47th Problem of Euclid

47th_Problem_of_Euclidhttp://www.phoenixmasonry.org/47th_problem_of_euclid.htm The 47th Problem of Euclid (A.K.A. The Pythagorean Theorem) The problem above is the 47th Problem of Euclid. It is an invention by an ancient Greek geometer, Pythagoras, who worked for many years to devise a method of finding the length of the hypothenuse of a right angle triangle. Pythagoras is credited with having first proved the rule successfully applied to the problem. The rule is that the square of the base added to the square of the altitude equals the square of the hypothenuse. The base of a right angle triangle is the side on which it rests, marked B 1 of 6 5/6/2009 11:22 AM Some say that the Greek mathematician and geometer Pythagoras. squared or multiplied by itself. In any case. 4.such as 3. 16. The hypothenuse is the connecting side of the triangle. and what was the significance of the problem which led to such a demonstration by the ancient philosopher? The knowledge contained in this proposition is at the bottom of all systems of measurement and every mechanic at the present day makes use of it consciously or unconsciously. which ranked as the highest kind of religious offering. 4. it was he who supplied the PROOF that the angle formed by the 3 : 4: 5 triangle is invariably square and perfect. described in Masonic lectures as "our worthy brother. All right angle triangles can be figured in the same manner. The altitude. equals 64. The altitude is the height and is marked A. that is a sacrifice of one hundred bulls. By adding these together we have 100. David J. Stab one stick in the ground and arrange a knot at the stick. 5 and 12. 8. This forces the creation of a 3 : 4 : 5 right triangle.org/47th_problem_of_euclid. whether it be the land surveyor blocking out a township. which we know is 10. 20. equals 36. stretch three divisions away from it in any direction and insert the second stick in the ground. The angle between the 3 units and the 4 units is of necessity a square or right angle. upon completing the proof.htm in the Figure above. but only multiples of the length of the three sides come even -.thin ones. which is the square of the hypothenuse. The ancient Egyptians used the string trick to create right angles when re-measuring their fields after the annual Nile floods washed out boundary markers. or the gardener 2 of 6 5/6/2009 11:22 AM . squared. the two ends joining.all whole numbers -." also went to Egypt and learned it there on his own. Compiled by: Wor. Their skill with this and other surveying methods led to the widely held (but false) belief that the Egyptians invented geometry (geo=earth.) Then get 3 sticks -. then place the third stick so that it falls on the knot between the 4-part and the 5-part division. It remains but to extract the square root of 100. The base. 6. (The divisions must be correct and equal or this will not work. and many others.is also known as "the Egyptian string trick.47th_Problem_of_Euclid http://www. Thales the Greek supposedly picked the string trick up while traveling in Egypt and took it back to Greece. It is also said that he actually sacrificed a hecatomb. and 5 -. metry=measuring).phoenixmasonry." The "trick" is that you take a string and tie knots in it to divide it into 12 divisions. therefore 10 is the length of the hypothenuse or third side of this right angle triangle. How is this forty-seventh proposition the foundation of all Masonry. marked C above. just strong enough to stick them into soft soil. of course. Lettelier for a Public Oration and Lecture The 47th problem of Euclid (called that because Euclid included it in a book of numbered geometry problems) in which the sides are 3. and then swing round the loose ends toward the west until they intersected and a right angled triangle was thus formed. he exclaimed Eureka. They were called. and if we will carry ourselves back in imagination to a time when this knowledge was still unknown. among the most distinguished. might well be one of the genuine secrets of a Master Mason.47th_Problem_of_Euclid http://www. He may not know anything about geometry. but the "rule of thumb" by which he works has been deduced from this proposition. To the operative mason it affords a means of correcting his square. It teaches Masons to be general lovers of the arts and sciences”. in Egypt. wireless TV or telephones.phoenixmasonry. it was Pythagoras who discovered it. harpedonaptae--meaning rope stretchers. The ancient temple builders in the long centuries before Christ were most punctilious in setting their temples due east and west. Each book contained many geometric propositions and explanations. we will realize that its discovery was an event of great importance in the history of architecture. The 47th problem was set out in Book 1. he erected this. which were called “Elements”. which is also known as “The Pythagorean Theorem”. The Discovery of the 47th problem of Euclid: Euclid wrote a set of thirteen books. and 5. These ancient temple builders. and whose sole duty it was to lay out the foundations of public edifices.htm measuring out his tennis court. 4. 4 divisions along the other. when. "I have found it. for if he wishes to test its accuracy he may readily do so by measuring off 3 divisions along one side. The knowledge of how to form a square without the possibility of error has always been accounted of the highest importance in the art of building. Why is it called by both these names? Although Euclid published the proposition. formed the square. This they secured by stretching a rope north and south divided divided into three parts in the proportion of 3. and in times when knowledge was limited to the few. To the practical builder the knowledge is invaluable. and the distance across must be 5 if the square is accurate. would be termed experts or specialists. in the joy of his heart.org/47th_problem_of_euclid. or Masonry. On this subject he drew out many problems and theorems. in modern phrase. and. We learn from the third degree lecture that: “ This wise philosopher (Pythagoras) enriched his mind abundantly in a general knowledge of things. or the carpenter calculating the pitch of a roof. and the centre was a point round which they could not err. and more especially in Geometry. by means of the centre. So exacting were they on this point that there was organized a set of men who. an epoch-making event to be ranked with such modern discoveries as those of the law of gravitation. and their next step was to get the east and west line exactly at right angles. They first laid out the north and south line by observation of the stars and the sun." and upon the discovery of which he is said to have sacrificed a hecatomb. in the Greek language signifying. Here also is the obvious answer to the question why it is customary at the erection of 3 of 6 5/6/2009 11:22 AM . (the Egyptian string trick again) fastening down the centre part by pegs. and in total Euclid published 465 problems. and space travel. Mark the two points where the bisecting line crosses the circle's circumference. To create a 1:1 square root of 2 right triangle. On soft ground.org/47th_problem_of_euclid. Using the compass again. the first two points -. also known as an isosceles right triangle. you need a compass and a straight edge -. The type of triangle most often used to demonstrate the 47th problem in Masonry is not the 3 : 4: 5 but the 1: 1 : square root of 2 form.where you marked the crossing of the bisecting diameter through the circle's circumference -. 4 of 6 5/6/2009 11:22 AM . specifically in the checkered floor and its tessellated border. Now connect the three points you have marked -and there is your 1 : 1 : square root of 2 right triangle. have we anything in our present ritual which might be relative in any way to this method of proving the square or obtaining a right angle without the possibility of error and which may have been connected with the instruction given in purely operative masonry. The question arises. as the jewel of office for a Past Master. To Freemasons. Then use the straight edge to bisect the circle through the center-point marked by the compass. These are the two "boundary" lines of conduct sometimes symbolized on Masonic tracing boards by the Two Saints John and sometimes referred to as indicators of the Summer and Winter Solstices. Therefore. of course.htm all stately and superb edifices to lay the foundation stone at the north-east angle of the building.familiar tools to the Craft. It is in this form that the Pythagorean theorem is most often visually encountered in Masonry. as a geometric proof on Lodge tracing boards. use the compass to inscribe a circle.47th_Problem_of_Euclid http://www. The square and the cube which are 1 unit on each side are of great symbolic meaning to Masons. whereon the feast days of those two saints occur. erect a perpendicular line that bisects this diameter-line and mark the point where the perpendicular touches the circle.phoenixmasonry. the bisection of the square into a pair of 1 : 1 : square root of 2 triangles has important Masonic connotations.can also be used to construct two further perpendicular lines. and in the form of some Masonic aprons. Th' assay so hard. we learn in the Third Degree that perfect knowledge is not to be attained on this side of the grave. No wonder that Pythagoras sacrificed an hecatomb! No wonder that Anderson speaks of this proposition as the foundation of all Masonry! The only wonder is that modern Freemasonry has lost sight of the importance of this symbol. we strive to attain it in the Second Degree as the summit of all knowledge. W e seek it in the First Degree under the symbolism of Light. the craft so long to lerne. The opening catechism of the Third Degree fits so accurately the process of forming a perfect square as used by the rope stretchers of ancient Egypt that the belief forms in the mind that we have here a fragment of the old operative instruction preserved in the mosaic of speculative Masonry. one element and one far-off divine event to which the whole creation moves. but everywhere it is taught as the unifying bond of the Craft. It is from the East and towards the West that one's steps are directed to find that which was lost.Chaucer.47th_Problem_of_Euclid http://www. Our consideration of the subject has brought us back again to the central point of modern Speculative Freemasonry--the knowledge of God--to which all our symbolism points. cementing us as a common brotherhood with a common Father." -. so sharp the conquering. even God--that God who ever lives and loves.phoenixmasonry. that point round which a Master Mason cannot err. one Law.org/47th_problem_of_euclid. "The lyf so short. 5 of 6 5/6/2009 11:22 AM .htm We also have a fragment of great interest in the ceremony of opening the Lodge in the Third Degree. one God. and it is with (by means of) the centre. 6 of 6 5/6/2009 11:22 AM .47th_Problem_of_Euclid http://www.phoenixmasonry.htm Museum Home Page Phoenixmasonry Home Page The Fine Print Copyrighted © 1999 . Inc.2007 Phoenixmasonry.org/47th_problem_of_euclid.