Hipparchus, the Nicaean 
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Hipparchus (in Greek: Hipparkhos, c. 190 BC. - c. 120 BC.) was a Greek astronomer, a geographer and a mathematician during the Hellenistic period. Hipparchus was born in Nicaea in Bithynia (today Iznik, Turkey) and, most probably, died on the island of Rhodes. As an astronomer he actively worked between 147-127 BC.. Hipparchus is considered, the most ancient observer among the astronomers and, by some, even the greatest over all the ancient astronomers. He was the first to be outlived by his accurate quantitative models relevant to the motion of the Sun and of the Moon. To reach such results, he certainly made use of the observations and, maybe, also of the mathematical techniques inherited from the Chaldeans of Babylonia. 

He developed trigonometry and constructed trigonometric tables. He also solved many problems related to spherical trigonometry. He tabulated the values for the chord function, that gives the chord’s length for each angle. He worked on a circle having a circumference 21,600 and a radius (rounded) 3,438 units. He tabulated the chords for angles with increments of 7.5°. In modern terms: the chord of an angle equals twice the sine of half the angle, that is to say: 
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With his solar and lunar theories and his trigonometry he, probably, was the first to develop a reliable method to predict solar eclipses. G. Toomer, a historian of astronomy, suggested that Ptolemy’s knowledge of eclipses and of other Babylonian observations in the Almagest originated from a list put down by Hipparchus. His other achievements include the discovery of precession, the compilation of the first complete star catalogue of the western world; maybe the invention of the astrolabe and of the armillary sphere that made its first appearance during his century. Hipparchus, probably, used it to draft much of his star catalogue. He also was the first to prove that the stereographic projection is conformal, and that it transforms circles on the sphere, that do not pass through the centre of the projection, to circles on the plane. This was the basis for the astrolabe. 

These are the sources, three centuries old, on which Claudius Ptolemaeus started to work, completing the studies of Hipparchus. The synthesis Ptolemy operated is heavily dependent on the discoveries made by Hipparchus in many areas. 
 
 
 
 

Hipparchus    this is the function tabulated by Hipparchus - image credit: Pighin

The eccentricity of the deferent satisfied, as nearly as possible, the observational requirement of the "first inequality". Such a deviation is naturally referred to a model having a perfectly centered deferent. The eccentric deferent, however, did not fully explain the motion observed. The longitudes in the apsides and in the quadratures suffice, while those at the octants do not. Therefore, an epicycle with appropriate dimension and rotation is requested to correct such discrepancies. 

The epicycle satisfied the observational requirement of the "second inequality". It was due to the fact that the Planet/the Moon was apparently slower when it was farther away. This, however, was also an absolute value. It was an approximate attempt to suit the variable motion on the ellipse, a non perfect attempt, as countless epicycles would be requested, the smaller ones infinitesimal, to improve the motion. We, therefore, are in presence of an "non-perfect" model. However, it worked adequately for fifteen hundred years, up to Copernicus, because the accuracy of the measurements was so inaccurate that deviations, at such a detailed level, were really non-influential. The main instrument of measure was the "cross-staff" also called "Jacob’s staff", a graduated cross with a short mobile arm. It had modest dimensions (and could be carried around and used by a single person quite easily). Its angular accuracy was: 0.5°; (equal to the Moon’s diameter). 
 

Claudius Ptolemaeus  image credit: Pighinimage credit: Pighin  "Ptolemy’s masterpiece", achieved with instruments incredibly cheap and simple... the orbit is true, not fictitious …..  modern telescopes use geocentric models, similar to Ptolemy’s, to follow planets and for long photographic exposures...

Copernicus, once again, revoked the studies and the reasons of the Islamic scholars, professing an Universe where the Sun was motionless, while the Earth moved (a Planet like all the others). This theory is "heliostatic", not "heliocentric": the Sun of Copernicus is a "mean Sun", therefore, an abstraction that finds no correspondence in observations. This is a Sun, that moving with uniform motion, covers its annual journey in a period of time same as that of the "astronomical Sun". The mean Sun is quite convenient to compute the ephemerides. Copernicus did not clearly indicate where the centre of the Universe was. For the learned, it obviously coincided with the centre of the Earth’s orbit, an empty point in space. The "Astronomical Sun" is eccentric: in Summer it is near; in Winter, far away. A well known event. Otto Neugebauer, one of the most well known experts of Ptolemy and of Copernicus, underlined how Copernicus did not add any further significance to the motion of the planets. He just placed them in a different system (from the center of the Earth, to the centre of their orbit). Copernicus, on the other hand, improved the Moon’s motion, giving to it a much more probable orbit than Ptolemy’s. 

From the quantitative point of view (ephemerides), the tables of Copernicus are not much better than Ptolemy’s. Computation methods are naturally much different. This was due to the fact that, during Ptolemy’s times, measures were taken with the help of a "cross-staff". Therefore, it was not easy to improve predictions. 
 
 

The decision to proceed as so originated from the observation that the conjunction between Jupiter and Saturn of 26 August 1563 was mistaken by a month, according to the known ephemerides. Tycho assumed that such discrepancy was due to inaccurate observations. He was the right man at the right time, as, in the meantime, bronze technology had been perfected. This technology allowed him to construct great and very accurate instruments. Brahe measures diligently nearly a thousand stars and observes the extraordinary details of the planets, especially those of Mars. 

Brahe did not accept the model of Copernicus, but proposed an equivalent one in which the Earth was centrally positioned. The Sun and the Moon orbited the Earth, and the other Planets orbited the Sun. In such a system the orbit of the Sun intersected that of the planets; in particular, Mars at times was nearer to the Earth (56 million kilometres) than the Sun (150 million kilometres ). Today we know this is true, but in Tycho’s time, this event looked very strange. Tycho made use of his measurements in his cosmological model, but was unable to find a correspondence, because his observations were thirty times more accurate than those measured in previous periods (1 minute, instead of thirty); therefore, his observations could not find a correspondence in models dimensioned on non trustable data. 

A way out from such a desperate mess was an impossible mission. He wanted to proceed independently; he jealously guarded his measurements, aware of their great value. He also wished to become part of history. This persuaded him to consider as his assistant an astronomer with great mathematical skills . Tycho died prematurely. After many disputes, his data ended up into the hands of his assistant who, in the meantime, was assigned his post. 
 
 

Tycho Brahe    Tycho’s "cyclopic quadrants" took the place of the portable "cross-staff", 1 metre long , ... accuracy was increased by 30 times

Brahe invited him to overlook his Tychonic system in the light of his discoveries. Galileo sent him an interesting answer, regretting the climate of derision and dullness towards the hypotheses of Copernicus and worrying because truth was neglected. 

Pythagoras, Plato and the cabbalistic learning provided the instruments and, with Kepler, shared the harmonic world, geometrically structured; a mythical and ideal universe, which Kepler retained valuable. 

Only the accurate measurements of Brahe constrained Kepler to abandon the theory of a perfect circular astral motion in favour of the elliptic motion, while trying to solve the problem of the irregular motion of Mars. A task on which he worked strenuously for ten long years, leaving us an enthusiastic description of his efforts. 

Nothing happened by chance, Kepler became successful due to his simple cosmological convictions that the Sun is a great rotating magnet that drags all the planets about. This action weakens with distance. The planets that are farther away are also the slowest. 

It is clear, from the above, Kepler referred to the "astronomical Sun", with his words the "observed Sun" not the "mean Sun". This was the "incipit" that drove him to the final solution. 
 
 

Using triangulations Mars-Earth-Sun, with Mars always in the same sidereal position, he found three points of the Earth’s orbit, thought to be circular in his times. By doing so, he obtained the value of the eccentricity of this orbit, which is half the value assumed by his predecessors. 

He modified the Copernican model using this new value and again tried to insert Tycho’s data: the error was in great part amended, but still a small inaccuracy was there: 8 arcminutes against the observed. Many people would have been satisfied, not Kepler. He knew the new data were accurate to one minute and thoroughly believed in his measurements. He realized that the shape of the orbit of Copernicus would never have suited his measurements. A new curve had to be studied. He did not take into consideration the ellipse. If that would have been the solution, it would have been already known to the ancients. 

After many attempts, he started to accept the idea that the ellipse could be the solution. He tried and, miraculously, everything worked into place. The "astronomical Sun" stood in a focus, outside of every symmetry. The myth of the uniform circular motion ended and left its place to a non uniform elliptic motion. He even expounded the law of motion. Measurements were sufficiently accurate and inaccurate, just that much to allow such an hypothesis. Galileo, by observing the Sun spots, discovered that the Sun rotated. This was of great support to Kepler. Jupiter’s satellites follow Keplerian ellipses. Therefore, the model has a general validity and fits both planets and satellites. 

In Kepler’s vision the Sun is active, while Planets are passive and do not perturb each other. In addition, Planets are just pointed shapes. A black mark still blurred this success: the Moon has an irregular motion that cannot be explained with the law of the ellipses. 
 
 
 

Using the spyglass, he was able to make the discoveries described in his tract Sidereus Nuncius (The Sidereal Messenger - 1610). Galileo became famous overnight. Cosimo II de’ Medici, Grand Duke of Tuscany, appointed him "primary mathematician at the University of Pisa", and "philosopher of his Serene Highness the Duke" without whatsoever obligation to lecture or to reside in Pisa. In this work, Galileo, gave news of the discoveries he unveiled with the use of the spyglass. 

At first, he was able to discern a great number of new stars. The previous picture of our Universe was changed. The Universe was becoming bigger and bigger. 

The spyglass had a great advantage: with an instrument that could magnify 20 times, the measurement’s accuracy was improved by the same factor. No correspondence to Kepler’s ellipses, also, arose within the Planets and not only for the Moon. Something new was somewhere in the air and its discovery was the task of an extraordinary unusual person. 

It was assumed that, maybe, planets perturb each other. It was also assumed that, getting near to a point, is not always the solution. Therefore, how can the problem be approached? 
 
 
 

 
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The famous conversation between Halley, Wren and Hooke took place at a meeting of the Royal Society in 1684. Newton became also involved in it. Sometime after, he published: "Philosophiae Naturalis Principia Matematica" (Mathematical Principles of Natural Philosophy).  

In this very important work he explained Kepler’s theory on the Copernican thesis, presuming the existence of a gravity which penetrated all the way to the centre of the Sun, decreasing in accordance to the inverse of the square of the distances and proportional to the quantity of solid matter of the planets. This work harmonized all previous theories, associating Copernicus, Descartes, Kepler and Galileo "in an unicum": finally the great book of nature was comprehensible. Starting from a corpuscolar vision of reality, mathematics was the instrument to understand its structure. 

In short, Newton, said: "everything attracts everything" and found the relevant law. The force of attraction between any two masses is proportional to the product of the two "quantities of matter" (masses) and is inversely proportional to the square of the distance. 

Starting from this point, Newton investigated the three laws of Kepler. The added value he was able to offer to this argument, thanks to his early studies in mathematics, was quite great. Newton found out how to serially expand the potency of a binomial in which one of the two addenda was sufficiently small. This allowed, once the sequence was reasonably broken up, to compute and to understand what kind of entities controlled the major perturbations. 

In the differential equations related with the universal gravity (that Newton’s infinitesimal analysis was capable to meet) quite often denominators showed binomials in which a term was multiplied by the eccentricity, which has extremely small values in the solar system. It was, therefore, possible to develop a numerical series and to break it up where precisely requested. 
 

Kepler’s ellipsis were no more closed, because, due to the perturbations, they slowly revolved and, in time, the apsidal line changed its direction. 

The Earth-Moon-Sun system, studied with the methods of the "restricted three-body problem", allowed to understand the non correspondence to the Keplerian ellipsis. Newton did not succeed to solve the problem "in toto", only because he used a much reduced number of terms of the polynomial series. The law, however, was correct. 

He, therefore, demonstrated that the Moon’s orbit revolved in 18.6 years (something that was well-known since ancient times, but had never been explained). It was demonstrated that evection was a function of the Moon’s elongation with respect to the Sun and to its anomalistic motion (a motion well-known to Hipparchus). 

A three body system showed to be intrinsically instable (at least, in the long term), except for particular solutions that will be discovered by Laplace. Consequently, Newton’s gravitation introduced chaos in the solar system. It was well-known that Jupiter, perturbing the orbits of the asteroids, lying between Jupiter and Mars, made them become instable, throwing them towards the inner solar system. Comets often have chaotic orbits, even if at times they may look regular. 

Many motions (e.g. precession) that before had been studied only through suitable kinematic motions of spheres, now find an exhaustive physic explanation. 

Today universal gravitation may be understood numerically without difficulty. Therefore today, it is possible to compute trustable ephemerides with simple means. A true revolution! 
 
 

Newton’s physic was based on the simple abstract idea of a physical point, a particle. Therefore, matter was discontinuous "a priori". It is well-known that Newton was a great supporter of the light’s corpuscles theory.  

How was it possible to explain mechanical phenomena in such a context? An "action-at-a-distance" was supposed to take place instantaneously among physical points. Newton’s method seemed to answer to everything that was related with mechanics. 

A problem, however, arose in the second half of the XIX century, when Maxwell’s electro-dynamic laws became known, as they could not be included, in a satisfying way, into the Newton’s system. 

Maxwell detailed these laws and discovered the differential equations that described the behaviour of electro-magnetic radiation. In particular, the electro-magnetic wave must move with a constant and finite velocity, the value of which is given by the measure of the frame of references of the surrounding environment. 

To get back the speed of light to constant and finite in all inertial references, the special theory of relativity (STR) was necessary.  

The general theory of relativity (GTR) was required to explain the advance of the precession of the perihelion of Mercury. The phenomenon was explained through the distortion of the surrounding space.  

Anyway, even Einstein had limits. He was not able to explain the "black hole". To understand it, quantum mechanics will be necessary. But this is a separate world and its synthesis (TOE – Theory of Everything) is still far away (GUT’s Grand Unified Theories). 
 
 

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