Without the solution to this puzzle, we would be living in medieval villages; with technology more primitive than that of the Amish of Pennsylvania; slavery AND semi-slavery wouldbe widespread; women and girls would be repressed, often abused, superstitution widespread. Whatwas the puzzle?

The SQUAREGATE PUZZLE seemed to say, "You can apply ARITHMETIC to the SIDES of THE SQUARE, butyou cannot apply ARITHMETIC to its DIAGONAL. So only GEOMETRY describes THE SQUAREGATE."

The term, magnitude,was created by the great Greek mathematician, Eudoxus of Cnidus (c.408-355 B.C.), a pupil of Archytas of Tarentum (c. 425-350 B.C.), a follower of Pythagoras.As a teacher in the Academy of Plato (427-327 B.C.), Eudoxus applied the notion of magnitudeto the composition of lines, angles, volumes in geometry and also to time. This restriction of time to geometry led to the notion that "motion is geometry set to time". Hence, the four beliefs developed in the Academy:

1. the diagonal of the square (in geometry) is not a magnitude describable by number (in arithmetic);
2. then geometry and arithmetic are incompatible;
3. then dynamical mechanics (the science of motion) and geometry are incompatible;
4. any person believing that dynamical mechanics and geometry are compatible is a follower of Pythagoras.

This prejudice (except among Islamic scholars) prevailed until the Renaissance. The great Galileo (1564-1642) was called a "Pythagorean" in advocating "a law of falling bodies".

The last three beliefs (above) derive from the first. The diagonal of a square is not a magnitude which is describable as a number. What does this mean?

We can explain by applying the socalled "Pythagorean formula", known centuries earlier to Babylonian priests.

Please consider a right triangle (one with an angle of 90 degrees). Let b denote the base of this triangle; let a denote the altitude or height of this triangle; let d denote the diagonal of this triangle. Then by the Pythagorean formula: b ² +a² = d². For example, let b = 3, a = 4.Then, 3² + 4² = 9 + 16 = 25 = d² . Then, 25 = 5²; then the triangle has a diagonal of lengthd = 5. Thus, the diagonal of this triangle can be measured by a number.

Now, please consider the right triangle in the unit square (diagrammed on the FrontPage of this Website). In this triangle, we have b = a = 1; then, 1² + 1² = 1 + 1 = 2 = d²; d = 2. Then the diagonal of "The SquareGate" triangle is the square root of two. What kind of number is this? The Pythagorean follower, Hippias (460-400 B.C.), appparently found a proof that the squareroot ofF 2 is not a fraction.

The ancient Babylonians apparently didn't know the answer to this question, for they bypassed it. They possessed a marvelous ALGORITHM for APPROXIMATING THE SQUARE ROOT OF ANY NUMBER. So they could find an approximating fraction that would satisfy any given case. Apparently, these practical Babylonian priests didn't bother with the theoretical question: "DOES SUCH A FRACTION EXACTLY EXIST?". But the Greeks, daring "to go theoretically where priests fear to tread", took up the challenge.

In The Elements of Geometry of Euclid (365-275 B.C.), numbers are acceptable if they translated, for exmple, into relations between segments of lines. Those numbers are acceptable in that first right triangle, above, because the relation between its line segments are commensurable. What does this mean?

I illustrate this below by diagramming (in red) a three-unit segment (as in the base of the first triangle considered above). Then I diagram (in blue) a four-unit segment (as in the altitudof that triangle). Then I diagram (in black) a five-unit segment (as in the triangle's diagonal).

|----^----^----|  |----^----^----^----|  |----^----^----^----^----|

Since 3 · 4 = 4 · 4 = 12, we can compare (below) four copies of the three-unit segment with three copies of the four-unit segment:

               |----^----^----|----^----^----|----^----^----|----^----^----|
               |----^----^----^----|----^----^----^----|----^----^----^----|

Please notice that these two extend equally, that is, they are congruent. This is the meaning of "commensurable": Two segments are commensurate if a multiple of one segment is congruent to a multiple of the other.

We now demonstrate -- because 3 · 5 = 5 · 3 = 15 that five instances of the 3-segment base is commensurable with three of the 5-segment diagonal:

|----^----^----|----^----^----|----^----^----|----^----^----|----^----^----|
|----^----^----^----^----|----^----^----^----^----|----^----^----^----^----|

Please notice, also, that the congruence of these two extensions indicate the commensurability of the base and diagonal of the SquareGate. And, since 4 ·5 = 5 · 4 = 20, we can compare five copies of the 4-segment altitude with four copies of the 5-segment diagonal:

|----^----^----^----|----^----^----^----|----^----^----^----|----^----^----^----|----^----^----^----|
|----^----^----^----^----|----^----^----^----^----|----^----^----^----^----|----^----^----^----^----|

The congruence of these two extensions indicates the commensurability of the altitude of the SquareGate with its diagonal.

The above demonstrations have been geometric. We can also write their interpretations in arithmetic:

• interpret "four copies of the 3-segment" as the fraction, 3/4;
• interpret "three copies of the 4-segment" as the fraction, 4/3;
• interpret "five copies of the 3-segment" as the fraction, 5/3;
• interpret "three copies of the 5-segment" as the fraction, 3/5;
• interpret "five copies of the 4-segment" as the fraction, 5/4;
• interpret "four copies of the 5-segment" as the fraction, 4/5;
• then we find that 4/3 · 3/4 = 1 = 5/3 · 3/5 = 5/4 · 4/5 = 1.

The arithmetical equivalence of the fractional products corresponds to the geometric commensurably of their corresponding extensions. Then geometric commensurbilty of segments corresponds to representation of these segments as a fraction. However, the "puzzle of the SquareGate" is that the relation of a side of the SquareGate to its diagonal cannot be represented by a fraction.

In Euclid's Elements of Geometry appears a proof (apparently due to Hippias) that the diagonal of a unit suqare is not a fraction. The critical notion, in the proof, is that every fraction can be reduced so that both numerator and denominator are not even numbers, otherwise the common factor of two can be divided out. (Remember! An even naturalnumber has the form, 2n, for some natural number n, and its square has the form, (2n)² = 4n²= 2(2n²). Similarly, an odd natural number has the form, 2n + 1, and its square has the form, (2n + 1) ² = 4n² + 4n + 1 = 2(n² + n) + 1.) The proof proceeds as follows:

1. Consider a/b = 2.
2. Then a = 2b.
3. Squaring both sides: a²2b².
4. The right-hand side has the form of an even number (twice some number), meaning that the left-hand number, a, is an even number.
5. To denote it as an even number, we write a c, for some natural number c.
6. Then we have (2c)² = 4c² = 2b².
7. Dividing out the common factor of two, we have: 2c² = b ².
8. Please notice that the left-hand side has the form of an even number (twice some number) , meaning that the squared-number on the right-hand side is an even number. But we saw above that only an even number has an even square, hence, number b must be an even number.
9. We now have the result that, if there is a fraction, a/b such that a/b = 2, then it must have the peculiar form that numerator and denominator are both even and cannot be reduced. There is no such number. The contradiction negates the assumption that the square root of two is afraction.
10. Hence, the diagonal of a unit square is incommensuable with either of its sides.

Eudoxus of Cnidus (cited above) developed a theory of proportions (in Book III of Euclid's Elements of Geometry) which permitted irrational numbers such as the square root of two. The "axiom of continuity" of Eudoxus indicated that, given the proportion of two magnitude, we can also give that multiple of one as a mulltiple of the other, ensuring that thesemagnitudes are commensurable. In particular, this axiom of Eudoxus allows the proportion between two spheres to be compared with two cubical structures erected on the diameter of eachsphere.

The great German mathemtician, Richard Dedekind (1845-1916), reformulated (in 1872) the idea of Eudoxus as the socalled "Dedekind cut":

1. A cut separates the rational numbers into two classes, "lower" and "upper", such that every number of the lower class is less than every number in the upper class.
2. If a representative of the lower class can be formulated in a fractional relation to a presentative of the opper class, then the cut itself is rationaal.
3. If not, the cut is irrational.

Dedekind thereby freed this study from geometry.

For our present case, we can assign to the upper class all numbers whose squares exceed two, and to the lower class all numbers whose square are less than two. The cut then is the square root oftwo.

We can show this by considering, seven digits, the approximation of the square root oftwo:

1. (1)² = 1 < 2 < 2² = 4;
2. (1.4)² = 1.96 < 2 < (1.5)² = 2.25;
3. (1.41)² = 1.9881 < 2 < (1.42)² = 2.0264;
4. (1.414)² = 1,999396 < 2 < (1.415)² = 2.002225;
5. (1.4142)² = 1.99996164 < 2 < (1.4143)² = 2,00024449;
6. (1.41421)² = 1.9999899924 < 2 < (1.41422)² = 2,0000182084;
7. (1.414213)² = 1.999998409369 < 2 < (1.414214)² = 2,000001237796;
8. etc.

Please notice that

1. as we augment the approximation by one digit,
2. its square approaches closer to two,
3. while its exceeder diminishes down towrd two,
4. and we approach the square root of two as the cut.

But the Eudoxian theory of proportiones motivated ancient Greek mathematicians to abandon the discotinuous or discrete structures of arithmetic for the continuous structures of geometry to describe relations between segments and such. And, since time was considered continuous, it was also separated from arithmetic. This meant that concepts of dynamical mechanics, such as speed, velocity, acceleration, force, etc., could not be defined in terms of arithmetic.

This situation continued in Western Europe until The Renaissance, when some adopted the advances of islamic scholars as described in the file, "Chronology", associated with this Website.

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In SUMMARY:

1. The Puzzle involves incommensurability of the diagonal with the side of a square.
2. But this puzzle can be resolved if a new kind of number could designate such "irrationals" asthe square root of two.
3. Irrationals eventually were accepted -- with all the consequences noted at the outset, creating today's civilization and the mechanical and electric-electronic slaves serving us!