In trigonometric terms, the Pythagorean theorem asserts that in a triangle ABC, the equality sin☪ + sin☫ = 1 is equivalent to the angle at C being right. Actually, for some people it came as a surprise that anybody could doubt the existence of trigonometric proofs, so more of them have eventaully found their way to these pages.
I must admit that, concerning the existence of a trigonometric proof, I have been siding with with Elisha Loomis until very recently, i.e., until I was informed of Proof #84. (In all, there were 100 "shorthand" proofs.) I'll give an example of their approach in proof #56. For example, the authors counted 45 proofs based on the diagram of proof #6 and virtually as many based on the diagram of #19 below. Counting possible variations in calculations derived from the same geometric configurations, the potential number of proofs there grew into thousands.
GEOMETRY THEOREMS LIST SERIES
In all likelihood, Loomis drew inspiration from a series of short articles in The American Mathematical Monthly published by B. Curiously, nowhere in the book does Loomis mention Euclid's VI.31 even when offering it and the variants as algebraic proofs 1 and 93 or as geometric proof 230. In the Foreword, the author rightly asserts that the number of algebraic proofs is limitless as is also the number of geometric proofs, but that the proposition admits no trigonometric proof. The book is a collection of 367 proofs of the Pythagorean Theorem and has been republished by NCTM in 1968. Dunham cites a book The Pythagorean Proposition by an early 20th century professor Elisha Scott Loomis. Euclid was the first (I.48) to mention and prove this fact. The converse states that a triangle whose sides satisfy a² + b² = c² is necessarily right angled. The Theorem is reversible which means that its converse is also true. In fact Euclid supplied two very different proofs: the Proposition I.47 (First Book, Proposition 47) and VI.31. Euclid's (c 300 B.C.) Elements furnish the first and, later, the standard reference in Geometry. Whether Pythagoras (c.560-c.480 B.C.) or someone else from his School was the first to discover its proof can't be claimed with any degree of credibility. The statement of the Theorem was discovered on a Babylonian tablet circa 1900-1600 B.C. Many of the proofs are accompanied by interactive Java illustrations. It's so basic and well known that, I believe, anyone who took geometry classes in high school couldn't fail to remember it long after other math notions got thoroughly forgotten.īelow is a collection of 118 approaches to proving the theorem. The theorem is of fundamental importance in Euclidean Geometry where it serves as a basis for the definition of distance between two points. In algebraic terms, a² + b² = c² where c is the hypotenuse while a and b are the legs of the triangle.
The Pythagorean (or Pythagoras') Theorem is the statement that the sum of (the areas of) the two small squares equals (the area of) the big one.
Both groups were equally amazed when told that it would make no difference. Which would you choose?" Interestingly enough, about half the class opted for the one large square and half for the two small squares. Then he asked, "Suppose these three squares were made of beaten gold, and you were offered either the one large square or the two small squares. He drew a right triangle on the board with squares on the hypotenuse and legs and observed the fact the the square on the hypotenuse had a larger area than either of the other two squares. and Other Philosophical Fantasies tells of an experiment he ran in one of his geometry classes. That is one of the secrets of success in life.'Ģ nd Movement in A Dance to the Music of Time 'You have given yourself the trouble to go into matters thoroughly, I see. 'An exceedingly well-informed report,' said the General.
0 kommentar(er)
