I’m supposed to be doing a (fairly basic) Open University Maths course this year.  Just because it seemed like a good idea at the time, which was in September when I’d spent the whole summer being nothing but a Mother.  Typically, I also decided to start advanced French conversations classes, and begin a new Philosophy course comparing the Continental and Analytical traditions.  What with a husband, two children, a dog, two houses, and two part-time jobs, inevitably I fail.  Inevitably.  Usually I get a viral illness which is my body screaming “Will you please slow down!”.  I’ve had a sore throat and felt exhausted for the last three weeks.  Which is why I had a party for twenty five people on Friday.  One day I will learn.  But I’m better now, thanks largely to a homeopathic remedy called ferrous phosphate.  This compound is used in garden chemicals such a slug killers.

Back to the Maths.

Unlike some Open University course, this one involves real, face-to-face tutorials with our tutor.  One a month on a Saturday morning.  Our tutor never takes off her hat and I spend quite a lot of the tutorial wondering why not, but that, too, is irrelevant.  This is about circumferences. 

Did you know that the circumference of the earth was first accurately measured by a Greek in  around 250BC? 

(Skip the next bit and jump to the blue unless you are very keen)

He did this using the following equation:

(360° ÷ θ) x (s)

In this calculation, (s) is the distance between two points that lie north and south of each other on the surface of the Earth. If you were to draw a line from each of these points to the center of the Earth, the angle formed between them would be θ.

Obviously, Eratosthenes could not go to the center of the Earth, so he got the angle measurement using the rays of the sun. At noon on the longest day of the year, the summer solstice, the sun shone directly into a deep well at Syene (which is now Aswan, Egypt), casting no shadow.

View of Earth with Alexandria and Syene marked.

Eratosthenes calculations were based on the assumption that Syene lay on the Tropic of Cancer and that Alexandria lay directly north of Syene on exactly the same line of longitude. Click on the image for larger view.

At the same time in Alexandria, Egypt, he found that the sun cast a shadow equivalent to about 1/50th of a circle or 7.12°. Eratosthenes combined this measurement with the distance between Syene and Alexandria, about 4,400 stades.

If we plug these numbers into the above equation, we get: (360°÷ 7.12°) which equals 50; and 50 x 4,400 equals 220,000 stades, or about 25,000 miles. The accepted measurement of the Earth’s circumference today is about 24,855 miles (Smith, 1988)



Eratosthenes method of calculating the Earths' circumference

Eratosthenes could not have been in both Alexandria and Syene at once.  Even calculating how far apart they were required a fair degree of estimation.  This is an account of how he found the information necessary for the calculation:

“Among the travelers’ tales circulating in Alexandria at the time 
was one about a well in Syene, up the Nile at the first cataract, 
where the Sun shone directly into its deep waters on the longest 
day of the year, June 21 [sic].  Nothing so strange had ever been 
seen in Alexandria.  To Eratosthenes the story meant that Syene 
(the Greek name for Aswan) must lie at the northern boundary of 
the tropics … 
With this in mind, Eratosthenes sensed that he had all he needed 
to measure the circumference of Earth.  …  If he could measure 
the angle of a shadow in Alexandria at noon on that day, Eratosthenes 
thought, he just might be able to measure the size of the Earth without 
ever leaving the grounds of the library. … 
From what Eratosthenes could learn from travelers, Syene was supposed 
to be due south of Alexandria, which was particularly convenient … 
if he determined the distance between the two he would know the exact 
length of an arc of the meridian … and therefore a part of the 
circumference of the Earth.  Eratosthenes was told that a camel caravan 
needed 50 days to make the journey and that camels usually traveled 
100 stadia a day.  The distance would thus be about 5,000 stadia. 
Royal surveyors, according to some versions of the story, may have 
paced off part of the route as an attempt to confirm the distance.”

(From  John Noble Wilford, The Mapmakers, p19-23)

Every tutorial begins with a nice little problem, just to get the left side of the brain woken up.  Below are two problems from the first and third tutorials.  The first is a well known problem, the second came from the Guardian on Saturday.  Answers in a while or you can find them yourselves on the internet.

First Problem

Imagine you have a very long piece of string.  It’s long enough to stretch all the way around the earth.  It’s a 40,074km piece of string.  You stretch it all round the world so that it fits snugly.  We won’t bother ourselves with the extra string you might need to knot it, because this is a theoretical maths problem, not real life.

Then you find an extra metre of string in your pocket and decide to insert it into the circle of string, so that it is now 1m longer.  Again, forget about knots and stuff.  Now your string is slightly off the ground because of the extra string you have added. 

How high off the ground would the string be?

Would it be any different if it was a much shorter piece of string around, say, a football?

Second problem

In a water cooler moment at the Department of Unclear Physics, cantankerous Professor Tantrum points out that if a 72 degree segment be discarded from a thin paper circle of radius 10cm, the remaining shape may be formed into a right circular cone by joining the two sides of the circle left by removing the segment.  What is the height of the cone?