The Pale Blue Dot? Chapter One: Pythagoras, Ptolemy, Copernicus, Kepler, Tycho and Newton; Standing the Test of Time

Originally Written By Thomas Perez. August 13, 2018 at 3:24AM. Copyright 2018. Updated 2020.

A. Pythagoras

Born 570-496BC was a Greek Philosopher and the eponymous founder of the Pythagorean movement. His political and religious teachings were well-known in Magna Graecia and influenced the philosophies of Plato and Aristotle. The proposed idea, or challenge if you will, that the Earth was a circular ball came into being c.500BC (6th Cent BC), with the likes of Pythagoras and his theorems.

According to many who support the absolute of theorems, their arguments are usually like the following:

“Theorems are statements that used to be conjectures, but now have been proven. To prove a theorem true one needs to show that it follows logically from the axioms. The Pythagorean Theorem is often cited as the first. There were many people who knew that a triangle with side lengths that follow the relationship a2+b2=c2 would be a right angle (the Chinese and the Egyptians for example), but no one had proven that it was true for all right angled triangles. This is what Pythagoras and his followers did.

So to sum up, there is no possible way to disprove the Pythagorean Theorem, or indeed any theorem. Theorems are absolute facts that only depend on the axioms that they are derived from. If one wanted to show a theorem to be false, one would need to show one of the axioms to be false, or at least not absolute. This has actually been done before. Google ‘Non-Euclidean Geometry’ and you will find examples of geometries where the Pythagorean Theorem is not true. However that does not invalidate the Pythagorean Theorem since it is only making a statement in Euclidean Geometry.”

For those that may not know what Pythagoras’s Theorem is, or if they simply forget what it consisted of; here is a picture:

“In mathematics, the Pythagorean Theorem, also known as Pythagoras’ Theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.” (Wiki).

What are conjectures and axioms, you may ask? A conjecture is an opinion or conclusion formed on the basis of incomplete information. An axiom is a statement or proposition that is regarded as being established, accepted, or self-evidently true. However, it must, or it should at least, be understood that if a conjecture is based upon uncompleted information, and an axiom is based upon the criteria of self evidence that is proven to be evidently true; then observational evidence should, in and of itself, be the final axiom. Observational evidences are easy to come by these days. More on this later. But for now let us keep our focus.

Now that you have heard, or at least refreshed your mind as to who was Pythagoras and what the Pythagorean Theorem is, allow me to show you how it did not work at one point. Bear with me. Pythagoras and his cult of Pythagorean had a secret club – a lodge, so to speak. In this club we had various mathematicians. They thought of new ways to understand whole numbers in relation to shapes of all sizes. But if someone, for I.e., brought a box that was 2 by 3½, the members of this secret cult society would say; “Three and a half?” “That’s not a number!” “Get out of our club!” Instead, they will make the units half the length and call it a 4 by 7, making everything fine, according to the Pythagorean. Even if your box is 2.718 by 6.28 you can just divide your units into thousandths and you will get a box that is a nice, even 2,718 by 6,280 – not a simple proportion, but the box still has a whole number proportion to it – hence, Pythagoras is pleased.

When we think of numbers, we think of them as being on a line. Numbers one way, zero, and negative numbers the other way, with numbers between them: fractions, rationals, filling in the gaps. But Pythagoras did not think about numbers like this at all. They were not points in a continuum, they were each their own separate being. In Pythagoras’s world, there is no number between 7 and 8 and so forth, and there is no number 3 over 2 (as in fractions). But 6 to 4 has the same relationship because the numbers share this evenness, which when accounted for makes it 3 to 2. To Pythagoras, the universe was made up of these relationships. Pythagoras loved whole numbers and proportions. However, in Pythagoras’s time there were no variables, equations, or formulas like we have today.

Mathematicians like lines too. They also like various numbers found between whole numbers. But numbers between whole numbers did not exist during Pythagoras’s time. Back in Pythagoras’s time, his theorem wasn’t ‘a squared plus b squared equals c squared,’ like it is today. It was ‘The squares of the legs of a right triangle have the same area as the square of the hypotenuse,’ all written out. And when he said “square” he meant ‘square.’

One leg’s square plus the other leg’s square equals the hypotenuse’s square. 3 literally squared plus 4 made into a square. They would cut out square boxes and fit them into each other. And when they did not fit, they would make an adjustment by cutting an even amount of boxes until they all fitted in a given triangle. So one box would have more pieces of boxes placed into them than another, until, whether more or less, the squares equaled that with the hypotenuse squares. To Pythagoras, it was just a matter of figuring out how many pieces to cut each side into. To him, there was a relationship between the length of one side and the length of another, and he wanted to find it on paper. But trouble began with the simplest right triangle – where both the legs are the same length, and where both the legs’ squares are equal. If the legs are both 1, then the hypotenuse is something that, when squared, gives 2.

So what is the square root of 2 and how do we make it into a number ratio? Square root 2 is very close to 1.4 which would be a whole number ratio of 10.14, but 10 squared + 10 squared is definitely not 14 squared, and a ratio of 1,000 to 1,414 is even closer, and it ratio of 100,000,000 to 142,421,356 is very close indeed, but still not exact, so what is it? Pythagoras wanted to find the perfect ratio. He knew it must exist. He wanted whole numbers only. However one day someone from his very own Pythagorean Brotherhood proved there wasn’t a ratio, the square root of 2 is irrational, that in decimal notation, once decimal notation was invented, the digits go on forever. Usually this proof is given algebraically, but the Pythagorean(s) did not know algebra. The individual who “happened to come along” was none other than Hippasus, and his observations concerning the pentagram.

Hippasus was a Pythagorean, and he was an excellent mathematician. That didn’t work out well for him. He noticed something about the pentagram. Namely that, if it’s divided up, there is a certain ratio between the carved up pieces. Hippasus took the measure of the length of the red side divided by the green side. It’s equal to the length of the green side divided by the blue side. And that’s equal to the length of the blue side divided by the purple side. And none of these things are expressible as the ratio of two whole numbers. They form the golden ratio, which, in decimals, is approximately 1.61803.

The golden ratio shows up in many works of art and architecture. It forms the background aesthetic of our lives to this day. Hippasus was obviously clever, but not that clever. He announced that he’d found a way to demolish Pythagoras’s religion on a boat populated by only himself, Pythagoras, and a lot of other Pythagorean(s) for company. Legend has it that Pythagoras tipped him over the side, drowned him, and swore the rest of the group to secrecy. It is doubtful this actually happened, especially since there is another rumor that Hippasus was killed not for coming up with the golden ratio, but with the square root of two – another irrational number. He realized there was no rational way to express the diagonal of a square with sides one unit long. The Pythagorean(s) still believed, and wanted to believe, that this irrationality was somehow false and the world was as they wanted it. So this proof stayed secret. My point? It shows that there’s more to the world than whole numbers and shame on the Pythagorean(s) who didn’t have the courage to admit it. My second point: Mathematical laws can indeed change. Nothing is absolute, something that French Philosopher, astronomer and mathematician Rene Descartes would discuss in his own treatise ‘La Géométrie’ (1637), a short tract included with the anonymously published ‘Discourse on Method.’

Moreover, this particular theorem was known by those before Pythagoras. He wasn’t the originator – this makes it less likely that he was the one that thought it up or proved it. The Babylonians knew this equation centuries earlier. There is no evidence that Pythagoras either discovered or proved it. Beyond that, the historical record suggests he was more interested in the numbers that underlie everyday life than he was in rigorous mathematical proofs, making it even more less likely. It is doubtful whether Pythagoras was really a mathematician as we understand the word. In fact, although genuine mathematical investigations were undertaken by later Pythagorean(s), the evidence suggests that Pythagoras was a mystic who believed that numbers underlie everything. He worked out, for instance, that perfect musical intervals could be expressed by simple ratios.

However, in order to fix the problem of the Theorem, and realizing that a non-right, or oblique, triangle has no right angles; another field was created called trigonometry – a subject whose rules are generally based on right triangles, but can still be used to solve a non-right triangle. However, to perform the proper ratio and decimal “irrationality” you will need different tools – the laws of sines and cosines. Simple, right? Basic high school math. Let us now apply the pythagorean theorem to a curved Earth and check, with our observational empirical pupils, whether this part of the presupposition concerning a curvature holds true, as opposed to a flat Earth model.

The equation M² x 8 = d is nothing new, as the picture demonstrates below…

But I wanted to try it myself. Below is a picture I drew. The large circle is the “spherical Earth.” The two triangles which make a square is the principle of the pythagorean theorem…

Here Is the Math I Did

a = 3963, b = 50, c = 3963.315405, d = .315405, d/m², .315405/50² = drop per mile, .315405/2500 = 000126162, convert into number of inches in a mile (63,360), 000126162 * 63,360 = Answer: The equation is 7.993624. So I came up with 7.993624. But here is a shortened version of both equations as seen below…

M² x .7993624 = d


M² x 8 = d

For those that may be confused, here is how the math is done…

a = (radius) of the Earth is 3963 miles – this is what the scientific community tells us. It is a given.

b = (distance between observer and object is, let just say) = 50 miles.

c = (radius to end point miles of distance between observer and object) = ???

d = (for drop below the horizon) = ???

Obviously “a” and “b” are a given. We needed to find out the answer for “c” and “d.” This is how it is done…

Using the pythagorean theorem, in order to find “c” we first have to take the square of “c” and move that to the left side to make it a square root…

√a² + b² = c

Then we take the radius and square it, plus the distance between the observer and the object, which is, in our case scenario, 50 miles and square it to 3963…

√(3963*3963) + (50*50) = c

Then you add the two numbers together…

√(15,705,369) + (2500) = c

After that, when you add those two numbers together, you get (15,707,869). Then you take the square root of this to get “c” which is 3963.315 miles. So now we found “c.” We now have the “c” radius to end point distance between observer and object.

Now we must find “d” – the drop below the horizon – that would be…

c – a = d

So let’s do the math to find “d”

3963.315405 – 3963 = d which is 315405 miles

Now that we have a, b, c and d; let us figure out how we got to 7.993635 or 8, when rounding it off. First we take the .315405 miles/50² drop in miles. Divide it by “b” 50 miles squared (50²)

d/m² = drop per mile

315405 miles/50² = drop per mile. The M² is our “b”

315405 miles/2500 = 000126162 which is the drop per mile along “b.”  For every mile in “b,” this would be the drop below the horizon

We now convert the number of inches in a mile which is 63,360. Then we…

000126162 * 63,360 = 7.993624.


M² x .7993624 = d


M² x 8 = d

This is “observational empirical evidence based upon pupil evidence” as stated above – which in turn is based upon the pythagorean theorem. The math can be applied to any distance in miles; whether 50, 100 or so on. When an observer at point “b” uses the technologies of today and brings back into focus an object that went supposedly under the horizon at point “d” isn’t that observer defying the pythagorean theorem of a curved/spherical Earth? Answer: Yes, they are. But how can this be? How can observations through terrestrial telescopes or various cameras, like the p900 from Nikon, bring back into view that which went over a curve? It shouldn’t, but it does. So therefore, the quote as cited above; “there is no possible way to disprove the pythagorean theorem, or indeed any theorem,” is correct, mathematically speaking that is; but it fails when we put the math to test. Objects seem to vanish under the horizon/curve due to perception. But they can easily be seen when one brings them back into view. This can only happen on a flat surface/plane. So the Earth, according to spherical equations, should drop down every .7993624 per mile, or every 8 inches, if we round it off that is. This is what the math tells us, but it doesn’t work observationally because there is no curve. 

However, I decided to see if the equation M²x8=d can be debunked. The only source I found is a website called, ‘Flat Earth Insanity.’ According to the blogger; “From this, is it easy to see that [8″×d²] is somewhat accurate for distances up to about 100 miles and then it is absolutely terrible after that.  And now we know why – they threw out all the little corrections in the Taylor series that keep it accurate in order to make a simple “rule” that was “good enough” for what they needed at the time.”

‘Omni,’ (as in the same folks at Omni’ Magazine) a more reputable source refutes the above citation, citing; “How large is the curvature of Earth, then? We don’t notice it in our everyday life, so it has to be quite small. Most sources consider 8 inches per mile as the most accurate estimate. It means that for every mile of the distance between you and a second object, the curvature will obstruct 8 inches of its height.

The first thing you can find with our Earth curvature calculator is the exact distance between you and the horizon. You only need to know two values: your eyesight level (in other words, the distance between your eyes and the ground) and the radius of the Earth. Input these numbers into the following equation:

a = √[(r + h)² – r²]


  • a stands for the distance to the horizon,
  • h is your eyesight level, and
  • r is the Earth’s radius, equal to 3959 miles


reference angle

reference angle

Look at the image above. It represents a situation analogical to the one with the ship: you can see a part of the object, but the rest of it is hidden behind the horizon. If you want to know the obstructed height, simply enter all necessary values into the Earth curvature calculator. You can also calculate the height manually:

  • Determine the distance between you (the observer) and the lowest point of the object that you can actually see. Let’s call this value d and assume it is equal to 25 miles.
  • Measure your eyesight level – that is, the height at which your eyes are. We will denote it with a letter h. We can assume it is equal to 6 feet, which is approximately 0.0011 miles.
  • Calculate the distance between you and the horizon a, using the aforementioned formula:

a = √[(r + h)² – r²] = √[(3959 + 0.0011364)² – 3959²] = 3 miles

  • Now, you can input these values to a second formula to find the height of the obstructed part of the object x: 

x = √(a² – 2ad + d² + r²) – r

x = √(3² – 2*3*25 + 25² + 3959²) – 3959

x = 0.0611 miles = 322.76 ft

However, what both sites fail to take into account is the fact that all obstructed objects can now be brought into full view; from top to bottom. That is possible only on a flat Earth. But if one were to ascribe to a flat Earth, then one would have to conclude that the Sun does not set under the horizon, or rise from it; but that it circles the Earth instead. Its light reaches areas where it is causing daylight and night where it is not. This is called geocentrism and Ptolemaism.

B. Claudius Ptolemy 

Born 98-168AD; Ptolemy was a Greco-Roman mathematician, astronomer, geographer, astrologer, and poet. Although he was a spherical ball Earth believer, he was, nevertheless, a geocentric adherent – having believed that Earth was at the center of the universe, or its occupied space, and that it was the Sun, Moon, stars and planets that revolved around the Earth. This view held sway for 1200 years, thanks to his treatise called the Almagest. Ptolemy contributed so much to the sciences, that his work is still looked upon in modern day academics.

The aforementioned Almagest – a 2nd century Greek language mathematical and astronomical treatise on the apparent motions of the stars and “planetary” paths. One of the most influential scientific texts of all time. Having its origin in Hellenistic Alexandria; and hence having its influence in the medieval Byzantine and Islamic worlds, and in Western Europe through the Middle Ages, and early Renaissance. The Almagest is the critical source of information on ancient Greek astronomy. It is still valuable to students of mathematics today because it documents the ancient Greek mathematician Hipparchus’s work. According to; C. M. Linton (2004). From Eudoxus to Einstein: a history of mathematical astronomy. Cambridge University Press. p. 52; “Hipparchus of Nicaea; c.190-120BC, was a Greek astronomer, geographer and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of precession of the equinoxes.”

Here is a picture of Hipparchus’s work on the geometric system of a geocentric Earth according to mathematicians that compared Ptolemy’s work with what, in all likelihood, might have been Hipparchus’s mathematical formula for the geocentric system and stellar parallax (the measuring of stars by calculating their distances).

Moreover, “It is certain, however, that in all his work Hipparchus showed a clear mind and a dislike for unnecessarily complex hypotheses. He rejected not only all astrological teaching but also the heliocentric views of the universe…”

Today the “Almagest is a peer reviewed academic journal that publishes contributions evaluating scientific developments. Almagest addresses the philosophical assumptions behind scientific ideas and developments and the reciprocal influence between historical context and these phenomena. The journal is abstracted in the Philosophy Research Index – STEP – Science and Technology in the European Periphery” Journal: STEP – Science and Technology in the European Periphery. Retrieved 2012-11-27.

However, Hipparchus’s work is lost. And because his works appear to have been lost, mathematicians use Ptolemy’s book as their source for Hipparchus’s work and ancient Greek trigonometry in general. Oh how I would love to see how Hipparchus, a geocentric adherent, used his calculations to place the Earth at the center of the universe, and the Sun orbiting around it. But it is, in all likelihood, lost for all time. It is also due to this precise reason that Ptolemy’s version stood firm for 1200 years until Copernicus. However, Tycho Brahe (1546-1601) had something to say about that – a contemporary to Johannes Kepler. More on Tycho and Kepler later.

C. Nikolai Copernicus

Born 1473-1543 was a Renaissance era mathematician and astronomer. Copernicus was no stranger to the Pythagorean Theorem or Ptolemy’s Almagest. But it is highly doubted that he considered the mathematics of Hipparchus’s work at all – most likely due to no one giving it a second thought until possibly Tycho Brahe. In short, Copernicus is responsible for undoing the work of Ptolemy (minus Hipparchus’s work), placing the Sun at the center of the Solar System, while placing the Earth as just another orbiting body – third from the Sun – going around a stationary Sun.

“It worked at least as well as Ptolemy’s spheres (circles; T. Perez) in explaining the apparent motion of the planets”…”some of Copernicus’s admirers argued that he had not really believed in a Sun-centered universe but had merely proposed it as a convenience for calculating the motions of the planets.” Cosmos. By Carl Sagan. Random House. New York; 1980. Page 53.

The 1st picture is a comparison between the two models. The 2nd picture is Copernicus’s heliocentric model. Pictures (Ibid; page 56, 58).

However, according to NASA; “Copernicus did not have the tools to prove his theories. By the 1600s, astronomers such as Galileo would develop the physics that would prove he was correct.”

But was Copernicus correct? More on that question below, but for now let us continue in historical order. The individual as quoted above by NASA, and by which of course we all know about, is none other than Galileo Galilei. Galileo 1564-1642, was an Italian polymath. Whether he supported the Copernican theory before his findings through the eye of his telescope, or due to the actual findings themselves, remains uncertain. Pointing his telescope toward Jupiter, he discovered the four major moons of Jupiter, and concluded that certain bodies, like Jupiter, had its own orbit around a body, independent of the Earth. If independent, then it is possible that the Copernican theory is correct; perhaps the Earth itself was an orbiting body, turning around a larger body: the Sun. And if so, how many other bodies might be out there, having their own independent orbiting structure. Well, it did not take him long to make another discovery: the phases of Venus. Not the planet itself, but its phases which can now be observed through his telescopes. The picture below demonstrates this…

But Galileo had to contend with the Tychonic System. But even after Galileo’s observation pertaining to the phases of Venus in 1610, most cosmological controversy then settled on variations of the Copernican systems and Tychonian systems introduced by Tycho Brahe.

D. Tycho Brahe

Born 1546-1601 was a Nobleman and astronomer. Tycho worked to combine what he saw as the geometrical benefits of the Copernican system with the philosophical benefits of the Ptolemaic system into his own model of the universe: The Tychonic System. “Tycho’s system was foreshadowed, in part, by that of Martians Capella (5th cent AD: A prose writer and early developer of the Liberal Arts – T.Perez) who described a system in which Mercury and Venus are placed on epicycles around the Sun, which circles the Earth. Copernicus, who cited Capella’s theory, even mentioned the possibility of an extension in which the other three of the six known planets would also circle the Sun.” “This was foreshadowed by the Irish Carolingian scholar Johannes Scotus Eriugena (9th cent AD: Irish Theologian, neoplatonist philosopher and poet), who went a step further than Capella by suggesting both Mars and Jupiter orbited the Sun as well” Others followed and supported the Tychonic system in the 15th century like; “Nilakantha Somayaji an Indian astronomer of the Kerala School of Astronomy and Mathematics who first presented a geo-heliocentric system where all the planets (Mercury, Venus, Mars, Jupiter and Saturn) orbit the Sun, which in turn orbits the Earth.”

Westman, Robert S. (1975). The Copernican achievement. University of California Press. p. 322.

Stanford Encyclopedia of Philosophy. “John Scottus Eriugena.” First published Thu Aug 28, 2003; substantive revision Sun Oct 17, 2004. Accessed April 30, 2014.

Ramasubramania, K. (1994). “Modification of the earlier Indian planetary theory by the Kerala astronomers (c. 1500 AD) and the implied heliocentric picture of planetary motion” (PDF). Current Science. 66: 784–90.

Joseph, George G. (2000), The Crest of the Peacock: Non-European Roots of Mathematics, p. 408, Princeton University Press

Ramasubramanian, K., “Model of planetary motion in the works of Kerala astronomers”, Bulletin of the Astronomical Society of India, 26: 11–31 [23–4].

Tycho’s model, established in 1588, is depicted below…

Tycho Brahe was a contemporary to Johannes Kepler.

E. Johannes Kepler

Born 1571-1630 Johannes Kepler was a “German astronomer who discovered three major laws of planetary motion, conventionally designated as follows: 1. the planets move in elliptical orbits (instead of spherically – T. Perez) with the Sun at one focus; 2. the time necessary to traverse any arc of a planetary orbit is proportional to the area of the sector between the central body and that arc (the “area law”); and 3. there is an exact relationship between the squares of the planets’ periodic times and the cubes of the radii of their orbits (the “harmonic law”). Kepler himself did not call these discoveries “laws,” as would become customary after Isaac Newton derived them from a new and quite different set of general physical principles. He regarded them as celestial harmonies that reflected God’s design for the universe. Kepler’s discoveries turned Nicklaus Copernicus’ Sun-centered system into a dynamic universe, with the Sun actively pushing the planets around in non-circular orbits. And it was Kepler’s notion of a physical astronomy that fixed a new problematic for other important 17th-century world-system builders, the most famous of whom was Newton.”

The “Harmonic Law” was something that they (Pythagoras, Copernicus, Kepler and Newton) all wanted to achieve through divine mathematics. They wanted to synergistically bring math, and the heavenly cosmos (the Earth, Sun, Moon, stars and planets) together as one. In them are the traits of Hermeticism – a philosophical religious belief coupled with what many saw as pseudoscience, accompanied with various thoughts, and practised by the Pythagorean(s). Kepler’s used the observations of Tycho himself to demonstrate that the orbits of the planets are ellipses and not circles, creating the modified Copernican system. However, the Tychonic system remained very influential in the late 16th and 17th centuries, due to the Roman Imperial Church’s ban on all heliocentric books; including anything by Copernicus, Galileo, Kepler and other authors. Galileo himself was placed under house arrest for the rest of his life. He was forced to recant his conclusions in his writings or face excommunication, or possible death. The Tychonic system was supported by the Church for two reasons. One, it fitted the religious ideologies of the time, and two, the math contained therein was still accurate. Things remained, more or less, the same. Then along came Issac Newton.

F. Sir Issac Newton

Born 1643-1727, Newton was a English mathematician, astronomer, theologian, author and physicist. Besides his work on universal gravitation (gravity), Newton developed the three laws of motion which form the basic principles of modern physics. His discovery of calculus led the way to more powerful methods of solving mathematical problems. But even in this, the Tychonic system was still standing in competition with its Copernican counterpart. The Tychonic system proved philosophically more intuitive than the Copernican system, as it reinforced common sense notions of how the Sun and the planets are mobile while the Earth is not. “The Tychonic system was an acceptable alternative as it explained the observed phases of Venus with a static Earth.” Pantin, Isabelle (1999). “New Philosophy and Old Prejudices: Aspects of the Reception of Copernicanism in a Divided Europe” page 262. However, a Copernican system would suggest the ability to observe stellar parallax which would be observed in the 1800’s. But to the Tychonians, parallax(s) are due to heavenly motions fluid in the heavens, which could accommodate intersecting circles. Parallax(s) lead to stellar aberration, discovered by James Bradley which proved, according to mainstream scientific data, that the Earth did in fact move around the Sun. This is when Tycho’s system fell out of use.

Seligman, Courtney. Bradley’s Discovery of Stellar Aberration.(2013).

“However, In the modern era, some of the modern geocentrists use a modified Tychonic system with elliptical orbits, while rejecting the concept of relativity.” They also challenge stellar aberration. So before you jump to conclusions pertaining to a heliocentric system, just remember its stronghold is only its ability to observe stellar aberrations. Without it, the system can fall. Moreover, if the system fails in its curvature of its Earth as demonstrated by the simple math above with modern day technologies, then perhaps the Earth is a flat disk, with the Sun, Moon, stars and planets (wandering stars) orbiting and circling around it.

Plait, Phil. (Sept. 14, 2010). Geocentrism Seriously? Discover Magazine.

Musgrave, Iam. (Nov. 14, 2010). Geo-xcentricities part 2; the view from Mars. Astroblog.