Galileo Galelei (1564-1642)   Historical Sketch

When Galileo's name is mentioned, many people think only of astronomy, his advocacy of the Copernican system, the Inquisition, his trial and prison sentence, and so on. And some people think of those things in addition to Galileo's scientific prowess outside of astronomy, but how many think of his contributions to mathematics?

Galileo Galelei was born in 1564 near the ancient city of Pisa when it was under Florentine rule. His father was a music teacher, and Galileo was his mother's first child. Galileo was sent to a monastery for his early education, and the youngster quite liked the monastic life. However, his father wanted the boy to enter the field of medicine, and so Galileo entered the University of Pisa for that purpose. But he was indifferent to such studies, preferring courses in mathematics and what is today called physical science. From 1582 to 1585, Galileo was a medical student in name but a mathematics student in fact---studying Euclid and Archimedes---and he left the University without completing a degree. Studying mathematics on his own and through correspondence with established mathematicians, and trying unsuccessfully to be appointed to teaching positions, Galileo was appointed in 1589 to the chair of mathematics at the University of Pisa in 1589, when he was but 25. Three years later, Galileo was appointed to a much more lucrative professorship of mathematics at the University of Padua, where he taught Euclidean geometry and geocentric astronomy to medical students (who needed such as an underpinning for the astrology they would use in medical treatments). During these early years at Pisa and Padua, Galileo studied motions, including the discovery of the isochronal behavior of a pendulum (the time expended for an oscillation of a pendulum is the same regardless of the amplitude of the swing), inventing the hydrostatic balance to measure the specific gravity of solids, demonstrating that bodies of different weights fall with equal velocity (the story that he dropped weights from Pisa's leaning tower are probably apocryphal), showing that the path of a projectile is parabolic (if air resistance is disregarded), built a telescope and observed mountains on the Moon, the satellites of Jupiter, existence of sunspots, the phases of Venus, and discovered that the Moon shines from reflected sunlight. His telescopic discoveries were rewarded with an appointment as mathematician extraordinary at the University of Florence. In 1637, Galileo became totally blind.

You can see that Galileo was mainly concerned with physics and astronomy, and he considered himself the creator of the subject of dynamics, since he had given to the study of dynamics a mathematical precision that had been deficient. His work with astronomy brought to life one of the ancient conics: the orbits of planets as realizations of the ellipse. He was also taken by ideas of the "infinitely small" and the "infinitely large," but he had trouble with the idea that there are as many square integers as there are integers.

Though much of Galileo's work could have been simplified by calculus as we know it today, he did use thought processes that could be labeled, maybe, precalculus. Because of this, astrologers of his time would likely have made much of the fact that Galileo died on the day that Newton was born.

1. Present an argument to show that there are exactly as many square positive integers as there are positive integers.
2. Describe other infinite collections of numbers that seem unequal in "size" but are not.
3. Galileo thought of a circle as a polygon with infinitely many sides of infinitesimal lengths. What do you think about this?