Thales of Miletus
Thales was born in Miletus around 624 BC and is sometimes considered the father of science, because he was the first to put forth the idea that physical phenomenons are caused by other physical phenomenons rather than by some supernatural intervention. He stressed that the workings of the world could be determined by applying logic and underlying theories. One of the things he tried to explain in this way is earthquakes. Until this point the accepted explanation in the greek world had been that earthquakes were caused by Poseidon striking the earth with his trident and thereby creating an earthquake.
Thales however was of the opinion that land floats on water and that it was the movement of this water that made earthquakes occur the same way the waves of an ocean would make a boat rock. Water was very important to Thales as it was also his belief that water was the original matter and that all else was in a way derived from water.
An anecdote told about Thales states that he was once asked to find out what was wrong with a donkey which was behaving oddly. The donkey was used to carry salt from a saltmine and one day as it was crossing a river, it had fallen and some of the salt had dissolved in the water, making the burden much lighter for the donkey to carry. After this day the donkey would let itself fall into the river each time it crossed. The owners, baffled by this odd and rather costly behaviour, took the donkey to be looked at for injuries that would make it fall, but none were found. Then Thales was asked and after observing the donkey he determined that it let itself fall on purpose and not due to any injury. So, he filled it's sacks with sponges instead of salt and as it was used to, the donkey fell in the water. But this time the sponges soaked up water, making the burden much heavier. After repeating this for a few days, the donkey stopped falling over and Thales had 'cured' it.
Thales however was of the opinion that land floats on water and that it was the movement of this water that made earthquakes occur the same way the waves of an ocean would make a boat rock. Water was very important to Thales as it was also his belief that water was the original matter and that all else was in a way derived from water.
An anecdote told about Thales states that he was once asked to find out what was wrong with a donkey which was behaving oddly. The donkey was used to carry salt from a saltmine and one day as it was crossing a river, it had fallen and some of the salt had dissolved in the water, making the burden much lighter for the donkey to carry. After this day the donkey would let itself fall into the river each time it crossed. The owners, baffled by this odd and rather costly behaviour, took the donkey to be looked at for injuries that would make it fall, but none were found. Then Thales was asked and after observing the donkey he determined that it let itself fall on purpose and not due to any injury. So, he filled it's sacks with sponges instead of salt and as it was used to, the donkey fell in the water. But this time the sponges soaked up water, making the burden much heavier. After repeating this for a few days, the donkey stopped falling over and Thales had 'cured' it.
Thales' Major Contributions
Thales' major contributions to mathemathics lie in the field of geometry. Some of his findings seem rather trivial today and even at the time they were not all new. What was new, was that Thales provided proof why these findings were true.
He also provided proof that:
- If a straight line is drawn through a circle and this line touches the center of the circle, then the area of the circle has been divided into two equally large part. Or to use other words, that drawing the diameter of circle will half it.
- If a triangle has two sides that are equally long (this is called an isosceles triangle), then the angles between these lines and the third line must always be the same.
- If two straight lines cross each other, then the angles on the opposing sides must always be equal to each other.
- If a line is drawn and two random angles chosen at either end of this line and thereby a triangle is created, then any line with this length and these angles at it's ends must be identical to the first triangle created.
Similar Triangles
Thales used similar triangles: Triangles with the same angles, but different side lengths, to determine the height of the pyramids in Egypt.
He did this by putting a stick into the ground so that it formed a 90° angle with the ground, then measured the length of the stick and it's shadow. Then he measured the length of the pyramid's shadow and added half the length of the pyramid's side to the shadow. This created two similar triangles and allowed him to use the relation between the stick and it's shadow, to calculate the relation between the shadow of the pyramid and it's height.
In this example: 1/1,2 = 0,833 This is the ratio between the height of the stick and the shadow it casts. Therefore the pyramid's height would be found by completing this calculation: 176*0,833 = 146.61 m |
He also used similar triangles and their relations to each other to find the distance to ships.
He would stand at one random spot (A) and draw a line to the ship, then stand at another random spot (B) and draw another line to the ship from there. Then he would draw a line at a right (90°) angle to A and continue this line until it would be possible to create another right angle from this line which would intersect with B. Then he would measure the line from A to the point where it intersects with the line between the ship and B, and the distance between C and the line where it intersects with the line between B and the ship. This would give him the ratio and from that point all he had to do was measure the line between B and C, and B and the place it intersects with the line between A and C. This builds upon his proof that if two straight lines cross each other, then the angles on the opposing sides must always be equal to each other.
The Slope of a Line
You might have noticed in the description of the measurement of the pyramid it was simply assumed that the sun's rays would hit the pyramid and the stick at the same angle and thereby create the similar triangle. This is of course because the sun is so much bigger than the earth and so much further away. Any slight difference in angle is entirely negligeble. But that also means that the sun's rays are always parallel as long as they are viewed at the same time of day. That means that the ratio between height and shadow will be the same for all things that stick out of the ground at a right angle.
This relation allows us to calculate the slope or gradient of the line created by the rays of the sun. The slope of a line is usually expressed in decimals while gradients are often expressed in percentages. To find the slope one simply uses this formula: "Height of structure” / "Length of shadow" It can of course also be used to find the slope of other things by relplacing the lenght of the shadow with the length of whatever one desires to find out. |
Measuring Angles
For those who are new to measuring angles, there is an easy tool that will help you do that. It's called a protractor. You use it by placing the protractor on top of the angle you wish to measure and making sure the line follows the line leading out to the 0 on the protractor. The point of measuring occurs where the cross with the circle inside it is.
In this example we can see that the angle that's being measured is the lower right angle and that it is 47° |