# Thales of Miletus Contribution in Mathematics Thales of Miletus is known to be one of the seven sages of ancient Greece. He was a philosopher, scientist, and mathematician. He was born around 624 BC and died in 545 BC. There is no confirmatory statement on how he died. Some say that he died from hunger, while others say that he died of heatstroke when he was playing games. Thales went to Egypt at a very young age to make his career as a philosopher and scientist. After completing his academics, he came back to the Greek city of Miletus where he opened up a school. But there are other sources as well, which claim that he never went to Egypt since Miletus had a permanent colony there. He managed to perform scientific experiments along with his school teaching. He was the first one who was involved in scientific philosophy. Instead of using mythology, Thales used realistic theories and hypotheses for elaborating natural phenomena to the world. According to Thales, “All things are full of God.” According to him, magnetic objects possess souls as magnetic objects can attract iron (in Greek view, the soul can move or change other things). However, no writings of Thales are available so it is a bit difficult to talk about all of his achievements. With the help of a geometrical approach, Thales estimated the height of a pyramid, and he also calculated the distance of a ship from the shore. He had also given several important theorems in geometry that we will study further. Not only mathematics, but his contribution to astronomy is also very significant. It is believed that Thales knew how to predict the solar eclipse. According to him, Earth is a flat surface that floats on water, and he explained the phenomena of earthquakes with the help of this hypothesis. Moreover, his understanding of geometry can be easily realized by his notion

apanta gar chorei (Μέγιστον τόπος· ἄπαντα γὰρ χωρεῖ.)

The greatest is space, for it holds all things.”

## 1. Thales’ Five Geometric Theorems

These theorems have been used and applied by every one of us. These theorems may sound very easy today but were considered to be of great significance at the time of Thales. These theorems help us to calculate many concepts of trigonometry when there is the inclusion of parallel lines and also to calculate proportions in geometric figures having parallel lines. These theorems are:-

1. A circle is bisected by any diameter.
2. The base angles of an isosceles triangle are equal.
3. The angles between two intersecting straight lines are equal.
4. Two triangles are congruent if they have two angles and one side equal.
5. An angle in a semicircle is a right angle.

## 2. Height of Great Pyramids

Thales is credited with measuring the height of the Great Pyramids of Cheops. When he visited Egypt, he encountered the great Pyramids of Giza. He enquired the Egyptian priests about the height of pyramids. After getting no response from them, he started measuring the height himself. Since no proof of his writings is available, there are several assumptions on how he measured the height of pyramids.

• One possibility is that he might have used a stick and inserted it into the ground. Then, he waited for the time when the shadow of the stick became equal to the length of the stick. At that moment of the day, the height of the shadow of the pyramid will be equal to the height of the pyramid.
• Another possibility is that he must have used the concept of similar triangles. ## 3. Thales’ Theorem of Interception

This theorem is also known as Basic Proportionality Theorem or Side Splitter Theorem. It tells us about the ratio of line segments formed when two intersecting lines are intercepted by a pair of parallel lines. Here, two intersecting lines XC and XB are intersecting at point X and are intercepted by two parallel lines ‘n’ and ‘m’. Then, according to this theorem,

|XD|:|DC|=|XA|:|AB| , |XC|:|XD|=|XB|:|XA| and |XC|:|DC|=|XB|:|AB| The converse of this theorem is also true, i.e., if the two intersecting lines are intercepted by two arbitrary lines and |XD|:|DC|=|XA|:|AB| holds then the two intercepting lines are parallel. Some applications of this theorem are as follows:-

• It helps in dividing a line segment into a given ratio. • It helps in estimating the height of the pyramid. • It helps in measuring the width of a river. • It helps in measuring the distance of the ship from the shore. ## 4. Thales’ Theorem

This theorem states that if A, B, and C are distinct points on a circle where the line AB is a diameter, then angle ABC is a right angle. This theorem is sometimes attributed to Pythagoras. But scientists believed that Thales proved this theorem by using results ” The base angles of an isosceles triangle are equal” and “sum of angles of a triangle is 180°”. That’s why this theorem is named after him.

Some applications of Thales’ Theorem are given below:-

• It can be used to draw tangents to a given circle that passes through a given point. In the circle k with centre O, a point P outside k. Join OP and a point H on OP such that OH is the radius of the circle, that intersects circle k at right angles represented by points T and T.´ • This theorem can also be used to find the centre of the circle by using a set square or any right-angled object. In this, the set square is placed at different points on the circumference. Then, the line joining the points where the two sides of the set square intersect the circumference, will give the diameter. Repeating this process one more time will give another diameter, and where these two diameters intersect will be the centre of the circle. 