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What did Newton originally say about Integration?

By Murray Bourne, 22 Jul 2010

isaac newton
Isaac Newton, age 46 [Source]

Most of us learn about math from modern textbooks, with modern notation and often divorced from the historical original. No wonder people think math is a modern invention that's only designed to torture students!

Isaac Newton wrote his ideas about calculus in a book called The Principia (or more fully, Philosophiae Naturalis Principia Mathematica, which means "Mathematical Principles of Natural Philosphy"). This was an amazing book for the time (first published in 1687), and included his Laws of Motion.

Principia cover
Cover of the Principia. [Source]

Newton wrote his Principia in Latin. It was common for mathematicians to write in Latin well into the 19th century, even though other scientists were writing (perhaps more sensibly) in their native tongues (or in commonly spoken languages like French, German and English).

Let's look at one small part (which he named "Lemma II") of Newton's work, from the first English translation made in 1729. You can see all of that translation here, thanks to Google Books (go to page 42 for Lemma II):

1729 English Translation of Principia

The problem below was very important for scientists in the late 17th century, since there were pressing problems in navigation, astronomy and mechanical systems that couldn't be solved with existing inefficient mathematical methods.

Some explanations before we begin:

  • A Lemma is a statement that has been proven, and it leads to a more extensive result.
  • In old English, the ∫ sign is an "S". The first word where this appears below is "in∫crib'd", which we would write as "inscribed". (Note the "s" used for plural nouns is the same as our 's".) The elongated S symbol ∫ came to be used as the symbol for "integration", since it is closely related to "sum".
  • "Dimini∫hed" is "diminished", and means "get smaller".
  • &c is "&c" which we would write these days as "etc" (et cetera)
  • "Augmented" means "get bigger".
  • "Ad infinitum" is Latin for "keep doing it until you approach infinity".

Newton's Principia

This is the diagram that is referred to in the above text.

area under a curve


Let's go through it one concept at a time, with appropriate tweaks to the diagram. We are trying to find the area between a curve abcdE and 2 lines Aa and AE.

Newton - area

area under curve

Newton 2

area rectangles


area under curve - rectangles


Newton Newton
Lower rectangles Upper rectangles

In other words, if we draw more and more thinner rectangles in the same manner, the area of the lower rectangles and the area of the upper rectangles will converge on the area under the curve. This is the area we need to find.

Below is the case where we have 25 rectangles. We can see the total areas of the rectangles is getting close to the area under the curve. Certainly the following ratio approaches 1, as Newton says.

lower rectangles : upper rectangles : area under the curve

rectangles - lower rectangles - upper
Lower rectangles Upper rectangles

This is a fundamental idea of calculus - find an area (or slope) for a small number of cases, increase the number of cases "ad infinitum", and conclude that we are approaching the desired answer.

You can explore this concept further (using an interactive graph) in the article on Riemann Sums.

Archimedes' contribution

This concept of finding areas of curved surfaces using infinite sums was not that new, since Archimedes was aware of it 2000 years ago. (See Archimedes and the Area of a Parabolic Segment.)

Learn math from primary sources

It is very interesting to see Newton's original notation and expression, even if it is via an English translation. The above, of course, is a very small part of Newton's original Principia.

We should learn (and teach) mathematics with a better understanding of why the math was developed, when it was developed and who developed it. We can't always use primary sources, obviously, but it is better to learn math with an understanding of its historical context rather than do it in a vacuum.

See the 28 Comments below.

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