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Early Proofs of the Pythagorean Theorem By Leonardo Da Vinci, Ptolemy, Thabit ibn Qurra, and Garfield
While scholars will argue about whether or not Pythagoras and his ancient school actually discovered the theorem that bears his name, it is still one of the most important theorems in mathematics. Evidence that the ancient Indians and Babylonians knew of its principles exists but no written proof of it surfaced until sometime later in Euclid’s Elements Book I Proposition 47 (Euclid 350-351). While many other proofs of Pythagoras have surfaced in the modern age, it is some of the proofs between Euclid and the present that bear interesting techniques and ideas that reflect the inner beauty of mathematical proofs.
While he may be known for his astronomy better, Claudius Ptolemy (b. 85 Egypt d. 165 Alexandria, Egypt) devised one of the first alternate proofs for the Pythagorean Theorem. His most famous volume of work, Almagest, is divided into 13 books and covers the mathematics of the motions of the planet. After introductory material, Book 3 dealt with his theory of the sun, Book’s 4 & 5 cover his theory of the moon, Book 6 examines ellipses, and Books 7 & 8 look at fixed stars as well as compile a catalog of them. The last five Books cover planetary theory where he “proves” mathematically the Geocentric Model by demonstrating how planets move in epicycles, or orbit in a circle about a fixed point, and this fixed point lies on an orbit about the Earth. While this model is certainly wrong, it explained the empirical data extremely well. Interestingly, he wrote one of the first books on astrology, feeling it was necessary to show the effects of the heavens upon people. Over the years, several notable scientists have criticized Ptolemy from plagiarism to bad science while others have come to defense and praised his efforts. The arguments show no signs of stopping anytime soon, so just enjoy his work for now and worry about who did it later (O’Connor “Ptolemy”).
His proof is as follows: Draw a circle and inscribe in it any quadrilateral ABCD and connect the opposite corners. Choose an initial side (in this case AB) and create ∠ ABE = ∠ DBC. Also, ∠’s CAB and CDB are equal because they both have the common side BC. From this, triangles ABE and DBC are similar since 2/3 of their angles are equal. We can now create the ratio (AE/AB) = (DC/DB) and rewriting that gives AE * DB = AB * DC . Adding ∠ EBD to the equation ∠ ABE = ∠DBC yields ∠ ABD = ∠ EBC. Since ∠ BDA and ∠ BCA are equal, having the common side AB, triangles ABD and EBC are similar. The ratio (AD/DB) = (EC/CB) follows and can be rewritten as EC * DB = AD * CB . Adding this and the other derived equation produces (AE + EC) * DB = AB * DC + AD * CB. Substituting AE + EC = AC gives the equation AC * BD = AB * CD + BC * DA. This is known as Ptolemy’s Theorem, and if the quadrilateral happens to be a rectangle, then all the corners are right angles and AB = CD, BC = DA, and AC = BD, yielding (AC)2 = (AB)2 + (BC)2 (Eli 102-104).
Thabit ibn Qurra
Many people had commented on the Pythagorean Theorem, but Thabit ibn Qurra (b. 836 in Turkey, d. 02.18.901 in Iraq) was one of the first to offer commentary on it and create a new proof for it also. A native of Harran, Qurra made many contributions to Astronomy and Math, including translating Euclid’s Elements to Arabic (in fact, most revisions of the Elements can be traced back to his work). His other contributions to Math include number theory on amicable numbers, the composition of ratios (“arithmetical operations applied to ratios of geometrical quantities”), generalized Pythagorean Theorem to any triangle, and discussions on parabolas, angle trisection and magic squares (which were the first steps towards integral calculus) (O’Connor “Thabit”).
His proof is as follows: Draw any triangle ABC, and from wherever you designate the top vertex (A in this case) draw lines AM and AN so that once drawn ∠AMB = ∠ ANC = ∠ A. Notice how this makes triangles ABC, MBA, and NAC similar. Using properties of similar objects yields the relationship (AB/BC) = (MB/AB) and from this we get the relation (AB)2 = BC * MB. Again, with properties of similar triangles, (AB/BC) = (NC/AC) and thus (AC)2 = BC * NC. From these two equations we arrive at (AC)2 + (AB)2 = BC * (MB + NC). This is known as Ibn Qurra’s Theorem. When the ∠ A is right, M and N fall on the same point and therefore MB + NC = BC and the Pythagorean Theorem follows (Eli 69).
Leonardo Da Vinci
One of history’s most interesting scientist who unveiled a unique proof for the Pythagorean Theorem was Leonardo Da Vinci (b. April 1453 Vinci, Italy, d. May 2 1519 Amboise, France). First an apprentice learning painting, sculpture, and mechanical skills, he moved to Milan and studied geometry, not working on his paintings whatsoever. He studied Euclid and Pacioli’s Suma, then began his own studies into geometry. He also discussed using lenses to magnify objects such as planets (otherwise known to us as telescopes) but never actually constructs one. He realized that the Moon was reflecting light from the sun and that during a lunar eclipse the reflected light from Earth reached the Moon and then traveled back to us. He tended to move often. In 1499, from Milan to Florence and in 1506, to Milan. He was constantly working on inventions, math, or science but very little time on his paintings while in Milan. In 1513 he moved to Rome, and finally in 1516 to France. (O’Connor “Leonardo”)
Leonardo’s proof is as follows: Following the figure, draw a triangle AKE and from each side construct a square, label accordingly. From the hypotenuse square construct a triangle equal to triangle AKE but flipped 180° and from the squares on the other sides of the triangle AKE also construct a triangle equal to AKE. Notice how a hexagon ABCDEK exists, bisected by the broken line IF, and because AKE and HKG are mirror images of one another about the line IF, I, K, and F are all collinear. To prove that quadrilaterals KABC and IAEF are congruent (thus having the same area), turn KABC 90° counterclockwise about A. This results in ∠ IAE=90° + α = ∠ KAB and ∠ ABC = 90° + β = ∠AEF. Also, the following pairs overlap: AK and AI, AB and AE, BC and EF, with all the angles between the lines still maintained. Thus, KABC overlaps IAEF, proofing that they are equal in area. Use this same method to show that the hexagons ABCDEK and AEFGHI are also equal. If one subtracts the congruent triangles from each hexagon, then ABDE = AKHI + KEFG. This is c2 = a2 + b2 , the Pythagorean theorem (Eli 104-106).
Amazingly, a U.S. president has also been the source of an original proof of the Theorem. Garfield was going to be a math teacher, but the world of politics drew him in. Before he rose to the presidency, he published this proof of the Theorem in 1876 (Barrows 112-3).
Garfield starts his proof with a right triangle that has legs a and b with hypotenuse c. He then draws a second triangle with the same measurements and arranges them so that both c's form a right angle. Connecting the two ends of the triangles forms a trapezium. Like any trapezium, its area equals the average of the bases times the height, so with a height of (a+b) and two bases a and b, A = 1/2*(a + b)*(a + b) = 1/2*(a + b)2. The area would also equal the area of the three triangles in the trapezium, or A = A1 + A2 + A3. The area of a triangle is half the base times the height, so A1 = 1/2*(a*b) which is also A2. A3 = 1/2(c*c) = 1/2*c2. Therefore, A = 1/2*(a*b) + 1/2*(a*b) + 1/2*c2 = (a*b) + 1/2*c2. Seeing this equal to the area of the trapezium gives us 1/2*(a + b)2 = (a*b) + 1/2*c2. Foiling out all of the left gives us 1/2*(a2 + 2*a*b + b2) = 1/2*a2 + (a*b) + 1/2*b2. Therefore (a*b) + 1/2*c2 = 1/2*a2 + (a*b) + 1/2*b2. Both sides have a*b so 1/2*a2 + 1/2*b2 = 1/2*c2. Simplifying this gives us a2 + b2 = c2 (114-5).
The period between Euclid and the modern era saw some interesting extensions and approaches to the Pythagorean Theorem. These three set the pace for the proofs that were to follow. While Ptolemy and ibn Qurra may not have had the Theorem in mind when they set about their work, the fact that the Theorem is included in their implications demonstrates how universal it is, and Leonardo shows how the comparison of geometric shapes can yield results. All in all, excellent mathematicians who do Euclid honor.
Barrow, John D. 100 Essential Things You Didn't Know You Didn't Know: Math Explains Your World. New York: W.W. Norton &, 2009. Print. 112-5.
Euclid, and Thomas Little Heath. The Thirteen Books of Euclid's Elements. New York: Dover Publications, 1956. Print.350-1
Maor, Eli. The Pythagorean Theorem: a 4000-year History. Princeton: Princeton UP, 2007. Print.
O'Connor, J J, and E F Robertson. "Leonardo Biography." MacTutor History of Mathematics. University of St Andrews, Scotland, Dec. 1996. Web. 31 Jan. 2011. http://www- history.mcs.st-and.ac.uk/Biographies/Leonardo.html
O'Connor, J J, and E F Robertson. "Ptolemy Biography." MacTutor History of Mathematics. University of St Andrews, Scotland, April. 1999. Web. 30 Jan. 2011. http://www- history.mcs.st-and.ac.uk/Biographies/Ptolemy.html
O'Connor, J J, and E F Robertson. "Thabit Biography." MacTutor History of Mathematics. University of St Andrews, Scotland, Nov. 1999. Web. 30 Jan. 2011. <http://www- history.mcs.st-and.ac.uk/Biographies/Thabit.html>.
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© 2011 Leonard Kelley