Neusis construction

In geometry, the neusis (νεῦσις; from Ancient Greek νεύειν (neuein) 'incline towards'; plural: νεύσεις, neuseis) is a special way to solve problems using shapes and measurements. This method was very important a long time ago, especially for Greek mathematicians. They used neusis to create exact lines and points that could not be made with just a ruler and compass alone.

Neusis works by sliding a ruler or a tool until it touches two lines or points in just the right way. This clever trick helped ancient thinkers solve difficult puzzles and make beautiful designs. Even though we have better tools today, learning about neusis helps us understand how smart and creative people in the past were when they studied shapes and space.

Geometric construction

The neusis construction is a way to fit a straight line of a certain length between two lines. One end of this line must touch the first line, and the other end must touch the second line, while the line is moved so it points toward a special point called the pole.

This method can be done with a special ruler that has marks on it. You move one mark along the first line until another mark touches the second line. Depending on whether the lines are straight or curved, this method can help solve different kinds of math problems.

Trisection of an angle by line–circle neusis

This method shows how to split an angle into three equal parts using a special drawing trick called neusis. You start with two lines that meet at a point, forming an angle. Then, you draw a circle and use a ruler to slide it until it touches both the circle and one of the lines in just the right way. By doing this, you can find a new point that helps you divide the original angle into three equal pieces.

Use of the neusis

Neuseis were important tools in ancient times because they could solve some geometry problems that couldn’t be solved with just a compass and straightedge. For example, they could split any angle into three equal parts and double the size of a cube. Famous mathematicians like Archimedes and Pappus of Alexandria used neuseis, and later Isaac Newton also used them. But over time, this method was used less and less.

Neusis helped in creating shapes called regular polygons with a certain number of sides. Some shapes, like those with 23 or 29 sides, couldn’t be made with neusis, while others, like the 3-, 4-, and 5-sided shapes, could be made with just a compass and straightedge. There are still many questions about which shapes can be made using neusis, especially those with 11, 25, or 31 sides.

Waning popularity

T. L. Heath, a historian of mathematics, suggested that the Greek mathematician Oenopides (around 440 BC) was the first to prefer using compass and straightedge tools instead of a special method called neusis. This idea might have been shared by Hippocrates of Chios (around 430 BC), who came from the same island as Oenopides and wrote one of the first organized geometry books. Later, around 300 BC, Euclid also avoided neusis in his famous book, The Elements.

As time went on, thinkers like Plato influenced ideas about geometry. They created a list of three types of geometric tools, from simplest to most complex. The simplest used just straight lines and circles. The next added curves like ellipses, parabolas, and hyperbolas. The most complex included methods like neusis. Over time, neusis was only used when the simpler methods couldn’t solve a problem. Using neusis when simpler tools would work was seen as a big mistake by later mathematicians like Pappus of Alexandria (around 325 AD).