A Simple Truth

A picture is worth a thousand words, so I will not say much. I just thought that it is interesting that when something is true, it can be demonstrated by many methods.

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Elisha S. Loomis and the Pythagorean Proposition

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 22^nd 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; 2^nd printing 1972.

National Council of Teachers of
Mathematics. 1968.
(Image courtesy of
Robert Blumenblatt)

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.

The other 368 proofs are left for the reader to enjoy.

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 22^nd 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.

“While he was living in Berea, Loomis began to study civil engineering and became the village engineer. He also served a three-year term as president of the Berea board of education. He continued to study while he taught, and this led to an A. M. degree from Wooster (Ohio) University in 1886 and a Ph.D. in metaphysics and social science from that same institution in 1888. The subject of his doctoral thesis was "Theism the Result of Completed Investigation."

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

ELISHA SCOTT LOOMIS, educator, author and man of affairs of Lakewood, was born on a farm in Wadsworth Township, Medina County, this state, September 18, 1852, son of the late Charles Wilson and Sarah (Oberholtzer) Loomis.  …  He was superintendent of schools at Shreve, Ohio, from 1876 to 1879; principal of Burbank Academy at Burbank, Ohio, from 1880 to 1881; principal of Richfield Central High School, Summit County, Ohio, from 1881 to 1885; professor of mathematics in Baldwin Wallace University from 1885 to 1895, and from 1895 up to the close of 1923 was head of the department of mathematics in West High School of Cleveland. At the last mentioned date he retired on school pension, having reached the limit established by Ohio law which provides that no teacher of the public schools shall hold position after having reached the age of seventy years. He taught his first school, beginning in April, 1873, and completed his last term of teaching in June, 1923, thus rounding out a full half century of successful teaching, and but for the intervention of the Ohio school law limiting the age of teachers for school teaching he would have continued his school work for an indefinite period, should he have so desired, for his physical and mental faculties are unimpaired, and "his spirit is willing."
http://www.onlinebiographies.info/oh/cuya/loomis-es.htm

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Truth to tell, I was disappointed to find a really nice write-up on Wikipedia.  I was hoping that this was still arcane knowledge.  As far as I can tell, it is not commonly taught in trigonometry classes.  We use the sine and cosine for problems with vectors; and that seems to be about it.  In point of fact, these constructions are the root and rock of computational trigonometry.  If you draw well and measure carefully, you can get two decimal places, or fractions to eighths.

Words have meaning.
If you ever suffered from a respiratory allergy,
then you know that it affects the "bay" the "sinus" cavities.
What we call the sine of an angle is the half-sine,
the semi-bay.
Similarly, our "tangent"
is the measure of the semi-tangent,
ignoring the reflection
below the part that we care about.

The trigonometric identities
come from the Pythagorean theorem.

In Feynman's Lost Lecture,  the professor allows that we do our maths with algebra and calculus.  We no longer rely on geometric constructions. Feynman had to create his own derivations for Newton's Laws simply because he could not follow Newton's easy claims about conic sections.  Newton used geometry to create the calculus.  However, calculus is such a powerful tool that we stopped learning the geometry that Newton knew. 

You can write out the algebraic statements
but a picture is worth a thousand words.

In Cosmos, Carl Sagan tells that Pythagoras fled from Samos because he could not tolerate the tyrant Polycrates, whom Sagan denigrates for having "started out as something like a caterer."  (Bold though he was, Sagan shared the anti-capitalist mentality.) But in The Ancient Engineers, de Camp tells us that engineers working for Polycrates bored through a mountain, starting at opposite ends, met in the middle, and were not off by a foot in a mile.  Lacking the positional notation of Arabic numerals, of course, they did all their calculations with geometry.

Descartes is credited with uniting algebra and geometry. 
His work reflects the seemingly intractable 
analytic-synthetic dichotomy of philosophers. 
In truth, it should have erased the distinction 
between the logically consistent and the empirically verifiable.
See the works of Gregory Browne on this blog here and here.

Long ago, at Curious Books in East Lansing, I found a old manual for apprentice carpenters that showed sines, cosines, and tangents to 32nds of an inch for triangles of given measurement.  Many times I have regretted not buying it then.  Even before that, when Laurel and I were first married, her father asked us what calculus was all about. We told him.  He asked, "Do you mean like this?" and took out a foot-sized caliper and showed us how he measured stair cases.  Rise over Run.  Ain't no doubt.

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