Galileo Galilei

 

Scholium: infinitesimals and instantaneous rates of change


    We have seen that Galileo understood the area under the graph of the velocity function v(t) between t = a and t = b to represent the distance traveled by the object in the time interval a < t < b(see note 16 from the previous commentary).  Since the area under the curve can be represented by an integral, we have the alternative formulation

=  [distance traveled in the time interval a < t < b]

Galileo justified this by conceiving of the region under question as made up of infinitesimally thin rectangles, essentially line segments, over which the velocity of the object is essentially constant.  In each of these intervals the distance covered satisfies Dd = v(tDt, where Dt is the infinitesimally small interval of time and Dd the corresponding increment of distance traveled; hence, adding all these increments gives the total distance traveled.

    What is significant about this process is that implicit in this explanation is a representation of the velocity of the object at a particular moment in time.  The claim made by Sagredo near the beginning of the diaglogue that "time is infinitely divisible" shows Galileo's readiness to associate a velocity to each moment of time.  This is clear from the geometric model he provides of the situation.  However, it was not possible for him to measure the velocity of an object at a particular time.  Modern devices like a speedometer were not available to him.  What he could measure was the average velocity of the object over an interval of time: if at second t1, the object had traveled a distance of d1 feet and at second t2 it had traveled a distance of d2 feet, then between t1 and t2 the object had moved on average at a speed of

feet per second, which is to say that at moments within this interval, the object may have moved faster or slower than this average speed ratio calculates.  However, Galileo is prepared to posit something further, that at each moment there is a quantity that measures the speed of the object then and just then, not over an interval of time.  At each point on the time axis of his "graph" he erects a line segment to the curve representing the velocities of the object over the duration of its motion; the length of each segment is the measure of the instantaneous velocity of the object at that moment.  He views this segment as an infinitesimally thin rectangle with dimensions v(t), its speed at time t, and Dt, an infinitesimally small duration of time.  Because the interval of time here is so small, he concludes that the velocity throughout this interval is constant, whence Dd = v(tDt.  This means however that v(t) = Dd/Dt.  While this formula is the same as the one given above for average velocity, the key difference is that the quantities in the numerator and denominator of this fraction are infinitesimally small; in particular,  Dt must be less than any positive quantity, for if it had a positive value, then we would be measuring instead the average velocity over some interval.  On the other hand, it is clear that Dt is not 0, for then Dd = 0 and the quantity 0/0 is meaningless.
    Galileo does not explain the mystery latent in these concepts.  This would have to wait for later thinkers.  Still, what we have here in the formula v(t) = Dd/Dt is a working definition of the new concept of instantaneous velocity.  It measures at a single point in time how fast the object moves exactly at that time.
    The same idea extends to the concept of acceleration.  If we measure at second t1 the speed v1 of the object in feet per second, at at some later time t2 find it to be moving at a speed of v2 feet per second, then between t1 and t2 the object has been accelerated on average at a rate of

feet per second per second, a measure of the average acceleration of the object over the given interval of time.  But we can determine the instantaneous acceleration on the object by computing instead the ratio a = Dv/Dt of infinitesimally small increments of velocity and time.
    In general, if y = f(x) is any function relating quantities x and y, we can measure the instantaneous rate of change of y with respect to change in x by computing the ratio Dy/Dx, which we also denote f´(x), and call the derivative of f at x.  (This modern term was first coined by Joseph-Louis Lagrange in  1797, over 160 years after Galileo wrote the text we have read here.)  Over the course of the rest of the seventeenth century, mathematicians would slowly come to terms with this idea of an instantaneous rate of change.  The concept of the derivative was the second of the two fundamental ideas whose properties formed the mathematical theory that would coalesce into calculus near the end of the century in the work of Newton and Leibniz.  As we move on through our readings we will see how this took place.
 
 

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last modified 11/6/00
Copyright (c) 2000.  Daniel E. Otero