Firstly that it is practically undecidable whether it's a math or a physics book due to its elegant formal mathematical style despite being about the natural world.

Secondly no coordinates, not even a mention.

I know principia is one of a kind masterpiece but it is famously hard to read mostly because it was written so long ago. is there some modernised version in the form of some paper or a textbook with similar style of proofs.

  • 1
    $\begingroup$ There is a reason why physicists have moved to often using coordinate-based derivations. They are both simpler and more powerful than geometrical ones. Newton’s way of doing things is mainly of historical interest to most physicists. $\endgroup$
    – G. Smith
    Feb 20, 2020 at 21:04
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    $\begingroup$ Don’t forget that it was Newton who (simultaneously with Leibnitz) invented calculus. The reason that he did not use calculus is his book is because it was too new for other physicists to understand at the time. His book could have been much shorter! $\endgroup$
    – G. Smith
    Feb 20, 2020 at 21:09
  • $\begingroup$ This is the point of my question. Is there an updated version that is shorter and has actual calculus but still retaining the purely geometric style $\endgroup$
    – Kugutsu-o
    Feb 20, 2020 at 21:13
  • $\begingroup$ On the first comment: how much do you know many geometric based derivations anyway? $\endgroup$
    – Kugutsu-o
    Feb 20, 2020 at 21:14
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    $\begingroup$ Elegant? Are you serious? Newton's Principia is torture. Newton chose that torturous path for two reasons: he viewed his calculus as so new that readers would view results based on it as suspect, and even more importantly, he couldn't see into the future so as to employ techniques that would not be developed until well after his death. Maxwell's Electrodynamics, which is a bit archaic, is less than half as old as Newton's Principia. Think of Newton's Principia as being akin to Beowolf, and Maxwell's Electrodynamics as being akin to The Canterbury Tales. Not that bad, but close. $\endgroup$ Feb 20, 2020 at 22:15

2 Answers 2


In a comment you said

This is the point of my question. Is there an updated version that is shorter and has actual calculus but still retaining the purely geometric style

From the question it wasn't clear to me that you were looking for a more modern perspective on Newton's Principia. What you're looking for is Chandrasekhar's book: Newton's Principia for the common reader.

It has both geometrical derivations eplained and expanded and their translation into modern calculus.

  • $\begingroup$ This is not bad. But still uses coordinates here and there. $\endgroup$
    – Kugutsu-o
    Feb 21, 2020 at 8:18
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    $\begingroup$ @Ezio It does when he writes down the proofs in modern mathematical language. But when he presents Newton's original proofs they are just geometrical, just paraphrased and expanded. You can skip the proofs done with modern calculus and only focus on Newton's original proofs. Maybe I'll add this to the answer directly. $\endgroup$
    – AnOrAn
    Feb 21, 2020 at 14:01

I'm not sure what the poster of this question Ezio is looking for exactly. He mentioned that he wants "an updated version that is shorter and has actual calculus but still retaining the purely geometric style" and that Newton's Principia is so elegant because it uses only geometry without coordinates.

To be accurate, Isaac Newton used a form of elementary coordinate geometry in the Principia. The words "ordinate(s)" and "abscissa(s)" are mentioned many times in the Principia in relation or in reference to corresponding geometrical figures.

Let's consider the following figure from Section 2 of Book 2 of the Principia (taken from the English translation by Andew Motte, and the first American edition, which can be found at the Internet Archive site):

enter image description here

Newton mentions abscissas CB,CD,CE, and ordinates BG,CH, and DI.

Here is another figure from Section 5 of Book 1, where Newton mentions the abscissa AD and the ordinate DG:

enter image description here

In other places, Newton mentions the word "axis" or "axes", sometimes in relation to lines representing the (Cartesian) coordinate axes of ellipses or conic sections.

So it can be seen that even Newton's Principia with its "purely geometric style" uses some form of coordinate geometry. I think it is not realistic to find an instructive book similar to the Principia with calculus and without any mention of coordinates or coordinate geometry.

In light of the considerations and remarks above, a helpful book would be Newton's Principia for the Common Reader by Chandrasekhar, as mentioned in the answer by AnOrAn.

Another helpful book published in 1855, using analytical methods and calculus with a style somewhat similar to Newton's Principia, is Analytical view of Sir Isaac Newton's Principia, by Henry Lord Brougham and E.J. Routh, which can be found online at this link.

Hope that helps.

  • $\begingroup$ That helps actually, thaugh not I wanted to see. Coordinate freeness is not necessary what I find elegant in and of itself althoght it is a major factor. For example, the Feynmans lost lecture, where he following newton's principia derives kepplers elipse law using as he says nothing, nothing but" infinite intelligence" that is. he is deriving it from only the basic things that he's got, without the conglomerate machinery of matrices, calculus etc.. This is what he calls an elementary derivation, despite it being pretty hard to follow. $\endgroup$
    – Kugutsu-o
    Apr 17, 2020 at 10:43
  • $\begingroup$ So it is elegant because it does not use the brute mechanical force of for example analisis on rectangular coordinates,etc.. Iv heard Newton using coordinates, but didn't know he was doing it in principia. $\endgroup$
    – Kugutsu-o
    Apr 17, 2020 at 10:46
  • $\begingroup$ @Ezio It is also to be noted that analytical geometry or coordinate geometry was being developed since the beginning of the 17th century. Newton knew about these developments and had read The Geometry of Descartes years before writing the Principia. $\endgroup$
    – E. Noujeim
    Apr 17, 2020 at 12:12

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