The Pythagorean Theorem:
x2 + y2 + z2 = w2
Charles William Johnson
In this essay, we shall consider the sum total of the values of two distinct series of perfect right triangles, and the resulting series of triangles that obtains.
Two series of perfect right triangles that we have been considering for the ancient reckoning systems concern those, whose measurements are:
3 • 4 • 5 and 5 • 1 2 • 13
Now, let us add up the corresponding values relating to each of these perfect right triangles:
Following the Pythagorean Theorem, and its poetic values, one would obtain:
In this manner, we would see that a relation of equivalency does not obtain from the simple sum and squares of the corresponding side measurements of the two series of perfect right triangles. On the other hand, the Pythagorean Theorem would assist us in this manner:
side measurements of the resulting right triangle would then be:
in terms of its two legs (6 • 16) and its hypotenuse (17.88854382).
However, if one wished to employ the numbers related to the two series of perfect right triangles (i.e., 8, 16, and 18), given the fact that these numbers are related to ancient reckoning systems, then, an adjustment would be in order:
22 + 82 + 162 = 182
4 + 64 + 256 = 324
324 = 324
In this manner, the actual values of the side measurements and the poetic values of the procedure have been brought into a relationship of equivalency. Their geometrical expression would then be that of an isosceles triangle.
In the above manner, one could have a proof similar to the Pythagorean Theorem, based upon a relationship of the squares of the terms:
22 + 82 + 162 = 182
x2 + y2 + z2 = w2
The previous illustration of the isosceles triangle (8 • 18 • 18), is not unlike many monumental structures found in ancient cultures around the world. The significance of such a shape and numerical procedure in the geometrical division of the side measurements could be related to the different ancient reckoning counts quite easily. The whole numbers of the divisions within the side measurements would facilitate computations and allow for relating different reckoning counts to one.
The basic unit value of this particular isosceles triangle would be two (2.0), as drawn in the previous illustration. This reminds us of the procedure that we employed in the extension of the Pythagorean Theorem to the cube for another distinctive perfect right triangle and/or isosceles triangle (12 • 10 • 10).
1³ + 6³ + 8³ = 9³
1 + 216 + 512 = 729
In an earlier essay, we have discussed how the previous illustration reminds us of the profile design of the Great Pyramid at Giza and the relationship of the entrance to the internal part of that pyramid. The basic unit of this pyramid would be that of one (1.0), as illustrated on the above side measurements and their division.
Note that the previous structure is simply represented by placing two 3 • 4 • 5 perfect right triangles back-to-back and doubling their side measurement values.
The interesting point regarding this particular triangle is that the difference between the real value (1 + 6 + 8 = 15) and the poetic value of equivalency (729) is now an ancient reckoning count:
In fact, in the previous two illustrations of isosceles triangles, the real numbers of the side measurements and the poetic values of equivalency may be employed to reflect ancient reckoning counts. The more significant point being that the actual side measurements and their divisions reflect ancient reckoning counts that could be easily doubled or trebled into order to obtain different series of ancient reckoning values.
The previously illustrated isosceles triangle, based upon the 3 • 4 • 5 perfect right triangle shall have implications for the analysis of the inverse squares law and its derivatives. We shall explore such possibilities in following extracts. For now, we shall simply list some of the numbers and triangles related to this particular law. The corresponding side measurements, as divided above, would then correspond in the following manner:
One may thus observe how the geometrical division of this particular isosceles triangle and its base perfect right triangle (3•4•5) reflect the ancient reckoning counts within their measured values. These numbers relate to the actual side measurements and their divisions. One may observe many ancient-reckoning counts (in bold numbers) make their appearance in this manner on the series of triangles with their sides divided in the manner illustrated in this essay.
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