THE SIEVE OF ERATOSTHENES

A useful rule for making  a table of prime numbers is attributed  to another famous Greek astronomer and mathematician, Erastosthenes (c.275-194 B.C.) To find all the prime numbers not greater than n, we write all the integers from 1 to n in a convenient tabulated form; from this table we delete every second number after 2, because they all are divisible by 2, every third number after 3, because they all are divisible by 3, every fifth number after 5, and so on; the numbers which remain after all the deletions are prime numbers less than or equal to n.

SETS

A mathematical set must be defined either by a list, a rule or a formula, so that its elements can be recognized without any doubt or ambiguity. If N is a set, we write n E N meaning that 'n is a member of the set N'. The set of all elements that are being considered in a problem is called the reference set (or universal set), and it is denoted by E or U. The empty set has no elements (compare it with the empty drawer of a desk); it is denoted by O. Any collection of elements of E, ranging from O to E it self, is called a subset of E.

ROTATIONAL SYMMETRY IN A PLANE

When we say that a figure has rotational symmetry we mean that it is possible to rotate it about a center in such a way that the whole figure is unchanged but the parts of it have been rotated. The number of ways in which this can be done is called the order of rotational symmetry, and it is usual to count the identity operation (rotation through 0 degree or 360 degree, and so on) as one of the symmetries. Any figure has the trivial rotational symmetry of order one about any point. The letter N, the parallelogram and many other familiar figures, have rotational symmetry of order two.

THE TRAPEZIUM

There is one more special type of quadrilateral which frequently occurs in practical work, the trapezium; this is a quadrilateral with just one pair of opposite sides parallel.

In the side figure, this is a trapezium ABCD with an axis of symmetry. Such a figure is called an isosceles trapezium.

BUSINESS CALCULATIONS

The custom of charging interest on loans must be as old as money itself. Merchants and moneylenders have always tried to make a profit and to relate the profit to the size of transaction; a big deal calls for a big profit.

Roman taxes were usually 1/20, 1/25, or 1/100 of the quantity involved, and these fractions may have led to the practice of working in hundredths, or percentages. During the middle ages calculation of profit or interest as so much in a hundred became the common practice and the phrase per cent was established. This phrase was abbreviated in many different ways; one of the abbreviations was P0c and it is thought that the modern symbol % may have been derived from this.

One of the earliest examples of the phrase ‘per cent’occurs in the Liber Abaci by Leonardo of Pisa, published in 1202, where the author uses it in a problem concerning a merchant who is to sell wool in Florence at a profit of 20 per cent

ALBERT EINSTEIN

Albert Einstein (pronounced /ˈælbərt ˈaɪnstaɪn/; German: [ˈalbɐt ˈaɪnʃtaɪn]; 14 March 1879 – 18 April 1955) was a theoretical physicist, philosopher and author who is widely regarded as one of the most influential and best known scientists and intellectuals of all time. A German-Swiss Nobel laureate, he is often regarded as the father of modern physics.^[3] He received the 1921 Nobel Prize in Physics "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect".^[4]

His many contributions to physics include the special and general theories of relativity, the founding of relativistic cosmology, the first post-Newtonian expansion, the explanation of the perihelion precession of Mercury, the prediction of the deflection of light by gravity (gravitational lensing), the first fluctuation dissipation theorem which explained the Brownian motion of molecules, the photon theory and the wave-particle duality, the quantum theory of atomic motion in solids, the zero-point energy concept, the semi-classical version of the Schrödinger equation, and the quantum theory of a monatomic gas which predicted Bose–Einstein condensation.

Einstein published more than 300 scientific and over 150 non-scientific works; he additionally wrote and commentated prolifically on various philosophical and political subjects.^[5] His great intelligence and originality has made the word "Einstein" synonymous with genius.

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SCALES

The scale of a map is often given in the form 1 : 100.000, or 1 : 25.000, that is as a ratio 1 : n. The scale 1 : 25.000 means that 1 cm on the map represents 25.000 cm or 0,25 km on the ground. When 1 : n is given the scale of a map it is called the representative fraction (R.F.) on the map.

THALES OF MILETUS

Thales of Miletus (pronounced /ˈθeɪliːz/; Greek: Θαλῆς, Thalēs; c. 624 BC – c. 546 BC) was a pre-Socratic Greek philosopher from Miletus in Asia Minor, and one of the Seven Sages of Greece. Many, most notably Aristotle, regard him as the first philosopher in the Greek tradition.^[1] According to Bertrand Russell, "Western philosophy begins with Thales."^[2] Thales attempted to explain natural phenomena without reference to mythology and was tremendously influential in this respect. Almost all of the other pre-Socratic philosophers follow him in attempting to provide an explanation of ultimate substance, change, and the existence of the world -- without reference to mythology. Those philosophers were also influential, and eventually Thales' rejection of mythological explanations became an essential idea for the scientific revolution. He was also the first to define general principles and set forth hypotheses, and as a result has been dubbed the "Father of Science".^[3][4]

In mathematics, Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. As a result, he has been hailed as the first true mathematician and is the first known individual to whom a mathematical discovery has been attributed.
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TRIANGLES AND QUADRILATERALS (Part 1)

Thales of Miletus, knew that the measurements of a triangle were fixed when the length of its base and the sizes of its basebase angles were known. He used this fact to determine the distance of a ship at sea. Pythagoras of Samos, who settled eventually in Crotona, develoved  the theory of congruent triangles. It was probably in his school at Crotona that the congruence theorems were first analysed and related to one another in some kind of logical order.

During the life time of Plato (429-348 B.C.) Athens became the chief center of mathematical studies. It held supremacy for about 150 years, when the first university in the world, at Alexandria, displaced it. Plato was born near Athens of rich and noble parents. He studied in Egypt, Cyrille, and Italy, returning to Athens about 380 B.C. About a mile outside Athens was a beautiful walled garden called Academia, after its owner Academos. Here Plato formed a school of students which came to be known as the Academy. Plato was a philosopher who believed in geometry as one of the foundations of a liberal education. The inscription over the entrance to the academy ran 'Let none ignorant of geometry enter my door'. Plato did great service to mathematics in making its foundations logical and secure. We still tend to regards the compasses and ruler as the chief mathematical instruments largely because he would allow no other instruments to be used.

CURRENCY

Most countries have a decimal currency, with a main unit divided into one hundred minor units. In the United Kingdom the main unit is the pound, United States is the dollar, Kingdom of Saudi Arabia is the real, and Indonesian Republic is Rupiah.

PYTHAGORAS OF SAMOS

Pythagoras of Samos (Greek: Ὁ Πυθαγόρας ὁ Σάμιος, O Pythagoras o Samios, "Pythagoras the Samian", or simply Ὁ Πυθαγόρας; c. 570-c. 495 BC^[1]) was an Ionian Greek philosopher and founder of the religious movement called Pythagoreanism. Most of our information about Pythagoras was written down centuries after he lived, thus very little reliable information is known about him. He was born on the island of Samos, and may have travelled widely in his youth, visiting Egypt and other places seeking knowledge. Around 530 BC, he moved to Croton, a Greek colony in southern Italy, and there set up a religious sect. His followers pursued the religious rites and practices developed by Pythagoras, and studied his philosophical theories. The society took an active role in the politics of Croton, but this eventually led to their downfall. The Pythagorean meeting-places were burned, and Pythagoras was forced to flee the city. He is said to have ended his days in Metapontum.

Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. He is often revered as a great mathematician, mystic and scientist, and he is best known for the Pythagorean theorem which bears his name. However, because legend and obfuscation cloud his work even more than with the other pre-Socratic philosophers, one can say little with confidence about his teachings, and some have questioned whether he contributed much to mathematics and natural philosophy. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors. Whether or not his disciples believed that everything was related to mathematics and that numbers were the ultimate reality is unknown. It was said that he was the first man to call himself a philosopher, or lover of wisdom,^[2] and Pythagorean ideas exercised a marked influence on Plato, and through him, all of Western philosophy.

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AVICENNA (IBNU SINA)

Abū ‘Alī al-Ḥusayn ibn ‘Abd Allāh ibn Sīnā, known as Abū Alī Sīnā^[1][2] (Persian: ابوعلی سینا، پورسینا) or, more commonly, Ibn Sīnā^[3] (Arabic: ابن سینا‎) or Pour Sina, but most commonly known in English by his Latinized name Avicenna (Greek: Aβιτζιανός, Avitzianós),^[4] (c. 980 - 1037) was a polymath of Persian^[5] origin and the foremost physician and philosopher of his time.^[6] He was also an astronomer, chemist, geologist, Hafiz, Islamic psychologist, Islamic scholar, Islamic theologian, logician, paleontologist, mathematician, Maktab teacher, physicist, poet, and scientist.^[7].

Ibn Sīnā studied medicine under a physician named Koushyar. He wrote almost 450 treatises on a wide range of subjects, of which around 240 have survived. In particular, 150 of his surviving treatises concentrate on philosophy and 40 of them concentrate on medicine.^[1][8] His most famous works are The Book of Healing, a vast philosophical and scientific encyclopaedia, and The Canon of Medicine,^[9] which was a standard medical text at many medieval universities.^[10] The Canon of Medicine was used as a text-book in the universities of Montpellier and Louvain as late as 1650.^[11]

Ibn Sīnā's Canon of Medicine provides a complete system of medicine according to the principles of Galen (and Hippocrates).^[12][13]

George Sarton, an early author of the history of science, wrote in the Introduction to the History of Science:

One of the most famous exponents of Muslim universalism and an eminent figure in Islamic learning was Ibn Sina, known in the West as Avicenna (981-1037). For a thousand years he has retained his original renown as one of the greatest thinkers and medical scholars in history. His most important medical works are the Qanun (Canon) and a treatise on Cardiac drugs^[14][15]. The 'Qanun fi-l-Tibb' is an immense encyclopedia of medicine. It contains some of the most illuminating thoughts pertaining to distinction of mediastinitis from pleurisy; contagious nature of phthisis; distribution of diseases by water and soil; careful description of skin troubles; of sexual diseases and perversions; of nervous ailments. 

Referred from:Wikipedia

PLATO

Plato (English pronunciation: /ˈpleɪtoʊ/; Greek: Πλάτων, Plátōn, "broad"^[2]; 428/427 BC^[a] – 348/347 BC), was a Classical Greek philosopher, mathematician, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the foundations of Western philosophy and science.^[3] Plato was originally a student of Socrates, and was as much influenced by his thinking as by his apparently unjust execution.

Plato's sophistication as a writer is evident in his Socratic dialogues; thirty-six dialogues and thirteen letters have been ascribed to him. Plato's writings have been published in several fashions; this has led to several conventions regarding the naming and referencing of Plato's texts.

Plato's dialogues have been used to teach a range of subjects, including philosophy, logic, rhetoric and mathematics.

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ARISTOTLE

Aristotle (Greek: Ἀριστοτέλης, Aristotélēs) (384 BC – 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, politics, government, ethics, biology, and zoology. Together with Plato and Socrates (Plato's teacher), Aristotle is one of the most important founding figures in Western philosophy. Aristotle's writings were the first to create a comprehensive system of Western philosophy, encompassing morality and aesthetics, logic and science, politics and metaphysics.

Aristotle's views on the physical sciences profoundly shaped medieval scholarship, and their influence extended well into the Renaissance, although they were ultimately replaced by Newtonian physics. In the zoological sciences, some of his observations were confirmed to be accurate only in the nineteenth century. His works contain the earliest known formal study of logic, which was incorporated in the late nineteenth century into modern formal logic. In metaphysics, Aristotelianism had a profound influence on philosophical and theological thinking in the Islamic and Jewish traditions in the Middle Ages, and it continues to influence Christian theology, especially Eastern Orthodox theology, and the scholastic tradition of the Catholic Church. His ethics, though always influential, gained renewed interest with the modern advent of virtue ethics. All aspects of Aristotle's philosophy continue to be the object of active academic study today. Though Aristotle wrote many elegant treatises and dialogues (Cicero described his literary style as "a river of gold"),^[1] it is thought that the majority of his writings are now lost and only about one-third of the original works have survived.^[2]

Despite the far-reaching appeal that Aristotle's works have traditionally enjoyed, today modern scholarship questions a substantial portion of the Aristotelian corpus as authentically Aristotle's own.

SOCRATES

Socrates (Greek: Σωκράτης, soˈkraːtɛːs, Sōkrátēs; c. 469 BC–399 BC^[1], in English pronounced /ˈsɒkrətiːz/) was a Classical Greek Athenian philosopher. Credited as one of the founders of Western philosophy, he is an enigmatic figure known chiefly through the accounts of later classical writers, especially the writings of his students Plato and Xenophon, and the plays of his contemporary Aristophanes. Many would claim that Plato's dialogues are the most comprehensive accounts of Socrates to survive from antiquity.^[2]

Through his portrayal in Plato's dialogues, Socrates has become renowned for his contribution to the field of ethics, and it is this Platonic Socrates who also lends his name to the concepts of Socratic irony and the Socratic method, or elenchus. The latter remains a commonly used tool in a wide range of discussions, and is a type of pedagogy in which a series of questions are asked not only to draw individual answers, but also to encourage fundamental insight into the issue at hand. It is Plato's Socrates that also made important and lasting contributions to the fields of epistemology and logic, and the influence of his ideas and approach remains strong in providing a foundation for much western philosophy that followed.

As one recent commentator has put it, Plato, the idealist, offers "an idol, a master figure, for philosophy. A Saint, a prophet of the 'Sun-God', a teacher condemned for his teachings as a heretic."^[3] Yet, the 'real' Socrates, like many of the other Ancient philosophers, remains at best enigmatic and at worst unknown.