about-integration/4878

ie: As the rectangles become thinner, the area of the inner rectangles and the area of the outer rectangles will converge on the area under the curve.

about-integration/4878

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.

1729 English Translation of Principia 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". integration/4878

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

originally-say-about-integration/4878 What did Newton originally say about Integration? 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!Source 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. Cover of the Principia. [Source]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 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". integration/4878

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.

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

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.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.) 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.

Related posts: What did Euclid really say about geometry?What did Euclid really say about geometry? Euclid's math textbook has been in use for over 2, Riemann SumsRiemann Sums You can investigate the area under a curve using an... Isaac Newton loses his fortuneIsaac Newton loses his fortune Isaac Newton lost buckets of money in a stock investment... Archimedes and the area of a parabolic segmentArchimedes and the area of a parabolic segment Archimedes had a good understanding of the way calculus works,... The IntMath Newsletter – Oct 2006The IntMath Newsletter – Oct 2006 In this newsletter: 1. Tom's money math problems 2. This... Posted by Murray in Learning, Mathematics - July 22nd, 2010.MurrayLearningMathematics 6 Comments on “What did Newton originally say about Integration?” WilliamWilliam says: 22 Jul 2010 at 9:03 pm Link to this comment St John’s College teaches math this way.Link to this commentteaches math this way. MurrayMurray says: 22 Jul 2010 at 10:08 pm Link to this comment Thanks for the link, William. They certainly do have a clear historical context in their math teaching!Link to this comment shaikh moulaali says: 25 Jul 2010 at 1:17 am Link to this comment 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.Link to this comment Thanks.

MurrayMurray says: 25 Jul 2010 at 7:51 am Link to this comment 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 PrincipiaLink to this comment1729 English Translation of Principia Haroon says: 10 Jan 2011 at 2:24 am Link to this comment “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), HaroonLink to this comment MurrayMurray says: 10 Jan 2011 at 1:16 pm Link to this comment Glad you found it useful, Haroon.Link to this comment 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.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.St John’s College uses in their historical approach Does anyone know about a good textbook that also includes a lot of historical context?

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".