On the Acceleration of Falling Bodies

...one unit of time, at the end of two, four; at the end of three, nine; at the end of four, sixteen, and so on. All of this may be represented in a certain large Triangle, whose sides and base are divided into equal parts. Lines are drawn between these points to divide the area into smaller triangles that are equal to each other and similar in shape. The parts of either side, starting from the top, are taken as the units of time. The bases of the triangles corresponding to these parts represent the degrees of speed. The intercepted triangles, or their areas, represent the spaces traveled. See also the sequence of the whole Letter or Dissertation. From page 3 to 9.

ARTICLES VI. VII. VIII. On the definition of uniformly accelerated motion.

Galileo defines uniformly accelerated motion original: "Motum æquabiliter acceleratum", which is the type that applies to falling heavy bodies, as that which starts from rest and acquires equal moments of speed celeritatis momenta: the incremental units of velocity gained during acceleration in equal units of time. However, the Reverend Father Pierre Cazré, a Jesuit mathematician and critic of Galileo rejects this. He contends instead that it should be defined according to the common opinion as that which acquires equal increases of speed over equal spaces. Although the uniformity of acceleration is clearly understood from Galileo's definition: the increasing speed relates to a line between the sides of the mentioned Triangle, increasing from the peak down to the base. This line increases uniformly because it acquires continuous equal additions corresponding to the equal parts of the sides, which represent time. From the definition approved by the Reverend Father, however, no such thing can be understood. Since he makes no mention of time, and takes the parts of the side of the triangle to be spaces and the intercepted triangles to be degrees of speed, it follows that if as many triangles are always added as there are parts of the side, a total triangle will be created whose area increases most unevenly from the peak to the base.