The town of Cambridge

The delightful town of Cambridge, England, lies on the banks of the little River Cam. It is famous for its University, and contains much of historical, architectural and artistic interest. The University has 31 colleges, all within easy walking distance of the town centre. Many of these have become well-known in their own right, but King’s College is perhaps best known, because of its magnificent white ‘perpendicular style’ Chapel, and its musical tradition. Along the banks of the river are beautiful parks and gardens known as ‘the Backs’. Something of the beauty of Cambridge is captured in this photograph of Clare College.                  

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The mathematical bridge

Spanning the River Cam behind Queens’ College is a little wooden footbridge known as the mathematical bridge.

It is said that this bridge was originally designed by mathematician Sir Isaac Newton, in such a way that it required no nails or other fastenings.

At a later date, the story goes, the bridge was dismantled for repair work, but could not then be reassembled without the use of bolts. This is how the visitor finds it today. For more on the history (and true facts!) of this bridge, see































Isaac Newton

Isaac Newton was born in 1642 in the hamlet of Woolsthorpe. As a youth he early demonstrated his creativity in devising clever mechanical models. At age 18 he entered Trinity College, Cambridge, and went on to a career of lecturing and research which lasted 18 years. In 1696 he was appointed Warden of the Mint, nine years later he was elected president of the Royal Society, and in 1705 he was knighted. He died in 1727. He published work in many fields: optics, gravitation, cubic curves, the binomial theorem, ... . His greatest work is his Principia Mathematica. However, he is best remembered today as one of the founders of the differential calculus. A new Isaac Newton Institute is located in Clarkson Road, close to the city centre.

 Project  Knowing about the lives of great mathematicians, and how different parts of mathematics came to be discovered really brings mathematics to life. Use your library or the Internet to discover more about mathematicians like Newton.

























The method of fluxions

Newton considered fluents (flowing quantities) and fluxions (their rates of change). (Today we speak of variables and their derivatives.) He knew that the slope of a line segment was found by taking the difference of the y-coordinates of the endpoints and dividing by the difference of the x-coordinates. Thus in the figure, the slope of segment PQ is y /x. (Here symbol is the capital Greek letter delta, and x stands for a small change in x.) Newton wondered if it is possible to find the rate of change (slope) at a point of a general curve. Suppose we take the illustrated curve with equation y = x2. Newton’s argument was:

y - x2 = 0
y + x - (x + x)2 = 0
y + y - x2 - 2x.x - (x)2 = 0
y - 2x.x - (x)2 = 0
y - 2x.x = 0
y /x = 2x		 P(x,y) lies on the curve.
Q lies on the curve.
Expanding.
Use the fact that y = x2.
x is small so ignore (x)2.
Dividing through by x 0.
Now take the limit as x tends to 0. The slope is 2x.

Although the method is simple, it led to much controversy. For when we divide through by x it is assumed that x 0. How then can we let x tend to 0 in taking the limit? The method gave correct answers, so continued to be used, but it was centuries later that mathematicians really understood the limit concept.

 






















The binomial theorem

Newton was aware of the common binomial expansion, say for integer powers of (1 + x). That is, if n is an integer,

Newton had the brilliant insight that this formula might also hold if n is a fraction.

  Try substituting n = in the above formula. You will notice that now the formula continues indefinitely as a series, and we require |x| < 1. Why is this? Evaluate the left hand side and the sum of the first few terms on the right for some small values of x. Do you think Newton got it right?