Ep. 38 Right Angles

This week I look at right angles and try to figure out who found them first. Come along for the ride this week as I ask, right angles, how did that happen? The meaning of right in right angle possibly refers to the Latin adjective rectus which means straight. erect, , upright,  In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. Right angles are everywhere, from your coffee table to the roads in your city. A triangle that has a right angle is also known as a right triangle, to which you can apply special properties such as the Pythagorean Theorem. Right angles are important to recognize and to understand, since they’re used everywhere from geometry to trigonometry to real-life applications. Babylonian trigonometry The Babylonians discovered their own unique form of trigonometry during the Old Babylonian period (1900-1600BCE), more than 1,500 years earlier than the Greek form. Remarkably, their trigonometry contains none of the hallmarks of our modern trigonometry - it does not use angles and it does not use approximation. The Babylonians had a completely different conceptualization of a right triangle. They saw it as half of a rectangle, and due to their sophisticated sexagesimal (base 60) number system they were able to construct a wide variety of right triangles using only exact ratios. Plimpton 322 Tablet We now know that the Babylonians studied trigonometry because we have a fragment of a one of their trigonometric tables. Plimpton 322 is a broken clay tablet from the ancient city of Larsa, which was located near Tell as-Senkereh in modern day Iraq. The tablet was written between 1822-1762BCE. In the 1920s the archaeologist, academic and adventurer Edgar J Banks sold the tablet to the American publisher and philanthropist George Arthur Plimpton. Plimpton bequeathed his entire collection of mathematical artefacts to Columbia University in 1936, and it resides there today in the Rare Book and Manuscript Library. In 1936 a clay tablet was excavated at Shush (Khuzistan region of Iran) some 350km from the ancient city of Babylon on which was inscribed a script that was only translated as late as 1950. The text provided confirmation that the Babylonians' measured angles using the figure of 360 to form a circle The inscription on the tablet shows the ratio of a perimeter of a regular hexagon to the circumscribed circle i.e. Six sides of a hexagon times their base of 60 = 360. The Babylonian approach is also much simpler because it only uses exact ratios. There are no irrational numbers and no angles, and this means that there is also no sin, cos or tan or approximation. The Babylonians discovered a method of finding Pythagorean triples, that is, sets of three whole numbers such that the square of one of them is the sum of the squares of the other two. The first known instrument for measuring angle was possibly the Egyptian Groma an instrument used in the construction of massive works such as the pyramids.  The Groma consisted of 4 stones hanging by cords from sticks set at right angles; measurements were then taken by the visual alignment of two of the suspended cords and the point to be set out. It was rather limited in is application due to the fact that it was only able to be used on fairly flat terrain and its accuracy limited by distance. Rule of 3-4-5 Throughout history, carpenters and masons have known a quick way to confirm if an angle is a true "right angle". It is based on the most widely known Pythagorean triple (3, 4, 5) and so called the "rule of 3-4-5". From the angle in question, running a straight line along one side exactly 3 units in length, and along the second side exactly 4 units in length, will create a hypotenuse (the longer line opposite the right angle that connects the two measured endpoints) of exactly 5 units in length.