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Pythagoras – 60 or Eight

Chapter 3 (Book 4)

Surely the person accompanying this study knows that only one ratio between the legs of a right triangle renders a whole number root of their squares. We will repeat this for the person who just “discovered” this page. Hence, only when the side lengths (legs) are parts 3 and 4, the sum of their squares will be 9+16 square parts, which is 25 parts and their square root is 5. If we go on increasing this proportion 3 by 3 = 4 by 4 and so on, then the root of their sum increases to 5. The same occurs if we add zeros, namely the square root of 30, 40 is 50. All the other proportions generate decimal numbers; or if the side lengths are whole numbers, then the square root of the hypotenuse is a decimal number or vice versa. The relationship of whole number side lengths is the only ratio that corresponds exactly to the Pythagorean Theorem a2+b2=c2. All other proportions show a surplus or a deficit on the square surfaces of the side lengths or hypotenuse regardless of how persistently to this very day we are taught that the formula of Pythagoras is applicable for every right triangle. One day computer simulation will confirm this assertion. This ratio is inscribed on the cover of an ancient tomb in South America in the form a right triangle in which one of its angles (the researchers wonder why) is in lack of a triangular part. Surely the relations between circles (ratio of descriptive and inscribed circles of the legs and the hypotenuse) would shed more light onto this knowledge, but as a beginning this geometrical approach will suffice but in this Book we will get back to this subject. Now we will simply keep to the naturally given sequences of quantitative relations only with the use of a compass and unmarked straightedge guided by a doubled sameness of angular magnitudes that give same results as the mentioned relations – exact angular sizes 37.5° and 52.5°. Now let’s go to ratio 30° and 60° – duplicated sameness only with compass and straightedge, in addition to two geometrical modes.

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Hence: straight line, on it a circle of random radius divided by the intersection of the straight line and its arc into 6 parts of circle’s intersections of the perpendicular through the center.

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Truncated from the intersection of the vertical by the same radius of side lengths of the described square of the circle and extended straight line from the center in the direction of the pole of division of lengths, which is then the radius of the hypotenuse and the vertical on its legs and centerline of the leg.

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Circle of radius of a half of the hypotenuse and another of the same from the vertex pole on the centerline plus one half.

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With descriptive circle of the hypotenuse radius and center (one end + other end) of hypotenuse we enable construction of the hypotenuse square. The square of the base leg is present from the start whilst the vertical hypotenuse is constructed.

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Now with compass we simply transmit the square of the leg to the surface area of the square of the hypotenuse. The leftover surface area remains.

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We transmit the width of the leftover to the square of the base leg. We now have three parts + leftover areas of the hypotenuse and the other leg.

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We transmit the width of the leg to the leftover area of the hypotenuse. Again we get leftover areas on both sides which we transmit with the compass. The width from the hypotenuse is transmitted to the part of the leg and then vice versa, the length to the hypotenuse part. A surplus surface area remains on the hypotenuse.

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If we were to transmit by “coverings” we would then get the surplus as a triangular part, however we will deal with such transmission when I analyze right triangles of different angular sizes.

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In this case one thing is certain – the case of angular size of hypotenuse gradient of 60°. The square surface of the hypotenuse reads an excess of surface area in proportion to the surface of the sum of squares of the legs. We did not analyze the magnitude of the surplus since the relations between circles might clarify it. It is possible that the surplus might even be greater, namely that the plotting with a compass still does not impart exact results but one thing is for sure. The surface of the square of the hypotenuse exceeds the sum of the squares of the legs, and this will one day, most probably in the near future, be carried out precisely, for if it was possible in ancient times and to us unknown civilizations, then our modern research and technology plus our genetically induced impulse should by all means succeed. Why do I mention the latter? As usual, I always follow the track of historical records and I bear in mind one of Solomon’s wise sayings that “what has been is what will be”. All this might provoke the scorn among some of our “glorious” academic guilds, but as I’ve already said – everything should be checked because the wise man verifies. Haughtiness is a common human hindrance – especially among the “learned”. We all know this. And haughtiness causes a sequence of reactions arising from our forgetting who and what we are, namely who we were and what were we and what will we be. And indeed man is, as it is written, a short life “grass” and therefore nowadays a couple more years are nil in comparison to the plenitude of some other natural coexistence – ancientness. But let’s get back to geometry for somewhere near “lies” the answer to the question: to what purpose is this just as it is? If we observe the shapes of some ancient artifacts we may conclude that they certainly had some beneficial and useable purpose. They say that “man never does anything if he expects no benefit from doing it”, whilst the other proverb reads that “history is our teacher”. The latter seems to suggest that it would be good to revise the natural sciences because in many respects they seem to be “more backward” that the sciences of times long past, which could be the result of a series of reasons that emerged in the new era and we will fall back even more and more if we finally do not undertake steps in this short lifetime of ours to leave something as a heritage to our present-day and future generations so they will know: this is wrong, and this is right because we have verified it. It is obvious that the need for constant verification is a necessity. Therefore we will once again corroborate this text, but this time in a different manner – by precipitation of the surface areas of the hypotenuse onto the base line since it should bring us the same results as well as the opposite analysis os the subtended angle, the angle of 30° (in the following chapter).

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Horizontal straight line. Circle divided by its radius starting from intersection of arc and straight line. Vertical line is the point of intersection between circle and center.

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Truncated and subtended semicircle from point of intersection between arc and vertical line. (square – descriptive circle)

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Direction of line of descriptive square and (initial) division into six parts. Gradient of angle of hypotenuse 60°.

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„Commingling” the squares of the side lengths (legs).

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Square of perpendicular leg from the center.

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Square of hypotenuse from the center. All in one (common denominator of square of hypotenuse and legs).

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Square of vertical side length in the square of the hypotenuse.

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Leftover from the base leg.

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The transmission of widths from surface areas of the legs to the leftover area of the hypotenuse, and then the lengths from the leftover of the hypotenuse to the leftover surfaces of the legs.

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The leftover (2 parts) stays on as two thirds of that one part (the above depicts the peak square of nine minus two = surface on square of hypotenuse). Logic tells us that the same is valid for angle of 30° (next chapter).

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Upon observing the development of the geometrical process of boiling down the right triangle 60° gradient of its hypotenuse, I thought I could step by step simply leave this construction be without comment, but for those who are not accustomed to this specific geometry, minor instructions have nevertheless still been retained at the bottom of each page as a necessity despite the fact that if one looks attentively and does not “spare” implementation of straight lines and circles, the structure process would by nature open up so that there would be no need to use the compass as a conveyor of magnitudes and by further logical analysis, our indication that we would “pass over it in silence” and just fervently stick to “overlapping” of surface areas and avoid circles and their relationships, I realized that in that way we would omit the numerous details that are present in our ancient heritage and have not yet been clarified. They could give us a series of replies to our how and why questions. Therefore in my next chapter I intend to demonstrate a portion that particularly refers to ancient buildings and also adverts to the planetary system, especially the distinct closeness between the cultures of ancient Mesopotamia (particularly Egypt) and South America and also many others. The symbols arising from all this (one of them being the new era’s first American dollar) imply that knowledge was transmitted and hidden and therefore I sometimes wonder why I’m going through all this trouble when the knowledge has been known and transmitted for centuries but is concealed and well “packed” in a veil of seemingly nonsensical initiations and so-called secret societies, apocryphal religions and who knows what other kinds of invented stories and fairy tales. Upon taking note of this from the other side, i.e the viewpoint of technological progress, the steps of mankind look absurdly small and even “primitive” – or is this simply what is known as “politicking”, and one of its reasons is the disunity of nations, races and religions and this has come down to the concentration from the Primates to the Highbrowed individuals in the same place at the same time, hence their common denomination is understandably undefined, and this issue can be described as Tolerance or Symbiosis – which again contradicts Darwin’s first theory (his second theory on symbiosis has never been published or is still being well-guarded in the archives of a historical realm). But this is not our theme, save for what has been said up to now and will be said from now on. Why am I saying this? Because I wonder, who will comprehend this, evaluate it and assert it? Only a large minority in the world of an enormous majority of inhabitants on this planet. Will that be Almighty God whom I know not since He is said to be a spirit? Regardless of the sequences of innovations on these pages, I ask my readers: Dose this make sense? Would it not be better to live a normal everyday life waiting for my human inevitability? This way (or just now) I don’t know what I am or what I am for! But, that’s until the next chapter – on 30 degrees.

CROATIA – RIJEKA December 7, 2013
AUTHOR: Tomo Perisha
ENGLISH: S.F. Drenovac (

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One Response to “Pythagoras – 60 or Eight”

  1. Ricardo says:

    Got it Mr.Tomo. Slowly I am learning. I found the right ratio. Its in the three Musketeers story. “The tip is the missing space”. Singular, Plural, Even, Odd. Its a way to read. There are many ways to read this geometry Mr.Tomo. GOD is teaching me thru you and all. Take care,I hope youre all doing alright. 1\2 e 2\1 Cheers.

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