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The enduring legacy of Galileo’s pioneering investigations was the establishment of a mathematical formulation as the basis for research in the physical sciences. The proving-ground for his revolutionary experimental-mathematical methodology was the little machine shop he had set up in his house in Padua. There Galileo perfected or reinvented two recent devices, the geometric compass and the telescope, issuing a small volume on each. Sidereus Nuncius (also available in an Octavo Edition) was published in Latin so as to acquaint the learned across Europe with his telescopic discoveries. Le Operazioni del Compasso Geometrico et Militare appeared in Galileo’s native Italian, as befitted an instructional manual originally composed to accompany the device. Patronage and politics ensured that a Renaissance scientist could no more ignore the study of ballistics than an architect could neglect the techniques of fortification. The so-called “geometrical and military compass” (or sector) was a mathematical device – a sort of calculating ruler based on the principle of proportional magnitudes – that brought speed and accuracy to computations about armaments and their trajectories. The use of Galileo’s device, however, extended well beyond the battlefield. It was widely used for rapid and complex calculations of proportion, volume, and square or cube roots, well before the advent of logarithmic tables – and eventually the slide-rule and computer. Galileo’s little book, privately printed in an edition of a mere sixty copies, is now extremely rare.
The original book imaged for this digital edition:
12 x 7 7/8 inches (305 x 200 mm)
The original book imaged for this digital edition:
12 x 7 7/8 inches (305 x 200 mm)
Calculating Cannoneers
In the second half of the sixteenth century, intense efforts were exerted to construct an instrument that could carry out, accurately and rapidly, all of the measurements and calculations necessary to the art of war. Special attention was focused on the compass – which was small and manageable yet geometrically similar to a triangle, the figure used to perform all proportional calculations. Galileo’s “geometric and military compass” belonged to this class of instruments; although it was designed mainly to carry out military operations, its functions were applicable to every other profession that required a knowledge of arithmetic and geometry. Accordingly, it was extremely useful to merchants, bankers, artisans, engineers, surveyors, and astronomers. Galileo’s invention was the final result of research that had engaged some of the leading mathematicians and military technicians of the sixteenth century, all tending toward a progressive transformation of the compass into a calculating instrument.
In the second half of the sixteenth century, intense efforts were exerted to construct an instrument that could carry out, accurately and rapidly, all of the measurements and calculations necessary to the art of war. Special attention was focused on the compass – which was small and manageable yet geometrically similar to a triangle, the figure used to perform all proportional calculations. Galileo’s “geometric and military compass” belonged to this class of instruments; although it was designed mainly to carry out military operations, its functions were applicable to every other profession that required a knowledge of arithmetic and geometry. Accordingly, it was extremely useful to merchants, bankers, artisans, engineers, surveyors, and astronomers. Galileo’s invention was the final result of research that had engaged some of the leading mathematicians and military technicians of the sixteenth century, all tending toward a progressive transformation of the compass into a calculating instrument.
Bristly Bruno
The development of the military compass was not without controversy. A heated argument broke out in Paris in 1585 between mathematician Fabrizio Mordente and the philosopher and Hermeticist Giordano Bruno. Mordente was trying to win the acclaim of the French court with his version of the compass. Bruno, who devoted no less than four dialogues to that instrument, insinuated that Mordente might have appropriated someone else’s invention without fully understanding its geometric potential. Mordente had accused Bruno of presumption, and the latter therefore reduced Mordente’s status from mathematician to technician, crediting himself with having discovered in that compass the principles of a new geometry (which would later guide his reflections on measuring the minimum and on atomistic mechanics).
The development of the military compass was not without controversy. A heated argument broke out in Paris in 1585 between mathematician Fabrizio Mordente and the philosopher and Hermeticist Giordano Bruno. Mordente was trying to win the acclaim of the French court with his version of the compass. Bruno, who devoted no less than four dialogues to that instrument, insinuated that Mordente might have appropriated someone else’s invention without fully understanding its geometric potential. Mordente had accused Bruno of presumption, and the latter therefore reduced Mordente’s status from mathematician to technician, crediting himself with having discovered in that compass the principles of a new geometry (which would later guide his reflections on measuring the minimum and on atomistic mechanics).
Galileo’s Gambit
In 1606 Galileo printed the manual of instructions for his compass, reproduced here, which in former years he had distributed to his students in an abbreviated, handwritten form as a guidebook to the use of the instrument. In the printed book he described thirty-two operations: seven for the arithmetic lines, seven for the geometric lines, six for the stereometric lines, five for the metallic lines, two for the polygraphic lines, four for the tetragonic lines, and one for the added lines. Nine more operations were described for the topographic use of the quadrant. To safeguard his rights, Galileo did not include a detailed illustration of the compass (the illustration in the copy reproduced here was taken from the second edition), or even to describe how the proportional scales were constructed. The book was sold only along with the instrument, of which about one hundred examples had been built for the European noblemen and sovereigns who were Galileo’s “pupils.” A copy of the book with two silver compasses was given to Cosimo de’ Medici, whom Galileo had the honor of personally instructing during that same summer of 1606.
In 1606 Galileo printed the manual of instructions for his compass, reproduced here, which in former years he had distributed to his students in an abbreviated, handwritten form as a guidebook to the use of the instrument. In the printed book he described thirty-two operations: seven for the arithmetic lines, seven for the geometric lines, six for the stereometric lines, five for the metallic lines, two for the polygraphic lines, four for the tetragonic lines, and one for the added lines. Nine more operations were described for the topographic use of the quadrant. To safeguard his rights, Galileo did not include a detailed illustration of the compass (the illustration in the copy reproduced here was taken from the second edition), or even to describe how the proportional scales were constructed. The book was sold only along with the instrument, of which about one hundred examples had been built for the European noblemen and sovereigns who were Galileo’s “pupils.” A copy of the book with two silver compasses was given to Cosimo de’ Medici, whom Galileo had the honor of personally instructing during that same summer of 1606.
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