The Numerical Prince

Two dots on a paper present an interesting picture. They may be joined to draw a portrait of a beautiful lady or of a lovely flower or even two sentences may be written conveying a profound truth. Another equally interesting thing that may be done with the two dots is to join them through the shortest distance and obtain a straight line. Although it appears simple, mathematically it is not so simple and is expressed by an equation

y=mx + c in an X-Y plane.

If two points are not joined through the shortest distance, the line will not be straight and therefore will have a curve and its mathematical expression will be  which is called a quadratic equation. This is common mathematics and is easily understood by school students. This expression is said to be developed in 300 BC by Euclid, who along with Pythagoras solved it geometrically. It took more than 2000 years to develop a new method of expression of quadratic equation.

If the difference of two numbers (a-b) or (b-a) is exactly divisible by m then a and b are said to be congruent with respect to modulus m. For example

100=2(mod 7) or 35=2(mod 11) and generally a=b(mod m)

If we take a quadratic equation, it may be written as =b(mod m) and it was invented by a 19 year old boy and a mathematician Carl Friederich Gauss. He did not stop here and went on to prove a beautiful reciprocity between the pair of congruences between

a =b(mod m) , a=m(mod b) where m and b are prime numbers and both are solvable and both become unsolvable otherwise. If b and m leave a remainder of 3 when divided by 4 then one is solvable and the other unsolvable. Being solvable means finding a solution for the variable “a”. This is called the law of quadratic reciprocity.

Gauss had a dream for mathematics and he worked towards advancement of the frontiers of the subject. It is said that after Archimedes, he was the first to contemplate mathematics for the sake of mathematics. In between, this purpose was lost in the blazing sun of Newton and the subject was conceived more as a hand maiden of Physics and Astronomy. Gauss had great admiration for Archimedes but he regarded the over sight of decimal system by Archimedes as the greatest calamity in the history of science and he says “To what heights would science now be raised if Archimedes had made that discovery”.

Gauss was born in April 30, 1777 in a Brunswick, Germany. His father was a gardener and a bricklayer and did not have any distinction other than fathering a genius. His mother Dorothea supported Gauss in all adversity coming from his father and took pride in her son from the day of his birth till the day of her death at the age of 97. The last twenty years of her life were spent at her son’s house and Gauss gave her a serene old age.

Gauss had great love for languages and he wanted to study philology. On March 30, 1796, he tossed a regular polygon of seventeen sides and its lucky fall induced Gauss to take up mathematics as his career. Philology lost Gauss forever on that day. This was exactly 30 days before he completed 20 years of age.

Gauss considered himself as “All Mathematician” and cultivated his greatest gifts to perfection. He had no desire for fame or money. His greatest joy was in unfolding of a mathematical equation. This freedom from personal ambition and his scientific serenity makes gifted him a less tumultuous life. All his ambition was for mathematics and this earned him respect and kindness of the Duke of Brunswick, Ferdinand. The only occasion when he vehemently opposed an issue was when Napolean asked him to contribute 2000 francs to his war chest. Not only he denied contribution but also refused to take any help from his friends and admirers nor did he appeal for Emperor’s clemency.

Hands in Prussia, Feet in Russia

 “Letters from a Father to His Daughter” echo strong sentiments among most young girls in India. A similar series of letters was written in about 1750s in Europe and was equally popular in the then society. Jawaharlal Nehru told stories of civilization and natural history to his daughter Indira Gandhi where as the earlier edition was written to give basic lessons in mechanics, physical optics, astronomy, sound etc. “letters to a German Princess” as it is known was intended for the niece of the Emperor Fredrick, the Great of Prussia, Princess Anhalt Dessau. These letters were circulated in seven languages in book form across several countries including the US. The writer of this book was surprisingly a mathematician, of an order to which belong Sir Issac Newton and Rene Descartes. His name was Leonhard Euler.

In 18th century, every nation attempted control over seas as it conferred great military advantage. The long and almost unending British hegemony over a large part of the world was chiefly due to its naval power.  Naval power essentially constituted in determining the coordinates of a ship in an ocean. In 1727, the prize problem of The Paris Academy was to determine mathematically the place where the mast should be placed on the deck of the ship. Leonhard Euler stood second in this competition losing only to Pierre Bouguer, Father of naval architecture. He later won this prize twelve times.

Euler introduced ideas in mathematics that we take as given today. The concept of function f(x) to denote the function f applied to argument  x,  modern notations of Sine, Cosine, Tan etc in trigonometry, popularized use of π to denote the ratio between circumference and diameter of a circle, began the use of ∑ for summation, i for imaginary numbers and e for the base of natural algorithm. He is also credited with the famous statement made to the philosopher Denis Diderot in the court of the Russian Empress Catherine the Great:

Sir, (a+b̂n)/n=x, hence God exists.

Leonhard Euler, a Swiss mathematician but a universal hero who is worshipped every day when any student anywhere in the world rejoices in mathematics. For Euler, the world existed for mathematics and If the world did not fit in to his mathematical equations, too bad for the world. He was a mathematician valued by two powerful Kings, Frederick of Prussia and Catherine the Great of Russia. They not only rewarded him handsomely monetarily for his mathematical work but also accorded great respect in their court. This helped him greatly in taking care of a family of 18 and also his life in last 17 years when he worked without eyesight.  Blindness made his ability keener and efforts more relentless, no wonder he was more prolific when blind.


Euler was a contemporary of several greats. He shared a love-hate relationship with Voltaire, the famous French author who continuously teased Euler by tying him into metaphysical knots, and French philosopher Denis Diderot who had to leave Great Catherine’s court after the famous repartee mentioned above. His life and work was also influenced by Daniel Bernoulli and Johann Bernoulli the famous physicists. He took over the position of the Head of Berlin Academy after Daniel Bernoulli and this position was refused bluntly by the great mathematician D’Alembert, when offered by King Frederick citing the offer as outrageous as it was inconceivable for him to put any other mathematician above Euler.

Euler was born in 1707, the year of Newton’s death. The time was ripe for Euler, the then mathematics comprised isolated branches and solutions to the problems therein but not a coherent whole. He unleashed the force of his genius on attempts at unification of mathematics. Today, time is again ripe, for another EULER.