The fact that over 300 proofs of the Pythagorean Theorem are known and an infinite number are possible is in and of the nature of the universe. When something is true, many other truths support it; and a truth leads to many others. Moreover, no truth can be supported by a fallacy; and a truth cannot lead to a fallacy.
The Pythagorean Proposition: Its Proofs Analyzed and Classified and Bibliography of Sources for Four Kinds of Proofs by Elisha Scott Loomis, Ph. D., LL.B., Masters and Wardens Association of the 22nd Masonic District of the Most Worshipful Grand Lodge of Free and Accepted Masons of Ohio, Cleveland, Ohio, 1927. National Council of Teachers of Mathematics, Washington, D.C. 20036, 1940, 1968; 2nd printing 1972.
Prof. Elisha S. Loomis divided proofs of the Pythagorean Theorem into four classes: algebraic, geometric, quaternionic, and dynamic.
Algebraic proofs show that for two numbers, each of them multiples of other numbers by themselves (squared), there exists a third number, also the product of some number by itself (a square) that is the sum of those. That is, for some integers A and B there is a C such that C*C equals A*A + B*B. Geometric proofs are achieved by constructions, folding, cutting, and superimpositions. Dynamic proofs come from the mathematics of engineering mechanics, from vector arithmetic, in which magnitude and direction together define a force.
For quaternions, I can only quote Wikipedia: “quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative." In other words A * B ≠B * A. The only explanation I know – perhaps not actually from quaternions – is that there are ways to flip a book (transformation with rotation) so that when you reverse the steps, the book is in a different orientation than when you started.
But I could follow the concepts supporting the algebraic and geometric proofs; and they are the foundation for the vector arithmetic.
The simplest algebraic proofs are attributed to Pythagoras and Euclid, but I would like to find them in the original because, in fact, those ancient savants did not know algebra. Nonetheless, citing Loomis, those proofs were explored and generalized by Artemas Martin (1891) and Leonard E. Dickson (1894).
Start with any odd number. Square it and subtract 1; then divide by 2; this is the second number. For the third, take the original, add 1 to it, and square the sum. That third number equals the sum of the squares of the other two.
Let n be odd; then n, ( n2 -1)/2 and (n2+1)/2 are three such numbers. For n2+ [(n2 -1)/2]2 = (4n2+ n4 – 2n2+1)/4 =(n2+1)/2]2 .
The algebraic proof is difficult to lay out with a word processor but is easy enough with paper and pencil.
The simplest geometric proof is achieved by starting with a right triangle. Drop a perpendicular from the right angle. The three triangles are similar. The ratios and proportions of their respective sides can be stated algebraically, and then be manipulated to show that the square of the original hypotenuse is equal to the sum of the squares of the other two sides.
Elisha Scott Loomis held a doctorate in mathematics from Wooster College in Ohio, earned a law degree and was admitted to the Ohio bar, worked as the town engineer in Beria, Ohio, and, after some other appointments and assignments became the chairman of the mathematics department at West High School in Cleveland, Ohio.
The first edition of this book includes a roster of the 22nd Masonic District of Ohio with their officers, including the Past Grand Masters, and the Masters and Wardens Association, about 150 names in all. I recognized many of the neighborhoods: Brooklyn, Denison, Collingwood, Windemere, Lakewood, Euclid, Sheffield. Other lodges had honorific names: Laurel, Pythagoras, Warren G. Harding, North Star, Halcyon.
The second edition had three printings, 1940, 1968, and 1972 by the National Council of Teachers of Mathematics. Their office is listed in Washington DC, but the books were printed in Ann Arbor. The dedications to the FAM lodges were removed in the reprinting.
You can find a biography of Elisha Scott Loomis in Wikipedia. The same information can be found in the front matter of the books. Other biographies have been archived online.
In 1895 Professor Loomis left Baldwin University to become head of the mathematics department at Cleveland West High School, where he remained until his retirement in 1923. In 1900 he earned the LL.B. degree from the Cleveland Law School and was admitted to the Ohio state bar. He served a 3-year term as president of the High School Teachers¼ Mathematics Club of Cleveland, and he was active in the Masonic lodge.” – Kullman ("Elisha Scott Loomis," The National Cyclopaedia of American Biography, vol. 15, New York, James T. White & Company, 1914, p. 186. Article by David E. Kullman, Miami University.)
|Title Page of First Edition|
PREVIOUSLY ON NECESSARY FACTS