Neusis construction facts for kids

Neusis construction, showing how the ruler is used.

The neusis is a special way of drawing shapes that was used by ancient Greek mathematicians. Back then, people didn't have easy ways to measure distances or use algebra. Instead, they learned about geometry by moving sticks of a certain length around on paper.

What is Neusis?

The word "neusis" comes from a Greek word meaning "to incline towards." This drawing method helps you fit a line of a certain length (let's call it a) between two other lines (l and m). The trick is that this line, or its extension, must also pass through a specific point P.

Imagine you have a special ruler called a 'Neusis Ruler'. This ruler can spin around point P, like a pin is holding it there. One end of the ruler has a mark (like a yellow eye in the picture). Another mark (a blue eye) shows the length a from the first mark.

To use it, you slide the yellow eye along line l. You keep moving it until the blue eye mark on the ruler touches line m. When this happens, you've found the correct position for your line. The picture shows this line as a dark blue bar.

In a neusis construction:

• Point P is called the pole.
• Line l is the directrix (or guiding line).
• Line m is the catch line.
• The length a is called the diastema, which means "distance" in Greek.

Why Was Neusis Used?

Neusis trisection of an angle. This method can divide an angle into three equal parts.

Neusis constructions were very important because they could solve geometry problems that were impossible to solve using only a compass and straightedge.

For example, it could help divide any angle into three equal parts (called "trisection of an angle"). It could also help draw shapes like a regular heptagon (a seven-sided polygon). Famous mathematicians like Archimedes (who lived from 287 to 212 BC) often used neusis.

However, over time, the Greeks started to use neusis less and less.

Why Did Neusis Become Less Popular?

Historians believe that a Greek mathematician named Oenopides (around 440 BC) was one of the first to prefer using only a compass and straightedge over neusis.

Another mathematician, Hippocrates of Chios (around 430 BC), might have helped spread the idea that mathematicians should avoid neusis if possible. He even wrote one of the first organized geometry textbooks. About 100 years later, Euclid also seemed to dislike neusis in his very famous book, The Elements.

Later, around the fourth century BC, the ideas of Plato became very popular. Plato's ideas led to a ranking of different ways to do geometric constructions:

• First group: Constructions using only straight lines and circles (with a compass and straightedge). These were seen as the most "abstract and noble."
• Second group: Constructions that also used conic sections, like ellipses, parabolas, and hyperbolas.
• Third group: Constructions that needed other tools, like neusis. These were seen as more "mechanical and earthly."

Because of this ranking, Greek mathematicians eventually thought that using neusis was only okay if the first two, "higher" methods couldn't solve the problem. Neusis became a last resort. The mathematician Pappus of Alexandria (around 325 AD) even said it was a "not inconsiderable error" to use neusis when other methods could have worked.

Related pages

• Squaring the circle

See also

In Spanish: Método neusis para niños