Tartaglia’s treatise on physics, specifically his theories on bodies in motion, gave rise to a generation of scientific investigation into the science that came to be known as ballistics. His observation-based theories helped pry sixteenth century physics away from Aristotelean thinking, which was entrenched in the Church-supported schools and universities, and toward an empirical, experimentally based physics approaching the modern scientific method.

The influence of the Greek philosopher Aristotle’s theory of motion and of other scientific concepts, left over from the study of the physical world in classical Greece, was pervasive throughout the Middle Ages. Attempts to devise new theories of motion took the form of commentaries on the works of Aristotle rather than being based on observation and description of the physical world. The mathematics used in these descriptions could be found in the geometry of the Elements, by the Alexandrian geometer Euclid (c. 330-c. 270 b.c.e.). Then, work in Italy in the middle of the sixteenth century led to a reevaluation of the basis for mathematical models of motion in the physical world. New Science, The (Tartaglia)
Tartaglia, Niccolò Fontana
Cardano, Gerolamo
Benedetti, Giovanni Battista
Cardano, Gerolamo
Benedetti, Giovanni Battista
Tartaglia, Niccolò Fontana

Aristotle (384-322 b.c.e.) had devoted a certain number of the works that circulated under his name to questions having to do with physics, although they were usually addressed from the standpoint of what might be called philosophy instead of mathematics or science. His concern was primarily to understand how motion and change were possible, in resisting the arguments of many of his contemporaries, who denied that possibility. By contrast, the tradition associated with the Greek mathematician Archimedes (c. 287-212 b.c.e.) started from the reality of certain physical processes and then tried to analyze them in terms of the mathematics known at the time, especially the geometry of Euclid. Both approaches to the study of motion continued through the Middle Ages, although the Aristotelean ideas received a larger share of attention and blessing from the Church. Those who studied questions of motion in the universities of Western Europe could be guaranteed a fair dose of Aristotelean doctrine.

It is therefore not surprising that the originator of the most lasting revolution in the study of motion outside the tradition of Aristotle was not the product of a university. Niccolò Fontana Tartaglia came from a family unable to bear the cost of formal education, so he was largely self-educated. That did not mean that he was unfamiliar with the extensive classical literature surrounding issues of motion, but he had no particular predisposition in favor of the Aristotelean view. Throughout the early part of the sixteenth century, various treatises, from both classical and medieval times, appeared in Italy, and Tartaglia was involved in bringing the work of Archimedes before the public. Tartaglia’s approach to the science of mechanics, which included the laws of motion, was based partly on his independence from tradition and partly on the need to try to resolve questions of pressing interest to those who had resources with which to support scholars.

Tartaglia’s La nova scientia (1537; The New Science, partial translation in Mechanics in Sixteenth-Century Italy: Selections from Tartaglia, 1969) first appeared in Latin in 1537 and then appeared in the vernacular within fifteen years afterward. In it, Tartaglia addressed one of the most compelling practical problems of the time: the study of the behavior of projectiles in motion, which came to be known as ballistics. These questions were key to understanding the operations of siege weapons, cannon, and firearms, or guns.

There had been plenty of practical discussion of gunnery previously, but not much of it had aspired to the dignity of a science. Tartaglia did not see any reason why the methods of mathematics could not be used to find solutions for the problems of gunnery, of which the most notable was the relationship between the angle at which a projectile was launched and the trajectory it followed. This was not an idle matter, with city walls to be bombarded in sieges, but it also could be fit into a mathematical framework. In Aristotelean accounts of motion, the fundamental curves were the straight line and the circle, so it had been assumed that the motion of a projectile could be analyzed as a mixture of those two. Just as the Aristotelean version of mechanics had been built into the system for planetary motion developed by the Greek astronomer Ptolemy (c. 100-c. 178), only to be replaced in 1543 by the system developed by the Polish astronomer Nicolaus Copernicus, so the Aristotelean theory of motion as applied to projectiles was rejected by Tartaglia, who recognized that circles and straight lines were not the best constructs for analyzing motion.

Tartaglia’s mathematical treatment of the path of a projectile arose from empirical observation. He observed that even if the projectile started off in a straight line, it began to curve and followed that curve for the rest of its flight. The curve was clearly not a circular arc, which left Tartaglia with the problem of determining what angle would produce the maximum range. Even though there was an error in Tartaglia’s mathematical analysis, he did obtain the correct value, namely, 45 degrees as the angle of inclination. Tartaglia did not have a theoretical model that explained the deviations from a straight line, but his empirical approach allowed the application of mathematics to this practical problem.

Gerolamo Cardano, a professor of mathematics in Milan and a rival of Tartaglia, held views on motion that were similar to those of Tartaglia. Unlike Tartaglia, however, Cardano was the product of the Italian university system and did not express his ideas as explicit deviations from Aristotle. Cardano asserted that if two spheres of different sizes were released at the same time, they would reach the ground at the same time. His mathematical argument for this theory was unconvincing, however, and Tartaglia—incensed over Cardano’s intellectual theft—impugned both his character and his mathematical competence. History, however, has recognized Tartaglia’s achievement and credits him as the father of ballistics Physics . Military;ballistics

The third member of the school of northern Italians who created the new science of ballistics was Giovanni Battista Benedetti, who claimed to have been a student of Tartaglia. He shared with Tartaglia the lack of a university education and in 1553 published a work on mechanics, De resolutione, that included a letter of dedication in which he asserted the “law of equal times of fall” that had been presented less clearly and argued for less effectively by Cardano. This law asserts that the time of descent for a body depends on the vertical distance traveled rather than the distance covered in other directions. The fact that the letter of dedication was to a priest is typical of the extent to which Tartaglia and Benedetti managed to stay on good terms with the Church in presenting notions contrary to the teachings of Aristotle. It is interesting to note that Cardano, by contrast, did spend some time in prison at the behest of the Inquisition. Perhaps his efforts to give an Aristotelean flavor to his novelties in the theory of motion were regarded with more alarm by the Church than the more practical speculations of Tartaglia and Benedetti.

The appearance of Tartaglia’s work and its influence on the school that included Cardano and Benedetti indicates a change from the intellectual and mathematical traditions of the past. Even though Tartaglia was familiar with the works of Euclid, he had a stronger interest in trying to predict the motion of projectiles than in trying to fit his observations into the geometry that Euclid presents. In particular, the idea that motion requires more than lines and circles for its analysis helped to remove the Aristotelean qualitative discussions from the center of the stage in favor of mathematical models.

As for the influence of Tartaglia’s work on the generations ahead, the outstanding example is certainly Galileo. When Galileo wrote about the two new sciences in Discorsie dimostrazioni matematiche intorno À due nuove scienze (1638; Dialogue Concerning Two New Sciences, 1665), he was echoing the title of Tartaglia’s work. In fact, Galileo in many ways was trying to perfect the ideas roughly sketched out by Tartaglia, Cardano, and Benedetti, by fitting them into a full world system. It was perhaps the attempt to make a world system out of his calculations that caused Galileo to follow Cardano into the clutches of an Inquisition reluctant to allow quite so much of Aristotelean physics to be abandoned. It is also clear that political protection was an important consideration for research into the motion of projectiles, with safety coming to those whose mathematical models helped their patrons to remain the victors.

• Clagett, Marshall. Archimedes in the Middle Ages. Madison: University of Wisconsin Press, 1964. Traces the mathematical stream from antiquity to the period in which Tartaglia started to work.
• DiCanzio, Albert. Galileo: His Science and His Significance for the Future of Man. Portsmouth, N.H.: ADASI, 1996. Brief sketches of the figures leading up to Galileo’s theory of motion, including Tartaglia and Benedetti.
• Drake, Stillman. Galileo at Work. Chicago: University of Chicago Press, 1978. Indicates how Galileo would have come across the works of Benedetti and Tartaglia.
• Drake, Stillman, and I. E. Drabkin, eds. Mechanics in Sixteenth-Century Italy. Madison: University of Wisconsin Press, 1969. The only book-length collection in English on this revolution in mechanics, with extensive translations from Tartaglia and Benedetti and an excellent essay by Drake.
• Field, J. V. The Invention of Infinity: Mathematics and Art in the Renaissance. New York: Oxford University Press, 1997. Investigates how practical considerations entered into theoretical mathematics during the sixteenth century.
• Swetz, Frank, et al., eds. Learn from the Masters. Washington, D.C.: Mathematical Association of America, 1995. Includes an essay by Swetz on Tartaglia’s modeling the flight of a cannonball with more mathematical details than are available elsewhere.

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c. 1478-1519: Leonardo da Vinci Compiles His Notebooks

Beginning 1490: Development of the Camera Obscura

c. 1510: Invention of the Watch

1530’s-1540’s: Paracelsus Presents His Theory of Disease

1543: Copernicus Publishes De Revolutionibus

1543: Vesalius Publishes On the Fabric of the Human Body

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1580’s-1590’s: Galileo Conducts His Early Experiments

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