The Pendulum Swings again: A Mathematical Reassessment of Galileo’s Experiments with Inclined Planes

After over 300 years of scrutiny, the subject of Galileo continues to be pursued with unabating intensity. Dava Sobel’s Galileo’s Daughter points to the popular interest in the man and his legacy. The Catholic Church, understandably interested in dispelling the notion that its censure of Galileo centuries ago is proof positive that religious faith and science as well as ecclesiastical authority and free pursuit of scholarship are irreconcilable, continues to offer explanations. New books, articles and conferences probe both in breadth and in depth the magnetic field charged by Galileo, science, and the Church. Galileo’s analysis of the physics of motion has also received considerable attention. In particular, a great deal has been written during the past thirty years about the structure and objectives of three experiments with inclined planes. Galileo had carried them out in Padua and recorded them in his working papers. The assessments of the three experiments differ widely in points of detail, but all regard them as sophisticated, ingenious, and remarkable. This article presents a new critical study of these experiments. Its conclusion is that one of the experiments is indeed a success, but that the other two fail and are abandoned because Galileo did not have a firm enough grip on the underlying physical principles and mathematical relationships. 1. Galileo on Motion. Galileo’s journey of discovery of the laws of motion is lengthy, twisted, and anything but smooth. We recall it only briefly. It is described in [10], [11] and, in compelling detail, in [38]. Galileo’s early efforts in Pisa (1589-1592) to understand the phenomenon of motion are the subject of the De motu manuscripts. Central to Galileo’s explanations are two underlying concepts, a basic one of uniform specific speed of fall and the auxiliary one of an impressed force, that account for the nonuniform motions actually observed. The uniform specific speed of the object depends on the medium in which the object moves and is determined via Archimedean hydrostatics by the difference in the densities between object and medium. The impressed force is something that is put into an object by an external mover. Once imparted, this impressed power decays I wish to thank Ernan McMullin for much information about Galileo and his scientific pursuits, and in particular for drawing my attention a number of years ago to the research of Stillman Drake on Galileo’s working papers. Thanks also go to Noel Swerdlow for very useful insights, and to Neil Delaney, David Kirkner, and Timothy O’Meara for many stimulating discussions about Galileo and his work. gradually and the speed of the body changes steadily until the impressed power is completely dissipated and the object assumes its natural constant speed specific to the medium in which the motion takes place. Galileo’s thinking during this time is still heavily influenced by Aristotelian and medieval elements. Only towards the end of the De motu does Galileo begin to accept the possibility that acceleration may be a fundamental feature of fall. During the Padua period (1597-1610) Galileo’s understanding of motion undergoes transition. His 1604 letter to Fra Paolo Sarpi provides important testimony: ”the spaces passed in natural motion are in proportion to the squares of the times taken, and consequently that the spaces traversed in equal successive time intervals are to the odd numbers ...” Diverted by his interests in astronomy and the well documented conflict with the Church, Galileo does not present the final synthesis of his theory of motion until 1638. The Discorsi [2] is the exposition of a learned, four day long conversation among Salviati who represents Galileo himself, Sagredo an open minded supporter of the new science, and Simplicio, an adherent of the old Aristotelian point of view. Days three and four of these discussions focus in vernacular Italian on the treatise De motu locali a book within a book in scholarly Latin that is Galileo’s own systematic treatment of motion. It is intended to be a clear and rigorous presentation in final form. Errors and missteps from earlier stages of Galileo’s thinking are visible in the probing comments of the protagonists. The discussion analyzes the motion of objects undergoing constant acceleration, both in the situation of free fall and along inclined planes. Galileo derives his conclusions deductively, often with geometric constructions. He expresses quantitative relationships between time, distance, velocity, and acceleration, in terms of proportions; indeed, he uses only proportions of magnitudes of the same kind, for example, distance to distance, velocity to velocity, but not distance to time. In particular, he does not have the real number system and decimal notation at his disposal and does not formulate his conclusions in terms of equations involving variables and constants. (The systematic development of all this was only in its infancy at this time.) Within his context, Galileo arrives at the following basic insights. 1. All bodies falling in a vacuum do so with the same constant acceleration. For a body falling from rest, the speed is proportional to the elapsed time. This is so both in the situation of free fall and for balls rolling on an inclined plane. 2. The law of fall, namely, that the distance covered by a body moving from rest (again, either in free fall or rolling on an inclined plane) is proportional to the square of the time of the motion. 3. The trajectory of a projectile has parabolic shape.