar to his. It was "provided with a few inscribed lines copied from mine, while other lines were omitted and replaced with other ones"⁹.

Referring perhaps to Capra and Simon Mayr, he says that on this occasion his emulators and his old enemy too hinted that he could have copied from the foreigner. As a result, Galileo had to cope with a confrontation at Giacomo Alvise Cornaro's house and, in front of many gentlemen, he finally managed to make the people present understand that it was Zugmesser who had copied from him.

Zugmesser became the mathematician of the archbishop Ernest ­the Elector of Koln- and several times in 1610 he met Martin Hisdale, who found him rather hostile to Galileo, because he considered himself offended by the text against Capra¹⁰.

Hasdale told Galileo that " the Mathematician of Koln" had said, " in Sir Cornaro's presence, Your Lordship confessed that his instrument was better than your own" and also that "in Your Lordship's instrument there was a lack which was not shown by his one"¹¹.

These assertions remained however totally discredited because, as one of the Padua witnesses, Pompeo de Conti da Pannichi¹² was in Koln and Hasdale wanted to set up a confrontation in the Elector's presence, Zugmesser "went always fleeing any danger".

Indeed, it is appropriate to make a deep analysis of this first episode of objection to the invention. It is necessary to mark that, apart from some verbal complaints, Zugmesser did not react to Galileo's claims, which however were circulating, and he did not even try to indicate any other origin of the instrument.

Galileo wrote about him: "five years ago he was in Padua, showing an instrument which was largely copied from mine; and, while he was leaving, he left illustrious Mr Michele Victor of Vustrou in Branswich () a few writings concerning the instrument's construction and a few of its uses, which, after him, came into M. Gasparo Pignani's possession, who was a very clever builder of any kind of mathematical instrument and an expert in mathematics and, as he made copies of them to somebody else, they went into Capra's possession() Moreover, I want to say that these writings are lacking not only in many operations () but also the whole description and the uses of the lines, called "Aggiunte" () and the poligraphic lines, as well as the bomber's sight and the division to measure inclines and the Quadrant division to measure by sight. Moreover, as my name is mentioned several times in these very brief treatises, it is clear that not only did the Flemish man know my writings, even though they had not been printed yet, but also he did not try to hide the truth, showing a better and more polite nature than Capra's"¹³.

It is fairly clear that the Flemish man (as Zugmesser was always called by Galileo) had just modified or, in some cases, tried to complete Galileo's instrument¹⁴.

A remark to conclude the "Zugmesser" subject: Galileo could object Capra chiefly the plagiarism of his publication, because of his adversary's vagueness about who invented the instrument. In the Introduction, for instance, Capra marks his decision to divulge both its use and making for the scholars' public utility, whereas at the same time other people are just debating about the invention and making copies for a high price. It is true that many ambiguous expressions, as they were found by Galileo, could persuade the reader about Capra's attribution of the invention of the instrument, but during a debate in front of the Riformatori, the accused does not dare in fact to claim even having ever made one.

The instrument was instead existent in the Zugmesser case, so Galileo was compelled to contest the invention, and he did so unhesitatingly.

Paolo Sarpi, Gianfrancesco Sagredo, Giacomo Alvise Cornaro and Giacomo Badouere believed and declared that Galileo was the inventor of the compass and surely also those that Galileo expected to come to Venice believed it: "I'm not lacking testimonies, whose declaration confirms that I can rightly demonstrate to Your Lordships that this is one of my old inventions, not stolen from anybody, and not Capra's new discovery¹⁵.

Cornaro's letter to Galileo refers to the testimonies: "Since I doubt that anybody from this Studio is coming there, I should suggest setting up another congress here, in Padua in the presence of the Signori Rettori of the Town. I spoke to Pilan, who told me that he bought Capra's book, and having carefully seen it, he found that it was copied from Your Lordship, Magini and that German, or Flemish man, and that there is nothing of himself. Therefore, one never says enough about that proud young man's boldness"¹⁶.

There is no need to quote all the fervent affirmations of paternity which can be easily found in Difesa, but the possibility of previous authors is shown in an important passage: "He (Capra) is claiming at the same time what he printed as his Maestro's work as well as Tycho Brahe's invention and he endlessly claims around Padua that I copied this invention from a book, printed in Germany and in the German language, which he is going to receive and show to everybody. () Therefore, this book will arrive and, as far as I know, Gromo will carry it. But it is necessary that Capra should take better care of this second one he will receive () in order to show it to anybody who does not believe just in his simple words"¹⁷.

Despite the irony of Galileo, who does not believe in the book's existence, in this case Capra was right, because a book had been printed by Levin Hulsio in 1604, another one by Philip Horcher in the following year, and a reduction compass attributed to Joost Burgi is described in both of them. However, it is a different thing from the compass of Galileo.

Now, having mentioned a first one, it is necessary to mention a series of other possible forerunners or inventors: Nicolo' Tartaglia and his bombers' sight, Fabrizio Mordente and Commandino and their compasses, Guidobaldo del Monte, Michel Coignet¹⁸, Thomas Hood and others.

Giovan Battista Venturi¹⁹ alleges that "a few pamphlets in German concerning geometrical instruments were published in Frankfurt by Hulsio: the third one, published in 1607, but already communicated and mentioned since 1603, includes a treatise concerning Giusto Birgio, the King's machinist's proportional compass. This is Commandino's compass a Centro Mobile colle Facce Piatte; one of its sides shows: 1- the right line's division into equal parts; 2- the circular line's division.

The other side shows:

1-proportiones homologorum augendo vel diminuendo, that is Galileo's geometrical lines;

2- -proportiones homologorum corporum augendo vel diminuendo, that is the stereometric ones;

3- the point corresponding to the centre, the other point is the periphery

4- the points use to transform each one of the six regular bodies into each other, called G, P, C, O, D, I, that is Globus, Piramis Cubus, Octaedrum, Dodecaedrum, Icosaedrum. One can see that Birgio had not copied from Galileo, but, drawing upon Commandino's compass, he had made several useful applications of his own and some of them were very similar to Galileo's. Even the principles of the compasses are not very different from each other: Picture V, Figure 1, EFGH shows Commandino's and Birgio's compass, whereas ABC is the compass of Galileo. At this point, on one side EK: KF = EG: HF and on the other side AM: AB=MN: BC. Accordingly, it can be said that the first compass's geometrical base is the same as the second compass; but the principle of the compass of Galileo appears to be more natural and simpler.

And yet, it seems that the proportional compass showing only two couples of arithmetic lines and the sines inscribed by Guidobaldo was actually used, since a similar instrument, as simple as this one, is mentioned in Speckle's military architecture; Clavio declares he saw a few similar instruments in Rome in 1604, and Henrion says that in 1614 he was shown one.

Nevertheless, except perhaps for the first two couples of lines, that is the arithmetic and geometrical lines, five other couples had been applied to the compass di centro fisso by Galileo on his own. The quadrant he added was commonly used as early as in the sixteenth century, being just a derivation from those which had been in use for a long time. Geometricians agreed in acknowledging the compass of Galileo to be subject to fewer aberrations and to have a more immediate and extensive use than Birgio's".

However, the study of relations of historical or logical dependence among different mathematical instruments or measurers similar to the compass suggested or made during the second half of the sixteenth century was not the subject of this paper. This paper is instead concerned with the issue of whether Galileo had full knowledge of a prior instrument that inspired him or his invention is totally original.

For instance, a total autonomy is not believed by Favaro: "It is not so unlikely that Galileo had some knowledge of the instrument. Surely he saw similar ones which led him to construct his own, since he was such an intimate friend of the marquis Guidobaldo Del Monte²⁰, and he in fact visited him at Pesaro as well"²¹.

It is therefore necessary to debate about the probability that Galileo's instrument was suggested by Guidobaldo.

A sufficient number of documents seem to exclude Guidobaldo from being the first source of the compass of Galileo or even of any of his rough versions. Guidobaldo had knowledge of Galileo's construction of the compass, because a copy was in the possession of Orazio, his son²². It is said that, as a real lord, Guidobaldo was not worried about claiming his priority. But what about Galileo: didn't he have to feel embarrassed towards someone had helped him a lot?

Instead, there had never been any grudge between them, and Galileo's relationship with the marquis's family was very close even after Guidobaldo's death, Alessandro, his son, writes on the 8th January 1607: "Since you're always been a good friend of our father Guid'Ubaldo, I cannot help letting you knowthat the other day, the day before Epiphany, at a quarter past eight, he passed awaySo, having lost somebody who loved Your Lordship so much, you will take part in our sorrow. We, me and my brothers, ask you to receive us as your servants with the same love as our father, as Your Lordship deserves"²³.

Finally, on June 15th 1610, Orazio Del Monte writes: "We are impatient to see anything about your geometrical instrument, as in your writings your Excellent Lordship promised to show something more in future"²⁴. One can see therefore that Orazio was in possess of both the extremely rare Compasso and Difesa.

In this letter Galileo is asked to look for a clever printer in Padua, because Orazio dal Monte would like to publish a few of his father's works, including a paper concerning "the construction of an instrument-provided with inscriptions-" he found.

It is difficult to believe that Orazio did not know whether his father had built a similar instrument many years before, since he himself appreciated the compass of Galileo, having a copy. It is even more difficult to imagine that, knowing that the first idea of the compass was his father's, and without receiving any merit for it, Orazio could entrust someone else who had used such a successful idea for himself.

A lot of confusion about the invention's priority was going on even during Favaro's time, and, even if he knew all this, he declared that "It has been wrongly said that Galileo is the inventor of the proportional compass"²⁵, as Galilei himself had recognised that "other similar instruments had been made and were circulating among scholars before his one".

A sentence from the introduction to Compasso (Ai discreti lettori) is mentioned by Favaro as proof for his declarations: "I will never say what has been reached by my work, leaving instead the judgement to those who understood it from me or will understand it in future, and most of all those who will see similar instruments discovered by other scholars, since the majority of the greatest discoveries included in my instrument have never been either experimented or imagined yet".

Is it right to give the se words the meaning Favaro intended? If this declaration is interpreted as an acknowledgement of the existence of other compasses, it will become incongruous with the solemn and formal one which can be read from Galileo's memorial to the Riformatori: "Therefore, I am the sole true legitimate inventor of both the instrument and any text about its operations already publicised, and nobody else has any part in them".

In fact, until 1607 nobody in Padua had any knowledge about any instrument similar to the compass, except for an instrument carried by Zugmesser. He was however defeated in such a miserable way that during his debates with Galileo, that Capra could not deny the attribution of the instrument to his adversary.

It is not possible to assert Favaro's position without belying Galileo, who declares in the presence of the highest academic authorities: "For ten years already, I have been studying very hard and I have reached some perfection in one of my mathematical instruments, thinking it up, inventing and improving it by myself. Its uses are important in number and quality in any domain of mathematics: the contemplative uses as well as the civil, military and mechanical ones. I esteemed it could be very useful to many people if they were given the instruments together with a clear and complete commentary about their uses"²⁶.

If possible, also all those who supported Galileo against Capra must not be belied.

What instruments, then was Galileo referring to? Even delimiting our research just to the catalogues of the Museum of Science History of Florence²⁷ a few instruments vaguely similar to the compass are found: their similarity consisting in their shape (hinged rules showing inscribed scales) or in their function: Baldassarre Lanci's instruments (for instance, a diastimeter, including two graduate compasses, an archimetro for triangulation and a 1557 compass); Antonio Bianchini's 1564 diastimeter compass; Haumphrey Cole's 1575 topographic ruler, whose scales can calculate surfaces and volumes and whose joint can be used as a clinometer, by blocking it and forming a right angle.

The total originality of Galileo's invention can be shown by analysing a few other considerations.

First of all the instrument's gradual evolution, based on objective documents: for instance, in 1601 Mercuriale writes to Galileo: "I must ask you again to finish your geometrical and military instrument at any cost"²⁸. The gradual improvement of the compass is asserted by Giovanfrancesco Sagredo and Jacques Badouère as well.

Further evidence for the importance of the compass's military use, applied by Galileo for his pupils' private education, is found at the end of Compasso: "Yet, as is said at the beginning, my actual intention is to speak only to military people and of just a few things other than those belonging to these professors, waiting for another occasionto publish information about the construction of the instrument together with a description of its uses".

His treatise Breve Istruzione all'Architettura Militare (May 25, 1593) is considered by Favaro to be a summary of Galileo's first-year lessons. As in other similar treatises, practical elements are given at the beginning; they consist mostly of how to divide lines, reproduce angles, construct regular polygons. "The way fortification maps must be described, by the measures and the proportions most suitable to the fortification desires" is part of the teaching program. "It is necessary to reduce those arms, or feet or perches our real fortification is measured by, into very small measures that can be understood from a little sheet of paper in front of us"²⁹. At this point, engineers have to cope with a problem consisting of having to increase or decrease measures when using different countries' units and with a problem of declaring how the scale is created and used. All these operations become easier by using the compass.

Not only did Galileo's treatises and lectures on fortifications include commentaries about the use of suitable instruments. Among those who described instruments found by themselves, the author of one of the more beautiful and instructive texts of the genre in his time deserves to be recalled³⁰: Le Fortificazioni di Buonaiuto Lorini, nobile Fiorentino, published at Venice in 1597. Reprinting it in 1609, Lorini added a sixth book "Showing how to measure distances and survey maps" and carefully describing³¹ the construction and use of a new instrument for measuring distances, called Mezzo Balestrino.

A quadrant is shown on his instrument: its functions are the same as those of the bombers' sight, since it can be used by "bombers to focus the artillery at the elevation required to hit the target". Its inscriptions are instead useful to "know the weight of the artillery's iron balls, and the depth of the valleys to draw any country's border or survey fortress maps. The original one by Lorini is reproduce in Figure 1, showing a more detailed description of the instrument. All these operations can be performed by the compass, and yet it is completely different. This is an example of what Galileo perhaps was referring to by "similar instruments."

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The decimal system is quite a recent system: in Galileo's time one braccio fiorentino was divided into twenty soldi and one soldo was divided into twelve piccioli or denari; one braccio veneto was divided into twelve ounces and one ounce into twelve punti. Non-decimal systems were used for weight, square, cubic measures and currencies as well; therefore, most times it was simpler to express the non-integer part by means of fractions.

In this way, however, even simple arithmetical operations were not too easy; and difficulties increased when extracting square or cubic roots.

For instance, Galileo's figures shown in Volume 72 of Manoscritti Galileiani, kept by the Biblioteca Nazionale of Florence³², include very simple operations whose results are wrong³³. Sometimes an error occurs when fractions are handled: for instance, sheet 176r shows 265 _ x 254 3/5 = 67570. It is interesting to try to see Galileo's procedure. It is possible that he first multiplied 265 by 264; then, without adding the partial products, he multiplied _ by 254 and wrote the result 127 writing units and tens in the respective columns' empty space and adding a new line, since there was no space for writing 1 in the hundreds column. Galileo should now multiply 265 by 3/5, but his attention was diverted (his errors are always of carelessness) and he multiplied 265 by _. Then he added the round result 133 by writing units, tens and hundreds into the respective columns' empty spaces: as a result, this calculation presents the same shape shown by as all the multiplications on the sheets of Manoscritti Galileiani, vol. 72.

Of course, the sheet shows 254 3/5 and not 254 _. Except for errors of carelessness, it is interesting to remark that, as well as in other cases, the exact result-which needed the multiplication of two fractions- is not considered important by Galileo.

Can the compass perform multiplication? The answer is yes: the multiplication of two numbers is included in Galileo's regola del tre. Why therefore does Galileo not use the compass?

In order to answer, 254 can be multiplied to 265 by an imaginary compass, trying to understand his procedure. These could be the instructions: open any ordinary compass in order that its legs are as distant from each other as 100 points from the arithmetical line. Open then the geometrical compass³⁴ in order that the ordinary compass's legs obliquely correspond to the points 254 of the two arithmetical lines; then close the ordinary compass's legs in order that their distance is just 26.5 points and use it to find the points of the oblique distance of 26.5 on the two arithmetical lines. As for performing divisions, the procedure is the same.

It is immediately clear that one cannot know more than three digits of the result: this can explain why the compass is not a very exact multiplying calculator (unless big figures are used) and why Galileo does not claim its use in this direction.

The compass is very useful indeed when extracting square or cubic roots, since in this case the figures are not simple for anybody not having a good mathematical education. These lines are exploited by Galileo for many operations.

Apart from particular uses, such as the rule to change currencies or calculate compound interest, the majority of these applications are geometrical ones, including different ways of measuring by sight, which are interesting from a practical point of view. The instrument is therefore useful when learning arithmetic or geometry.

In his work on the compass, the Jesuit Paolo Casati³⁵ explains its possible uses: "it is a work useful to geometricians, landsurveyors, civil and military architects, painters, sculptors, and anybody using pictures, but also to bombers, sergeants at war, merchants and so on, because of the simplicity of the arithmetical operations". In his Prefazione, after having mentioned the difficulty in finding texts concerning the use of the compass, Casati affirms he "had never had any chance of seeing any author except for Galilei: twenty-two years ago, in the Collegio Romano's bookshop I found a text concerning this subject, which I could not understand then.".

A century later, Francesco Maria Gaudio writes³⁶: "Ab eadem doctrina pendet constructio Sectori Geometici, qui Circinum proportionum, aut Geometricus, vel etiam Compassus appellari solet". A little below, Gaudio's assertion reveals the didactic value still attributed to the Compass of Galileo after a century and a half after its invention: "this instrument's explanation and use embraces almost the whole of geometry and its commentary will consist without any doubt in a very big part of our Geometria Pratica. It is enough to allude to the principle it derives from, which had to be in its inventor's mind when he imagined it".

As for the compass's scientific value, one cannot deny that the most interesting lines are the metallic ones. In order to know the density values inscribed by Galileo along them, one could need a compass surely attributed to Galileo, otherwise the distances on the figure printed in Padua in 1640 should be measured assuming that it is a careful commentary. The only explicit thing in this text is the ratio between the volumes of the two spheres of the same weight, one of gold and the other one of silver. The indicated ratio 0.6 could be a round figure, because in the density table associated to the Bilancetta the ratio is instead 0.54, the same as in modern measures.

Mattia Bernaggeri's and Capra's treatises are very interesting for what they say about the metallic rules, widening the debate considerably, especially the former. Galilei as well wondered at Capra's indication of the density value of argento vivo, as mercury was called.

None of them, however, noticed an important error Galileo had made.

This is publicised by Pierre Petit, in a very interesting little book printed³⁷ in 1634 ­even if a part of it had already been written in 1625- in the proposition VII beginning on page 113, under the title: "Erreurs de Galilée, de Berneggerus et de Henrion"³⁸. It is worth mentioning a long passage for its interest: "In his translation from an Italian treatise about the use of the proportional compass of Galileo, on pages 32 and 33 Mathias Berneggerus declares that, in order to know the size of mixed bodies composed of different metals, the difference of these simple metals' size should be divided accordingly. The alloy's proportion and this division point should be taken".

For instance, this is his own example: if C is the diameter of copper and CE the diameter of tin, one must find the size of a mixed body composed of three parts of copper and two of tin. He says that in order to do it the difference DE should be divided in such a way that two parts are contained in DF and three parts in BE. Accordingly, the line CF is the size (or diameter) of the mixed body of metals.

According to the proportion of 3 and 2 mentioned before. Even if it is confirmed by his translator, it seems that one should not believe that Galileo really said it, yet, one of our ordinary writers who easily takes the best he finds³⁹ from other has no doubts and considers this false document real, as Berneggerus says. These are his own words, on page 143 of the 1626 and 1631 texts about the use of the proportional compass: "one can do the same using two metals mixed together, and an alloy composed of one half of silver and the other half of copper: the distance between the two characters (that is, chemical signs) of silver and copper should be divided into two equal parts, then one should operate on this division point exactly on the same way used for the simple metals. Yet, if an alloy composed of one part of copper and two of silver is required, the distance should be divided into three equal parts and the first division point, that is the point nearer to the silver, must be used for the alloy composed of one part of copper every two in silver. For an alloy of one part of silver every two in copper, it is the point nearer to the copper which is required." This opinion can be denied by means of his own numbers: for instance, having given the diameter of gold of 730 parts and the diameter of silver of 895 parts, he concludes that the diameter of a body consisting in an alloy composed of one half of gold and the other one of silver will be the half of their difference, that is 165; that is by adding half of the difference (82.5) to 730 (gold), the result is 812.5".

Two more pages are needed by Pierre Petit for the explanation of the error and his conclusion: in the case of an alloy of equal parts of gold and silver, if three spheres of gold, silver and of the alloy must have the same weight, then the cube of the alloy's diameter is the same as half the sum of the cubes of the diameter of the spheres of gold and silver. That is why the alloy's diameter is 820 and not 812.5, as was claimed by Henrion and Galilei⁴⁰.

One can ask whether Galileo ever saw this error and, in this case, why he did not try to reprint Compasso, Having the chance to correct it and publicise further applications, which were not included in the 1606 edition.

The big quantity of publications on the compass printed during these years can be a possible reason. The debate with Capra had been enough for Galileo: surely he did not want to be involved in new debates, because his being famous could be useful to anyone wanting to become famous by means of contestations.

For instance, Henrion claimed his priority by dating his researches on the compass back to 1606: "Therefore I declare the use of this compass mine because I did it without having seen any other compass and in fact nobody had published anything yet, and the most useful operations of the compass are not known by these who published anything immediately after it, Two years ago, Mister Gunter, professor of astronomy at Gresham College in London, printed a book in English concerning the use of the compass , showing what I had published twelve years before".

The text mentioned by Henrion is Edmund Gunter, The description and use of the Sector, London 1623.

A brief list of other publications is suggested by Venturi⁴¹:

1604. Hulsii Levini, Beschreibung und Unterricht des Jobst Burgi proportional-Cirkels, Frankfurt.

1605. Philippi Horcher, Libri tres, in quibus primo constructio circini proportionum edocetur. Deinde explicatur quomodo eodem mediante circino, tam quantitates, continuae, quam discretae, inter se addi, subduci, multiplicari, et divid, brevissimo compendio posint, Maguntiae.

1607. Balthesaris Caprae, Usus et fabrica circini cuisdam proportionis, Patavi.

1608. Leonhard Zubler, Nova Geometria Pyrobolia, Zurich.

1610. Faulhabers, Proportional-Zirkel(in his treatise concerning new discoveries in geometry and perspective), Ulmae.

1610. Georgius Gelgemayers, Unterricht von proportional cirkel. Laugingen, 1610. Reprint:Augsburg 1611, Ulm 1615 and 1617.

1613. Galilaeis de Galilaeis, De proportionum instrumentoTractatus, a Mathia Berneggero ex italica in latinam linguam translatus: adjectis etiam notis illlustratus, quibus & artificiosa Instrumenti fabrica, & usus ulterior exponitur, Argentorati. (Circulating again in 1635 with a new title page).

1615. Christ Laurenbergii, Clavis instrumentalis; oder arithm. Geom. Proportional-Instrument. Leipzig.

1618. D. Henrion, Usage du compas de proportion. Paris. Reprint: 1624 and so further. Between 1630 and 1681 more than twenty editions were made.

1619 Georgius Gelgemayers, Centiloqium circini proportionum, Numberg.

1623. Adriani Metii, Praxis nova geometrica per usum circini proportionalis, Franeckerae.

1626 Mich. Cornette, La géometrie reduite en une facile pratique par deux instruments, dont un est le pantometre ou compas de proportion, Paris.

1526. Nicolaus Barthelt, Instrumentum instrumentorum mathematicorum, Berlin. Reprint: Rostoch 1627.

1526. Wolffangus Lochman, Instrumentum instrumentorum, Alten Stettin. Reprint: Rostoch 1627

1534. P. Petit, construction et usage du compas de proportion, Paris.

This was enough for Galileo to renounce reprinting his book: he was thinking of something else!

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Collecting and linking information concerning the geometrical compass can help us to see clearly what economic and scientific reasons caused Galileo's request to the grand duke for a useful final homecoming to Tuscany, since Galileo tried to approach his Prince by means of young Cosimo's mathematical education with the use of the compass⁴².

Having recognised the importance of the role of the compass in the "strategy for his return", even the reason for the criticism of Capra becomes more easily understandable, as Galileo's plan could have been jeopardised by the instrument and his writing's plagiarism.

Galileo started his contacts for a gradual approach over a long period, from September 27th 1604, when the contract for his lectures on Mathematics in Padua was over, until August 6th 1606, when he was conferred his chair again, after almost two years waiting.

In June 1605 Christine of Lorena received Galileo 's news about his intention of dedicating his work concerning the use of the compass to the prince of Tuscany. It was a clever move: he got her approval and an invitation to spend the summer at court.

In this way contact was established and, later, strengthened by the esteem Galileo deserved from the royal family.

What reasons made him wish for a total change in his life? Galileo 's need to carry out a great intellectual project that was clear in his mind, even if he was not young anymore, was the most important of them. He decided not to teach nor to give private lessons to dozens of student anymore and use all his free time for researching, experimenting and writing.

He knew that all this time was required in order to realise his new science, an "immense concept full of philosophy, astronomy and geometry" he had brought to life, but he could not foresee that he had still to wait for thirty years and to face many battles before realising it.

A very meaningful promise is evident in his dedication of the compass to Cosimo: "Your Lordship will enjoy my mathematical joke, as I call it, and it will suit your education, it is one of the fruits which Divine Providence let me pick up from my labour in these really princely domains, and they will increase together with my age".

Galileo came back with full honours to his chair at Pisa which he had left in 1592, but only pro forma, because all his requests was taken into considerations by the new grand Duke Cosimo⁴³, who not only conferred on him the title of Primario Matematico dello Studio, (so that he was paid using University funds) "without being obliged to live at Pisa nor performing lectures except for when you like to or we explicitly and extraordinarily ask"⁴⁴, but, more important, the title of "truly Primario Matematico e Filosofo". Foreseeing the controversies, he would have to cope with, this title was so useful to Galileo that in the negotiations for his return he had asked explicitly for it, having declared a longer period for his studies in philosophy than in mathematics.

It is surprising that Galileo published his first work, Le operazioni del compasso geometrico e militare, in 1606, after a twenty-year experience in teaching and researching. It was a two-copy edition, and it was never reprinted by him again. It was an edition printed at home and dedicated "first of all to my Lord, the young Prince of Tuscany, and then to other Lords who were waiting for it".

He printed it after forty-two years! About the Paduan period, he writes: "I have been spending twenty years the best of my life, in giving everybody a little of the ability in my profession that I received from God and my labours" .

It is easy to understand how pained he felt because of Baldassarre Capra's fault: "who is stealing from us by fraud and bold usurpation the honour, fame and glory we deserve, not inherited from nature nor destiny or case, but a fruit of our studies, labours and long eves".

Printing the Compasso Geometrico e Militare as the fruit of his labour, Capra attributed the instrument to himself, calling in fact Galileo a bold usurper even if Galileo had dedicated the Compass to his lord The Prince of Tuscany just a year before.

His Difesa contro alle calunnie ed imposture di Baldessar Capra is the first example of Galileo's wonderful fervent writing, an effective, polemical, lucid and at the same time well-balanced piece of writing: a few pages are a real chef-d'oeuvre announcing Il Saggiatore, written to defend and restore his honour against anybody claiming the attribution of the Compass.

These episodes may be considered unimportant, but they occur in a very important period of Galileo's life and the previous and the following events are related in different ways to these writings and the instruments described.

Galileo's psychology cannot be easily understood without considering his economic difficulties. The starting point is very remote: his father's death⁴⁶. His decision to leave Pisa and go to Padua is a clear example of a period when economic aspects were more urgent than scientific ones: it was certainly due to his family's need, which he had to think about, since he was the eldest son.

The circumstances of his going to Padua are very carefully told by Favaro⁴⁷ , but he is wrong in assuming that Galileo accepted a salary just a little higher than what he used to earn at Pisa: he had tone more reason to see as justified Galileo's decision to go to Padua, compared with Viviani and Arrighini, the first Galileo's biographers.

The papers of the Studio di Pisa (1589 and 1590) show that Galileo earned 60 florins, and the 1589-1592 cash books show that Galileo received 60 ducats⁴⁸. Therefore, when he was in Pisa, Galileo received 420 Florentine Lira⁴⁹.

It is useful to recall that 41 professors are mentioned in the 1589-1590 paper and a good 10 of them received just 45 ducats per year, whereas the minimum salary was that of two professors of civil law: the one who used to teach in the morning received 650 ducats, the one who taught in the afternoon 600 ducats. A salary consisting of 60 ducats was not too low for a young teacher at his first experience of teaching, considering that he used to give private lessons as well.

However, these are the lowest salaries: in these years brick-layers at Florence used to earn two Lira per day, that is from 80 to 90 ducats per year, whereas hodmen and farm labourers earned half of that sum.

In Padua instead, Galileo received in his first year 180 Florins, that is 900 Venetian Lira⁵⁰.

Favaro was wrong in calculating the Venetian Lira value as half of the Florentine Lira: in fact, the Venetian Lira and the Florentine Lira were two fictitious currencies, having been in use for ages in accounting in order to know the value of different currencies in proportion to the quantity of gold and silver contained and the value of the two metals. Since the same quantity of gold and silver was contained in both Venetian and Florentine currencies, they received the same value in Lira, and maintained it for a long time⁵¹.

It is easy to see from these papers that Galileo's reasons for his departure from Pisa were mostly economic, and the reasons mentioned by Viviani and Gherardini were not the most important: it seems that they were hiding the actual economic factor on purpose, since none of them alluded to it.

The supposition that the content of their stories comes from Galileo is not too far wrong: could the grand Duke's Mathematician and Philosopher confess without any embarrassment that Ferdinand and the University of Pisa had missed an illustrious professor just for a few coins? This hypothesis can find some ground in Galileo's carefulness towards the Medici family: he never forgot he was their subject.

Vivani tells that "full of envy, many would-be philosophers, his emulators, acted against him and made a very famous hate him because of his judgement about the invention to empty the wet dock (Darsena) of Leghorn: Galileo foresaw a bad event-which occurred in fact- through mechanical grounding and philosophical freedom; thus, he took into consideration the chair at the University of Padua, which had been free for a long time since Giuseppe Moletti's death, and following the marquis Guidubaldo's advice and with the grand Duke 's permission he decided to move before that his adversaries started to receive fruits from his fall"⁵².

In fact, Galileo's unhappiness was immediately evident. Guidobaldo del Monte, who helped him in getting a chair at Pisa, wrote to Galileo (April 10th 1590) "I would like to see you happier and considered as you deserve. I do not have any news from Venice, but I will try to have some and I will let you know".

In the middle of his first year of teaching, Galileo was already trying to move. At the beginning of the second academic year, Guidobaldo asks: "I would like to know if you have ever obtained an increase of salary, as I wish and you deserve". Finally, another commentary on the subject (February 21st, 1592): "I regret to see Your Lordship not considered as you deserve and I regret even more that you do not have any hope. If you go to Venice this summer, I will invite you to come and see me. Since I cannot bear seeing you treated in this way I will help you. I will do anything I can"⁵³.

Therefore, Galileo's main worries were still economic ones. Had his difficulties been those suggested by his first biographers (that is, his colleagues' widespread envy and Giovanni de' Medici's hostility), Guidobaldo could have helped him, as he was highly esteemed by the Medici family. Yet, a great increase in salary (that is, to be treated as he deserved) was even more difficult to obtain, as well as well-paid private lessons, which could easily e found instead in Padua, where a chair was free.

The marquis had carried out his studies in Padua, which means that he had very important friends in the Republic; moreover, he was one of his admirers. Yet⁵⁴, what helpful friends had Galileo at Florence?

Giovan Batista Pinelli, one of marquis Guidobaldo's influential friends, was sent to Venice at the end of 1592 and in a few days he obtained the chair of Mathematics for Galileo, free since 1588 because eof Giuseppe Moletti's death⁵⁵: "They told me that you will surely receive 200 florins and that "he" will be here tomorrow or the day after: so, Your Lordship can go and see him and thank him for his goodness and application for a move can be made"⁵⁶. The person Pinelli was referring to was Giovanni Michel, a solicitor and one of the three Riformatori: Galileo had to hurry up in sending the 26th September verdict.

Being sure of his professorship at Padua Galileo let Ferdinand de' Medici know it through Giovanni Uguccioni. A letter written by the Resident of Tuscany to Belisario Vinta (15th September 1592) says: "I am in Padua, I came here together with Mr Galileo Galilei, a lecturer on Mathematics at Pisa who arrived at Venice fifteen days ago. Yesterday, talking with me in a coach he said that, while he was at Venice, he was offered a chair at the University of Padua where he would earn about 200 scuds each year, but he answered that he cannot decide anything because he is in the grand Duke's service, therefore I think I will come there to discuss about it with Your Kingship".

The grand Duke did not keep Galileo at Pisa, since transforming he quantity of florins into the same number of ducats made the salary appear three times higher; consequently, he left him free to leave the town. This was an easily understandable decision if one considers that the young professor had many adversaries, not only his colleagues, but even Giovanni de' Medici.

Nevertheless, Galileo's economic situation in Padua had never been completely satisfactory and he had to give private lessons in order to make ends meet. During his first few years, when he was still young, he could find time for his research and studies, but not enough.

The endless increase in the cost of living was one of the reasons for his difficulties: one can briefly mention the phenomenon of the price revolution which took place in every European country over the second half of the sixteenth century, with the beginning of an inflation reaching its peak in the period 1590-1610, with big increases in the cost of living.

Each state had its economic problems, but Venice 's difficulties were even bigger. A deep crisis had entered the Republic: first a fire at the Arsenal (1569), then a trade crisis after the war against the Turks ended in the loss of Cyprus (1571), then the 1575 pestilence which depopulated the University of Pad⁵⁷ and finally the interdiction from Paul V.

After Francesco de' Medici's death, instead, a new policy appeared to foresee a period of economic boom for Florence.

Nevertheless, Galileo faced his economic problems by increasing his earnings by giving private lessons to dozens of students⁵⁸. The production of scientific instruments started as well: Galileo made a good mechanical construction of not only his geometrical compasses, but also bussole, instruments for drawing and even balances. His students were given the compass at cost price, but he received rich presents from the wealthier students.

Yet his sister Livia's marriage weighed on his shoulders, as did his sister Virginia as well: in 1601 he had to take care of Livia's dowry: 800 ducats immediately and 200 each year for five years. He was obliged to help his brother Michelangelo as well. Moreover, he was responsible for his mother at Florence and for the natural family that was born from his relationship with Marina Gamba: his first child was born in 1600, the second in the following year, and finally Vincenzio in 1606, the sole son he was able to recognise.

Thanks to his charming personality and his open character, Galileo did not have any difficulty in developing a vast network of relationships in Padua and Venice, involving acquaintances as well as cultivated and wealthy friends: that however required quite an expensive standard of living.

Ten years before, Girolamo Mercuriale wrote: "His talent's home was the Studio di Padova"⁵⁹. Galileo's economic situation was a problem, and had an immediate influence on his scientific production. it is true that he was well-known for his professorship and his correspondence with many scientists and famous men, but private lessons were an obstacle for his studies⁶⁰.

For some time already he had started to work on a few subjects which the chef-d'oeuvres of his maturity were built on, but he kept the results hidden.

Galileo abhorred the prostitute slavery of selling his labours' fruits for any adventurer's arbitrary price. He wished to carry out the work he had started "even all the freedom I have here is not enough for me, since many hours and the best ones during the day are spent in satisfying everybody else's requests". He had several intentions "and I would find many other inventions, if I had more rest and freedom to dedicate myself to the most important work"⁶¹. Galileo aimed to come back to the Granduca di Toscana's services , since he could obtain these facilitations only from an absolute prince.

Unfortunately, on September 27 1604, Galileo's second year as a lecturer on mathematics at Pad came to an end⁶². Galileo did not succeed in obtaining any increase in salary. A few months before he was told that Vincenzo Gonzaga wanted him in his service at Mantova, offering 300 ducats per year plus victuals for him and one servant. Galileo's requests⁶³ were 500 ducats plus three rations of victuals. The Serenissimo Duke of Mantova, who had asked Galileo for the instrument 's commentary two years before, and had then given him a necklace and a medal bearing his effigy⁶⁴, had to give up, "since it is right that you are free to do what you want"⁶⁵. Favaro⁶⁶ declares that Galileo took the duke's offer into consideration, most of all in order to get an improvement of his conditions in the case of a new conferring of his professorship.

A month after having received the permission to print Compasso, on August 5th 1606, his long-desired nomination for six years more (as usual) arrived, and his increase in salary consisted of two hundred florins, thanks to the grand duke's good offices⁶⁷ as well; Galileo had started in 1592 with 180 florins, he obtained 320 florins in 1598, and at this point he arrived at 520 florins.

The actual value of Galileo's salary has already been shown, but in fact, it was not enough for him, even considering periodic increases.

In 1601 he was given a valuable suggestion by his friend Girolamo Mercuriale: "I'm asking you to come at any cost, because the Prince⁶⁸ is twelv3 years old and I think he is able to do any mathematical thing Your Excellency can show him; the boy has got a good intellect and memory and the most curious mind you can imagine: so, I believe Your Excellency can use your talent and perhaps it will be useful to you. I must remind you again to finish in the meanwhile the military and geometric instrument at any cost, in order to take it to San Giovanni at Florence next year, where I will be, too. On the first occasion I will do that service due to Our Highnesses and, if you were so kind to send me a brief description of what you are going to do to the Prince, together with its use and functions, I would show it to Their Highnesses and I am sure that the Prince will have fun"⁶⁹.

It is not difficult to understand Galileo's situation: his projects were ambitious and justified by many important theoretic and experimental results achieved, therefore he needed to dedicate all his time to his studies but, as he says: "usually one cannot obtain any salary without being a public servant, even in a splendid and generous Republic, () I cannot hope to obtain this sum from anyone other than an absolute prince"⁷⁰.

He had gathered enough information to write three works⁷¹ whose titles he had already decided: two books De sistemate seu constitutione universi, "an immense concept full of philosophy, astronomy and geometry" which was called twenty years later Dialogo sopra ai massimi sistemi del mondo, three books De motu locali, later included in Discorsi e dimostrazioni matematiche intorno a due nuove scienze- very important for Galileo, who called the content " a totally new science, since none of the many symptoms of the natural and violent movements , as I demonstrate, by, has been discovered by anybody else,; therefore, I can rightly call it a new science that I have discovered from its very first principles" and, finally, three books: Le Mecaniche.

The grand Duke's favour, indeed, his friendship was needed by Galileo. Four years after Mercuriale, Vincenzo Giugni writes⁷²: "Hearing the praises our instrument and its usefulness is deserving, I decided to talk about it to our Ladyship, mentioning in the best way I could Your Excellency's intention to dedicate the instrument and its reason to our Highness the Prince; and I added as well that you could decide to come here this summer and be ready to show the grand Prince the great instrument 's uses, madam Her Highness answered that you will be treated as your virtues deserve. So, come, you will be happily received".

In order to give to this letter the right value it is enough to know that on 28th October 1604 Giugni's son, Niccolò, had arrived at Galileo's house in order to stay for a dozen days. He came back to Florence in April of the next year. It was him who praised the instrument and his Maestro in front of his father and, since Galileo had decided to publish Compasso, he asked for permission to dedicate it to Cosimo.

Galileo was officially invited on 15th August 1605: "madam Her Highness wishes Your Lordship to come here, for both the Prince's virtuous entertainment and her recovery thanks to the healthy air of Pratolino"⁷³. However, at the beginning the results were not very encouraging, so Cosimo's preceptor writes, at the end of 1605: "As for our Prince's studies, since you wanted to know if he knows any mathematics, I can say that after your departure he has not seen or used the instrument, not because he does not like sciences, but because there is nobody who remembers any operations and the Court has been moving to and fro, and other obstacles; but, as soon as we arrive at Pisa something will be done. Meanwhile, please, work at your book and try to finish its printing: this text will be a good help for the Prince and a good memorial"⁷⁴.

The grand duke's appreciation and liking were immediately evident. He acted⁷⁵ effectively to help Galileo in what was still worrying him: a delay in conferring his professorship in mathematics.

Ferdinand I 's words confirmed the probability of Galileo's not distant return to Florence: "I really want to help Galileo because of his talent; therefore, please, ask Vinta to put in good words in the letter to the Resident"⁷⁶.

Three years later, Vinta reported Cristina di Lorena's words: "Since Galileo is the first and the most precious mathematician in the Christian world, the grand Duke and I wish him to spend this summer with us, if it does not disturb him, and help our son the Prince in his exercises in mathematics, that he likes so much"⁷⁷.

Galileo staked everything on Compasso Geometrico. He did not have anything else finished or better than that. His other works were less important: apart from a few leaflets which have been lost, he had written Breve Istruzione all'Architettura Militare, Trattato di Fortificazione⁷⁸ and a text that he probably used in his lessons, Le Mecaniche⁷⁹.

At this point Le Mecaniche were already overtaken by his research; as for the Trattato di Fortificazione, even if it is known that Galileo intended to write books concerning soldiers, nothing shows his intention to print the text in its present form: actually, it seems very different from the contemporary texts written by very experienced military engineers, which composed his documentation⁸⁰.

Instead, great success had been achieved by Il Compasso, and more than 100 instruments had been constructed and sold or given as a present to Princes and Lords of different countries, and his pupils spread it across Europe, often together with commentaries about the way it is used. Galileo staked everything on the compass.

His need for more spare time for his research and carrying out great projects already started has been already mentioned, but there is another scientific reason, not mentioned by Galileo, which was surely important for his determination to leave the University of Padua: his desire to divulge his new cosmological theory, which he was a convinced assertor of, and was writing a book about, whose title had already been decided: De sistemate, seu constitutione universi.

This proposal was not only scientific, but also ideological; the adversaries of his work would have been not only peripatetics, who were so powerful in all universities, but also great theologians: in a period of Catholic restoration, an idea like this could be considered extravagant and wrong.

Galileo wanted the ecclesiastical authorities to accept the Copernican theory; but the Church's neutrality could not be obtained by staying in the interdicted Venice, in Paolo Sarpi and Giovanfrancesco Sagredo's circle, in a republic which had banished Jesuits.

Instead, the Medici family was bound to the Church, Paolo V was Tuscan and Galileo was in good relations with the very influential father Clavius, the greatest expert in Mathematics and Astronomy in the order of Sant'Ignazio.

For that reason as well, it was time to come back to Tuscany: only from that place could he cope with his adversaries' bad acts, falsely showing zeal and charity, not just as a simple professor anymore, but as a famous scientist and a courtesan loved by his Prince.

It is difficult to identify the most powerful reason among the multiple scientific and economic needs determining his pressure to come back to his country.

Up to this point the importance of the compass ­ both of the instrument and its manual- in the realisation of Galileo's hopes has been shown. Surely, the great scientist could not foresee that his forthcoming destiny was to make a telescope and make astonishing discoveries in the skies.

After his astronomical observations in 1609, Galileo saw the end of his difficulties: Venice offered him a salary of 1000 Florins for all his life, but he could get it only after receiving his professorship, that is from 19th September 1610, and, moreover, with "no possibility of any increase in salary"⁸¹.

However, because of Galileo's dissatisfaction, his negotiations with the grand Duke went on, and the grand Duke decided to call him back and offered him a salary of 1000 Scuds per year, equivalent to 7000 Florentine Lira. Belisario Vinta communicated it to him on June 5th, and he marked other advantages which could later be obtained as well: "by living and talking with Their Highnesses, they will have knowledge and proof of your valuable talent in many domains, and they will add love and esteem to your merit and will give you favours and honours. Yet, if this is enough for Your Lordship, you will have to explicitly express it in your letters, in order to write a petition under your name, and Her Highness's decree and, when your Lordship wants to publish it: in the meanwhile, it will be kept as secret as possible⁸².

He obtained 2000 Lira more and a lot of spare time: at Florence he could stop giving private lessons ­and receiving higher earnings- because he was promised "favours and honours"! After his diploma of nomination, signed on July 10th, Galileo was ready for his departure, which was delayed only because of illness and the death of a dear servant of his, however at the beginning of September he was already at Florence, probably without having taken leave of the Venetian government⁸³.

He never returned to the Venetian State. Paolo Sarpi, Giovanfrancesco Sagredo and other Venetian friends were disgusted with him⁸⁴. A period of Galileo's life was over.

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