Did you know . . .?

We have the plague to thank for the binomial theorem! In 1665, plague was raging in England, and Isaac Newton, a new (and undistinguished) graduate of the University of Cambridge, was forced to spend most of the next two years in the relative safety of his family's country manor in Woolsthorpe.

It turned out that solitude and free time was just the stimulous Newton's creative brain needed. In that 18-month period of retreat, he came up with his proof and extension of the binomial theorem, invented calculus (which he called his "method of fluxions"), discovered the law of universal gravitation, and proved that white light is composed of all colors. All of this before the age of 25!

(Image source: http://www.pestcontrolportal.com/Pests/rattus.asp)

Isaac Newton (1642-1727).

(Illustration source: http://www-groups.dcs.st-andrews.ac.uk./~history/PictDisplay/Newton.html)

Newton was not the first to describe a formula for binomial expansions, or multiplying out any expression of the form (a + b)n. We know, for example, that an Islamic mathematician named al-Karaji (d. 1029) constructed a table of binomial coefficients up to (a+b)5 (that is, Pascal's triangle), and later Muslim mathematicians credited him with discovering the formula for the expansion of (a + b)n. Furthermore, in a now lost work, Omar Khayyam (1048-1131) apparently gave a method for finding nth roots based on the binomial expansion and binomial coefficients.

Ancient Indian and Chinese mathematicians also knew the binomial theorem. And in Europe, already a century before Newton's birth, Blaise Pascal's Treatise on the Arithmetical Triangle provided a handy way to generate binomial coefficients. All of these methods for binomial expansion, however, work only for positive integer values of n.

What Newton discovered was a formula for (a+b)n that would work for all values of n, including fractions and negatives:

(a+b)n = an + nan-1b + [n(n-1)an-2b2] / 2! + [n(n-1)(n-2)an-3b3] / 3! + . . . + bn

For -1<n<1, this formula produces an infinite, converging series. Newton's consideration of infinite series and the notion of limit through the binomial theorem led directly to his development of calculus.

Newton's mathematical genius was not apparent when he was a child. As a boy, Newton had been a bit of a tinkerer, but his studies at Trinity College, Cambridge, were largely focused on the law. It is said that he became interested in mathematics only in 1663, when he picked up an astrology book at a local fair and couldn't understand the mathematics in the book. That led him to read trigonometry (which he also couldn't grasp) and finally back to Euclid's Elements. A Cambridge professor in 1664 pronounced his mastery of Euclid insufficient, however. Obviously, Newton made a lot of progress in a very short time!

(Image source: http://www.newton.org.uk/cambridge/Trinity.html)

In their own words:

"Threatening my [step-]father and mother Smith to burn them and the house over them." --From a list of his sins made by Isaac Newton at age 19.

"Plato is my friend, Aristotle is my friend, but my best friend is truth." --Head of Newton's Quaestiones Quaedam Philosophicae (Certain Philosophical Questions), ca. 1664.

"The latest authors, like the most ancient, strove to subordinate the phenomena of nature to the laws of mathematics. "

"I know not what I appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell, whilest the great ocean of truth lay all undiscovered before me." --Quoted in D Brewster, Memoirs of Newton

(Source for quotations: http://www-groups.dcs.st-andrews.ac.uk./~history/Mathematicians/Newton.html)

"Nature and Nature's laws lay hid in night; God said, Let Newton be! and all was light." --Alexander Pope (Quotation source: http://www.treasure-troves.com/bios/Newton.html)

Newton's REAL birthdate and other trivia: http://www.seanet.com/~ksbrown/kmath121.htm

Newton's anagram of calculus: http://www.seanet.com/~ksbrown/kmath414.htm

Newton and the dangers of experimentation: http://www.seanet.com/~ksbrown/kmath471.htm

Newton and the apple: http://www.newton.org.uk/essays/Apple.html

http://www-groups.dcs.st-andrews.ac.uk./~history/Mathematicians/Newton.html

http://www.maths.tcd.ie/pub/HistMath/People/Newton/RouseBall/RB_Newton.html

http://www.newton.org.uk/

http://www.rit.edu/~flwstv/newton.html

http://www.treasure-troves.com/bios/Newton.html

http://www.krysstal.com/binomial.html

http://es.rice.edu/ES/humsoc/Galileo/Catalog/Files/newton.html

http://www.aimsedu.org/Math_History/Samples/Newton/newton1.html (Scroll down to "anecdotes.")

http://web.clas.ufl.edu/users/rhatch/pages/01-Courses/current-courses/08sr-newton.htm

http://www.inetarena.com/~pdx4d/ocn/binomial.html

By Laura Smoller, UALR Department of History.

June 2001.