The Essential Galileo (60 page)

Let us represent by the line
AB
the time in which the space
CD
is traversed by a body that starts from rest at
C
and is uniformly accelerated; let the final and highest value of the speed gained during the interval
AB
be represented by the line
EB
, drawn at right angles to
AB;
draw also the line
AE;
then all lines drawn from equidistant points on
AB
and parallel to
BE
will represent the increasing values of the speed, beginning with the instant
A.
Let the point
F
bisect the line
EB;
draw
FG
parallel to
BA
, and
GA
parallel to
FB
, thus forming a parallelogram
AGFB
, whose area will be equal to that of the triangle
AEB
, and whose side
GF
bisects the side
AE
at the point
I.

Now, if the parallel lines in the triangle
AEB
are extended to
GI
, then the aggregate of all the parallels contained in the quadrilateral is equal to the aggregate of those contained in the triangle
AEB;
for those in the triangle
IEF
are equal to those contained in the triangle
GIA
, while those included in the trapezium
AIFB
are common. Furthermore, each and every instant of time in the time interval
AB
has its corresponding point on the line
AB
, from which points the parallels drawn in and limited by the triangle
AEB
represent the increasing values of the growing velocity; and the parallels contained within the rectangle represent the values of a speed that is not increasing but constant. [209] Hence it appears that the moments of speed acquired by the moving body may be represented, in the case of the accelerated motion, by the increasing parallels of the triangle
AEB
, and in the case of the uniform motion, by the parallels of the rectangle
GB;
for, what the moments of speed may lack in the first part of the accelerated motion (the deficiency of the moments being represented by the parallels of the triangle
AGI
) is made up by the moments represented by the parallels of the triangle
IEF
. Therefore, it is clear that equal spaces will be traversed in equal times by two bodies, one of which starts from rest and moves with uniform acceleration, while the other moves with a uniform speed whose moment is one-half the maximum moment of speed under the accelerated motion. QED.

Theorem 2, Proposition 2: If a body falls from rest with a uniformly accelerated motion, then the spaces traversed are to each other as the squares of the time intervals employed in traversing them.

Let the time beginning with any instant
A
be represented by the straight line
AB
, in which are taken any two time intervals
AD
and
AE.
Let
HI
represent the distance through which the body, starting from rest at
H
, falls with uniform acceleration. If
HL
represents the space traversed during the time interval
AD
, and
HM
that covered during the interval
AE
, then the space
HM
stands to the space
HL
in a ratio that is the square of the ratio of the time AE to the time AD
;
or we may say simply that the distances
HM
and
HL
are related as the squares of
AE
and
AD.

Draw the line
AC
making any angle whatever with the line
AB;
and from the points
D
and
E
, draw the parallel lines
DO
and
EP;
of these two lines,
DO
represents the greatest velocity attained during the time interval
AD
, while
EP
represents the maximum velocity acquired during the time
AE.
But it has just been proved that so far as distances traversed are concerned, it is precisely the same whether a body falls from rest with a uniform acceleration or whether it falls during an equal time interval with a constant speed that is one-half the maximum speed attained during the accelerated motion. It follows therefore that the distances
HM
and
HL
are the same as would be traversed during the time intervals
AE
and
AD
by uniform velocities equal to one-half those represented by
EP
and
DO
respectively. If, therefore, one can show that the distances
HM
and
HL
are in [210] the same ratio as the squares of the time intervals
AE
and
AD
, our proposition will be proven. But in the fourth proposition of the first section above,
²²
it has been shown that the spaces traversed by two bodies in uniform motion bear to one another a ratio that is equal to the product of the ratio of the velocities by the ratio of the times; and in the present case the ratio of the velocities is the same as the ratio of the time intervals, for the ratio of one-half
EP
to one-half
DO
, or of
EP
to
DO
, is the same as that of
AE
to
AD;
hence the ratio of the spaces traversed is the same as the squared ratio of the time intervals. QED.

It also clearly follows that the ratio of the distances is the square of the ratio of the final velocities, that is, of the lines
EP
and
DO
, since these are to each other as
AE
to
AD
.

Corollary 1: Hence it is clear that if we take any number of consecutive equal intervals of time, counting from the beginning of the motion, such as
AD, DE, EF, FG,
in which the spaces
HL, LM, MN, NI
are traversed, these spaces will bear to one another the same ratio as the series of odd numbers, 1
,
3, 5, 7.

For this is the ratio of the differences of the squares of the lines which exceed one another by equal amounts and whose excess is equal to the smallest of these same lines; or we may say that this is the ratio of the differences of the squares of the natural numbers beginning with unity. Therefore, whereas after equal time intervals the velocities increase as the natural numbers, the increments in the distances traversed during these equal time intervals are to one another as the odd numbers beginning with unity.

S
AGR.
    Please suspend the reading for a moment, since there just occurs to me an idea which I want to illustrate by means of a diagram in order that it may be clearer both to you and to me. Let the line
AI
represent the lapse of time measured from the initial instant
A;
through
A
draw the straight line
AF
making any angle whatever; join the terminal points
I
and
F;
divide the time
AI
in half at
C;
draw
CB
parallel to
IF
. Let us consider
CB
as the maximum value of the velocity that increases from zero at the beginning in simple proportionality to the segments (inside the triangle
ABC
) of lines drawn parallel to
BC
; or what is the same thing, let us suppose the velocity to increase in proportion to the time; then I admit without question, in view of the preceding argument, that the space traversed by a body falling in the aforesaid manner will be equal to the space traversed by the [211] same body during the same length of time traveling with a uniform speed equal to
EC
, or half of
BC.
Further let us imagine that the body has fallen with accelerated motion so that at the instant
C
it has the velocity
BC.
It is clear that if the body continued to descend with the same speed
BC
, without acceleration, it would in the next time interval
CI
traverse double the distance covered during the interval
AC
with the uniform speed
EC
, which is half of
BC.
But since the falling body acquires equal increments of speed during equal increments of time, it follows that the velocity
BC
, during the next time interval
CI
, will be increased by an amount represented by the parallels of the triangle
BFG
, which is equal to the triangle
ABC
. Thus, if one adds to the velocity
GI
half of the velocity
FG
, the maximum increment of speed acquired by the accelerated motion and determined by the parallels of the triangle
BFG
, one will have the uniform velocity
IN
with which the same space would have been traversed in the time
CI.
And since this speed
IN
is three times as great as
EC
, it follows that the space traversed during the interval
CI
is three times as great as that traversed during the interval
AC.
Now, let us imagine the motion extended over another equal time interval
IO
, and the triangle extended to
APO;
it is then evident that if the motion continues during the
interval IO
, at the constant rate
IF
acquired by acceleration during the time
AI
, the space traversed during the interval
IO
will be four times that traversed during the first interval
AC
, because the speed
IF
is four times the speed
EC.
But if we enlarge our triangle so as to include
FPQ
, which is equal to
ABC
, still assuming the acceleration to be constant, we shall add to the uniform speed an increment
RQ
, equal to
EC;
then the value of the equivalent uniform speed during the time interval
IO
will be five times that during the first time interval
AC;
therefore, the space traversed will be quintuple that during the first interval
AC.
It is thus evident by this simple computation that a moving body starting from rest and acquiring velocity at a rate proportional to the time, will, during equal intervals of time, traverse distances that [212] are related to each other as the odd numbers beginning with unity, 1, 3, 5; or considering the total space traversed, that covered in double time will be quadruple that covered during unit time; in triple time, the space is nine times as great as in unit time. And in general the spaces traversed are in the squared ratio of the times, i.e., in the ratio of the squares of the times.

S
IMP.
    In truth, I find more pleasure in this simple and clear argument of Sagredo than in the Author's demonstration, which to me appears rather obscure; thus, I am convinced that matters are as described, once having accepted the definition of uniformly accelerated motion. But as to whether this acceleration is that which nature employs in the case of falling bodies, I am still doubtful. So it seems to me, not only for my own sake but also for all those who think as I do, that this would be the proper moment to introduce one of those experiments— and there are many of them, I understand—which correspond in several ways to the conclusions demonstrated.

S
ALV.
    The request which you make, like a true scientist,
²³
is a very reasonable one. For this is the custom—and properly so—in those sciences where mathematical demonstrations are applied to natural phenomena; this is seen in the case of perspective, astronomy, mechanics, music, and others, which by sense experience confirm the principles that become the foundations of the entire superstructure. I hope therefore it will not appear to be a waste of time if we discuss at considerable length this first and most fundamental question upon which hinge numerous consequences; of these we have in this book only a small number, placed there by the Author, who has done so much to open a pathway hitherto closed to minds of a speculative turn. As far as experiments go, they have not been neglected by the Author; and often, in his company, I have myself performed the tests to ascertain that the acceleration of naturally falling bodies is that above described.