Science in the Qur'an
The use of math to validate theories has always been used in Physics, as a computation in most cases will somewhat fit what is occurring by shuffling numbers, but if the theory is wrong, the set of equations are useless and stagnation within the field will occur as progress to expand knowledge slows to a crawl. Quantum mechanics is a very powerful theory which has led to an accurate description of the micro-physical mechanisms. It is founded on a set of postulates from which the main processes pertaining to its application domain are derived. A challenging issue in physics is therefore to exhibit the underlying principles from which these postulates might emerge.
The theory of scale relativity consists of generalizing to scale transformations the principle of relativity, which has been applied by Einstein to motion laws. It is based on the giving up of the assumption of spacetime coordinate differentiability, which is usually retained as an implicit hypothesis in current physics. Even though this hypothesis can be considered as mostly valid in the classical domain (except possibly at some singularities), it is clearly broken by the quantum-mechanical behavior. It has indeed been pointed out by Feynman that the typical paths of quantum mechanics are continuous but nondifferentiable. Even more, Abott and Wise have observed that these typical paths are of fractal dimension DF = 2.
This is the reason why we propose that the scale relativity first principles, based on continuity and giving up of the differentiability hypothesis of the coordinate map, be retained as good candidates for the founding of the quantum-mechanical postulates. We want to stress here that, even if coordinate differentiabilty is recovered in the classical domain; nondifferentiability is a fundamental property of the geometry that underlies the quantum realm.
To deal with the scale relativistic construction, one generally begins with a study of pure scale laws, i.e., with the description of the scale dependence of fractal paths at a given point of space (spacetime). Structures are therefore identified, which evolve in a so-called 'scale space' that can be described at the different levels of relativistic theories (Galilean, special relativistic, general relativistic).
The next step, which we consider here, consists of studying the effects on motion in standard space that are induced by these internal fractal structures.
Scale relativity, when it is applied to microphysics, allows us to recover quantum mechanics as a non-classical mechanics on a nondifferentiable, therefore fractal spacetime. Since we want to limit our study to the basic postulates of nonrelativistic quantum mechanics (first quantization), we focus our attention on fractal power law dilations with a constant fractal dimension DF = 2, which means to work in the framework of 'Galilean' scale relativity.
Now, we come to a rather subtle issue. What is the set of postulates needed to completely describe the quantum-mechanical theory? It is all the more tricky to answer this question that some of the postulates usually presented as such in the literature can be derived from others.
THE POSTULATES OF QUANTUM MECHANICS
The postulates listed below are formulated within a coordinate realization of the state function, since it is in this representation that their scale relativistic derivation is the most straightforward. Their momentum realization can be obtained by the same Fourier transforms which are used in standard quantum mechanics, as well as the Dirac representation, which is another mathematical formulation of the same theory, can follow from the definition of the wavefunctions as vectors of a Hilbert space upon which act Hermitian operators representing the observables corresponding to classical dynamical quantities.
The set of statements we find in the literature as 'postulates' or 'principles' can be split into three subsets: the main postulates which cannot be derived from more fundamental ones, the secondary postulates which are often presented as 'postulates' but can actually be derived from the main ones, and then statements often called 'principles' which are well known to be as mere consequences of the postulates.
1. MAIN POSTULATES
1. Complex state function. Each physical system is described by a state function which determines all can be known about the system. The coordinate realization of this state function, the wavefunction is an equivalence class of complex functions of all the classical degrees of freedom generically noted r, of the time t and of any additional degrees of freedom such as spin s which are considered to be intrinsically quantum mechanical.
Two wavefunctions represent the same state if they differ only by a phase factor (this part of the 'postulate' can be derived from the Born postulate, since, in this interpretation, probabilities are defined by the squared norm of the complex wavefunction and therefore the two wavefunctions differing only by a phase factor represent the same state). The wavefunction has to be finite and single valued throughout position space, and furthermore, it must also be a continuous and continuously differentiable function.
2. Schrodinger equation. The time evolution of the wavefunction of a non-relativistic physical system is given by the time-dependent Schrodinger equation
,
where the Hamiltonian Ĥ is a linear Hermitian operator, whose expression is constructed from the correspondence principle.
3. Correspondence principle. To every dynamical variable of classical mechanics there corresponds in quantum mechanics a linear, Hermitian operator, which, when operating upon the wavefunction associated with a definite value of that observable (the eigenstate associated to a definite eigenvalue), yields this value times the wavefunction. The more common operators occurring in quantum mechanics for a single particle are listed below and are constructed using the position and momentum operators.
Position Multiply by
Momentum
Kinetic energy
Potential energy Multiply by
Total energy
Angular momentum
More generally, the operator associated with the observable A which describes a classically defined physical variable is obtained by replacing in the 'properly symmetrized' expression of this variable the above operators for r and p. This symmetrization rule is added to ensure that the operators are Hermitian and therefore that the measurement results are real numbers.
However, the symmetrization (or Hermitization) recipe is not unique. As an example, the quantum-mechanical analogue of the classical product can be either or . The different choices yield corrections of the order of some ¯h power and, in the end, it is the experiments that decide which is the correct operator. This is clearly one of the main weaknesses of the axiomatic foundation of quantum mechanics, since the ambiguity begins with second orders, and therefore concerns the construction of the Hamiltonian itself.
4. Von Neumann's postulate. If a measurement of the observable A yields some value ai , the wavefunction of the system just after the measurement is the corresponding eigenstate (in the case that ai is degenerate, the wavefunction is the projection of ψ onto the degenerate subspace).
5. Born's postulate: probabilistic interpretation of the wavefunction. The squared norm of the wavefunction |ψ|2 is interpreted as the probability of the system of having values (r, s) at time t. This interpretation requires that the sum of the contributions |ψ|2 for all values of (r, s) at time t be finite, i.e., the physically acceptable wavefunctions are square integrable. More specifically, if ψ(r, s, t) is the wavefunction of a single particle,is the probability that the particle lies in the volume element located at r at time t. Because of this interpretation and since the probability of finding a single particle somewhere is 1, the wavefunction of this particle must fulfil the normalization condition
2. SECONDARY POSTULATES
One can find in the literature other statements which are often presented as 'postulates' but which are mere consequences of the above five 'main' postulates. We examine below some of them and show how we can derive them from these 'main' postulates.
1. Superposition principle. Quantum superposition is the application of the superposition principle to quantum mechanics. It states that a linear combination of state functions of a given physical system is a state function of this system. This principle follows from the linearity of the Ĥ operator in the Schrodinger equation, which is therefore a linear second order differential equation to which this principle applies.
2. Eigenvalues and eigenfunctions. Any measurement of an observable A will give as a result one of the eigenvalues a of the associated operator Â, which satisfy the equation
3. Expectation value. For a system described by a normalized wavefunction ψ, the expectation value of an observable A is given by
This statement follows from the probabilistic interpretation attached toψ, i.e., from Born's postulate.
4. Expansion in eigenfunctions. The set of eigenfunctions of an operator Âforms a complete set of linearly independent functions. Therefore, an arbitrary state ψ can be expanded in the complete set of eigenfunctions of Â(Âψn = anψn), i.e., as
where the sum may go to infinity. For the case where the eigenvalue spectrum is discrete and non-degenerate and where the system is in the normalized state ψ, the probability of obtaining as a result of a measurement of A the eigenvalue an is |cn|2. This statement can be straightforwardly generalized to the degenerate and continuous spectrum cases.
Another more general expression of this postulate is 'an arbitrary wavefunction can be expanded in a complete orthonormal set of eigenfunctions ψn of a set of commuting operators An'. It writes
while the statement of orthonormality is
where is the Kronecker symbol.
5. Probability conservation. The probability conservation is a consequence of the Hermitian property of Ĥ. This property first implies that the norm of the state function is time independent and it also implies a local probability conservation which can be written (e.g., for a single particle without spin and with normalized wavefunction ψ) as
where
6. Reduction of the wave packet or projection hypothesis. This statement does not need to be postulated since it can be deduced from other postulates. It is actually implicitly contained in von Neumann's postulate.
3. DERIVED PRINCIPLES.
1. Heisenberg's uncertainty principle. If P and Q are two conjugate observables such that their commutator equals iâ„, it is easy to show that their standard deviations P and Q satisfy the relation
whatever the state function of the system. This applies to any couple of linear (but not necessarily Hermitian) operators and, in particular, to the couples of conjugate variables: position and momentum, time and energy. Moreover, generalized Heisenberg relations can be established for any couple of variables.
2. The spin-statistic theorem. When a system is composed of many identical particles, its physical states can only be described by state functions which are either completely antisymmetric (fermions) or completely symmetric (bosons) with respect to permutations of these particles, or, identically, by wavefunctions that change sign in a spatial reflection (fermions) or that remain unchanged in such a transformation (bosons). All half-spin particles are fermions and all integer-spin particles are bosons.
Demonstrations of this theorem have been proposed in the framework of field quantum theory as originating from very general assumptions. The usual proof can be summarized as follows: one first shows that if one quantizes fermionic fields (which are related to half-integer spin particles) with anticommutators one gets a consistent theory, while if one uses commutators, it is not the case; the exact opposite happens with bosonic fields (which correspond to integer spin particles), one has to quantize them with commutators instead of anticommutators, otherwise one gets an inconsistent theory. Then, one shows that the (anti)commutators are related to the (anti)symmetry of the wavefunctions in the exchange of two particles. However, this proof has been claimed to be incomplete but more complete ones have been subsequently proposed.
3. The Pauli exclusion principle. Two identical fermions cannot be in the same quantum
state. This is a mere consequence of the spin-statistic theorem.
SCIENCE IN THE HOLY QUR'AN
Al-Mighty Allah has given a lot of sign; about 1400 years ago regarding the movement of planets and stars in the Holy Qur'an. It is depends on us whether to sit down and relax or to seek the wonder of Allah's creation.
Surah Al 'Imran verse 27:
"You make the night to enter into the day, You make the day to enter into the night (i.e. increase and decrease in the hours of the night and the day during winter and summer), and You bring the living out of the dead, and You bring the dead out of the living. And you give wealth and sustenance to whom You will, without limit (measure or account)"
also in Surah Al-Anbiya' verse 33:
"And He it is Who has created the night and the day, and the sun and the moon,
each in an orbit floating"
Another one is in Surah Yasin verse 40:
"It is not for the sun to overtake the moon nor does the night outstrip the day.
They all float, each in an orbit"
Titius-Bode Law
http://milesmathis.com/bode.jpg
Bode's Law, also known as the Titius-Bode Law is one of the most famous unexplained laws in the Solar System. The Titius-Bode Law or Rule is the observation that orbits of planets in the solar system, the distances of the planets from the Sun follow a simple arithmetic rule quite closely. The relationship was first pointed out by Johann D. Titius in 1766 and was formulated as a mathematical expression by J. E. Bode in 1778.
The first mention of a series approximating Bode's Law is found in David Gregory's "The Elements of Astronomy", published in 1715. In it, he says, "…supposing the distance of the Earth from the Sun to be divided into ten equal parts, of these the distance of Mercury will be about four, of Venus seven, of Mars fifteen, of Jupiter fifty two, and that of Saturn ninety five" A similar sentence, likely paraphrased from Gregory, appears in a work published by Christian Wolff in 1724.
Titius and Bode experimented with my formulas until they found one that closely fit the orbital profile of our solar system. It was a great achievement for their time, but does not accurately predict all planetary orbits in this, or any solar system in the universe. It was a mathematical representation of what he observed in their point in time.
The law relates the mean distances of the planets from the sun to a simple mathematic progression of numbers. Translated into astronomical units (AU), where one AU is the mean distance of the Earth from the Sun, the law amounts to this. Make a sequence of numbers:
0, 3, 6, 12, 24...
With the exception of the first number, the other are simple twice the value of the preceding number.
Add 4 to each number:
4, 7, 10, 16, 28…
Then divide by 10.
The law can be written as
a = (n + 4) / 10
where n=0, 3, 6, 12, 24…
The modern formulation is that the mean distance a of the planet from the Sun is, in astronomical units (AUearth = 147.597 * 106 km):
a = 0.4 + 0.3 * k
where k = 0, 1, 2, 4, 8, 16… (sequence of powers of 2 and 0).
There is no solid theoretical explanation of the Titius-Bode Law.
Distance from Sun (AU)
Double
x
( x +4 ) / 10
Mercury
0.387
0.25
0
0.4
Venus
0.723
0.5
3
0.7
Earth
1
1
6
1
Mars
1.524
2
12
1.6
(Ceres)
2.767
4
24
2.8
Jupiter
5.203
8
48
5.2
Saturn
9.539
16
96
10
(Uranus)
19.19
32
192
19.6
(Neptune)
30.06
64
384
38.8
(Pluto)
39.53
128
768
76.4
Noted that Ceres is the asteroid belt.
http://www.astro.cornell.edu/academics/courses/astro2201/images/bodes_moons.gif
All well and good, except that there was a big gap between Mars and Jupiter. Titius and Bode decided to skip a number, making Jupiter a particularly good fit. This law was sometimes taken to predict that a planet would be found between Mars and Jupiter. Within a few years (1781), Uranus was discovered by Sir William Herschel, and it fit right into the law. This discovery made the law respectable, and the hunt for the missing planet began.
In 1801, Giuseppe Piazzi discovered the minor planet Ceres, at just the right distance. Ceres was incredibly tiny for a planet. To date, more than 9000 minor planets (asteroids) have been discovered. At first it was thought that a planet was destroyed by a collision, at the distance from the Sun. Now it is thought that the gravity of Jupiter prevented planet from forming the fragments there.
The hypothesis correctly predicted the orbits of Ceres and Uranus, but failed as a predictor of Neptune and Pluto's orbit. The first explanation is just a guess, but it is a bad guess since orbital resonances have been given to gravity, but no one has ever shown a mechanical cause of any "gravitational resonance". Resonances cannot be caused by gravity, and no one in history has shown that they can.
http://upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Titus-Bode_law.svg/350px-Titus-Bode_law.svg.png
Comparison of Bode's Law with Actual Distances
Bode's Law does become more inaccurate as we move out to the margins of the Solar system. Perhaps one reason for this is that it does not take into account a planet's mass. More recent versions of the law have been elaborated during the XX sec., as for example the Blagg Law (1913) and the Richardson Law (around 1943).
In these last versions the law is able to describe not only the planetary distances within the solar system, including planets like Neptune and Pluto, but also can be successfully applied to the systems of satellites orbiting Jupiter, Saturn and Uranus. The agreement between the predicted and the observed distances of the various satellites from the central body is really astonishing, of the order of a few percents.
The main feature shared by these modern versions of the Titius-Bode Law is that the rule can be expressed, if we neglect second order corrections, by an exponential relation as
r = ae2λn,
where the factor 2 is introduced for convenience reasons and n = 1, 2, 3, …
For the Solar system we have
2λ = 0.53707, e2λ ≃ 1.7110,
a = 0.21363 A. U.
The amazing thing found by Blagg was that the geometric progression ratio e2λ is roughly the same both for the Solar system, and also for the satellite systems of Jupiter (e2λ ≃ 1.7277), Saturn (e2λ ≃ 1.5967), Uranus (e2λ ≃ 1.4662). Of course the parameter a, which is linked to the radius of the first orbit, will take case by case the opportune values.
A plenty of theories have been developed during the last 240 years in order to explain the Titius-Bode Law. There have been dynamical models connected with the theory of the origin of the Solar system, electromagnetic theories, gravitational theories, nebular theories.
QUANTUM APPROACH OF PLANETARY
ORBIT DISTANCE
It is known that quantum mechanics exhibits fractality at dF = 2, and an extensive report has been written on this subject. Moreover, a fractal solution of time-dependent Schrodinger equation has been suggested some time ago by Datta (1997). On the other side, if one takes a look at planetesimals in the case of planetary system formation, interstellar gas and dust in the case of star formation, the description of the trajectories of these bodies is in the shape of non-differentiable curves, and we obtain fractal curves with fractal dimension 2.
This coincidence between fractality of quantum mechanics and fractal dimension of astrophysical phenomena seems to suggest that we can expect to use quantum mechanical methods such as wave mechanics and periodic orbit quantization to analyze astrophysical phenomena.
BOHR MODEL OF THE HYDROGEN ATOM
The electron orbits in the Bohr model for the hydrogen atom are supposed circular (this will be held for planetary orbits also). The two main equations are the equation for the force (i.e. the equation of motion)
where m and e are the mass and charge of the electron; and the quantization condition on the (z-component) of the angular momentum
In the Bohr model, all the orbits belong to the same plane, and this is also taken for true in the planetary models. From the two equations above, one easily derives
The first equation is the law of electron distance from the nucleus in the Bohr model. With this law, from the classical expression for the total energy we get the energy spectrum of the bound orbits.
MODEL a la BOHR FOR A PLANETARY SYSTEM
It is a model for the "quantization" of a planetary system. The model acquires its discrete, or "quantum", properties from a modification of the Bohr quantization rule for the angular momentum. The equations here proposed, for a generic planet of mass m, orbiting a central body of mass M, are
where n = 1; 2; 3; : : : and s is a constant. Some comments are immediately required:
Because of the principle of equivalence the masses m on the LHS and on the RHS of eq. (3) cancel out each other.
The constant s in the RHS of the second of the equation has the dimensions of an action per unit mass. It plays the role of Ñ› and it must be understood as an action typical of the planetary system under consideration. It is not possible to use Ñ› itself, because this would fix the wrong initial radius in the Titius-Bode law, that is the constant in .
The constant λ is the one obtained from the observation (2λ = 0.53707 for the Sun, 2λ = 0.54677 for Jupiter, 2λ = 0.46794 for Saturn, 2λ = 0.38271 for Uranus).
In the second of the equation we quantize the angular momentum per unit mass. This is somewhat a consequence of the principle of equivalence. If we did not do so, we would obtain a law for where the scale of distance changes from a planet to another, as the planetary masses change. We should in fact remind that not all the planets have the same mass, as instead the electrons have.
From the equation one immediately gets
which is the Titius-Bode law if we identify
We can also compute the energy spectrum for the i-th planet from the equation above
where n = 1, 2, 3,…
As we see, the energy of the i-th planet is not properly quantized by itself. This is because the mass changes in general with the planet, and this would imply different sets of energy levels for different planets. Instead, the energy per unit mass
is exactly quantized, i.e. it is a quantity which depends on only (apart from the general constants ). Therefore, the energy levels per unit mass are valid for the whole set of planetary orbits.
Also here some comments are needed, in order to complete the explanation given before.
The constant can be computed in terms of the mass of the central body and of the parameter (remind that the radius of the first orbit is )
This constant is not the same for all the planetary systems (Sun, Jupiter, Saturn, Uranus). In fact, if it were so, this would imply that the parameter should be in inverse proportion to the mass of the central body, which is not true. Therefore the constant is not universal, like â„, but it depends on the planetary system under consideration.
If the quantization rule had been written with the mass of the planet, namely
this would have implied for
that is, the parameter would change from planet to planet, contrary to the generality of the Titius-Bode law, which maintains the same parameters within the same planetary system.
The quantization rule does not allow us to compute some known experimental constant, as instead it happens in the case of the Bohr model of the hydrogen atom, where the Rydberg constant was computed from the model. Nevertheless, a semiclassical quantum language is introduced.
It should be noted also that a condition like presents some difficulties for a wave interpretation. In fact, the Bohr quantization condition for the H-atom can be easily interpreted in terms of de Broglie's stationary matter waves
while and is an integer. The quantity can be interpreted as a wavelength of a stationary wave just because n is an integer. The analog condition in our model yields (for a given planet of mass )
The number is not an integer, in general. Hence is difficult to interpret as a wavelength of a stationary wave.
Moreover, even using a de Broglie-like relation the wavelength of the matter wave associated to the planet has to be of the same order of the parameter a. In fact
In principle, this could create interference phenomena in the probability amplitudes, but these phenomena are not observed at classical level in planetary systems. We must therefore postulate a unknown mechanism which suppresses these interferences of probability waves.
From this last observation, it appears clearly that the model we are building is not actually a quantum model, in the sense of ordinary quantum theory. Rather, it resembles some quantum-like properties, mainly the quantization of the orbital radii.
In spite of all these difficulties, we shall see that a wave equation can still be written in coherence with the condition, and this wave equation will be able to describe the main features of planetary systems.
BOHR-SOMMERFELD QUANTIZATION RULES
Periodic orbit quantization as suggested by Bohr-Sommerfeld is used in order to analyze quantization in astrophysical phenomena, which is a planetary orbit distance. It is known that Bohr-Sommerfeld quantization rules can be deduce from Burger's turbulence, and such an approach leads to a subfield in physics known as quantum turbulence. Therefore turbulence phenomena can also yield quantization, which also seems to suggest that turbulence and quantized vortice is a fractal phenomenon.
Bohr-Sommerfeld quantization rules for planetary orbit distances have the same result with a formula based on macroscopic Schrodinger equation.
Begin with Bohr-Sommerfeld's conjecture of quantization of angular momentum. For the wavefunction to be well defined and unique, the momenta must satisfy Bohr-Sommerfeld's quantization condition:
,
for any closed classical orbit Г. For the free particle of unit mass on the unit sphere the left-hand side is:
,
Where T = is the period of the orbit. Hence the quantization rule amounts to quantization of the rotation frequency (the angular momentum): ω = . Then we can write the force balance relation of Newton's equation of motion:
Using Bohr-Sommerfeld's hyphothesis of quantization of angular momentum, a new constant g was introduced:
.
Just like in the elementary Bohr theory (just before Schrodinger), this pair of equations yields a known simple solution for the orbit radius for any quantum number of the form:
,
or
, *
Where 0 represents orbit radii (semimajor axes), quantum number (=1, 2, 3,…), Newton gravitation constant, and mass of the nucleus of orbit, and specific velocity, respectively. In equation above we denote:
The value of m and g in equation above are adjustable parameters.
Interestingly, we can remark here that equation * is exactly the same with what is obtained by Nottale using his Schrodinger-Newton formula. Therefore here we can verify that the result is the same, either one uses Bohr-Sommerfeld quantization rules or Schrodinger-Newton equation. The applicability of equation * includes that one can predict new exoplanets (extrasolar planets) with remarkable result.
Furthermore, one can find a neat correspondence between Bohr-Sommerfeld quantization rules and motion of quantized vortice in condensed-matter systems, especially in superfluid helium. In this regards, a fractional Schrodinger equation has been used to derive two-fluid hydrodynamical equations for describing the motion of superfluid helium in the fractal dimension space. Therefore, it appears that fractional Schrodinger equation corresponds to superfluid helium in fractal dimension space.
Therefore, we can conclude that while our method as described herein may be interpreted as an oversimplification of the real planetary migration process which took place sometime in the past, at least it could provide us with useful tool for prediction. Now we also provide new prediction of other planetoids which are likely to be observed in the near future (around 113.8AU and 137.7 AU). It is recommended to use this prediction as guide to finding new objects (in the inner Oort Cloud).
What we would like to emphasize here is that the quantization method does not have to be the true description of reality with regards to celestial phenomena. As always this method could explain some phenomena, while perhaps lacks explanation for other phenomena. But at least it can be used to predict something quantitatively, i.e. measurable (exoplanets, and new planetoids in the outer solar system etc.). In the mean time, a correspondence between Bohr-Sommerfeld quantization rules and Gutzwiller trace formula has been shown in, indicating that the Bohr-Sommerfeld quantization rules may be used also for complex systems. Moreover, a recent theory extends Bohr-Sommerfeld rules to a full quantum theory.
MAN BEHIND THE SCENE
JOHANN ELERT BODE
bode2.GIF 200px-Johann_Daniel_Titius.jpg
Johann Elert Bode Johann Daniel Titius
Johann Elert Bode was born on January 19, 1747 in Hamburg, Germany. He became a member of the Berlin Academy of Sciences and director of the Berlin Observatory. Together with Johann Heinrich Lambert, he founded the German language ephemeris, the Astronomisches Jahrbuch oder Ephemeriden [Astronomical Yearbook and Ephemeris] in 1774, later called simply Astronomisches Jahrbuch and then Berliner Astronomisches Jahrbuch, which he continued to publish until his death in 1826. In 1774, Bode started to look for nebulae and star clusters in the sky, and observed 20 of them in 1774-5. Among them are three original discoveries, M81 and M82 which he both discovered on December 31, 1774, and M53, discovered on February 3, 1775, as well as a newly cataloged asterism.
Bode merged his discoveries and other observed objects with those from other catalogs he had access, namely the existing objects and most of the asterisms and non-objects from Hevelius' catalog, the sufficiently northern objects from Lacaille's catalog, most of the 45 objects in the first 1771 edition of Messier's catalog, and some others, to a "Complete Catalog of hitherto observed Nebulous Stars and Star Clusters" of an overall 75 entries, which he published in 1777 in the "Astronomisches Jahrbuch" for 1779. Unfortunately, he added a large number of non-existing objects without verification, in particular from Hevelius, so that over 20 of his objects don't exist. In the years following, he discovered two more objects: His original discovery of M92 occurred on December 31, 1777, and he found M64 on April 4, 1779, only 12 days after Edward Pigott had first discovered it. These two discoveries were announced along with the publication of Koehler's catalog in 1779 in the Astronomisches Jahrbuch for 1782. Consequently, he continued to compile catalogs and atlasses, and in his 1782 "Vorstellung der Gestirne," publishes own independent rediscoveries of open clusters M48 (NGC 2548) and IC 4665 in Ophiuchus. On January 6, 1779, Johann Elert Bode discovered the comet of that year (C/1779 A1, 1779 Bode).
Bode was greatly interested in the new planet discovered by William Herschel in March 1781. While Herschel always referred to this planet as "Georgium Sidus" to honor King George III of England, Bode proposed the name "Uranus" which was soon adopted by the rest of the world. Bode collected virtually all observations of this planet by various astronomers, published many of them in the Astronomisches Jahrbuch, and found that Uranus had been observed before its discovery on a number of occasions, among them an observation of Tobias Mayer from 1756 and earliest by Flamsteed, in December 1690, cataloged as "star" 34 Tauri.
In 1801 Bode published his famous and popular star atlas, Uranographia, where he reproduced or introduced a number of new and strange constellaitons, including "Officina Typographica," "Apparatus Chemica," "Globus Aerostaticus," "Honores Frederici," "Felis," and "Custos Messium," all of which have not survived and vanished from modern star charts.
In 1825, after almost 40 years, Bode retired from the post of a director of the Berlin Observatory, and was succeeded by J.F. Encke. Johann Elert Bode died on November 23, 1826 in Berlin, Germany.
In 1768, Bode published his popular book, "Anleitung zur Kenntnis des gestirnten Himmels" [Instruction for the Knowledge of the Starry Heavens], which was printed in a number of editions. In this book, he stressed an empirical law on planetary distances, originally found by J.D. Titius (1729-96), now called "Bode's Law" or "Titius-Bode Law".>
ISLAMIC SCHOLAR IN THE FIELD
OF MATHEMATICS
The Muslim mind has always been attracted to the mathematical sciences in accordance with the "abstract" character of the doctrine of Oneness which lies at the heart of Islam. The mathematical sciences have traditionally included astronomy, mathematics itself and much of what is called physics today. In astronomy the Muslims integrated the astronomical traditions of the Indians, Persians, the ancient Near East and especially the Greeks into a synthesis which began to chart a new chapter in the history of astronomy from the 8th century onward.
The Almagest of Ptolemy, whose very name in English reveals the Arabic origin of its Latin translation, was thoroughly studied and its planetary theory criticized by several astronomers of both the eastern and western lands of Islam leading to the major critique of the theory by Nasir al-Din al-Tusi and his students, especially Qutb al-Din al-Shirazi, in the 13th century.
The Muslims also observed the heavens carefully and discovered many new stars. The book on stars of 'Abd al-Rahman al-Sufi was in fact translated into Spanish by Alfonso X el Sabio and had a deep influence upon stellar toponymy in European languages. Many star names in English such as Aldabran still recall their Arabic origin. The Muslims carried out many fresh observations which were contained in astronomical tables called Zij.
One of the acutest of these observers was al-Battani whose work was followed by numerous others. The Zij of al-Ma'mun observed in Baghdad, the Hakimite Zij of Cairo, the Toledan Tables of al-Zarqali and his associated, the II-Khanid Zij of Nasir al-Din al-Tusi observed in Maraghah, and the Zij of Ulugh-Beg from Samarqand are among the most famous Islamic astronomical tables. They wielded a great deal of influence upon Western astronomy up to the time of Tycho Brahe.
The Muslims were in fact the first to create an astronomical observatory as a scientific institution, this being the observatory of Maraghah in Persia established by al-Tusi. This was indirectly the model for the later European observatories. Many astronomical instruments were developed by Muslims to carry out observation, the most famous being the astrolabe. There existed even mechanical astrolabes perfected by Ibn Samh which must be considered as the ancestor of the mechanical clock.
Astronomical observations also had practical applications including not only finding the direction of Makkah for prayers, but also devising almanacs (the word itself being of Arabic origin). The Muslims also applied their astronomical knowledge to questions of time-keeping and the calendar. The most exact solar calendar existing to this day is the Jalali calendar devised under the direction of 'Umar Khayyam in the 12th century and still in use in Persia and Afghanistan.
As for mathematics proper, like astronomy, it received its direct impetus from the Quran not only because of the mathematical structure related to the text of the Sacred Book, but also because the laws of inheritance delineated in the Quran require rather complicated mathematical solutions. Here again Muslims began by integrating Greek and Indian mathematics.
The first great Muslim mathematician, al-Khwarazmi, who lived in the 9th century, wrote a treatise on arithmetic whose Latin translation brought what is known as Arabic numerals to the West. To this day Guarismo, derived from his name, means figure or digit in Spanish while algorithm is still used in English. Al-Khwarzmi is also the author of the first book on algebra. This science was developed by Muslims on the basis of earlier Greek and Indian works of a rudimentary nature.
The very name algebra comes from the first part of the name of the book of al-Khwarazmi, entitled Kitab al-jabr wa'l-muqabalah. Abu Kamil al-Shuja' discussed algebraic equations with five unknowns. The science was further developed by such figures as al-Karaji until it reached its peak with Khayyam who classified by kind and class algebraic equations up to the third degree.
The Muslims also excelled in geometry as reflected in their art. The brothers Banu Musa who lived in the 9th century may be said to be the first outstanding Muslim geometers while their contemporary Thabit ibn Qurrah used the method of exhaustion, giving a glimpse of what was to become integral calculus. Many Muslim mathematicians such as Khayyam and al-Tusi also dealt with the fifth postulate of Euclid and the problems which follow if one tries to prove this postulate within the confines of Eucledian geometry.
Another branch of mathematics developed by Muslims is trigonometry which was established as a distinct branch of mathematics by al-Biruni. The Muslim mathematicians, especially al-Battani, Abu'l-Wafa', Ibn Yunus and Ibn al-Haytham, also developed spherical astronomy and applied it to the solution of astronomy and applied it to the solution of astronomical problems.
The love for the study of magic squares and amicable numbers led Muslims to develop the theory of numbers. Al-Khujandi discovered a particular case of Fermat's theorem that "the sum of two cubes cannot be another cube", while al-Karaji analyzed arithmetic and geometric progressions such as: 13+23+33+...+n3=(1+2+3+...+n)2. Al-Biruni also dealt with progressions while Ghiyath al-Din Jamshid al-Kashani brought the study of number theory among Muslims to its peak.
In the field of physics the Muslims made contributions in especially three domains. The first was the measurement of specific weights of objects and the study of the balance following upon the work of Archimedes. In this domain the writings of al-Biruni and al-Khazini stand out.
Secondly they criticized the Aristotelian theory of projectile motion and tried to quantify this type of motion. The critique of Ibn Sina, Abu'l-Barakat al-Baghdadi, Ibn Bajjah and others led to the development of the idea of impetus and momentum and played an important role in the criticism of Aristotelian physics in the West up to the early writings of Galileo.
Thirdly there is the field of optics in which the Islamic sciences produced in Ibn al-Haytham (the Latin Alhzen) who lived in the 11th century, the greatest student of optics between Ptolemy and Witelo. Ibn al-Haytham's main work on optics, the Kitab al-manazir, was also well known in the West as Thesaurus opticus.
Ibn al-Haytham solved many optical problems, one of which is named after him, studied the property of lenses, discovered the Camera Obscura, explained correctly the process of vision, studied the structure of the eye, and explained for the first time why the sun and the moon appear larger on the horizon. His interest in optics was carried out two centuries later by Qutb al-Din al-Shirazi and Kamal al-Din al-Farisi. It was Qutb al-Din who gave the first correct explanation of the formation of the rainbow.
It is important to recall that in physics as in many other fields of science the Muslims observed, measured and carried out experiments. They must be credited with having developed what came to be known later as the experimental method.
After hearing the history about Islamic civilization, we know that most of the knowledge we known today is all came from the Islamic scientist and scholar. Contribution from them have been manipulated by the West makes our Islamic scholar not to be known. For the next section, we will focus on various Islamic scholar with contribute in mathematics.
MOHAMMAD BIN MUSA kw.jpg
(AL-KHAWARIZMI)
Abu Abdullah Mohammad Ibn Musa al-Khawarizmi was born at Khawarizm (Kheva), south of Aral Sea. Very little is known about his early life, except for the fact that his parents had migrated to a place south of Baghdad. The exact dates of his birth and death are also not known, but it is established that he flourished under Al-Mamun at Baghdad through 813-833 and probably died around 840 C.E.
To celebrate the 1200th birth anniversary of Muhammad bin Musa Al-Khawarizmi the former USSR issued this postal stamp pictured on top. The terms Algebra and Algorithm are familiar to all of us but how many have heard of their founder Mohammed Al-Khawarizmi. In Geography he revised and corrected Ptolemy's view and produced the first map of the known world in 830 CE. He worked on measuring the volume and circumference of the earth, and contributed to work related to clocks, sundials and astrolabes.
It is believed that this is a copy of Al-Khawarizmi's arithmetic text, which was translated into Latin in the twelfth century by Adelard of Bath (an English scholar). Al-Khawarizmi left his name to the history of mathematics in the form of Algorism (the old name for arithmetic).
His Work
Al-Khawarizmi was a mathematician, astronomer and geographer. He was perhaps one of the greatest mathematicians who ever lived, as in facts he was the founder of several branches and basic concepts of mathematics. In the words of Phillip Hitti:
"He influenced mathematical thought to a greater extent than any other mediaeval writer."
His work on algebra was outstanding, as he not only initiated the subject in a systematic form but he also developed it to the extent of giving analytical solutions of linear and quadratic equations, which established him as the founder of Algebra. Hisab Al-jabr wAl-muqabala, contains analytical solutions of linear and quadratic equations and its author may be called one of the founders of analysis or algebra as distinct from geometry.
He also gives geometrical solutions (with figures) of quadratic equations, for example X2 + 1OX = 39, an equation often repeated by later writers. The 'Liber ysagogarum Alchorismi in artem astronomicam a magistro A. [Adelard of Bath] compositus!' deals with arithmetic, geometry, music, and astronomy; it is possibly a summary of Al-Khawarzmi's teachings rather than an original work. His astronomical and trigonometric tables, revised by Maslama Al-Majrti (Second half of tenth century), were translated into Latin as early as l126 by Adelard of Bath. They were the first Muslim tables and contained not simply the sine function but also the tangent (Maslama's interpolation).
His arithmetic synthesized Greek and Hindu knowledge and also contained his own contribution of fundamental importance to mathematics and science. Thus, he explained the use of zero, a numeral of fundamental importance developed by the Arabs. Similarly, he developed the decimal system so that the overall system of numerals, 'algorithm' or 'algorism' is named after him.
In addition to introducing the Indian system of numerals (now generally known as Arabic numerals), he developed at length several arithmetical procedures, including operations on fractions. It was through his work that the system of numerals was first introduced to Arabs and later to Europe, through its translations in European languages. He developed in detail trigonometric tables containing the sine functions, which were probably extrapolated to tangent functions by Maslamati.
He also perfected the geometric representation of conic sections and developed the calculus of two errors, which practically led him to the concept of differentiation. He is also reported to have collaborated in the degree measurements ordered by Al-Mamun which were aimed at measuring of volume and circumference of the earth.
His Books
Several of his books were translated into Latin in the early 12th century. In fact, his book on arithmetic, Kitab Al-Jam'a wal-Tafreeq bil Hisab Al-Hindi, was lost in Arabic but survived in a Latin translation. His astronomical tables were also translated into European languages and, later, into Chinese. His geography captioned Kitab Surat-Al-Ard,(The Face of the Earth) together with its maps, was also translated. In addition, he wrote a book on the Jewish calendar Istikhraj Tarikh Al-Yahud, and two books on the astrolabe. He also wrote Kitab Al-Tarikh and his book on sun-dials was captioned Kitab Al-Rukhmat, but both of them have been lost.
A Servant of God
Al-Khawarizmi emphasised that he wrote his algebra book to serve the practical needs of the people concerning matters of inheritance, legacies, partition, law suits and commerce. He considered his work as worship to God.
Quotation from Al-Khawarizmi:
That fondness for science, ... that affability and condescension which God shows to the learned, that promptitude with which he protects and supports them in the elucidation of obscurities and in the removal of difficulties, has encouraged me to compose a short work on calculating by aljabr and al-muqabala , confining it to what is easiest and most useful in arithmetic. [al-jabr means "restoring", referring to the process of moving a subtracted quantity to the other side of an equation; al muqabala is "comparing" and refers to subtracting equal quantities from both sides of an equation.]
THABIT IBN QURRA
(836-901 C.E)
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Thabit Ibn Qurra Ibn Marwan al-Sabi al-Harrani was born in the year 836 C.E. at Harran (present Turkey). As the name indicates he was basically a member of the Sabian sect, but the great Muslim mathematician Muhammad Ibn Musa Ibn Shakir, impressed by his knowledge of languages, and realising his potential for a scientific career, selected him to join the scientific group at Baghdad that was being patronised by the Abbasid Caliphs.
There, he studied under the famous Banu Musa brothers. It was in this setting that Thabit contributed to several branches of science, notably mathematics, astronomy and mechanics, in addition to translating a large number of works from Greek to Arabic. Later, he was patronised by the Abbasid Caliph al-M'utadid. After a long career of scholarship, Thabit died at Baghdad in 901 C.E.
Thabit's major contribution lies in mathematics and astronomy. He was instrumental in extending the concept of traditional geometry to geometrical algebra and proposed several theories that led to the development of non-Euclidean geometry, spherical trigonometry, integral calculus and real numbers. He criticised a number of theorems of Euclid's elements and proposed important improvements. He applied arithmetical terminology to geometrical quantities, and studied several aspects of conic sections, notably those of parabola and ellipse. A number of his computations aimed at determining the surfaces and volumes of different types of bodies and constitute, in fact, the processes of integral calculus, as developed later.
Thabit ibn Qurra was a native of Harran and a member of the Sabian sect. The Sabian religious sect were star worshippers from Harran often confused with the Mandaeans. Of
course being worshipers of the stars meant that there was strong motivation for the study of astronomy and the sect produced many quality astronomers and mathematicians. The sect, with strong Greek connections, had in earlier times adopted Greek culture, and it was common for members to speak Greek although after the conquest of the Sabians by Islam, they became Arabic speakers. There was another language spoken in southeastern Turkey, namely Syriac, which was based on the East Aramaic dialect of Edessa. This language was Thabit ibn Qurra's native language, but he was fluent in both Greek and Arabic.
Some accounts say that Thabit was a money changer as a young man. This is quite possible but some historians do not agree. Certainly he inherited a large family fortune and must have come from a family of high standing in the community. Muhammad ibn Musa ibn Shakir, who visited Harran, was impressed at Thabit's knowledge of languages and, realising the young man's potential, persuaded him to go to Baghdad and take lessons in mathematics from him and his brothers (the Banu Musa).
In Baghdad Thabit received mathematical training and also training in medicine, which was common for scholars of that time. He returned to Harran but his liberal philosophies led to a religious court appearance when he had to recant his 'heresies'. To escape further persecution he left Harran and was appointed court astronomer in Baghdad. There Thabit's patron was the Caliph, al-Mu'tadid, one of the greatest of the 'Abbasid caliphs.
At this time there were many patrons who employed talented scientists to translate Greek text into Arabic and Thabit, with his great skills in languages as well as great mathematical skills, translated and revised many of the important Greek works. The two earliest translations of Euclid's Elements were made by al- Hajjaj. These are lost except for some fragments.
There are, however, numerous manuscript versions of the third translation into Arabic which was made by Hunayn ibn Ishaq and revised by Thabit. Knowledge today of the complex story of the Arabic translations of Euclid's Elements indicates that all later Arabic versions develop from this revision by Thabit.
In fact many Greek texts survive today only because of this industry in bringing Greek learning to the Arab world. However we must not think that the mathematicians such as Thabit were mere preservers of Greek knowledge. Far from it, Thabit was a brilliant scholar who made many important mathematical discoveries.
Although Thabit contributed to a number of areas the most important of his work was in mathematics where he :-
... played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-euclidean geometry. In astronomy Thabit was one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics.
We shall examine in more detail Thabit's work in these areas, in particular his work in number theory on amicable numbers. Suppose that, in modern notation, S(n) denotes the sum of the aliquot parts of n, that is the sum of its proper quotients. Perfect numbers are those numbers n with S(n) = n while m and n are amicable if S(n) = m, and S(m) = n.
In Book on the determination of amicable numbers Thabit claims that Pythagoras began the study of perfect and amicable numbers. This claim, probably first made by Iamblichus in his biography of Pythagoras written in the third century AD where he gave the amicable numbers 220 and 284, is almost certainly false. However Thabit then states quite correctly that although Euclid and Nicomachus studied perfect numbers, and Euclid gave a rule for determining them:-
... neither of these authors either mentioned or showed interest in
[amicable numbers].
Thabit continues :-
Since the matter of [amicable numbers] has occurred to my mind,
and since I have derived a proof for them, I did not wish to write
the rule without proving it perfectly because they have been neglected
by [Euclid and Nicomachus]. I shall therefore prove it after
introducing the necessary lemmas.
After giving nine lemmas Thabit states and proves his theorem: for n > 1, let pn= 3.2n-1 and qn= 9.22n-1-1. If pn-1, pn, and qn are prime numbers, then a = 2npn-1pn and b = 2nqn are amicable numbers while a is abundant and b is deficient. Note that an abundant number n satisfies S(n) > n, and a deficient number n satisfies S(n) < n.
More details are given in "Notes on Thabit ibn Qurra and his rule for amicable numbers",where the authors conjecture how Thabit might have discovered the rule. In "Thabit ibn Qurra and the pair of amicable numbers 17296,18416 " Hogendijk shows that Thabit was probably the first to discover the pair of amicable numbers 17296, 18416.
Another important aspect of Thabit's work was his book on the composition of ratios. In this Thabit deals with arithmetical operations applied to ratios of geometrical quantities. The Greeks had dealt with geometric quantities but had not thought of them in the same way as numbers to which the usual rules of arithmetic could be applied. The authors of "Remarks on the treatise of Thabit ibn Qurra (Russian), in Phys. Math. Sci. in the East 'Nauka'"and "A treatise of Thabit ibn Qurra on composite ratios (Russian), in Phys. Math. Sci. in the East 'Nauka'" stress that by introducing arithmetical operations on quantities previously regarded as geometric and non-numerical, Thabit started a trend which led eventually to the generalisation of the number concept.
Thabit generalised Pythagoras's theorem to an arbitrary triangle (as did Pappus). He also discussed parabolas, angle trisection and magic squares. Thabit's work on parabolas and paraboliods is of particular importance since it is one of the steps taken towards the discovery of the integral calculus. An important consideration here is whether Thabit was familiar with the methods of Archimedes. Most authors believe that although Thabit was familiar with Archimedes' results on the quadrature of the parabola, he did not have either of Archimedes' two treatises on the topic. In fact Thabit effectively computed the integral of x :-
The computation is based essentially on the application of upper and lower integral sums, and the proof is done by the method of exhaustion: there, for the first time, the segment of integration is divided into unequal parts.
Thabit also wrote on astronomy, writing Concerning the Motion of the Eighth Sphere. He believed (wrongly) that the motion of the equinoxes oscillates. He also published observations
of the Sun. In fact eight complete treatises by Thabit on astronomy have survived and the article "Tabit b. Qurra and Arab astronomy in the 9th century" describes these. The author writes:-
When we consider this body of work in the context of the beginnings of the scientific movement in ninth-century Baghdad, we see that Thabit played a very important role in the establishment of astronomy as an exact science (method, topics and program), which developed along three lines: the theorisation of the relation between observation and theory, the 'mathematisation' of astronomy, and the focus on the conflicting relationship between 'mathematical' astronomy and 'physical' astronomy.
An important work Kitab fi'l-qarastun (The book on the beam balance) by Thabit is on mechanics. It was translated into Latin by Gherard of Cremona and became a popular work on mechanics. In this work Thabit proves the principle of equilibrium of levers. He demonstrates that two equal loads, balancing a third, can be replaced by their sum placed at a point halfway between the two without destroying the equilibrium.
After giving a generalisation Thabit then considers the case of equally distributed continuous loads and finds the conditions for the equilibrium of a heavy beam. Of course Archimedes considered a theory of centres of gravity, but in some books the author argues that Thabit's work is not based on Archimedes' theory.
Finally we should comment on Thabit's work on philosophy and other topics. Thabit had a student Abu Musa Isa ibn Usayyid who was a Christian from Iraq. Ibn Usayyid asked various questions of his teacher Thabit and a manuscript exists of the answers given by Thabit, this manuscript being discussed in "Thabit Qurra's conception of number and theory of the mathematical infinite". Thabit's concept of number follows that of Plato and he argues that number exist, whether someone knows them or not, and they are separate from numerable things. In other respects Thabit is critical of the ideas of Plato and Aristotle, particularly regarding motion. It would seem that here his ideas are based on an acceptance of using arguments concerning motion in his geometrical arguments. Thabit also wrote:-
... logic, psychology, ethics, the classification of sciences, the grammar of the Syriac language, politics, the symbolism of Plato's Republic ... religion and the customs of the Sabians.
His son, Sinan ibn Thabit, and his grandson Ibrahim ibn Sinan ibn Thabit, both were eminent scholars who contributed to the development of mathematics. Neither, however, reached the mathematical heights of Thabit.
OMAR AL-KHAYYAM
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Ghiyath al-Din Abul Fateh Omar Ibn Ibrahim al-Khayyam was born at Nishapur, the provincial capital of Khurasan around 1044 C.E. (c. 1038 to 1048). Persian mathematician, astronomer, philosopher, physician and poet, he is commonly known as Omar Khayyam. Khayyam means the tent-maker, and although generally considered as Persian, it has also been suggested that he could have belonged to the Khayyami tribe of Arab origin who might have settled in Persia.
Little is known about his early life, except for the fact that he was educated at Nishapur and lived there and at Samarqand for most of his life. He was a contemporary of Nidham al-Mulk Tusi. Contrary to the available opportunities, he did not like to be employed at the King's court and led a calm life devoted to search for knowledge. He travelled to the great centres of learn- ing, Samarqand, Bukhara, Balkh and Isphahan in order to study further and exchange views with the scholars there. While at Samarqand he was patronised by a dignatory, Abu Tahir. He died at Nishapur in 1123-24.
Khayyam played on the meaning of his own name when he wrote:-
Muslim Scholars and Scientists
Khayyam, who stitched the tents of science,
Has fallen in grief's furnace and been suddenly burned,
The shears of Fate have cut the tent ropes of his life,
And the broker of Hope has sold him for nothing!
The political events of the 11th century played a major role in the course of Khayyam's life. The Seljuq Turks were tribes that invaded southwestern Asia in the 11th century and eventually founded an empire that included Mesopotamia, Syria, Palestine, and most of Iran. The Seljuq occupied the grazing grounds of Khorasan and then, between 1038 and 1040, they conquered all of north-eastern Iran. The Seljuq ruler Toghrïl Beg proclaimed himself sultan at Nishapur in 1038 and entered Baghdad in 1055. It was in this difficult unstable military empire, that Khayyam grew up.
Khayyam studied philosophy at Naishapur and one of his fellow students wrote that he was:-
... endowed with sharpness of wit and the highest natural powers..
However, this was not an empire in which those of learning, even those as learned as Khayyam, found life easy unless they had the support of a ruler at one of the many courts. Even such patronage would not provide too much stability since local politics and the fortunes of the local military regime decided who at any one time held power. Khayyam himself described the difficulties for men of learning during this period in the introduction to his (Maqalat fi al-Jabr wa al-Muqabila) Treatise on Demonstration of Problems of Algebra (see for example):-
I was unable to devote myself to the learning of this algebra and the continued concentration upon it, because of obstacles in the vagaries of time which hindered me; for we have been deprived of all the people of knowledge save for a group, small in number, with many troubles, whose concern in life is to snatch the opportunity, when time is asleep, to devote themselves meanwhile to the investigation and perfection of a science; for the majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do not use what they know of the sciences except for base and material purposes; and if they see a certain person seeking for the right and preferring the truth, doing his best to refute the false and untrue and leaving aside hypocrisy and deceit, they make a fool of him and mock him.
However Khayyam was an outstanding mathematician and astronomer and, despite the difficulties which he described in this quote, he did write several works including Problems of
Arithmetic, a book on music and one on algebra before he was 25 years old. In 1070 he moved to Samarkand in Uzbekistan which is one of the oldest cities of Central Asia.
There Khayyam was supported by Abu Tahir, a prominent jurist of Samarkand, and this allowed him to write his most famous algebra work, (Maqalat fi al-Jabr wa al-Muqabila) Treatise on Demonstration of Problems of Algebra from which we gave the quote above. We shall describe the mathematical contents of this work later in this biography.
Toghril Beg, the founder of the Seljuq dynasty, had made Esfahan the capital of his domains and his grandson Malik-Shah was the ruler of that city from 1073. An invitation was sent to Khayyam from Malik-Shah and from his vizier Nizam al-Mulk asking Khayyam to go to Esfahan to set up an Observatory there. Other leading astronomers were also brought to the Observatory in Esfahan and for 18 years Khayyam led the scientists and produced work of outstanding quality. It was a period of peace during which the political situation allowed Khayyam the opportunity to devote himself entirely to his scholarly work.
During this time Khayyam led work on compiling astronomical tables and he also contributed to calendar reform in 1079. This sonar calndar become necessary in view of the revenue collections and other administrative matters that were to be performed at different times of the year. Khayyam introduced a calendar that was remarkably accurate, and was named as Al- Tarikh-al-J
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