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 D*t*,
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 2*d* and 2*d*' would have to be 2*v*,
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 D*t*.
Viewing the speed as constant over that small interval, the distance D*d*covered
in that interval obeys the rule D*d*
= *v*·D*t*.
Of course, this also corresponds to the area of an infinitesimally thin
parallelogram (actually rectangles) with "width" D*t
*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.

(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,

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
*n*^{2}.
That this is true can be deduced from the fact that the difference between
succesive square numbers is (*n*+1)^{2}-*n*^{2}
= 2*n* + 1, the *n*th odd number. See the exercises.

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last modified 11/5/00

Copyright (c) 2000. Daniel E. Otero