# Concerning Pi, again

For Pi Day this year (2018), I will provide some results related to this interesting mathematical constant. These results are mostly inspired or taken from answers I gave at quora.com about  $\pi$ or about similar mathematical topics.

The millionth decimal digit of π is 1 (verified with Mathematica).

The 10 millionth decimal digit of π is found to be 7, and the 100 millionth decimal digit of π is 2.

The billionth decimal digit of $\pi$ (in base 10) is 9 (verified with Mathematica).

The 2 billionth decimal digit of π is found to be 0 (this result takes a longer time to compute with Mathematica).

Here are some (repeated) number sequences or numeric strings found among the first 2 billion decimal digits of π.

The numeric string 777777777 appears at the 24, 658, 601 st decimal digit of π :

9919408245759718530477777777724846769425931046864

The numeric string 111111111 appears at the 812, 432, 526 th decimal digit of π :

2891450444990691713511111111100399876875718824885

Here are also two numeric strings from 1 to 9 in increasing order and decreasing order:

The numeric string 123456789 appears at the 523, 551, 502 nd decimal digit of π:

7260489917323889207212345678922486448188070486710

The numeric string 987654321 appears at the 719, 473, 323 rd decimal digit of π :

5221398663241526793698765432160793011913242320510

The numeric strings above can be calculated or found with the help of the following link or web page:

Irrational Numbers Search Engine

The numerical value of $\pi^{\pi}$ to 1000 decimal digits is equal to:

36.46215960720791177099082602269212366636550840222881873870933592293407436888169990462007987570677485436814688343670070542736699139359264431565675267180230917777595737242260530320050233549595161382594571885422222305402433199779769167302876444780028452117394296018175249159350019492001619423210110480018557258718860782819839215304503453543238476218257664861595609057280314341958390400811991506636066295817900302292747422204210046403709493285441101884797707466358510710362803891181156618083260884536505255311095948029552909133361385823497120761861157606574436205295895657736468959837126404885207348833917602169536002174958035720670509318770633086434935593202425189496008880555048213388792769304015639745480898380866639428337794250284522113741878602793251848366623602214652151453276098545038540482041645516919082097210265379423765817354600472953993840487842116366153365057093066926223775915023204727672609958737278566633593210689698807508602353552267490321670730929373240451651980750157959708689483469068

Two expressions involving π and infinite sums:

Representation of π in continued fraction form:

The sum of π and e, the base of natural logarithms, is equal to:

$\pi + e=\displaystyle \sum _{k=0}^{\infty } \frac{(3 k)^2+1}{(3 k)!}+2 i \ln \left(\frac{1-i}{1+i}\right)i$

The letter i  represents the imaginary unit of complex numbers.

Another expression involving π, e, and an infinite product:

$\pi = \displaystyle 2 e \prod _{k=1}^{\infty } \left(\frac{2}{k}+1\right)^{(-1)^{k+1} k}$

And here is an identity relating the Golden Ratio, π, e, and the imaginary unit i:

$\displaystyle \varphi=e^{i\pi/5}+e^{-i\pi/5}=\frac{1+\sqrt{5}}{2}$

The value of π can be deduced from the identity above:

$\displaystyle\pi =5 i \ln \left(\frac{1}{2} \left(\varphi -\sqrt{\varphi ^2-4}\right)\right)$

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# Concerning the relationship between science and philosophy

This post consists of the  elements of an answer I wrote at quora.com ; the question there was: “Is philosophy the top of all kinds of sciences?”

I think it would be convenient to distinguish between the general term “science”, referring to the state or fact of knowing, or to knowledge acquired by study and learning, and the modern meaning of “science”, mostly referring to mathematics and to the exact sciences using the rules of the scientific method (astronomy, physics…).

Philosophy and science were not separate in Antiquity.

In the original sense, philosophy meant the love, study, or pursuit of wisdom, or the knowledge of things and their causes, theoretical as well as practical.

Pythagoras was a mathematician, and at the same time it is said that he was the first one to call himself a philosopher, or “lover of wisdom”.

Plato was a philosopher who recommended the knowledge and the study of geometry. In The Republic, Plato thought that the best ruler was the king-philosopher.

Aristotle studied nature and wrote works about physics, biology, logic, etc, from a philosophical point of view.

According to the OED:

“In the Middle Ages, ‘the seven (liberal) sciences’ was often used synonymously with ‘the seven liberal arts’, for the   group of studies comprised by the Trivium (Grammar, Logic, Rhetoric) and the Quadrivium (Arithmetic, Music, Geometry, Astronomy).”

The expression Natural philosophy was frequently used for centuries :

“Natural philosophy or philosophy of nature (from Latin philosophia naturalis) was the philosophical study of nature and the physical universe that was dominant before the development of modern science. It is considered to be the precursor of natural science.

From the ancient world, starting with Aristotle, to the 19th century, the term “natural philosophy” was the common term used to describe the practice of studying nature. It was in the 19th century that the concept of “science” received its modern shape with new titles emerging such as “biology” and “biologist”, “physics” and “physicist” among other technical fields and titles; institutions and communities were founded, and unprecedented applications to and interactions with other aspects of society and culture occurred. Isaac Newton‘s book Philosophiae Naturalis Principia Mathematica (1687), whose title translates to “Mathematical Principles of Natural Philosophy”, reflects the then-current use of the words “natural philosophy”, akin to “systematic study of nature”. Even in the 19th century, a treatise by Lord Kelvin and Peter Guthrie Tait, which helped define much of modern physics, was titled Treatise on Natural Philosophy (1867).

In the last few centuries, alchemy separated from chemistry, astrology separated from astronomy, and there was also a certain separation between philosophy on one side, and mathematics and the exact sciences on the other side.

Mathematics became progressively the most prominent and the essential scientific discipline, it is acknowledged as the language of science and of the physical world.

Philosophy is nowadays often regarded as a reflection, view or study of the general principles of a particular branch of knowledge, or activity. There is a philosophy of science, philosophy of mathematics, philosophy of education ,etc.

Some theories or views related to epistemology (which is concerned with the general theory and the study of knowledge) and philosophy, such as rationalism, empiricism, and positivism, share a number of principles with the scientific approach to events and phenomena.

(Source of the image above: Wikimedia Commons)

A scientist or a physicist can also be a philosopher. Important thinkers can be philosophers and create philosophical systems, but modern philosophers must take into account the advances, discoveries and theories in modern science. A historical example would be Immanuel Kant elaborating his philosophical system and philosophical ideas at the end of the eighteenth century in light of and in relation to the exact sciences known at that time, especially Euclidean geometry and Newtonian classical physics and mechanics.

# A poem I wrote years ago

I was fifteen- soon to be sixteen- years old ; I had been reading (important) books about science, physics, philosophy , and other similar topics,  and all those ideas in my head intermingled and inspired me to write a poem involving particle physics and particle collisions and combining elements of science and philosophy .

I wrote the poem in French , using the French alexandrine poetic meter of twelve syllables, but I didn’t follow the poetic rules very closely.

I will provide the final version of the poem here , with a line by line English translation. Different people have different tastes and opinions , I hope it will be liked .

The hydrogen-1 atom mentioned in the title of the poem is also called “protium” , but this last word is not much used in French. Protium is the most common hydrogen isotope, having one proton ( and one electron) and no neutrons.

A proton is supposed to be talking or telling the story in the poem . I think I was a little inspired by the poem ” Le Bateau ivre ” by Arthur Rimbaud .

Here it is :

Bombardement d’atomes par un proton d’hydrogène 1H
Bombardment of atoms by a proton of protium 1H

Synchrotrons , canons à électrons, cyclotrons
Synchrotrons, electron guns, cyclotrons

Soyez prêts, particules, deutons, neutrons, hélions
Be prepared, particles, deutons/deuterons, neutrons, helions

En attendant que les hommes préparent les canons
Until men prepare the guns

Le moment est arrivé, l’appareil frappe
The moment has come, the apparatus strikes

Dans son coeur vidé moi, le proton j’attrape
In its emptied heart I , the proton take

Le coup et je vais croiser les atomes en grappe
The blow and I go meet the atoms in clusters

Je fuis dans l’espace et le temps calculables
I flee in computable space and time

Ma vitesse est vertigineuse, incroyable
My speed is vertiginous, incredible

Non pas celle de la lumière, infranchissable
Not that of light, insurmountable

C’est le lieu de la relativité impie
It is the place of impious Relativity

Masses, longueurs, lois de la physique varient
Masses, lengths, physical laws vary

Ma trajectoire déterminée sera suivie
My particular/determined path will be followed

Par d’autres microcosmes malheureux
By other unfortunate microcosms

Le trajet est terminé, le choc a eu lieu
The journey is over, the shock/collision occurred

Je donne la vie à de nouveaux corps heureux
I give life to new happy/fortunate bodies

Quanta de matière utilisés pour la paix
Quanta of matter used for peace

Dans le monde de la science un pas est fait
In the world of science a step/discovery has been made

L’humanité en marche en connaît les bienfaits
Humanity in motion/advancing knows the benefits (of this discovery)

# Books about physics, astrophysics and astronomy regarded as important classics

This post is mostly inspired (with some additions and modifications) from and answer I wrote at quora.com .

I will try to give a list of famous , influential books or classic books having a significant historical importance in the fields of physics , astrophysics and astronomy . It’s a somewhat extensive list but it’s not exhaustive.

Starting with Antiquity :

Then advancing to more recent times:

Below is a page from the Astronomia Nova (in 1609) showing the three models of planetary motion known in the seventeenth century (free image from Wikipedia) :

• Recherches sur la théorie des quanta (Researches on the quantum theory) , and The Current Interpretation of Wave Mechanics: A Critical Study , by Louis de Broglie .
• Collected papers , The interpretation of Quantum Mechanics , and Statistical Thermodynamics , by Erwin Schrödinger .
• The Physical Principles of the Quantum Theory , by Werner Heisenberg .
• Books and papers by Paul Dirac , such as and Lectures on Quantum Field Theory .
• Space, Time and Gravitation: An Outline of the General Relativity TheoryThe Internal Constitution of Stars , and The Nature of the Physical World , by Arthur Eddington .
• Problems of Cosmology and Stellar Dynamics , An Introduction to the Kinetic Theory of Gases , and  The Growth of Physical Science , by James Hopwood Jeans .
• The Theory of Sound , by John William Strutt, 3rd Baron Rayleigh .
• Problems of Atomic Dynamics , Atomic Physics , Principles of Optics , Experiment and Theory in Physics , and A General Kinetic Theory of Liquids , by Max Born .
• Books and papers by David Bohm , such as Quantum Theory , Causality and Chance in Modern Physics , The Undivided Universe.

Some more recent well known , insightful and/or widely used books would include :

• The Large Scale Structure of Space-Time , by Stephen Hawking and George F. R. Ellis .
• Speakable and Unspeakable in Quantum Mechanics , by John Stewart Bell .
• Classical-Mechanics , by Herbert Goldstein .
• Classical Electrodynamics , by J.D. Jackson .
• Galactic Dynamics , by Binney and Tremaine .
• The Quantum Theory of Motion: an account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics , by Peter Holland .
• Photons and Atoms: Introduction to Quantum Electrodynamics , by Claude Cohen-Tannoudji , Gilbert Grynberg and Jacques Dupont-Roc .
• Introduction to Elementary Particles , by D.J. Griffiths .
• Condensed Matter Field Theory , by Alexander Altland .
• The Standard Model and Beyond , by Paul Langacker .
• The Road to Reality , by Roger Penrose .
• Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law , by Peter Woit .
• The Trouble With Physics: The Rise of String Theory, The Fall of a Science, and What Comes Next , by Lee smolin .
• Farewell to Reality: How Modern Physics Has Betrayed the Search for Scientific Truth , by Jim Baggott .

Additional relevant links :

https://en.wikipedia.org/wiki/Hi…

https://en.wikipedia.org/wiki/Hi…

Astronomy in the medieval Islamic world

Indian astronomy

Chinese astronomy

# Books about mathematics regarded as noteworthy classics

This post is taken (with some modifications from an answer I gave at quora.com  .

Classic books or classics may refer to great and historically important books , or to books widely popular , read and used ,or both . I will try to mention both types of books.
I will cite first a number of books which I think are of primary importance in the history of mathematics , and therefore are generally regarded as classics . This is not a exhaustive list .
Let’s start with some books from Antiquity :
Moving a few centuries later to modern times :
Some recent modern and well known math books that may be regarded as classics :
See also the following links :

# Question: what is the square root of 36 ? – Part Two

I will continue with answers and results equal to the square root of 36 ( originally answered at quora.com ). This time the results are mostly related to physics.

With physics one has to take into account the units and the corresponding dimensions of the equations and of the constants.

6 and the square root of 36 are dimentionless numbers , so the result must be dimentionless .
If the result is a simple fraction with numerator and denominator , then the units usually cancel out.
In other cases when one deals with logarithms one should multiply with the inverse dimensions to get a dimentionless result.
In one or two results where I didn’t look for the inverse units  I multiplied the equation with a quantity I called (U) ,  which represents the inverse of the units by which one should multiply the result to get a dimentionless number.

Here are the results :

One possible way to explain what I have done here is the following:
If some people , living on an isolated fictitious island or on an another hypothetical planet , attached a great importance to and had a fixation on  the square root of 36 (or the number 6) for one reason or another , and got accustomed to the use of 6 as a fundamental constant ,  unit or number , then they would have likely  tried to construct a system of measurement  based on the number 6 , and  to express physics and math formulas ,equations , constants and rules in relation to 6.

After all , 6 or $\sqrt{36}$ is equal to :

• The floor of  $2 \pi$ :
$2 \pi \approx 6.2831853071795864769$ ;$6=\lfloor 2 \pi \rfloor$
• $\frac{1}{60}$ of the circumference of a circle in degrees.
• It is also  one tenth of 60 seconds which make up a minute , one tenth of 60 minutes which make up an hour , one fourth of 24 hours which equal a day on Earth , one half of  12 months which make up a year , etc.
• A peculiar ‘hexacentric’ system , so to speak.

Or this can be seen as a (creative) exploration of or exercise in advanced math and physics in order to express many equations , formulas and constants in relation to the number 6 (or $\sqrt{36}$ ) .
Or whatever.

Apologies to Isaac Newton , Leonhard Euler , Bernhard Riemann , Einstein , Stokes , Coulomb , Avogadro , Lagrange , and others (wherever they may be) , for playing around with their equations , formulas , constants , and/or functions.

And one more addiction to this answer :

Does the future of humanity depend on answering what is the square root of  36 , or not?
Have philosophers from Antiquity to the present overlooked this fundamental question , which goes beyond the Kantian categories of space and time set out in his Critique of Pure Reason , and beyond Nietzsche’s Beyond Good and Evil , ushering the transmutation of all values and a defining moment for a new era  in the history of Humankind?
It’s just a square root , for common sense’s sake (or is it?).

Anyway , enough philosophizing.

Here are ( $3=\frac{\sqrt{36}}{2}$ ) more answers to $\sqrt{36}$ , this time with images :

$\sqrt{36}$  is equal to :

The number subjected to a geometric rotation in the following image (done with Mathematica and some Photoshop) :

The number expressing the power and the coefficients in the equation of the curve in the polar plot below :

The number expressing the degree of the root  and the power of the variables in the 3D plot below :

The rotated number  and the polar plotted curve in the first two images  above seem to exhibit symmetry.
Symmetry is an very important property in science , math , physics , equations , nature , and wherever it is found.

Online sources and reference works related to what I have written in this answer can be found in my pages about Science booksand philosophy books in this site/blog.

Some other online sources:

http://en.wikipedia.org/wiki/Category:Mathematics

http://en.wikipedia.org/wiki/Category:Physics

# 2015 the year , the number , and some of their characteristics

The year 2015 has begun (wish it will be a happy year for everybody) and here are some properties of the corresponding number (worked out with the help of Mathematica).

First , a list of the representations of 2015  in various number base notations:

2015 in roman numerals: MMXV;

2015 can be represented as:

$2015=2^{11}-33$

$2015=\frac {92^4 - 1} {35553}$

Here is a big one (obtained with Mathematica by solving $x^3 + x^2 + x = 2015$ ):

2015=5 x 13 x 31

The last relation above can be  derived from the trigonometric equation:

I have mentioned these types of trigonometric relations in a previous post.

The number of prime numbers up to 2015 is 305 , the last  prime number before 2015 is 2011 ,  and the prime number which comes after 2015 is 2017.

The last Mersenne prime number before 2015 is:
$127=2^{7}-1$

The Mersenne prime number which comes after 2015 is:
$8191=2^{13}-1$

The last Sophie Germain prime (n is a Sophie Germain prime if 2n + 1 is also prime) before 2015 is 2003 , and the Sophie Germain prime which comes after 2015 is 2039.

The Riemann prime counting function is given by:

where li(x) is the logarithmic integral and  μ(n) is the Möbius function.

And below is a graph showing the natural distribution and the number  of prime numbers π(n) less than or equal to n (blue colored curve , here n=2015) along with the distribution of primes given by the Riemann counting function (orange colored curve) up to 2015 :

Moving to other fields of science , I will mention briefly a few events taking place in the year 2015:

In the field of space exploration , the New Horizons spacecraft will get closest to the dwarf planet Pluto in July 2015.

In the following image (made with the help of Starry Night and Photoshop) the longer red arrow near the name of a planet indicates the direction towards the place where the planet is located on its orbit.

The Dawn spacecraft will arrive at the dwarf planet Ceres between Mars and Jupiter in March 2015.

Concerning astronomy and sky events , there will be a solar eclipse in March 20 , a lunar eclipse in April 4 , a partial solar eclipse in September 13 , and a lunar eclipse in September 28.

And that is it for now.

# Rosetta’s path and the encounter with comet 67P/Churyumov–Gerasimenko

Here is a video animation I have  made of the Rosetta spacecraft on it way to  comet . It shows the path of Rosetta (in yellow) since its launch in 2004 until it approaches and meets the comet (orbit in blue). The animation was made using the Starry Night astronomy program.

The same  video with shorter length , less effects and a slightly better resolution can be viewed at this link.

The coordinates of comet 67P  were not prebuilt in my version of Starry Night , so I had to program the orbital elements of Churyumov–Gerasimenko myself. After researching I used the following  elements :

Eccentricity (e): 0.640980 ;  pericenter or perihelion distance (q): 1.243230 AU ;

Node : 50.1423 °  ;  argument of pericenter ( w ) : 12.7854°  ;

Inclination (i): 7.0402°  ; pericenter time (Tp) : 2457247.5683  ;

Epoch : 2456967.5  ;  Magnitude : 11 ;

North pole right ascension: 69.00°  ;  North pole declination: 64° ;

Rotation rate : 2 rotations/day  .

Here is also a slideshow of the position and orbit of Rosetta at different dates . In the last three images the position of the Dawn spacecraft is shown as well. Dawn entered the orbit of the Vesta asteroid (or protoplanet)  in July 20 , 2011 , and completed  its fourteen month survey mission of Vesta in late 2012. Dawn is getting nearer to the dwarf planet (or large asteroid) Ceres and will arrive there in early 2015. The images were created and prepared with the Redshift astronomy software.

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Last but not least , here is a an image I’ve prepared ( with Starry Night and Photoshop ) showing the path and orbit of the Rosetta spacecraft from its launch in 2004 to the end of 2014 , including the orbit of comet 67P/Churyumov–Gerasimenko ,  with annotations recounting briefly the important steps of the Rosetta mission.

# About languages I know and languages I’m getting to know

I have a problem. Could be a problem , or not.The thing is , I read a lot , and in more than one language.

At present I know three languages very well , and I’ve read literary books and various kinds of books in these languages.But I have also other books and coursebooks for learning other languages , some are in print form and others are ebooks along with online learning material I’ve downloaded . Audio files are generally  helpful while learning a new language.The three languages I know best are English , French and Arabic . I’ve studied and learned other  languages intermittently , depending on the circumstances or on how much free time I had.

For example , a few years ago I met a Russian girl and I had a relationship with her , and that gave me the incentive to learn Russian.Then I met a German girl , and that encouraged me to start learning German using a coursebook I had bought two years earlier.I have a Spanish language coursebook as well , so I’ve studied recently the first few chapters of this book.Of course the Spanish language is interesting and useful since it is spoken not only in Spain but in most of Central and South America and in many countries around the globe.

I have additionally a Portuguese language coursebook waiting to be read ,  and a Chinese language textbook for beginners and a Chinese language learning software I haven’t used yet . Now  the Portuguese and Spanish languages are more or less related to each other , but learning Chinese could be more difficult as it is a non-alphabetic language and uses characters such as logograms , pictograms and ideograms . I’ll have to see if or when I will have enough time to start learning these languages.

Moving to another group or type of languages , I’ve been using and working with the Mathematica software and the (Wolfram) Mathematica programming language for more than fifteen years , and I have occasionally used in the past programming languages such as QBasic and the BASIC-like language of the TI-92 Plus calculator. I have also studied web programming for a certain period of time  , and I got to know well a number of programming (and scripting ) languages. These include HTML , PHP (and the MySQL database management system which uses the SQL query language) and JavaScript.

However , I have to say that among the languages I know the most important one is a language called Mathematics.This is the essential and fundamental language of the exact sciences , and according to good old Galileo , the mathematical language is the one in which the great book of Nature and the Universe is written.

And if you are of the opinion that Mathematics is not as important as other languages for communicating and connecting people , think again.
From Astronomy and Astronautics to Physics and Chemistry to electronic devices and components (which use disciplines such as solid-state physics , circuit and signal analysis , differential and difference equations , Fourier analysis , etc) , to computers (which use fields such as Boolean algebra , binary logic ,  and combinatorics)  , to Biology , Psychology , Economics and business studies , Mathematics and applied math are indispensable to any  study or research in science and engineering . Programming languages are more or less related to applied , discrete and computational math and to mathematical logic. Even the social and human sciences use math and statistics to  get a reputation for exactness , precision and scientificity .

# Some more thoughts about education

Children who have the possibility to finish school at an early age and to start their higher education at a very young age are often called child prodigies or gifted children.They are
frequently treated as curiosities or rare people with an acute intelligence and uncommon abilities .These kids may have special aptitudes but there is a plausible explanation for their situation .To put it simply , either they have fast stages of intellectual development and they were noticed and helped by their parents , family, teachers and/or professionals in order to have an accelerated education , skip grades and go to college at a very young age (probably 9 or 11 or 12) , or they have  average (or slightly higher than average) developmental stages and were also taught , helped and trained by their  parents , teachers and education professionals in ways which allowed them to finish school early and to enter the university precociously.

I think  the way these precocious or gifted children are taught ,  accelerated education and the opportunity to complete one’s studies early and to start university studies at a younger age ought to be given and extended progressively to all children .This way fast learners will not be left behind or neglected , and generation after generation more and more kids and young people will have faster stages of growth and will be able to assimilate more information at younger ages . When lowering the age of entry to the university , some youngsters could  start higher education at the age of 11 or 13 , but the minimum age could be set at 10 or 9 .

Moreover , a smaller , restricted acceleration could be applied  to higher education and university studies . For example , becoming an engineer requires four years of study in certain countries and five years in other countries . The five-year programs could be condensed into four years by appropriate methods such as adding hours to the four years of study , condensing some courses and/or adding summer courses . Medical studies usually take eight or ten years or even longer to be completed . By applying the convenient acceleration and condensation of courses and studies the eight years of study could be reduced to seven and the ten to nine and so forth , without loss of knowledge or qualification for the future doctor or medical practitioner. Similarly predoctoral and master’s degree studies could be reduced from five to four years or from six years to five.

Hence in the long run everybody would benefit from this reform and acceleration of education , and the gap between youngsters considered to be average or normal and child prodigies would be less wide ,  which I consider to be  a good thing , providing more educational equality and efficiency .

A short remark concerning one aspect of education: Nowadays young people are generally becoming sexually aware (and sometimes active) at younger ages compared with older generations , and this fact should be taken into consideration in education.

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I’ll add a final remark here : The general system  of education followed in a large number of countries today was formed about two or three centuries ago . In a country such as France  a major reform in education took place , coinciding with the period of the French Revolution , and with the  time when the first  successful endoatmospheric human flights were made using
hot air balloons  . One thing to note is that the Montgolfier brothers , who flew the first balloon , were elevated to the nobility as a reward . Nowadays the old aristocratic nobility
titles are gone in France , so people are instead rewarded by  honorary degrees ,which can be considered as  contemporary nobility titles . Many of the thinkers and scientists who
contributed to the reform of  the education system in France two centuries ago , such as Condorcet and Laplace , had acquired old nobility  titles (such as ‘marquis’) , titles related
to the ‘Ancien Régime’ , and were forming the new system of instruction which would provide the new (education related)  nobility titles in France and elsewhere . Perhaps nobility
titles in one form or another will frequently or regularly succeed one another with time , but there should always be room for reassessment , open-mindedness , change and reform . Wilhelm von Humboldt witnessed and was influenced by the French Revolution . He attempted to reform and reorganize the Prussian and German educational system , and founded the University of Berlin in 1810 . In the following decades of the nineteenth century , educators in the United states and other countries emulated and implemented the Prussian education system.
The rest of the  nineteenth and most of the twentieth centuries witnessed generally lesser scattered reforms in education , the elaboration of educational and psychological theories , and were the scene of  a progressive worldwide adoption of the educational ideas and institutions developed in European nations before , during and after the Enlightenment period and the  French Revolution .This epoch  coincided with the development of aerodynamics,
aeronautics and heavier-than-air aircraft , and with the beginnings of astronautics and space exploration .

The following idea is worth considering: While the existent educational system , with its diplomas and degrees and its requirements to study or act in a certain way and to start
higher education at the average age of eighteen , has worked well and has been sufficient for people who move or travel on planet Earth and in its surroundings, I think the reform of
the educational system  I have proposed and written about here , i.e accelerating education , skipping  or condensing  grades and lowering the age of entry to college or the university , will
prove to be important , more efficient  , and necessary in this era of planetary globalization , and more so  in the future , as humans in the age of space travel  intend or attempt to go to Mars and  others planets , traveling in the solar system beyond the Earth-Moon system and the immediate vicinity of planet Earth.