# Newton's view on Line generation

• Vinay080
In summary, the excerpt from Sir Isaac Newton's treatises on the quadrature of curves and analysis by equations of an infinite number of terms reveals his preference for a motion-based view of mathematical quantities. He argues that lines are not made up of individual parts, but are instead described by a continued motion of points. This concept of motion is seen in nature and was also used by the ancients in their teachings on reflection. Newton does not provide a clear reason for his preference for the motion view, but it is likely influenced by his predecessors and the concept of continuously varying motion being intuitive. Further research on Newton's views on points and his three different views on calculus may provide more insight into his reasoning.
Vinay080
Gold Member
The following passage has been extracted from the John Stewart's English translated version of the "Sir Issac Newton's two Treatises: Of the Quadrature of Curves, and Analysis by equations of an infinite number of terms" http://archive.org/details/sirisaacNewtons00stewgoog:
I consider mathematical quantities in this place not as consisting of parts; but as described by a continued motion. Lines are described, and there by generated not by the apposition of parts, but by the continued motion of points; superficies's by the motion of lines; Solids by the motion of superfices's; Angles by the rotation of the sides; Portion of time by a continual flux: and so in other quantities. These geneses really take place in nature of things, and are daily seen in the motion of bodies. And after this manner the ancients, by drawing moveable right lines along immoveable right lines taught the genesis of reflection...

Here Newton doesn't provide any reason on why he wants to describe lines to be generated by the "continued" motion rather than by the appositon of parts (= points??). Is there any reason for his preference for motion view?

And I noticed that Newton doesn't define point. I don't understand whether he is following Euclid's method of having some of the terms to be undefined, or some other philosophy. I want to know Newton's view on mysterious points. I will be really happy if sources on this regard (Newton's view on points) is provided.

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Meaning of Apposition from "The New Oxford American Dictionary": The positioning of things or the condition of being side by side or close together. So, I interpret apposition of parts to be positioning of points/parts side by side or close together to form a line.

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References to the full latin text:

- In Whiteside collection, [vol VIII]: http://books.google.es/books?id=EqlWllD_H8MC&pg=PA106&lpg=PA106&dq=partibus

I have already asked this question on all other websites, but I am not satisfied. As the question is so significant for me, to understand the calculus, I don't mind in asking this again here. Thank you.

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I find Newton trying to give reason for choosing "motion" view, but everywhere it is blurry, I don't understand his deep roots in reasoning. For example, in 2nd paragraph, he says "I sought a method of determining quantities...I fell by degrees upon the method of Fluxions...". And in 12th paragraph, "..and I was willing to show that, in the method of fluxions, there is no necessity of introducing figures of infinitely small in geometry..."

I even read few pages of Issac Barrow's book "Geometrical Lectures", in which he describes some of the methods of generation of magnitude, and he even says to have given a copy of his book to Newton. In that book he claims the generation of magnitude by motion to be of "primary and chief" because "without motion nothing can be generated or produced", his reasoning seems to be not so strong to me.

I am now reading the article by Philip Kitcher, in which he claims Newton to have three different views (infinitesimals, fluxions, ultimate ratios) in three different times.

I think that it is important to keep in mind that calculus was not put on a mathematically rigorous footing until well after Newton and you may not find any deeper reason why Newton chose this description except for that it was used by some of his predecessors (infinitesimals and 'ultimate ratios' were also used prior to Newton). This is from chapter 16 of A History of Mathematics by Katz:

For Newton, the basic ideas of calculus had to do with motion. Every variable in an equation was to be considered, at least implicitly, as a distance dependent on time. Of course, this idea was not new with Newton, but he did make the idea of motion fundamental: “I consider quantities as though they were generated by continuous increase in the manner of a space over which a moving object describes its course.”12 The constant increase of time itself Newton considered virtually an axiom, for he gave no definition of time. What he did define was the concept of fluxion: The fluxion x ̇ of a quantity x dependent on time (called the fluent) was the speed with which x increased via its generating motion. In his early works, Newton did not attempt any further definition of speed. The concept of continuously varying motion was, Newton believed, completely intuitive.

You may also want to check out Euler as a Physicist by Suisky. I recall that there was a chapter in that text that went into some detail about Newton, but I don't have a copy on hand.

Vinay080

## What is Newton's view on line generation?

Isaac Newton believed that lines were fundamental to understanding the physical world. He saw lines as the basis for all geometric shapes and used them to explain the laws of motion and gravitation.

## How did Newton contribute to the development of line generation?

Newton's theory of calculus and his laws of motion were essential in the development of line generation. He used these principles to study curves and lines, and his work laid the foundation for modern calculus and analytic geometry.

## What is the significance of Newton's view on line generation?

Newton's view on line generation revolutionized the way we think about space and geometry. His laws of motion and gravitation, along with his theories of calculus, allowed for a deeper understanding of the physical world and paved the way for future advancements in mathematics and science.

## What challenges did Newton face in developing his theories on line generation?

One of the main challenges Newton faced was the lack of advanced mathematical tools at the time. He had to develop his own methods and techniques to solve complex problems, which required a great deal of time and effort. He also faced criticism from other scientists and mathematicians who were skeptical of his ideas.

## How has Newton's view on line generation influenced modern science and mathematics?

Newton's theories and discoveries laid the foundation for modern physics and mathematics. His work on line generation has been applied to various fields, including engineering, astronomy, and computer graphics. His ideas continue to be studied and used in new ways, making him one of the most influential scientists in history.

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