= [distance traveled in the time interval a < t
< b]
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(t)·Dt, 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.
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(t)·Dt.
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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