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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.

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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?

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.

Even at the time of Newton, a century after Galileo, they phrased the theorems in terms of proportions, rather than real numbers, functions etc.

In general, to read and appreciate the old authors like Galileo, one has to study some of the previous development to have an idea of the background these people had.

Remark. There is a funny paper, N. J. Wildberger, Real fish, real numbers, real jobs, The Mathematical Intelligencer, March 1999, Volume 21, Issue 2, pp 4–7, which tries to explain to modern students how deep, abstract and counter-intuitive the concept of real number really is. When I was a student, a rigorous theory of real numbers was taught in the mathematics departments of universities. Nowadays it is not, as too complicated for first-year students taking too much time etc. So students are just forced to believe that there is some rigorous theory, and trained to use the results.

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  • $\begingroup$ U said '' to read and appreciate the old authors like Galileo, one has to study some of the previous development to have an idea of the background these people had.'' Could u please suggest the pre-requisites in order to easily comprehend the works of the great authors like Galileo,Newton etc. The reason why I want to read such texts is simply because(as u have also mentioned) these days texts are written is such a fashion that one ends up indoctrinating the concepts in their minds without knowing the hows and whys of the concept and they end up doing no science virtually. $\endgroup$ – user596245 Oct 26 '18 at 15:06

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