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.
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
threedimensional 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, ( n^{2}
1)/2 and (n^{2}+1)/2 are three such numbers. For n^{2}+ [(n^{2}
1)/2]^{2} = (4n^{2}+ n4 – 2n^{2}+1)/4 =(n^{2}+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.
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 3year
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.)
Amazon Ripoff
https://www.amazon.com/gp/offer
listing/0873530365/
ref=dp_olp_used_mbc?
ie=UTF8&condition=used

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/loomises.htm
A caveat regarding Loomis’s books for sale on Amazon: Despite the
exorbitant price ($150), this book is clearly some kind of photocopy, rebound with a
GBC or similar coil. It is a cheap, modern violation. Other sellers apparently are offering the real thing, the 1940 edition or lawful reprints, at reasonable prices.
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