# Doubt regarding “Two New Science” by Galileo Galilei

I was reading "Two New Sciences" by Galileo Galilei and therein was a chapter named third day which deals with motion. When I started reading that chapter I eventually came across the theorem 1 wherein I could not comprehend two things.

ONE: Why do we need to prove such a thing since the statement of the theorem is the direct consequence of this relation speed = distance/time?

TWO: Why are the speed and time represented as geometric lines, what does that mean?

and as i proceeded to the theorem 2 3, and 4 the same two things were bewildering me a lot and I was no more able to digest the further text in the book.

Please help me comprehending these two things and I feel that there's something of great importance which I don't know but need to know, which will eventually account for my understanding of the doubt.

NOTE: I started reading the book from the chapter third day and haven't read the chapters prior to it.

This is exactly what he tries to prove here. The difference between you and Galileo is that you were taught some concepts in your childhood, which Galileo was not. For example, the concept of a (real) number. The ancients did not have this concept. They only had rational numbers. When they discovered that $$\sqrt{2}$$ is not a rational number, hey had to abandon the whole concept of number in measuring and substitute it by geometric concepts (length). They developed a highly sophisticated "theory of proportions" to deal with these lengths. This was the mathematics available at the time of Galileo. The modern concept of real number was slowly developing (with some major developments at the time of Galileo, but reached sufficient clarity and was widely accepted only in 19th century. And nowadays we are indoctrinated (I cannot say "taught", the subject is too subtle) with this concept in elementary school.