Many cultures have contributed to the development of modern science. Let’s take a moment and look at the roots of mathematics.
Mesopotamia (present-day Iraq)
Mesopotamians made important contributions to the foundations of medicine, astronomy, physics and mathematics. The Sumerians of Mesopotamia and the Elamites invented writing, probably as a means of keeping accounts during the ancient Elamite trade. Startlingly, the astronomy and mathematics of the Mesopotamians was advanced. Like ours, their number system was positional, based on six and sixty. We still use it as seen in how we measure angles in degrees, and how we measure time in hours, minutes and seconds.
The Mesopotamians knew about square roots and cube roots and knew how to solve quadratic equations. They also knew about exponentials and logarithms. Their geometry was not more advanced than their algebra. Although they knew some of the properties of circles and triangles, they were not systematic in proving them.
As early as 4,000 B.C., Egyptians were the first people to make books, in the form of scrolls. The building of pyramids and the surveying needs due to the periodic flooding of the Nile led the Egyptians to develop geometry (meaning “earth measurement ” in Greek). For example they knew that a 3, 4, and 5 units triangle is a right triangle, and that the sum of the areas of the squares formed on the two short sides is equal to the area of the square formed on the longest side. However, they didn’t know this applied to all right triangles, a great theory later discovered by Pythagoras.
Thales of Miletus
Thales, born of a Phoenecean mother, travelled extensively in Egypt and Mesopotamia and brought Egyptian geometry to Greece, which he also contributed to. His student, Anaximander also helped to bring Mesopotamian and Egyptian science to Greece. Anaximander was the first to try to draw a map of the world. He imagined the earth to be cylindrical rather than spherical.
Pythagoras was a student of Anaximander. He discovered that musical harmonics are governed by mathematics. His greatest contribution is the Pythagorean Theorem, the most important single theorem in mathematics.
C2 = a2 + b2
Together with his followers, he discovered that the square root of two is an irrational number, that is, it cannot be expressed as the ratio of two integers.
He was probably educated at Plato’s Academy in Athens, but later worked at the museum in Alexandria. Euclid arranged the theorems of Greek geometry in a logical order. An order so elegant that it can hardly be improved. He wrote “Elements of Geometry”, which has proved to be the most successful textbook of all time.
He was the director of the great library at Alexandria, Egypt. He made a precise measurement of the radius of the earth. He correctly concluded that most of the earth’s surface is covered by water.
Up there with Newton and Gauss, Archimedes is one of the greatest mathematicians of all time. One of Archimedes’ great contributions to mathematics was his development of methods for ﬁnding the areas of plane ﬁgures bounded by curves, as well as methods for ﬁnding the areas and volumes of solid ﬁgures bounded by curved surfaces.
With his use of the doctrine of limits, Archimedes showed that the ratio between the volume of a sphere inscribed in a cylinder to the volume of the cylinder is 2/3, and that the area of the sphere is 2/3 the area of the cylinder. This pleased him so much that he asked a sphere and a cylinder, together with the ratio 2/3, be engraved on his tomb.
Archimedes must be credited with the invention of differential and integral calculus. Unfortunately, he was not able to pass his invention of calculus to other mathematicians of his time because there was no union between geometry and algebra. This had to wait for Descartes, Fermat, newton and Leibniz.
He reunited algebra and geometry, which had been separated ever since the Pythagoreans abandoned algebra after their shocking discovery of irrational numbers, a discovery so contrary to their religion that they kept it secret and renounced algebra. Descartes’ algebraic geometry paved the way for the rediscovery of calculus by Fermat, Newton, and Leibniz. Cartesian coordinates are named after him.
Isaac Newton was born on 25th Dec, 1642, the year in which Galileo died. While in Cambridge, he first showed his mathematical genius by extending the binomial theorem, which had previously been studied by Pascal and Wallis. Newton’s work on binomial coefficients was foreshadowed by that of the French mathematician Blaise Pascal (1623-1662), inventor of “Pascal’s triangle”.
Newton developed his binomial theorem into differential calculus. What we today call “derivatives” he called “fluxions”. He applied his method of fluxions to mechanics, deducing that the force with which the sun attracts a planet is the square of the distance between the planet and the sun, using the three Kepler laws of planetary motion.
Huygens and Leibniz
Huygens, the Dutch physicist, was the first person to estimate numerically the distance to a star. Another important invention made by him is the pendulum clock.
German philosopher and mathematician Gottfried Wilhelm Leibniz (1646-1716) was among the friends of Christian Huygens. He introduced determinants into mathematics, independently invented the calculus and invented a calculating machine which could multiply and divide as well as adding and subtracting.
Daniel Bernoulli (1700-1782) is sometimes called the “father of mathematical physics” because of the far-reaching importance of his work with partial differential equations.
He developed the famous wave equation, which we now call a partial differential equation. He developed his wave equation to describe the motion of a vibrating string. Bernoulli realized that the sum of any two solutions to his wave equation is also a solution.
Daniel Bernoulli’s superposition principle is a mathematical proof of a property of wave motion noticed by Huygens, the fact that many waves can propagate simultaneously through the same medium without interacting.
Leonhard Euler (1707-1783) was the most proliﬁc mathematician in history. His memory and his powers of concentration were amazing. Many of his important results were obtained during the last period of his life, when he was totally blind.
Euler’s identities make it easy to derive relationships between trigonometric functions. He was able to show, using the calculus of variations, which he helped to invent, that the equilibrium conﬁguration of a chain hanging between two ﬁxed supports is described by a hyperbolic cosine.
During the centuries that separated Archimedes from Newton, the developing union between geometry and algebra was lost, but through the work of Descartes, Newton, Leibniz, the Bernoulli’s, Euler and many others, both diﬀerential and integral calculus were rediscovered and turned into practical tools that form part of the foundation of the modern world.
Jean-Baptiste Fourier (1768-1830) founded a branch of mathematics now known as Fourier analysis. Its generalizations have great importance for many branches of theoretical science and engineering.
In 1926, the physicist Erwin Schrodinger wrote down a diﬀerential equation that governs the motion of very small particles such as electrons moving in an atom.
Using the Schrodinger equation, one can analyze in a very exact way the allowed states of atoms. These allowed states are found to be closely analogous to the harmonics of vibrating strings, studied by Pythagoras many centuries earlier.
There are other great mathematicians who have not been mentioned here, the likes of Joseph-Louis Lagrange, Pierre-Simon Laplace, Adrien-Marie Legendre and many others. But these are outstanding in the development of modern science and technology. Their work is the reason why technology has developed to where we see it today. Not forgetting the works of Albert Einstein, Stephen Hawking and other modern scientists, it is fair to say that more is yet to come in uncovering the secrets of the universe. Mathematics is key.