The reason that Galileo quickly became a highly respected member of the academic circle in Padua was largely due to his brilliance as a lecturer, where he displayed his acute intelligence as well as his considerable linguistic skills. He had students from the Italian states and from further afield. Some came from the highest echelons of society, like the exiled Swedish Prince, who was sent there by his uncle, King Sigismund of Poland9. The prince even lived under Galileo's roof for a while, and Galileo gave him lessons in Italian.
The vague boundaries between subjects meant that Galileo by no means limited himself to pure mathematics. He lectured on astronomy - but did not reveal his belief in the motion of the Earth. Instead he recited the traditional, Ptolemaic counter-arguments: birds would be left behind as the Earth revolved beneath them, objects dropped from a tower would land far away from the tower's foot.
Mathematics was a "utilitarian discipline" with many applications. Galileo even lectured on military engineering, one of the subjects his teacher
Ricci at the grand ducal court had also mastered. Galileo gave two lectures on it. The first was on "the art of fortifying cities" - the second, logically enough, on how such fortified cities were to be conquered!
Providing lodging and tuition to aristocratic students supplemented his income. Galileo was perennially short of money. He was paying dowry instalments, convent fees, music lessons and the living expenses of his mother and brother. Furthermore, he was a man in his best years, and life was not all experimenting and teaching. He soon made friends, good friends, both in Padua and in Venice.
The practical Galileo, the designer and craftsman, always ready to link theoretical calculations to empirical experience, did not hide his light under a bushel. During his first years at Padua he developed a remarkable instrument for calculating and sighting, compasso geometrico militare. In translation this means something like "the geometric and military compass".
It was partially based on the proportional sector, an instrument used to transfer dimensions from one scale to another. Guidobaldo del Monte had constructed one. Gradually, it became reasonably common for painters to use proportional sectors, as these more easily allowed them to find the dimensional correlation between their models and what was to appear on the canvas. It must have been a really huge proportional sector of the del Monte type that once got the painter Caravaggio arrested on a street in Rome - on the spur of the moment the officer took it to be some kind of weapon!
The other prototype Galileo used was the plumb-line and square that was inserted in cannon barrels for calculating elevation, so that the projectiles would land where they were supposed to. But his fully developed instrument had a far wider application.
The geometric and military compass is a fine piece of bronze workmanship. Its two feet are about ten inches long, and one has an integral limb that can be further extended. The feet are joined by a curved cross-piece, and at the apex where the feet are hinged, a plumb-line can be attached. Feet and cross-piece are etched with lines and scales on both sides.
The instrument is geometric. Galileo kept to his Euclidean roots. All calculations that can be carried out with the help of the compass are approximations, based on the comparison of parts of lines and triangles. They are founded on proportionality, not on any absolute, given unit of measurement. (There were none in existence, even the commonly used braccio varied from town to town.) As a unit of length was needed as a basis for the quantifying
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of the proportions, Galileo used the more or less private measure of the punto, plural punti, approx. 0.94 of a millimetre.
The compass was an amazingly versatile instrument. In military use it could of course serve to measure cannon elevation, but one could also estimate distance and difference in levels with it. It could be used as an astronomical quadrant for fixing the position of stars in navigation.
Its purely geometrical functions included the calculating of inscribed and proscribed circles to polygons; but one could also use it to find the radius of a circle with the same area as various rectangular polygons - estimated only, since "squaring the circle", as we know, is one of the insoluble problems of mathematics. Most interesting of all, perhaps, was that with a given polygon, for example a square, one could easily calculate the sides of a new polygon with n times larger area. If one selects a suitable square, this provides a direct method of finding - or at the very least of estimating - square roots (expressed as one side of the square n, they can be measured in punti). It can therefore be claimed that Galileo's compass was the first proper mechanical calculator.
Making such an instrument required great precision and took a long time. Galileo solved that problem by employing a craftsman, an instrument maker who had worked at Venice's famous shipyard, the Arsenal. The man moved in with the professor in Padua - with his entire family on a board and lodging basis. Galileo made a little money this way. The compasses sold for five scudi, which did not give much profit once the bronze had been paid for and the instrument maker had his wages. But it was complicated enough for the user to need thorough instruction. Galileo gave private tuition in its use - for a sizeable fee: twenty scudi.
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