Alex

Is Beauty Quantifiable ?

144 posts in this topic

A minor correction either you write a^2 + b^2 = c^2

OR

(a^2 + b^2)^(1/2) = sqrt (a^2 + b^2) = c

[text omitted]

Bob Kolker

Bob:

Please check my math for I think that I permitted an error that we both may have missed.

We left the discussion with the equation as written:

a^2 + b^2 = c^2

however, shouldn't it have been:

a^2 + 2ab + b^2 = c^2

?

Extra information:

The equation is known as the "Application of Areas" because the area of the square comprised of,

a^2 + 2ab + b^2

exactly fits within the square comprised of,

c^2 .

The "Application of Areas" was graphically, using coincident shapes, considered by the AGGs to be an inferior method of proof. Where the concept worked was where finite numerical values for the measured magnitudes, e.g., a,b,c, were employed. The proofs of the AGG geometrical theorems were then considered to be precisely valid. That was the real consequence of the discovery of the "Application of Areas" principle.

Mathematics may have begun or may have been conceptually "separated" from the conceptual and graphical geometry of the Pythagoreans. A universal principle always then would have a finite result. The system of logical proof was retained while the precision of numerical measurement was added to the equations. The AGGs had been ignoring arithmetic for some time, and considered it to be merely practical and not universally elitist. With the discovery of the "AoA" arithmetic was integrated into geometrical science. Numbers and not only figures and specific relationships were then one of the legitimate finite results of equations. That is one of the discoveries that is at the begininnings of the science of mathematics.

Also, the parts of the "AoA" within c^2 that were,

a^2 + 2ab,

formed a right angle shape, e.g., an "L", and they were together called the "gnomon" [or, sun-dial tower].

The parts of the "AoA" within c^2 that were,

b^2,

or, the remainder from, c^2 - a^2 - 2ab,

formed a square shape, and that was called the "defect".

My question is this.

The "defect" is a concept that also is found in the plays of the AGs. It refers to the quality of the play that is universal and that is implied by the concretes that were given by the perfomance.

That was also the beginning of the Conceptual Arts, or the arts that had implied universal meaning or content that was specifically integrated into a work of art and for the purpose of intellectual and esthetic emotional contemplation [e.g., painting, sculpture, and literature; rather than for the purpose of direct perceptual experience as in the Formal Arts, e.g., architecture, and dance.]

Who first concieved of the relationship of the "gnomon" and "defect" in philosophy and playwriting, or other artistic creations?

Inventor

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Forum and Bob:

I failed to correctly differentiate the explanations for the formulas,

a^2 + b^2 = c^2

and,

a^2 + 2ab + b^2 = c^2

.

The formula given by Bob is correct for the Pythagorean solution

to right triangles, and the additional formula for the Application

of Areas should simply be considered as an additional post.

Once the reader has in mind which of the two formulas

that are described in this or that post are the current context of

discussion there should be no misunderstanding.

The appropriate discussions of each principle are correct, however.

[Also, due to the different principles involved, a table of finite

values, or definitions, for a, b, c, in each instance may not

necessarily be the same.]

Sorry for the misunderstanding.

Inventor

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There is no record that Pythagoras ever said any such thing. Beyond their limited geometry Pythagoreans' mysticism committed all kinds of mystic fallacies in speculating about the nature of the universe.

One can prove that two is not the square of the ratio of two integers.

Dr. Harry Binswanger stated his own definition and explanation of a straight line in his lectures on science. The lectures are offered for sale by ARI.

On another note, we may only wish that the mysticism of Pythagoras had not been recorded and that his system of geometry had instead been recorded.

Continuing the thought that, "One can prove that two is not the square of the ratio of two integers," I would say, for example, that, "One can prove that the square root of two is the sum of the squares of two integers (for straight lines of unit lengths placed at right angles to one another)."

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Dr. Harry Binswanger stated his own definition and explanation of a straight line in his lectures on science. The lectures are offered for sale by ARI.

On another note, we may only wish that the mysticism of Pythagoras had not been recorded and that his system of geometry had instead been recorded.

Continuing the thought that, "One can prove that two is not the square of the ratio of two integers," I would say, for example, that, "One can prove that the square root of two is the sum of the squares of two integers (for straight lines of unit lengths placed at right angles to one another)."

Inventor

Incorrect. An integer is rational. The square of an integer is rational. The sum of rational numbers is rational. Therefore the square root of two is not the sum of two rational numbers.

Proofs: Let n be an integer then n = n/1

Let n/m be a rational number where n, m integers. (n/m)^2 = n^2/m^2. but n^2 and m^2 are integers. So (n/m)^2 is rational by definition of rational. Let n/m and j/k be rational numbers.

n/m + j/k = (n*k + m*j)/m*k but the numerator is the sum of integers which is an integer and m*k is an integer so n/m + j/k is rational.

Q.E.D.

Has Dr. Binswanger ever published a mathematical article in any refereed mathematical journal?

Bob Kolker

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Dr. Harry Binswanger stated his own definition and explanation of a straight line in his lectures on science. The lectures are offered for sale by ARI.

On another note, we may only wish that the mysticism of Pythagoras had not been recorded and that his system of geometry had instead been recorded.

Continuing the thought that, "One can prove that two is not the square of the ratio of two integers," I would say, for example, that, "One can prove that the square root of two is the sum of the squares of two integers (for straight lines of unit lengths placed at right angles to one another)."

Inventor

Incorrect. An integer is rational. The square of an integer is rational. The sum of rational numbers is rational. Therefore the square root of two is not the sum of two rational numbers.

Proofs: Let n be an integer then n = n/1

Let n/m be a rational number where n, m integers. (n/m)^2 = n^2/m^2. but n^2 and m^2 are integers. So (n/m)^2 is rational by definition of rational. Let n/m and j/k be rational numbers.

n/m + j/k = (n*k + m*j)/m*k but the numerator is the sum of integers which is an integer and m*k is an integer so n/m + j/k is rational.

Q.E.D.

You've got me on that one. I had to laugh that I realized that I was dumfounded and got confused with the ratios and non-ratios.

I would have said only that the AGGs, including Pythagoras. would likely have said in a lemma, in the custom of his time, that,

Given a construction between parallel straight lines placed one unit apart [which was the common practice for the AGGs from Pythagoras to Archimedes].

Construct a medial line from a point on one parallel line to the second at right angles to the first line.

Construct a second medial line parallel to the first at a distance of one unit.

Construct a straight line to connect the far end point of the first medial line at the second parallel line with the start point of the second medial line at the first parallel line.

Two right triangles are formed. [definition of right triangle]

The hypotenuse is the straight line connecting the base and medial lines.

The base and medial straight line sides of the right triangle have unit lengths.

The length of the hypotenuse is the side of the area of the square placed on that line, wherein, [according to a previous proof]

The square on the hypotenuse is equal in area to the sum of the squares on the base and medial lines of the first right triangle.

If the squares on the base and medial lines of the first right triangle are each the product of a unit length,

The squares, therefore, have unit areas.

The sum of the squares is two units of area. [one plus one equals two]

The area of the square on the hypotenuse is equal to the sum of the areas on the base and medial lines of the first triangle.

The sum of the areas on the base and medial lines of the first triangle is two units.

The area of the square on the hypotenuse is equal to two units.

Therefore, the length of the hypotenuse of the first right triangle is equal to the root of the square comprised of two units of area.

Here, no points are specified. and only locations relative to a point place on the first of two parallel straight lines are specified. The only length given is a unit.

Reflecting upon the definition of a point, Euclid may have intended to say more that,

A point is that which has no part and has location only.

I suspect that a translation error was made in The Elements in that the first part only was provided, and that says, "A point is that which has no part."

The definition is incomplete: that is only the genus of the full definition.

The differentia is provided elsewhere in The Elements as a lemma, "A point is that which has location only."

The term "that" is the symbol for the genus.

"That", the genus, is a concept of a geometric entity that exists in mind only.

"That" may be demonstrated virtually in logic or in actuality by means of specifying a location, which location is the differentia of the definition.

To modernize the definition we may say that,

A point is the basic scientific concept of a location that is expressed as a unit entity. [definition by this author]

The word, "only" in Euclid's lemma is a qualification that the entity cannot be more than a unit entity. That means that the location cannot be a line, length, construction, or other more complex geometric concept or entity.

Where Euclid says, "A point is that which has no part," he means that he is expressing the idea of a universal scientific concept.

Where Euclid says, "A point is that which has location only," he means that the entity of the concept exists, for visualization and demonstration purposes only, at a specific chance or selected finite location.

Since a point has location only it also has no other properties or attributes, for example, it has no breadth.

The concept of location implies a universal that exists as an identification of places in physical reality in a system of locations that correspond to physical entities, or their equivalent in a systgem of ideas. The concept, location, implies a universe of things, or ideas, that exist in a measurable system of relationships.

Has Dr. Binswanger ever published a mathematical article in any refereed mathematical journal?

Bob Kolker

Don't know. Check with the HBL or ARI.

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The equation is known as the "Application of Areas" because the area of the square comprised of,

a^2 + 2ab + b^2

exactly fits within the square comprised of,

c^2 .

Did you mean that the other way around? I.e., the square whose side length is c fits within the square whose side length is a+b. If the square whose side length is c is rotated such that its four corners exactly touch the sides of the square whose side length is a+b, the result is four right triangles in the "a+b" square, at its corners, having sides a and b and hypotenuse c. It can be seen that the area of the inner square, c^2, is equal to the area of the outer square, side length a+b, minus the areas of the four right triangles at the corners:

c^2 = (a+b )^2 - 4 x 0.5 x ab = (a^2 + 2ab + b^2) - 2ab = a^2 + b^2.

In my high school geometry textbook (published in the 1950s), this is presented as an algebraic proof of the Pythagorean Theorem.

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Dr. Harry Binswanger stated his own definition and explanation of a straight line in his lectures on science. The lectures are offered for sale by ARI.

On another note, we may only wish that the mysticism of Pythagoras had not been recorded and that his system of geometry had instead been recorded.

Continuing the thought that, "One can prove that two is not the square of the ratio of two integers," I would say, for example, that, "One can prove that the square root of two is the sum of the squares of two integers (for straight lines of unit lengths placed at right angles to one another)."

Inventor

Incorrect. An integer is rational. The square of an integer is rational. The sum of rational numbers is rational. Therefore the square root of two is not the sum of two rational numbers.

...

Has Dr. Binswanger ever published a mathematical article in any refereed mathematical journal?

I don't think he has, but what difference does it make to this discussion? He is a professional philosopher with a background in science, including a BS in physics from MIT. The lecture I think inventor is referring to is a lecture on the philosophy of selected concepts in science (which I have not heard), and is not relevant to publication in a "refereed mathematical journal" -- and probably is also not relevant to whatever it is inventor is talking about. You should not assume that Harry Binswanger (or anyone else) endorses his fallacious claims like two integers can sum to the square root of two -- as an "example" of something unspecified. This has nothing to do with Objectivism.

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The equation is known as the "Application of Areas" because the area of the square comprised of,

a^2 + 2ab + b^2

exactly fits within the square comprised of, c^2.

Did you mean that the other way around?

No. However, you've put your finger on the wonderful coincidence that Pythagoras was so interested in.

What you say works where lengths a = b, in a + b = c. An additional case is needed where lengths, a > b, or, b > a.

Notice that the AGGs used points to define the location, identity, and size of magnitudes [which term, magnitude, meant 'scientific concept' to them], rather than named line lengths.

Also, where, within a proposition, you see a list of lines, e.g.,

a: e _____________ f

b: g ___ y

c: x ________________ y

the figure may not have been drawn.

The naming of lengths, e.,g., a,b,c, is a modern practice in order that numerical values from algebraic relationships made be always indicated. The AGGs kept the concepts abstract and universal throughout the propositions, and throughout all The Elements. Where line, x,y, was mentioned, it meant to them, the relationship of the summation of, e,f, and, g,y. The essential identity of the entity and the relationships given due to the properties of the entities was never discarded. Non-essential characteristics, e.g., color, or solidity, were ignored where irrelevant for the discussion.

Entities in Euclid's "The Elements" were graphically notated as lines having properties, without the more general figure having been supplied. and the graphic line length entities, e.g., xy, often represented complex arguments, e.g. [in algebra], =,

a^2 + 2ab + b^2,

or, to the AGG in point designations for values,

ef^2 + 2ef, gy + gy^2 = xy^2 .

Still, the square (area) on straight line or numerical length, ef, added to the square (area) on the straight line or numerical length, gy, plus the two rectangles (areas), e.g., ef, gy, equals the square (area) on the straight line or numerical length of, xy.

The important transition that occurred sometime between possibly, Eudoxus, and Euclid, was that the figures, or general overall scheme or relationship of concepts, were kept in mind and not drawn graphically in all instances. The complete figure for the proposition at the beginning of a series of related propositions may have been fully drawn, unless the series of propositions was based upon on a relationship of ideas that had been previously discussed.

When reading Euclid, it is most helpful to imagine the graphic diagrams of Pythagoras, e.g., the type that would have been drawn in the sand on the classroom floor, and, especially the diagrams of Pythagoras' Hekatompedon geometrical and educational system, and to fit the points called out in the proposition, for example, onto the diagram.

Even Aristotle alludes to the Hekatompedon system, and he dabbles in some geometry and philosophical commentary, himself. More significantly, Aristotle converted the "Theory of Equiproportionality" of Eudoxus into the systematization of proofs and deductive logic with the integration of his own theory of genus and differentia definitions and hierarchical concepts.

In modern algebra the conceptual relationships are imagined often without the geometric figure being imagined. The imagination of the geometric figure for proofs is a Pythagorean teaching technique. While drawing was at the base of many of the early discoveries of geometry, and in the days preceding Thales, geometry was accomplished or taught by shipboard navigators by reference to the stellar constellations and named stars..

[Notice that the name of one of Archimedes' books was, The Sand Rekoner."]

The line lengths drawn in propositions, e.g., that were shown above ,were a particularization in graphic form only of relationships and/or finite values for line lengths, sums, squares, areas, for example. Those lines drawn in The Elements were not only straight line lengths placed at certain locations in the figure imagined. The AGGs held that once the proposition had been proved, more than one figure may have been possible for the same conceptualization of relationships and validation.

Then who did the drawing for the AGs that made the geometry and mathematics of the scientists useful? The music scientists, shipwrights, navigators, city and civil planners, mappers, potters, and architects, for example.

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[Notice that the name of one of Archimedes' books was, The Sand Rekoner."]

The Sand Reckoner had nothing to do with figures drawn in the sand. It had to do with estimating the number of grains of sand that would fill the universe (as was known to Greek astronomers). Archimedes developed an exponential system for expressing very large numbers. Here is the blurb from the Wiki article on -The Sand Reckoner-.;

"The Sand Reckoner (Greek: Αρχιμήδης Ψαµµίτης, Archimedes Psammites) is a work by Archimedes in which he set out to determine an upper bound for the number of grains of sand that fit into the universe. In order to do this, he had to estimate the size of the universe according to the then-current model, and invent a way to talk about extremely large numbers. The work, also known in Latin as Archimedis Syracusani Arenarius & Dimensio Circuli, is about 8 pages long in translation, is addressed to the Syracusan king Gelo II (son of Hiero II), and is probably the most accessible work of Archimedes; in some sense, it is the first research-expository paper."

Bob Kolker

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[Notice that the name of one of Archimedes' books was, The Sand Rekoner."]
The Sand Reckoner had nothing to do with figures drawn in the sand. It had to do with estimating the number of grains of sand that would fill the universe ...

I see that Bob has drawn a line in the sand against these highly imaginative assertions.

Archimedes developed an exponential system for expressing very large numbers. Here is the blurb from the Wiki article on -The Sand Reckoner-.;

"The Sand Reckoner (Greek: Αρχιμήδης Ψαµµίτης, Archimedes Psammites) is a work by Archimedes in which ...

More useful for the forum than counting grains of sand is that the copy and paste with Greek letters shows how to display mathematical symbols here on the Forum. Archimedes would have loved it and may have gone on to publish "The HTML Reckoner" about how many bytes it would take to fill the universe.

I had seen such copying of special characters work before but hadn't looked closely to see how to do it. The Forum software works with html escape codes d; where d is a decimal number. Use the translations given here or here. But you have to use the numerical code version, like 8747 for ∫. You can't use the name codes, like ∫, which don't work, and neither do sub, sup or overline for superscripts, subscripts or overlining to extend the radical √, for which there apparently are no numeric formatting codes. But you can copy expressions with exponents: the overline is translated to parentheses and the exponents are retained: √(a²+b²)

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[Notice that the name of one of Archimedes' books was, The Sand Rekoner."]

The Sand Reckoner had nothing to do with figures drawn in the sand. It had to do with estimating the number of grains of sand that would fill the universe (as was known to Greek astronomers). [...]

Bob Kolker

My error. I had thought that "The Sand Rekoner" title referred to Euclid, who was killed by a Roman Soldier while defending his drawing in sand. Not so, apparently.

On HPO I wrote a note regarding the finite example of a large number given by Archimedes. The number he gave was a number that was commonly known to AGGs as a number conceived of by Pythagoras. Recalling what I read years ago on the topic, apparently a student had asked him what was the largest number. Archimedes replied that it was [one miriad times one miriad times one miriad times eight].

To the AGGs, one miriad was 10,000 units or entities.

I won't quote chapter and verse to prove what I say. Suffice it to say that to experts a certain level of credibility is often expected. Technical works require quotations and source materials, however, checkability, is required. No fancy assertions e.g., claiming the Big Bang originated from a primordial pea, are considered to be respectable claims. That is simply uncheckable.

Why Pythagoras? In his system of geometry, all developments are drawn from a sort of 3D mental template, first from a point, then a straight line, then a parallel to the line at unit distance from the first, then a series of squares made of medial lines, omitting all the root rectangles for this demonstration, e.g., x^.5. Then extending the squares out to 10 squares, for the number 100^.5, and adding squares until one miriad units is reached. Then imagine all the units squared to find more units, and then complete all the squares in the model to complete a greater square of one miriad on a side. That's in the xy directions. Then add a z dimension. That's a miriad times a miriad miriad more units or points.

That's part of what Archimedes said. He said that number that he was expressing was 8 times that large. So far we have created a cube of units that is one miriad on a side, per Archimedes' specification, and that started at an origin point of the imaginary model. Addressing the given number 8, simply add 7 more one miriad^3 cubes. Place them so that each of the 8 cubes are separately constructed beginning from the origin point first given.

The result is a super cube that is two miriads on a side.

That is Archimedes' great number, and it was a matter of selection.

The number was constructed in accordance with Pythagoras' geometric construction system.

LLLLLLLLLLL

For extra information lets reflect upon the epistemological entity called the sphere. All good student geometers will have done their "circles", and those proofs will have been placed for demonstrations' sake in the above described geometric model format. Measurements are omitted except where specified.

Of course the spheres are transparent, and because they exist in mind only, they may be seen to be invisible. The spheres, and the selected elements of the Hekatompdon demonstration model exist in the mind only as ideas, or to us or Aristotle, as epistemological entities.

The first sphere has a selected radius of one unit starting at the first point placed. Imagine spheres placed at all radii up to radii of one miriad units.

The concept of "magnitude" to the AGGs meant scientific concept or principles, and it is due to mistranslation that the "magnitude" of the AGGs, or of Pythagoras, was interpreted to mean finite numbers.

The selected spheres, all elements, and demonstrated functional relationships are epistemological only. They are not physical.

To Pythagoras the universe was made of and functioned in accordance with epistemological facts and principles.

The universe of ideas to Aristotle was made of "spheres", or "magnitudes", again, selected principles and functionings, for purposes of discussion.

A large number to Archimedes was, e.g., 8(10000)^3.

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A large number to Archimedes was, e.g., 8(10000)^3.

Inventor

When you consider how cumbersome the Greek numbering system was (they used their alphabet to represent numbers based on a linear tally - just like Roman numerals or the Hebrew number system) this was a breakthrough.

Unfortunately Archimedes did not have the zero. If he had, he very well might have (re)invented the positional number system (the Chinese had it at the time of Archimedes as did the mathematicians of India).

Bob Kolker

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The equation is known as the "Application of Areas" because the area of the square comprised of,

a^2 + 2ab + b^2

exactly fits within the square comprised of,

c^2 .

[text omitted]

c^2 = (a+b )^2 - 4 x 0.5 x ab = (a^2 + 2ab + b^2) - 2ab = a^2 + b^2.

In my high school geometry textbook (published in the 1950s), this is presented as an algebraic proof of the Pythagorean Theorem.

I'm not strong in math, however, it seems to me that there are so many errors in the equations that science stopped.

1. Formal error - proof.

a. No proper format for a proof has been followed, e.g., equations are proved by reasoning from universals to particulars, and no particulars have been found.

b. No proper first and middle premises and conclusion have been placed.

2. Formal error - ambiguous structure.

a. Its a run-on sentence that has three [=] verbs.

3. Formal error:

a. The terms are not equal as required by the [=] sign.

b. If it were broken down into three premises required for a proof it might be:

c^2 = (a+b )^2 - 4 x 0.5 x ab

= (a^2 + 2ab + b^2) - 2ab

= a^2 + b^2

b. The first premise is not true. [Make a=b, a=1, solve.]

c. The second premise is not true. [The term, "-2ab" is arbitrary.]

d. The conclusion is wrong. [it lacks the term, "+2ab".]

e. Substitutions of numbers in the premises do not yield consistent results when, e.g., a=b, a=2 [or, e.g., a=b, a=1]

2. Logical errors.

a. The first premise is so not true that it cannot equal the conclusion. That leads to the fallacy of Non Sequitur.

b. The first premise with its numbers is more particular than the conclusion; that's a fallacy of distribution.

c. Both the second premise and the conclusion should equal "c^2", and they do not.

d. Nor does the second premise equal the conclusion, as specified by the "=" sign.

If I were more skilled in math, I would have seen the faults and I wouldn't have bothered. I may have been time-trolled.

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[Notice that the name of one of Archimedes' books was, The Sand Rekoner."]
The Sand Reckoner had nothing to do with figures drawn in the sand. It had to do with estimating the number of grains of sand that would fill the universe (as was known to Greek astronomers). [...]

Bob Kolker

My error. I had thought that "The Sand Rekoner" title referred to Euclid, who was killed by a Roman Soldier while defending his drawing in sand. Not so, apparently...

No, it is not so, and neither is that description of Euclid's death (in Egypt ca. 265BC), the nature of which is unknown. Archimides himself, not Euclid, was killed by a Roman soldier for continuing to work when the soldier demanded that he accompany him (in Syracuse 212BC). Archimedes did not entitle his Sand Reckoner based on Euclid's death -- or his own previous death.

I won't quote chapter and verse to prove what I say. Suffice it to say that to experts a certain level of credibility is often expected. Technical works require quotations and source materials, however, checkability, is required. No fancy assertions e.g., claiming the Big Bang originated from a primordial pea, are considered to be respectable claims. That is simply uncheckable...

Whatever this statement is supposed to mean, for all his lengthy posts and his interest in this subject matter, inventor is not an expert in this field. I do not intend this in a personal way, but as a matter of content his continuing posts on this topic are characterized by mathematical falsehoods, unintelligible technical claims employing ambiguous terminology, mis-attributions in history, and over-enthusiastic speculations attributing undocumentable 'history' presented as fact about the person of Pythagoras in particular -- about whom very little is in fact known. Much of this is unintelligible, contextless rambling (which I am sure is not his intent) in which one cannot discern what he is talking about, or where he is going with it, with no clue as to what the source or point of it is. He can and should pursue and enjoy any interest he chooses -- and this one in particular -- but no one unfamiliar with the historical and mathematical subject matter should, if anyone is, be tempted by the notion that these posts represent expertise, or grant the credibility inventor claims for them. (If this is not already obvious, remember the fallacy "Muddy waters look deep" -- not that this is his intention.) In particular, in the context of this Forum, unless and until he can demonstrate otherwise, the content of his posts should not be construed as having any intellectual relation to Objectivism or any Objectivist he mentions.

Inventor appreciates the early accomplishments of the Greeks, as does Bob and many others among us, but understanding and presenting them requires far more objectivity and care for fact than has been demonstrated in inventor's posts, which I hope he will choose to correct. Those interested in a reliable account of ancient Greek mathematics should consult reputable historical sources such as Sir Thomas Heath's A Manual of Greek Mathematics, 1931, re-issued as a relatively inexpensive Dover paperback Greek Mathematics in 1963 (only $3.75 in 1973!).

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[...]

Whatever this statement is supposed to mean, for all his lengthy posts and his interest in this subject matter, inventor is not an expert in this field. I do not intend this in a personal way, but as a matter of content his continuing posts on this topic are characterized by mathematical falsehoods, unintelligible technical claims employing ambiguous terminology, mis-attributions in history, and over-enthusiastic speculations attributing undocumentable 'history' presented as fact about the person of Pythagoras in particular -- about whom very little is in fact known. Much of this is unintelligible, contextless rambling (which I am sure is not his intent) in which one cannot discern what he is talking about, or where he is going with it, with no clue as to what the source or point of it is. He can and should pursue and enjoy any interest he chooses -- and this one in particular -- but no one unfamiliar with the historical and mathematical subject matter should, if anyone is, be tempted by the notion that these posts represent expertise, or grant the credibility inventor claims for them. (If this is not already obvious, remember the fallacy "Muddy waters look deep" -- not that this is his intention.) In particular, in the context of this Forum, unless and until he can demonstrate otherwise, the content of his posts should not be construed as having any intellectual relation to Objectivism or any Objectivist he mentions.

Inventor appreciates the early accomplishments of the Greeks, as does Bob and many others among us, but understanding and presenting them requires far more objectivity and care for fact than has been demonstrated in inventor's posts, which I hope he will choose to correct. [...]

I should speak lines in my defense here, however, my mood is not there.

I made two errors regarding my mentions of "The Sand Reckoner". I'm sorry that I didn't check the facts. I thank you for bring that to my attention and for the mention of the Heath book.

The issue is whether my remarks are on topic. The thread is entitled, Is Beauty Quantifiable?" The "Golden Mean" and other primary principles are basic to an appreciation of beauty, and in the field of design and architecture, for example, the study of the basics of geometry and mathematics created by the AGGs are some of the foundations of our knowledge. AG philosophers also dealt with human values and art. All of that is appropriate on this thread.

You've made some sweeping unsupported generalizations about my ideas, and you've too often failed to bring forth any single fact or logical rebuttal regarding any single sentence that I have written. That is, one premise at a time.

The debasing of my character in an Objectivist context is unwarranted, and you owe me an apology. In the future if you fail to criticize specifics and instead provide a high-handed lambasting of my character and conclusions I will bring the matter to the attention of the moderator.

And don't tell me you can prove with mathematics that the Big Bang exists as an identification of a supposed origin of the universe. [That's supposed to be funny.] The universe is quite continuous of existence as it is.

Lighten up a little.

Inventor

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[...]

Whatever this statement is supposed to mean, for all his lengthy posts and his interest in this subject matter, inventor is not an expert in this field. I do not intend this in a personal way, but as a matter of content his continuing posts on this topic are characterized by mathematical falsehoods, unintelligible technical claims employing ambiguous terminology, mis-attributions in history, and over-enthusiastic speculations attributing undocumentable 'history' presented as fact about the person of Pythagoras in particular -- about whom very little is in fact known. Much of this is unintelligible, contextless rambling (which I am sure is not his intent) in which one cannot discern what he is talking about, or where he is going with it, with no clue as to what the source or point of it is. He can and should pursue and enjoy any interest he chooses -- and this one in particular -- but no one unfamiliar with the historical and mathematical subject matter should, if anyone is, be tempted by the notion that these posts represent expertise, or grant the credibility inventor claims for them. (If this is not already obvious, remember the fallacy "Muddy waters look deep" -- not that this is his intention.) In particular, in the context of this Forum, unless and until he can demonstrate otherwise, the content of his posts should not be construed as having any intellectual relation to Objectivism or any Objectivist he mentions.

Inventor appreciates the early accomplishments of the Greeks, as does Bob and many others among us, but understanding and presenting them requires far more objectivity and care for fact than has been demonstrated in inventor's posts, which I hope he will choose to correct. [...]

I should speak lines in my defense here, however, my mood is not there.

I made two errors regarding my mentions of "The Sand Reckoner". I'm sorry that I didn't check the facts. I thank you for bring that to my attention and for the mention of the Heath book.

The issue is whether my remarks are on topic. The thread is entitled, Is Beauty Quantifiable?" The "Golden Mean" and other primary principles are basic to an appreciation of beauty, and in the field of design and architecture, for example, the study of the basics of geometry and mathematics created by the AGGs are some of the foundations of our knowledge. AG philosophers also dealt with human values and art. All of that is appropriate on this thread.

You've made some sweeping unsupported generalizations about my ideas, and you've too often failed to bring forth any single fact or logical rebuttal regarding any single sentence that I have written. That is, one premise at a time.

Numerous falsehoods in your posts have been rebutted throughout this entire thread. This is not restricted to the two most recent blunders, and it includes the baseless assertion that the golden mean is "basic to an appreciation of beauty", which is false. But much of what your writing here is so rambling and vague that that there is nothing specific enough to know what it means let alone discuss or rebut it. You are not an expert on this subject in either knowledge or ability to present it. Irrelevance to the topic of the thread is the least of it.

The debasing of my character in an Objectivist context is unwarranted, and you owe me an apology. In the future if you fail to criticize specifics and instead provide a high-handed lambasting of my character and conclusions I will bring the matter to the attention of the moderator.

No one has debased your character. No one has said anything like that. Your rambling posts, full of errors in both history and mathematics -- which you are apparently unaware of on all counts -- have nothing to do with Objectivism and should not be mistaken as such by readers.

And don't tell me you can prove with mathematics that the Big Bang exists as an identification of a supposed origin of the universe. [That's supposed to be funny.] The universe is quite continuous of existence as it is.

Lighten up a little.

I said nothing about your "Big Bang" assertions, which have nothing to do with this discussion, the thread or Greek geometry. We have no way of knowing which of your irrelevant insertions are supposed to be a "joke" and which are supposed to be serious pronouncements. As long as you are posting here it must be presumed that you are responsible for what you write.

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b. If it were broken down into three premises required for a proof it might be:

c^2 = (a+b )^2 - 4 x 0.5 x ab

= (a^2 + 2ab + b^2) - 2ab

= a^2 + b^2

b. The first premise is not true. [Make a=b, a=1, solve.]

c. The second premise is not true. [The term, "-2ab" is arbitrary.]

d. The conclusion is wrong. [it lacks the term, "+2ab".]

e. Substitutions of numbers in the premises do not yield consistent results when, e.g., a=b, a=2 [or, e.g., a=b, a=1]

The first equation is a true and correct formulation for the area of a square whose side length is 'c', residing within a larger square whose side length is (a+b ), if the inner square (side length c ) is rotated in exactly the right way so as to create four right triangles at the corners of the larger square. Each of the right triangles has legs a and b, and hypotenuse c. The equation is true for any positive real value of a and b.

The second equation is just an algebraic reduction of the first. It is not a second premise in a syllogism.

The third equation is just an algebraic reduction of the second.

I have to wonder if Inventor and I are talking about the same geometric shape. To repeat: my discussion is for a larger square, side length (a+b ), having a smaller square, side length c, entirely inside of it.

Also, my equations are not a complete proof. The complete proof, as presented in my geometry textbook, also involves reasoning from geometric principles to show that the four a:b:c triangles do, indeed, form a square of side length (a+b ) when arranged in the right way, and that the area in the middle, consisting of four sides of length c, is indeed a square and not some other kind of quadrilateral (especially in its corner angles).

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I have to wonder if Inventor and I are talking about the same geometric shape.

No, we are not.

The formula,

c^2 = a^2 + 2ab + b^2

is for the Application of Areas.

Note that the square on c is exactly the same as the square on a+b.

You have brought forth one of the diagrams for the proof of one case of the solution to the Hypotenuse of a Right Triangle,

x^2 = y^2 + z^2

Where x is the hypotenuse and y,z are sides of a right triangle, and in most diagrams the square on x is rotated.

The two formulas are not the same.

A diagram helps.

I'm an expert designer-architect-inventor, a non-mathematician, and a would-be writer-geometer.

Thanks for your comments.

Inventor

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b. If it were broken down into three premises required for a proof it might be:

c^2 = (a+b )^2 - 4 x 0.5 x ab

= (a^2 + 2ab + b^2) - 2ab

= a^2 + b^2

b. The first premise is not true. [Make a=b, a=1, solve.]

c. The second premise is not true. [The term, "-2ab" is arbitrary.]

d. The conclusion is wrong. [it lacks the term, "+2ab".]

e. Substitutions of numbers in the premises do not yield consistent results when, e.g., a=b, a=2 [or, e.g., a=b, a=1]

The first equation is a true and correct formulation for the area of a square whose side length is 'c', residing within a larger square whose side length is (a+b ), if the inner square (side length c ) is rotated in exactly the right way so as to create four right triangles at the corners of the larger square. Each of the right triangles has legs a and b, and hypotenuse c. The equation is true for any positive real value of a and b.

The second equation is just an algebraic reduction of the first. It is not a second premise in a syllogism.

The third equation is just an algebraic reduction of the second.

I have to wonder if Inventor and I are talking about the same geometric shape. To repeat: my discussion is for a larger square, side length (a+b ), having a smaller square, side length c, entirely inside of it.

According to a post from last Jan 11 he is talking about a square of side c = a + b, with smaller squares of side a and b touching at their corners in the interior of the larger square of side c. He seems to be trying to explain, out of the blue and for unspecified motives, the Greek's method of "application of areas" which he misuses as meaning his particular quadratic equation with particular variable names. If you can't follow what he is talking about you are not alone. You have to already know what it is to untangle the disconnected statements, many of them wrong, misleading, meaningless or nonsequiturs. He apparently does not understand the simple equations and straightforward general algebraic reductions you noted above, confusing them as "premises" about something else he has in mind.

The Greek's "applications of areas" was a method for constructing figures in different configurations, constrained to be a given area. The constructions were the geometric equivalent of our algebraically solving different kinds of quadratic equations, which algebraic concepts they did not have. If you are interested in pursuing this you can read competent classic accounts (by Heath) here and here.

The construction you were talking about for the Pythagorean theorem immediately precedes those discussions. It is the same or equivalent one used in Bronowski's Ascent of Man TV series in the

on the Pythagorean Theorem.

But this has nothing to do with the mythology behind the alleged role of the 'golden ratio' in esthetics, which misuse is more like Pythagorean number mysticism and has nothing to with mathematical science by Greeks or anyone else. The mathematics related to that ratio is an entirely different subject.

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According to a post from last Jan 11 he [inventor] is talking about a square of side c = a + b, with smaller squares of side a and b touching at their corners in the interior of the larger square of side c. He seems to be trying to explain, out of the blue and for unspecified motives, the Greek's method of "application of areas" which he misuses as meaning his particular quadratic equation with particular variable names. If you can't follow what he is talking about you are not alone. You have to already know what it is to untangle the disconnected statements, many of them wrong, misleading, meaningless or nonsequiturs. He apparently does not understand the simple equations and straightforward general algebraic reductions you noted above, confusing them as "premises" about something else he has in mind.

The Greek's "applications of areas" was a method for constructing figures in different configurations, constrained to be a given area. The constructions were the geometric equivalent of our algebraically solving different kinds of quadratic equations, which algebraic concepts they did not have. If you are interested in pursuing this you can read competent classic accounts (by Heath) here and here.

The construction you were talking about for the Pythagorean theorem immediately precedes those discussions.

Thanks for the explanation and excellent references on "Application of Areas" and Pythagorean Geometry. I see now that 'c' in AoA is being used simply as a name for a+b, i.e., c = a + b, a square whose side length is (a + b ), whereas I was using 'c' as the hypotenuse of a right triangle having legs a and b. Somehow the discussion got AoA mixed up with the Pythagorean Theorem (juxtaposed as they are even in Heath). Thanks again for masterfully sorting it all out.

Incidentally, I also found some excellent discussion of constrained-area analyses of the Pythagorean Theorem in the Wkipedia article on "Pythagorean Theorem."

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The equation is known as the "Application of Areas" because the area of the square comprised of,

a^2 + 2ab + b^2

exactly fits within the square comprised of,

c^2 .

Did you mean that the other way around? I.e., the square whose side length is c fits within the square whose side length is a+b. If the square whose side length is c is rotated such that its four corners exactly touch the sides of the square whose side length is a+b, the result is four right triangles in the "a+b" square, at its corners, having sides a and b and hypotenuse c. It can be seen that the area of the inner square, c^2, is equal to the area of the outer square, side length a+b, minus the areas of the four right triangles at the corners:

c^2 = (a+b )^2 - 4 x 0.5 x ab = (a^2 + 2ab + b^2) - 2ab = a^2 + b^2.

In my high school geometry textbook (published in the 1950s), this is presented as an algebraic proof of the Pythagorean Theorem.

Regarding the Pythagorean Theorem, e.g., c^2 = a^2 + b^2 :

Later than the textbook reference you provided there was a question raised regarding the above terms that the textbook states and whether the term premises is appropriate.

I haven't read the book, however, you do say "proof". That means that either a deductive logical demonstration of a proposition is made from the premises, or the Application of Areas method of proof is used.

The series of statements given really isn't in the form of a logical proof. The middle term is not omitted, and the last premise doesn't follow from more universal premises.

One other reason the statement isn't a proof is that each of the three premises are numerically equal to c^2. That is a mere statement of equivalencies formed by algebraic manipulation. Its like a run on sentence, and the thing being proved, the third premise being equal to c^2, is present in each premise. That is a violation of the principles of deductive logical proofs, and it would be an example of the fallacy of Post Hoc Ergo Propter Hoc.

Its not a proof. Its more like a run-on sentence style of manipulation that is commonly used in mathematics to skirt the necessity for a logical proof from universal premises.

Regarding the Application of Areas, e.g., c^2 = a^2 + 2ab + b^2 :

I'll try to add clarity regarding these geometric concepts.

In the time of Pythagoras there were two basic types of proof. The proof by deduction in a demonstration from more universal concepts or facts, and the Application of Areas method of proof.

The logical demonstration was inconsistently used during Pythagoras time, and it wasn't until Eudoxus Theory of Equiproportionality, and later, Aristotle's theories of definitions, the hierarchical classification of knowledge using the genus and differentia of concepts, and his systematization of deductive logic, that logical proofs were the standard of truth and fact.

Prior to Aristotle the Application of Areas was commonly used. There were errors of thinking that resulted, and certain types of new work in solving certain types of geometric or mathematical problems stopped.

For example, The Duplication of the Cube, and The Cube of the Sphere, were not possible or provable until proofs involving properties and principles of entities or elements were used.

The AoA method is the method of validating a geometric idea by means of setting up the terms so that one figure will exactly fit another. It was merely that a fit is a fit, and other properties of the things so fitted were not always considered in the proof of a proposition. For example, in,

c^2 = a^2 + 2ab + b^2 ,

the geometer using the AoA method was able to say that, c^2, was equal to, a^2 + 2ab + b^2, because the drawings of the entities, e.g., the square drawn on, c^2, exactly appeared to fit the square drawn on the length, a+b. The square figures had the same origin and side straight lines. The squares were known figures, but the other properties of the equal entities were found to be more fundamental and led to a form of logical proof based upon universals and facts.

AGGs became aware that squares were not the only entities that were roadblocks to further work, for example, the area of a square did not have to be limited to squares, and could be equal to that of a circle. Or, the number of the area of a circle could be given a numerical value and be expressed a straight line that had the same number. These straight lines are commonplace in Euclid for example, and they are used for many numerical property purposes, and other purposes, e.g., the relationships of lesser and greater. Properties was the name of the game after Pythagoras.

For example, a proof that was based upon the properties of some however unspecified numerical values of line lengths was more exact and could be proved by logical methods, e.g, by alternate paths of calculation or values. Pythagoras found that drawings were not the total deal, and that properties, including numerical values, were more important than the fits, e.g., the congruencies, of the parts being equated. Things that were equal no longer had to be congruent, or even numerically equal, for that matter, for the epistemological entities that were the elements and principles had many possible types of properties and functionings that could be defined. Pythagoras started a process of more sophisticated and reliable proof, and Aristotle made a more exact science of all that.

The proposition that was known as the Application of Areas became famous as the step from which geometry could precisely move from drawings and fits for proofs to a more exact science. The drawing for the AoA was the same as the diagram for that needed for a properties-based logical method of proof.

Pythagoras did it all in teaching demonstrations

Just because one thing is numerically equal to another thing does not mean that they have equal properties.

Logical proof finds that the line length that has the numerical value of length c squared will equal the line length that has the numerical value of length a that has been extended by the numerical value of length b, the sum squared.

What's next?

The diagram for the Golden Mean Division of a Straight Line.

That too is a scheme that is constructed between parallel lines, and that was in use prior to Pythagoras in the design of the Great Pyramid of Kheops.

Inventor

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For example, in,

c^2 = a^2 + 2ab + b^2 ,

the geometer using the AoA method was able to say that, c^2, was equal to, a^2 + 2ab + b^2, because the drawings of the entities, e.g., the square drawn on, c^2, exactly appeared to fit the square drawn on the length, a+b.

Change "c^2" to "c", and, adding, "straight line length", the latter should read:

" . . . e.g., the square drawn on straight line length, c, exactly appeared to fit the square drawn on the straight line length, a+b.

Inventor

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The equation is known as the "Application of Areas" because the area of the square comprised of,

a^2 + 2ab + b^2

exactly fits within the square comprised of,

c^2 .

Did you mean that the other way around? I.e., the square whose side length is c fits within the square whose side length is a+b. If the square whose side length is c is rotated such that its four corners exactly touch the sides of the square whose side length is a+b, the result is four right triangles in the "a+b" square, at its corners, having sides a and b and hypotenuse c. It can be seen that the area of the inner square, c^2, is equal to the area of the outer square, side length a+b, minus the areas of the four right triangles at the corners:

c^2 = (a+b )^2 - 4 x 0.5 x ab = (a^2 + 2ab + b^2) - 2ab = a^2 + b^2.

In my high school geometry textbook (published in the 1950s), this is presented as an algebraic proof of the Pythagorean Theorem.

Regarding the Pythagorean Theorem, e.g., c^2 = a^2 + b^2 :

Later than the textbook reference you provided there was a question raised regarding the above terms that the textbook states and whether the term premises is appropriate.

I haven't read the book, however, you do say "proof". That means that either a deductive logical demonstration of a proposition is made from the premises, or the Application of Areas method of proof is used.

The series of statements given really isn't in the form of a logical proof. The middle term is not omitted, and the last premise doesn't follow from more universal premises.

One other reason the statement isn't a proof is that each of the three premises are numerically equal to c^2. That is a mere statement of equivalencies formed by algebraic manipulation. Its like a run on sentence, and the thing being proved, the third premise being equal to c^2, is present in each premise. That is a violation of the principles of deductive logical proofs, and it would be an example of the fallacy of Post Hoc Ergo Propter Hoc.

Its not a proof. Its more like a run-on sentence style of manipulation that is commonly used in mathematics to skirt the necessity for a logical proof from universal premises.

There is no question about the incorrectness of calling algebraic derivations "premises". They are not. There is no question that algebraic derivations from one step to the next constitute proof. These analyses are so simple that there is no requirement to restate all the rules of algebra every time they are applied. The contention that this is a "fallacy" and the equivalent of a "run-on sentence" shows that you do not understand these simple algebraic derivations.

Regarding the Application of Areas, e.g., c^2 = a^2 + 2ab + b^2 :

I'll try to add clarity regarding these geometric concepts.

In the time of Pythagoras there were two basic types of proof. The proof by deduction in a demonstration from more universal concepts or facts, and the Application of Areas method of proof.

You are not clarifying anything. There is almost no written record of Pythagoras' work, let alone "two types of proof". "Application of Areas" is not a "type of proof" in contrast to "deduction". It is an application of deduction for a certain class of geometrical problem.

The logical demonstration was inconsistently used during Pythagoras time, and it wasn't until Eudoxus Theory of Equiproportionality, and later, Aristotle's theories of definitions, the hierarchical classification of knowledge using the genus and differentia of concepts, and his systematization of deductive logic, that logical proofs were the standard of truth and fact.

Deductive proofs are not the "standard of truth and fact". Truth is correspondence with fact.

Prior to Aristotle the Application of Areas was commonly used. There were errors of thinking that resulted, and certain types of new work in solving certain types of geometric or mathematical problems stopped.

For example, The Duplication of the Cube, and The Cube of the Sphere, were not possible or provable until proofs involving properties and principles of entities or elements were used.

That is not a meaningful statement. The Greeks lacked the ability to solve certain kinds of problems because their geometric axioms were inadequate for them. "Application of areas" was used to solve certain kinds of problems. It had nothing to do with "errors of thinking" "prior to Aristotle".

The AoA method is the method of validating a geometric idea by means of setting up the terms so that one figure will exactly fit another. It was merely that a fit is a fit, and other properties of the things so fitted were not always considered in the proof of a proposition.

That is not what the method of application of areas means. Did you read the references previously given?

For example, in,

c^2 = a^2 + 2ab + b^2 ,

the geometer using the AoA method was able to say that, c^2, was equal to, a^2 + 2ab + b^2, because the drawings of the entities, e.g., the square drawn on, c^2, exactly appeared to fit the square drawn on the length, a+b. The square figures had the same origin and side straight lines. The squares were known figures, but the other properties of the equal entities were found to be more fundamental and led to a form of logical proof based upon universals and facts.

"More fundamental" for what? The identification of a length of a line segment being the sum of the lengths of two segments of which it is comprised is a simple observation of numerical addition of quantities, in the Greek's geometric form, as one step in a longer, more complex chain of reasoning. The geometric constructions were abstractions that referred to an unlimited number of instances; the constructions were "universal". Other "properties" do not "lead to a form of logical proof".

AGGs became aware that squares were not the only entities that were roadblocks to further work, for example, the area of a square did not have to be limited to squares, and could be equal to that of a circle. Or, the number of the area of a circle could be given a numerical value and be expressed a straight line that had the same number. These straight lines are commonplace in Euclid for example, and they are used for many numerical property purposes, and other purposes, e.g., the relationships of lesser and greater. Properties was the name of the game after Pythagoras.

Squares were not "roadblocks". The Greeks used line segments to represent quantitites because they used geometry, not algebra.

For example, a proof that was based upon the properties of some however unspecified numerical values of line lengths was more exact and could be proved by logical methods, e.g, by alternate paths of calculation or values. Pythagoras found that drawings were not the total deal, and that properties, including numerical values, were more important than the fits, e.g., the congruencies, of the parts being equated. Things that were equal no longer had to be congruent, or even numerically equal, for that matter, for the epistemological entities that were the elements and principles had many possible types of properties and functionings that could be defined. Pythagoras started a process of more sophisticated and reliable proof, and Aristotle made a more exact science of all that

Pythagoras started a process but no one knows what he said, if anything, about "properties" versus "fits". A "congruency" between geometrically "similar" figures is not a "fit".

The proposition that was known as the Application of Areas became famous as the step from which geometry could precisely move from drawings and fits for proofs to a more exact science. The drawing for the AoA was the same as the diagram for that needed for a properties-based logical method of proof.

"Application of areas" was not a "step" from "drawings" to "proofs". It was a method of proof for a certain class of problems.

Pythagoras did it all in teaching demonstrations

What "all" are you talking about? Very little is known about what he did in the early stages of Greek geometry.

Just because one thing is numerically equal to another thing does not mean that they have equal properties.

Who said it does?

Logical proof finds that the line length that has the numerical value of length c squared will equal the line length that has the numerical value of length a that has been extended by the numerical value of length b, the sum squared.

This is a trivial fact that does not require pontification. If c = a+b then they represent the same number.

What's next?

The diagram for the Golden Mean Division of a Straight Line.

"Next"? There is no connection.

That too is a scheme that is constructed between parallel lines, and that was in use prior to Pythagoras in the design of the Great Pyramid of Kheops.

Inventor

There is no evidence that the golden ratio was used for any such esthetic design or that it has any such objective meaning.

What is your purpose in these posts? They appear to be intended as some kind of instruction to us, with no stated reason for it and no credibility because they are filled with errors, ambiguities and non sequiturs. Where are you getting this from and why are you writing it here on the Forum?

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