Real Mathematics: Geometry – #7

Ancient Greek philosophers used only compass and an unmarked ruler to come up with incredible results in geometry. In this article, I’ll be talking about what kinds of methods were used in geometry’s holy book Euclid’s Elements in order to cut a segment into as many equal parts as we’d like.

So far, I’ve showed in my previous articles how to bisect any segment and any angle with just using a compass and an unmarked ruler.

How about trisecting a given segment?

Trisecting a segment is shown in Euclid’s Elements, Book 6, Proposition 9. Let me try to show you Euclid’s method in a nutshell.

Define any two points A and B on a plane and construct a line segment between them. Choose a point C which is not on the line segment AB.

Draw a line from the point A passing through C.

Use compass to mark the points D, E and F on the line AC such that AD, DE and EF are equal line segments.

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Connect F and B. Then draw line segments from points D and E which should be parallel to BF.

Thus, line segment AB is cut into three equal parts. Actually, using Euclid’s method, we can divide AB into as many equal parts as we’d like.

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It might seem easy at first sight but when you think about it, there are a few information we need in order to use Euclid’s method for trisecting a line segment. If you are careful enough, you might have realized that one should know how to draw a parallel to any given line. In order to do that, we should know the method from Elements Book I, Proposition 31, which requires how to create a specific angle on a random point (Elements Book I, Proposition 23). And it doesn’t end here: Proposition 23 requires knowledge of proposition 22 from the same book.

Suddenly Euclid’s ingenious methods seem a bit complicated.

Muslim Ingenious

Elements were written more than 2400 years ago and humanity should thank a few Muslim scholars for its existence today!

Al-Nayrizi (865-922) was one of the first Muslim philosophers who read and commented on Elements. Actually you could still buy his commentary on Elements even today on amazon.com even though it costs a little over 200 USD.

Al-Nayrizi has come up with a truly magnificent method for cutting a segment into equal parts.

Let’s start with drawing a straight line AB. Al-Nayrizi’s method requires only the following knowledge: Drawing lines through A and B respectively which are perpendicular to the line AB.

If we’d like to cut AB into n equal parts, we should mark n-1 equal segments on these perpendicular lines. We could easily do that using a compass. Let’s assume that we’d like to cut AB into four equal parts. That means we need 4-1=3 equal segments on the perpendicular lines.

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Now all we have to do is to connect the dots as shown below.

In the end we managed to cut AB into four equal parts.

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M. Serkan Kalaycıoğlu

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