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The Pythagorean Theorem – A Rectification



Stephen Hawking: „Pythagora is not the creator of the theorem that the sum of the squares of the length of the sides of a right angle is equal to the square of the length of the hypotenuse.” Of course this angered me, but more because of his theorizing about God and the afterlife in the chapter of his review of Pythagora, but… it is well known that only “common sense can bring results”. I use this occasion to apologize to S. Hawking although I don’t know the motive behind his conviction, but he is right. Was it a premonition that something was wrong, I know not, but it would have been normal that mathematicians since the time when extracting the square was inaugurated should have logically had such a premonition, or even worse, they were aware of the mistake but simply ignored it. But worst of all, they taught us, as well as the generations before us, to act like ‘parrots’(I hope real parrots won’t get me wrong)with regard to such an obvious error. After this chapter you will comprehend it and see it, and you yourself may put it to the test. Although I have announced that I can no longer be in the service of the “spirit” due to plainly financial reasons (cost of the English translation and of my own ignorance in maintaining the website and that also costs), I am nevertheless obliged to release this for the present and future generations. Therefore, what should have been perceived or observed? In any sum of the squares of the legs of any right triangle, the extract of its square gives the length of the hypotenuse. And the length of a hypotenuse is never a whole number but a decimal number with hundredths, thousandths, etc as their decimal remainder. Both you and I were previously taught: whole numbered squares above the lengths of the legs and above the hypotenuse and that’s that! As for the hundredths, thousandths, etc forget them, „you can stuff it“ as folks often say. Wasn’t it essential to notice that the “squares” of the length of the hypotenuse of some other size? Bur, to make a long story short, you will see this outlined, in a simple manner, this time on two schoolbook-geometry examples. We will “cover” the surface areas of the squares. The other logic was presumed “suspicious”. Was Pythagora in his time able to precisely outline his theorem when even today we have not yet achieved the possibility to do this with preciseness? But let’s take a look, step by step, (this time with measures)but without markings and everything will become clear to an extent that you yourself can go further on.

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The first example: Unequal legs
Sraight line. Perpendicular symmetry on the straight line. Right angle measurement on base line 7 cm. – leg of base line.
The other leg 3 cm.

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Right-angled triangle 7×7=49 and 3×3=9 sum = 58
Extracted square of 58 = 7.6157731 = hypotenuse

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Delineated square areas of legs and of hypothenuse.
Squares and their sizes.

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Carry over the square of baseline leg to the square of the hypotenuse.

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With compass we carry-over the remaining width to the square of the other leg.
4 parts + remainder.

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Now, from the square of the leg we return 4 parts to the square of the hypotenuse. Visibly we can already anticipate that the remainder of the square of the hypotenuse is larger than the remainder of the square of the leg.

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But let’s check. Take the length of a part of the remainder of the hypotenuse and carry it over to the remainder of the leg.

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And we then return it to the remainder of the hypotenuse. The outlined surface area is a surplus on the square of the hypotenuse.
Conclusion: the area of the square of the hypotenuse is larger than the sum of the squares of the legs of the right-angled triangle of unequal legs.

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The second example: Right-angled triangle of equal legs.
Legs – 6 hypotenuse = 6×6 = 36×2 = 72
= extracted square of 72 = 8.4852813

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Squares of the legs (are constructed first for the purpose of easier construction of the square of the hypotenuse).

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Straight lines to which the length of the hypotenuse is carried over, with a small verification with a compass.

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This is a geometrical presentation of the surface areas of a regular right-angled triangle or a triangle of equal sides or the areas of the squares of the legs and the square of the hypotenuse.

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In the same way as in the first example, we carry over the square of the leg to the square of the hypotenuse. The remainder is on it.

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The width of the remainder is conveyed to the square of the other leg of two parts plus its remainder. Now we carry over the width of the remainder onto the remainder of the square of the hypotenuse.

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Two parts and a remainder. The width of remainder is conveyed to the (hypotenuse)remainder on the leg.
(These two examples are just one of some ten examples of the conclusion in reference to various methods: cutting out, overlaying, drawing. In any case, you can do-it-yourself.

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This time a “surplus” was evidenced on the square of the leg.
Conclusion: the sum of the squares of the legs is larger than the surface of the square of the hypotenuse of a right-angled triangle of equal legs.

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On these two examples we observed that the surface area of a regular right-angled triangle of equal legs evidenced a shortage on the square area of the hypotenuse. The matter is reverse in the case of unequal legs. We then observe a shortage of the sum of squares of the legs in relation to the square of the hypotenuse. All this logically leads to investigating of the theorem that says that the sum of the squares of the length of the sides (legs) is equal to the square of the hypotenuse. However, this is not valid even in the case of a right-angled triangle, in other words of equal sides, just as I have thought from the start – but thinking and doing are two different but similar things. Furthermore, it is not valid in the case of right-angled triangles of unequal legs, with the exception of a one-and-only case, and it is found on the baseline-leg as the angle and should be looked for between a 33° and 40° inclination of the hypotenuse. This method of overlaying of surface areas is not bad from an amateurish outlook, but for the hundredth time I must express my regret that the computer technology has not involved itself, and I am also sorry that certain suspicious „natural laws“ are being taken by us for granted instead of being subjected to verification in various ways, in particular when mathematics are in question (and this goes for geometry), as well as for physics which is the materialization of numbers and of geometrical drawing, even though my opinion regarding natural science also include biology and chemistry as a union and I consider that they should not be separately going their own ways. The ancient divine laws of Mount Horeb of 3300 years ago speak to us about this. However, what I am talking about is the fact that nobody believes in those laws any more. If there was belief, then there would be those who would simply say: “Get back on your feet, in the name of God” and nothing more. Way back in 1977, I verified this truth myself, but in the meantime I also became a “Doubting Thomas” in spite of the fact that I was named after Thomas Aquinas, but that was the custom in those days. But in the same way that doubt occurred to S. Hawking about there being nothing in the afterlife, which made me angry, because we don’t have “the faintest idea” about this, yet at the same time he incited me to look into Pythagoras Theorem. And lo and behold! That doubt brought about my undertaking the real experiment of cutting out surface areas and overlaying them on one another, and then this gave birth to the result. There is another word that makes me mad, and that word is “impossible”. Today, when we beg for energy, in front of our “noses” there is a mechanism that generates more than it spends, but our present-day pundits say that such a thing is “impossible”, even after all the stones of knowledge gathered during the centuries. But I can’t do it alone. Anyway, to my forture or misfortune, I have managed to sell a part my little worldly dream: the last parcel of land on which I intended to spend my old-age days after my children grow up on the resplendent island on this sea coast, in a small boat in which I would sail out into the bay and in silence with a fish line between my fingers go fishing in the old-fashioned way where one can feel the biting of the fish under ones fingers… now just a turned down boyish dream, no longer possible. However this will certainly enable me to soon release a few more chapters (even though besides this one two more are already written and drawn), but in secrecy since I am faced with the ultimatum on the part of those “who eat at my table”. Solomon would say “nothing new under the vault of heaven”. So I will leave you to continue your research to find which right-angled triangle is it that really abides by the theorem with its equality of surface areas, but once again I stress that computer technology would be a welcome innovation for a precise system of overlaying.
However I am now preoccupied with another problem: a ground zero right-angled triangle and just why? But don’t hold it against me and greetings to you all!

RIJEKA, CROATIA September 19, 2013
Author: T. Periša
WebMaster: SLIM
English : S.F. Drenovac


As I have who knows how many times already mentioned in theses more than 3000 pages, when the Spirit opens its doors then at first sight one cannot grasp everything at once, but rather cursorily. But when one takes a better look, one cannot help seeing it. Why am I mentioning this? It is because I wish to spare you the agony of searching. The answers, the unbelievable answers of ancient geometry, or but the basic, miraculous part has been perceived by me and I will bring it to you in the forthcoming chapter. The truth is that Pythagora and the Pythagoreans are not the creators of all of this, but it is possible that they saw, comprehended and for some reason decided to be silent about it. Something similar to what we can find in the legends of the prophecies of the ancient biblical prophets. Many oracles were sealed “for the distant future”. It is without a doubt that everything in this world has its reasons, its meaning and its time, its cycle charted within the entire natural universe, whether we acknowledge it or not. It is probable that premature cognition and knowledge could be harmful and interrupt the natural flow of things. Be it as it may, this same perception did not give me the peace of mind to stop, but rather to continue, to search and find and see geometrically how simple things are. And this assembly of simplicities fills this fatigued human mind of ours with enthusiasm. Therefore, spare yourself the agony, and actually computer simulation is not necessary because sacred, divine, natural geometry has a solution for everything.

Rijeka, Croatia September 21, 2013
Author: T. Perisha

2 Responses to “The Pythagorean Theorem – A Rectification”

  1. Mladen P says:

    Poštovani g. Periša! Zahvalan sam na vašim objašnjenjima svete geometrije. Podržavam vaše dijelovanje. I sam proučavam suštinu živoza, prirode itd ali potpuno upoštivajuci zaboravljenu nauku i duhovnu strukturu svega postojećeg.
    Želim Vam puno zdravlja i dobre volje za dalji rad. Mladen

  2. Tomo says:

    Hvala Vama i lijep pozdrav!

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