[Math Lair] The Pythagorean Proposition: Foreword

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According to Hume, (England's thinker who interrupted Kant's "dogmatic slumbers"), arguments may be divided into:  (a) demonstrations; (b) proofs; (c) probabilities.

By a demonstration, (demonstro, to cause to see), we mean a reasoning consisting of one or more catagorical propositions "by which some proposition brought into question is shown to be contained in some other proposition assumed, whose truth and cer­tainty being evident and acknowledged, the proposi­tion in question must also be admitted certain. The result is science, knowledge, certainty." The knowl­edge which demonstration gives is fixed and unalter­able. It denotes necessary consequence, and is synonymous with proof from first principles.

By proof, (probo, to make credible, to demon­strate), we mean 'such an argument from experience as leaves no room for doubt or oposition'; that is, evidence confirmatory of a proposition, and adequate to establish it.

The object of this work is to present to the future investigator, simply and concisely, what is known relative to the so-called Pythagorean Proposi­tion, (known as the 47th proposition of Euclid and as the "Carpenter's Theorem") and to set forth certain established facts concerning the algebraic and geo­metric proofs and the geometric futures pertaining thereto.

It established that:

First, that there are but four kinds of demonstrations for the Pythagorean proposition, viz.:

  I. Those based upon Linear Relations (im­plying the Time Concept)--the Algebraic Proofs.

 II. Those based upon Comparison of Areas (implying the Space Concept)--the Geometric Proofs.

III. Those based upon Vector Operation (im­plying the Direction Concept)--the Quaternionic Proofs.

 IV. Those based upon Mass and Velocity (im­plying the Force Concept)--the Dynamic Proofs.

Second, that the number of Algebraic proofs is limitless.

Third, that there are only ten types of geo­metric figures from which a Geometric Proof can be deduced.

This third fact is not mentioned nor implied by any work consulted by the author of this treatise, but which, once established, becomes the basis for the classification of all possible geometric proofs.

Fourth, that the number of geometric proofs is limitless.

Fifth, that no trigonometric proof is possi­ble.

By consulting the Table of Contents any in­vestigator can determine in what field his proof falls, and then, by reference to the text, he can find out wherein it differs from what has already been established.

With the hope that this simple exposition of this historically renowned and mathematically funda­mental proposition, without which the science of Trig­onometry and all that it implies would be impossible, may interest many minds and prove helpful and sugges­tive to the student, the teacher and the future orig­inal investigator, to each and to all who are seeking more light, the author, sends it forth.

Some Noted Proofs
For more information on this work, see The Pythagorean Proposition by Elisha Scott Loomis.