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17TH CENTURY MATHEMATICS. DESCARTES

Descartes has been dubbed the "Father of Modern Philosophy", but he was also one of the key figures in the Scientific Revolution of the 17th Century, and is sometimes considered the first of the modern school of mathematics.

As a young man, he found employment for a time as a soldier (essentially as a mercenary in the pay of various forces, both Catholic and Protestant). But, after a series of dreams or visions, and after meeting the Dutch philosopher and scientist Isaac Beeckman, who sparked his interest in mathematics and the New Physics, he concluded that his real path in life was the pursuit of true wisdom and science.

Back in France, the young Descartes soon came to the conclusion that the key to philosophy, with all its uncertainties and ambiguity, was to build it on the indisputable facts of mathematics. To pursue his rather heretical ideas further, though, he moved from the restrictions of Catholic France to the more liberal environment of the Netherlands, where he spent most of his adult life, and where he worked on his dream of merging algebra and geometry.

In 1637, he published his ground-breaking philosophical and mathematical treatise The “Discourse on Method” is now considered a landmark in the history of mathematics. Following on from early movements towards the use of symbolic expressions in mathematics this work introduced what has become known as the standard algebraic notation, using lowercase a, b and c for known quantities and x, y and z for unknown quantities. It was perhaps the first book to look like a modern mathematics textbook, full of a's and b's, x2's, etc.

Descartes is perhaps best known today as a philosopher who espoused rationalism and dualism. His philosophy consisted of a method of doubting everything, then rebuilding knowledge from the ground, and he is particularly known for the often-quoted statement “Cogito ergo sum”(“I think, therefore I am”).

17TH CENTURY MATHEMATICS. LEIBNIZ

The German polymath Gottfried Wilhelm Leibniz occupies a grand place in the history of philosophy. Like many great thinkers before and after him, Leibniz was a child prodigy and a contributor in many different fields of endeavour.

But, between his work on philosophy and logic and his day job as a politician and representative of the royal house of Hanover, Leibniz still found time to work on mathematics. He was perhaps the first to explicitly employ the mathematical notion of a function to denote geometric concepts derived from a curve, and he developed a system of infinitesimal calculus, independently of his contemporary Sir Isaac Newton. He also revived the ancient method of solving equations using matrices, invented a practical calculating machine and pioneered the use of the binary system.

Unlike Newton, however, he was more than happy to publish his work, and so Europe first heard about calculus from Leibniz in 1684, and not from Newton (who published nothing on the subject until 1693). When the Royal Society was asked to adjudicate between the rival claims of the two men over the development of the theory of calculus, they gave credit for the first discovery to Newton, and credit for the first publication to Leibniz.

You can watch a video  Newton vs. Leibniz - The Controversy Over the Discovery of the Calculus to know some facts:

Leibniz is also often considered the most important logician between Aristotle in Ancient Greece and George Boole and Augustus De Morgan in the19th Century. Even though he actually published nothing on formal logic in his lifetime, he enunciated in his working drafts the principal properties of what we now call conjunction, disjunction, negation, identity, set inclusion and the empty set.

SHAPES

Geometry is the branch of mathematics that describes shapes.

Geometry can be divided into:

Plane Geometry is about flat shapes like lines, circles and triangles ... shapes that can be drawn on a piece of paper.

Solid Geometry is about three dimensional objects like cubes, prisms, cylinders and spheres.

There are two main types of solids, "Polyhedra", and "Non-Polyhedra":

A polyhedron is a solid with flat faces (from Greek poly- meaning "many" and -edron meaning "face").

Non-Polyhedra is a solid, witch any surface is not flat

Sphere: A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball.

The shape of the Earth is very close to that of an oblate spheroid, a sphere flattened along the axis from pole to pole such that there is a bulge around the equator.

Hexagons: Hexagons are six-sided polygons, closed, 2-dimensional, many-sided figures with straight edges.

For a beehive, close packing is important to maximise the use of space. Hexagons fit most closely together without any gaps; so hexagonal wax cells are what bees create to store their eggs and larvae. 

Cones: A cone is a three-dimensional geometric shape that tapers smoothly from a flat, usually circular base to a point called the apex or vertex.

Volcanoes form cones, the steepness and height of which depends on the runniness (viscosity) of the lava. Fast, runny lava forms flatter cones; thick, viscous lava forms steep-sided cones.

Parallel lines: In mathematics, parallel lines stretch to infinity, neither converging nor diverging.

These parallel dunes in the Australian desert aren't perfect - the physical world rarely is.

Fibonacci spiral: If you construct a series of squares with lengths equal to the Fibonacci numbers (1,1,2,3,5, etc) and trace a line through the diagonals of each square, it forms a Fibonacci spiral.

Many examples of the Fibonacci spiral can be seen in nature, including in the chambers of a nautilus shell.

Watch the video to know how to draw

Fibonacci Spiral

MATHS AND NATURE

"The laws of nature are but the mathematical thoughts of God"

                                                                                      - Euclid

Mathematics is everywhere in this universe. We seldom note it. We enjoy nature and are not interested in going deep about what mathematical idea is in it. Here are a very few properties of mathematics that are depicted  in nature.

SYMMETRY

Symmetry is everywhere you look in nature.

Symmetry is when a figure has two sides that are mirror images of one another. It would then be possible to draw a line through a picture of the object and along either side the image would look exactly the same. This line would be called a line of symmetry. 

There are two kinds of symmetry.

One is bilateral symmetry in which an object has two sides that are mirror images of each other. 

The human body would be an excellent example of a living being that has bilateral symmetry.

The other kind of symmetry is radial symmetry. This is where there is a center point and numerous lines of symmetry could be drawn. 

The most obvious geometric example would be a circle.

Not all objects have symmetry; if an object is not symmetrical, it is called asymmetric.

Symmetry in mathematics

Symmetry occurs in many areas of mathematics. Symmetry comes from a Greek word meaning 'to measure together' and is widely used in the study of geometry. Mathematically, symmetry means that one shape becomes exactly like another when you move it in some way: turn, flip or slide. For two objects to be symmetrical, they must be the same size and shape, with one object having a different orientation from the first. 

The mathematical study of symmetry is systematized and formalized in the extremely powerful and beautiful area of mathematics called group theory.

Symmetry can be present in the form of coefficients of equations as well as in the physical arrangement of objects. By classifying the symmetry of polynomial equations using the machinery of group theory, for example, it is possible to prove the unsolvability of the general quintic equation.

You can see a video lesson about symmetry on ed.ted.com/lessons

ARCHIMEDES. FATHER OF MATHEMATICS

CONTRIBUTIONS TO MATHEMATICS

Archimedes, a Greek mathematician is considered one of the three great mathematicians along with Isaac Newton and Carl Fredrick Gauss. . His greatest contributions to mathematics were in the area of Geometry. Archimedes was also an accomplished engineer and an inventor. He discovered the method to determine the area and volumes of circles, spheres and cones. Archimedes invented the water screw, a machine for raising water to bring it to fields. His crane was reportedly used in warfare during the Roman siege of his home, Syracuse. Another invention was a miniature planetarium, a sphere whose motion imitated that of the earth, sun, moon, and the five planets that were then known to exist.

A FAMOUS STORY 

There are many stories about how Archimedes made his discoveries. A famous one tells how he uncovered an attempt to cheat King Hieron. 

The king ordered a golden crown and gave the crown's maker the exact amount of gold needed. The maker delivered a crown of the required weight, but Hieron suspected that some silver had been used instead of gold. He asked Archimedes to think about the matter. One day Archimedes was considering it while he was getting into a bathtub. He noticed that the amount of water overflowing the tub was proportional (related consistently) to the amount of his body that was being immersed (covered by water). This gave him an idea for solving the problem of the crown. He was so thrilled that he ran naked through the streets shouting, "Eureka!" (Greek for "I have discovered it!"). 

There are several ways Archimedes may have determined the amount of silver in the crown. One likely method relies on an idea that is now called Archimedes's principle. It states that a body immersed in a fluid is buoyed up (pushed up) by a force that is equal to the weight of fluid that is displaced (pushed out of place) by the body. Using this method, he would have first taken two equal weights of gold and silver and compared their weights when immersed in water. Next he would have compared the weight of the crown and an equal weight of pure silver in water in the same way. The difference between these two comparisons would indicate that the crown was not pure gold.

 You can see his story on ed.ted.com/lessons

To see another story go ed.ted.com/lessons

Euclid. What's the point of Geometry?

Euclid (Greek Eukleides) was the most prominent mathematician of Greco-Roman antiquity. He was born c. 300 BCE in Alexandria, Egypt). Euclid best known for his treatise on geometry, his work 'the Elements'. Today we are going to know what some geometrical terms are ...

EUCLID’S FAMOUS QUOTES:

“The laws of nature are but the mathematical thoughts of God”

“ There is no other Royal path which leads to geometry”.

THE ELEMENTS:

The Elements is divided into 13 books.

• The first 6 books deals with plane geometry.
• Books  7to 9 deals with number theory.
• Book 10 deals with the theory of irrational numbers .
• Books 11 to 13 deals with three-dimensional geometry .

Euclid's Elements is remarkable for the clarity with which the theorems are stated and proved.

Study ed.ted.com/lessons to know How to prove a mathematical theory

EUCLID'S OTHER WORKS :       

• ON DIVISION deals with plane geometry.
• The book DATA discusses plane geometry and contains propositions.
• PHAENOMENA is a work by what we call today as applied mathematics, concerning the  geometry of spheres for use in astronomy. 
• THE OPTICS, corrects the belief held at the time that the sun and other heavenly bodies are actually the size they appear to be to the eye.
• CONICS was a work on conic sections.