23rd December: The Start of Geometry
The oldest suggestions of an ordered system of measurements go back to the ancient Babylonians, who developed methods of land surveying embodying calculations of the area of simple geometric figures bounded by straight lines and area of circles. This reflected in the name “geometry”, whose literal meaning is Earth measuring.
The assumption of Babylonian astronomers that the year had 360 days is very likely the origin of our system of measuring angles in degrees; the fact that the angles of equilateral triangles are 60 degrees may explain, in part, the hexadecimal method of counting. The Babylonians laid the first foundations of the science of geometry by their purposeful study of the properties of circles. ( the image – a replica from the British museum of a Babylonian ‘math text book’ that depicts the Pythagorean theorem thousands of years before the time of Pythagoras)
Unlike the Egyptians, whose interest in geometry lay exclusively in the practical considerations of land measurement the Greeks devoted their energies to a systematic study of geometrical figures and their properties to establish a new science. Greatest among ancient Greek scientists were Plato and his pupil Aristotle. Other ancient scientists are Eudoxus (theory of proportion and the invention of the method of exhaustion to find approximate areas and volumes of curvilinear forms and shapes – a forerunner of integral calculus), Euclid, Achimedes and many more.
The Italian architect Leone Alberti was the first who, when discussing perspective in art, formulated ideas that pointed the way toward a future science of projective geometry. Girard Desargues, in his turn, was the first professional mathematician who, exploring the logical foundations of Euclidean geometry, initiated a formal study which extended Euclid’s methods to projective geometry. Jean Victor Poncelet in 1822 published a treatise which revitalized interest in the subject of projective geometry as a science that treats form and position distinct from size.
The magic square is of Chinese origin, being first mentioned in a manuscript from the time of Emperor Yu around 2200 BC. This square has 3 x 3 cells, each with Chinese characters equivalent to 1 through 9, and giving the sum 15 in all directions.
It was inevitable that a square with these “magical”, and therefore mysterious, qualities should appear to astrologers and cranks of all descriptions.
Thus, a square of one cell containing the digit 1 was considered to represent the eternal perfection of God, as explained to the lay mind by one Cornelius Agrippa (1486-1535), an astrologer by profession.
The unfortunate fact that a magic square with 2 x 2 cells cannot be constructed was considered proof of the imperfection of the four elements: air, water, fire and water; some self-styled Great Thinkers attributed the failure to Original Sin.
The first magic square to appear in the Western world was, in all probability, the one depicted in the upper right-hand corner of a copperplate engraving called “Melancolia I” by the German artist-mathematician Albrecht Durer, who also managed to enter the year of engraving – 1514 – in the two middle cells of the bottom row.
Combinatorics traces its history back to ancient times when it was often closely associated with number mysticism, as in the Chinese book “I Ching” from 2200 BC.
In the Western world, interest in combinatorial mathematics was awakened in the 17th and 18th centuries and largely stimulated by questions being raised about odds in gambling and other games of chance, which led to the creation and development of probability theory.
Galileo Galilei studied the relative probability of various sums of points occurring when rolling dice. Other important contributors in the field of probability theory were Pierre de Fermat, Blaise Pascal, Leibniz, and Nicolaus and Daniel Bernoulli.
In 1736, Leonard Euler solved the celebrated problem of the Konigsberg bridges – the question whether it would be possible to make a tour of the city and return to the starting point by crossing all of its seven bridges just once; this marks the beginning of graph theory.
Combinatorial geometry includes problems of covering, packing and symmetry. A celebrated conjecture in packing theory, posed in 1611 by Kepler, is that the most compact way to stack spheres is into four-sides pyramids. This method has been used for the display of fruit long before Kepler’s time.
27th December: Mathematical Analysis part 1
Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians.
Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno’s paradox of the dichotomy (the paradox of Achilles and the Tortoise). Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. The explicit use of infinitesimals appears in Archimedes’ The Method of Mechanical Theorems, a work rediscovered in the 20th century.
In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would later be called Cavalieri’s principle to find the volume of a sphere in the 5th century. The Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle’s theorem in the 12th century.
28th December: Mathematical Analysis part 2
In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and the Taylor series, of functions such as sine, cosine, tangent and arctangent. Alongside his development of the Taylor series of the trigonometric functions, he also estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series. His followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century.
The modern foundations of mathematical analysis were established in 17th century Europe. Descartes and Fermat independently developed analytic geometry, and a few decades later Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis, and generating functions. During this period, calculus techniques were applied to approximate discrete problems by continuous ones.
29th December: Mathematical Analysis part 3
In the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano’s work did not become widely known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation. He formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the formal theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis. The contributions of these mathematicians and others, such as Weierstrass, developed the (ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis.
30th December: Mathematical Analysis part 4 (final)
In the middle of the 19th century Riemann introduced his theory of integration. The last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the “epsilon-delta” definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the “gaps” between rational numbers, thereby creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the “size” of the set of discontinuities of real functions.
Also, “monsters” (nowhere continuous functions, continuous but nowhere differentiable functions, space-filling curves) began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue solved the problem of measure, and Hilbert introduced Hilbert spaces to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.
Source: Mathematics: From the Birth of Numbers by Jan Gullberg
Hope you enjoyed this series of posts. I totally loved doing the event on Facebook. Also, I have created a category on Google+ for it, which will be updated with other interesting history facts on the way ^_^
Don’t forget that maths is everywhere! Enjoy! ~LThMath~