The Evolution Of A Mathematical Instrument – I

Before the publication of Sidereus Nuncius in 1610 launched him into European scientific superstardom, Galileo Galilei (1564–1591) was a comparatively unknown professor for mathematics at the University of Padua. He had a reputation in Northern Italy as a highly intelligent, somewhat arrogant, argumentative scholar with a decided taste for the good life. Professor for mathematics was at the very bottom of the academic status pyramid and comparatively poorly paid. Galileo had always lived beyond his means and money was tight. This was exacerbated in the first decade of the seventeenth century, when he had his mistress, Marina Gamba, and three children–Virginia (b.1600), Livia (b.1601), Vincenzo (b. 1606)–to support. However, his financial problems didn’t start with the founding of his family. 

When Galileo first took up his position in Padua in 1592, he was already under financial duress. Following the death of his father, Vincenzo Galilei in 1591, as the eldest surviving son, Galileo become head of the family and financial responsible for his three surviving younger siblings. The youngest, his brother Michelangolo, was a professional musician, who was always broke and relied on handouts from his older brother. Galileo’s two sisters both required doweries in order to  get married and Galileo was forced to supply the necessary money. The young professor was financially in dire straits.

As a solution to his money problems, Galileo followed a widespread tradition amongst academics and took the student sons of wealthy families into his house as lodgers, supplying them with board, lodging and private tuition for suitable remuneration. In 1597, he launched a new money making endeavour, he designed and manufactured a proportional compass, also known as a military compass, or in England as a sector, which he then sold for a very healthy profit. 

Although the design was by Galileo the actual production was carried out by the instrument maker Marc’Antonio Mazzoleni (b.?–1632). Galileo had hired Mazzoleni away from the Arsenale in Venice, as his personal instrument maker in 1597. In 1599, Mazzoleni moved into Galileo’s house with his wife and daughter. His wife became Galileo’s cook and housekeeper. Mazzoleni lived in Galileo’s house for four years, but continued to be his instrument maker after he moved out. He even continued after Galileo returned to Florence in 1610. Mazzoleni also helped Galileo produce other instruments, including hydrostatic balances, air thermometers, magnets and magnetic compasses for ships, and various kinds of drawing compasses for engineers and architects. He is known to have produced at least one hundred proportional compasses for Galileo.

The manufacture and sale of his proportional compass was a very lucrative enterprise for Galileo. As well as the actual instrument, which sold for a very healthy profit, Galileo offered the purchasers tuition in its multiple applications. He also sold a book of instructions in its use to the new owners. This instruction manual was in the form of a handwritten manuscript without illustrations. Without an instrument to refer to it was almost useless. Selling the manuals individually with the instruments, rather than putting out a printed book meant that nobody could simply gain access to Galileo’s design and potentially ruin his business. However, in 1606, Galileo did put out a limited printed edition of the manual under the title Le operazioni del compasso geometrico et militare, which led to the earliest of Galileo’s many heated disputes. 

Baldassarre Capra (1580–1626), the son of Count Marco Aurelio Capra an impoverished aristocrat, was a physician and astronomer, who had studied medicine in Padua. Capra received private tuition in mathematics from Simon Marius (1573–1625) and together they observed the supernova of 1604. Capra attacked Galileo in his pamphlet Consideratione astronomica circa la nova et portentosa Stella che nell’anno 1604 a dì 10 ottobre apparse for not acknowledging Capra’s priority in observing the supernova in his three lectures at the university on the topic. Galileo responded anonymously in his usual robust style. The stage was set for a major clash. 

Capra and his father had examined Galileo’s proportional compass in 1602, and in 1605 he borrowed an instrument from his friend Giacomo Alvise Cornaro an aristocratic intellectual, who was Galileo’s next door neighbour. Capra also visited Mazzoleni’s workshop, where he spent time studying the instrument. In 1607, Capra published the tract Usus et fabrica circini cuiusdam proportionis, per quem omnia fere tum Euclidis, tum mathematicorum omnium problemata facile negotio resolvitur, basically a Latin translation of Galileo’s own manual, dedicated to Cornaro, claiming to having invented the proportional compass and thereby accusing Galileo of plagiarism. 

Galileo, being Galileo, did not take this lying down and asked the university rectors to adjudicate the matter. At the hearing, at which Cornaro spoke on Galileo’s behalf, Capra got crucified. He was found guilty and all copies of the book was ordered to be burnt, but by then several had escaped into general circulation. In 1607, Galileo published Difesa contro alle calunnie et imposture di Baldessar Capra, which showed how Capra’s accusations were false.

As a prominent member of the Simon Marius Society, I should point out that in 1623, in his Il Saggiatore, as part of his virulent and highly insulting attack on Marius for having dared to observe the Moons of Jupiter, Galileo also accused Marius of having been behind Capra’s false claims and in fact of having written Capra’s Latin tract, this despite the fact  that Marius had already left Italy in 1605, a year before Galileo had published his own manual. Unfortunately, all over the web, even on supposedly serious websites, people still repeat Galileo’s unfounded and false accusations and calumnies without bothering to actually check the facts.

One of the many Galileo myths that get repeated ad nauseum is that he invented the proportional compass, although they always use the term military compass. Quite simply, he didn’t. The proportional compass has quite a lot of history tied up with it and I shall be looking at that history in a series of blog posts of which this is the first. Another popular claim, made by those who know that the invention story is a myth, is that Galileo perfected the proportional compass. This is also not true. Galileo made a very sophisticated and advanced model of the instrument but one can’t talk of perfecting. Firstly, In England both Thomas Hood and Edmund Gunter made equally sophisticated models of the compass independently of Galileo. Secondly, different models of the compass fulfil different functions depending on the type of scales engraved upon the arms. 

So, what is a proportional compass and why is it called a “proportional” compass? The proportional compass is an analogue calculating device widely in use from the sixteenth through to the nineteenth century. It usually consists of two rulers of equal length joined a one end by a hinge that allows then to be opened up to 180° to form a straight line. The rulers are engraved with a number of scales which are used to make calculation is various areas in arithmetic, geometry, and trigonometry. Different instruments had different scales. The  instrument was employed in various practical fields such as gunnery, surveying and navigation to find direct, easy solution to mathematical problems in those fields.

It is called a called a proportional compass because it utilises the theory of proportion explicated in The Elements of Euclid. First laid out generally in Book V in relation to magnitudes rather than numbers in order to avoid irrationals, in Book VI the theory is extended to geometrical figures. Propositions 2 and 4 in Book VI provided the theory on which the proportional compass is based.

Book VI, Proposition 2:

If a straight line be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally : and, if the sides of the triangle be cut proportionally, the line joining the points of section will be parallel to the remaining side of the triangle.^[1]

Book VI, Proposition 4:

In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides which subtend the equal angles.^[2]

The rulers of a proportional compass form the sides of an isosceles triangle and when opened a line between the two sides parallel to the line joining the tips of the rulers created two equiangular isosceles triangles creating the conditions for Euclid’s two propositions. This reliance on Euclid’s theory of proportion from The Elements explains the title of Capra’s plagiarism of Galileo’s pamphlet. 

Illustration taken from CIRCINUS: Proportional Compasses from the 15th to the 18th Century, Arithmeum Press, 2024

The proportional compass has a prehistory that stretches back to at least Ancient Rome and as noted above I shall be taking a look at that history in the following posts.

^[1] Euclid The Thirteen Books of The Elements, translated with introduction and commentary by Thomas L. Heath, Vol. 2 (Books III–IX), Dover, 1956, p. 194

^[2] Heath, p. 200