
Oresme's Mean Speed Theorem (left), bears a distinct resemblence
to Galileo's later law of free fall.
Indeed, the work of Oresme and other 14th century logicians provides
a basis for Galileo's later boast that it is possible to "deduce
the law of free fall without recourse to observation." 

Consider a graph of speed versus time
(left; click any diagram on this page to view a larger version).
If the speed increases by one unit for each unit of time; that is, if
the speed increases by a constant amount per unit of time, then one is
dealing with uniform acceleration. 

Since distance equals speed multiplied by time, the
distance traveled in the first unit of time will be 1 unit of distance,
or 1 triangle (left).
After the second unit of time (T2), the distance will be 4 units, or
4 triangles (count them, right). 


After three units of time (T3), the distance traveled will be 9 triangles
(left, count them).
16 triangles at time 4 (right, count them). 


25 triangles at time 5 (left).
And a distance of 36 triangles will be traveled after 6 units of time
(T6, right).
Count them to be sure! 


We can summarize this logical exercise by saying that the
distances traveled by a body moving with uniform acceleration
increase as the square of the times,
or D is proportional to the square of t.
This is Galileo’s law of falling bodies.
The red rectangle at left contains only square numbers. That is, at the
first time, T1, 1 unit of distance has been traveled
(look within the red rectangle). At T2, 4 units of distance
have been traveled. 9 triangles are covered at T3, 16
at T4, 25 at T5, and 36 at T6. 

Another way of saying this is that uniform acceleration proceeds
as the odd numbers beginning from 1. That is, in the first unit
of time, a distance of 1 triangle is traveled (left).
Within the second unit of time, a distance of 3 triangles are traveled
(right). 


5 triangles are crossed during the next unit of time (left);
then 7 (right);
then 9 (below left);
then 11 units of distance are traveled during the last period of time
(below center),
for a total of 36 triangles (below right). 





The Mean Speed of a body
in motion equals the initial speed (zero in this case) plus the final
speed, divided by 2. It is represented by a red horizontal line at left.
Oresme noted that the same number of triangles occur below the mean speed
line as occur in uniform acceleration (compare the diagram at left for Mean
Speed with the one above right for Uniform Acceleration). 

Therefore, the distance traveled by
a body moving with uniform accleration is the same as the distance it
would travel if moving constantly at its average speed. This
is Oresme's Mean Speed Theorem. Or, as Lindberg expressed
it, a body moving with uniformly accelerated motion covers the same distance
in a given time as if it were to move for the same duration with a uniform
speed equal to its mean (or average) speed" (p. 300).
(The average speed of a body in uniform acceleration is represented in
the graph by the point of intersection between the red mean speed line
and the blue area for uniform acceleration). 

On the left is Oresme’s diagram of the Mean Speed Theorem in
a modern edition.
Only the orientation is different from the corresponding page from Galileo's
Discorsi (1638, right).
The Mean Speed Theorem applies to all cases generally where there is
uniform acceleration. Galileo was not the first to specifically apply
the Mean Speed Theorem to falling bodies as an example of uniform acceleration:
In the 15th century, the Spaniard logician Domingo de Soto
applied Oresme's Mean Speed Theorem specifically to falling bodies in
a work which Galileo and many of his contemporaries read. 
