What did Newton originally say about Integration?
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.
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".
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".
This is the diagram that is referred to in the above text.
Explanation
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.
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| 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
 |
 |
| 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.
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22 Jul 2010 at 9:03 pm [Comment permalink]
St John’s College teaches math this way.
22 Jul 2010 at 10:08 pm [Comment permalink]
Thanks for the link, William. They certainly do have a clear historical context in their math teaching!
25 Jul 2010 at 1:17 am [Comment permalink]
OH my God.
How nicely and lucidly explained the original theory of Sir Issac Newton.
We would like to read the English translation of Principia,please suggest us.
Thanks.
25 Jul 2010 at 7:51 am [Comment permalink]
Hi Shaikh. Glad you enjoyed the article. There is a link to the English translation in the article, but here it is again for convenience: 1729 English Translation of Principia
10 Jan 2011 at 2:24 am [Comment permalink]
“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.”
This is so true. I’ve had the same issues with the ahistorical instruction of mathematics. Would you know any texts which go through mathematics historically (I mean develop it along its historical course)?
Thanks (and kudos for all this work you’ve put in),
Haroon
10 Jan 2011 at 1:16 pm [Comment permalink]
Glad you found it useful, Haroon.
Sadly, authors seem to separate textbooks from history of math. An Amazon search didn’t find much, but this one looks interesting: Historical Connections in Mathematics: Resources for Using History of Mathematics in the Classroom.
Another possibility is to find out what resources St John’s College uses in their historical approach.
Does anyone know about a good textbook that also includes a lot of historical context?
1 Feb 2011 at 3:13 am [Comment permalink]
Thank YOU and Thank YOU SO MUCH.
10 Mar 2011 at 1:34 am [Comment permalink]
Thanks for the help interpreting the Principia. My son is writing a paper comparing Newton’s caluclus method to Leibniz’, and we have been trying to figure out if Newton ever performed integration by substitution. He seems to have generally taken the route of converting transcendental functions to polynomials (using infinite series)then using term-wise integration. The question is, was Leibniz alone in coming up with the very powerful tool of using substitution? Certainly, his notation seems more suited for this.
10 Mar 2011 at 2:34 pm [Comment permalink]
@Kathy: I don’t profess to be an expert in this area, but I do find it interesting.
It appears unlikely Newton provided a formula for integration by substitution. A history of analysis By Hans Niels Jahnke suggests he would have more likely provided examples of how it could be done. Newton’s notation was often quite difficult, of course. Leibniz was much more interested in making readable and meaningful mathematics (as am I!)
If I find more definitive statements on this issue, I’ll let you know.
6 Sep 2011 at 11:36 am [Comment permalink]
If you want mathematics taught with a historical backdrop, check out any book by Morris Kline. Full of history, background and application.
His books are downright cheap as textbooks. Seems that since his teaching methods were ditched towards the end of his own life, they have become completely scrapped by academia.
6 Sep 2011 at 8:18 pm [Comment permalink]
Thanks for reminding me of Kline, Aaron. Good tip!
19 Oct 2011 at 1:51 pm [Comment permalink]
I just read through this article and then saw the first comment and had to laugh. I go to St. John’s College and was just trying to get some ideas rolling for a paper on Newton’s integration. The diagrams of the areas with 25 rectangles are really helpful aides!