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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.

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, they had to abandon the whole concept of number in measuring and substitute for it 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. $\endgroup$ Oct 26, 2018 at 15:06
  • $\begingroup$ @dRIFT sPEED: The most basic source is Euclid. I would say that everyone doing mathematics before 18 century read Euclid. The rest of the reading list may depend on the author that you try to study. $\endgroup$ Jun 13, 2022 at 13:15
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ONE: As others have stated, real numbers were questionable at the time, so to overcome this deficency, numbers were constructed based on magnitudes.

In Euclidean geometry, magnitudes can only be compared (i.e., =,>,<) to like magnitudes, and from this, additive and subtractive operations are permitted, which in turn permits us to multiply and divide magnitudes by quantities of these magnitudes, i.e numbers. As as such, these operations increase/decrease the magnitude in kind by quantities of that number; for example, a length is n-multiples of a another length, or a length is m-parts of another length.

We can't say the same about any other operations that result in another kind of magnitude since they don't have a result that is comparable (=,>,<) to the original magnitudes, thus making it absurd. Thus, it is forbidden to mix (i.e., add/subtract/multiply/divide) magnitudes with other magnitudes since dividing angles by areas is just as absurd as adding lines to areas, and multiplying areas by areas. Since magnitudes cannot be mixed, Euclid would object to the meaning of "v = x/t". It's easy to recognize that "x" is a magnitude of spatial length, but what does that make of "v" and "t"?

To further this point, Euclid doesn't say that an area of a rectangle equates to length x width, rather, he states ratios of the areas of two rectangles are the same as the product of the ratio of lengths and the ratio of widths. Nor does Euclid say that the area of a square is the multiplication of side by side (i.e, s^2), but rather that the ratio of the areas between two squares is the same as the product of the ratios of the sides. Thus, "the area from squaring the sides" is to be meant as "the area resulting from a square constructed with equal lengths". This also means that Pythagoras's theorem is equating areas using squares that are constructed using the sides of the triangle and not directly a multiplication of a number assigned to the sides. (See Euclid Elements Book I-Prop 47 and Book VI-Prop 23)

However, suppose we're able to correlate a comparison of a set of magnitudes to another comparison of a set of magnitudes. In that case, we can define "ratios", which further permit us to relate magnitudes to other magnitudes (See Euclid Book 5-Def 5).

Using this rigorous type of math, Galileo is showing us how these "exotic" magnitudes of speeds, transit-distances, and transit-times are relatable through ratios. He starts by defining what uniform speed is, then provides us the underlying axioms that would support this definition. Galileo recognizes that these axioms provide correlations between comparisons, which sets him up to apply the theory of ratios. Notice how Theorem I is a near direct regurgitation of Euclid Book 5 Definition 5?

Through Theorems 1-6, he shows every possible way to combine ratios with speed, space, and time - all without using any algebraic algorithms (e.g., cross multiplying or any other equation balancing). Notice that, on Theorem 6, if we assign the first motion to be all "1's", (e.g., v1=1, x1=1, t1=1), then the ratio collapses to our velocity equation "v2=x2/t2"

TWO: Good question, that's because Euclid's theory of ratios is powerful in that it allows us to relate magnitudes to any other kinds of magnitudes, so long as the ratios of their own kind remain the same. Therefore, we're permitted to represent a ratio of times or speeds as a ratio of lines so long as they both adhere to the definition of a ratio (i.e., Euclid Book V-Def 5) - we can see this being done with areas and lines in Euclid Book VI-Prop 1, which shows that the ratio of areas are the same as the ratio of lines of triangles/parallelograms.

However, it should be noted that, though we can represent distance, speed, and time each as lines, it remains questionable whether we can add/subtract these lines from each other (e.g., nothing justifies subtracting a distance line from a speed line) or even construct upon them to build shapes (e.g., parallelograms and triangles), or rather, at least not yet. Galileo is acutely aware of this, which is why for Theorems 1-6 for uniform motion he is very careful to keep these sets of lines for speeds, time, and distance separate without physical construction upon each other. His primary concern is to show that ratios between these lines exist in every possible configuration without constructing upon them.

What's curious is what happens in Theorems 4-6: Galileo shows how ratios of one speed, time, and distances are also the "compound/composite/product" of the others (e.g., the ratio of distances is the same as the "compound" of the ratios of speed and time). Compounding ratios is roughly the same as "multiplying ratios to each other," but keep in mind multiplying only applies to numbers, not magnitudes. Compounding is a strange beast in that it isn't formally defined by Euclid but rather inferred by Book VI-Prop 23 and the definitions of duplicate/triplicate ratios, which are viewed as specific cases of the former. A major takeaway from Book VI-Prop 23 is that it says that two parallelograms are in ratio as the compound ratios of their sides, which will be important later for Galileo.

Since Galileo is able to show the compounding relationship between speed, time, and distance, and since parallelograms and their sides produce the same compound ratios (i.e., VI.23), then this allows Galileo to construct parallelograms (and triangles, since they are half a parallelogram) with speed, time, and distance. In other words, Galileo uses theorems 1-6 in conjunction with VI.23 in order to justify the triangle and parallelogram constructions of Theorem 1 of uniform acceleration!

As an interesting side note, while Galileo does draw a figure using a right triangle and a rectangle for Theorem 1 of uniform acceleration, and the most common translation (Alfonso de Salvio and Henry Crew) says they are drawn at right angles, Galileo in the original Latin/Italian text does not explicitly require this! The angle between speed and time can be any angle (but not in line with each other since this could imply they are summable) and the parallelogram can still be drawn since the compounding relationship still holds true according to VI.23. We can't do that with our modern version x=v*t since areas in algebra are defined as square-like areas!

RESPONSE TO NOTE: The 3rd day can be read independently from Days 1 and 2. The latter generally discusses statics (i.e., the perception of weight, the balances of weights, and structure), which is hardly needed for the 3rd day.

Lastly, in response to:

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.

To understand Galileo and Newton, you need Euclid's Elements Books 1-7ish, and primarily just books 1 (defines magnitudes and basic operations), 5 (theory of ratios), and 6 (application of the theory). After that, Conics of Appolonius, but you'd be better off with less antiquated versions of it through T.L. Heath or W.H. Besant who both reorganize it to logically flow clearer. Personally, I like Besant since it picks up straight from Euclid's theory of proportions instead of beginning with intersections of a cone.

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