ISAAC NEWTON Math References

* BOOKS on Newton-math

* WEBSITES on Newton-math

  with Newton-math articles

Sources that offer an expanded
(or alternative) explanation of
Newton's math results and methods.
(in English, rather than Latin or other languages)

Diagram for Theorem 1 of Book 1, Principia

Inertial (constant velocity)
flight in equal times :

    triangle areas SAB = SBc (because the
    triangles have equal bases and same height).

Then imagine this inertial flight being
pulled in by a continuously acting
'centripetal' force to a common center S.

See figure to the right.

Same diagram (of a discretized orbit)

Centripital force effect,
in the same time :

    triangle areas SBc = SBC (because the
    triangles have equal bases and same height).

[BC is resultant of vectors BV
(centripetal) and Bc (inertial).]

Therefore triangle areas SAB = SBC (both = SBc).
Similarly, SBC = SCD = SDE = SEF = ...
equal area triangles in equal times.

Thus, as time interval goes to zero,
a radial from the force center S
to an orbital curve sweeps out equal areas
in equal times (think of two collections of
extremely thin triangles) ---
no matter where on the orbit --- and no matter
what the form of the centripetal force,

not just 'inverse-square' force.

(Newton, the generalist, at work.)
(Orbit could be elliptical, parabolic,
hyperbolic, even spiral.)

See figure to the left.

You can imagine the two equal areas to be
composed of an equal number of
extremely thin 'equal-time-thin-triangles' and
extremely thin 'equal-time-fatter-triangles' ---
in the 'slower' and 'faster' areas, respectively.

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In 2016 I started trying to wade through the 'Principia' (full name 'Philosophiae Naturalis Principia Mathematica') of Isaac Newton.

I bought the English translation of the 3rd (1726) version of the 'Principia' as translated into English by Bernard Cohen and Anne Whitman (with help from Julia Budenz), 1999, University of California Press. (I got the version without Cohen's 'Guide'.)

I found that this is a tough slog (as many others in the 1600's and 1700's found it to be), so I bought the book Magnificent Principia by Colin Pask.

By reading in bits and pieces ('baby steps') from both Cohen-Whitman and Pask, I found I could gradually absorb more and more of what the amazing man deduced and 'synthesized'.

As Pask points out (on page 167 of his book):

    "There is no doubt that dressing up ideas from calculus in geometrical form made it virtually impossible for many to read the Principia."

Furthermore, Pask quotes William Whewell from his 1837 'History of the Inductive Sciences":

    "... we gaze at it [the Principia] with admiring curiosity, as on some gigantic implement of war, which stands idle among the memorials of ancient days, and makes us wonder what manner of man he was who could wield as a weapon what we can hardly lift as a burden."

On the other hand, Pask occasionally points out, as on page 184 of his book 'Magnificent Principia':

    "I suspect that in retrospect, many people will feel that [in some steps], Newton exhibited the physics of ... motion better than we see it through [today's vector-calculus equations]."

Pask leaves many of Newton's propositions/theorems/corollaries/lemmas undiscussed. I find that I may find answers to many of my questions by looking at still other authors who have tried to make Newton's writings understandable to a wider audience.

Many books on Newton have nary a geometric diagram and nary an algebraic expression in them --- no mentions of ratios and 'ultimate ratios'. Those are not books that will help me.

However, in some university libriaries, I have found some books that I may eventually buy for further reading in bits and pieces (more baby steps). Namely:

Chandrasekhar works his way through almost all the propositions and corollaries of the Principia. De Gandt takes an approach that might be more digestable for many readers --- an approach that may be a good introduction to the more exhaustive approach of Chandrasekhar.

A few years before the first edition of the 'Principia' was published in 1687, Newton wrote a smaller work (about 9 pages), titled 'De motu corporum in gyrum' (On the motion of bodies in an orbit). De Gandt (in French) essentially provides an explanation of 'De motu'.

For more information, here are some 'general web searches' on these authors and titles:

Newtonian scholars who are not afraid to show some mathematics in their works on Newton include

To give an idea of why it is important that many people on this earth should understand something of the contributions of Newton to the body of human knowledge, here are several quotes from Einstein:

  • "The whole evolution of our ideas about the processes of nature, with which we have been concerned so far, might be regarded as an organic development of Newton's ideas."

  • "We have to realize that before Newton, there existed no self-contained system of physical causality which was somehow capable of representing any of the deeper features of the empirical world."

  • In 1927, on the 200th anniversary of the death of Isaac Newton, Einstein wrote: "we feel impelled at such a moment to remember this brilliant genius, who determined the course of western thought, research, and practice like no one else before or since."


After I retired (in 2005) --- and, in particular, after 2009 (when I switched from Microsoft operating systems to Linux on my desktop and laptop computers), I started on a retirement hobby of developing software in the Tcl-Tk programming language.

Many of the programming projects that I put on my 'to-do' list involved bringing many of the classical mathematical results (theorems) of Thales, Euclid, Archimedes, Newton, Euler, etc. 'to life' --- via animated or interactive presentation of the mathematical results (geometry, number theory, sequences, series, fractals, whatever) on a 'Tk canvas'.

The 'Newton-math' books (and other sources) on this page are meant to help give me ideas for 'classical mechanics' projects to undertake --- in particular, 'integration of ordinary differential equations' --- with depiction of motions on a Tk 'canvas'.

To augment the lists of links on this page, I have a a page of math PDF's --- mostly public domain documents gathered from the site --- or from the site.

Also, I have a page of mathematician names, in chronological order (a 'timeline' of mathematicians) --- with links to more information on each mathematician.

For my software projects, I have bought many math and math-physics books. Many of my 'hard-copies' are listed on a 'math book inventory' page. (This page is probably way out of date, because I am continually adding to my collection/inventory, without updating that page.)

If the lists on this page and the other pages of this site do not satisfy my need for more software projects, I may wish to scan the list of physicists at Wikipedia.

In addition, here is a page of physics-related lists at Wikipedia.

And here is a list of mathematicians at Wikipedia. In particular, here is a page for mathematicians by nationality, and here is a page for mathematicians by century.

More mathematicians (and physicists) can be found via the Galileo Project of Rice University. A convenient list of many of those mathematicians' names on one page (including Kepler and Newton) is here.

This Should I become a mathematician? thread at provides many interesting source books on mathematics and mathematicians.

A few more books with the intent of popularizing math (i.e. taking math to the masses) can be found at the Wikipedia pages on Popular_mathematics and Recreational_mathematics.


The book titles (and some of the author names) will usually be a link to more information on the book (and author). Often, the book-title link will be to (for reader reviews, in particular) --- and the author link may be to Wikipedia.

Alternatively, the book-title link may be to a site like Google Books, which may have some excerpts from the book --- and the author link may be to the author's web site.

In the first releases of this page, many of the links to journals and magazines may be 'general web searches', not links to the website of the journal or magazine. This is because many of these sites only offer their articles for a fee. I see their need to cover their costs, but, in order to support STEM programs, it would be nice if they would put their old publications in the public domain --- say articles more than 20 years old --- or articles by people who are no longer alive --- or articles by people (or their legal survivors) who have released their articles under a 'Creative Commons' license.


If you are looking for some particular information, you can use the text search function of your web browser.

For example, if you are looking for information on a topic such as 'geometry', 'calculus', or 'trigonometry', enter a key-string such as 'geom', 'calc', or 'trig' in the text search entry field of your web browser.


The wheels of thought that Newton set into motion offer many occasions for wonderment at the logical and geometric beauties of mathematical-physics subject matter.

Kepler said (ref: page 55 of Pask's book):

    "Astronomers should not be granted excessive licence to conceive anything they please without reason: on the contrary, it is also necessary for you to establish the probable causes of your Hypotheses which you recommend as the true causes of Appearances. Hence, you must first establish the principles of your Astronomy in a higher science, namely Physics or Metaphysics."


    "My goal is to show that the heavenly machine is not a kind of divine living being but similar to a clockwork insofar as almost all the manifold motions are taken care of by one single absolutely simple magnetic bodily force, as in a clockwork all motion is taken care of by a simple weight. And indeed I also show how this physical representation can be presented by calculation and geometrically."

Newton responded to this challenge and took Kepler's laws many steps further. Newton must have felt that he went a long way toward the goal he laid out for himself (ref: page 92 of Pask's book):

    "... the difficulty of philosophy [what we call today 'physics'] seems to consist in this --- from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena."

Newton demonstrated that an inverse-square 'law' of gravitational attraction explained elliptic (and parabolic and hyperbolic) planetary/celestial orbits. And he went on to show how that same 'law' could explain many phenomena:

  • motions of moons
    (of Earth and Jupiter and Saturn)

  • motions of pendulums
    (at various altitudes)

  • motions of projectiles and pendulums
    (taking into account air/fluid resistance)

  • the phenomena of tides
    (mainly due to lunar and solar gravity)

  • motions of comets
    (predicting the month of their return)

  • for more, see his 'Book III'.

Newton did not 'feign a hypothesis' to explain the cause of gravity. Like Kepler, he had considered magnetic or electrical causes, but could not resolve the issues that arose.

Even physicists today (2016) are struggling with that problem. Although he could not answer all the questions that he raised, it is no wonder that England buried him as if he were a military hero. (Voltaire was impressed with the funeral.)

I look forward to following in the footsteps of his thoughts.

One of the few portraits available of a youthful Newton.

Many of the following books were mentioned in the references for the book 'Magnificent Principia' by Colin Pask.

Most of these books are less than 70 years old. Most are not in the public domain, so most are not available at 'ebook' sites like and in formats such as PDF or EPUB.

I generally provide two links for each of these books --- one to (mainly for the reader reviews) and one to a general web search on the author name(s) and keywords in the title of the book. The general web search might reveal alternate sources for the book.

Some books on classical mechanics that
do not exhibit Newton's methods:

Many of the following journals and magazines were mentioned in the references for the book 'Magnificent Principia' by Colin Pask. The publications that had about 5 or more articles on Newton's mathematics are indicated below by a double asterisk (**).

Many of these journals do not allow open (free) access to their articles --- even articles that are more than 50 years old. So rather than providing a link to their web sites below, I provide a link to a web search on the journal/magazine name and keywords such as 'newton math'.

If I ever find that some of the articles on Newton's mathematics are openly available from these journals, I may provide links to specific articles (of the journals) below.

  • Newton Project
    (A website directed by Professor Rob Iliffe with the goal of compiling Newton's works into one electonic edition.)

  • Stanford Encyclopedia of Philosopy
    (has some articles on Newton and the Principia --- for example, article by George E. Smith, 2007)

  • National Library of Israel
    (Many of the documents from a famous box of Newton's documents that sat essentially unopened at Cambridge for about 200 years were bought in a 1936 auction by this library. Mostly NON-mathematical papers.)

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Page history

Page was created 2016 May 24.

Page was changed 2016 Jun 06.
(Added an equal-areas image.)

Page was changed 2019 Mar 11.
(Added css and javascript to try to handle text-size for smartphones, esp. in portrait orientation.)

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