Text: # in naturally accelerated motion, velocity increases uniformly with time, that the distance traversed is proportional to the square of the time elapsed and thus, as Galileo says, the distances traversed in successive seconds are as the odd number series 1, 3, 5, 7Š. ------------------- Galileo Galilei Commentary on the text 1. That is, gravitationally accelerated motion. 2. "Equable motion" refers to constant speed. 3. Note the contrast made between philosophical speculation about phenomena (the Aristotelian model of science) and Galileo's procedure of testing hypothesis with experimentation. 4. "Equable motion" occurs when equal distances are covered in equal times; we denote the ratio of distance by time--which is therefore constant throughout the motion of the object regardless of how long a distance or time is considered--the speed of the object. In modern terminology, velocity (which is directed speed, so that an object has negative velocity if it moves in an opposite direction) measures the rate of increase (or decrease) of distance over time. In symbols, since equable motion occurs when velocity is distance divided by time, v = d/t. 5. "Natural acceleration", that is gravitation, corresponds to uniform (or constant) acceleration, in which velocity increases at a constant rate. In particular, this is different from "equable motion". In modern terminology, acceleration measures this rate of increase (or decrease) of velocity over time, so for uniform acceleration, this is measured as velocity divided by time: a = v/t. Galileo goes on to explain that velocity increases with time in this situation, so that v = at. In most descriptions of this phenomenon today, we denote by g the constant acceleration due to gravity, and write v = gt. These ideas are formulated in the defintion he proposes below. 6. Sagredo, the Aristotelian philosopher, is prone to "picturing" things in his mind, in constrast to the progressive scientist, Salviati, who makes frequent recourse to experimentation to support his claims. 7. Sagredo is troubled by the relation v = gt. If t is very small ("instants of time closer and closer to the first [instant] of its moving from rest"), then since g is constant, so must v be very small ("there will be no degree of speed, however small ... that the moveable will not be found to have"). What is of interest here is his--and by extension, Galileo's--assumption that time is "infinitely divisible". This willingness to include the notion of infinity, especially the infinitely small, also marks Galileo as a modern thinker. Salviati's response to these comments is an attempt to put at ease any worry that an object can have an arbitrarily small speed. 8. Shades of Zeno! Simplicio brings up the same objections that Zeno does in the Achilles paradox. 9. Salviati tries to resolve the objection by pointing out that time is continuous, not discrete. 10. Here Sagredo describes the Aristotelian understanding that the object thrown up into the air has been imparted with a force that is "consumed" as it rises until it is overpowered by the natural force of its heaviness to fall to the earth. 11. Note the critical jab levelled at the Aristotelians, describing their speculations as "fantasies". He continues by arguing for testing these hypotheses against real phenomena. 12. Sagredo is now proposing that the velocity of a falling object increases with distance, so that v = kd for some constant k. Salviati will admit that "our Author himself did not deny to me", that is, that this proposition was once believed by Galileo to be true. But he will demonstrate how this supposition is false. Then, poor Simplicio chimes in late by saying that he thinks it is true also. 13. If velocity increased with distance, then the speed of an object falling 4 bracchia would be double the speed after it fell only 2 bracchia. But we consider the time of fall to be made up of (infinitely many!) instants of time Dt, each of which equals the ratio v/(d' -d), v being the nearly constant speed of the object as it covers the interval d' -d between "distance markers" d and d', then since the velocity of the object between 2d and 2d' would have to be 2v, the ratios would be equal. Hence, the object would have to cover the first 2 bracchia in the same time as it covers the 4, which would imply that it instantaneously moves through the last 2 bracchia, contrary to experience! 14. This is the so-called mean speed law, except that it is usually expressed in terms of the distances traversed rather than the time of travel, as here: the distance traversed by a uniformly accelerated object over an interval of time is equal to the distance traversed by an object moving with constant speed over the same interval of time whose speed is equal to half the greatest speed of the first object. 15. This significant passage tells us that Galileo has formulated a geometric model to represent the physical phenomenon he is describing. The figure he draws here implicilty depicts a coordinate system in which the t-axis is the vertical line AB (time increasing downwards) and the perpendicular line through A is a v-axis (velocity increasing to the left). As time proceeds from the start of the motion of the object, its speed increases linearly, producing line AE as its "graph". Each horizontal line segment drawn from AB represents the speed of the object at a different time during its fall. Galileo views the triangle ABE as describing the motion of the falling object. Similarly, the segments of constant length AG produce the "graph" GF (for Galileo, the parallelogram AGFE) to represent the motion of an object moving with constant speed over the same interval of time. 16. Galileo is equating the areas of the triangle ABE and the parallelogram AGFE, each viewed as an "aggregate of all parallels". That is, he conceives of the triangle as being made up of infinitely many line segments, all parallel to BE, each representing a different "momentum of speed" at a distinct and particular "instant of time", and the parallelogram in a similar way as the aggregate of all the parallel segments between AG and BF. These segments represent the motion of the object at a particular speed v over an infinitesimally small interval of time Dt. Viewing the speed as constant over that small interval, the distance Ddcovered in that interval obeys the rule Dd = v·Dt. Of course, this also corresponds to the area of an infinitesimally thin parallelogram (actually rectangles) with "width" Dt and height v. Since the entire triangle ABE is the aggregate of the infinitesimally thin rectangles, the total distance covered by the object from start to finish is the area of ABE. Similarly, the area of AGFE represents the total distance covered by the other object moving at constant speed. Galileo notes that the two regions have equal area, hence they cover the same distance. This is true regardless how speed varies with time: if v(t) is a function that represents the speed of an object at time t, where t varies between the times t = a and t = b, then the interval from a to b is made up of infinitely many infinitesimally small "instants" Dt, during each of which the speed is essentially constant. The distance traversed at each instant is Dd = v·Dt and is represented geometrically as the area of an infinitesimally thin rectangle at t of length v. The total distance covered by the object over the entire motion is then equal to the total area of the various rectangles, i.e., the area under the curve v = v(t) between t = a and t = b. Therefore, (distance travelled between t = a and t = b) = . 17. So if an object under uniform acceleration covers distances d and D in times t and T respectively, If we call the common constant ratio in the second proportion k, then we find that d = kt2 for all times t. Galileo's Theorem II is the equivalent of the statement that the distance traveled by the object increases as the square of the time. Indeed, we can say more for the object in free-fall: since the acceleration is constant, v = gt, and Theorem I applies to assert that the distance traveled between time t = 0 and t = 1 is the area of the triangle with base 1 and height v = g·1 = g. That is, d = (1/2)g. On the other hand, d = k·12 = k, so k = (1/2)g. That is, the distance travelled in time t is d = (1/2)gt2. 18. This is the same "graph" as in the statement of Theorem I. 19. This is a reference to the fact that 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16, and that in general, the sum of the first n odd numbers equals n2. That this is true can be deduced from the fact that the difference between succesive square numbers is (n+1)2-n2 = 2n + 1, the nth odd number. See the exercises. http://cerebro.cs.xu.edu/math/math147/02f/galileo/galileo3rdnotes.html See pdf in Galileo folder.

See Also: GRAVITY; ACCELERATION

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