Methods

time. He did not employ algebraic symbols or equations,
or, except for tangents, the concepts of trigonometry. His numbers were
always expressed as positive integers, never as decimals. Calculus, discovered
later by Newton and Gottfried Leibniz, was not available. To make calculations
he relied on ratios and proportionalities, as defined in Euclid’s Elements. His
reasoning was mostly geometric, also learned from Euclid.
Galileo’s mathematical style is evident in his many theorems on uniform and
accelerated motion; here a few are presented and then “modernized” through
translation into the language of algebra. The first theorem concerns uniform motion:
If a moving particle, carried uniformly at constant speed, traverses two distances,
the time intervals required are to each other in the ratio of these
distances.
For us (but not for Galileo) this theorem is based on the algebraic equation s
vt, in which s represents distance, v speed, and t time. This is a familiar calculation.
For example, if you travel for three hours (t 3 hours) at sixty miles per
hour (v 60 miles per hour), the distance you have covered is 180 miles (s 3
60 180 miles). In Galileo’s theorem, we calculate two distances, call them
s1 and s2, for two times, t1 and t2, at the same speed, v. The two calculations are
s1 vt1 and s2 vt2Dividing the two sides of these equations into each other, we get the ratio of
Galileo’s theorem,
t1 s1 .
t2 s2
Here is a more complicated theorem, which does not require that the two
speeds be equal:
If two particles are moved at a uniform rate, but with unequal speeds, through
unequal distances, then the ratio of time intervals occupied will be the product
of the ratio of the distances by the inverse ratio of the speeds.
In this theorem, there are two different speeds, v1 and v2, involved, and the two
equations are
s1 v1t1 and s2 v2t2.
Dividing both sides of the equations into each other again, we have
s1 v1 t1 .
s2 v2 t2
To finish the proof of the theorem, we multiply both sides of this equation by
and obtain
v2
v1
t1 s1 v2 .
t2 s2 v1
On the right side now is a product of the direct ratio of the distances and the
s1
s2
inverse ratio of the speeds , as required by the theorem.
v2
v1
These theorems assume that any speed v is constant; that is, the motion is not
accelerated. One of Galileo’s most important contributions was his treatment of
uniformly accelerated motion, both in free fall and down inclined planes. “Uniformly”
here means that the speed changes by equal amounts in equal time intervals.
If the uniform acceleration is represented by a, the change in the speed
v in time t is calculated with the equation v at. For example, if you accelerate
your car at the uniform rate a 5 miles per hour per second for t 10 seconds,
your final speed will be v 5 10 50 miles per hour. A second equation,
, calculates s, the distance covered in time t under the uniform accelera-
at2
s
2
tion a. This equation is not so familiar as the others mentioned. It is most easily
justified with the methods of calculus, as will be demonstrated in the next
chapter.
The motion of a ball of any weight dropping in free fall is accelerated in the
vertical direction, that is, perpendicular to Earth’s surface, at a rate that is conventionally represented by the symbol g, and is nearly the same anywhere on
Earth. For the case of free fall, with a g, the last two equations mentioned are
v gt, for the speed attained in free fall in the time t, and for the corgt2
s
2
responding distance covered.
Galileo did not use the equation , but he did discover through experi-
gt2
s
2
mental observations the times-squared (t2) part of it. His conclusion is expressed
in the theorem,
The spaces described by a body falling from rest with a uniformly accelerated
motion are to each other as the squares of the time intervals employed in traversing
these distances.
Our modernized proof of the theorem begins by writing the free-fall equation
twice,
gt2 gt2 s 1 and s 2, 1 2 2 2
and combining these two equations to obtain
s t2 1 1 . s t2 2 2
In addition to his separate studies of uniform and accelerated motion, Galileo
also treated a composite of the two in projectile motion. He proved that the
trajectory followed by a projectile is parabolic. Using a complicated geometric
method, he developed a formula for calculating the dimensions of the parabola
followed by a projectile (for example, a cannonball) launched upward at any
angle of elevation. The formula is cumbersome compared to the trigonometric
method we use today for such calculations, but no less accurate. Galileo demonstrated
the use of his method by calculating with remarkable precision a detailed
table of parabola dimensions for angles of elevation from 1 to 89 .
In contrast to his mathematical methods, derived mainly from Euclid, Galileo’s
experimental methods seem to us more modern. He devised a system of units
that parallels our own and that served him well in his experiments on pendulum
motion. His measure of distance, which he called a punto, was equivalent to
0.094 centimeter. This was the distance between the finest divisions on a brass
rule. For measurements of time he collected and weighed water flowing from a
container at a constant rate of about three fluid ounces per second. He recorded
weights of water in grains (1 ounce 480 grains), and defined his time unit,
called a tempo, to be the time for 16 grains of water to flow, which was equivalent
to 1/92 second. These units were small enough so Galileo’s measurements of
distance and time always resulted in large numbers. That was a necessity because
decimal numbers were not part of his mathematical equipment; the only way he
could add significant digits in his calculations was to make the numbers larger.