# Galileo's Background on Trajectories

It is impossible to speak of Galileo's experiments on projectiles without first ex amining the prevailing science of the time, which was dominated strongly by Aristotle. Even Bonamicio, Galileo's teacher, was a subscriber to Aristotle's ideas (Koyre, 10). Aristotle divided motion into two categories: that of natural motion, which is the motion of a body to its natural place, and violent motion, which is caused by a force. Natural motion occurs because the substance of a body seeks to bring it to its natural place (for example, a rock's natural place is the Earth and thus falls towar ds the Earth) (Koyre, 6). Violent motion, though, requires a continuous action of an external motive force. There is therefore no such thing as an action at a distance. Once the motive force is removed, motion ceases.

The problem with this theory is that it is contradicted by the everyday occurrence of throwing a rock. Once the rock leaves the hand, it should stop, but instead it goes on until hitting the ground (Koyre, 7). Aristotle explains this by introducing the concept of the medium, such a s air or water, assisting the violent motion. Parts of the surrounding medium take up the place to the rear of the moving rock and push it along. But why does the rock eventually stop? Aristotle would say that because the moving body impresses an unnat ural force on the medium, the medium then opposes as well as assists the motion. This rather illogical explanation was gradually realized to be innaccurate, and scientists had to struggle for a new explanation.

On the road to that new explanation we find Niccolo Tartaglia, the founder of modern gunnery. He published his theories in 1531, which were a great help both to theorists and to gunners (Hall, 38). He also divided motion into violent and natural, and claimed that any trajectory consisted of rectilinear and curved portions. The violent motion at the beginning of the blast is in a straight line at an oblique angle. The natural motion of the ball toward the Earth takes over and this transition results in a curve. He denied the fact that the violent motion at an oblique angle would at all times be completely straight, instead saying that because of gravity working on it constantly, no part of the trajectory would be a completely straight line, but would appear very similar to one (Hall, 40).

Going even farther in denying Aristotle's explanation of trajectory motion is Bennedetti, who thought Aristotle's theory was worthless (Koyre, 21). Roughly a contemporary of Galileo, Bennedetti claimed the medium hinders, not aids, the motion. He states that the medium cannot be both the resistor and the motive force (Koyre, 21). He also introduced the ideas of impetus, which became very important to Galileo as well. Impetus is a quality transferred to a body which enables it to have motion. The longer the body is impressed with this impetus, the more it acquires (Koyre, 22). Galileo clarified and made the impetus theory more coherent.

Using the example of heat and sound, Galileo explained impetus. For example, once you transfer heat to a body, it is hot and remains so until the heat has dissipated just as the motion remains until the medium resists and dissipates it (Koyre, 28). If you strike a bell, it is given a sonorous quality that is transferred from the hand to the bell, though the sound continues even after the hand is removed. The sonorous quality is not natural to the bell, but is rather impressed upon it.

But what did Galileo himself know of parabolic trajectories at the time he performed the experiments we examined, r oughly around 1609? He knew the law of falling bodies; that they fall with a uniformly accelerated motion, though he didn't know the rate or the reason (Hall, 86). He didn't buy into the rectilinear and curved path proposed by Aristotle, believing that gravity prevented the motion from ever being in a straight line (Koyre, 196). However, it is unclear whether at this time he discovered that trajectories follow a parabolic path, althought many influential historians, including Stillman Drake, believe so . In his Dialogue Concerning Two World Systems he proposes the idea that the path would follow the circumference of a circle, but six years later in his Discourses Concerning Two New Sciences he states conclusively that trajectories do inde ed follow parabolic paths (Koyre, 199).

He was not, however, the first to publish mathematical proof of parabolic trajectories. His former student, Bonaventura Cavalieri, a Jesuit, published in 1632 a book called Speccio Ustoria, where he mat hematically demonstrates them (Hall, 86). Galileo was hurt and upset by this, claiming that Cavalieri had stolen his thunder, his careful research of the last 30 years or so of his life. Cavalieri later apologized for not telling Galileo about the publi shing beforehand, and the two were eventually reconciled.

The vagueness of the notes on Galileo's experiments regarding the exact order and purpose of the parabolic trajectory experiments that tested both the laws of motion and the parabolic path itself were never clearly outlined. As a result, there have been numerous discrepancies amongst historians of science as to the exact order, methods, and purpose of each experiment. What we have compiled is a brief summary of the major ideas of the orde r, methods, and purpose of the experiments of four of the more prominent Galilean scientific historians; Stillman Drake and James MacLachlan, R.H. Naylor, and David Hill. Nevertheless, all of these historians agree that, unlike previous skeptics who ques tioned Galileo's rights to the actual discovery of parabolic trajectory, Galileo discovered parabolic trajectory by 1608 and proved it mathematically by 1609. (Drake and MacLachlan, p.106) In addition to the analysis of other historian's interpretations , we have included our own impressions on Galileo's experiments. Before going through all the arguments we will first review the basic experimental design of the three main folios; f.81, f.114, and f.116. (see diagrams) Folios 81 and 114 are both believed to have used similar apparatuses which produced reproducible oblique angles. Folio 81 was probably run by keeping the same angle and varying the preprojection roll distance and the vertical drop of the ball from the end of the ramp. Such data could have provided evidence for geometric proof of the parabolic trajectory. Folio 114 was probably designed with a constant angle and vertical drop distance while only varying the preprojection roll in hopes of confirming Galileo's new hypothetical speed la w (will discuss later). Finally, f.116 most likely used an inclined plane with a horizontal deflection at the bottom that forced the ball to leave the ramp parallel to the ground. This experiment could have used the same angle and vertical drop while v arying the preprojection roll in hopes of confirming the double distance and the speed law.

Drake asserted that, by 1604, Galileo had developed the time squared law in which "the free fall the time elapsed from rest are as the smaller distance is to the mean proportional between two distances falling." (Drake, p.294) Drake's view of Galileo's experiments placed f.81 first followed by f.116. According to Drake, Galileo was trying to confirm the double distance rule in f.116, that in equal interva ls of time the distance traveled in the horizontal will be double the initial free fall. (Naylor, "Gallileo's theory," p.568) Drake asserts that discovering parabolic trajectory may not have even been an objective of these experiments and it was only thr ough observing the experiments that Galileo noticed the parabolic path. (Drake and MacLachlan, p.103) Unfortunately, Galileo could not conclusively prove the double distance rule as the horizontal travel fell short of what was expected because he was una ware that friction retarded the roll by 2/7. So the acceleration factor of 5/7 applies to the acceleration of the rolling body compared to a body in free fall. (Drake and MacLachlan, p. 106) Puzzled by the results, Drake claims that Galileo tried to rectify the situation by conducting f.114, where there was no horizontal deflection at the end of the run. (Drake and MacLachlan, p.109) In order to prove parabolic trajectory, Drake believed that Galileo needed to know the law of free fall, that distance f allen is proportional to the square of the time elapsed, and the restricted principle of inertia which stated that the relation between motion imparted to a body and the behavior of the body following an impulse. (Drake and MacLachlan, p.102) Drake goes so far to state that f.114 was designed to establish the restricted principle of inertia while f.117 was used to establish the parabolic curve (Drake and MacLachlan, p.107).

Naylor believes that the earliest trajectory experiments were performed i n 1605 when Galileo still held that velocity increased with the square root of the distance fallen. He attacks Drake's presumption that Galileo had not considered parabolic trajectories before f.81 as he claims Drake based his analysis on poorly supporte d assumptions and not the experimental procedure itself. Furthermore, he insists that the experiments run on f.116 required some previous knowledge about parabolic trajectory. Naylor was the historian on which we based our f.81 experiment. He reconstru cted f. 81 using an inclined plane at angles of 20, 7, and 3.5 and believed that the experiment proved that, geometrically, the trajectory was parabolic in form. (Naylor, "Galileo's theories," p.553) It was during this time when Galileo still believed th at speed was directly proportional to free fall time and it was before Galileo knew that velocity was proportional to the square root of the distance. (Naylor, "Galileo's theory," p.552) However, Galileo's new discoveries of the conservation of momentum and parabolic trajectory led to the development of f.114, f.107, and f.117 to try and relate the recent ideas back to 1604 ideas that velocity was proportional to distance and that horizontal speed was proportional to the vertical drop. To Galileo's surp rise, the new discovery that horizontal motion is conserved in projectile motion made his previous notion that velocity increased with distance incorrect. (Naylor, "Galileo's theory," 561) The old hypothesis was replaced by the belief that average veloci ty increased due to increases in vertical momentum. (Naylor, "Galileo's theory," 563) Naylor believes that the goal of f.114, an experiment he believed to be closely related to f.116, was to study the double distance rule while f.116 was designed to con firm the double distance law and test the double velocity laws as 116 was a direct extension of the results in f.114. (Hill, "Dissecting," p.658) Naylor agrees with Drake on the fact that Galileo could not prove the double distance rule, despite its valid ity, due to the underestimation of the effects that rolling down a ramp would have in comparison to actual free fall. (Naylor, "Galileo's theory," p.569)

Hill was the last historian we read and he, of course, has his own ideas as to the order and s ignificance of Galileo's major projectile studies. Hill maintains that f.114 followed shortly after f.81r and was probably conducted at the same time using the same equipment in order to test different principles. (Hill, "Dissecting," p.658) Folio 116 w as a work that was performed later as a result of previous trajectory experiments. Hill's analysis of f.81 criticizes Naylor's approach of using three different inclines because, in order to replicate Galileo's numbers, one of Naylor's inclined planes ha d to be curved, something that seemed unnecessary for Galileo to do in an experiment. Furthermore, Naylor's results do not create a situation in which all the curves imply semi-parabolic trajectories. (Hill, "Dissecting," p.650) Hill, on the other hand, proposes that Galileo might have performed the experiment using different inclines each adjusted to a 250 punti horizontal distance for the lowest vertical level. (Hill, "Dissecting," p.649) Hill went so far as to analyze Galileo's data and determine th e three most likely angles that Galileo used for the experiment. (Hill, "Dissecting," p.656) His assessment of f.114 upholds the belief that Galileo maintained a constant angle and vertical drop while changing the length of the preprojection roll. (Hill, "Dissecting," p. 658) Hill also disagrees with Drake who performed the experiment using a thirty degree incline which produced preprojection roll ratios that Hill argues were obscure and should have been written on the folio if they were actually used. (Hill, "Dissecting," p.659) Furthermore, Hill questions why Galileo changed the height in f.114 to 500 punti if Drake and MacLachlan were correct in stating that 114 followed 116. (Hill, "Dissecting," p.658) Although Naylor was also able to make a func tional reproduction of f.114, he used obscure preprojection roll ratios and the question remains as to why there was no indication of a connection between f.114 and f.116. (Hill, "Dissecting," p. 661) Hill rectifies all the inconsistencies in Naylor's an d Drake's analyses by showing that f.81 and f.114 are related in that they each have the same preprojection ratio of 1:2:3:4:5:6:7, one that wouldn't have to be written down. In addition, he supplies compelling evidence for the close relation of experime nt 81r and 114v by showing that it is possible to use the 81r procedure to reproduce the 114v data. (Hill, "Dissecting," p.661) Hill believes that 114v was an initial test of Galileo's new speed law, altered from the 1604 version, in which speed is propo rtional to the square root of the distance covered. Unfortunately, Hill claims that the oblique angle used in f.114 caused the results to be short of the speed law by approximately ten percent. (Hill, "Dissecting," p.662) Eventually, Hill maintains that Galileo must have recognized the trend of the f.114 data and assessed the problem of the oblique angle, which subsequently led to experiment 116v. In order to alleviate the problems with the oblique angle, 116v was designed to produce a projection that lacked horizontal acceleration and this was done by deflecting the ball horizontally before leaving the inclined plane. (Hill, "Dissecting," p. 662) The results recorded on f.116 confirm the hypothesis in that they provide strong confirmation of the new speed law, something f.114 could not do. Hill points to Galileo's knowledge and use of the law of chords as a means to derive this new speed law. (Hill, "Galileo," p.283) Furthermore, Hill relates f.116 back to f.81 by realizing the fact that although n o evidence exists to prove it, f.116 like f.81 provides proof that semi-parabolas can be produced from horizontal projections. (Hill, "Dissecting," p.664) While Hill and Naylor both agree that the chief focus of f.116 was to confirm the new speed law, D rake believed it only led to the idea of a speed law, and actually, was a test for horizontal inertia. (Hill, "Dissecting," p.662)

After having reviewed the assessments we feel that Hill has created the most sensible evaluation based on Galileo's di agrams. Drake's hypothesis is weakened by the discrepancies of the height differences between f.114v and f.116v and the definitive proof of the speed law in 116v. This leads to question why Galileo would retest similar principles in f.114v assuming that Galileo did actually discover the speed law and the double distance law with f.116v before doing f.114v. Naylor's assessment is weakened by the use of curved ramps to reproduce data, and the poorly supported hypothesis that f.114v and f.116v are closely related. Furthermore, logic dictates that most of the parabolic trajectory theories that Galileo developed were required for the development of f.116v. 116 data provides compelling evidence that 116 could only be produced after the necessary discoverie s in earlier experiments. Hill's evaluation is most convincing due to his ratio argument of the ball drops on f.116v, and his proof of the close relation of f.81 and f.114v. However, the fact remains that all these historians support the idea that Gal ileo, experimentally, proved the speed law and the parabolic trajectory of falling objects.