Kevin T. Kelly
Department of Philosophy
Carnegie Mellon University
Several ancient traditions lead to Galileo's law of accelerated motion (following Drake and Drabkin 1969)
Aristote: Questions on Mechanics. By early Aristotelians, but not by Aristotle himself. Qualitative, dynamical. Levers, oars, rudders, wedges, pulleys, vortex motion, and balance.
Archimedes: On Plane Equilibrium and On Bodies in Water. Rigorous geometrical treatment of statics problems.
Hero: Pneumatics, Automata, Mechanics, Collections. Theoretical mechanics.
Technological: Vitruvius, Oribasius. construction and use of actual machinery in mining and construction.
By the 13th century, an original Medieval approach had developed.
Medieval Weight Science: Balance, steelyard, hydrostatics. Problems stimulated by commerce in the 14th c. Dynamical, Aristotelian principles explored in Archimedean, axiomatic form. Aristotle viewed the lever in terms of speed and force. The medievals analyzed it in terms of weight and vertical displacement (i.e., work).
Philosophy of Motion: universitiy commentaries on Aristotle's Physics. Aristotelian principles developed into disagreement with particular aristotelian views. Graphical representation of velocity and acceleration, instantaneous velocities, mean speed theorem, theory of impetus.
University: Latin commentaries on Aristotle. Mathematics has low status.
Technology: practical problems required quantitative focus. Vernacular literacy in rising technical class.
Typical dissemination pattern: Archimedes' On Bodies in Water
1269: William of Moerbeke produces Latin translation.
1543: Tartaglia publishes corrupt manuscript of Moerbeke's translation in Venice.
1544: Greek text published in Basel.
1551: Tartaglia publishes Italian translation and commentary.
Self-taught son of a post rider. Private geometry tutor. Published Archimedes and Euclid in Italian, solved cubic equations.
1537: Nova Scientia. Theory of projectile motion based on mixture of Aristotelian ideas. Projectile problem was posed bya professional artilleryman. He suppressed publication for moral reasons. Projectile motion composed of segments of (approximately) straight and circular motion. In the following explanation, the speed makes the canonball light, causing it to float in the air until friction slows it down.
The great speed is the true cause that holds the ball's motion to straightness, if that is possible. And similarly, the loss of speed in the air is the true ause that makes it tend and decline in its motion curvedly toward the earth, and the more it loses its speed there,thegreater becomes its declining curvature. And all this happens because for any heavy body driven violently through the air, the faster it goes, the less heavy it is in that motn, and hrefor the straighter it oes trough the air ecause the air more easily sustains a body the lighter it is. Also, in making its effect in such motion, it assumes much greater heaviness than its own, and therefore the faster a heavy body goes in violent motion, the greater effect it makes.... [Questions and Inventions, 1546, question 23].
Upper class, studied geometry with Tartaglia, mostly taught by father. Court mathematician in Parma and Turin.
1552: Archimedian theory of free fall: two bodies of the same density will fall at the same speed irrespective of weight. Natural motion in a medium is proportional to the difference between the specific gravity of the body and that of the medium. This eliminated the Aristotelian paradox of infinite speed in vacuum. Resistance is different from natural buoyant free-fall and is proportional to surface area.
1554 Had Galileo's conceptual argument against Aristotle's proportionality of speed to weight: Take two cannonballs of the same size and material (Drake and Drabkin; p. 158).. They fall at the same rate. Now connect the two with a fine filament of negligible weight. Now we have one object of twice the weight, which should fall twice as fast. But that is absurd (imagineyourself cutting the wire).
Sources
Stillman Drake and I.E. Drabkin (1969) Mechanics in Sixteenth-Century Italy. Milwaukee: University of Wisconsin Press.
University man, knew medieval philosophy of motion. Also hung around with the engineers, so familiar with Archimedes.
1590: De Motu: unpublished dialogue written prior to professorship. Gives Benedetti's in principle argument against Aristotle's free-fall law (Drake and Drabkin; p. 371). Presents a buoyancy theory of fall like Benedetti's. Acceleration is an accident due to bleeding off of impressed force (like cooling off) (Drake and Drabkin; p. 372). To see the equable motion proportional to the difference of specific gravities of the projectile and the medium, one must wait for the impressed force to bleed off completely. Notice how Galileo's Platonic rhetoric against appearances serves his early theory:
The fact that it is contrary to the view o man doesnot concern me, as long as it is in harmony with reason and experience, even though at times experience seems rather to point to the opposite. For, if a stone falls from a high tower, its speed is observed always to be increasing. But this happens because the stone is very heavy in comparison with the medium through which it moves, i.e., the air. Since it starts its fall with an impressed force equal to its weight, it starts, of course, with a great deal of impressed force; and the motion from the tower's height is insufficient for the using up of all this impressed force. ... But if we were to take an object that has weight, but whose weight did not so very far exceed the weight of air, we should then surely see with our own eyes that, a little after the begining of its oto, the body would move uniformly....
Galileo probably dropped this theory when he arrived at the law of inclined planes and related it to free fall as a limiting case. Then the "accidental" and the "essential" exactly switched, but the Platonic rhetoric in favor of abstraction continued. Equable motionin a medium was now accidental and violent. Acceleration was natural. This is one of the most amazing reversals in scientific history.
How does the new theory work? Let
Consider the case in which v(t) does not depend on t. Then we say v(t) = v. Everybody knows that distance travelled in this case is rate times elapsed time: s = vt. Going forty miles per hour for two hours, one travels eighty miles.
In other words, s is the area of a rectangle with base t and height v. This is the same as saying that s is the area under the graph of v(t) in terms of t.
Now suppose that the velocity changes linearly through time: v(t) = kt. Abstracting from the preceding, degenerate case, we say that s is the area under the graph of v(t) in terms of t. But the graph of v(t) = kt is just a line through the origin with slope (rise over run) = k. Thus, the area under the curve up to t is just the area of a right triangle of altitude v(t) and base t, so s(t) = 1/2(vt). But v(t) = kt, so we have
s(t) = 1/2ktt = (k/2)t^2.
What about the business of the distances travelled in successive times being proportional to the odd numbers 1,3,5,...? Using the above formula, we have
s(1) = (k/2)
s(2) -s(1) = (4k/2)-(k/2) = (3k/2)
s(3) - s(2) = (9k/2)-(4k/2) = (5k/2) ...
Choosing k/2 as the constant of proportionality, we have the ratios 1:3:5:...
Without electronic timekeeping and measurement devices, this theory could not be tested accurately using free-fall. Galileo tested the theory using inclined planes equipped with bells separated by odd number ratios. The bells rang at even time intervals, as measured by an appropriately adjusted pendulum, providing an accurate test. Incidentally, Galileo invented the pendulum-regulated clock, revolutionizing the science of mechanics.
The geometry of this argument was already worked out already in the 14th century by the university scholar Nicole Oresme, in his commentary on Aristotle's Physics. True to his Aristotelian calling, Oresme presented this rule as an account of change in general, and did not bother to correct Aristotle's account of free-fall in particular.
Notice that neither Oresme nor Galileo (writing 300 years later) had a precise definition of instantaneous velocity and acceleration in terms of derivatives. That had to wait until Newton. Also, Galileo's account did not account for varying acceleration through distance, which is necessary to unify terrestrial free-fall with planetary motion. That also had to wait until Newton.
Galileo recognized that what he previously thought was natural motion in a medium according to the buoyancy theory was actually the effect of increasing air resistance with speed, so that the body would eventually stop accelerating. Thus, the natural/violent distinction exactly reversed in his mind.
Galileo accounted for projectile motion by composing uniform rectilinear motion with free fall to produce a parabolic trajectory. His argument for the uniform horizontal motion is ingeneous. A ball will roll up a ramp as far as it rolls down. By making the second ramp shallower and shallower, the ball rolls farther and farther. Making the ramp completely horizontal must then, as the limiting case, yield an infinitely long motion, ignoring "accidental" friction. Galileo made great use of these a priori arguments from continuity.
It is not entirely clear from Galileo's writing what he took this uniform rectilinear motion to be. He knew that a genuine rectilinear motion tangent to the earth's surface would force the body to roll uphill, away from the Earth's center. Thus, the rectilinear inertia seems to be circular inertia cocentric with the Earth. This is just the "natural circular motion" of Copernicus applied to projectile motion. Such circular inertia explained why objects are not left behind by a moving Earth. Galileo was thoroughly committed to Aristotle's distinction between natural circular and rectlinear motion. His theory of trajectories made them a composition of the two motions. The only correction to aristotle was (1) to impart heavenly circular motion to cannonballs and (2) to correct Aristotle's theory of free-fall with the medieval account of acceleration!
Galileo was opposed to Kepler's physical approach to planetary motion which deviated from the Copernican circular scheme, as we shall see. For an unambiguous statement of rectilinear inertia, one must wait until Renee Descartes.